Asymptotic Behaviors of an Opinion Dynamical System without or with a Leader

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顺应论英文介绍

顺应论英文介绍Adaptationism, also known as the "adaptive paradigm," is a theoretical framework that aims to explain how organisms evolve and adapt to their environment. It suggests that the traits and behaviors of organisms are primarily a result of natural selection, which favors traits that increase an organism's chances of survival and reproductive success.Adaptationism assumes that organisms are adapted to their environment through the process of natural selection acting on random variations in traits. It suggests that the traits that are most advantageous for survival and reproduction are more likely to be passed on to future generations, leading to the evolution of new species or changes within existing species.Proponents of adaptationism argue that this framework provides a powerful and comprehensive explanation for the diversity of life on earth. They believe that understanding how organisms adapt to their environment can help us better predict and manage the impacts of environmental changes, such as climate change or introduction of invasive species.Critics of adaptationism, however, caution against its tendency to overly focus on natural selection as the sole driver of evolution. They argue that other mechanisms, such as genetic drift or gene flow, also play important roles in shaping the genetic makeup of populations.Overall, adaptationism offers a useful perspective for studying and understanding the evolutionary processes that have shaped thebiological world. Its insights continue to inform research in areas such as evolutionary biology, ecology, and conservation biology.。

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1 The saddle-point method
The saddle-point method is the method of choice in this domain. We describe it in the general case where it used to get an estimate of z n ] (z ), and will then comment on the implications of taking (z ) = f d (z ) with d tending to in nity. One starts from Cauchy's theorem 1 I (z ) dz; (z ) = 2i z n+1 where the contour contains the origin and no other singularity. A suitable contour is a circle through a particular point called saddle-point, which is a root of the derivative of the integrand (see 10] for a good intuitive justi cation of this): 0( ) n + 1 = : (1) () If everything goes well, there is such a point on the positive real axis, it is a maximum of the integrand on the circle, locally the integrand behaves as a Gaussian and the other parts of the integral are negligible. If is the only maximum of f on the circle of radius , h(z ) = log (z ) ? (n + 1) log z and h00( ) 6= 0 (su cient conditions for this will be given below), then one can see that h(z ) = h( ) + h00 ( )u2=2 de nes two functions u(z ) analytic at z = that do not vanish on

