Counterexamples to a conjecture of Lemmermeyer

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收敛的固定测量法英语

收敛的固定测量法英语Convergent Fixed Measurement MethodMeasurement is a fundamental aspect of scientific inquiry, as it allows us to quantify and compare various phenomena. In the realm of science and engineering, the ability to accurately measure physical quantities is of utmost importance. One such method that has gained significant attention is the convergent fixed measurement method. This approach aims to provide a reliable and consistent way of obtaining precise measurements, ultimately contributing to the advancement of scientific knowledge and technological development.The convergent fixed measurement method is based on the principle of minimizing the variation in repeated measurements of a particular quantity. The underlying idea is that by taking multiple measurements under controlled conditions and using a consistent set of procedures, the results will converge towards a true value, thereby reducing the impact of random errors and systematic biases.One of the key advantages of the convergent fixed measurement method is its ability to account for the inherent uncertainties associated with measurement processes. In any measurement, thereare various sources of error, such as instrument limitations, environmental factors, and human errors. The convergent fixed measurement method addresses these challenges by employing statistical techniques to analyze the distribution of the measured values and identify the most reliable and representative result.The process of implementing the convergent fixed measurement method typically involves several steps. First, the measurement setup and procedures are carefully designed to ensure consistent and controlled conditions. This may include the selection of appropriate measurement instruments, the establishment of a stable environment, and the development of standardized protocols for data collection.Next, multiple measurements of the same quantity are performed, with the number of measurements determined based on the desired level of statistical confidence and the expected variability of the results. The collected data is then analyzed using statistical tools, such as mean, standard deviation, and confidence intervals, to determine the most representative value and its associated uncertainty.One of the key aspects of the convergent fixed measurement method is the iterative nature of the process. If the initial set of measurements does not converge within the desired tolerance,additional measurements may be performed, or the measurement setup and procedures may be refined to improve the consistency and reliability of the results.The convergent fixed measurement method has found widespread application in various fields, including physics, chemistry, engineering, and materials science. In these domains, accurate and precise measurements are crucial for understanding fundamental principles, validating theoretical models, and developing new technologies.For example, in the field of materials science, the convergent fixed measurement method is often used to characterize the mechanical properties of materials, such as tensile strength, hardness, and fatigue resistance. By applying this approach, researchers can obtain reliable data that can be used to optimize material design, improve manufacturing processes, and develop new materials with enhanced performance.Similarly, in the field of physics, the convergent fixed measurement method is employed in experiments involving fundamental constants, such as the speed of light, the Planck constant, and the gravitational constant. Precise measurements of these quantities are essential for validating and refining our understanding of the underlying laws of nature.In conclusion, the convergent fixed measurement method is a powerful tool that enables researchers and engineers to obtain accurate and reliable measurements, ultimately contributing to the advancement of scientific knowledge and technological progress. By systematically addressing the sources of measurement uncertainty and employing statistical techniques, this method provides a robust and reproducible approach to data collection and analysis, making it an indispensable part of the scientific and engineering toolkit.。

A FINITE ELEMENT METHOD FOR CRACK GROWTH WITHOUT REMESHING

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt.J.Numer.Meth.Engng.46,131–150(1999)A FINITE ELEMENT METHOD FOR CRACKGROWTH WITHOUT REMESHINGNICOLAS MO ES†,JOHN DOLBOW‡AND TED BELYTSCHKO∗;§Department of Mechanical Engineering,Northwestern University,2145Sheridan Road,Evanston,IL60208-3111,U.S.A.SUMMARYAn improvement of a new technique for modelling cracks in theÿnite element framework is presented.A standard displacement-based approximation is enriched near a crack by incorporating both discontinuousÿelds and the near tip asymptoticÿelds through a partition of unity method.A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed.This technique allows the entire crack to be represented independently of the mesh,and so remeshing is not necessary to model crack growth.Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique.Copyright?1999John Wiley&Sons,Ltd.KEY WORDS:ÿnite elements;fracture1.INTRODUCTIONThe modelling of moving discontinuities with theÿnite element method is cumbersome due to the need to update the mesh to match the geometry of the discontinuity.Several newÿnite element techniques have been developed to model cracks and crack growth without remeshing.These in-clude the incorporation of a discontinuous mode on an element level[1],a moving mesh technique [2],and an enrichment technique forÿnite elements based on a partition-of-unity which involves minimal remeshing[3].In Belytschko and Black[3],curved cracks were treated by mapping the straight crack enriched ÿeld.This is not readily applicable to long cracks or three dimensions.In this paper we improve the method by incorporating a discontinuousÿeld across the crack faces away from the crack tip.The method incorporates both the discontinuous Haar function and the near-tip asymptotic functions through a partition of unity method.A crack is grown by redeÿning the tip location and∗Correspondence to:Ted Belytschko,Department of Mechanical Engineering,Northwestern University,2145Sheridan Road,Evanston,IL60208-3111,U.S.A.E-mail:t-belytschko@†Research Associate‡DOE Computational Science Graduate Fellow§Walter P.Murphy Professor of Mechanical EngineeringContract=grant sponsor:O ce of Naval ResearchContract=grant sponsor:Army Research O ceContract=grant sponsor:DOE Computational Science Graduate Fellowship ProgramCCC0029-5981/99/250131–20$17.50Received2February1999 Copyright?1999John Wiley&Sons,Ltd.132N.MO ES,J.DOLBOW AND T.BELYTSCHKOand by adding new crack segments.The addition of a discontinuousÿeld allows for the entire crack geometry to be modelled independently of the mesh,and completely avoids the need to remesh as the crack grows.The present technique exploits the partition of unity property ofÿnite elements identiÿed by Melenk and BabuÄs ka[4],which allows local enrichment functions to be easily incorporated into a ÿnite element approximation.A standard approximation is thus‘enriched’in a region of interest by the local functions in conjunction with additional degrees of freedom.For the purpose of fracture analysis,the enrichment functions are the near-tip asymptoticÿelds and a discontinuous function to represent the jump in displacement across the crack line.The method di ers from the work of Oliver[5],who introduces step functions into the displace-mentÿeld and then treats the e ects on an element level by a multiÿeld approach with an assumed strainÿeld.In the method described in this paper,the displacementÿeld is actually global,but the support of the enrichment functions are local because they are multiplied by nodal shape functions. This paper is organized as follows.In the next section we present the strong and weak form for linear elastic fracture mechanics.The discrete equations are given in Section3,which also describes the incorporation of enrichment functions to model cracks.Several numerical examples are given in Section4.Finally,Section5provides a summary and some concluding remarks.2.PROBLEM FORMULATIONIn this section,we brie y review the governing equations for elasto-statics and give the associated weak form.Speciÿcally,we consider the case when an internal line is present across which the displacementÿeld may be discontinuous.erning equationsConsider the domain bounded by .The boundary is composed of the sets u; t,and c,such that = u∪ t∪ c as shown in Figure1.Prescribed displacements are imposed on u, while tractions are imposed on t.The crack surface c(lines in2-D and surfaces in3-D)is assumed to be traction-free.The equilibrium equations and boundary conditions are∇·A+b=0in (1a)A·n=t on t(1b)A·n=0on c+(1c)A·n=0on c−(1d) where n is the unit outward normal.In the above,A is the Cauchy stress,and b is the body force per unit volume.In the present investigation,we consider small strains and displacements.The kinematics equa-tions therefore consist of the strain–displacement relationU=U(u)=∇s u(2) where∇s is the symmetric part of the gradient operator,and the boundary conditionsu=u on u(3) Copyright?1999John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)FINITE ELEMENT METHOD FOR CRACK GROWTH133Figure1.Body with internal boundary subjected to loadsThe constitutive relation is given by Hooke’s law:A=C:U(4)where C is the Hooke tensor.2.2.Weak formThe space of admissible displacementÿelds is deÿned byU={v∈V:v=u on u v discontinuous on c}(5)where the space V is related to the regularity of the solution.The details on this matter when the domain contains an internal boundary or re-entrant corner may be found in BabuÄs ka and Rosenzweig[6]and Grisvard[7].We note that the space V allows for discontinuous functions across the crack line.The test function space is deÿned similiarly asU0={v∈V:v=0on u v discontinuous on c}(6)The weak form of the equilibrium equations is given byA:U(v)d =b·v d +tt·v d ∀v∈U0(7)Using the constitutive relation and the kinematics constraints in the weak form,the problem is toÿnd u∈U such thatU(u):C:U(v)d =b·v d +tt·v d ∀v∈U0(8)It is shown in Belytschko and Black[3]that the above is equivalent to the strong form(1), including the traction-free conditions on the two crack faces.In contrast to boundary element techniques,this enables the method to be easily extended to non-linear problems.Copyright?1999John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)134N.MO ES,J.DOLBOW AND T.BELYTSCHKOFigure2.Finite element mesh near a crack tip,the circlednumbers are element numbersFigure3.Regular mesh without a crack3.DISCRETIZATIONIn this section,we present the construction of approximations which are discontinuous across a given line or surface and include the asymptotic near-tipÿelds.After examining a simple case, criteria for selecting the enriched nodes for an arbitrary mesh and crack geometry are given.We also discuss the modiÿcations necessary to accurately integrate the weak form,and review the numerical procedure to calculate the stress intensity factors.3.1.Crack modelling using discontinuous enrichmentThe model consists of a standardÿnite element model and a crack representation which is independent of the elements.In order to introduce the notion of discontinuous enrichment,weÿrst consider a simple case of an edge crack modelled by four elements as shown in Figure2.For convenience,the local co-ordinate system is aligned with the crack tip.We wish to illustrate how an equivalent discrete space can be constructed with the mesh shown in Figure3and the addition of a discontinuousÿeld.Theÿnite element approximation associated with the mesh in Figure2isu h=10i=1u i i(9)where u i is the displacement at node i and i is the bilinear shape function associated with node i. Each shape function i has a compact support!i given by the union of the elements connected to node i.Deÿning a and b asa=u9+u102;b=u9−u102(10) We can express u9and u10in terms of a and bu9=a+b;u10=a−b(11) Copyright?1999John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)FINITE ELEMENT METHOD FOR CRACK GROWTH 135Figure 4.Crack not aligned with a mesh,the circled nodes are enriched with the discontinuous function Figure 5.Crack not aligned with a mesh,the circled nodes are enriched with the discontinuous function and the squared nodes with the tip enrichment functions.En-richment with only the discontinuous function shortens the crack to point pThen replacing u 9and u 10in terms of a and b in (9)yieldsu h =8i =1u i i +a ( 9+ 10)+b ( 9+ 10)H (x )(12)where H (x )is referred to here as a discontinuous,or ‘jump’function.This is deÿned in the local crack co-ordinate system asH (x;y )= 1for y¿0−1for y¡0(13)such that H (x )=1on element 1and −1on element 3,respectively.If we now consider the mesh in Figure 3, 9+ 10can be replaced by 11,and a by u 11.The ÿnite element approximation now readsu h =8i =1u i i +u 11 11+b 11H (x )(14)The ÿrst two terms on the right-hand side represent the classical ÿnite element approximation,whereas the last one represents the addition of a discontinuous enrichment.In other words,when a crack is modeled by a mesh as in Figure 2,we may interpret the ÿnite element space as the sum of one which does not model the crack (such as Figure 3)and a discontinuous enrichment.The previous derivation provides insight into the extension of the technique for the case when the crack does not align with the mesh.The key issues are the selection of the appropriate nodes to enrich,and the form of the associated enrichment functions.In terms of enrichment with the jump function,we adopt the convention that a node is enriched if its support is cut by the crack into two disjoint pieces.This rule is seen to be consistent with the previous example,in which only node 11was enriched.Figure 4illustrates the application of this rule when the crack is not aligned with the element edges,in which the circled nodes are enriched with the jump function.Copyright ?1999John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)136N.MO ES,J.DOLBOW AND T.BELYTSCHKO Figure 6.An arbitrary crack placed on a mesh Figure 7.Local axes for the polar co-ordinates at the two crack tipsIn a more general case such as that shown in Figure 5,the crack tip will not coincide with an element edge,and in this instance the discontinuity cannot be adequately described using only a function such as H (x ).The jump enrichment of the circled nodes in this case only provides for the modelling of the discontinuity up until point p .To seamlessly model the entire discontinuity along the crack,the squared nodes are enriched with the asymptotic crack tip functions with the technique developed in Belytschko and Black [3].For example,for the discretization shown in Figure 5,the approximation takes the formu h = i ∈I u i i + j ∈J b j j H (x )+ k ∈K k 4 l =1c l k F l (x ) (15)in which J is the set of circled nodes and K the set of squared nodes.The functions F l (x )are deÿned as {F l (r;Â)}≡ √r sin Â ;√r cos Â ;√r sin Â sin (Â);√r cos Â sin (Â) (16)where (r;Â)are the local polar co-ordinates at the crack tip.Note that the ÿrst function in (16),√r sin (Â=2),is discontinuous across the crack faces whereas the last three functions are continuous.The function H (x )is given by (13)where the local axes are taken to be aligned with the crack tip as in Figure 2.We now generalize to the case of an arbitrary crack,as shown in Figure 6.The approximation takes the form:u h = i ∈I u i i + j ∈J b j j H (x )+ k ∈K 1 k 4 l =1c l 1k F 1l (x ) + k ∈K 2 k 4 l =1c l 2k F 2l (x ) (17)where K 1and K 2are the sets of nodes to be enriched for the ÿrst and second crack tip,respectively.The precise deÿnition of these two sets as well as the set J will be given further.The functions F 1l (x )and F 2l (x )are identical to the ones given in (16),with (r 1;Â1)and (r 2;Â2)being deÿned in the local crack tip system at tips 1and 2,respectively as shown in Figure 7.The jump function H (x )is deÿned as follows.The crack is considered to be a curve parametrized by the curvilinear co-ordinate s ,as in Figure 8.The origin of the curve is taken to coincide with one of the crack tips.Given a point x in the domain,we denote by x ∗the closest point on the Copyright ?1999John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)FINITE ELEMENT METHOD FOR CRACK GROWTH137Figure8.Illustration of normal and tangential co-ordinates for a smooth crack:(a)and for a crack with a kink;(b)x∗is the closest point to x on the crack.In both of the above cases,the jump function H(x)=−1crack to x.At x∗,we construct the tangential and normal vector to the curve,e s and e n,with the orientation of e n taken such that e s∧e n=e z.The sign of the step function H(x)is then given by the sign of the scalar product(x−x∗)·e n.In the case of a kinked crack as shown in Figure8(b), where no unique normal but a cone of normals is deÿned at x∗,H(x)=1if the vector(x−x∗) belongs to the cone of normals at x∗and−1otherwise.We now turn to the deÿnitions of J,K1and K2.We shall denote by x1and x2the location of the crack tips1and2,respectively,and by C the geometry of the crack.The sets K1and K2consist of those nodes for which the closure of the nodal support contains crack tip1or2, respectively.The set J is the set of nodes whose support is intersected by the crack and do not belong to K1or K2.K1={k∈I:x1∈!k}(18)K2={k∈I:x2∈!k}(19)J={j∈I:!j∩C=∅;j=∈K1;j=∈K2}(20) Remarks.(1)From the deÿnition of the sets K1;K2,and J,we see that any node whose support isintersected by the crack will be enriched by a discontinuous function:of H type for the nodes in J and of F type for the nodes in K1and K2.So,the displacement is allowed to be discontinous along the full extent of the crack.(2)The treatment of a free edge crack is similar,with either set K1or K2being empty.(3)The set K1and=or K2can be enlarged to include all nodes within a characteristic radiusof the associated crack tip,in which region the asymptotic near-tipÿelds are assumed to dominate the solution.(4)The case of multiple cracks is obtained by considering the proper sets J;K1and K2foreach crack.(5)For the case when multiple crack segments are enriched with the near-tipÿelds,a mappingis used to align the discontinuity with the crack geometry[3].Copyright?1999John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)138N.MO ES,J.DOLBOW AND T.BELYTSCHKOFigure9.Crack on a uniform mesh(left)and on a non-uniform mesh(right).The circled nodes are enriched by the jump function whereas the squared nodes are enriched by the crack tip functionsFigure9illustrates the sets J(circled nodes),K1(squared nodes near tip1)and K2(squared nodes near tip2)for a uniform and non uniform mesh.3.2.Extension to higher-orderÿnite elementsIn the previous section,we have shown how a discontinuity is introduced into aÿnite element approximation with a local enrichment.We considered that theÿnite element approximation prior to the introduction of the discontinuity was of theÿrst order,i.e.one with linear displacements for triangular elements and bilinear for quadrilaterals.The extension to higher-orderÿnite element approximations is straightforward.Consider hierarchical based p-orderÿnite elements[8]:aÿrst-orderÿnite element approximation uses the classical nodal shape functions whereas a higher-order approximation introduces additional edge and element(bubble)shape functions.The enrichment strategy must therefore consider these additional edge and internal degrees of freedom.Let D be the set of all the degrees of freedom for a given mesh and order of approximation. Each degree of freedom u i;i∈D,is a vector with a x and y displacement components.Theÿnite element approximation,prior to the introduction of the discontinuity,is given byu h=i∈Du i i(21)where i is the shape function associated with the i th degree of freedom.In order to select the degrees of freedom to be enriched by the Haar function,we extend the rule introduced in the previous section:‘a node is enriched if its support is cut by the crack’to ‘a degree of freedom is enriched if it’s support is cut by the crack’.The support of an edge degree of freedom is made of the element(s)connected to it and the support of an element degree of freedom is the element itself.More precisely,the subset D ⊂D of the degrees of freedom to be enriched by the Haar function must have their support cut by the crack but must not contain any of the crack tips.Formally,in the case of a crack with two tips:D ={i∈D:!i∩C=∅;x1=∈!i;x2=∈!i}(22) Theÿnite element approximation after the introduction of the discontinuity is thenu h=i∈Du i i+j∈Db j j H(x)+k∈K1k4l=1c l1k F1l(x)+k∈K2k4l=1c l2k F2l(x)(23)Copyright?1999John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)FINITE ELEMENT METHOD FOR CRACK GROWTH 139Figure 10.Generation of subpolygons for the quadrature of the weak form in:(a)those elements cut by a crack.The polygons (b)formed from the intersection of the crack and the element geometries are triangulated as in (c)to create the element subdomainswhere K 1={k ∈˜D :x 1∈!k };K 2={k ∈˜D :x 2∈!k }(24)and ˜D ⊂D is the subset of nodal degrees of freedom.We therefore keep a nodal-based enrichment for the near-tip enrichment.From the expression for approximation (23),we notice that the jump displacement across the crack is now described by higher-order functions along the crack faces.In addition,we have not assumed a uniform p order of approximation.3.3.Numerical integration of the weak formFor elements cut by the crack and enriched with the jump function H (x ),we make a modiÿcation to the element quadrature routines in order to accurately assemble the contribution to the weak form on both sides of the discontinuity.As the crack is allowed to be arbitrarily oriented in an element,the use of standard Gauss quadrature may not adequately integrate the discontinuous ÿeld.If the integration of the the jump enrichment is indistinguishable from that of a constant function,spurious singular modes can appear in the system of equations.In this section,we present the modiÿcations made to the numerical integration scheme for elements cut by a crack.The discrete weak form is normally constructed with a loop over all elements,as the domain is approximated by = ee (25)where e is the element subdomain.For elements cut by a crack,we deÿne the element subdomain to be a sum of a set of subpolygons whose boundaries align with the crack geometry e = ss (26)In two dimensions,the triangles shown in Figure 10work well.Simpler schemes,such as the trapezoids used by Fish [9]may also perform adequately.It is emphasized that the subpolygons are only necessary for integration purposes;no additional degrees of freedom are associated with their construction.In the integration of the weak form,the element loop is replaced by a loop over the subpolygons for those elements cut by the crack.3.4.Crack growth and stress intensity factor evaluationIn this section,we brie y review the criterion used to specify the direction of crack growth.In addition,we describe the domain form of the interaction integral for the extraction of mixed-mode stress intensity factors.Copyright ?1999John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)140N.MO ES,J.DOLBOW AND T.BELYTSCHKO Under general mixed-mode loadings,the asymptotic near-tip circumferential and shear stresses take the following form: ÂÂ rÂ =K I √2 r 14 3cos (Â=2)+cos (3Â=2)sin (Â=2)+sin (3Â=2) +K II √2 r 14−3sin (Â=2)−3sin (3Â=2)cos (Â=2)+3cos (3Â=2) (27)Among the criteria for determining the growth direction are:(1)the maximum energy release rate criterion [10],(2)the maximum circumferential stress criterion or the maximum principal stress criterion [11]and (3)the minimum strain energy density criterion [12].In this paper,we use the maximum circumferential stress criterion,which states that the crack will propagate from its tip in a direction Âc so that the circumferential stress ÂÂis maximum.The circumferential stress in the direction of crack propagation is a principal stress.Therefore,the critical angle Âc deÿning the radial direction of propagation can be determined by setting the shear stress in (27)to zero.After a few manipulations,the following expression is obtained:1√2 r cos Â2 12K I sin(Â)+12K II (3cos(Â)−1) =0(28)This leads to the equation deÿning the angle of crack propagation Âc in the tip co-ordinate system.K I sin(Âc )+K II (3cos(Âc )−1)=0(29)Solving this equation givesÂc =2arctan 14(K I =K II ± (K I =K II )2+8)(30)The stress intensity factors are computed using domain forms of the interaction integrals [13;14].For completeness these are discussed here.The coordinates are taken to be the local crack tip co-ordinates with the x 1-axis parallel to the crack faces.For general mixed-mode problems we have the following relationship between the value of the J -integral and the stress intensity factorsJ =K 2I E ∗+K 2II E ∗(31)where E ∗is deÿned in terms of material parameters E (Young’s modulus)and (poisson’s ratio)as E ∗= E plane stress E 1−2plane strain (32)Two states of a cracked body are considered.State 1,( (1)ij ; (1)ij ;u (1)i ),corresponds to the present state and state 2,( (2)ij ; (2)ij ;u (2)i ),is an auxiliary state which will be chosen as the asymptotic ÿelds for Modes I or II.The J -integral for the sum of the two states is J (1+2)= 12( (1)ij + (2)ij )( (1)ij + (2)ij ) 1j −( (1)ij + (2)ij )@(u (1)i +u (2)i )@x 1 n j d (33)Expanding and rearranging terms givesJ (1+2)=J (1)+J (2)+I (1;2)(34)Copyright ?1999John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)Figure 11.Conventions at crack tip.Domain A is enclosed by ,C +,C −,and C 0.Unit normal m j =n j on C +,C −,and C 0and m j =−n j onFigure 12.Elements selected about the crack tip forcalculation of the interaction integralwhere I (1;2)is called the interaction integral for states 1and 2I(1;2)= W (1;2)1j − (1)ij @(u 2i )@x 1− (2)ij @(u 1i )@x 1 n j d (35)where W (1;2)is the interaction strain energyW (1;2)= (1)ij(2)ij = (2)ij (1)ij (36)Writing equation (31)for the combined states gives after rearranging termsJ (1+2)=J (1)+J (2)+2E∗(K (1)I K (2)I +K (1)II K (2)II )(37)Equating (34)with (37)leads to the following relationship:I (1;2)=2E∗(K (1)I K (2)I +K (1)IIK (2)II )(38)Making the judicious choice of state 2as the pure Mode I asymptotic ÿelds with K (2)I =1gives mode I stress intensity factor for state 1in terms of the interaction integralK (1)I =2E∗I (1;ModeI)(39)Mode II stress intensity factor can be determined in a similiar fashion.The contour integral (35)is not in a form best suited for ÿnite element calculations.We therefore recast the integral into an equivalent domain form by multiplying the integrand by a su ciently smooth weighting function q (x )which takes a value of unity on an open set containing the crack tip and vanishes on an outer prescribed contour C 0.Then for each contour as in Figure 11,assuming the crack faces are traction free and straight in the region A bounded by the contour C 0,the interaction integral may be written asI (1;2)=CW (1;2) 1j − (1)ij @u 2i 1− (2)ij @u 1i1 qm j d (40)Copyright ?1999John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)Figure13.Weight function q on the elementswhere the contour C= +C++C−+C0and m is the unit outward normal to the contour C.Now using the divergence theorem and passing to the limit as the contour is shrunk to the crack tip, gives the following equation for the interaction integral in domain form:I(1;2)=A(1)ij@u2i@x1+ (2)ij@u1i@x1−W(1;2) 1j@q@x jd A(41)where we have used the relations m j=−n j on and m j=n j on C0,C+and C−.For the numerical evaluation of the above integral,the domain A is set from the collection of elements about the crack tip.In this paper,weÿrst determine the characteristic length of an element touched by the crack tip and designate this quantity as h local.For two-dimensional analysis, this quantity is calculated as the square root of the element area.The domain A is then set to be all elements which have a node within a ball of radius r d about the crack tip.Figure12shows a typical set of elements for the domain A with the domain radius r d taken to be twice the length h local.Figure13shows the contour plot of the weight function q for these elements.The q function is taken to have a value of unity for all nodes within the ball r d,and zero on the outer contour.The function is then easily interpolated within the elements using the nodal shape functions.4.NUMERICAL EXAMPLESIn this section,we present several numerical examples of cracks and crack growth under the assumptions of plane strain two-dimensional elasticity.We begin with a simple example of an edge crack to demonstrate the robustness of the discretization scheme,and then present results for more complicated geometries.In all of the following examples,the material properties are assumed to be that of glass with young’s modulus100kpsi,and poisson ratio0·3.The calculation of the stress intensity factors is performed with the domain form of the interaction integral as detailed in the previous section.Copyright?1999John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)Figure14.The geometry of the edge crack problem for the robustness and shear studies.The parameters are a=W=1=2;L=W=16=7;W=7Figure15.Zoom of the mesh in the vicinity of the crack tip,with:(a)the initial conÿguration and x; y shown.The en-richment is also shown for;(b)theÿnal conÿguration.The circled nodes are enriched with the jump function and the squarednodes with the near-tip functions4.1.Robustness analysisConsider the geometry shown in Figure14:a plate of width w and height L with an edge crack of length a.We analyse the in uence of the location of the crack with respect to the mesh on the K I stress intensity factor when the position of the crack is perturbed by x in the X-direction and y in the Y-direction.The model is a uniform mesh of24×484-noded quadrilateral elements. In this study,several di erent discretizations are obtained depending on the position of the crack with respect to the mesh.Two cases are shown in Figure15.Depending on the location of the crack tip,the total number of degrees of freedom varies from2501to2541.The purpose of the study is to compare the accuracy of the solution when the crack is aligned with the mesh to the case when it is slightly o set.The exact solution for this problem is given by[15]K I=C √a (42)where C is aÿnite-geometry correction factor:C=1·12−0·231 aW+10·55aW2−21·72aW3+30·39aW4(43)Copyright?1999John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng.46,131–150(1999)。