Curvelets a surprisingly effective nonadaptive representation of objects with

Curvelets–A Surprisingly EﬀectiveNon adaptive Representation For Objects with Edges Emmanuel J.Cand`e s and David L.DonohoAbstract.It is widely believed that to eﬃciently represent an otherwisesmooth object with discontinuities along edges,one must use an adaptiverepresentation that in some sense‘tracks’the shape of the discontinuityset.This folk-belief—some would say folk-theorem—is incorrect.Atthe very least,the possible quantitative advantage of such adaptation isvastly smaller than commonly believed.We have recently constructed atight frame of curvelets which provides stable,eﬃcient,and near-optimalrepresentation of otherwise smooth objects having discontinuities alongsmooth curves.By applying naive thresholding to the curvelet transformof such an object,one can form m-term approximations with rate of L2approximation rivaling the rate obtainable by complex adaptive schemeswhich attempt to‘track’the discontinuity set.In this article we explainthe basic issues of eﬃcient m-term approximation,the construction ofeﬃcient adaptive representation,the construction of the curvelet frame,and a crude analysis of the performance of curvelet schemes.§1.IntroductionIn many important imaging applications,images exhibit edges–discontinu-ities across curves.In traditional photographic imaging,for example,this occurs whenever one object occludes another,causing the luminance to un-dergo step discontinuities at boundaries.In biological imagery,this occurs whenever two diﬀerent organs or tissue structures meet.In image synthesis applications,such as CAD,there is no problem in deal-ing with such discontinuities,because one knows where they are and builds the discontinuities into the representation by specially adapting the representation —for example,inserting free knots,or adaptive reﬁnement rules.In image analysis applications,the situation is diﬀerent.When working with real rather than synthetic data,one of course doesn’t‘know’where these edges are;one only has a digitized pixel array,with potential imperfections caused by noise,by blurring,and of course by the unnatural pixelization of the underlying continuous scene.Hence the typical image analyst onlySaint-Malo Proceedings1 XXX,XXX,and Larry L.Schumaker(eds.),pp.1–10.Copyright o c2000by Vanderbilt University Press,Nashville,TN.ISBN1-xxxxx-xxx-x.All rights of reproduction in any form reserved.2 E.J.Cand`e s and D.L.Donoho has recourse to representations which don’t‘know’about the existence andgeometry of the discontinuities in the image.The success of discontinuity-adapting methods in CAD and related imagesynthesisﬁelds creates a temptation for an image analyst–a temptation tospend a great deal of time and eﬀort importing such ideas into image analysis.Almost everyone we know has yielded to this temptation in some form,whichcreates a possibility for surprise.Oracles and Ideally-Adapted RepresentationOne could imagine an ideally-privileged image analyst who has recourse toan oracle able to reveal the positions of all the discontinuities underlying theimage formation.It seems natural that this ideally-privileged analyst coulddo far better than the normally-endowed analyst who knows nothing aboutthe position of the discontinuities in the image.To elaborate this distinction,we introduce terminology borrowed fromﬂuid dynamics,where‘edges’arise in the form of fronts or shock fronts.A Lagrangian representation is constructed using full knowledge of theintrinsic structure of the object and adapting perfectly to that structure.•Inﬂuid dynamics this means that theﬂuidﬂow pattern is known,and one constructs a coordinate system which‘ﬂows along with the particles’,with coordinates mimicking the shape of theﬂow streamlines.•In image representation this could mean that the edge curves are known, and one constructs an image representation adapted to the structure of the edge curves.For example,one might construct a basis with disconti-nuities exactly where the underlying object has discontinuities.An Eulerian representation isﬁxed,constructed once and for all.It isnonadaptive–having nothing to do with the known or hypothesized detailsof the underlying object.•Inﬂuid dynamics,this would mean a usual euclidean coordinate system, one that does not depend in any way on theﬂuid motion.•In image representation,this could mean that the representation is some ﬁxed coordinate representation,such as wavelets or sinusoids,which does not change depending on the positions of edges in the image.It is quite natural to suppose that the Lagrangian perspective,whenit is available,is much more powerful that the Eulerian one.Having theprivilege of‘inside information’about the position of important geometriccharacteristics of the solution seems a priori rather valuable.In fact,thisposition has rather a large following.Much recent work in computationalharmonic analysis(CHA)attempts toﬁnd bases which are optimally adaptedto the speciﬁc object in question[7,10,11];in this sense much of the ongoingwork in CHA is based on the presumption that the Lagrangian viewpoint isbest.In the setting of edges in images,there has,in fact,been considerableinterest in the problem of developing representations which are adapted tothe structure of discontinuities in the object being studied.The(equivalent)Curvelets3 concepts of probing and minimum entropy segmentation are old examples of this: wavelet systems which are speciﬁcally constructed to allow discontinuities in the basis elements at speciﬁc locations[8,9].More recently,we are aware of much informal unpublished or preliminary work attempting to build2D edge-adapted schemes;we give two examples.•Adaptive triangulation aims to represent a smooth function by partition-ing the plane into a sequence of triangular meshes,reﬁning the meshes at one stage to createﬁner meshes at the next stage.One represents the underlying object using piecewise linear functions supported on individ-ual triangles.It is easy to see how,in an image synthesis setting,one can in principle develop a triangulation where the triangles are arranged to track a discontinuity very faithfully,with the bulk of reﬁnement steps allocated to reﬁnements near the discontinuity,and one obtains very ef-fective representation of the object.It is not easy to see how to do this in an image analysis setting,but one can easily be persuaded that the development of adaptive triangulation schemes for noisy,blurred data is an important and interesting project.•In an adaptively warped wavelet representation,one deforms the under-lying image so that the object being analyzed has all its discontinuities aligned purely horizontal or vertical.Then one analyzes the warped ob-ject in a basis of tensor-product wavelets where elements take the form ψj,k(x1)·ψj ,k (x2).This is very eﬀective for objects which are smooth apart from purely horizontal and purely vertical discontinuities.Hence, the warping deforms the singularities to render the the tensor product scheme very eﬀective.It is again not easy to see how adaptive warping could work in an image analysis setting,but one is easily persuaded that development of adaptively warped representations for noisy,blurred data is an important and interesting project.Activity to build such adaptive representations is based on an article of faith:namely,that Eulerian approaches are inferior,that oracle-driven Lagrangian approaches are ideal,and that one should,in an image analysis setting,mimic Lagrangian approaches,attempting empirically to estimate from noisy,blurred data the information that an oracle would supply,and build an adaptive representation based on that information.Quantifying Rates of ApproximationIn order to get away from articles of faith,we now quantify performance,using an asymptotic viewpoint.Suppose we have an object supported in[0,1]2which has a discontinuity across a nice curveΓ,and which is otherwise smooth.Then using a standard Fourier representation,and approximating with˜f F m built from the best m nonzero Fourier terms,we havef−˜f F m 22 m−1/2,m→∞.(1)4 E.J.Cand`e s and D.L.Donoho This rather slow rate of approximation is improved upon by wavelets.The approximant˜f W m built from the best m nonzero wavelet terms satisﬁesf−˜f W m 22 m−1,m→∞.(2) This is better than the rate of Fourier approximation,and,until now,is the best published rate for aﬁxed non-adaptive method(i.e.best published result for an‘Eulerian viewpoint’).On the other hand,we will discuss below a method which is adapted to the object at hand,and which achieves a much better approximation rate than previously known‘nonadaptive’or‘Eulerian’approaches.This adaptive method selects terms from an overcomplete dictionary and is able to achievef−˜f A m 22 m−2,m→∞.(3) Roughly speaking,the terms in this dictionary amount to triangular wedges, ideallyﬁtted to approximate the shape of the discontinuity.Owing to the apparent trend indicated by(1)-(3)and the prevalence of the puritanical belief that‘you can’t get something for nothing’,one might suppose that inevitably would follow theFolk-Conjecture/[Folk-Theorem].The result(3)for adaptive representa-tions far exceeds the rate of m-term approximation achievable byﬁxed non-adaptive representations.This conjecture appeals to a number of widespread beliefs:•the belief that adaptation is very powerful,•the belief that the way to represent discontinuities in image analysis is to mimic the approach in image synthesis•the belief that wavelets give the bestﬁxed nonadaptive representation.In private discussions with many respected researchers we have many times heard expressed views equivalent to the purported Folk-Theorem.The SurpriseIt turns out that performance almost equivalent to(3)can be achieved by a non adaptive scheme.In other words,the Folk-Theorem is eﬀectively false.There is a tight frame,ﬁxed once and for all nonadaptively,which we call a frame of curvelets,which competes surprisingly well with the ideal adaptive rate(3).A very simple m-term approximation–summing the m biggest terms in the curvelet frame expansion–can achievef−˜f C m 22≤C·m−2(log m)3,m→∞,(4) which is nearly as good as(3)as regards asymptotic order.In short,in a problem of considerable applied relevance,where one would have thought that adaptive representation was essentially more powerful than ﬁxed nonadaptive representation,it turns out that a newﬁxed nonadaptive representation is essentially as good as adaptive representation,from the point of view of asymptotic m-term approximation errors.As one might expect, the new nonadaptive representation has several very subtle and distinctive features.Curvelets5 ContentsIn this article,we would like to give the reader an idea of why(3)represents the ideal behavior of an adaptive representation,of how the curvelet frame is constructed,and of the key elements responsible for(4).We will also attempt to indicate why curvelets perform for singularities along curves the task that wavelets perform for singularities at points.§2.A Precedent:Wavelets and Point SingularitiesWe mention an important precedent–a case where a nonadaptive scheme is roughly competitive with an ideal adaptive scheme.Suppose we have a piecewise polynomial function f on the interval[0,1], with jump discontinuities at several points.An obvious adaptive representation is toﬁt a piecewise polynomial with breakpoints at the discontinuities.If there are P pieces and each polynomial is of degree≤D,then we need only keep P·(D+1)coeﬃcients and P−1 breakpoints to exactly represent this mon sense tells us that this is the natural,and even,the ideal representation for such a function.To build this representation,we need to know locations of the discontinu-ities.If the measurements are noisy or blurred,and if we don’t have recourse to an oracle,then we can’t necessarily build this representation.A less obvious but much more robust representation is to take a nice wavelet transform of the object,and keep the few resulting nonzero wavelet coeﬃcients.If we have an N-point digital signal f(i/N),1≤i≤N,and we use Daubechies wavelets of compact support,then there are no more than C·log2(N)·P·(D+1)nonzero wavelet coeﬃcients for the digital signal.In short,the nonadaptive representation needs only to keep a factor C log2(N)more data to give an equally faithful representation.We claim that this phenomenon is at least partially responsible for the widespread success of wavelet methods in data compression settings.One can build a single fast transform and deal with a wide range of diﬀerent f,with diﬀerent discontinuity sets,without recourse to an oracle.In particular,since one almost never has access to an oracle,the nat-uralﬁrst impulse of one committed to the adaptive viewpoint would be to ‘estimate’the break points–i.e.to perform some sort of edge detection.Un-fortunately this is problematic when one is dealing with noisy blurred data. Edge detection is a whole topic in itself which has thousands of proposed so-lutions and(evidently,as one can see from the continuing rate of publication in this area)no convincing solution.In using wavelets,one does not need edge detectors or any other prob-lematic schemes,one simply extracts the big coeﬃcients from the transform domain,and records their values and positions in an organized fashion.We can lend a useful perspective to this phenomenon by noticing that the discontinuities in the underlying f are point singularities,and we are saying that wavelets need in some sense at most log(n)coeﬃcients to represent a point singularity out to scale1/n.6 E.J.Cand`e s and D.L.DonohoIt turns out that even in higher dimensions wavelets have a near-ideal ability to represent objects with point singularities.The two-dimensional object fβ(x1,x2)=1/((x1−1/2)2+(x2−1/2)2)βhas,forβ<1/2,a square-integrable singularity at the point(1/2,1/2)and is otherwise smooth.At each level of the2D wavelet pyramid,there are eﬀec-tively only a few wavelets which‘feel’the point singularity,other coeﬃcients being eﬀectively negligible.In approximation out to scale1/n,only about O(log(n))coeﬃcients are required.Another approach to understanding the representation of singularities, which is not limited by scale,is to consider rates of decay of the countable coeﬃcient sequence.Analysis of wavelet coeﬃcients of fβshows that for any desired rateρ,the N-th largest coeﬃcient can be bounded by CρN−ρfor all N.In short,the wavelet coeﬃcients of such an object are very sparse.Thus we have a slogan:wavelets perform very well for objects with point singularities in dimensions1and2.§3.Failure of Wavelets on EdgesWe now brieﬂy sketch why wavelets,which worked surprisingly well in repre-senting point discontinuities in dimension1,are less successful dealing with ‘edge’discontinuities in dimension2.Suppose we have an object f on the square[0,1]2and that f is smooth away from a discontinuity along a C2curveΓ.Let’s look at the number of substantial wavelet coeﬃcients.A grid of squares of side2−j by2−j has order2j squares intersectingΓ. At level j of the two-dimensional wavelet pyramid,each wavelet is localized near a corresponding square of side2−j by2−j.There are therefore O(2j) wavelets which‘feel’the discontinuity alongΓ.Such a wavelet coeﬃcient is controlled by| f,ψj,k1,k2 |≤ f ∞· ψj,k1,k2 1≤C·2−j;and in eﬀect no better control is available,since the object f is not smoothwithin the support ofψj,k1,k2[14].Therefore there are about2j coeﬃcients ofsize about2−j.In short,the N-th largest wavelet coeﬃcient is of size about 1/N.The result(2)follows.We can summarize this by saying that in dimension2,discontinuities across edges are spatially distributed;because of this they can interact rather extensively with many terms in the wavelet expansion,and so the wavelet representation is not sparse.In short,wavelets do well for point singularities,and not for singularities along curves.The success of wavelets in dimension1derived from the fact that all singularities in dimension1are point singularities,so wavelets have a certain universality there.In higher dimensions there are more types of singularities,and wavelets lose their universality.For balance,we need to say that wavelets do outperform classical meth-ods.If we used sinusoids to represent an object of the above type,then weCurvelets7 have the result(1),which is far worse than that provided by wavelets.For completeness,we sketch the argument.Suppose we use for‘sinusoids’the complex exponentials on[−π,π]2,and that the object f tends smoothly to zero at the boundary of the square[0,1]2,so that we may naturally extend it to a function living on[−π,π]2.Now typically the Fourier coeﬃcients of an otherwise smooth object with a discontinuity along a curve decay with wavenumber as|k|−3/2(the very well-known example is f=indicator of a disk,which has a Fourier transform described by Bessel functions).Thus there are about R2coeﬃcients of size≥c·R−3/2,meaning that the N-th largest is of size≥c·N−3/4,from which(1)follows.In short:neither wavelets nor sinusoids really sparsify two-dimensional objects with edges(although wavelets are better than sinusoids).§4.Ideal Representation of Objects with EdgesWe now consider the optimality result(3),which is really two assertions.On the one hand,no reasonable scheme can do better than this rate.On the other hand,a certain adaptive scheme,with intimate connections to adaptive triangulation,which achieves it.For more extensive discussion see[10,11,13].In talking about adaptive representations,we need to deﬁne terms care-fully,for the following reason.For any f,there is always an adaptive repre-sentation of f that does very well:namely the orthobasisΨ={ψ0,ψ1,...} withﬁrst elementψ0=f/ f 2!This is,in a certain conception,an‘ideal representation’where each object requires only one nonzero coeﬃcient.In a certain sense it is a useless one,since all information about f has been hidden in the deﬁnition of representation,so actually we haven’t learned anything. Most of our work in this section is in setting up a notion of adaptation that will free us from fear of being trapped at this level of triviality. Dictionaries of AtomsSuppose we are interested in approximating a function in L2(T),and we have a countable collection D={φ}of atoms in L2(T);this could be a basis,a frame, aﬁnite concatenation of bases or frames,or something even less structured.We consider the problem of m-term approximation from this dictionary, where we are allowed to select m termsφ1,...,φm from D and we approximate f from the L2-closest member of the subspace they span:˜f=P roj{f|span(φ1,...,φm)}.mWe are interested in the behavior of the m-term approximation errore m(f;D)= f−˜f m 22,where in this provisional deﬁnition,we assume˜f m is a best approximation of this form after optimizing over the selection of m terms from the dictionary.However,to avoid a trivial result,we impose regularity on the selection process.Indeed,we allow rather arbitrary dictionaries,including ones which8 E.J.Cand`e s and D.L.Donoho enumerate a dense subset of L2(T),so that in some sense the trivial result φ1=f/ f 2e m=0,∀m is always a lurking possibility.To avoid this possibility we forbid arbitrary selection rules.Following[10]we proposeDeﬁnition.A sequence of selection rules(σm(·))choosing m terms from a dictionary D,σm(f)=(φ1,...,φm),is said to implement polynomial depth search if there is a singleﬁxed enumeration of the dictionary elements and aﬁxed polynomialπ(t)such that terms inσm(f)come from theﬁrstπ(m)elements in the dictionary.Under this deﬁnition,the trivial representation based on a countable dense dictionary is not generally available,since in anyﬁxed enumeration, a decent1-term approximation to typical f will typically be so deep in the enumeration as to be unavailable for polynomial-depth selection.