New upper bounds on the chromatic number of a graph

Corollary 7. Let G be a graph. Then χ(G) ≤ 1 α(G) (ω (G) + ∆(G) + 1) + κ(G) + 1 − . 2 4
2
Chromatic Excess
Deﬁnition 8. Let G be a graph. The chromatic excess of G is deﬁned to be η (G) = max |H | − 3χ(H ).
∆(G)+2 , 2
Proof. Assume the former does not hold. Apply Corollary 7 to get upper bounds on α and ω in terms of κ. Now use Ramsey Theory. Conjecture 15. There exists a constant C > 0 such that χ>
Corollary 5. Let G be a graph. Then, for any induced subgraph H of G, χ(G) ≤ 1 5κ(G) + 3χ(H ) − |H | (ω (G) + ∆(G) + 1) + . 2 4
Corollary 6. Let G be a graph and K a cut-set in G. Then χ(G) ≤ 1 4χ(G[K ]) + α(G[K ]) + 3 − α(G) (ω (G) + ∆(G) + 1) + . 2 4
New upper bounds on the chromatic num6632v1 [math.CO] 25 Jun 2006

ECCC

We consider the conjecture stating that a matrix with rank o(n) and ones on the main diagonal must contain nonzero entries on a 2 2 submatrix with one entry on the main diagonal. We show that a slightly stronger conjecture implies that an explicit linear transformation cannot be computed by linear size and logarithmic depth circuits. We prove some partial results supporting the conjecture.
ECCC
TR97-043
FTP: WWW: Email:
ftp.eccc.uni-trier.de:/pub/eccc/ http://www.eccc.uni-trier.de/eccc/ ftpmail@ftp.eccc.uni-trier.de with subject ’help eccc’
gates computing linear functions over a given eld.
Although both a random matrix and a matrix whose entries are di erent indeterminates have rigidity even larger than required by Theorem 1 (close to n2 ), very little is known about explicit matrices. The best known lower bounds on the rigidity of explicit matrices are of the form ( nr2 log n ) 3], r which gives only linear lower bounds on RM ("n). It seems that Hadamard matrices have large rigidity over the real eld, but the best bound is so far only (n2 =r2 ), due to Alon (unpublished). Another relation of this type was found by Razborov 8]. He proved that some weaker bounds on rigidity would imply that a f0; 1g matrix de nes a function which is not in the communication complexity version of the polynomial hierarchy. The existing lower bounds on the rigidity are, however, insu cient even for that. As an intermediate step, Smolensky (and independently one of the authors) proposed to prove a large lower bound on the rigidity of a Toeplitz matrix with indeterminates. He suggested to make the problem of proving a lower bound easier by allowing the changed values to be just linear functions of the indeterminates. The proof in 7] can be easily adapted to show that the 'triangle conjecture' implies a lower bound of the type needed in Theorem 1 for a Toeplitz matrix with indeterminates. (This is true even if one allows the changes to be polynomials.) In this paper we come up with an explicit matrix which has the rigidity required by Theorem 1, just assuming a slightly stronger conjecture (Section 2). In the current situation where we lack any methods for proving large lower bounds on rigidity, this gives a concrete program for proving such bounds, provided that the Triangle Conjecture is true. Even if the conjecture fails, this matrix may be a good candidate for large rigidity. We also show a construction of circulant matrices which gives nonlinear bounds on the rigidity RM ("n), for every xed " > 0 assuming only the Triangle Conjecture (Section 3). Such a bound is su cient for proving a nonlinear lower bound on the size of series-parallel circuits computing the transformation M . In the following sections we present some supporting evidence for and results on the conjecture. We have tested the conjecture for some small symmetric matrices, in the case of rank over GF 2]. For the sizes n 32 we have veri ed that every symmetric matrix with ones on the main diagonal and rank n=4+1 contains a triangle. There is a unique, up to isomorphisms, family of symmetric matrices of rank n=4 + 2 with ones on the main diagonal and without triangles. Such matrices do exist for every n, but we cannot prove that they are extremal. They will be described in Section 5. We shall also show a decomposition for symmetric matrices over GF 2] with at least one 1 on the main diagonal, which simpli es either the search for counterexamples or a possible proof (Section 2

Insight Problem Solving A Critical Examination of the Possibility

The Journal of Problem Solving • volume 5, no. 1 (Fall 2012)56Insight Problem Solving: A Critical Examination of the Possibilityof Formal TheoryWilliam H. Batchelder 1 and Gregory E. Alexander 1AbstractThis paper provides a critical examination of the current state and future possibility of formal cognitive theory for insight problem solving and its associated “aha!” experience. Insight problems are contrasted with move problems, which have been formally defined and studied extensively by cognitive psychologists since the pioneering work of Alan Newell and Herbert Simon. To facilitate our discussion, a number of classical brainteasers are presented along with their solutions and some conclusions derived from observing the behavior of many students trying to solve them. Some of these problems are interesting in their own right, and many of them have not been discussed before in the psychologi-cal literature. The main purpose of presenting the brainteasers is to assist in discussing the status of formal cognitive theory for insight problem solving, which is argued to be considerably weaker than that found in other areas of higher cognition such as human memory, decision-making, categorization, and perception. We discuss theoretical barri-ers that have plagued the development of successful formal theory for insight problem solving. A few suggestions are made that might serve to advance the field.Keywords Insight problems, move problems, modularity, problem representation1 Department of Cognitive Sciences, University of California Irvine/10.7771/1932-6246.1143Insight Problem Solving: The Possibility of Formal Theory 57• volume 5, no. 1 (Fall 2012)1. IntroductionThis paper discusses the current state and a possible future of formal cognitive theory for insight problem solving and its associated “aha!” experience. Insight problems are con-trasted with so-called move problems defined and studied extensively by Alan Newell and Herbert Simon (1972). These authors provided a formal, computational theory for such problems called the General Problem Solver (GPS), and this theory was one of the first formal information processing theories to be developed in cognitive psychology. A move problem is posed to solvers in terms of a clearly defined representation consisting of a starting state, a description of the goal state(s), and operators that allow transitions from one problem state to another, as in Newell and Simon (1972) and Mayer (1992). A solu-tion to a move problem involves applying operators successively to generate a sequence of transitions (moves) from the starting state through intermediate problem states and finally to a goal state. Move problems will be discussed more extensively in Section 4.6.In solving move problems, insight may be required for selecting productive moves at various states in the problem space; however, for our purposes we are interested in the sorts of problems that are described often as insight problems. Unlike Newell and Simon’s formal definition of move problems, there has not been a generally agreed upon defini-tion of an insight problem (Ash, Jee, and Wiley, 2012; Chronicle, MacGregor, and Ormerod, 2004; Chu and MacGregor, 2011). It is our view that it is not productive to attempt a pre-cise logical definition of an insight problem, and instead we offer a set of shared defining characteristics in the spirit of Wittgenstein’s (1958) definition of ‘game’ in terms of family resemblances. Problems that we will treat as insight problems share many of the follow-ing defining characteristics: (1) They are posed in such a way as to admit several possible problem representations, each with an associated solution search space. (2) Likely initial representations are inadequate in that they fail to allow the possibility of discovering a problem solution. (3) In order to overcome such a failure, it is necessary to find an alternative productive representation of the problem. (4) Finding a productive problem representation may be facilitated by a period of non-solving activity called incubation, and also it may be potentiated by well-chosen hints. (5) Once obtained, a productive representation leads quite directly and quickly to a solution. (6) The solution involves the use of knowledge that is well known to the solver. (7) Once the solution is obtained, it is accompanied by a so-called “aha!” experience. (8) When a solution is revealed to a non-solver, it is grasped quickly, often with a feeling of surprise at its simplicity, akin to an “aha!” experience.It is our position that very little is known empirically or theoretically about the cogni-tive processes involved in solving insight problems. Furthermore, this lack of knowledge stands in stark contrast with other areas of cognition such as human memory, decision-making, categorization, and perception. These areas of cognition have a large number of replicable empirical facts, and many formal theories and computational models exist that attempt to explain these facts in terms of underlying cognitive processes. The main goal58W. H. Batchelder and G. E. Alexander of this paper is to explain the reasons why it has been so difficult to achieve a scientific understanding of the cognitive processes involved in insight problem solving.There have been many scientific books and papers on insight problem solving, start-ing with the seminal work of the Gestalt psychologists Köhler (1925), Duncker (1945), and Wertheimer (1954), as well as the English social psychologist, Wallas (1926). Since the contributions of the early Gestalt psychologists, there have been many journal articles, a few scientific books, such as those by Sternberg and Davidson (1996) and Chu (2009), and a large number of books on the subject by laypersons. Most recently, two excellent critical reviews of insight problem solving have appeared: Ash, Cushen, and Wiley (2009) and Chu and MacGregor (2011).The approach in this paper is to discuss, at a general level, the nature of several fun-damental barriers to the scientific study of insight problem solving. Rather than criticizing particular experimental studies or specific theories in detail, we try to step back and take a look at the area itself. In this effort, we attempt to identify principled reasons why the area of insight problem solving is so resistant to scientific progress. To assist in this approach we discuss and informally analyze eighteen classical brainteasers in the main sections of the paper. These problems are among many that have been posed to hundreds of upper divisional undergraduate students in a course titled “Human Problem Solving” taught for many years by the senior author. Only the first two of these problems can be regarded strictly as move problems in the sense of Newell and Simon, and most of the rest share many of the characteristics of insight problems as described earlier.The paper is divided into five main sections. After the Introduction, Section 2 describes the nature of the problem solving class. Section 3 poses the eighteen brainteasers that will be discussed in later sections of the paper. The reader is invited to try to solve these problems before checking out the solutions in the Appendix. Section 4 lays out six major barriers to developing a deep scientific theory of insight problem solving that we believe are endemic to the field. We argue that these barriers are not present in other, more theo-retically advanced areas of higher cognition such as human memory, decision-making, categorization, and perception. These barriers include the lack of many experimental paradigms (4.1), the lack of a large, well-classified set of stimulus material (4.2), and the lack of many informative behavioral measures (4.3). In addition, it is argued that insight problem solving is difficult to study because it is non-modular, both in the sense of Fodor (1983) but more importantly in several weaker senses of modularity that admit other areas of higher cognition (4.4), the lack of theoretical generalizations about insight problem solv-ing from experiments with particular insight problems (4.5), and the lack of computational theories of human insight (4.6). Finally, in Section 5, we suggest several avenues that may help overcome some of the barriers described in Section 4. These include suggestions for useful classes of insight problems (5.1), suggestions for experimental work with expert problem solvers (5.2), and some possibilities for a computational theory of insight.The Journal of Problem Solving •Insight Problem Solving: The Possibility of Formal Theory 592. Batchelder’s Human Problem Solving ClassThe senior author, William Batchelder, has taught an Upper Divisional Undergraduate course called ‘Human Problem Solving” for over twenty-five years to classes ranging in size from 75 to 100 students. By way of background, his active research is in other areas of the cognitive sciences; however, he maintains a long-term hobby of studying classical brainteasers. In the area of complex games, he achieved the title of Senior Master from the United States Chess Federation, he was an active duplicate bridge player throughout undergraduate and graduate school, and he also achieved a reasonable level of skill in the game of Go.The content of the problem-solving course is split into two main topics. The first topic involves encouraging students to try their hand at solving a number of famous brainteasers drawn from the sizeable folklore of insight problems, especially the work of Martin Gardner (1978, 1982), Sam Loyd (1914), and Raymond Smullyan (1978). In addition, games like chess, bridge, and Go are discussed. The second topic involves presenting the psychological theory of thinking and problem solving, and in most cases the material is organized around developments in topics that are covered in the first eight chapters of Mayer (1992). These topics include work of the Gestalt psychologists on problem solving, discussion of experiments and theories concerning induction and deduction, present-ing the work on move problems, including the General Problem Solver (Newell & Simon, 1972), showing how response time studies can reveal mental architectures, and describing theories of memory representation and question answering.Despite efforts, the structure of the course does not reflect a close overlap between its two main topics. The principal reason for this is that in our view the level of theoreti-cal and empirical work on insight problem solving is at a substantially lower level than is the work in almost any other area of cognition dealing with higher processes. The main goal of this paper is to explain our reasons for this pessimistic view. To assist in this goal, it is helpful to get some classical brainteasers on the table. While most of these problems have not been used in experimental studies, the senior author has experienced the solu-tion efforts and post solution discussions of over 2,000 students who have grappled with these problems in class.3. Some Classic BrainteasersIn this section we present eighteen classical brainteasers from the folklore of problem solving that will be discussed in the remainder of the paper. These problems have de-lighted brainteaser connoisseurs for years, and most are capable of giving the solver a large dose of the “aha!” experience. There are numerous collections of these problems in books, and many collections of them are accessible through the Internet. We have selected these problems because they, and others like them, pose a real challenge to any effort to • volume 5, no. 1 (Fall 2012)60W. H. Batchelder and G. E. Alexander develop a deep and general formal theory of human or machine insight problem solving. With the exception of Problems 3.1 and 3.2, and arguably 3.6, the problems are different in important respects from so-called move problems of Newell and Simon (1972) described earlier and in Section 4.6.Most of the problems posed in this section share many of the defining characteristics of insight problems described in Section 1. In particular, they do not involve multiple steps, they require at most a very minimal amount of technical knowledge, and most of them can be solved by one or two fairly simple insights, albeit insights that are rarely achieved in real time by problem solvers. What makes these problems interesting is that they are posed in such a way as to induce solvers to represent the problem information in an unproductive way. Then the main barrier to finding a solution to one of these problems is to overcome a poor initial problem representation. This may involve such things as a re-representation of the problem, the dropping of an implicit constraint on the solution space, or seeing a parallel to some other similar problem. If the solver finds a productive way of viewing the problem, the solution generally follows rapidly and comes with burst of insight, namely the “aha!” experience. In addition, when non-solvers are given the solu-tion they too may experience a burst of insight.What follows next are statements of the eighteen brainteasers. The solutions are presented in the Appendix, and we recommend that after whatever problem solving activity a reader wishes to engage in, that the Appendix is studied before reading the remaining two sections of the paper. As we discuss each problem in the paper, we provide authorship information where authorship is known. In addition, we rephrased some of the problems from their original sources.Problem 3.1. Imagine you have an 8-inch by 8-inch array of 1-inch by 1-inch little squares. You also have a large box of 2-inch by 1-inch rectangular shaped dominoes. Of course it is easy to tile the 64 little squares with dominoes in the sense that every square is covered exactly once by a domino and no domino is hanging off the array. Now sup-pose the upper right and lower left corner squares are cut off the array. Is it possible to tile the new configuration of 62 little squares with dominoes allowing no overlaps and no overhangs?Problem 3.2. A 3-inch by 3-inch by 3-inch cheese cube is made of 27 little 1-inch cheese cubes of different flavors so that it is configured like a Rubik’s cube. A cheese-eating worm devours one of the top corner cubes. After eating any little cube, the worm can go on to eat any adjacent little cube (one that shares a wall). The middlemost little cube is by far the tastiest, so our worm wants to eat through all the little cubes finishing last with the middlemost cube. Is it possible for the worm to accomplish this goal? Could he start with eating any other little cube and finish last with the middlemost cube as the 27th?The Journal of Problem Solving •Insight Problem Solving: The Possibility of Formal Theory 61 Figure 1. The cheese eating worm problem.Problem 3.3. You have ten volumes of an encyclopedia numbered 1, . . . ,10 and shelved in a bookcase in sequence in the ordinary way. Each volume has 100 pages, and to simplify suppose the front cover of each volume is page 1 and numbering is consecutive through page 100, which is the back cover. You go to sleep and in the middle of the night a bookworm crawls onto the bookcase. It eats through the first page of the first volume and eats continuously onwards, stopping after eating the last page of the tenth volume. How many pieces of paper did the bookworm eat through?Figure 2.Bookcase setup for the Bookworm Problem.Problem 3.4. Suppose the earth is a perfect sphere, and an angel fits a tight gold belt around the equator so there is no room to slip anything under the belt. The angel has second thoughts and adds an inch to the belt, and fits it evenly around the equator. Could you slip a dime under the belt?• volume 5, no. 1 (Fall 2012)62W. H. Batchelder and G. E. Alexander Problem 3.5. Consider the cube in Figure 1 and suppose the top and bottom surfaces are painted red and the other four sides are painted blue. How many little cubes have at least one red and at least one blue side?Problem 3.6. Look at the nine dots in Figure 3. Your job is to take a pencil and con-nect them using only three straight lines. Retracing a line is not allowed and removing your pencil from the paper as you draw is not allowed. Note the usual nine-dot problem requires you to do it with four lines; you may want to try that stipulation as well. Figure 3.The setup for the Nine-Dot Problem.Problem 3.7. You are standing outside a light-tight, well-insulated closet with one door, which is closed. The closet contains three light sockets each containing a working light bulb. Outside the closet, there are three on/off light switches, each of which controls a different one of the sockets in the closet. All switches are off. Your task is to identify which switch operates which light bulb. You can turn the switches off and on and leave them in any position, but once you open the closet door you cannot change the setting of any switch. Your task is to figure out which switch controls which light bulb while you are only allowed to open the door once.Figure 4.The setup of the Light Bulb Problem.The Journal of Problem Solving •Insight Problem Solving: The Possibility of Formal Theory 63• volume 5, no . 1 (Fall 2012)Problem 3.8. We know that any finite string of symbols can be extended in infinitely many ways depending on the inductive (recursive) rule; however, many of these ways are not ‘reasonable’ from a human perspective. With this in mind, find a reasonable rule to continue the following series:Problem 3.9. You have two quart-size beakers labeled A and B. Beaker A has a pint of coffee in it and beaker B has a pint of cream in it. First you take a tablespoon of coffee from A and pour it in B. After mixing the contents of B thoroughly you take a tablespoon of the mixture in B and pour it back into A, again mixing thoroughly. After the two transfers, which beaker, if either, has a less diluted (more pure) content of its original substance - coffee in A or cream in B? (Forget any issues of chemistry such as miscibility).Figure 5. The setup of the Coffee and Cream Problem.Problem 3.10. There are two large jars, A and B. Jar A is filled with a large number of blue beads, and Jar B is filled with the same number of red beads. Five beads from Jar A are scooped out and transferred to Jar B. Someone then puts a hand in Jar B and randomly grabs five beads from it and places them in Jar A. Under what conditions after the second transfer would there be the same number of red beads in Jar A as there are blue beads in Jar B.Problem 3.11. Two trains A and B leave their train stations at exactly the same time, and, unaware of each other, head toward each other on a straight 100-mile track between the two stations. Each is going exactly 50 mph, and they are destined to crash. At the time the trains leave their stations, a SUPERFLY takes off from the engine of train A and flies directly toward train B at 100 mph. When he reaches train B, he turns around instantly, A BCD EF G HI JKLM.............64W. H. Batchelder and G. E. Alexander continuing at 100 mph toward train A. The SUPERFLY continues in this way until the trains crash head-on, and on the very last moment he slips out to live another day. How many miles does the SUPERFLY travel on his zigzag route by the time the trains collide?Problem 3.12. George lives at the foot of a mountain, and there is a single narrow trail from his house to a campsite on the top of the mountain. At exactly 6 a.m. on Satur-day he starts up the trail, and without stopping or backtracking arrives at the top before6 p.m. He pitches his tent, stays the night, and the next morning, on Sunday, at exactly 6a.m., he starts down the trail, hiking continuously without backtracking, and reaches his house before 6 p.m. Must there be a time of day on Sunday where he was exactly at the same place on the trail as he was at that time on Saturday? Could there be more than one such place?Problem 3.13. You are driving up and down a mountain that is 20 miles up and 20 miles down. You average 30 mph going up; how fast would you have to go coming down the mountain to average 60 mph for the entire trip?Problem 3.14. During a recent census, a man told the census taker that he had three children. The census taker said that he needed to know their ages, and the man replied that the product of their ages was 36. The census taker, slightly miffed, said he needed to know each of their ages. The man said, “Well the sum of their ages is the same as my house number.” The census taker looked at the house number and complained, “I still can’t tell their ages.” The man said, “Oh, that’s right, the oldest one taught the younger ones to play chess.” The census taker promptly wrote down the ages of the three children. How did he know, and what were the ages?Problem 3.15. A closet has two red hats and three white hats. Three participants and a Gamesmaster know that these are the only hats in play. Man A has two good eyes, man B only one good eye, and man C is blind. The three men sit on chairs facing each other, and the Gamesmaster places a hat on each man’s head, in such a way that no man can see the color of his own hat. The Gamesmaster offers a deal, namely if any man correctly states the color of his hat, he will get $50,000; however, if he is in error, then he has to serve the rest of his life as an indentured servant to the Gamesmaster. Man A looks around and says, “I am not going to guess.” Then Man B looks around and says, “I am not going to guess.” Finally Man C says, “ From what my friends with eyes have said, I can clearly see that my hat is _____”. He wins the $50,000, and your task is to fill in the blank and explain how the blind man knew the color of his hat.Problem 3.16. A king dies and leaves an estate, including 17 horses, to his three daughters. According to his will, everything is to be divided among his daughters as fol-lows: 1/2 to the oldest daughter, 1/3 to the middle daughter, and 1/9 to the youngest daughter. The three heirs are puzzled as to how to divide the horses among themselves, when a probate lawyer rides up on his horse and offers to assist. He adds his horse to the kings’ horses, so there will be 18 horses. Then he proceeds to divide the horses amongThe Journal of Problem Solving •Insight Problem Solving: The Possibility of Formal Theory 65 the daughters. The oldest gets ½ of the horses, which is 9; the middle daughter gets 6 horses which is 1/3rd of the horses, and the youngest gets 2 horses, 1/9th of the lot. That’s 17 horses, so the lawyer gets on his own horse and rides off with a nice commission. How was it possible for the lawyer to solve the heirs’ problem and still retain his own horse?Problem 3.17. A logical wizard offers you the opportunity to make one statement: if it is false, he will give you exactly ten dollars, and if it is true, he will give you an amount of money other than ten dollars. Give an example of a statement that would be sure to make you rich.Problem 3.18. Discover an interesting sense of the claim that it is in principle impos-sible to draw a perfect map of England while standing in a London flat; however, it is not in principle impossible to do so while living in a New York City Pad.4. Barriers to a Theory of Insight Problem SolvingAs mentioned earlier, our view is that there are a number of theoretical barriers that make it difficult to develop a satisfactory formal theory of the cognitive processes in play when humans solve classical brainteasers of the sort posed in Section 3. Further these barriers seem almost unique to insight problem solving in comparison with the more fully developed higher process areas of the cognitive sciences such as human memory, decision-making, categorization, and perception. Indeed it seems uncontroversial to us that neither human nor machine insight problem solving is well understood, and com-pared to other higher process areas in psychology, it is the least developed area both empirically and theoretically.There are two recent comprehensive critical reviews concerning insight problem solving by Ash, Cushen, and Wiley (2009) and Chu and MacGregor (2011). These articles describe the current state of empirical and theoretical work on insight problem solving, with a focus on experimental studies and theories of problem restructuring. In our view, both reviews are consistent with our belief that there has been very little sustainable progress in achieving a general scientific understanding of insight. Particularly striking is that are no established general, formal theories or models of insight problem solving. By a general formal model of insight problem solving we mean a set of clearly formulated assumptions that lead formally or logically to precise behavioral predictions over a wide range of insight problems. Such a formal model could be posed in terms of a number of formal languages including information processing assumptions, neural networks, computer simulation, stochastic assumptions, or Bayesian assumptions.Since the groundbreaking work by the Gestalt psychologists on insight problem solving, there have been theoretical ideas that have been helpful in explaining the cog-nitive processes at play in solving certain selected insight problems. Among the earlier ideas are Luchins’ concept of einstellung (blind spot) and Duncker’s functional fixedness, • volume 5, no. 1 (Fall 2012)as in Maher (1992). More recently, there have been two developed theoretical ideas: (1) Criterion for Satisfactory Progress theory (Chu, Dewald, & Chronicle, 2007; MacGregor, Ormerod, & Chronicle, 2001), and (2) Representational Change Theory (Knoblich, Ohls-son, Haider, & Rhenius, 1999). We will discuss these theories in more detail in Section 4. While it is arguable that these theoretical ideas have done good work in understanding in detail a few selected insight problems, we argue that it is not at all clear how these ideas can be generalized to constitute a formal theory of insight problem solving at anywhere near the level of generality that has been achieved by formal theories in other areas of higher process cognition.The dearth of formal theories of insight problem solving is in stark contrast with other areas of problem solving discussed in Section 4.6, for example move problems discussed earlier and the more recent work on combinatorial optimization problems such as the two dimensional traveling salesman problem (MacGregor and Chu, 2011). In addition, most other higher process areas of cognition are replete with a variety of formal theories and models. For example, in the area of human memory there are currently a very large number of formal, information processing models, many of which have evolved from earlier mathematical models, as in Norman (1970). In the area of categorization, there are currently several major formal theories along with many variations that stem from earlier theories discussed in Ashby (1992) and Estes (1996). In areas ranging from psycholinguistics to perception, there are a number of formal models based on brain-style computation stemming from Rumelhart, McClelland, and PDP Research Group’s (1987) classic two-volume book on parallel distributed processing. Since Daniel Kahneman’s 2002 Nobel Memorial Prize in the Economic Sciences for work jointly with Amos Tversky developing prospect theory, as in Kahneman and Tversky (1979), psychologically based formal models of human decision-making is a major theoretical area in cognitive psychology today. In our view, there is nothing in the area of insight problem solving that approaches the depth and breadth of formal models seen in the areas mentioned above.In the following subsections, we will discuss some of the barriers that have prevented the development of a satisfactory theory of insight problem solving. Some of the bar-riers will be illustrated with references to the problems in Section 3. Then, in Section 5 we will assuage our pessimism a bit by suggesting how some of these barriers might be removed in future work to facilitate the development of an adequate theory of insight problem solving.4.1 Lack of Many Experimental ParadigmsThere are not many distinct experimental paradigms to study insight problem solving. The standard paradigm is to pick a particular problem, such as one of the ones in Section 3, and present it to several groups of subjects, perhaps in different ways. For example, groups may differ in the way a hint is presented, a diagram is provided, or an instruction。