(Of course, one can make this statement quantitative,using information-theoretic ideas).More fundamentally,our deﬁnition not only forbids trivialities,but it allows us to speak of optimal dictionaries and get meaningful results.Starting now,we think of dictionaries as ordered,having aﬁrst element,second element, etc.,so that diﬀerent enumerations of the same collection of functions are diﬀerent dictionaries.We deﬁne the m-optimal approximation number for dictionary D and limit polynomialπase m(f;D;π)= f−˜f m 22,where˜f m is constructed by optimizing the choice of m atoms among theﬁrst π(m)in theﬁxed enumeration.Note that we use squared error for comparison with(1)-(3)in the Introduction.Approximating Classes of FunctionsSuppose we now have a class F of functions whose members we wish to ap-proximate.Suppose we are given a countable dictionary D and polynomial depth search delimited by polynomialπ(·).Deﬁne the error of approximation by this dictionary over this class bye m(F;D,π)=maxe m(f;D,π).f∈FWe mayﬁnd,in certain examples,that we can establish boundse m(F;D,π)=O(m−ρ),m→∞,for allρ<ρ∗.At the same time,we may have available an argument showing that for every dictionary and every polynomial depth search rule delimited by π(·),e m(F;D,π)≥cm−ρ∗,m≥m0(π).Then it seems natural to say thatρ∗is the optimal rate of m-term approxi-mation from any dictionary when polynomial depth search delimited byπ(·).Curvelets9Starshaped Objects with C 2Boundaries We deﬁne Star-Set 2(C ),a class of star-shaped sets with C 2-smooth bound-aries,by imposing regularity on the boundaries using a kind of polar coor-dinate system.Let ρ(θ):[0,2π)→[0,1]be a radius function and b 0=(x 1,0,x 2,0)be an origin with respect to which the set of interest is star-shaped.With δi (x )=x i −x i,0,i =1,2,deﬁne functions θ(x 1,x 2)and r (x 1,x 2)byθ=arctan(−δ2/δ1);r =((δ1)2+(δ2)2)1/2.For a starshaped set,we have (x 1,x 2)∈B iﬀ0≤r ≤ρ(θ).Deﬁne the class Star-Set 2(C )of sets by{B :B ⊂[110,910]2,110≤ρ(θ)≤12θ∈[0,2π),ρ∈C 2,|¨ρ(θ)|≤C },and consider the corresponding functional class Star 2(C )= f =1B :B ∈Star-Set 2(C ) .The following lower rate bound should be compared with (3).Lemma.Let the polynomial π(·)be given.There is a constant c so that,for every dictionary D ,e m (Star 2(C );D ,π)≥c 1m 2log(m ),m →∞.This is proved in [10]by the technique of hypercube embedding.Inside the class Star 2(C )one can embed very high-dimensional hypercubes,and the ability of a dictionary to represent all members of a hypercube of dimension n by selecting m n terms from a subdictionary of size π(m )is highly limited if π(m )grows only polynomially.To show that the rate (3)can be achieved,[13]adaptively constructs,for each f ,a corresponding orthobasis which achieves it.It tracks the boundary of B at increasing accuracy using a sequence of polygons;in fact these are n -gons connecting equispaced points along the boundary of B ,for n =2j .The diﬀerence between n -gons for n =2j and n =2j +1is a collection of thin triangular regions obeying width ≈length 2;taking the indicators of each region as a term in a basis,one gets an orthonormal basis whose terms at ﬁne scales are thin triangular pieces.Estimating the coeﬃcient sizes by simple geometric analysis leads to the result (3).In fact,[13]shows how to do this under the constraint of polynomial-depth selection,with polynomial Cm 7.Although space constraints prohibit a full explanation,our polynomial-depth search formalism also makes perfect sense in discussing the warped wavelet representations of the Introduction.Consider the noncountable ‘dic-tionary’of all wavelets in a given basis,with all continuum warpings applied.Notice that for wavelets at a given ﬁxed scale,warpings can be quantized with a certain ﬁnite accuracy.Carefully specifying the quantization of the warping,one obtains a countable collection of warped wavelets,for which polynomial depth search constraints make sense,and which is as eﬀective as adaptive triangulation,but not more so .Hence (3)applies to (properly interpreted)deformation methods as well.10 E.J.Cand`e s and D.L.Donoho§5.Curvelet ConstructionWe now brieﬂy describe the curvelet construction.It is based on combining several ideas,which we brieﬂy review•Ridgelets,a method of analysis suitable for objects with discontinuities across straight lines.•Multiscale Ridgelets,a pyramid of windowed ridgelets,renormalized and transported to a wide range of scales and locations.•Bandpass Filtering,a method of separating an object out into a series of disjoint scales.We brieﬂy describe each idea in turn,and then their combination.RidgeletsThe theory of ridgelets was developed in the Ph.D.Thesis of Emmanuel Cand`e s(1998).In that work,Cand`e s showed that one could develop a system of analysis based on ridge functionsψa,b,θ(x1,x2)=a−1/2ψ((x1cos(θ)+x2sin(θ)−b)/a).(5)He introduced a continuous ridgelet transform R f(a,b,θ)= ψa,b,θ(x),f with a reproducing formula and a Parseval relation.He also constructed frames, giving stable series expansions in terms of a special discrete collection of ridge functions.The approach was general,and gave ridgelet frames for functions in L2[0,1]d in all dimensions d≥2–For further developments,see[3,5].Donoho[12]showed that in two dimensions,by heeding the sampling pat-tern underlying the ridgelet frame,one could develop an orthonormal set for L2(I R2)having the same applications as the original ridgelets.The orthonor-mal ridgelets are convenient to use for the curvelet construction,although it seems clear that the original ridgelet frames could also be used.The ortho-ridgelets are indexed usingλ=(j,k,i, , ),where j indexes the ridge scale,k the ridge location,i the angular scale,and the angular location; is a gender token.Roughly speaking,the ortho-ridgelets look like pieces of ridgelets(5) which are windowed to lie in discs of radius about2i;θi, = /2i is roughly the orientation parameter,and2−j is roughly the thickness.A formula for ortho-ridgelets can be given in the frequency domainˆρλ(ξ)=|ξ|−12(ˆψj,k(|ξ|)w i, (θ)+ˆψj,k(−|ξ|)w i, (θ+π))/2.are periodic wavelets for[−π,π), Here theψj,k are Meyer wavelets for I R,wi,and indices run as follows:j,k∈Z Z, =0,...,2i−1−1;i≥1,and,if =0, i=max(1,j),while if =1,i≥max(1,j).We letΛbe the set of suchλ.The formula is an operationalization of the ridgelet sampling principle:•Divide the frequency domain in dyadic coronae|ξ|∈[2j,2j+1].•In the angular direction,sample the j-th corona at least2j times.•In the radial frequency direction,sample behavior using local cosines.The sampling principle can be motivated by the behavior of Fourier trans-forms of functions with singularities along lines.Such functions have Fourier transforms which decay slowly along associated lines through the origin in the frequency domain.As one traverses a constant radius arc in Fourier space,one encounters a ‘Fourier ridge’when crossing the line of slow decay.The ridgelet sampling scheme tries to represent such Fourier transforms by using wavelets in the angular direction,so that the ‘Fourier ridge’is captured neatly by one or a few wavelets.In the radial direction,the Fourier ridge is actu-ally oscillatory,and this is captured by local cosines.A precise quantitative treatment is given in [4].Multiscale RidgeletsThink of ortho-ridgelets as objects which have a “length”of about 1and a “width”which can be arbitrarily ﬁne.The multiscale ridgelet system renor-malizes and transports such objects,so that one has a system of elements at all lengths and all ﬁner widths.In a light mood,we may describe the system impressionistically as “brush strokes”with a variety of lengths,thicknesses,orientations and locations.The construction employs a nonnegative,smooth partition of energyfunction w ,obeying k 1,k 2w 2(x 1−k 1,x 2−k 2)≡1.Deﬁne a transportoperator,so that with index Q indicating a dyadic square Q =(s,k 1,k 2)of the form [k 1/2s ,(k 1+1)/2s )×[k 2/2s ,(k 2+1)/2s ),by (T Q f )(x 1,x 2)=f (2s x 1−k 1,2s x 2−k 2).The Multiscale Ridgelet with index µ=(Q,λ)is thenψµ=2s ·T Q (w ·ρλ).In short,one transports the normalized,windowed ortho-ridgelet.Letting Q s denote the dyadic squares of side 2−s ,we can deﬁne the subcollection of Monoscale Ridgelets at scale s :M s ={(Q,λ):Q ∈Q s ,λ∈Λ}.Orthonormality of the ridgelets implies that each system of monoscale ridgelets makes a tight frame,in particular obeying the Parseval relationµ∈M s ψµ,f 2= f 2L 2.It follows that the dictionary of multiscale ridgelets at all scales,indexed byM =∪s ≥1M s ,is not frameable,as we have energy blow-up:µ∈M ψµ,f 2=∞.(6)。