Modular verification of software components in C

Modular Veriﬁcation ofSoftware Components in CSagar Chaki Edmund Clarke Alex GroceCarnegie Mellon University{chaki|emc|agroce}@Somesh Jha,University of Wisconsinjha@Helmut Veith,Technische Universit¨a t M¨u nchenveith@in.tum.de(Invited Paper)This research was supported by the ONR under Grant No.N00014-01-1-0796,by the NSF under Grant R-9803774, CCR-0121547and CCR-0098072,by the ARO under Grant No.DAAD19-01-1-0485,the Austrian Science Fund Project NZ29-INF,the EU Research and Training Network GAMES and graduate student fellowships from Microsoft and NSF.Any opinions,ﬁndings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reﬂect the views of NSF or the United States Government.AbstractWe present a new methodology for automatic veriﬁcation of C programs againstﬁnite state machine speciﬁcations.Our approach is compositional,naturally enabling us to decompose the veriﬁcation oflarge software systems into subproblems of manageable complexity.The decomposition reﬂects themodularity in the software design.We use weak simulation as the notion of conformance between theprogram and its speciﬁcation.Following the counterexample guided abstraction reﬁnement(CEGAR)paradigm,our tool MAGICﬁrst extracts aﬁnite model from C source code using predicate abstractionand theorem proving.Subsequently,weak simulation is checked via a reduction to Boolean satisﬁability.MAGIC is able to interface with several publicly available theorem provers and SAT solvers.We reportexperimental results with procedures from the Linux kernel and the OpenSSL toolkit.Index TermsSoftware Engineering,Formal Methods,Veriﬁcation.I.I NTRODUCTIONState machines have been recognized repeatedly as an important artifact in the software development process;in fact,variants of state machines have been proposed for virtually all software engineering methodologies,including,most notably,Statecharts[1]and the UML[2]. The sustained success of state machines in software engineering stems from the fact that state machines provide for both a concise mathematical theory,and an intuitive semantics of system behavior which naturally allows for visualization,hierarchy,and abstraction.Traditionally,state machines have been mainly used in the design phase of the software life-cycle;they are intended to guide and constrain the implementation and the test phase,and may later be reused for documentation purposes.In most cases,however,the assertion that a state machine safely abstracts an existing implementation is kept implicit and informal.With the rise of Internet-based technologies,the signiﬁcance of state machines has only increased.In particular,security protocols and communication protocols are naturally speciﬁed in terms of state machines[3],[4],[5].Similar applications of state machines can be found in other safety-critical domains including medicine and aerospace.Moreover,the dramatic change of focus from relatively monolithic systems to highly distributed and heterogeneous systems whose development cycles are interdependent,callsfor new speciﬁcation methodologies;for example,on August2002,IBM,Microsoft,and BEA announced the publication of three speciﬁcations(WS-Coordination,WS-Speciﬁcation, BPEL4WS[6])which”collectively describe how to reliably deﬁne,create and connect multiple business processes in a Web services environment”.We foresee state machines being used for contracts describing software capabilities.In both cases–protocol speciﬁcation and distributed computation–we observe that state machines are no longer just tools for internal use,but are being introduced increasingly into the public domain.In this paper,we describe our tool MAGIC(M odular A nalysis of pro G rams I n C)[7]which is capable of verifying whether a state machine(or,more precisely,a labeled transition system)is a safe abstraction of a C procedure;the C procedure in turn may invoke other procedures which are themselves speciﬁed in terms of state machines.Our approach has a number of tangible beneﬁts:•Utility.The capability of MAGIC to formally verify the correctness of state-machine speci-ﬁcations closes an evident gap in many software development methodologies,most notably, but not only,for security-related system features.In the future,we envision that tools based on ideas from MAGIC will assist the contracting process with third party software providers.•Compositionality.MAGIC veriﬁcation can be used early on during the development cycle,as speciﬁcations can be plugged in for missing system positionality evidently fosters concurrent development by independent groups of developers.•Complexity.State-space explosion[3]remains the bottleneck of most automated veriﬁcation tools.Due to compositionality,the size of the individual system parts to be veriﬁed by MAGIC remains manageable,as demonstrated by our experiments.Moreover,the veriﬁcation process in MAGIC is reduced to computing a weak simulation relation betweenﬁnite state systems, for which we can provide highly efﬁcient algorithms.•Flexibility.Internally,MAGIC uses several theorem provers and SAT solvers.The open design of MAGIC facilitates the easy integration of new and improved tools from this quickly developing area.Consequently,we believe that MAGIC like tools have the potential to become indispensable in the software engineering process.In the rest of this section we describe the technical contributions of this paper.beled Transition Systems as Speciﬁcation MechanismIn the literature,several variants of state machines have been investigated;purely state-based formalisms such as Kripke structures[3]are often used to model and specify systems.For the MAGIC framework,however,we employ labeled transition systems(LTS),which are similar to Kripke structures but for the fact that state transitions are labeled by actions.From a theoretical point of view the presence of actions does not increase the expressive power of LTS over Kripke structures.In our experience,however,it is more natural for designers and software engineers to express the desired behavior of systems using a combination of states and actions.For example,the fact that a lock has been acquired or released can be expressed naturally by lock and unlock actions.In the absence of actions,the natural alternative is to introduce a new variable indicating the status of the lock,and update it accordingly.The LTS approach certainly is more intuitive,and allows both for a simpler theory and for an easier speciﬁcation process. Some sample LTSs used in our framework are shown in Figure4.A formal deﬁnition will be given in Section III.The use of LTSs is also motivated by work in concurrency.Process algebras like CCS[8], CSP[9]and theπ-calculus[10]have been used widely to formally reason about message passing concurrent systems.In these formalisms,actions are crucial for modeling the sending and receiving of messages across channels.Process algebras lead very naturally to LTSs.Thus, even though we currently only analyze sequential programs,we believe that the use of LTSs will facilitate a smooth transition to concurrent message-passing programs in the future.B.Procedure AbstractionsThe goal of MAGIC is to verify whether the implementation of a system is safely abstracted by its speciﬁcation.To this end,MAGIC veriﬁes individual procedures against the respective LTS. In our implementation,it is possible to handle a group of procedures with a dag-like call graph as a single procedure by inlining;therefore,for simplicity,we speak only of single procedures in this paper.In practice,it often happens that single procedures perform quite different tasks for certain settings of their parameters.In our approach,this phenomenon is accounted for by allowing multiple LTSs to represent a single procedure.The selection among these LTSs is achieved byguards,i.e.,formulas which describe the conditions on the procedure parameters under which a certain LTS is applicable.This gives rise to the notion of procedure abstraction(PA);formally a PA for a procedure proc is a tuple d,l where:•d is the declaration for proc,as it appears in a C headerﬁle.•l is aﬁnite list g1,M1 ,..., g n,M n where each g i is a guard formula ranging over the parameters of proc,and each M i is an LTS with a single initial state.The procedure abstraction expresses that proc conforms to one LTS chosen among the M i’s. More precisely,proc conforms to M i if the corresponding guard g i evaluates to true over the actual arguments passed to proc.We require that the guard formulas g i be mutually exclusive so that the choice of M i is unambiguous.positionalityThe general goal of MAGIC is to prove that a user-deﬁned PA for proc is valid.The role of PAs in this process is twofold:1)A target PA is used to describe the desired behavior of the procedure proc.2)To assist the veriﬁcation process,we employ valid PAs(called the assumption PAs)forlibrary routines used by proc.Thus,PAs can be seen both as conclusions and as assumptions of the veriﬁcation process. Consequently,our methodology yields a scalable and compositional approach for verifying large software systems.Figure1illustrates this by depicting the call graph of an implementation and the steps involved in verifying it.In order to verify baz we need only assumption PAs for the other library routines.For bar we additionally use the PA for baz as an assumption PA while for foo we employ the PAs of both bar and baz as assumptions.Note that due to the sound compositional principles on which MAGIC is based upon,no particular ordering of these veriﬁcation steps is required.Assumption PAs are not only important for compositionality,they are in fact essential for handling recursive library routines.Since MAGIC inlines all library routines for which assumption PAs are unavailable,it would be unable to proceed if the assumption PA for a recursive library routine was absent.Without loss of generality we will assume throughout this paper that the targetPA contains only one guard G Spec and one LTS M Spec.To achieve the result in full generality,.the described algorithm can be iterated for each guard of M SpecD.Algorithms and Tool DescriptionThe MAGIC tool follows the CEGAR paradigm[11],[12],[13],[14]that can be summarized as follows:•Step1:Model Creation.Extract an LTS M Imp from proc using the assumed PAs,the guard G Spec and a set of predicates.In MAGIC,the model is computed from the controlﬂow graph(CFG)of the program in combination with an abstraction method called predicate abstraction[12],[15],[16].To decide properties such as equivalence of predicates,we use theorem provers.The details of this step are described in Section IV.•Step2:Veriﬁcation.Check if M Spec safely abstracts M Imp.If this is the case,the veriﬁcation successfully terminates;otherwise,extract a counterexample and perform step3.In MAGIC, the veriﬁcation step amounts to checking whether a weak simulation relation(cf.Section III) holds between M Spec and M Imp.We reduce weak simulation to the satisﬁability of a certain Boolean formula,thus utilizing highly efﬁcient SAT procedures.The details of this step are described in Section V.•Step3:Validation.Check whether the counterexample extracted in step2is valid.If this is the case,then we have found an actual bug and the veriﬁcation terminates unsuccessfully.Otherwise construct an explanation for the spuriousness of the counterexample and proceed to Step4.•Step4:Reﬁe the explanation from the previous step to construct an improved set of predicates.Return to Step1to extract a more precise M Imp using the new set of predicates instead of the old one.The new predicate set is constructed in such a way as to guarantee that all spurious counterexamples encountered so far will not appear in any future iteration of this loop.At its current stage of development,MAGIC can perform all the above steps in an automated manner.The input to MAGIC consists of(i)a set of preprocessed ANSI-Cﬁles representing proc and(ii)a set of speciﬁcationﬁles containing textual descriptions of M Spec,G Spec and a set of predicates for abstraction.The textual descriptions of LTSs are given using an extended version of the FSP notation by Magee and Kramer[17].For example,the LTS Do A shown in Figure4 is described textually as follows:A1=(a->A2),A2=(return{}->STOP).E.Tool OverviewThe schematic in Figure2explains the software architecture of MAGIC.Model Creation is handled by Stage I of the program.In this stage,the inputﬁles are parsed and the controlﬂow graph(CFG)of the C program is constructed.Simpliﬁcations are made so that the resulting CFG only has simple statements and side-effect free expressions.Finally,M Imp is extracted from the annotated CFG using the assumed PAs,G Spec and the predicates.As described later,this process requires the use of theorem provers.MAGIC can interact with several public domain theorem provers,such as Simplify[18],CVC[19],ICS[20],CVC Lite[21],and CPROVER[22].Veriﬁcation is performed in Stage II.As mentioned above,weak simulation here is reduced to a form of Boolean satisﬁability.MAGIC can interface with several publicly available SAT solvers, such as Chaff[23],FGRASP[24]and SATO[25].We also have our own efﬁcient SAT solver implementation which leverages the speciﬁc nature of SAT formulas that arise in this stage toVerificationUnsuccessful Fig.2.Overall architecture of MAGIC.deliver better performance than the public domain solvers.The veriﬁcation process is presented in Section V in more detail.If the veriﬁcation step fails,MAGIC generates an appropriate counterexample and checks its validity in Stage III.If the counterexample is found to be spurious,an improved set of predicates is computed in Stage IV and the entire process is repeated from Stage I.Stages III and IV are completely automated and require the use of theorem provers.In this paper we focus on model creation and veriﬁcation;details about counterexample validation and abstraction reﬁnement are presented elsewhere[26].The rest of this paper is organized as follows:In Section II we present related work.This is followed in Section III by some basic deﬁnitions that are used in the rest of this article.In Section IV we describe in detail the model construction procedure used in MAGIC to extract LTS models from C programs.Section V describes how we check weak simulation between M Spec and M Imp using Boolean satisﬁability.In Section VI we present a broad range of benchmarks and results that we have used to evaluate MAGIC.Finally,in Section VII we give an overviewof various ongoing and future research directions that are relevant to MAGIC.II.R ELATED W ORKDuring the last years advances in veriﬁcation methodology as well as in computing power have promoted renewed interest in software veriﬁcation.The resulting systems–most notably Bandera[27]and Java PathFinder[28],[29],ESC Java[30],SLAM[31],BLAST[32]and MC[33],[34]–are increasingly able to handle industrial software.Among the six mentioned systems,theﬁrst three focus on Java,while the last three all deal with C.Java veriﬁcation is quite different from C,because object orientation,garbage collection and the logical memory model require speciﬁc analysis methods.Among the C veriﬁcation tools,MC(which stands for meta-compilation)has a distinguished place because it amounts to a form of pattern matching on the source code,with surprisingly good results for scanning relatively simple errors in large amounts of code.SLAM and BLAST are closely related tools,whose technicalﬂavor is most akin to ours.SLAM is primarily optimized to analyze device drivers,and is going to be included in the Windows development cycle.In contrast to SLAM which uses symbolic algorithms,BLAST is an on-the-ﬂy reachability analysis tool.MAGIC is the only tool which uses LTS as speciﬁcation formalism,and weak simulation as the notion of conformance.This choice reﬂects the area of security currently being our primary application domain.Except for MC and ESC Java,the above-mentioned tools are based on variations of model checking[3],[35],and they all require abstraction methods to alleviate the state explosion problem,most notably data abstraction[36]and the more generally predicate abstraction[16]. The abstraction method used in SLAM and BLAST is closest to ours.However,due to compositionality,we can afford to invest more computing power into computing abstractions, and are therefore able to improve on Cartesian abstraction[37].Generally,we believe that the form of compositionality provided by MAGIC is unique among existing software veriﬁcation systems.Virtually all systems that use abstraction interface with theorem provers for various purposes. The software architecture of MAGIC is designed as to facilitate the integration of various theorem provers.In addition,MAGIC is the only tool which leverages the enormous success of SAT procedures in hardware veriﬁcation[38]in software veriﬁcation.SAT procedures have been successfully used for checking validity of software speciﬁcations(expressed in a relationalcalculus)[39],[40],[41].III.D EFINITIONSIn this section we present some basic deﬁnitions that will be used in the rest of this article.beled Transition SystemsA labeled transition system(LTS)M is a4-tuple S,init,Σ,T ,where(i)S is aﬁnite non-empty set of states,(ii)init∈S is the initial state,(iii)Σis aﬁnite set of actions(alphabet), and(iv)T⊆S×Σ×S is the transition relation.We assume that there is a distinguished state STOP∈S which has no outgoing transitions, i.e.,∀s∈S,∀a∈Σ,(STOP,a,s)∈T.We will write s a−→t to mean(s,a,t)∈T and denotethe set{t|s a−→t}by Succ(s,a).B.ActionsIn accordance with existing practice,we use actions to denote externally visible behaviors of systems being analyzed,e.g.acquiring a lock.Actions are atomic,and are distinguished simply by their names.Often,the presence of an action indicates a certain behavior which is achieved by a sub-procedure in the implementation.Since we are analyzing C,a procedural language, we model the termination of a procedure(i.e.,a return from the procedure)by a special class of actions called return actions.Every return action r is associated with a unique return value RetVal(r).Return values are either integers or void.We denote the set of all return actions whose return values are integers by IntRet and the special return action whose return value is void by VoidRet.All actions which are not return actions are called basic actions.A distinguished basic action τdenotes the occurrence of an unobservable internal event.In this article we only consider procedures that terminate by returning.In particular,we do not handle constructs like setjmp and longjmp.Furthermore,since LTSs are used to model procedures,any LTS S,init,Σ,T must obey the following condition:∀s∈S,s a−→STOP iff a is a return action.C.Conformance via Weak SimulationIn the context of LTS,simulation[8]is the natural notion of conformance between a speciﬁcation LTS and an implementation pared to conformance notions based on trace containment[11],simulation has the additional advantage that it is computationally less expensive to check.Among the many technical variants of simulation[8],we choose weak simulation as our notion of conformance because it allows for asynchrony between the LTSs, i.e.,one step of the speciﬁcation LTS may correspond to multiple steps of the implementation. This feature of weak simulation is crucial to our approach,because one step in M Spec typically corresponds to multiple steps in M Imp.D.Weak SimulationLet M1= S1,init1,Σ,T1 and M2= S2,init2,Σ,T2 be two LTSs with the same alphabet.A relation R⊆S1×S2is called a weak simulation iff if obeys the following two conditions for all s1∈S1,t1∈S1and s2∈S2:1)If(s1,s2)∈R,a=τand s1a−→t1then there exists t2∈S2such that s2a−→t2and(t1,t2)∈R.2)If(s1,s2)∈R and s1τ−→t1then at least one of the following two conditions hold:a)(t1,s2)∈Rb)There exists t2∈S2such that s2τ−→t2and(t1,t2)∈RWe say that LTS M2weakly simulates M1(denoted by M1 M2)if there exists a weak simulation relation R⊆S1×S2such that(init1,init2)∈R.E.Algorithm for Computing Weak SimulationThe existence of a weak simulation relation between M1and M2can be checked efﬁciently by reducing the problem to an instance of Boolean satisﬁability[42].Interestingly the SAT instances produced by this method always belong to a restricted class of SAT formulas known as the weakly negated HORN formulas.In contrast to general SAT(which has no known polynomial time algorithm),satisﬁability of weakly negated HORN formulas can be solved in linear time[43]. As part of MAGIC,we have implemented an online linear time HORNSAT algorithm[44].MAGIC can also interface with public domain general SAT solvers like Chaff[23],FGRASP[24]and SATO[25].IV.M ODEL C ONSTRUCTIONLet M Spec= S Spec,init Spec,ΣSpec,T Spec and the assumption PAs be{PA1,...,PA k}.In this section we show how to extract M Imp from proc using the assumption PAs,the guard G Spec and the predicates.The extraction of M Imp relies on several principles:•Every state of M Imp models a state during the execution of proc;consequently every state is composed of a control and data component.•The control components intuitively represent values of the program counter,and are formally obtained from the CFG of proc.•The data components are abstract representations of the memory state of proc.These abstract representations are obtained using predicate abstraction.•The transitions between states in M Imp are derived from the transitions in the controlﬂow graph,taking into account the assumption PAs and the predicate abstraction.This process involves reasoning about C expressions,and therefore requires the use of a theorem prover.S0:int x,y=8;S1:if(x==0){S2:do a();S4:if(y<10){S6:return0;}else{S7:return1;}}else{S3:do b();S5:if(y>5){S8:return2;}else{S9:return3;}}Fig.3.A simple proc we use as a running example.Without loss of generality,we can assume that there are onlyﬁve kinds of statements in proc: assignments,call-sites,if-then-else branches,goto and return.In our implementation, we use the CIL[45]tool to transform arbitrary C programs into the above format.Note that call-sites correspond to library routines called by proc for which assumed PAs are available.We assume the absence of indirect function calls and pointer dereferences in the lhs’s of assignments.In reality,MAGIC handles these constructs by using aliasing information conservatively [26].We denote by Stmt the set of statements of proc and by Exp the set of all pure (side-effect free)C expressions over the variables of proc .As a running example of proc ,we use the C program shown in Figure 3.It invokes two library routines do a and do b .Let the guard and LTS list in the assumption PA for do a be TRUE ,Do A .This means that under all invocation conditions,do a is safely abstracted by theLTS Do A .Similarly the guard and LTS list in the assumption PA for do b is TRUE ,Do B .The LTSs Do A and Do B are described in Figure 4.Also we use G Spec =TRUE and M Spec =Spec (shown in Figure 4).STOP Do_B STOP C3C5C4C7C6C9C8C11C10C13C12STOPDo_Aa b return{}return{}A1A2B1B2C1C2ττττa ττττreturn{2}return{0}b ττSpec Fig.4.The LTSs in the assumption PAs for do a and do b .The VoidRet action is denoted by return {}.A.Initial abstraction with control ﬂow automataThe construction of M Imp begins with the construction of the control ﬂow automaton (CFA)of proc .The states of a CFA correspond to control points in the program.The transitions between states in the CFA correspond to the control ﬂow between their associated control points in the program.Thus,a CFA of a program is a conservative abstraction of the program’s control ﬂow,i.e.it allows a superset of the possible traces of the program.Formally the CFA is a 4-tuple S CF ,I CF ,T CF ,L where:•S CF is a set of states.•I CF ∈S CF is an initial state.•T CF ⊆S CF ×S CF is a set of transitions.•L :S CF \{FINAL }→Stmt is a labeling function.S CF contains a distinguished FINAL state.The transitions between states reﬂect the ﬂow of control between their labeling statements:L (I CF )is the initial statement of proc and (s 1,s 2)∈T CF iff one of the following conditions hold:•L (s 1)is an assignment,call-site or goto with L (s 2)as its unique successor.•L (s 1)is a branch with L (s 2)as its then or else successor.•L (s 1)is a return statement and s 2=FINAL .Example 1:The CFA of our example program is shown in Figure 5.Each non-ﬁnal state is labeled by the corresponding statement label (the FINAL state is labeled by FINAL ).Henceforth we will refer to each CFA state by its label.Therefore the states of the CFA in Figure 5are S0...S9,final with S0being the initial state.y = 8x == 0return 0a()y < 10y > 5b()1{p , p }21{p }1{p }{p }2{p }21{p , p }2return 1return 2return 3{ }{ }{ }{ }{ }S0S1S2S3S4S5S6S8S9S7FINAL FINAL Fig.5.The CFA for our example program.Each non-FINAL state is labeled the same as its corresponding statement.The initial state is labeled S0.The states are also labeled with inferred predicates when P ={p 1,p 2}where p 1=(y <10)andp 2=(y >5).B.Predicate inferenceSince the construction of M Imp from proc involves predicate abstraction,it is parameterized by a set of predicates P .The main challenge in predicate abstraction is to identify the set P that is necessary for proving the given property.In our framework we require P to be a subset of the branch predicates in proc .Therefore we sometimes refer to P or subsets of P simply as a set of branches.The construction of M Imp associates with each state s of the CFA a ﬁnite subset of Exp derived from P ,denoted by P s .The process of constructing the P s ’s from Pis known as predicate inference and is described by the algorithm PredInfer in Figure6.Note that P FINAL is always∅.The algorithm uses a procedure for computing the weakest precondition WP[11],[46],[47] of a predicate p relative to a given statement.Consider a C assignment statement a of the form v=e.Letϕbe a pure C expression(ϕ∈Exp).Then the weakest precondition ofϕwith respect to a,denoted by WP[a]{ϕ}is obtained fromϕby replacing every occurrence of v inϕwith e. Note that we need not consider the case where a pointer appears in the lhs of a since we have disallowed such constructs from appearing in proc.Input:Set of branch statements POutput:Set of P s’s associated with each CFA stateInitialize:∀s∈S CF,P s:=∅Forever doFor each s∈S CF doIf L(s)is an assignment statement and L(s )is its successorFor each p ∈P s add WP[L(s)]{p }to P sElse If L(s)is a branch statement with condition cIf L(s)∈P,then add c to P sIf L(s )is a‘then’or‘else’successor of L(s)P s:=P s∪P sElse If L(s)is a call-site or a‘goto’statement withsuccessor L(s )P s:=P s∪P sElse If L(s)returns expression e and r∈ΣSpec∩IntRetAdd the expression(e==RetVal(r))to P sIf no P s was modified in the For loop,then exitFig.6.The algorithm PredInfer that MAGIC uses for predicate inference.The weakest precondition is clearly an element of Exp as well.Note that PredInfer may not terminate in the presence of loops in the CFA.However,this does not mean that our approach is incapable of handling C programs containing loops.In practice,we force termination ofPredInfer by limiting the maximum size of any P ing the resulting P s’s,we can compute the states and transitions of the abstract model as described later.Irrespective of whether PredInfer was terminated forcefully or not,M Imp is guaranteed to be a safe abstraction of proc.We have found this approach to be very effective in practice.A similar algorithm was proposed by Dams and Namjoshi[48].Example2:Consider the CFA described in Example1.Suppose P contains the two branches S4and S5.Then PredInfer begins with P S4={(y<10)}and P S5={(y>5)}.From this it obtains P S2={(y<10)}and P S3={(y>5)}.This leads to P S1={(y<10),(y>5)}. Then P S0={WP[y=8]{y<10},WP[y=8]{y>5}}={(8<10),(8>5)}.Since we ignore predicates that are trivially TRUE or FALSE,P S0=∅.Since the return actions in Spec have return values{0,2},P S6={(0==0),(0==2)},which is again∅.Similarly, P S7=P S8=P S9=P FINAL=∅.Figure5shows the CFA with each state s labeled by P s.C.Predicate valuation and concretizationSo far we have described a method for computing the initial abstraction(the CFA)and a set of predicates associated with each location in the program.The states of the abstract system M Imp correspond to the various possible valuations of the predicates in each location.Formally,for a CFA node s suppose P s={p1,...,p k}.Then a valuation V of P s is a function from P s to theset{TRUE,FALSE}.Alternately,one can view the valuation V as a Boolean vector v1,...,v k of size k where each v i is the result of applying the function V to the predicate p i.Let V s be the set of all predicate valuations of P s.Note that the size of V s is exponential in the size of P s.The predicate concretization functionΓs:V s→Exp is deﬁned as follows.Given a valuation V={v1,...,v k}∈V s,Γs(V)= k i=1p v i i where p TRUE i=p i and p FALSE i=¬p i. As a special case,if P s=∅,then V s={⊥}andΓs(⊥)=TRUE.Example3:Consider the CFA described in Example1and the inferred predicates as explained in Example2.Recall that P S1={(y<10),(y>5)}.Suppose V1={0,1}and V2={1,0}. ThenΓS1(V1)=(¬(y<10))∧(y>5)andΓS1(V2)=(y<10)∧(¬(y>5)).D.States of M ImpEach state s∈S CF gives rise to a set of states of M Imp,denoted by IS s.In addition,M Imp has an unique initial state INIT.The deﬁnition of IS s consists of the following sub-cases:。

AlgebraicGraphTheory

Algebraic Graph TheoryChris Godsil(University of Waterloo),Mike Newman(University of Ottawa)April25–291Overview of the FieldAlgebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example,automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects.One of the oldest themes in the area is the investigation of the relation between properties of a graph and the spectrum of its adjacency matrix.A central topic and important source of tools is the theory of association schemes.An association scheme is,roughly speaking,a collection of graphs on a common vertex set whichﬁt together in a highly regular fashion.These arise regularly in connection with extremal structures:such structures often have an unex-pected degree of regularity and,because of this,often give rise to an association scheme.This in turn leads to a semisimple commutative algebra and the representation theory of this algebra provides useful restrictions on the underlying combinatorial object.Thus in coding theory we look for codes that are as large as possible, since such codes are most effective in transmitting information over noisy channels.The theory of association schemes provides the most effective means for determining just how large is actually possible;this theory rests on Delsarte’s thesis[4],which showed how to use schemes to translate the problem into a question that be solved by linear programming.2Recent Developments and Open ProblemsBrouwer,Haemers and Cioabˇa have recently shown how information on the spectrum of a graph can be used to proved that certain classes of graphs must contain perfect matchings.Brouwer and others have also investigated the connectivity of strongly-regular and distance-regular graphs.This is an old question,but much remains to be done.Recently Brouwer and Koolen[2]proved that the vertex connectivity of a distance-regular graph is equal to its valency.Haemers and Van Dam have worked on extensively on the question of which graphs are characterized by the spectrum of their adjacency matrix.They consider both general graphs and special classes,such as distance-regular graphs.One very signiﬁcant and unexpected outcome of this work was the construction,by Koolen and Van Dam[10],of a new family of distance-regular graphs with the same parameters as the Grassmann graphs.(The vertices of these graphs are the k-dimensional subspaces of a vector space of dimension v over theﬁniteﬁeld GF(q);two vertices are adjacent if their intersection has dimension k1.The graphs are q-analog of the Johnson graphs,which play a role in design theory.)These graphs showed that the widely held belief that we knew all distance-regular graphs of“large diameter”was false,and they indicate that the classiﬁcation of distance-regular graphs will be more complex(and more interesting?)than we expected.1Association schemes have long been applied to problems in extremal set theory and coding theory.In his(very)recent thesis,Vanhove[14]has demonstrated that they can also provide many interesting results in ﬁnite geometry.Recent work by Schrijver and others[13]showed how schemes could used in combination with semidef-inite programming to provide signiﬁcant improvements to the best known bounds.However these methods are difﬁcult to use,we do not yet have a feel for we might most usefully apply them and their underlying theory is imperfectly understood.Work in Quantum Information theory is leading to a wide range of questions which can be successfully studied using ideas and tools from Algebraic Graph Theory.Methods fromﬁnite geometry provide the most effective means of constructing mutually unbiased bases,which play a role in quantum information theory and in certain cryptographic protocols.One important question is to determine the maximum size of a set of mutually unbiased bases in d-dimensional complex space.If d is a prime power the geometric methods just mentioned provide sets of size d+1,which is the largest possible.But if d is twice an odd integer then in most cases no set larger than three has been found.Whether larger sets exist is an important open problem. 3Presentation HighlightsThe talks mostlyﬁtted into one of four areas,which we discuss separately.3.1SpectraWillem Haemers spoke on universal adjacency matrices with only two distinct eigenvalues.Such matrices are linear combinations of I,J,D and A(where D is the diagonal matrix of vertex degrees and A the usual adjacency matrix).Any matrix usually considered in spectral graph theory has this form,but Willem is considering these matrices in general.His talk focussed on the graphs for which some universal adjacency matrix has only two eigenvalues.With Omidi he has proved that such a graph must either be strong(its Seidel matrix has only two eigenvalues)or it has exactly two different vertex degrees and the subgraph induced by the vertices of a given degree must be regular.Brouwer formulated a conjecture on the minimum size of a subset S of the vertices of a strongly-regular graph X such that no component of X\S was a single vertex.Cioabˇa spoke on his recent work with Jack Koolen on this conjecture.They proved that it is false,and there are four inﬁnite families of counterexamples.3.2PhysicsAs noted above,algebraic graph theory has many applications and potential applications to problems in quantum computing,although the connection has become apparent only very recently.A number of talks were related to this connection.One important problem in quantum computing is whether there is a quantum algorithm for the graph isomorphism problem that would be faster than the classical approaches.Currently the situation is quite open.Martin Roetteler’s talk described recent work[1]on this problem.For our workshop’s viewpoint,one surprising feature is that the work made use of the Bose-Mesner algebra of a related association scheme; this connection had not been made before.Severini discussed quantum applications of what is known as the Lov´a sz theta-function of a graph.This function can be viewed as an eigenvalue bound and is closely related to both the LP bound of Delsarte and the Delsarte-Hoffman bound on the size of an independent set in a regular graph.Severini’s work shows that Lov´a sz’s theta-function provides a bound on the capacity of a certain channel arising in quantum communication theoryWork in quantum information theory has lead to interest in complex Hadamard matrices—these are d×d complex matrices H such that all entries of H have the same absolute value and HH∗=dI.Both Chan and Sz¨o ll˝o si dealt with these in their talks.Aidan Roy spoke on complex spherical designs.Real spherical designs were much studied by Seidel and his coworkers,because of their many applications in combinatorics and other areas.The complex case languished because there were no apparent applications,but now we have learnt that these manifest them-selves in quantum information theory under acronyms such as MUBs and SIC-POVMs.Roy’s talk focussedon a recent 45page paper with Suda [12],where (among other things)they showed that extremal complex designs gave rise to association schemes.One feature of this work is that the matrices in their schemes are not symmetric,which is surprising because we have very few interesting examples of non-symmetric schemes that do not arise as conjugacy class schemes of ﬁnite groups.3.3Extremal Set TheoryCoherent conﬁgurations are a non-commutative extension of association schemes.They have played a sig-niﬁcant role in work on the graph isomorphism problem but,in comparison with association schemes,they have provided much less information about interesting extremal structures.The work presented by Hobart and Williford may improve matters,since they have been able to extend and use some of the standard bounds from the theory of schemes.Delsarte [4]showed how association schemes could be used to derive linear programs,whose values provided strong upper bounds on the size of codes.Association schemes have both a combinatorial structure and an algebraic structure and these two structures are in some sense dual to one another.In Delsarte’s work,both the combinatorial and the algebraic structure had a natural linear ordering (the schemes are both metric and cometric)and this played an important role in his work.Martin explained how this linearity constraint could be relaxed.This work is important since it could lead to new bounds,and also provide a better understanding of duality.One of Rick Wilson’s many important contributions to combinatorics was his use of association schemes to prove a sharp form of the Erd˝o s-Ko-Rado theorem [15].The Erd˝o s-Ko-Rado theorem itself ([5])can certainly be called a seminal result,and by now there are many analogs and extensions of it which have been derived by a range of methods.More recently it has been realized that most of these extensions can be derived in a very natural way using the theory of association schemes.Karen Meagher presented recent joint work (with Godsil,and with Spiga,[8,11])on the case where the subsets in the Erd˝o s-Ko-Rado theorem are replaced by permutations.It has long been known that there is an interesting association scheme on permutations,but this scheme is much less manageable than the schemes used by Delsarte and,prior to the work presented by Meagher,no useful combinatorial information had been obtained from it.Chowdhury presented her recent work on a conjecture of Frankl and F¨u redi.This concerns families F of m -subsets of a set X such that any two distinct elements of have exactly λelements in common.Frankl and F¨u redi conjectured that the m -sets in any such family contain at least m 2 pairs of elements of X .Chowdhury veriﬁed this conjecture in a number of cases;she used classical combinatorial techniques and it remains to see whether algebraic methods can yield any leverage in problems of this type.3.4Finite GeometryEric Moorhouse spoke on questions concerning automorphism groups of projective planes,focussing on connections between the ﬁnite and inﬁnite case.Thus for a group acting on a ﬁnite plane,the number of orbits on points must be equal to the number of orbits on lines.It is not known if this must be true for planes of inﬁnite order.Is there an inﬁnite plane such that for each positive integer k ,the automorphism group has only ﬁnitely many orbits on k -tuples?This question is open even for k =4.Simeon Ball considered the structure of subsets S of a k -dimensional vector space over a ﬁeld of order q such that each d -subset of S is a basis.The canonical examples arise by adding a point at inﬁnity to the point set of a rational normal curve.These sets arise in coding theory as maximum distance separable codes and in matroid theory,in the study of the representability of uniform matroids (to mention just two applications).It is conjectured that,if k ≤q −1then |S |≤q +1unless q is even and k =3or k =q −1,in which case |S |≤q +2.Simeon presented a proof of this theorem when q is a prime and commented on the general case.He developed a connection to Segre’s classical characterization of conics in planes of odd order,as sets of q +1points such that no three are collinear.There are many analogs between ﬁnite geometry and extremal set theory;questions about the geometry of subspaces can often be viewed as q -analogs of questions in extremal set theory.So the EKR-problem,which concerns characterizations of intersecting families of k -subsets of a ﬁxed set,leads naturally to a study of intersecting families of k -subspaces of a ﬁnite vector space.In terms of association schemes this means we move from the Johnson scheme to the Grassmann scheme.This is fairly well understood,with thebasic results obtained by Frankl and Wilson[6].But inﬁnite geometry,polar spaces form an important topic. Roughly speaking the object here is to study the families of subspaces that are isotropic relative to some form, for example the subspaces that lie on a smooth quadric.In group theoretic terms we are now dealing with symplectic,orthogonal and unitary groups.There are related association schemes on the isotropic subspaces of maximum dimension.Vanhove spoke on important work from his Ph.D.thesis,where he investigated the appropriate versions of the EKR problem in these schemes.4Outcome of the MeetingIt is too early to offer much in the way of concrete evidence of impact.Matt DeV os observed that a conjecture of Brouwer on the vertex connectivity of graphs in an association scheme was wrong,in a quite simple way. This indicates that the question is more complex than expected,and quite possibly more interesting.That this observation was made testiﬁes to the scope of the meeting.On a broader level,one of the successes of the meeting was the wide variety of seemingly disparate topics that were able to come together;the ideas of algebraic graph theory touch a number of things that would at ﬁrst glance seem neither algebraic nor graph theoretical.There was a lively interaction between researchers from different domains.The proportion of post-docs and graduate students was relatively high.This had a positive impact on the level of excitement and interaction at the meeting.The combination of expert and beginning researchers created a lively atmosphere for mathematical discussion.References[1]A.Ambainis,L.Magnin,M.Roetteler,J.Roland.Symmetry-assisted adversaries for quantum state gen-eration,arXiv1012.2112,35pp.[2]A.E.Brouwer,J.H.Koolen.The vertex connectivity of a distance-regular graph.European bina-torics30(2009),668–673.[3]A.E.Brouwer,D.M.Mesner.The connectivity of strongly regular graphs.European binatorics,6(1985),215–216.[4]P.Delsarte.An algebraic approach to the association schemes of coding theory.Philips Res.Rep.Suppl.,(10):vi+97,1973.[5]P.Erd˝o s,C.Ko,R.Rado.Intersection theorems for systems ofﬁnite sets.Quart.J.Math.Oxford Ser.(2),12(1961),313–320.[6]P.Frankl,R.M.Wilson.The Erd˝o s-Ko-Rado theorem for vector binatorial Theory,SeriesA,43(1986),228–236.[7]D.Gijswijt,A.Schrijver,H.Tanaka.New upper bounds for nonbinary codes based on the Terwilligeralgebra and semideﬁnite binatorial Theory,Series A,113(2006),1719–1731. [8]C.D.Godsil,K.Meagher.A new proof of the Erd˝o s-Ko-Rado theorem for intersecting families of per-mutations.arXiv0710.2109,18pp.[9]C.D.Godsil,G.F.Royle.Algebraic Graph Theory,Springer-Verlag,(New York),2001.[10]J.H.Koolen,E.R.van Dam.A new family of distance-regular graphs with unbounded diameter.Inven-tiones Mathematicae,162(2005),189-193.[11]K.Meagher,P.Spiga.An Erdos-Ko-Rado theorem for the derangement graph of PGL(2,q)acting onthe projective line.arXiv0910.3193,17pp.[12]A.P.Roy,plex spherical Codes and designs,(2011),arXiv1104.4692,45pp.[13]A.Schrijver.New code upper bounds from the Terwilliger algebra and semideﬁnite programming.IEEETransactions on Information Theory51(2005),2859–2866.[14]F.Vanhove.Incidence geometry from an algebraic graph theory point of view.Ph.D.Thesis,Gent2011.[15]R.M.Wilson.The exact bound in the Erds-Ko-Rado binatorica,4(1984),247–257.。

Computing crossing numbers in quadratic time

Computing Crossing Numbers in Quadratic TimeMartin GroheNovember29,2002AbstractWe show that for everyﬁxed there is a quadratic time algorithm that decides whether a given graph has crossing number at most and,if this is the case,computes a drawing of the graph into theplane with at most crossings.1.IntroductionHopcroft and Tarjan[13]showed in1974that planarity of graphs can be decided in linear time.It is natural to relax planarity by admitting a small number of edge-crossings in a drawing of the graph.The crossingnumber of a graph is the minimum number of edge crossings needed in a drawing of the graph into the plane.Not surprisingly,it is NP-complete to decide,given a graph and a,whether the crossing numberof is at most[12].On the other hand,for everyﬁxed there is a simple polynomial time algorithm deciding whether a given graph has crossing number at most:It guesses pairs of edges that cross1 and tests if the graph obtained from by adding a new vertex at each of these edge crossings is planar.The running time of this algorithm is.Downey and Fellows[7]raised the question of whether the crossing-number problem isﬁxed parameter-tractable,that is,whether there is a constant such that for everyﬁxed the problem can be solved in time.We answer this question positively with. In other words,we show that for everyﬁxed there is a quadratic time algorithm deciding whether a given graph has crossing number at most.Moreover,we show that if this is the case a drawing of into theplane with at most crossings can be computed in quadratic time.It is interesting to compare our result to similar results for computing the genus of a graph.(The genus of a graph is the minimum taken over the genera of all surfaces such that can be embedded into.) As for the crossing number,it is NP-complete to decide if the genus of a given graph is less than or equal to a given[18].For aﬁxed,atﬁrst sight the genus problem looks much harder.It is by no means obvious how to solve it in polynomial time;this has been proved possible by Filotti,Miller,and Reif[10].In1996, Mohar[14]proved that for every there is actually a linear time algorithm deciding whether the genus of a given graph is.However,the fact that the genus problem isﬁxed-parameter tractable was known earlier as a direct consequence of a strong general theorem due to Robertson and Seymour[17]stating that all classes of graphs that are closed under taking minors are recognizable in cubic time.Recall that a minor of a graph is a graph obtained from a subgraph of by contracting edges.It is easy to see that the class of all graphs of genus at most is closed under taking minors.Unfortunately the class of all graphs of crossing number at most is not closed under taking minors. So in general Robertson and Seymour’s theorem cannot be applied to compute crossing numbers.An exception is the case of graphs of degree at most3;Fellows and Langston[9]observed that for such graphs Robertson and Seymour’s result immediately yields a cubic time algorithm for computing crossing numbers.2Although we cannot apply Robertson and Seymour’s result directly,the overall strategy of our algorithm is inspired by their ideas:The algorithmﬁrst iteratively reduces the size of the input graph until it reaches a graph of bounded tree-width,and then solves the problem on this graph.For the reductionstep,we use Robertson and Seymour’s Excluded Grid Theorem[16]together with a nice lemma due to Thomassen[19]stating that in a graph of bounded genus(and thus in a graph of bounded crossing number) every large grid contains a subgrid that,in some precise sense,lies“ﬂat”in the graph.Such aﬂat grid does not essentially contribute to the crossing number and can therefore be contracted.For the remaining problem on graphs of bounded tree-width we apply a theorem due to Courcelle[4]stating that all properties of graphs that are expressible in monadic second-order logic are decidable in linear time on graphs of bounded tree-width.Let me remark that the hidden constant in the quadratic upper bound for the running time of our algo-rithm heavily depends on.As a matter of fact,the running time is,where is at least doubly exponential.Thus our algorithm is only of theoretical interest.2.PreliminariesGraphs in this paper are undirected and loop-free,but they may have multiple edges.3The vertex set of a graph is denoted by,the edge set by.We always assume that.For graphs and we let andboth endpoints of are contained in.A subgraph of a graph is a graph withand.We formally treat paths and cycles in a graph as subgraphs of this graph(as opposed to, say,sequences of vertices).Paths and cycles are always simple,that is,they have no self-intersections. 2.1.Topological Embeddings.A topological embedding of a graph into a graph is a mapping that associates a vertex with every and a path in with every in such a way that:–For distinct vertices,the vertices and are distinct.–For distinct edges,the paths and are internally disjoint(that is,they have at most their endpoints in common).–For every edge with endpoints and,the two endpoints of the path are and ,and for all.We let be the subgraph of consisting of the images of the vertices and edges of under. Formally,.2.2.Drawings and Crossing Numbers.A drawing of a graph is a mapping that associates with every vertex a point and with every edge a simple curve in in such a way that:–For distinct vertices,the points and are distinct.–For distinct edges,the curves and have at most one interior point in common (and possibly their endpoints).–For every edge with endpoints and,the two endpoints of the curve are and ,and for all.–At most two edges intersect in one point.Formally,for all .We let.An with is called a crossing of.The crossing number of is the number of crossings of.A drawing of crossing number0is called plane.The crossing number of a graph is the minimum taken over the crossing numbers of all drawings of.A graph of crossing number0is called planar.Figure 1.The hexagonal grids2.3.Hexagonal Grids.For ,we let be the hexagonal grid of radius .Instead of giving a formal deﬁnition,we refer the reader to Figure 1to see what this means.The principal cycles of are the the concentric cycles,numbered from the interior to the exterior (see Figure 2).C 1C C 32Figure 2.The principal cycles of2.4.Flat Grids in a Graph.For graphs ,an -component (of )is either a connected component of together with all edges connecting with and their endpoints in or an edge in whose endpoints are both in together with its endpoints.The vertices in the intersection of an -component with are called the vertices of attachment of the component.Let be a graph and a topological embedding.The interior of is the subgraph (remember that is the outermost principal cycle of ).A proper -component is an -component that has at least one vertex of attachment in the interior of .The topological embedding is ﬂat if the union of with all its proper components is a planar graph.We shall use the following theorem due to Thomassen [19].Actually,Thomassen stated the result for the genus of a graph rather than its crossing number.However,it is easy to see that the crossing number of a graph is an upper bound for its genus.Theorem 1(Thomassen [19]).For allthere is an such that the following holds:If is a graph of crossing number at most anda topological embedding,then there is a subgrid such that the restriction of to is ﬂat.2.5.Tree-Width.We assume that reader is familiar with the notion tree-width (of a graph).It is no big problem if not;we never really work with tree-width,but just take it as a black box in Theorems 2–4.Robertson and Seymour’s deep Excluded Grid Theorem [16]states that every graph of sufﬁciently large tree-width contains the homeomorphic image of a large grid.We use the following algorithmic version of this theorem.Theorem2.(Robertson and Seymour[17],Bodlaender[1],Perkovi´c and Reed[15]).Let.Then there is a and a linear time algorithm that,given a graph,either(correctly)recognizes that the tree-width of is at most or computes a topological embedding.Robertson and Seymour[17]gave a quadratic time algorithm,but they pointed out that it can be im-proved to linear time using Bodlaender’s[1]linear time algorithm for computing tree-decompositions. However,this improvement is not entirely straightforward:Let usﬁx a constant.The essential part of Robertson and Seymour’s algorithm for the problem stated in the theorem is a quadratic time algorithm that,given a graph,returns a tree-decomposition of of width at most if the tree-width of the input graph is at most.Furthermore,the algorithm returns a“counterexample”subgraph of of tree-width larger than and at most if the tree-width of is greater than.This counterexample is very important here because the subgraph is then used toﬁnd the topological embedding of into.Bodlaender[1]gave a linear time algorithm computing a tree-decomposition of width at most if the tree-width of the input graph is at most,but his algorithm does not return a counterexample if the tree-width of is greater than.Perkovi´c and Reed[15]extended Bodlaender’s algorithm in such a way that it still works in linear time,but does return a counterexample if the tree-width of the input graph is greater than.2.6.Courcelle’s Theorem.Courcelle’s theorem states that properties of graphs deﬁnable in Monadic Second-Order Logic MSO can be checked in linear time on input graphs of bounded tree-width.In this logical context we consider graphs as relational structures of vocabulary,where and are unary relation symbols interpreted by the vertex set and edge set of a graph and is a binary relation symbol interpreted by the incidence relation.To simplify the notation,for a graph we let and call the universe of.I assume that the reader is familiar with the deﬁnition of MSO.However,for those who are not I have included it in Appendix A.Theorem3(Courcelle[4]).Let and letbe an MSO-formula.Then there is a linear time algorithm that,given a graph of tree-width at most and,,decides whether.We shall also use the following strengthening of Courcelle’s theorem,a proof of which can be found in [11]:Theorem4.Let and letbe an MSO-formula.Then there is a linear time algorithm that,given a graph of tree-width at most and,,decides if there exist,such thatand,if this is the case,computes such elements and sets.3.The AlgorithmFor an,a graph,and a subset of forbidden edges,an-good drawing of with respect to is a drawing of of crossing number at most such that no forbidden edges are involved in any crossings,that is,for every crossing of we have.Weﬁx a for the whole section.We shall describe an algorithm that solves the following gener-alized-crossing number problem in quadratic time:Input:Graph and subset.Problem:Decide if has a-good drawing with respect to.Later,we shall extend our algorithm in such a way that it actually computes a-good drawing if there exists one.Our algorithm works in two phases.In theﬁrst,it iteratively reduces the size of the input graph until it obtains a graph whose tree-width is bounded by a constant only depending on.Then,in the second phase,it solves the problem on this graph of bounded tree-width.3.1.Phase I.We let and choose sufﬁciently large such that for every graph of crossing number at most and every topological embedding there is a subgrid such that the restriction of to isﬂat.Such an exists by Theorem1.Then we choose with respect to according to Theorem2such that we have a linear time algorithm that,given a graph of tree-width greater than,ﬁnds a topological embedding.We keepﬁxed for the rest of the section.Lemma5.There is a linear time algorithm that,given a graph,either recognizes that the crossing number of is greater than,or recognizes that the tree-width of is at most,or computes aﬂat topological embedding.Proof:Weﬁrst apply the algorithm of Theorem2.If it recognizes that the tree-width of the input graph is at most,we are done.Otherwise,it computes a topological embedding.By our choice of ,we know that either the crossing number of is greater than or there is a subgrid such that the restriction of to isﬂat.For each we can decide whether isﬂat by a planarity test,which is possible in linear time[13].Our algorithm tests whether isﬂat for all.Either itﬁnds aﬂat,or the crossing number of is greater than.4Since is aﬁxed constant,the overall running time is linear.Let be a graph and a topological embedding.For,we let be the subgrid of bounded by the th principal cycle.We let be the subgraph of consisting ofand all-components all of whose vertices of attachment are in.Moreover,we let be the subgraph of consisting of and all-components all of whose vertices of attachment are in .In particular,we call the subgraph the kernel of,the boundary of the kernel,and the interior of the kernel(see Figure3).Lemma6.Let be a graph,,and let be a-good drawing of with respect to of minimum crossing number.Let be aﬂat topological embedding.Then none of the edges of the kernel of is involved in any crossing of.To understand the signiﬁcance of this lemma,note that theﬂatness of the topological embedding guarantees that the graph is planar for all.However,this does not necessarily mean that the restriction of the speciﬁc drawing to is plane.The lemma implies that at least the restriction of to the kernel is plane(the actual statement of the lemma is slightly stronger).Proof:For,the th ring of is the subgraph of consisting of and(the images of the th and th principal cycle)and the images of all edges in connecting these two cycles(see Figure4).Then for with,the graphs and are disjoint. Recall that.Since at most two edges are involved in any crossing,by the pigeonhole-principle there is an such that none of the edges in is involved in any crossing of. Let,,,and.Then and are both connected planar graphs.Let be the-component that contains.Thus consists of,all edges connecting to,and the endpoints of these edges.(Recall the deﬁnition of an-component of a graph from the beginning of Subsection2.4.)Note that the vertices of attachment of are all on.Figure3.Aﬂat grid in a graph,its kernel,and the boundary of the kernelFigure4.The ring in a gridWithout loss of generality we can assume that is connected.Then the graph consists of the boundary and one connected component EXT that contains the exterior part of the grid(in particular the cycle—recall that)and the rest of.consists of the cycle and possibly additional -components.Let usﬁrst consider the restriction of to.Claim1:There exists a connected component of such that the-images of all vertices of attachment of are on the boundary of.Proof:Since is a cycle and none of the edges of is involved in any crossing of,is a simple closed curve in the plane.Furthermore,since EXT is a connected graph,EXT must be entirely contained in one connected component of,say,in the exterior.Let be the interior of.Then must be contained in(except for its vertices of attachment,which are on the boundary of).To see this,suppose for contradiction that is not contained in.Since is a connected graph and none of the edges of the ring is involved in any crossing of,either is contained in the-image one of the hexagons of the ring,or it is contained in the exterior of the ring.But this can both not happen,because the vertices of attachment of are on,and they are not contained in the boundary of a single hexagon.So is contained in.Now if(that is,if there are no components),or if maps all components to the exterior of,then is a connected component of such that the-images of all vertices of attachment of are on the boundary of.But in general,may map some of the components to .Suppose for contradiction that there are two connected component of such that the boundary of both contains the image of a vertex of attachment of.Since is connected,there is a path connecting these two vertices,and must intersect.This contradicts the fact that noneof the edges of is involved in any crossing of.Claim2:The restriction of to is plane.Proof:Suppose for contradiction that this is not the case.Then two edges of must cross under.Let be a plane drawing of the planar graph.Let be the the connected component ofthat contains.Without loss of generality we can assume that is not the exterior component,that is, is homeomorphic to an open disc.Similarly we can assume that the set of Claim1is homeomorphic to an open disc.Let be the vertices of attachment of,and let be a homeomorphism fromto.We deﬁne a new drawing of by letting on and on.Then no edges of are involved in any crossing of,thus the crossing number of is smaller than that of.This contradicts the minimality of the crossing number of and proves Claim2.Claim3:None of the edges of is involved in any crossing of.Proof:Since the restriction of to is plane,and since none of the edges of is involved in any crossing,the only possible crossing involving an edge of would be between an edge of EXT and an edge of.But since and EXT are embedded into different components of(we showed this in the proof of Claim1),each such crossing would induce a crossing with an edge of.So there cannot be any such crossings,and Claim3is proved.To complete the proof of the lemma,we just recall that the kernel is a subgraph of.Now we are ready to describe the main reduction step our algorithm performs.Lemma7.There is a linear time algorithm that,given a graph and an edge set,either recognizes that the crossing number of is greater than,or recognizes that the tree-width of is at most,or computes a graph and an edge set such that,and has a-good drawing with respect to if,and only if,has a-good drawing with respect to.Proof:Weﬁrst apply the algorithm of Lemma5.If it tells us that the crossing number of is greater than or that the tree-width of is at most,there is nothing else we need to do.So suppose the algorithm returns aﬂat topological embedding.Let be the kernel of,its interior,and its boundary.Let the graph obtained from by contracting the connected subgraph to a single vertex(see Figure5).5Figure5.The transformation from a graph toLet be the union of with the set of all edges of and all edges incident with the new vertex.I claim that has a-good drawing with respect to if,and only if,has a-good drawing with respect to.The forward direction of this claim follows from Lemma6.For the backward direction we observethat every-good drawing of with respect to can be turned into a-good drawing of with respect to by embedding the planar graph into a small neighborhood of.Clearly,given and,the graph and the edge-set can be computed in linear time.Moreover .This yields the desired algorithmIterating the algorithm of the lemma,we obtain:Corollary8.There is a quadratic time algorithm that,given a graph,either recognizes that the crossing number of is greater than or computes a graph and an edge set such that the tree-width of is at most and has a-good drawing with respect to if,and only if,has a-good drawing with respect to.3.2.Phase II.If the algorithm has not found out that the graph has crossing number greater than in Phase I,it has produced a graph of tree-width at most and a set such that has a-good drawing with respect to if,and only if,has a-good drawing with respect to.In Phase II,the algorithm has to decide whether has a-good drawing with respect ing Courcelle’s Theorem3,we prove that this can be done in linear time.To this end,we shallﬁnd an MSO-formula such that for every graph and every setwe have if,and only if,has a-good drawing with respect to.We rely on the well-known fact that there is an MSO-formula planar saying that a graph is planar.(Actually,this is quite easy to see:just says that neither contains nor as a topological subgraph.Also see[6].)planarFor a graph and edges that do not have an endpoint in common we let be the graph obtained from by deleting the edges and and adding a new vertex and four edges connecting with the endpoints of the edges of and in(see Figure6).Observe that for everyFigure6.A graph with selected edges and the resultinga graph has an-good drawing with respect to an edge set if,and only if,either has an-good drawing with respect to or there are edges that do not have an endpoint in common such that has an-good drawing with respect to.Lemma9.For every MSO-formula there exists an MSO-formula such that for all graphs,edge sets and edges that do not have an endpoint in common we have:This lemma can easily be proved by a standard technique from logic,the method of syntactic interpre-tations.6For readers not familiar with syntactic interpretations,a direct proof of the lemma can be found in Appendix B.Using this lemma,we inductively deﬁne,for every,formulas and such that for every graph and edge set we havehas an-good drawingwith respect toand for all,,and edges that do not have an endpoint in common we havehas an-gooddrawing with respect toWe letplanarand,for,This completes our proof that there is a quadratic time algorithm deciding if a graph has a good drawing with respect to a set.puting a Good Drawing.So far we have only proved that there is a quadratic time algorithm deciding if a graph has a good drawing.It is not hard to modify this algorithm so that it actually computes a drawing:For Phase I,we observe that if we have a good drawing of with respect to then we can easily construct a good drawing of with respect to.So we only have to worry about Phase II.By induction on,for every we deﬁne a linear-time procedure DRAW that,given a graph of tree-width at most and a subset,computes an-good drawing of with respect to(if there exists one).DRAW just has to compute a plane drawing of.For,we apply Theorem4to the MSO-formulaIt yields a linear time algorithm that,given a graph and an,computes two edgessuch that(if such edges exist).It follows immediately from the deﬁnition of that if,and only if,are in that do not have an endpoint in common andhas an-good drawing with respect to.Given and,the procedure DRAW applies this linear-time algorithm to compute edges such thatThen it applies DRAW to the graph to compute an-good drawing of a graphwith respect to.It modiﬁes this drawing in a straightforward way to obtain an-good drawing of with respect to.Remark10.In the conference version of this paper I claimed that the use of Courcelle’s theorem in our proof can be avoided in favor of a direct algorithm that employs“the usual dynamic programming tech-niques”.Although this is true—after all,Courcelle’s algorithm also employs these dynamic programming techniques,so we can simply take the formula constructed above and extract an algorithm from Courcelle’s proof—it is not as simple as I thought then.The formula we construct essentially says:“There exists edges such that if we cross with (for)the graph does not contain a-minor or a-minor.”This statement involves a non-trivial quantiﬁer alternation,which is what makes the translation to an algorithm difﬁcult.At this point,I think that the route through logic and Courcelle’s theorem is essential for our proof.3.4.Uniformity.Inspection of our proofs and the proofs of the results we use shows that actually there is one algorithm that,given a graph with vertices and a non-negative integer,decides whether the crossing number of is at most in time for a suitable function.Unfortunately,grows extremely fast,at least doubly exponentially.To see this,note that the tree-width we derive a from the excluded grid theorem is exponential in.Testing whether a graph has tree-width requires time exponential in,that is,doubly exponential in.To give an upper bound on the growth of,we mainly have to analyse the running time of the algorithm we get out of Courcelle’s theorem.Its dependence on the formula is-fold exponential in the tree-width and the formula length,where is the number of quantiﬁer alternations of the formula.The straightforward bound on the number of quantiﬁer alternations of our formula saying that the crossing number is at most is3(independent of).There may be slight improvements reducing the running time by one or two exponentials,but as we saw our approach will not give us a running time whose dependence on is less than doubly exponential.4.ConclusionsWe have proved that for every there is a quadratic time algorithm deciding whether a given graph has crossing number at most.The running time of our algorithm in terms of is enormous,which makes the algorithm useless for practical purposes.This is partly due to the fact that the algorithm heavily relies on graph minor theory.However,knowing the crossing number problem to beﬁxed-parameter tractable may help toﬁnd better algorithms that are practically applicable for small values of.This has happened in a similar situation for the vertex cover problem.Theﬁrst proof[9]showing theﬁxed-parameter tractability of vertex cover used Robertson and Seymour’s theorem that classes of graphs closed under taking minors are recognizable in cubic time.Starting from there,much better algorithms have been developed;by now,vertex cover can be (practically)solved for a quite reasonable problem size(see[2]for a state-of-the-art algorithm).Although I do not expect there to be such a simple algorithm for deciding whether the crossing number of a graph is at most as there is for deciding whether there is a vertex cover of size at most,I conjecture that there is a more elementary algorithm for the crossing number problem whose running time is.Appendix A:Monadic Second Order LogicWeﬁrst explain the syntax of MSO:We have an inﬁnite supply of individual variables,denoted by et cetera,and also an inﬁnite supply of set variables,denoted by,et cetera.Atomic MSO-formulas(over graphs)are formulas of the form,,,,and,where are individual variables and is a set variable.The class of MSO-formulas is deﬁned by the following rules:–Atomic MSO-formulas are MSO-formulas.–If is an MSO-formula,then so is.–If and are MSO-formulas,then so are,,and.–If is an MSO-formula and is a variable(either an individual variable or a set variable),thenand are MSO-formulas.Let be a graph.Recall that.A-assignment is a mapping that associates an element of with every individual variable and a subset of with every set variable.We inductively deﬁne what it means that a graph together with a-assignment satisﬁes an MSO-formula(we write ):–,,,and endpoint of,–,–and,and similarly for,meaning“or”,and,meaning“implies”.–there exists ansuch thatdenotes the assignment with for all.–for all,,and similarly for meaning“for all”.It is easy to see that the relation only depends on the values of at the free variables of,that is,those variables not occurring in the scope of a quantiﬁer or.We writeto indicate that the free individual variables of are among and the free set variables are among .Then for a graph and,we writeif for every assignment with and we have.A sentence is a formula without free variables.For example,for the sentencewe have if,and only if,is2-colorable.Appendix B:Proof of Lemma9For the reader’s convenience,we repeat the statement of the Lemma:For every MSO-formula there exists an MSO-formula such that for allgraphs,edge sets and edges that do not have an endpoint incommon we have:(1)Proof:Let be a graph,,and let be edges that do not have an endpoint in common.Let be the endpoints of and the endpoints of.We deﬁne a graph as follows:–The vertex set of is the set–The edge set of is the set–A vertex is an endpoint of an edge if。