Asymptotic analysis

1
Asymptotic analysis of a thin layer device with Tresca's contact law in elasticity
Summary :
In this paper, we consider a thin elastic layer between a rigid body and an elastic one. A Tresca law is assumed between the two elastic bodies. The Lame coe cients of the thin layer are assumed to vary with respect to its height ". This dependence is shown to be of primary importance in the asymptotic behavior of the device, a critical case leading to a non classical contact law when deleting the bond.
2 Stating the problem
2.1 The model problem
For sake of simplicity the study is made for a 2-dimensional problem but it is also valid for a 3-dimensional one. We consider the device de ned by g
A line of research in the study of friction between two bodies involves introducing a very thin third body between them. This third body, despite being often made by fragments of the two rst bodies, can exhibit very di erent mechanical properties than those of the initial bodies. A very similar situation occurs when taking into account the e ects of a thin layer which has been bonded onto the surface of a body to prevent wear caused by the contact with another solid. It is therefore of interest to study the asymptotic behavior of the thin layer between the two bodies, assuming various contact laws between them. This kind of con guration in which the geometric data exhibit di erent magnitudes can be found in many mechanical situations such as plates, reinforcement problems and lubrication. Rescaling the coordinate through the thin region is often carried out and an expansion procedure follows 1] 3] 5] 6] 7] 14]. In the present situation, the con guration is similar to that of a bond between two bodies. So far it seems that such problems have been treated, both in the elastic area or in the simpler context of the thermic area, only by assuming perfect adhesion conditions between all three bodies. A lot of papers have been devoted to such studies with various geometries, behavior laws and more or less rigorous approaches. In 1] 10], the asymptotic method has been used for a particular value of the Lame coe cients which have been set proportional to the height " of the joint. In this case, the expansion procedure with respect to the powers of " allows new asymptotic boundary conditions between the rst bodies to be obtained; these conditions replaced the vanishing joint. Another way to carry out the study without using the rescaling procedure may be found in 2]. The idea is to plunge the domain of the thin layer into a xed one and to use particular test functions to obtain the limit problem in the real domain. In this study, a range of variation for the Lame coe cients of the layer with respect to " is proposed, each of them leading to a di erent behavior for the device. Some generalisations appear in 13]. In the present paper, we study the behavior of a device constituted of three bodies, one of them sliding on a thin layer, itself bonded to a rigid support. The Lame coe cients of the thin layer are assumed to vary with respect of ". The 3

Asymptotic Theory

A.1 Convergence in probability (laws of large numbers)
3
Theorem A.4 Markov’s strong law of large numbers. If {zj } is sequence of independent random variables with E [zj ] = µj < 1+ E |zj µj | and if for some > 0, < then z n µn converges almost j 1+ n n surely to 0, where z n = n1 zj and µn = n1 µj .
nLeabharlann A frequently employed special case is convergence in quadratic mean. Theorem A.1 Convergence in quadratic mean (or mean square). If xn has mean µn and variance 2 n such that ordinary limits of µn and
n j =1
Almost sure convergence implies convergence in probability (but not necessarily the converse).
A.1.2 Applications of convergence
Deﬁnition A.3 Consistent estimator. An estimator ˆ of parameter is a consistent estimator i p lim ˆ = .

成考学位英语考试

1、What is the primary purpose of writing a business email?A. To express personal emotions.B. To provide detailed instructions on a hobby.C. To communicate professionally and efficiently in a work setting.D. To discuss current events with friends. (答案：C)2、Which of the following is NOT a common feature of academic writing?A. Use of formal language.B. Inclusion of personal anecdotes.C. Clear organization and structure.D. Citation of sources. (答案：B)3、In which situation would you use a SWOT analysis?A. When creating a personal journal entry.B. When evaluating the strengths, weaknesses, opportunities, and threats of a business or project.C. When writing a fictional short story.D. When planning a casual social gathering. (答案：B)4、What does "ROI" stand for in the context of business and finance?A. Return On InvestmentB. Random Order InputC. Rapid Online InquiryD. Resource Optimization Initiative (答案：A)5、Which of these is an example of active listening in a business meeting?A. Interrupting the speaker to share your own opinion.B. Checking your phone while the other person is talking.C. Nodding and providing verbal cues to show understanding.D. Thinking about your next response without paying attention to the speaker. (答案：C)6、What is the main goal of a marketing strategy?A. To increase production costs.B. To decrease the quality of products.C. To identify and satisfy customer needs and wants.D. To limit competition in the market. (答案：C)7、Which of the following is a key element of effective time management?A. Procrastinating tasks until the last minute.B. Prioritizing tasks based on urgency and importance.C. Multitasking constantly without focus.D. Avoiding planning and spontaneously tackling tasks as they arise. (答案：B)8、In project management, what does the acronym "SMART" stand for when setting goals?A. Specific, Measurable, Achievable, Relevant, Time-boundB. Simple, Modern, Accessible, Reliable, TimelyC. Strategic, Minimal, Attractive, Responsive, TechnologicalD. Swift, Meticulous, Ambitious, Resourceful, Tactical (答案：A)9、Which of the following best describes the concept of "supply and demand" in economics?A. The relationship between the quantity of a product available and the desire for that product.B. The process of producing goods and services.C. The study of how money is created and managed.D. The analysis of government spending and taxation. (答案：A)10、What is the purpose of a feasibility study in starting a new business?A. To determine the company's annual revenue.B. To assess the legal requirements for operating the business.C. To evaluate the practicality and potential success of the business idea.D. To design the company's logo and branding. (答案：C)。