盖革-弥勒计数器

Introduction to Geiger CountersA Geiger counter(Geiger-Muller tube)is a device used for the detection and measurement of all types of radiation:alpha,beta and gamma radiation.Basically it consists of a pair of electrodes surrounded by a gas.The electrodes have a high voltage across them.The gas used is usually Helium or Argon.When radiation enters the tube it can ionize the gas.The ions(and electrons)are attracted to the electrodes and an electric current is produced.A scaler counts the current pulses, and one obtains a”count”whenever radiation ionizes the gas.The apparatus consists of two parts,the tube and the(counter+power supply). The Geiger-Mueller tube is usually cylindrical,with a wire down the center.The (counter+power supply)have voltage controls and timer options.A high voltage is established across the cylinder and the wire as shown in theﬁgure.When ionizing radiation such as an alpha,beta or gamma particle enters the tube, it can ionize some of the gas molecules in the tube.From these ionized atoms,an electron is knocked out of the atom,and the remaining atom is positively charged. The high voltage in the tube produces an electricﬁeld inside the tube.The electrons that were knocked out of the atom are attracted to the positive electrode,and the positively charged ions are attracted to the negative electrode.This produces a pulse of current in the wires connecting the electrodes,and this pulse is counted.After the pulse is counted,the charged ions become neutralized,and the Geiger counter is ready to record another pulse.In order for the Geiger counter tube to restore itself quickly to its original state after radiation has entered,a gas is added to the tube.For proper use of the Geiger counter,one must have the appropriate voltage across the electrodes.If the voltage is too low,the electricﬁeld in the tube is too weak to cause a current pulse.If the voltage is too high,the tube will undergo continuous discharge,and the tube can be ually the manufacture recommends the correct voltage to use for the rger tubes require larger voltages to produce the necessary electricﬁelds inside the tube.In class we will do an experiment to determine the proper operating voltage.First we will place a radioactive isotope in from of the Geiger-Mueller tube.Then,we will slowly vary the voltage across the tube and measure the counting rate.In theﬁgure I have inclded a graph of what we might expect to see when the voltage is increased across the tube.For low voltages,no counts are recorded.This is because the electricﬁeld is too weak for even one pulse to be recorded.As the voltage is increased,eventually one obtains a counting rate.The voltage at which the G-M tube just begins to count is called the starting potential.The counting rate quickly rises as the voltage is increased.For our equipment,the rise is so fast,that the graph looks like a”step”12potential.After the quick rise,the counting rate levels oﬀ.This range of voltages is termed the”plateau”region.Eventually,the voltage becomes too high and we have continuous discharge.The threshold voltage is the voltage where the plateau region begins.Proper operation is when the voltage is in the plateau region of the curve.For best operation,the voltage should be selected fairly close to the threshold voltage, and within theﬁrst1/4of the way into the plateau region.A rule we follow with the G-M tubes in our lab is the following:for the larger tubes to set the operating voltage about75Volts above the starting potential;for the smaller tubes to set the operating voltage about50volts above the starting potential.In the plateau region the graph of counting rate vs.voltage is in general not completelyﬂat.The plateau is not a perfect plateau.In fact,the slope of the curve in the plateau region is a measure of the quality of the G-M tube.For a good G-M tube,the plateau region should rise at a rate less than10percent per100volts.That is,for a change of100volts,(∆counting rate)/(average counting rate)should be less than0.1.An excellent tube could have the plateau slope as low as3percent per100 volts.Eﬃciency of the Geiger-counter:The eﬃciency of a detector is given by the ratio of the(number of particles of radiation detected)/(number of particles of radiation emitted):ε≡number of particles of radiation detectednumber of particles of radiation emitted(1)This deﬁnition for the eﬃciency of a detector is also used for our other detectors. In class we will measure the eﬃciency of our Geiger counter system andﬁnd that it is quite small.The reason that the eﬃciency is small for a G-M tube is that a gas is used to absorb the energy.A gas is not very dense,so most of the radiation passes right through the tube.Unless alpha particles are very energetic,they will be absorbed in the cylinder that encloses the gas and never even make it into the G-M tube.If beta particles enter the tube they have the best chance to cause ionization. Gamma particles themselves have a very small chance of ionizing the gas in the tube. Gamma particles are detected when they scatter an electron in the metal cylinder around the gas into the tube.So although the Geiger counter can detect all three types of radiation,it is most eﬃcient for beta particles and not very eﬃcient for gamma particles.Our scintillation detectors will prove to be much more eﬃcient for detecting speciﬁc radiation.Some of the advantages of using a Geiger Counter are:31.They are relatively inexpensive2.They are durable and easily portable3.They can detect all types of radiationSome of the disadvantages of using a Geiger Counter are:1.They cannot diﬀerentiate which type of radiation is being detected.2.They cannot be used to determine the exact energy of the detected radiation3.They have a very low eﬃciencyResolving time(Dead time)After a count has been recorded,it takes the G-M tube a certain amount of time to reset itself to be ready to record the next count.The resolving time or”dead time”,T,of a detector is the time it takes for the detector to”reset”itself.Since the detector is”not operating”while it is being reset,the measured activity is not the true activity of the sample.If the counting rate is high,then the eﬀect of dead time is very important.In our experiments in Phy432L,we will estimate the dead time by examining the discharge pulse of the tube.If there is time,we can also examine the series of times between successive Geiger Counter pulses.This is somewhat diﬀerent than the conventional methods used in other student labs which don’t have the capability to examine the discharge or measure the time between successive pulses.Below,I discuss the conventional method for both correcting for dead time and measuring it. Correcting for the Resolving time:We deﬁne the following variables:T=the resolving time or dead time of the detectort r=the real time that the detector is operating.This is the actual time that the detector is on.It is our counting time.tr does not depend on the dead time of the detector,but on how long we actually record counts.t l=the live time that the detector is operating.This is the time that the detector is4able to record counts.t l depends on the dead time of the detector. C=the total number of counts that we record.n=the measured counting rate,n=C/t rN=the true counting rate,N=C/t lNote that the ratio n/N is equal to:n N =C/t rC/t i=t lt r(2)This means that the fraction of the counts that we record is the ratio of the”live time”to the”real time”.This ratio is the fraction of the time that the detector is able to record counts.The key relationship we need is between the real time,live time,and dead time.To a good approximation,the live time is equal to the real time minus C times the dead time T:t l=t r−CT(3) This is true since CT is the total time that the detector is unable to record counts during the counting time t r.We can solve for N in terms of n and T by combining the two equations above.First divide the second equation by t r:t l t r =1−CTt r=1−nT(4)From theﬁrst equation,we see that the left side is equal to n/N:nN=1−nT(5) Solving for N,we obtain the equation:N=n1−nT(6)This is the equation we need to determine the true counting rate from the measured one.Notice that N is always larger than n.Also note that the product nT is the key parameter in determining by how much the true counting rate increases from the measured counting rate.For small values of the nT,the product nT(unitless)is the fractional increase that N is of n.For values of nT<0.01dead time is not important, and are less than a1%eﬀect.Dead time changes the measured value for the counting rate by5%when nT=0.05.The product nT is small when either the counting rate n is small,or the deat time T is small.5Measuring the Resolving TimeWe can get an estimate of the resolving time of our detector by performing the following measurements,called the ”two source”method for estimating detector dead time.First we determine the counting rate with one source alone,call this counting rate n 1.Then we add a second source next to the ﬁrst one and determine the counting rate with both sources together.Call this counting rate n 12.Finally,we take away source 1and measure the counting rate with source 2alone.We call this counting rate n 2.You might think that the measured counting times n 12should equal n 1plus n 2.If there were no dead time this would be true.However,with dead time,n 12is less than the sum of n 1+n 2.This is because with both sources present the detector is ”dead”more often than when the sources are being counted alone.The true counting times do add up:N 12=N 1+N 2(7)since these are the counting rates corrected for dead time.Substituting the expres-sions for the measured counting times into the above equation gives:n 121−n 12T =n 11−n 1T +n 21−n 2T(8)An approximate solution to these equations is given byT ≈n 1+n 2−n 122n 1n 2(9)You can imagine the diﬃculties in obtaining a precise value for T using the ”two source”method.One needs to be very careful that the positions of source 1and 2with respect to the detector alone is the same as the positions of these sources when they are measured together.Also,since n 12is not much smaller than n 1+n 2,one needs to measure all three quantities very accurately.For this one needs many counts,since the relative statistical error equals 1/√N tot ,where N tot is the total number of counts.For suﬃcient accuracy one needs to use an active source for a long time.The values that we usually obtain in our experiments range from 100to 500µsec.The dead time of the G-M tube is also available from the manufacturer,and are between 100and 300µsec.As the G-M tube ages,the dead time can increase.Although dead time will not play a big role in our experiments,one always needs to consider it and make the appropriate corrections.6。

数学外文+中文翻译

SIAM J. DISCRETE MATH.V ol. 26, No. 1, pp. 193–205ROMAN DOMINATION ON 2-CONNECTED GRAPHS∗CHUN-HUNG LIU†AND GERARD J. CHANG‡Abstract. A Roman dominating function of a graph G is a function f: V (G) →{0, 1, 2} such that whenever f(v) = 0, there exists a vertex u adjacent to v such that f(u) = 2. The weight of f is w(f) = . The Roman domination number of G is the minimum weight of a Roman dominating function of G Chambers,Kinnersley, Prince, and West [SIAM J. Discrete Math.,23 (2009), pp. 1575–1586] conjectured that ≤[2n/3] for any 2-connected graph G of n vertices.This paper gives counterexamples to the conjecture and proves that≤max{[2n/3], 23n/34}for any 2-connected graph G of n vertices. We also characterize 2-connected graphs G for which = 23n/34 when 23n/34 > [2n/3].Key words. domination, Roman domination, 2-connected graphAMS. subject classifications. 05C69, 05C35D O I. 10.1137/0807330851. Introduction. Articles by ReVelle [14, 15] in the Johns Hopkins Magazine suggested a new variation of domination called Roman domination; see also [16] for an integer programming formulation of the problem. Since then, there have been several articles on Roman domination and its variations [1, 2, 3, 4, 5, 7, 8, 9, 10,11, 13, 17, 18, 19]. Emperor Constantine imposed the requirement that an army or legion could be sent from its home to defend a neighboring location only if there was a second army which would stay and protect the home. Thus, there are two types of armies, stationary and traveling. Each vertex (city) that has no army must have a neighboring vertex with a traveling army. Stationary armies then dominate their own vertices; a vertex with two armies is dominated by its stationary army, and its open neighborhood is dominated by the traveling army.In this paper, we consider (simple) graphs and loopless multigraphs G with vert ex set V (G) and edge set E(G). The degree of a vertex v∈V (G) is the number of edges incident to v. Note that the number of neighbors of v may be less than degGv in a loopless multigraph. A Roman dominating function of a graph G is a function f:V(G) →{0, 1, 2} such that whenever f(v) = 0, there exists a vertex u adjacent to v such that f(u) = 2. The weight of f, denoted by w(f), is defined as.For any subgraph H of G, let w(f,H) =. The Roman dominationnumber of G is the minimum weight of a Roman dominating function.Among the papers mentioned above, we are most interested in the one by Chambers et al. [2] in which extremal problems of Roman domination are discussed.In particular, they gave sharp bounds for graphs with minimum degree 1 or 2 and boundsof + and . After settling some special cases, they gave the following conjecture in an earlier version of the paper [2].Conjecture (Chambers et al. [2]). For any 2-connected graph G of n vertices, ≤[2n/3]。

Two-dimensional Quantum Field Theory, examples and applications

Abstract The main principles of two-dimensional quantum ﬁeld theories, in particular two-dimensional QCD and gravity are reviewed. We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum ﬁeld theory. The dynamics of conﬁnement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. We describe in detail how the eﬀective action of string theory in non-critical dimensions can be represented by Liouville gravity. By comparing the helicity amplitudes in four-dimensional QCD to those of integrable self-dual Yang-Mills theory, we extract a four dimensional version of two dimensional integrability.
2 48 49 52 54 56
5 Four-dimensional analogies and consequences 6 Conclusions and Final Remarks

The importance of Non-theorems and Counterexamples in Program Verification. University of E