asymptotic analysis缩写

asymptotic analysis缩写Asymptotic analysis is a mathematical method used to analyze the behavior of an algorithm as the input size approaches infinity. It is widely used in computer science and engineering to compare different algorithms and design efficient algorithms. In this article, we will discuss the basics of asymptotic analysis and its important concepts.Firstly, we need to understand the importance of asymptotic analysis. In computer science and engineering, we often deal with huge datasets and complex algorithms. Therefore, it is important to know how the algorithm will behave as the input size becomes large. Asymptotic analysis helps us to estimate the computational time and space complexity of an algorithm for large inputs. This estimation can help us to choose the best algorithm for a given problem.Asymptotic analysis is based on the concept of limits. A limit is the value a function approaches as the input value approaches a certain point. We use big O, big Omega, and big Theta notations to express the growth rate of a function. These notations give us a rough idea about the behavior of the function.Big O notation: The big O notation gives the upper bound of the running time of an algorithm. We say that algorithm A has a time complexity of O(f(n)) if the running time of the algorithm does not exceed a constant multiple of f(n) for large n. For example, if the running time of the algorithm A is less than or equal to 2n^2+3n+4, we can say that the time complexity of the algorithm A is O(n^2).Big Omega notation: The big Omega notation gives the lower bound of the running time of an algorithm. We say that algorithm A has a time complexity of Omega(f(n)) if the running time of the algorithm is not less than a constant multiple of f(n) for large n. For example, if the running time of the algorithm A is greater than or equal to n^2/2, we can say that the time complexity of the algorithm A is Omega(n^2).Big Theta notation: The big Theta notation gives the tight bounds of the running time of an algorithm. We say that algorithm A has a time complexity of Theta(f(n)) if the running time of the algorithm is between a constant multiple of f(n) and another constant multiple of f(n) for large n. For example, if the running time of the algorithm A is between 5n^2+3n+4 and 7n^2+5n+6, we can say that the time complexity of the algorithm A is Theta(n^2).Asymptotic analysis also covers the space complexity of an algorithm. We use the same notations to express the growth rate of the space usage of an algorithm. For example, if the space used by the algorithm A is less than or equal to 3n+4, we can say that the space complexity of the algorithm A is O(n).In conclusion, asymptotic analysis is an important concept in computer science and engineering. It helps us to estimate the computational time and space complexity of an algorithm for large inputs. By using the big O, big Omega, and big Theta notations, we can compare different algorithms and choose the best algorithm for a given problem.。

cfd中的asymptotic theory

cfd中的asymptotic theory[cfd中的asymptotic theory]Asymptotic theory plays a vital role in Computational Fluid Dynamics (CFD) as it provides a mathematical framework to understand the behavior of fluid flow in various situations. In this article, we will explore the fundamentals of asymptotic theory in CFD and its applications.What is asymptotic theory?Asymptotic theory is a branch of mathematics that deals with the behavior of functions as a variable approaches a particular value. It provides a systematic way to approximate complex functions and simplify their analysis. In CFD, asymptotic theory is used to solve fluid flow problems by approximating complex equations and simplifying their solutions.The foundation of asymptotic theory in CFD lies in the concept of a small parameter. A small parameter is a dimensionless quantity that represents the scale of a physical phenomenon relative to other quantities in the problem. It allows us to analyze thebehavior of the fluid flow as the small parameter tends to zero or infinity.The key idea behind asymptotic theory is that we can approximate the solution to a problem by expanding it as a series in powers of the small parameter. The resulting series is then truncated at an appropriate order to obtain an approximate solution. This approach is particularly useful when the problem involves a wide range of scales, such as in turbulent flows or flows with thin boundary layers.Applications of asymptotic theory in CFD1. Boundary layer theory: Boundary layers are thin layers of fluid that form near solid boundaries in a fluid flow. They play a crucial role in many engineering applications, such as aerodynamics and heat transfer. Asymptotic theory allows us to derive simplified equations for the boundary layer and study its behavior.By assuming the small parameter to be the ratio of the boundary layer thickness to the characteristic length scale of the problem, we can derive the famous Prandtl's boundary layer equations.These equations provide a simplified description of the flow near the boundary and have been widely used to analyze various flow problems.2. Perturbation methods: Perturbation methods involve solving a problem by treating the small parameter as a small perturbation to an already known solution. This approach is particularly useful when the small parameter is not related to a physical scale but appears in the governing equations due to simplifications or assumptions.For example, in the study of unsteady flows, the small parameter can be the ratio of the unsteadiness time scale to a characteristic time scale of the problem. By assuming a known steady solution and perturbing it in terms of the small parameter, we can develop a systematic procedure to obtain the unsteady solution.3. Homogenization theory: Homogenization theory is concerned with problems where the governing equations have oscillatory coefficients or rapidly varying parameters. These problems arise in various fields, such as porous media flow and composite materials.Asymptotic theory provides a powerful tool to derive effective (homogenized) equations that capture the behavior of the system on a macroscopic scale. By assuming the small parameter as the ratio of the periodicity length to the characteristic length scale of the problem, we can develop an asymptotic expansion to obtain the macroscopic equations.ConclusionAsymptotic theory is an essential tool in CFD for studying complex fluid flow problems. It allows us to approximate the solution to a problem by expanding it as a series in powers of a small parameter and truncating the series at an appropriate order. This approach provides simplified equations that capture the essential behavior of the flow and enable efficient computational analysis. Through examples such as boundary layer theory, perturbation methods, and homogenization theory, we have seen how asymptotic theory finds applications in various areas of CFD. By using asymptotic theory, researchers can gain valuable insights into the behavior of fluid flow and develop efficient numericalmethods for practical engineering applications.。