The Importance of Non-theorems andCounterexamples in Program VeriﬁcationGraham SteelSchool of Informatics,University of Edinburgh,Edinburgh,EH89LE,Scotland.graham.steel@/gsteelAbstract.We argue that the detection and refutation of non-theorems,and the discovery of appropriate counterexamples,is of vital importanceto the Grand Challenge of a Program Veriﬁer.1IntroductionIn this essay,we make a case for the inclusion of non-theorem(i.e.incorrect conjecture)detection and counterexample generation as a core theme in the research program of the Grand Challenge of a Program Veriﬁer.We will argue that:–Research in program veriﬁcation technology will be hindered if counterex-ample generation research does not catch up and keep pace.We will give the reasons for this in§2.–Detecting false conjectures and generating counterexamples to them is a fascinating scientiﬁc challenge in itself.We will argue this in§3.–Deduction based approaches for veriﬁcation,which oﬀer perhaps the best chance of achieving the goals of the grand challenge,must improve in their handling of non-theorems if they are to compete with model checking ap-proaches,which are already able to give counterexamples to false veriﬁcation conditions.Our arguments will be followed in§4by a look at previous and current research on the topic,including our own eﬀorts in the Mathematical Reasoning Group at the University of Edinburgh.Since the areas of application are the same,and since the problems of program veriﬁcation and counterexample generation are in a sense‘dual’,it seems logical that they should be thought of as part of the same Grand Challenge.Note that we fully support the view that theﬁnal goal of the project must be a program veriﬁer,not a system that justﬁnds more and more bugs.However,we will argue in this paper that to achieve this goal,the dual problem of non-theorem detection and counterexample generation must be given due attention.2The Importance of Counterexample GenerationEven the most diligent and expert programmer will very rarely write a bug-freeﬁrst version of a program.The majority of calls to a program veriﬁer will therefore involve an incorrect program.If we want program veriﬁcation tools to become widely used,they must deal with buggy programs in a competent manner.Simply presenting failed veriﬁcation conditions or open subgoals is not suﬃcient,since it leaves the user with no idea whether a bug has been found, or whether the veriﬁcation system is simply unable to dispose of that particular proof obligation.What is required is a system which can not only detect an incorrect conjecture,but also supply a counterexample to allow the user to locate theﬂaw.Non-theorem detection is also important to the internal processes of a veriﬁ-cation tool.Automated proof attempts,particularly when induction is involved, will frequently require the conjecturing of lemmas and generalisations.Often these conjectures will be false,and it is vital that we detect these cases and prune our search space appropriately.There is a further,pragmatic argument for the inclusion of counterexample generation within the Grand Challenge.Great scientiﬁc problems,from landing a man on the moon to producing a computer to beat a grand master at chess,have been solved by making iterative improvements to a prototype and learning from failures.To apply this methodology to our Grand Challenge,we must encourage participation from industry,in order to ensure a supply of case study material,to get feedback on the tools we produce,and where appropriate,to try to inﬂuence software engineering practice.For this,we need to ensure that we are able to make aﬁnancial argument for the use of our tools even when they are at a prototype stage,and unable to deliver a fully veriﬁed end product.Being able to detect and present counterexamples that allow bugs to be identiﬁed is a way of ensuring some payback to our industrial partners.The current industrial preference for model checking over theorem proving can partly be explained by the ability of model checkers to present counterexample traces.3The Challenge of Counterexample GenerationThe non-theorems that arise in program veriﬁcation can sometimes be easy toﬁnd.They often occur close to the base case of a recursive data type,and evaluating the conjecture at some small values can be suﬃcient to detect the bug.However,there are a large number of cases where the counterexample is a far more subtle object,and it is here that the scientiﬁc challenge lies.An example of a success story in this area is the application of formal methods to discovering attacks on security protocols.Here,the attacks are counterexamples to conjectures about the security of a system.The counterexamples may be quite large,for example up to14messages in[18],and may require an intruder to exploit quite subtleﬂaws in the protocol.Further challenges remain:for example, taking into account the mathematical properties of the cryptofunctions in use.Other problems are well outside the scope of current techniques.For example, the book‘Numerical recipes in C’contains Knuth’s code for a(pseudo-)random number generator,[14,p.283].The code is designed to return aﬂoating point number between0and1.However,if certain very large seed values are used, the code can return a number outside of this range.With seed value752005 7091,13out of theﬁrst10,000calls return a value very signiﬁcantly outside the required range.The large seed value may seem ridiculous,but if you were seeding your generator on the number of seconds since1Jan1970,this value would have occurred as a seed during1993.The generation of these counterexamples remains well beyond the capabilities of current tools,and presents a substantial scientiﬁc challenge.4Survey of Counterexample GenerationGiven how important the detection and presentation of counterexamples is to applications of formal methods,it is surprising how comparatively little attention it has received.A community of researchers interested in the subject is now beginning to emerge,with the second‘Workshop on Disproving’due to be held at CADE20052.Much past research has focused on fast generation of counterexamples to con-jectures which are in some sense‘obviously’false.Model generators like MACE, [9],and FINDER,[17],can be used to enumerateﬁnite domains to search for counterexamples.The Isabelle theorem prover,[11],now includes a tactic to search for small counterexamples,called quick-check,[5].For inﬁnite data struc-tures with recursive deﬁnitions,methods have been proposed by Protzen,[15], and Reif,[16].Both can deal with small problems quite eﬀectively,but are not suitable for large counterexamples,or for domains that cannot be easily enu-merated.Model generation has also been proposed as a method for refuting non-theorems,[1,20].Model checking can be a very eﬀective method forﬁnding counterexamples to false veriﬁcation conditions,particularly inﬁnite domains.As we have al-ready remarked,the fact that model checking can produce counterexamples as well as provide guarantees is one reason for its increasing popularity in industry. Many researchers are now working on extending model checkers to non-ﬁnite domains,using‘lazy’and‘on-the-ﬂy’techniques to construct the inﬁnite models as they are checked(e.g.[4]),with good results.Another large branch of current model checking research is in counterexample guided abstraction reﬁnement,[6, 3].Here,the processing of counterexamples is used to guide management of the level of detail that is taken into account when attempting veriﬁcation of a program,balancing tractability of the model checking problem against over-abstraction.Theorem provers can be used to check the feasibility of counterexam-1Knuth stated in the speciﬁcation that the seed value can be any(large)number under1000000000.This is a known bug-the Numerical Recipes in C website contains a patch toﬁx the code.2http://www.cs.chalmers.se/~ahrendt/cade20-ws-disproving/ple traces,[2].A further current direction involves passing unproven conjectures from a theorem prover to a model checker to search for counterexamples,[13].Our work in Edinburgh has lead to the development of the Coral system3, which refutes incorrect inductive conjectures using aﬁrst-order theorem prover adapted to follow the‘proof by consistency strategy’of Comon and Nieuwenhuis, [7].Its major successes have been in theﬁeld of protocol analysis,where it has been used to discover6previously unknown attacks on3group protocols,[18, 19].The protocols are modelled inductively following Paulson,[12],and the attacks found as counterexamples to security conjectures.Coral proved to be particularly suitable for group protocols because they can be formalised very naturally in an inductive model.This is something that rival approaches,such as model checking,struggle with.In future,we plan to experiment with Coral in other areas of formal veriﬁcation.Many of the big problems remain unsolved.For example,how to deal ade-quately with arithmetic,or how to explore very large or non-trivially enumerable spaces for counterexamples.Some current directions include trying to use more information from failed proof attempts to guide the counterexample search,[8], and to suggest patches for incorrect conjectures,[10].5SummaryWe have argued that disproof and counterexample generation are vital areas for research in the development of practical program veriﬁcation systems.There remain many exciting open problems,and milestones to pass,such as the auto-matic generation of counterexamples for Knuth’s random number bug,described above.Non-theorem detection therefore deserves to be included as a core theme in the Grand Challenge of a Program Veriﬁer.References1.W.Ahrendt.Deductive search for errors in free data type speciﬁcations using modelgeneration.In A.Voronkov,editor,18th Conference on Automated Deduction, volume2392of Lecture Notes in Computer Science,pages211–225.Springer,2002.2.T.Ball,B.Cook,hiri,and L.Zhang.Zapato:Automatic theorem proving forpredicate abstraction reﬁnement.In Rajeev Alur and Doron Peled,editors,CAV, volume3114of Lecture Notes in Computer Science,pages457–461.Springer,2004.3.T.Ball and S.Rajamani.The SLAM toolkit.In G´e rard Berry,Hubert Comon,andAlain Finkel,editors,CAV,volume2102of Lecture Notes in Computer Science, pages260–264.Springer,2001.4. D.Basin,S.M¨o dersheim,and L.Vigan`o.An on-the-ﬂy model-checker for securityprotocol analysis.In Proceedings of the2003European Symposium on Research in Computer Security,pages253–270,2003.Extended version available as Technical Report404,ETH Zurich.3/gsteel/coral5.Stefan Berghofer and Tobias Nipkow.Random testing in isabelle/hol.In2ndInternational Conference on Software Engineering and Formal Methods(SEFM 2004),pages230–239,2004.6.Edmund M.Clarke,Orna Grumberg,Somesh Jha,Yuan Lu,and Helmut Veith.Counterexample-guided abstraction reﬁnement for symbolic model checking.Jour-nal of the Association for Computing Machinery,50(5):752–794,2003.on and R.Nieuwenhuis.Induction=I-Axiomatization+First-Order Con-rmation and Computation,159(1-2):151–186,May/June2000.8.L.A.Dennis.The use of proof planning critics to diagnose errors in the basecases of recursive programs.In W.Ahrendt,P.Baumgartner,and H.de Nivelle, editors,IJCAR2004Workshop on Disproving:Non-Theorems,Non-Validity,Non-Provability,pages47–58,2004.9.W.McCune.A Davis Putnam program and its application toﬁniteﬁrst ordermodel search.Technical report,Argonne National Laboratory,1994.10.R.Monroy.Predicate synthesis for correcting faulty conjectures:The proof plan-ning paradigm.In Automated Software Engineering,pages247–269,2003.11.L.C.Paulson.The foundation of a generic theorem prover.JAR,5:363–397,1989.12.L.C.Paulson.The Inductive Approach to Verifying Cryptographic Protocols.Jour-nal of Computer Security,6:85–128,1998.13.L.Pike,P.Miner,and W.Torres.Model checking failed conjectures in theoremproving:a case study.Technical Report NASA/TM–2004–213278,NASA Lang-ley Research Center,November2004.Available at / ~lepike/pub_pages/unproven.html.14.William H.Press,Saul A.Teukolsky,William T.Vetterling,and Brian P.Flannery.Numerical Recipes in C:The Art of Scientiﬁc Computing.Cambridge University Press,1992.15.M.Protzen.Disproving conjectures.In D.Kapur,editor,11th Conference onAutomated Deduction,pages340–354,Saratoga Springs,NY,USA,June1992.Published as Springer Lecture Notes in Artiﬁcial Intelligence,No607.16.W.Reif,G.Schellhorn,and A.Thums.Flaw detection in formal speciﬁcations.InIJCAR’01,pages642–657,2001.17.J.Slaney.FINDER:Finite Domain Enumerator.Australian National Univer-sity,1995.Available from ftp://.au/pub/papers/slaney/finder/ finder.ps.gz.18.G.Steel and A.Bundy.Attacking group multicast key management protocols usingCORAL.Electronic Notes in Theoretical Computer Science(ENTCS),125(1):125–144,2004.Also available as Informatics Research Report EDI-INF-RR-0241.Pre-sented at the ARSPA workshop2004.19.G.Steel,A.Bundy,and M.Maidl.Attacking a protocol for group key agreement byrefuting incorrect inductive conjectures.In D.Basin and M.Rusinowitch,editors, Proceedings of the International Joint Conference on Automated Reasoning,num-ber3097in Lecture Notes in Artiﬁcial Intelligence,pages137–151,Cork,Ireland, July2004.Springer-Verlag Heidelberg.20.T.Weber.Bounded model generation for isabelle/hol.In IJCAR2004Workshopon Disproving-Non-Theorems,Non-Provability,pages27–36,Cork,Ireland,2004.。

Problem 1.1 Prove that

OPEN PROBLEMSA.StoimenowCurrent version:May31,1999First version:January13,1999 Contribution to problem session of the International Knot Theory Meeting,August7-15,1998,Delphi,Greece1Vassiliev invariants and some conjectures on braid representa-tionsThe origin of the problems I’d like to propose is the followingProblem1.1Prove thatdim Vassiliev invariants of degree k C kfor some constant C 1.As the usual approach of counting CDs[1]does not look very promising,in[7]I started a different method which gave solutions in cases where Vassiliev invariants are restricted to some knot classes, e.g.arborescent knots.For many of these classes it might well be that they cover all knots,and this is the origin of the problems I propose here.Hence,any one of the conjectures in the following implies a positive solution of the problem and all they are of the form“some class covers all knots”or equiv-alently“any knot has the property used to describe this class”.The reasons why these conjectures solve the problem are explained in[5,6].I start with a conjecture related to a conjecture of Przytycki[3](quoted,inter alia,in Rob Kirby’s problem book[2]).A diagram is called matched if its crossings can be paired up so as to look likeorwhere orientation of the strands is arbitrary.So,contrarily to Przytycki,I allow parallel orientation of the strands.A matched diagram according to Przytycki’s deﬁnition I call reversely matched.That is,a diagram is reversely matched if it is matched and the orientation of the strands in any pair is reverse.The method works for knots with matched diagrams:in[5]it was proved that if one restricts Vassiliev invariants to the class M of knots with matched diagrams thendim Vassiliev invariants of degree k on M C kBut possibly Przytycki’s conjecture[3],that there are knots with no reversely matched diagram,is true,so I there is some caution with the following21Vassiliev invariants and some conjectures on braid representationsConjecture 1.1Any knot has a matched diagram.To get possibly more realistic problems,we turn to braids.Trying a different approach,it was possible in [5]to construct large classes of knots via braids,for which the exponential bound holds (in the same sense as for knots with matched diagrams).A problem related to the ﬁrst such class is like a braid version of conjecture 1.1:Conjecture 1.2Any knot is a closed braid γB n with γσ1σn1r∏i 1σpik i with n r k i1and all p i even.Several weaker versions of conjecture 1.2sufﬁce:replace σk i by βk i j i with βk j being the positive band (in Rudolph’s [4]sense)interchanging strands k and j ,s.t.for all i we have k i j i C for some global constant C ,or more generallyreplace σpi k i by any pure braid with crossing number (that is,the minimal length of words in the Artin generators representing it)C for some global constant C .Conjecture 1.3Any knot is a closed braid γB n withγr∏i 1σr i p i(1)for some n r N with p i 0r i Z ,such that even (index)generators appear with exactly one odd power (that is,for each even 1p n !i with p i p and r i odd).Or more generally the following weaker property sufﬁces:c i i N with c i 1c i 1C for some global constant C with C and c i independent of γ,s.t.k :σc k appears with exactly one odd power in (1)(that is,for each k N !i with p i c k and r i odd).Conjecture 1.4Any knot is a closed braid of the form (1)such that 1ir :p i1p i2Cfor some global constant C N .These are the braids presentable by (stacking up)at most C “blocks”as in ﬁgure 1(blocks are braids of arbitrary but equal strand number,where all generators commute with each other,i.e.there are no σi and σi 1therein).Figure 1:The block corresponding to σ22σ36σ18In any of the previous problems ‘any knot’may be replaced by ‘any alternating knot’or ‘any closedpositive braid knot’.References[1]S.V.Chmutov and S.V.Duzhin,An upper bound for the number of Vassiliev knot invariants,Jour.ofKnot Theory and its Ramiﬁcations,3(2)(1994),141–151.[2]R.Kirby,Problems of low-dimensional topology,book available on/˜kirby.[3]J.Przytycki,t k moves on links,in“Braids”,Santa Cruz,1986(J.S.Birman and A.L.Libgober,eds.),Contemp.Math.78,615–656.[4]L.Rudolph,Braided surfaces and Seifert ribbons for closed braids,Comment.Math.Helv.58(1983),1–37.[5] A.Stoimenow,The braid index and the growth of Vassiliev invariants,preprint.[6],Onﬁniteness of Vassiliev invariants and a proof of the Lin-Wang conjecture via braid-ing polynomials,preprint.[7],Gaußsum invariants,Vassiliev invariants and braiding sequences,preprint.2Genus and Homﬂy polynomialProblem2.1Decide whether some of the following13knots(recorded in Dowker-Thistlethwaite notation[DT])has a Seifert surface of genus maxdeg∆.Such a knot would be a counterexample to an inequality conjectured by Morton[Mo]:2g mindeg v P. Unfortunately,for any of them(or any other possible counterexample)surfaces of the Seifert algo-rithm of genus maxdeg∆cannot exist.The knots below are the only examples in Thistlethwaite’s tables[HTW]with16crossings,for which the conjectured inequality of Morton is false when replacing the genus by any of its lower bounds1,maxdeg∆,σ2and the maximal bound of Bennequin’s inequality applied to all the diagrams of the knot that Thistlethwaite’s tool knotfind generates from the diagram included in the table.The list is also available on[St2]or,also in diagram form,on request.maxdeg∇2maxdeg∆,mindeg v P and maxdeg z P and the doubled Bennequin inequality bound are recorded at the start of each entry.1649492146644816218-22-2461428-12-1032302026 16575572466441012-1628-22-6-26-30-24-14-20-32-18-28 16575703466441012-1628-24-6-28-30-32-26-14-22-20-18 16585206466441012-1828-22-26-6-32-30-14-28-16-24-20 16585619466441012-1828-24-26-6-30-32-28-16-14-22-20 16592788466441012-2028-24-26-28-6-32-30-18-16-14-22 166146504664410-1416-302-18-2422-12-28-6-32-8-20-26 16788939688641214-16-22210-20-26-6-32-28-30-18-24-8 168516006886412181622226106-32828301424-20 168838416886412-20-16-22218-28-832-10-6-30-14-24-26 161007218246241426-18-24-22212-3028-32-1016206-8 161116868688668142-18-28424-123026321620-1022 16111839768866814222-1842426-30-283216-12-2010 Problem2.2A similar appealing inequality to the one conjectured by Morton is maxdeg z P2g.Is it always true?Possible counterexamples are the4knots onﬁgure9of[St].References[DT] C.H.Dowker and M.B.Thistlethwaite,Classiﬁcation of knot projections,Topol.Appl.16(1983), 19–31.[HTW]J.Hoste,M.Thistlethwaite and J.Weeks,Theﬁrst1,701,936knots,Math.Intell.20(4)(1998),33–48. [Mo]H.Morton,Problems,in“Braids”,Santa Cruz,1986(J.S.Birman and A.L.Libgober,eds.),Contemp.Math.78,557–574.[St] A.Stoimenow,Knots of genus two,preprint.[St2],web document rmatik.hu-berlin.de/˜stoimeno/ mexs.dat.3Q polynomial and Vassiliev invariantsProblem3.1The Q polynomial contains the Vassiliev knot invariant of degree2:in[K],Kanenobu proved the relation Q2V1for the evaluation at2of theﬁrst derivative of the Q polynomial [BLM,Ho].Are the evaluations at2of the higher derivatives of Q also Vassiliev invariants?The n-th one of degree2n?By derivating Kanenobu’s formula[K2,theorem1]at u1the derivatives of Q evaluated at2 are polynomials in the derivatives of V evaluated at1on rational links and hence on them Vassiliev invariants.References[BL]J.S.Birman and X-S.Lin,Knot polynomials and Vassiliev’s invariants,Invent.Math.111(1993)225–270.[BLM]R.D.Brandt,W.B.R.Lickorish and lett,A polynomial invariant for unoriented knots and links, Inv.Math.74(1986),563–573.[Ho] C.F.Ho,A polynomial invariant for knots and links–preliminary report,Abstracts Amer.Math.Soc.6(1985),300.[K]T.Kanenobu,An evaluation of theﬁrst derivative of the Q polynomial of a link,Kobe J.Math.,5(1988), 179–184.[K2],Relations between the Jones and Q polynomials of2-bridge and3-braid links,Math.Ann.285(1989),115–124.Acknowledgement.I would wish to thank to J.Birman and H.Morton for their helpful remarks.。

Fully Coherent X-ray Pulses from a Regenerative Amplifier Free Electron Laser

a r X i v :p h y s i c s /0602076v 1 [p h y s i c s .a c c -p h ] 12 F eb 2006SLAC-PUB-11598Fully Coherent X-ray Pulses from a Regenerative Ampliﬁer FreeElectron LaserZhirong Huang and Ronald D.Ruth Stanford Linear Accelerator Center,Stanford University,Stanford,CA 94309(Dated:January 30,2006)AbstractWe propose and analyze a novel regenerative ampliﬁer free electron laser (FEL)to produce fully coherent x-ray pulses.The method makes use of narrow-bandwidth Bragg crystals to form an x-ray feedback loop around a relatively short undulator.Self-ampliﬁed spontaneous emission (SASE)from the leading electron bunch in a bunch train is spectrally ﬁltered by the Bragg reﬂectors and is brought back to the beginning of the undulator to interact repeatedly with subsequent bunches in the bunch train.The FEL interaction with these short bunches not only ampliﬁes the radiation intensity but also broadens its spectrum,allowing for eﬀective transmission of the x-rays outside the crystal bandwidth.The spectral brightness of these x-ray pulses is about two to three orders of magnitude higher than that from a single-pass SASE FEL.PACS numbers:41.50.+h,41.60.CrAn x-ray free electron laser(FEL)based on self-ampliﬁed spontaneous emission(SASE) is an importantﬁrst step towards a hard x-ray laser and is expected to revolutionize the ultrafast x-ray science(see,e.g.,Refs.[1,2]).Despite its full transverse coherence,a SASE x-ray FEL starts up from electron shot noise and is a chaotic light temporally.Two schemes have been proposed to improve the temporal coherence of a SASE FEL in a single pass conﬁguration.A high-gain harmonic generation(HGHG)FEL uses available seed lasers at ultraviolet wavelengths and reaches for shorter wavelengths through cascaded harmonic generation[3].In this process,the ratio of electron shot noise to the laser signal is ampliﬁed by at least the square of the harmonic order and may limit itsﬁnal wavelength reach to the soft x-ray region[4].Another approach uses a two-stage SASE FEL and a monochromator between the stages[5].The SASE FEL from theﬁrst undulator is spectrallyﬁltered by a monochromator and is then ampliﬁed to saturation in the second undulator.This approach requires an undulator system almost twice as long as a single-stage SASE FEL.Another seeding scheme,a regenerative ampliﬁer FEL(RAFEL),has been demonstrated in the infrared wavelength region[6]and discussed in the ultraviolet wavelength region[7].It consists of a small optical feedback and a high-gain FEL undulator.In the hard x-ray region, perfect crystals may be used in the Bragg reﬂection geometry for x-ray feedback[8,9]and have been demonstrated experimentally for x-ray photon storage(see,e.g.,Ref.[10,11]). In this paper,we propose and analyze a novel x-ray RAFEL using narrow-bandwidth,high-reﬂectivity Bragg mirrors.The basic schematic is shown in Fig.1.Three Bragg crystals are used to form a ring x-ray cavity around a relatively short undulator.Alternative backscat-tering geometry with a pair of crystals may also be used.SASE radiation from the leading electron bunch in a bunch train is spectrallyﬁltered by the Bragg reﬂectors and is brought back to the beginning of the undulator to interact with the second bunch.This process continues bunch to bunch,yielding an exponentially growing laserﬁeld in the x-ray cavity. The FEL interaction with these short bunches not only ampliﬁes the radiation intensity but also broadens its spectrum.The downstream crystal transmits the part of the radiation spectrum outside its bandwidth and feeds back theﬁltered radiation to continue the ampli-ﬁcation pared to a SASE x-ray FEL that typically requires more than100m of undulator distance,this approach uses a signiﬁcantly shorter undulator but a small number of electron bunches to generate multi-GW x-ray pulses with excellent temporal coherence. The resulting spectral brightness of these x-ray pulses can be another two to three orders ofundulatorBragg mirrorx-raychicaneFIG.1:(Color)Schematic of an x-ray RAFEL using three Bragg crystals.magnitude higher than the SASE FEL.We ﬁrst consider a one-dimensional (1-D)model of the narrow-bandwidth RAFEL to describe its main characteristics such as the temporal proﬁle,the round-trip power gain and the maximum extraction eﬃciency.At the beginning of the n th undulator pass,the radiation ﬁeld is represented by E n (t ),where t is the arrival time relative to the longitudinal center of the electron bunch.The radiation ﬁeld at the exit of the undulator isE a n (t )≈E n (t )g (t )+δE n (t ),(1)where δE n (t )is the SASE signal of the n th electron bunch.When the radiation slippage length is much smaller than the electron bunch length,we can assume the electric ﬁeld gain factor g (t )is a function of the local beam current which can be approximated byg (t )≈g 0exp−t 22πe−iωt∞−∞dt ′E a n (t ′)e iωt ′f (ω−ωr ),(3)where f (u )=r exp (−u 2/4σ2m )is a Gaussian spectral ﬁlter function with the rms intensitybandwidth σm ,and ωr is the central frequency of the ﬁlter with the power reﬂectivity |r |2≤1.For a high-gain ampliﬁer after a few passes,the seed signal dominates over the SASE,so that we can neglect the second term on the right side of Eq.(1).Integrating Eq.(3)overthe frequency yieldsE n+1(t)= ∞−∞dt′rσmπe−iωr(t−t′)e−σ2m(t−t′)2g(t′)E n(t′).(4) Since there is no initial seed signal,E1(t)=0,andE2(t)= ∞−∞dt′rσmπe−iωr(t−t′)e−σ2m(t−t′)2δE1(t′)(5) is the spectrallyﬁltered SASE from theﬁrst pass that seeds the second pass.For n≫1,we look for an exponentially growing solutionE n(t)=Λn A(t)e−iωr t.(6) Eq.(4)is then transformed to an integral equation:ΛA(t)= ∞−∞dt′K(t,t′)A(t′),(7)with the kernelK(t,t′)=rσmπe−σ2m(t−t′)2g(t′).(8)Since both r and g(t′)may be complex,K(t,t′)is in general not a hermitian kernel. We expect that a Gaussian fundamental mode will have the largest gain|Λ0|,i.e.,A0(t)=exp −t24σ2x0 =g0r2σmσxa1+4σ2mσ2xa exp −σ2m t22σ2x0+σ2τis the rms x-ray pulse duration at the undulator end(see Eq.(14)).The self-consistent solution of Eq.(10)isσ2x0=1+4σ2mσ2xa1+8σ2mσ2τ+11+8σ2mσ2τ−1Thus,the round-trip power gain isG eﬀ≡|Λ0|2=G0R4σ2mσ2xa1+8σ2mσ2τ−1.(12)1+8σ2mσ2τ+1where G0=|g0|2is the peak FEL gain,and R=|r|2is the peak reﬂectivity of the feedback system.Regenerative ampliﬁcation requires that G eﬀ>1.Note that G eﬀdepends on the time-bandwidth productσmστ,but not onσm orστseparately.Theﬁltered radiation power at the undulator entrance for n≫1is thenP n(t)=|E n|2=P0G n eﬀexp −t22σ2xa ,(14) withσxa given by Eq.(11).If we neglect any absorption in the crystal,the part of the radiation energy(with frequency content mainly outside the feedback bandwidth)may be transmitted with the maximum eﬃciencyη= P a n(t)dt− P n+1(t)dt4σ2mσ2xaFIG.2:X-ray reﬂectivity of a100-µm-thick diamond(400)crystal for8-keV,π-polarized radiation. Fig.2.The crystals may be bent slightly to provide the necessary focusing of theﬁltered radiation at the undulator entrance.In order to accelerate a long bunch train in the SLAC linac,we use the entire rf macropulse available without the rf pulse compression(SLED).The maximum LCLS linac energy,with-out the SLED,is about10GeV.Table I lists the beam and undulator parameters that are typical for x-ray FELs such as the LCLS,except that the length of the undulator is only 20m instead of more than100m planned for the LCLS.We perform the three-dimensional (3-D)GENESIS[13]FEL simulation that shows the maximum power gain G0≈39after the20-m undulator,with the fwhm relative gain bandwidth about2×10−3(see Fig.3).The LCLS accelerator and bunch compressor systems are expected to generate a bunch current proﬁle which is moreﬂattop than Gaussian,with aﬂattop duration T=100fs[1].If we takeστ≈T/2.35andσm≈∆ωm/2.35in Eq.(12),we obtain the round-trip gain G eﬀ≈16 under these parameters.We have developed a1-D FEL code that simulates the regenerative ampliﬁcation process. The electron rms energy spread is increased in the1-D code to3.8×10−4so that the1-D FEL gain matches the3-D FEL gain G0=39determined by parameters in Table I.The simulation using aﬂattop current proﬁle and a nearlyﬂattop crystal reﬂectivity curve shows that the round-trip gain G eﬀ≈14in the exponential growth stage and that the RAFEL reaches saturation within10x-ray passes.For a total x-ray cavity length of75m(25m for each of three cavity arms in Fig.1),the duration of the10-bunch train is about2.25µs,well within the3.5-µs uncompressed rf pulse length even after taking into account the structureTABLE I:Parameters for an x-ray RAFEL. Parameter Symbol ValueFIG.3:Power gain factor predicted from GENESIS simulation as a function of the relative fre-quency detune.FIG.4:(Color)Average radiated energy(blue solid line)and relative rms energyﬂuctuation(green dashed line)at the undulator end.ation transmitted through the end crystal(the noisy part of the blue solid curve in Fig.5) can be separated from the narrow-bandwidth signal by another monochromator following the transmission as demonstrated in Fig6.The total x-ray energy dose absorbed by the undulator-end crystal(FEL plus spontaneous radiation)is estimated to be two orders of magnitude smaller than the melting dose level for diamond.Finally,Fig.4also shows that the shot-to-shot radiation energyﬂuctuates up90%in the exponential growth stage but quickly reduces to about5%at the end of the10th pass.Although a monochromator may also be used in a saturated SASE FEL to select a single longitudinal mode,the radiation power will be reduced by the ratio of the SASE bandwidth to the monochromator bandwidth,FIG.5:(Color)Temporal proﬁle of the reﬂected(green dashed line)and transmitted(blue solid line)FEL power at the end of10th pass.FIG.6:Temporal proﬁle of theﬁnal transmitted FEL power after passing a monochromator with a fwhm bandwidth2∆ωm/ωr=8×10−6toﬁlter out the SASE radiation.and theﬁltered radiation energy stillﬂuctuates100%.While we consider a ring x-ray cavity with60◦Bragg reﬂection for illustration,the RAFEL scheme and its analysis presented in the paper is equally applicable to a backscattered x-ray cavity with90◦Bragg reﬂection.The round-trip time of such a cavity is only two thirds of the ring cavity shown in Fig.1,allowing for50%more electron bunches in a bunch train of the same duration to participate in the RAFEL process.The reﬂectivity at exactly90◦Bragg reﬂection for cubic crystals such as diamond may be complicated by multiple-wave diﬀraction and has not been studied here.Crystals with lower structure symmetry such as sapphire may provide the necessary high reﬂectivity in backscattering as demonstrated inRef.[11].In summary,we have described a narrow-bandwidth regenerative ampliﬁer FEL(RAFEL) at the hard x-ray wavelength region using Bragg crystals that produces nearly transform limited x-ray pulses in both transverse and longitudinal pared to a SASE x-ray source that possesses a typical bandwidth on the order of10−3,the bandwidth of an x-ray RAFEL can be more than two orders of magnitude smaller,resulting in a factor of a few hundred improvement in spectral brightness of the radiation source.The use of multiple bunches in a bunch train for regenerative ampliﬁcation allows for a relatively short undulator system and may be adapted in the LCLS using the SLAC s-band linac.Since superconducting rf structures can support a much longer bunch train in an rf macropulse, an x-ray RAFEL based on a superconducting linac may require a much lower single pass gain and hence relax some of beam and jitter requirements provided that the additional radiation damage to the x-ray optics is tolerable.Therefore,the method described in this paper is a promising approach to achieve a fully coherent x-ray laser.We thank J.Hastings and J.Arthur for useful discussions on x-ray optics.Z.H.thanks K.-J.Kim for general discussions and for providing Refs.[8,9].This work was supported by Department of Energy contracts DE–AC02–76SF00515.[1]LCLS Conceptual Design Report,SLAC-R-593,(2002).[2]TESLA XFEL Technical Design Report(Supplement),TESLA-FEL-2002-09(2002).[3]L.-H.Yu,Phys.Rev.A44,5178(1991).[4] E.Saldin,E.Schneidmiller,and M.Yurkov,mun.202,169(2002).[5]J.Feldhaus et al.,mun.140,341(1997).[6] D.Nguyen et al.,Nucl.Instrum.Methods A429,125(1999).[7] B.Faatz et al.,Nucl.Instrum.Methods A429,424(1999).[8]R.Colella and A.Luccio,mun.50,41(1984).[9] B.Adams and G.Materlik,in Proceedings of the1996Free Electron Lasers,II–24,(Elsevier,Amsterdam,1997).[10]K.-D.Liss et al.,Nature404,371(2000).[11]Yu.Shvyd’ko et al.,Phys.Rev.Lett.90,013904(2003).[12]M.S.del Rio and R.Dejus,“XOP-X-ray oriented programs”,http://www.esrf.fr/computing/scientiﬁc/xop/.[13]S.Reiche,Nucl.Instrum.Methods A429,243(1999).11。