OrganizationBehavior组织行为学

OrganizationBehavior组织⾏为学Organization BehaviorChapter1/doc/ad25610c4a7302768e9939d1.html anizational behavior (OB)：A field of study that investigates the impact that individuals, groups and structure have on behavior within organization, for the purpose of applying such knowledge toward improving an organization’s effectiveness2. Managers doManagement functionPlanningA process that includes defining goals, establishingstrategy(策略), and developing plans to coordinate（调整）activitiesControllingMonitoring activities to ensure they are beingaccomplished as planned and correcting anysignificant deviations（背离）.OrganizingDetermining what tasks are to be done, who is todo them, how the tasks are to be grouped, whoreports to whom, and where decisions are to bemade.LeadingA function that includes motivating employees,directing others, selecting the most effectivecommunication channels, and resolving conflictsManagement role: 1) interpersonal(⼈际⾓⾊)Figurehead(头⾯⼈物) Leader(领导者) Liaison(联络⼈)2) informational(信息传递者)Monitor(监控者) Disseminator(传递者) Spokesperson(发⾔⼈)3) Decisional(决策⾓⾊)Entrepreneur(创业者) Disturbance handler(混乱处理者)Resource allocator(资源分配者) Negotiator(谈判者)Management skills: 1) Technical skillsThe ability to apply specialized knowledge or expertise（专门技术）.2) Human skillsThe ability to work with, understand, and motivate other people, both individually and in groups.3) Conceptual skillsThe mental ability to analyze and diagnose(诊断) complex situations. 3. Effective versus Successful Managerial Activities 1) Traditional managementDecision making, planning, and controlling2) CommunicationExchanging routine(例⾏的) information and processing paperwork3) Human resource managementMotivating, disciplining, managing conflict, staffing(⼈员指挥), and training4) NetworkingSocializing, politicking(政治活动), and interacting(相互影响) with others4. Challenges and Opportunities for OB1) Responding to GlobalizationIncreased foreign assignmentsWorking with people from different culturesOverseeing movement of jobs to countries with low-cost labor2) Managing Workforce Diversity(差异，多样性)Embracing diversityChanging demographics（⼈⼝）Implications for managersRecognizing and responding to differences3) Improving Quality and ProductivityQuality management (QM)Process reengineering4) Responding to the Labor ShortageChanging work force demographicsFewer skilled laborersEarly retirements and older workers5) Improving Customer ServiceIncreased expectation of service qualityCustomer-responsive cultures6) Improving People Skills7) Empowering(授权) People8) Stimulating(刺激) Innovation(改⾰) and Change9) Coping with “Temporariness(临时性)”10) Working in Networked Organizations11) Helping Employees Balance Work/Life Conflicts12) Improving Ethical(伦理的) Behavior5. Independent and dependent variables1) individual-level variables个体⽔平变量⼈们带着不同的特点进⼊组织，这些特点将影响到他们在⼯作中的⾏为。

马万彪男教授(博士),出生于1961年

马万彪男教授（博士），出生于1961年。

1986.8-1993.9: 内蒙古师范大学数学系任助教、讲师、副教授；1993.10-2001.3: 日本国立京都大学数理工学专攻访问学者、日本国立静冈大学大学院读博士、日本大阪府立大学工学部数理工学科共同研究员、日本国立静冈大学工学部系统工学科副教授； 2001.5-现在: 北京科技大学应用学院数力系任副教授、教授。

一、获奖情况1、1989 内蒙古自治区优秀自然科学论文一等奖2、1992 内蒙古师范大学优秀科研成果特等奖（斯力更、马万彪）3、1992 内蒙古自治区科学技术技进步一等奖（斯力更、马万彪）二、教学改革项目1、《线性代数优秀课程建设》，北京科技大学，主要参加者。

2、《线性代数精品课程建设》，北京科技大学，主要参加者。

三、科研项目1、Conley 理论及其在微分方程中的应用（中科院数学所主持），国家自然科学基金，第一参加者，2003-20062、动力系统的定性分析，留学回国基金，主持，中华人民共和国教育部， 2001年5月-2003年12月3、泛函微分方程及生物数学中一些非线性应用动力学系统的稳定性研究，北科大校基金，主持，2005-2006四、科研文章[1] E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic models with distributed time delay, Nonl. Anal., 47(2001), 4107-4115. (SCI，ISTP)[2] Y. Saito, W. Ma and T. Hara, Necessary and sufficient conditions for permanence of a Lotka-Volterra discrete systems with delays, J. Math. Anal. Appl., 256(2001), 162-174.(SCI)[3] Y. Saito, T. Hara and W. Ma, Harmless delays for permanence and impersistence of a Lotka-Volterra discrete predator-prey system, Nonl. Anal., 50(2002), 705-715. (SCI，EI)[4] W. Ma, Y.Takeuchi, T.Hara and E.Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J., 54(2002), 581-591. (SCI)[5] T. Amemiya and W. Ma, Global asymptotic stability of nonlinear delayed systems of neutral type, Nonl. Anal., 54(2003), 83-91.(EI, SCI)[6] W. Ma and Y. Takeuchi, Asymptotic properties of a delayed SIR epidemic model with density dependent birth rate, Disc.Conti.Dyna.Sys.-B, 4(2004), 671-678. (SCI)[7] M. Yamaguchi, Y. Takeuchi and W. Ma, Population dynamics of sea bass and young sea bass, Disc. Conti.Dyna.Sys.-B, 4(2004), 833-840. (SCI)[8] W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Letters, 17(2004),1141-1145.(SCI)[9] G. Fu, W. Ma and S. Ruan, Qualitative analysis of a Chemostat modelwith inhibitory exponential substrate uptake, Chaos, Solitons and Fractals., 23(2005), 873-886.(SCI)[10] M. Song, W. Ma and Y. Takeuchi, Asymptotic properties of a revised SIR epidemic model with density dependent birth rate and time delay, Dyn. Cont. Discrete Impul. Sys., 13, 2(2006), 199-208. (SCI)[11] H. Shi，and W. Ma, n improved model of T cell development in the thymus and its stability analysis , Math. Biosci. Eng., 3, 1(2006), 237-248. （SCI）[12]付桂芳，马万彪, 由微分方程所描述的微生物连续培养动力系统－(I), 微生物学通报,31, 5(2004), 136-139. (核心期刊)[13]付桂芳，马万彪，由微分方程所描述的微生物连续培养动力系统-（II），微生物学通报，31，6（2004），128-131．（核心期刊）[14]靳欣,马万彪,胸腺细胞发育的非线性动力系统模型的定性分析,数学的实践与认识，2006．（校稿，核心期刊）五、兼职情况1993起任美国数学会 [Math.Rev.] 评论员； 2004年起兼任中国数学会生物数学分会秘书长，[生物数学学报]常务编委。