A counterexample to the hot spots conjecture

a rX iv:mat h /9833v2[mat h.PR]1Jan1999Annals of Mathematics,149(1999),309–317A counterexample to the “hot spots”conjecture By Krzysztof Burdzy and Wendelin Werner*Abstract We construct a counterexample to the “hot spots”conjecture;there ex-ists a bounded connected planar domain (with two holes)such that the second eigenvalue of the Laplacian in that domain with Neumann boundary condi-tions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain.1.Introduction The “hot spots”conjecture says that the eigenfunction corresponding to the second eigenvalue of the Laplacian with Neumann boundary conditions attains its maximum and minimum on the boundary of the domain.The conjecture was proposed by J.Rauch at a conference in 1974.Our paper presents a counterexample to this conjecture.Suppose that D is an open connected bounded subset of R d ,d ≥1.Let {ϕ1,ϕ2,...}be a complete set of L 2-orthonormal eigenfunctions for the Lapla-cian in D with Neumann boundary conditions,corresponding to eigenvalues 0=µ1<µ2≤µ3≤µ4≤....The ﬁrst eigenfunction ϕ1is constant.Theorem 1.There exists a planar domain D with two holes (i.e.,con-formally equivalent to a disc with two slits )such that the second eigenvalue µ2is simple (i.e.,there is only one eigenfunction ϕ2corresponding to µ2,up to a multiplicative constant )and such that the eigenfunction ϕ2attains its strictmaximum at an interior point of D .There remains a problem of proving the conjecture under additional as-sumptions on the geometry of the domain.Since our method does not seem to be able to generate a counterexample with fewer than two holes,it is natural to ask if this failure has causes of fundamental nature.310KRZYSZTOF BURDZY AND WENDELIN WERNERProblem1.Does the“hot spots”conjecture hold in all planar domains which have at most one hole?Using a probabilistic coupling argument,Ba˜n uelos and Burdzy(1999) proved the conjecture for some“long and thin”(not necessarily convex)planar domains and for some convex planar domains with a line of symmetry.We know of only one other published result on the conjecture;it is contained in a book by Kawohl(1985).We are grateful to David Jerison and Nikolai Nadirashvili for telling us about their forthcoming results.They include a proof of the“hot spots”con-jecture for convex planar domains and a diﬀerent counterexample.See the introduction to Ba˜n uelos and Burdzy(1999)for a detailed review of various aspects of the“hot spots”conjecture,and a complete reference list. Our techniques are very close to those introduced in that paper so we will be rather brief and we ask the reader to consult that paper for more details.We would like to thank Rodrigo Ba˜n uelos and David Jerison for very useful advice,and the anonymous referee for suggesting a short proof of Lemma1. The second author had the pleasure of being introduced to the problem by JeﬀRauch,the proposer,at E.N.S.Paris in1995.2.Domain constructionBefore describing precisely our domain D,let us now give a short intuitive argument that provides some heuristic insight into our counterexample.Con-sider a planar domain that looks like a bicycle wheel with a hub,at least three very very thin spokes and a tire.Consider the heat equation in that domain with Neumann boundary conditions and an initial temperature such that the hub is“hot”and the tire is“cold.”Due to the fact that the cold arrives in the hub only via the spokes,the“hottest spot”of the wheel will be pushed towards the center of the hub.This implies that the second Neumann eigenfunction in the domain attains its maximum near the center of the hub and therefore not on the boundary of the domain.For technical reasons that will become apparent in the proof,our domain D does not quite look like a bicycle wheel,but it does have a“hub,”three “spokes”and a“tire.”We will use0by the angle2π/3.We will use the point-to-set mapping T x={σ(x),σ∈G}.Typically,T x contains six points.The meaning of T K for a set K is self-evident.A COUNTEREXAMPLE TO THE“HOT SPOTS”CONJECTURE311Supposeε∈(0,1/200)is a very small constant whose value will be chosen later in the proof.Let us name a few points in the plane,A1=03/7),A3=(5,1/100),A4=(11/2,1/200),A5=(6,ε),A6=(13/2,1/200),√3), A7=(7,1/100),A8=(8,8A10=(235,0).Let D1be the domain whose boundary is a polygon with consecutive vertices A1,A2,A3,A4,A5,A6,A7,A8,A9,A10and A1.Let D2be the closure of T D1and let D3be the interior of D2.Finally,we obtain D be removing the line segment between(−18,0)and(−16,0)from D3.We will show that D has the properties stated in Theorem1.The domain D3has three holes while D has only two,because of the cut between(−18,0)and(−16,0).Letα1andα2be the minimum and maximum of the angles betweenA j A j+1,j=1,2,...,9,and the horizontal axis.We have chosen the vectors−−−−→points A j,j=1,...,10,in such a way thatα2−α1<π/2;this fact will be useful at the end of the proof,when we apply results of Ba˜n uelos and Burdzy (1999).3.The second eigenvalue is smallIn this section,we will prove the following result.Lemma1.For everyδ>0,there existsε0>0such that for allε∈(0,ε0), the second Neumann eigenvalueµ2in the domain D=D(ε)deﬁned in Section 2is not greater thanδ.Proof.Recall the points A j deﬁned in Section2.Let A11=(6+100ε/(1−200ε),0);this point lies at the intersection of the line containing A4and A5and the horizontal axis.Suppose thatε<1/1600.Let d1(z)denote the Euclidean distance between z and A11and deﬁne for all z∈D1,the function f1as follows.•f1(z)=0if|z|>6or if d1(z)<400ε,•f1(z)=log(d1(z)/400ε)/log(1/800ε)if|z|≤6and d1(z)∈[400ε,1/2],•f1(z)=1if|z|<6and d1(z)>1/2.Extend f1into a continous function on D invariant under G.The function f1is equal to1in the inside region of D,it is equal to0in the exterior region and it slopes from1to0in the inside parts of the narrow channels connecting the two regions.Note that f1satisﬁes the Neumann boundary conditions on∂D.312KRZYSZTOF BURDZY AND WENDELIN WERNERIt is clear that when ε→0,the two quantities D |f 1|and D |f 1|2re-main bounded and bounded away from 0,and it is elementary to check that D |∇f 1|2→0when ε→0.Similarly,deﬁne a function f 2that is equal to 1in the exterior domain,to 0in the inside part of D and that slopes to 0in the exterior parts of the narrow necks connecting the two.More precisely,let A 12lie at the intersection of the horizontal line and the line containing A 5and A 6,let d 2(z )=dist(z,A 12)and for z ∈D 1,•f 2(z )=0if z <6or if d 2(z )<400ε,•f 2(z )=log(d 2(z )/400ε)/log(1/800ε)if |z |>6and d 2(z )∈[400ε,1/2],•f 2(z )=1if |z |>6and d 2(z )>1/2.Extend f 2in a continuous fashionto D so that it is invariant under the action of G .Both D |f 2|and D |f 2|2remain bounded and bounded away from 0when ε→0.It is easy tocheck that f 2satisﬁes the Neumann boundary conditions and that D |∇f 2|2→0when ε→0.Finally,deﬁne for all ε>0,the function f on D byf (z )=f 1(z ) D f 2.The function f satisﬁes the Neumann boundary conditions.As the supports of f 1and f 2are disjoint,it is clear that D |f |2remains bounded and bounded away from 0when ε→0,and that D |∇f |2→0when ε→0.Since f is orthogonal to the constant function 1(i.e.,to the lowest eigenfunction)because D f =0,we conclude that the second Neumann eigenvalue µ2in D satisﬁes0<µ2≤D |∇f |2A COUNTEREXAMPLE TO THE“HOT SPOTS”CONJECTURE313Recall that s denotes the symmetry with respect to the horizontal axis, and deﬁne for j=3,4,5,6,7the line segmentsK j=)and the exterior one E.Also,let M o i (M o e)denote the part of D between the line segments K3and K5(K5and K7). Put M e=T M o e and M i=T M o i.In the rest of the paper,X t=(X1t,X2t)will denote reﬂected Brownian motion in D(with normal reﬂection on∂D).For all U⊂D,τU will denote theﬁrst hitting time of U by X;i.e.,τU=inf{t>0:X t∈U}.Deﬁne Z t=|X1t−6|.As long as X t stays in M o,the process Z t is a one-dimensional Brownian motion reﬂected at0,with some local time push always pointing away from0,due to the normal reﬂection of X t on the boundary of D.Hence,there is some p1>0,independent ofε<1/200,such that Z t may reach1within1/2unit of time,for any starting point of X t inside M o,with probability greater than2p1.In other words,if X0∈M o then with probability greater than2p1,the process X t will hit K3∪K7before time t=1/2.By symmetry,the process will be more likely to hit K3ﬁrst,if it starts to the left of K5(i.e.in M o i),and it will be more likely to hit K7ﬁrst if it starts in M o e.314KRZYSZTOF BURDZY AND WENDELIN WERNERThe same analysis applies to the other two“bridges”of D.Hence,there exists p1>0such that for allε∈(0,1/200),for all x∈M i and all x′∈M e, P(τI<1/2|X0=x)>p1and P(τE<1/2|X0=x′)>p1.Suppose thatγ⊂D is a connected set of diameter greater than10−10 such thatγ∩I=∅.It is easy and elementary to prove the following:There exists p2>0such that for allε∈(0,1/200),for all x∈I,for allγ⊂D satisfying the above conditions,P(τγ<1/2|X0=x)≥P(τγ<1/2,τT K>1|X0=x)>p2.4Note that p2is independent ofεas the second probability in the last formula depends only on the connected component of D\T K4containing0A COUNTEREXAMPLE TO THE“HOT SPOTS”CONJECTURE315 and consequently thatµ2≥−log(1−p1p2).Note that p1and p2are indepen-dent ofε<1/200.Hence,combining this with Lemma1shows that for small enoughε,one never has{Γ∩I=∅andΓ∩E=∅}.The other two cases,namely{Γ∩E=∅andΓ∩I=∅}and{Γ∩E =∅andΓ∩I=∅},can be dealt with in the same way.Hence,for smallε,Γ⊂M.Remark.In almost exactly the same way,one could prove that the nodal line is in fact conﬁned to an arbitrarily small neighbourhood of T K6whenεis suﬃciently small,but Lemma2is suﬃcient for our purposes.In the rest of the paperε>0is assumed to be small enough so that the nodal line of any second Neumann eigenfunction in D is a subset of M.5.The second eigenvalue is simpleOur proof of the fact that the second eigenvalue is simple is based on an almost trivial argument.However,this argument seems to be so useful that we state it as a lemma.It originally appeared in the proofs of Propositions2.4 and2.5of Ba˜n uelos and Burdzy(1999).Lemma3.Suppose that there exists z0∈D such that the nodal line of any second Neumann eigenfunction does not contain z0.Then the second eigen-value is simple.Proof.Suppose thatϕ2and ϕ2are two independent eigenfunctions cor-responding toµ2.By assumption,ϕ2(z0)=0and ϕ2(z0)=0so the functionx→ϕ2(x) ϕ2(z0)−ϕ2(z0) ϕ2(x)is a nonzero eigenfunction corresponding toµ2.Since it vanishes at z0,we obtain a contradiction.The lemma applies to our domain D becauseΓ⊂M.6.Gradient direction for the second eigenfunctionThisﬁnal part of the proof follows the arguments of Ba˜n uelos and Burdzy (1999)so closely that we will only present a sketch and refer the reader to that paper for more details.Let A denote the disc B(0316KRZYSZTOF BURDZY AND WENDELIN WERNERWe set u(t,y)=u(t,x)for all y∈T x and then extend the function u(t,x)to all x∈D by continuity.Due to the fact that u satisﬁes the Neumann boundary conditions in D1and the symmetry,it is clear that u(t,x)solves the Neumann heat equation in D with the initial condition u(0,x)=1A(x).Since the nodal line ofϕ2is conﬁned to M,the sign of1A(x)ϕ2(x)is constant.We conclude thatc2= D u(0,x)ϕ2(x)dx= Aϕ2(x)dx=0,and so the second eigenfunction coeﬃcient c2is nonzero in the eigenfunction expansion for u(t,x),u(t,x)=c1+c2ϕ2(x)e−µ2t+....With no loss of generality,we can assume that c2>0,choosing the sign ofϕ2 accordingly.But(see,e.g.,Proposition2.1of Ba˜n uelos and Burdzy(1999)), u(t,x)=c1+c2ϕ2(x)e−µ2t+R(t,x),x∈D,t≥0,where R(t,x)converges to0as t→∞faster than e−µ2t,uniformly in x∈D. Hence,if we can show that for someﬁxed x,y∈D and all t>0we have u(t,x)≥u(t,y)then we must also haveϕ2(x)≥ϕ2(y).Recall thatα1andα2denote the minimum and maximum of the angles between vectors−−−−→A j A j+1,j=1,2,...,9,and the horizontal axis and that α2−α1<π/2.In view of this fact,the arguments of Theorems3.1and3.2(see also Example3.2)of Ba˜n uelos and Burdzy(1999)can be easily adjusted to our domain D1and imply that with our choice of the initial condition for u(t,x), we have u(t,x)≥u(t,y)whenever the angle between the vector−→xy and the horizontal axis lies within(α2−π/2,α1+π/2).Hence,we haveϕ2(x)≥ϕ2(y) for all such x,y∈D1.In particular,for every x∈D1,ϕ2(0D1\{0y and−→yx with the horizontal axis belong to (α2−π/2,α1+π/2).Ifϕ2(0)=ϕ2(y)=ϕ2(x)for all y∈F x.The remark following Corollary(6.31)in Folland(1976)may be applied to the operator∆+µ2to conclude that the eigenfunctions are real analytic and therefore they cannot be constant on an open set unless they are constant on the whole domain D.We see thatϕ2attains its strict maximum in .Since the same argument applies to every setσ(D1)for all σ∈G,the functionϕ2attains its strict maximum in D at0A COUNTEREXAMPLE TO THE“HOT SPOTS”CONJECTURE317University of W ashington,Seattle,W AE-mail address:burdzy@Universit´e Paris-Sud,Orsay,FranceE-mail address:wendelin.werner@math.u-psud.frReferences[1]R.Ba˜n uelos and K.Burdzy,On the“hot spots”conjecture of J.Rauch,to appear in J.of Functional Analysis,1999.[2]R.Courant and D.Hilbert,Methods of Mathematical Physics,Interscience PublishersInc.,New York,1953.[3]G. B.Folland,Introduction to Partial Diﬀerential Equations,Mathematical Notes,Princeton Univ.Press,Princeton,1976.[4] B.Kawohl,Rearrangements and Convexity of Level Sets in PDE,Lecture Notes inMathematics1150,Springer-Verlag,New York,1985.(Received March3,1998)。

超限检测英文

超限检测1.Pair platforms overrun testing control system and its application;双秤台超限检测控制系统及其应用更多例句>>2) highway ultralimit detection公路超限检测1.Solutions are proposed to the problems in current systems for highway ultralimit detection and charging.通过对上海市实施超限检测收费系统的设计,介绍了公路超重收费系统的组成,重点阐述了其中称重系统的构成及设计,还针对目前公路超限检测收费系统存在的问题,提出了解决的办法。

3) inspection of out-of-gauge of ends端部超限检测4) automobile overweight detecting system超限检测系统1.In the paper, the fixed type automobile overweight detecting system is introduced.在对固定式汽车超限检测系统进行介绍的基础上,论述了系统的程序设计方法和程序结构。

更多例句>>5) vehicle overloading detection车辆超限超载检测6) freight train gauge-exceeding detecting system货车超限检测系统1.In view of the freight train image collected with the sky as background,the wavelet modulus maxima technique was applied to detect the freight train edge in the freight train gauge-exceeding detecting system based on image processing.针对图像法货车超限检测系统中采集的货运火车图像均以天空为背景,采用基于小波变换模极大值边缘检测技术检测货车边缘。

Theorem

Building counterexamples to generalizations for rational functions of Ritt’s decompositionTheoremJaime Gutierrez a,1David Sevilla b,1a Dpto.de Matem´a ticas,Estad´ıstica y Computaci´o n,Universidad de Cantabria,E–39071Santander,Spainb Dpt.of Computer Science and Software Engineering,University of Concordia,Montreal,CanadaAbstractThe classical Ritt’s Theorems state several properties of univariate polynomial de-composition.In this paper we present new counterexamples to theﬁrst Ritt theorem, which states the equality of length of decomposition chains of a polynomial,in the case of rational ly,we provide an explicit example of a rational function with coeﬃcients in Q and two decompositions of diﬀerent length. Another aspect is the use of some techniques that could allow for other coun-terexamples,namely,relating groups and decompositions and using the fact that the alternating group A4has two subgroup chains of diﬀerent lengths;and we pro-vide more information about the generalizations of another property of polynomial decomposition:the stability of the baseﬁeld.We also present an algorithm for com-puting theﬁxing group of a rational function providing the complexity over the rational numberﬁeld.1IntroductionThe starting point is the decomposition of polynomials and rational functions in one variable.First we will deﬁne the basic concepts of this topic.Deﬁnition1If f=g◦h,f,g,h∈K(x),we call this a decomposition of f in K(x)and say that g is a component on the left of f and h is a component on the right of f.We call a decomposition trivial if any of the components is a unit with respect to decomposition.1Partially supported by Spain Ministry of Science grant MTM2004-07086 Preprint submitted to Elsevier Science18May2006Given two decompositions f=g1◦h1=g2◦h2of a rational function,we call them equivalent if there exists a unit u such thath1=u◦h2,g1=g2◦u−1,where the inverse is taken with respect to composition.Given a non–constant f,we say that it is indecomposable if it is not a unit and all its decompositions are trivial.We deﬁne a complete decomposition of f to be f=g1◦···◦g r where g i is indecomposable.The notion of equivalent complete decompositions is straight-forward from the previous concepts.Given a non–constant rational function f(x)∈K(x)where f(x)=f N(x)/f D(x) with f N,f D∈K[x]and(f N,f D)=1,we deﬁne the degree of f as deg f=max{deg f N,deg f D}.We also deﬁne deg a=0for each a∈K.Remark2From now on,we will use the previous notation when we refer to the numerator and denominator of a rational function.Unless explicitly stated,we will take the numerator to be monic,even though multiplication by constants will not be relevant.Theﬁrst of Ritt’s Theorems states that all the decomposition chains of a polynomial that satisﬁes a certain condition have the same length.It is well known that the result is not true for rational functions,see for example[]. Here we explore new techniques related to this,and include a counterexample in Q(x).Another result in this fashion states that if a polynomial is indecomposable in a certain coeﬃcientﬁeld,then it is also indecomposable in any extension of thatﬁeld.This is also false for rational functions,see[4]and[1].We look for bounds for the degree of the extension in which we need to take the coeﬃcients if a rational function with coeﬃcients in Q has a decomposition in a larger ﬁeld.In this paper we present a computational approach to this question and our conclusions.In Section2we study how to compute bounds for the minimalﬁeld that contains all the decompositions of a given rational function.In Section3we introduce several deﬁnitions and properties of groups related to rational func-tions,which we use in Section4to discuss the number of components in the rational case.In particular,we present an algorithm for computingﬁxing2group of a rational function and we provide the complexity over the rationalnumberﬁeld.Finally,in Section4we present an example of a degree12ratio-nal function with coeﬃcients in Q and two decompositions of diﬀerent length;as far as we know this is theﬁrst example in Q of this kind.2Extension of the coeﬃcientﬁeldSeveral algorithms for decomposing univariate rational functions are known,see for instance[18]and[1].In all cases,the complexity of the algorithmgrows enormously when the coeﬃcientﬁeld is extended.A natural questionabout decomposition is whether it depends on the coeﬃcientﬁeld,that is,the existence of polynomials or rational functions that are indecomposable inK(x)but have a decomposition in F(x)for some extension F of K.Polynomials behave well under certain conditions,however in the rational case this is nottrue.We will try to shed some light on the rational case.Deﬁnition3f∈K[x]is tame when char K does not divide deg f.The next theorem shows that tame polynomials behave well under extensionof the coeﬃcientﬁeld,see[8].It is based on the concept of approximate rootof a polynomial,which always exists for tame polynomials,and is also the keyto some other structural results in the tame polynomial case.Theorem4Let f∈K[x]be tame and F⊇K.Then f is indecomposable inK[x]if and only if it is indecomposable in F[x].The next example,presented in[1],shows that the previous result is false forrational functions.Example5Letf=ω3x4−ω3x3−8x−1 2ω3x4+ω3x3−16x+1whereω∈Q butω3∈Q\{1}.It is easy to check that f is indecomposable in Q(x).However,f=f1◦f2wheref1=x2+(4−ω)x−ω2x2+(8+ω)x+ω,f2=xω(xω−2)xω+1.We can pose the following general problem:Problem6Given a function f∈K(x),compute a minimalﬁeld F such that3every decomposition of f over an extension of K is equivalent to a decompo-sition over F.It is clear that,by composing with units in F(x)⊇K(x),we can always turn a given decomposition in K(x)into one in F(x).Our goal is to minimize this, that is,to determineﬁelds that contain the smallest equivalent decompositions in the sense of having the smallest possible extension over K.Given a decomposition f=g(h)of a rational function in K(x),we can write a polynomial system of equations in the coeﬃcients of f,g and h by equating to zero the numerator of f−g(h).The system is linear in the coeﬃcients of g.Therefore,all the coeﬃcients of g and h lie in some algebraic extension of K.Our goal is toﬁnd bounds for the degree of the extension[F:K]where F contains,in the sense explained above,all the decompositions of f.One way toﬁnd a bound is by means of a result that relates decomposition and factorization.We state the main deﬁnition and theorems here,see[9]for proofs and other details.Deﬁnition7A rational function f∈K(x)is in normal form if deg f N> deg f D and f N(0)=0(thus f D(0)=0).Theorem8(i)Given f∈K(x),if deg f<|K|then there exist units u,v such that u◦f◦v is in normal form.(ii)If f∈K(x)is in normal form,every decomposition of f is equivalent to one where both components are in normal form.We will analyze the complexity ofﬁnding the units u and v later.Theorem9Let f=g(h)with f,g,h in normal form.Then h N divides f N and h D divides f D.This result provides the following bound.Theorem10Let f∈K(x)and u1,u2be two units in K(x)such that g= u1◦f◦u2is in normal form.Let F be the splittingﬁeld of{g N,g D}.Then any decomposition of f in K (x),for any K ⊃K,is equivalent to a decomposition in F(x).PROOF.By Theorems8and9,every decomposition of g is equivalent to another one,g=h1◦h2,where the numerator and denominator of h2divide those of g,thus the coeﬃcients of that component are in F.As the coeﬃcients of h1are the solution of a linear system of equations whose coeﬃcients are4polynomials in the coeﬃcients of g and h2,they are also in F.We also have u1,u2∈K(x),therefore the corresponding decomposition of f lies in the same ﬁeld.2This bound,despite being of some interest because its generality and simplic-ity,is far from optimal.For example,for degree4we obtain[F:K]≤3!·3!= 36.As we will show next,we can use computational algebra techniques,in particular Gr¨o bner bases,toﬁnd good bounds for diﬀerent degrees of f.The following theorem completes Example5.Theorem11Let f∈Q(x)of degree4.If f=g(h)with g,h∈Q(x),there exists aﬁeld K with Q⊂K⊂Q and a unit u∈K(x)such that g(u−1),u(h)∈K(x)and[K:Q]≤3.PROOF.Without loss of generality we assumef=x4+r3x3+r2x2+r1xs3x3+s2x2+s1x+1,g=x2+axbx+1,h=x2+cxdx+1.Then we have the following system of polynomial equations:ac−r1=0,2d+bc−s1=0,c2+cad+a−r2=0,d2+bcd+b−s2=0,2c+ad−r3=0,bd−s3=0.Let I⊂C[r1,r2,r3,s1,s2,s3,a,b,c,d]be the ideal generated by these ing elimination techniques by means of Gr¨o bner bases(see for example[3]),weﬁnd polynomials in I involving r1,r2,r3,s1,s2,s3and each of the variables a,b,c,d:{a3−r2a2+r1r3a−r21,b3−s2b2+s1s3b−s23,2c3−2r3c2+12r23+12r1s1c−14r1r3s1+14r21s3,2d3−2s1d2+12s21+12r3s3d−14r3s1s3+14r1s23⊂I.Now it is clear that,given a degree3function,the coeﬃcients of its components have degree at most3over Q each.But in fact there is aﬁeld of degree3that5contains all of them:ac −r 1=0⇒c ∈Q (a ),2c +ad −r 3=0⇒d ∈Q (a,c )=Q (a ),bd −s 3=0⇒b ∈Q (d )⊆Q (a ).The well–known Extension Theorem (see [3])may be used to extend the points in the variety deﬁned by J =I ∩C [r 1,r 2,r 3,s 1,s 2,s 3].This can also be used to study other questions related to the variety,for example the existence of functions with one decomposition in Q (x )and other in a proper extension of Q .The ideal J determines a variety in C 6of dimension 4(in fact,the equations deﬁning I provide a parametrization of the variety),which represents the set of decomposable functions of degree 4(in normal form).The four polynomials in the diﬀerent variables showed above allow the successive application of Extension Theorem to prove that each point of J is the image of some point in the parametrization.That is,each point in the variety corresponds to a function of degree 4that is decomposable in C (x ).From Example 5we can deduce that,after normalization,not every point of J corresponds to a function that can be decomposed in Q (x ).Remark 12In the polynomial case of degree 4we havef =x 4+r 3x 3+r 2x 2+r 1x ,g =x 2+ax ,h =x 2+cx ∈C (x ).The equations of the ideal I ⊂C [r 1,r 2,r 3,a,c ]determined by this areI =(2c −r 3,c 2+a −r 2,ac −r 1),I ∩C [r 1,r 2,r 3]= 18r 33−12r 2r 3+r 1.It is clear that,from the initial equations and by Extension Theorem,for each point there exist corresponding values for a ,c .However,if we choose the coeﬃcient ﬁeld to be Z 2,the ideal is generated by I =(r 3,c 2+a +r 2,ac +r 1)and the variety is I ∩C [r 1,r 2,r 3]=(r 3).We cannot apply Extension Theorem with the generators we have for I ;in fact,the polynomial x 4+x 2+x is in the variety,but the corresponding equations are c 2+a +1=0,ac +1=0which are false for any values a,c ∈Z 2,so this polynomial is indecomposable in Z 2[x ].63Fixing group andﬁxedﬁeldIn this section we introduce several simple notions from classical Galois theory. LetΓ(K)=Aut K K(x)(we will write simplyΓif there can be no confusion about theﬁeld).The elements ofΓ(K)can be identiﬁed with the images of x under the automorphisms,that is,with M¨o bius transformations(non–constant rational functions of the form(ax+b)/(cx+d)),which are also the units of K(x)under composition.Deﬁnition13(i)Let f∈K(x).We deﬁne G(f)={u∈Γ(K):f◦u=f}.(ii)Let H<Γ(K).We deﬁne Fix(H)={f∈K(x):f◦u=f∀u∈H}. Example14(i)Let f=x2+1x2∈K(x).Then G(f)=x,−x,1x,−1x.(ii)Let H={x,ix,−x,−ix}⊂Γ(C).Then Fix(H)=C(x4).These deﬁnitions correspond to the classical Galois correspondences(not bi-jective in general)between the intermediateﬁelds of an extension and the subgroups of its automorphism group,as the following diagram shows: K(x)←→{id}||K(f)−→G(f)||Fix(H)←−H||K←→ΓRemark15As K(f)=K(f )if and only if f=u◦f for some unit u,we have that the application K(f)→G(f)is well–deﬁned.Next,we state several interesting properties of theﬁxedﬁeld and theﬁxing group.Theorem16Let H be a subgroup ofΓ.(i)H is inﬁnite⇒Fix(H)=K.7(ii)H isﬁnite⇒K Fix(H),Fix(H)⊂K(x)is a normal extension,and in particular Fix(H)=K(f)with deg f=|H|.PROOF.(i)It is clear that no non–constant function can beﬁxed by inﬁnitely many units,as these mustﬁx the roots of the numerator and denominator.(ii)We will show constructively that there exists f such that Fix(H)=K(f) with deg f=|H|.Let H={h1=x,...,h m}.LetP(T)=mi=1(T−h i)∈K(x)[T].We will see that P(T)is the minimum polynomial of x over Fix(H)⊂K(x).A classical proof of L¨u roth’s Theorem(see for instance[17])states that any non–constant coeﬃcient of the minimum polynomial generates Fix(H),and we are done.It is obvious that P(x)=0,as x is always in H.It is also clear that P(T)∈Fix(H)[T],as its coeﬃcients are the symmetric elementary polynomials in h1,...,h m.The irreducibility is equivalent to the transitivity of the action of the group on itself by multiplication.2Theorem17(i)For any non–constant f∈K(x),|G(f)|divides deg f.Moreover,for any ﬁeld K there is a function f∈K(x)such that1<|G(f)|<deg f.(ii)If|G(f)|=deg f then K(f)⊆K(x)is normal.Moreover,if the extension K(f)⊆K(x)is separable,thenK(f)⊆K(x)is normal⇒|G(f)|=deg f.(iii)Given aﬁnite subgroup H ofΓ,there is a bijection between the subgroups of H and theﬁelds between Fix(H)and K(x).Also,if Fix(H)=K(f),there is a bijection between the right components of f(up to equivalence by units) and the subgroups of H.PROOF.(i)Theﬁeld Fix(G(f))is between K(f)and K(x),therefore the degree of any generator,which is the same as|G(f)|,divides deg f.For the second part,take8for example f=x2(x−1)2,which gives G(f)={x,1−x}in any coeﬃcient ﬁeld.(ii)The elements of G(f)are the roots of the minimum polynomial of x over K(f)that are in K(x).If there are deg f diﬀerent roots,as this number equals the degree of the extension we conclude that it is normal.If K(f)⊂K(x)is separable,all the roots of the minimum polynomial of x over K(f)are diﬀerent,thus if the extension is normal there are as many roots as the degree of the extension.(iii)Due to Theorem16,the extension Fix(H)⊂K(x)is normal,and the result is a consequence of the Fundamental Theorem of Galois.Remark18K(x)is Galois over K(that is,the only rational functionsﬁxed byΓ(K)are the constant ones)if and only if K is inﬁnite.Indeed,if K is inﬁnite,for each non–constant function f there exists a unit x+b with b∈K which does not leave itﬁxed.On the other hand,if K isﬁnite thenΓ(K) isﬁnite too,an the proof of Theorem16provides a non–constant rational function that generates Fix(Γ(K)).Algorithms for computing several aspects of Galois theory can be found in [16].Unfortunately,it is not true in general that[K(x):K(f)]=|G(f)|;there is no bijection between intermediateﬁelds and subgroups of theﬁxing group of a given function.Anyway,we can obtain partial results on decomposability. Theorem19Let f be indecomposable.(i)If deg f is prime,then either G(f)is cyclic of order deg f,or it is trivial.(ii)If deg f is composite,then G(f)is trivial.PROOF.(i)If1<|G(f)|<deg f,we have K(f) K(Fix(G(f))) K(x)and any generator of K(Fix(G(f)))is a proper component of f on the right.Therefore, G(f)has order either1or deg f,and in the latter case,being prime,the group is cyclic.(ii)Assume G(f)is not trivial.If|G(f)|<deg f,we have a contradiction as in(i).If|G(f)|=deg f,as it is a composite number,there exists H G(f) not trivial,and again any generator of Fix(H)is a proper component of f on the right.Corollary20If f has composite degree and G(f)is not trivial,f is decom-posable.9Now we present algorithms to eﬃciently computeﬁxedﬁelds andﬁxing groups.The proof of Theorem16provides an algorithm to compute a generator of Fix(H)from its elements.Algorithm1INPUT:H={h1,...,h m}<Γ(K).OUTPUT:f∈K(x)such that Fix(H)=K(f).A.Let i=1.pute the i-th symmetric elementary functionσi(h1,...,h m).C.Ifσi(h1,...,h m)∈K,returnσi(h1,...,h m).If it is constant,increase i and return to B.Analysis.The algorithm merely needs computing the product n i=1(T−h i) using O(log n)multiplications,so it is eﬃcient both in theory and practice.Example21LetH=±x,±1x,±i(x+1)x−1,±i(x−1)x+1,±x+ix−i,±x−ix+i<Γ(C).ThenP(T)=T12−x12−33x8−33x4+1x2(x−1)2(x+1)2(x4+2x2+1)T10−33T8+2x12−33x8−33x4+1x2(x−1)2(x+1)2(x4+2x2+1)T6−33T4−x12−33x8−33x4+1x2(x−1)2(x+1)2(x4+2x2+1)T2+1.Thus,Fix(H)=Cx12−33x8−33x4+1x2(x−1)2(x+1)2(x4+2x2+1).H is isomorphic to A4.It is known that A4has two complete subgroup chains of diﬀerent lengths:{id}⊂C2⊂V⊂A4,{id}⊂C3⊂A4.10In our case,{x}⊂{±x}⊂±x,±1x⊂H,{x}⊂x,x+ix−i,i(x+1)x−1⊂H.Applying our algorithm again we obtain the followingﬁeld chains:C(f)⊂Cx2+1x2⊂C(x2)⊂C(x),C(f)⊂C−i(t+i)(1+t)t(−t+i)(−1+t)⊂C(x).As there is a bijection in this case,the corresponding two decompositions are complete.In order to compute theﬁxing group of a function f we can solve the system of polynomial equations obtained fromfax+bcx+d=f(x).This can be reduced to solving two simpler systems,those given byf(ax+b)=f(x)and fax+bx+d=f(x).This method is simple but ineﬃcient;we will describe another method that is faster in practice.We need to assume that K has suﬃciently many elements.If not,we take an extension of K and later we check which of the computed elements are inΓ(K) by solving simple systems of linear equations.Theorem22Let f∈K(x)of degree m in normal form and u=ax+bcx+dsuchthat f◦u=f.(i)a=0and d=0.(ii)f N(b/d)=0.(iii)If c=0(that is,we take u=ax+b),then f N(b)=0and a m=1. (iv)If c=0then f D(a/c)=0.11PROOF.(i)Suppose a =0.We can assume u =1/(cx +d )=(1/x )◦(cx +d ).But if we consider f (1/x ),its numerator has smaller degree than its denominator.As composing on the right with cx +d does not change those degrees,it is impossible that f ◦u =f .Also,as the inverse of u is dx −b −cx +a,we have d =0.(ii)Letf =a m x m +···+a 1x b m −1x m −1+···+b 0.The constant term of the numerator of f ◦u isa mb m +a m −1b m −1d +···+a 1bd m −1=d m f N (b/d ).As d =0by (i),we have that f N (b/d )=0.Alternatively,0=f (0)=(f ◦u )(0)=f (u (0))=f (b/d ).(iii),(iv)They are similar to the previous item.We can use this theorem to compute the polynomial and rational elements of G (f )separately.Algorithm 2INPUT :f ∈K (x ).OUTPUT :G (f )={w ∈K (x ):f ◦w =f }.A .Compute units u,v such that f =u ◦f ◦v is in normal form.Let m =deg f .Let L be an empty list.B .Compute A ={α∈K :αm =1},B ={β∈K :f N (β)=0}andC ={γ∈K :fD (γ)=0}.C .For each (α,β)∈A ×B ,check if f (αx +β)=f (x ).In that case add ax +b to L .D .For each (β,γ)∈B ×C ,let w =cγx +βc x +pute all values of c for which f ◦w =f .For each solution,add the corresponding unit to L .E .Let L ={w 1,...,w k }.Return {v ◦w i ◦v −1:i =1,...,k }.Analysis .It is clear that the cost of the algorithm heavily depends on the complexity of the best algorithm to compute the roots of a univariate poly-nomial in the given ﬁeld.We analyze the bit complexity when the ground12ﬁeld is the rational number Q.We will use several well–known results about complexity,those can be consulted in the book[7].In the following,M denotes a multiplication time,so that the product of two polynomials in K[x]with degree at most m can be computed with at most M(m)arithmetic operations.If K supports the Fast Fourier Transform, several known algorithms require O(n log n log log n)arithmetic operations. We denote by l(f)the maximum norm of f,that is,l(f)= f ∞=max|a i| of a polynomial f= i a i x i∈Z[x].Polynomials in f,g∈Z[x]of degree less than m can be multiplied using O(M(m(l+log m)))bit operations,where l=log max(l(f),l(g)).Now,suppose that the given polynomial f is squarefree primitive,then we can compute all its rational roots with an expected number of T(m,log l(f))bit operations,where T(m,log l(f))=O(m log(ml(f))(log2log log m+(log log l(f))2log log log l(f))+m2M(log(ml(f))).We discuss separately the algorithm steps.Let f=f N/f D,where f N,f D∈Z[x]and let l=log max(l(f N),l(g D))and m=deg f.Step A.Let u∈Q(x)be a unit such that g N/g D=u(f)with deg g N> deg g D.Such a unit always exists:–If deg f N=deg f D.Let u=1/(x−a),where a∈Q veriﬁes deg f N−a deg f D<deg f N.–If deg f N<deg f D,let u=1/x.Now,let b∈Z such that g D(b)=0.Then h N/h D=g N(x+b)/g D(x+b) veriﬁes h D(0)=0and the rational function(x−h(0))◦h N/h D is in normal form.Obviously,the complexity in this step is dominated on choosing b.In the worst case,we have to evaluate the integers0,1,...,m in g D.Clearly,a complexity bound is O(M(m3l)).Step pute the set A can be done on constant time.Now,in order to compute the complexity,we can can suppose,without loss of generality,that f N and f D are squarefree and primitive.Then the bit complexity to compute both set B and set C is T(m,ml).Step C.A bound for the cardinal of A is4and m for the cardinal of B.Then, we need to check4m times if f(αx+β)=f(x)for each each(α,β)∈A×B. So,the complexity of this step is bounded by O(M(m4l)).13Step D.In the worst case the cardinal of B×C is m2.This step requires to compute all rational roots of m2polynomials h(x)given by the equation: f◦w=f,for each(β,γ)∈B×C,where w=cγx+βc x+1.A bound for the degree ofh(x)is m2.The size of the coeﬃcients is bounded by ml,so a bound for total complexity of this step is m4T(m2,lm2).Step E.Finally,this step requires substituting at most2m rational functions of degree m and the coeﬃcients size is bounded by lm3.So,abound for the complexity is O(M(m4l)).We can conclude that the complexity of this algorithm is dominated by that of step D,that is,m4T(m2,lm2).Of course,a worst bound for this is O(m8l2). The following example illustrates the above algorithm:Example23Letf=(−3x+1+x3)2x(−2x−x2+1+x3)(−1+x)∈Q(x).We normalize f:let u=1x−9/2and v=1x−1,thenf=u◦f◦v=−4x6−6x5+32x4−34x3+14x2−2x 27x5−108x4+141x3−81x2+21x−2is in normal form.The roots of the numerator and denominator of f in Q are{0,1,1/2}and {1/3,2/3}respectively.The only sixth roots of unity in Q are1and−1;as char Q=0there cannot be elements of the form x+b in G(f).Thus,there are two polynomial candidates:−x+1/3,−x+2/3.A quick computation reveals that none of themﬁxes f.Let w=cβx+αc x+1.Asα∈{0,1,1/2}andβ∈{1/3,2/3},another quickcomputation shows thatG(f)=x,−x+1−3x+2,−2x+1−3x+114andG(f)=v·G(f)·v−1=x,11−x,x−1x.From this group we can compute a proper component of f as in the proof of Theorem19,obtaining f=g(h)withh=−3x+1+x3(−1+x)x,g=x2x−1.In the next section we will use these tools to investigate the number of com-ponents of a rational function.4Ritt’s Theorem and number of componentsOne of the classical Ritt’s Theorems(see[13])describes the relation among the diﬀerent decomposition chains of a tame polynomial.Essentially,all the decompositions have the same length and are related in a rather simple way.Deﬁnition24A bidecomposition is a4-tuple of polynomials f1,g1,f2,g2such that f1◦g1=f2◦g2,deg f1=deg g2and(deg f1,deg g1)=1.Theorem25(Ritt’s First Theorem)Let f∈K[x]be tame andf=g1◦···◦g r=h1◦···◦h sbe two complete decomposition chains of f.Then r=s,and the sequences (deg g1,...,deg g r),(deg h1,...,deg h s)are permutations of each other.More-over,there exists aﬁnite chain of complete decompositionsf=f(j)1◦···◦f(j)r,j∈{1,...,k},such thatf(1)i=g i,f(k)i=h i,i=1,...,r,and for each j<k,there exists i j such that the j-th and(j+1)-th decompo-sition diﬀer only in one of these aspects:(i)f(j)ij ◦f(j)ij+1and f(j+1)i j◦f(j+1)i j+1are equivalent.15(ii)f(j)ij ◦f(j)ij+1=f(j+1)i j◦f(j+1)i j+1is a bidecomposition.PROOF.See[13]for K=C,[5]for characteristic zeroﬁelds and[6],[15]for the general case.2Unlike for polynomials,it is not true that all complete decompositions of a rational function have the same length,as shown in Example21.The paper [10]presents a detailed study of this problem for non tame polynomial with coeﬃcients over aﬁniteﬁeld.The problem for rational functions is strongly related to the open problem of the classes of rational functions which commute with respect to composition,see[14].In this section we will give some ideas about the relation between complete decompositions and subgroup chains that appear by means of Galois Theory.Now we present another degree12function,this time with coeﬃcients in Q, that has two complete decomposition chains of diﬀerent length.This function arises in the context of Monstrous Moonshine as a rational relationship be-tween two modular functions(see for example the classical[2]for an overview of this broad topic,or the reference[12],in Spanish,for the computations in which this function appears).Example26Let f∈Q(x)be the following degree12function:f=x3(x+6)3(x2−6x+36)3 (x−3)3(x2+3x+9)3.f has two decompositions:f=g1◦g2◦g3=x3◦x(x−12)x−3◦x(x+6)x−3==h1◦h2=x3(x+24)x−3◦x(x2−6x+36)x2+3x+9.All the components except one have prime degree,hence are indecomposable; the component of degree4cannot be written as composition of two components of degree2.If we compute the groups for the components on the right in Q we have:G Q(f)=G Q(g2◦g3)=G Q(g3)= 3x+18x−3,x,G Q(h2)={x}.16However,in C:G C(f)= 3αix+18αix−3,3αi x−18−18αix−3αi,3αi x+18x+3αi+3, 3x+18αix−3αi,3x+18x−3,αi x,x,G C(g2◦g3)= 3αix−18−18αix−3αi,3x+18x−3,x,G C(g3)= 3x+18x−3,x,G C(h2)= 3αix+18x+3αi+3,xwhereαi,i=1,2are the two non-trivial cubic roots of unity.In order to obtain the function in Example21,we used Theorem17,and in particular the existence of a bijection between the subgroups of A4and the intermediateﬁelds of a function that generates the correspondingﬁeld.The existence of functions with this property has been known for some time,as its construction from any group isomorphic to A4is straightforward.On the other hand,the example above is in Q(x),but there is no bijection between groups and intermediateﬁelds.In general,there are two main obstructions for this approach.On one hand, there is no bijection between groups andﬁelds in general,as the previous example shows for Q.On the other hand,only someﬁnite groups can be subgroups of PGL2(K).The onlyﬁnite subgroups of PGL2(C)are C n,D n, A4,S4and A5,see[11].In fact,this is true for any algebraically closedﬁeld of characteristic zero(it suﬃces that it contains all roots of unity).Among these groups,only A4has subgroup chains of diﬀerent length.This is even worse if we consider smallerﬁelds as the next known result shows:Theorem27Everyﬁnite subgroup of PGL2(Q)is isomorphic to either C n or D n for some n∈{2,3,4,6}.Indeed these all occur,unfortunately none of them has two subgroup chains of diﬀerent lengths,so no new functions can be found in this way.5ConclusionsIn this paper we have presented several counterexamples to the generalization of theﬁrst Ritt theorem to rational functions.We also introduced and analyzed17。

Woodall’s conjecture

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Figure 2: The ﬁgure shows (colored lines) a cut of size 3 and three pairwise
Woodall’s conjecture
Paulo Feoﬁloff
p.br/~pf
//, updated //
Abstract Woodall’s conjecture asserts the following about every directed graph: if every directed cut of the graph has k or more edges then the graph has k or more mutually disjoint dijoins. Here, a dijoin is a set J of arcs such that any vertex is connected to any other by a path all of whose forward-directed arcs are in J . This talk is a little survey of the counterexamples to a generalized version of the conjecture.

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Lemma 2 Let G = G(m, n) := a, b|am, bn, [b, a] = a4 , where m and n are powers of 2, m ≥ 16, n ≥ 4, m ≤ 4n. Then G has order mn, G′ = a4 is cyclic of
It is interesting to note that 4 of the 18 known counterexamples of order 128 (namely numbers 45, 100, 841, 886) have quotients isomorphic to our counterexamples of order 64, but that in each case the kernel is not characteristic.
1991 Mathematics Subject Classiﬁcation. 11R37, 20D15. The ﬁrst author was partially supported by NSF grant DMS 96-22590.
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NIGEL BOSTON AND CHARLES LEEDHAM-GREEN
1. Introduction.
Let K be a quadratic number ﬁeld, H be its Hilbert class ﬁeld, and H2 its Hilbert 2-class ﬁeld (in other words, the maximal everywhere unramiﬁed abelian and 2-abelian extensions of K respectively). There has been much work recently regarding the possible structure of Gal(H/K) and Gal(H2/K) (which are isomorphic to the ideal class group of K and its 2-Sylow subgroup respectively). For instance, the conjectures of Cohen and Lenstra [2] suggest that every abelian group (respectively abelian 2-group) should appear as Gal(H/K) (respectively Gal(H2/K)) for a suitable choice of K.
2. The Counterexamples.
It is straightforward to ﬁnd the smallest counterexamples by a systematic check using the MAGMA databases [1] of groups of order dividing 256. It turns out that all groups of order dividing 32 satisfy (∗). Of the 267 groups of order 64, exactly 2 fail to satisfy (∗). These are numbers 28 and 46 in the database. They can be given by respective compact presentations:
It is possible to give human proofs that the groups above fail to satisfy (∗). For instance, the following argument shows that there exist inﬁnitely many counterexamples to Lemmermeyer’s conjecture.
Using this idea, we obtain at least 2 further examples of groups failing (∗), these of order 256. Numbers 45 and 100 in the database of groups of order 128 have nuclear rank 1 (in the terminology of [3]) and each have 1 (terminal) descendant. All other groups mentioned above have nuclear rank 0.
Lemma 1 Suppose that G is a ﬁnite 2-group and N a characteristic subgroup. If G/N fails (∗), then so does G.
Proof Suppose, on the contrary, that G embeds with index 2 in a group Γ generated by involutions. Then N is normal in Γ and G/N embeds with index 2 in Γ/N , which is generated by involutions.
A natural next question is to ask for nonabelian generalisations of this. For instance, one might ask what ﬁnite groups (respectively ﬁnite 2-groups) occur as the Galois group over K of extensions obtained by iterating the construction of H (respectively H2). Note that such extensions are Galois over Q, not just over K. If we consider any unramiﬁed extension L of K with Galois group G such that L is Galois over Q, then G has the following property:
arXiv:math/9808146v1 [math.NT]REXAMPLES TO A CONJECTURE OF LEMMERMEYER
Nigel Boston and Charles Leedham-Green
January 7, 1998
Abstract. We produce inﬁnitely many ﬁnite 2-groups that do not embed with index 2 in any group generated by involutions. This disproves a conjecture of Lemmermeyer and restricts the possible Galois groups of unramiﬁed 2-extensions, Galois over Q, of quadratic number ﬁelds.
G1 = a, b|a16, b4, [b, a] = a4 , G2 = a, b|a16, b4, [b, a] = a−2 . The method employed is to calculate, for each of the 2328 groups of order 128, the subgroup generated by all involutions. Those groups for which this subgroup is the whole group are saved and their maximal subgroups computed. This produced a list of 7007 groups of order 64. These fall into 265 isomorphism classes. Checking these against the 267 groups of order 64 yields the 2 isomorphism classes missed, namely G1 and G2 above. Similarly one can work with the maximal subgroups of groups of order 256 with ≤ 4 generators, generated by involutions. Of the 162 groups of order 128 with precisely 2 generators, exactly 11 fail to satisfy (∗), namely numbers 45, 87, 88, 98, 100, 101, 130, 144, 145, 146, and 158 in the database. Of the 833 groups of order 128 with precisely 3 generators, exactly 7 fail to satisfy (∗), namely numbers 836, 841, 842, 886, 887, 912, and 971 in the database. It is not computationally feasible to search for all the groups of order 128 with precisely 4 generators failing (∗).
(∗) G embeds as a subgroup of index 2 in a group generated by elements of order 2 (“involutions”).