人工智能的负责任使用

THE RESPONSIBLE USE OF ARTIFICIAL INTELLIGENCEPOSITIONING & MESSAGINGAt Motorola Solutions, we believe that AI-powered solutions applied purposefully can complement human decision making and cultivate public trust while preserving individual rights. More speciﬁcally, we understand that the human aspect — the individual using the technology and the citizens impacted by its use in society — should always be at the center of each innovation. We consistently hold ourselves to the highest ethical standards in our work and this is no different when building new technologies and innovating with AI.This paper is intended to provide employees visibility into how AI is being leveraged responsibly at each level of the product development process. Additionally, there are a number of use cases included to illustrate how AI can assist in accelerating both the public safety and enterprise workﬂow.FUNDAMENTAL TENETS – MISSION-CRITICAL AI BY MOTOROLA SOLUTIONSAll of our AI technology is created to be equitable,transparent, understandable, accountable, secure and reliable. Additionally, we adopt four fundamental tenets to guide the development and use of AI technology:●Human in the Loop:We believe that AI should be used to assist and accelerate human decision making, not replace it. Our AIsystems are advisory in nature and will never take consequential actions on their own. Using AI-generated guidance, we can help customers make better decisions faster.●Workﬂow-Focused Application:We identify authorized and speciﬁc areas in the workﬂow where AI can deliverimprovements, enabling users to accomplish the same tasks faster and more successfully. We are applying proven AItechnology in a focused and collaboratively developed way, to narrowly-deﬁned and well-understood use cases.●Disciplined Innovation:We apply proven AI and machine learning technologies that are carefully trained,thoroughlycharacterized and extensively tested.This results in AI-powered solutions that are far less likely to behave unexpectedly in the ﬁeld.●Oversight:We established the Motorola Solutions Technology Advisory Committee (MTAC) to serve as a ‘technicalconscience’ and advise the company on ethics, limitations and implications of speciﬁc product technologies.Additionally, we consult with objective third parties to provide an outside-in point of view to guide our decisions.ENSURING MISSION-CRITICAL AI IS APPLIED RESPONSIBL YWe are committed to ensuring that mission-critical AI is applied responsibly through each step of the product/solution innovation process, from ideation to customer deployment.Applying Human-Centered Design to Our Products●We apply a human-centered approach to all our products,which means we place our customers, their communities and theirunderlying challenges at the center of our design process. When incorporating AI-enabled features in our products, thisapproach ensures that the ideal user experience is supported by the appropriate technical solutions.●We thoughtfully design the speciﬁc interactions where users make AI-assisted decisions as a part of their workﬂow, ensuringthat information presented is meaningful, understandable and preserves user control.●We apply our knowledge of our users’ workﬂows to consider precisely when, how and why users experience AI-poweredfeatures in our solutions, enabling the appropriate use of the technology through experiential and technological guardrails.●We enforce these principles consistently and rigorously throughout our product development process. We identify risks early,deﬁne a strategy informed by empathy, and continuously improve solutions with user feedback.Training Our Products With Machine Learning:●We recognize that machine learning algorithms operate as a function of the data to which they are exposed,and as such, thealgorithms perform differently if they learn purely on the basis of their ﬁelded environment since individual customers will not have a breadth and diversity of training data.●We do not allow our products to learn in customer operational settings when the consequences of inaccuracy are signiﬁcantor there is considerable complexity in properly training an algorithm (e.g., facial recognition). We thoroughly test andcharacterize all applications of AI and, where appropriate,we train the algorithms ourselves. We may allow our products to employ machine learning to adapt to speciﬁc customer dynamics (e.g., camera scene speciﬁcs or patterns of life to identify unusual events) or when localization is required (e.g.,training on natural language vernacular that is speciﬁc to a particular user group).Providing Transparency Into Products●We explain all of our AI-based applications in plain,understandable language to give users guidance on their intended use aswell as how the applications perform under applicable operating conditions.●We participate in open and objective industry benchmarks,such as NIST’s Facial Recognition Vendor Test.Managing Data Within Our Products●We carefully secure and manage all of the data that we use in creation of our AI capabilities to ensure that we know its usesin our products as well as its provenance for quality and compliance purposes.●When we accept data from our customers for this purpose(e.g., machine learning algorithmic training), we do so under anexplicit contractual agreement.●We rigorously secure data, de-identify it to the greatest possible extent, carefully maintain its provenance and adhere toinstructions from our customers.Validating Our Products●We consult with objective third parties to provide an outside-in point of view to guide our decisions including industry groupslike IJIS (Integrated Justice Information System),university engagements like MIT, and customer research.●We employ former practitioners in order to provide an end-user perspective. We also have comprehensive quantitativecustomer research programs in place with our customer and user communities.Deploying Our Products and Supporting Our Customers●We build compliance controls and audits into our products,enabling our customers to adhere to local laws and regulationsas well as enforce their own policies. We train our customers on those product controls and how to be compliant withindustry standards and regulations.●Customers can enforce responsible policies and accountability by instituting end-to-end compliance controls across theentire workﬂow. For example, a user must be authenticated and have an active case number attached with actions,useractions are logged and checked against an agency's policies.●We collect operational feedback from ﬁelded products when possible in order to identify performance issues as well as anyinconsistent or undesirable behavior. This allows us to improve accuracy by reﬁning and retaining the model with enhanced data sets, and our cloud-based system allows us to rapidly deploy changes universally across our database.Controlling to Whom We Sell Our Products●We have a robust export control compliance program for ensuring that we only sell to countries and customers allowed bythe U.S. government. Our processes are applied on a transaction (contemplated sale) and product capability basis thatprecludes commerce with the U.S. embargoed countries list, tests against third party tools and services for screeningcountries, organizations and individuals, and assesses all contemplated products against applicable control lists published by the U.S. Department of State and Department of Commerce.●We apply our corporate code of business conduct, which all employees are trained on regularly, and we will not engage inbusiness transactions that would violate that standard.●We carefully vet and monitor our channel partners and resellers to ensure that they are compliant with these standards. ACCELERATING THE WORKFLOW WITH AIWe believe when leveraged responsibly, artiﬁcial intelligence can vastly improve the workﬂow for both businesses and public safety. Challenge:Multitasking leads to as much as a 40%drop in productivity (Harvard Business Review)How AI Helps:AI actively monitors cameras for unusual situations, such as the appearance of smoke or individuals matching the description of missing or abducted persons, allowing video analysts to verify potential items of interest instead of scanning endless video feeds.Challenge:240M 911 calls received each year (National Emergency Number Association)How AI Helps:AI automatically transcribes and translates speech to text – including recognizing key terms like “heart attack” –allowing the call taker to focus on the response and streamlining 911 call center interactions.Challenge:70% of oﬃcers prefer requesting license plate checks verbally vs. looking them up manually.(2016 Virtual Personal Assistant Online Survey)How AI Helps:AI uses natural language recognition to perform standard queries helping oﬃcers save time and stay safer in the ﬁeld by operating “eyes up and hands free.”Challenge:27,000 camera feeds collecting video every day in a large cityHow AI Helps:AI rapidly searches historical video for missing citizens or persons of interest, speeding up case resolution, especially in major incident scenarios.Challenge:94.2% of high schools employ security cameras in varying degrees.However, oftentimes the data lags and the cameras are only used as a reactionary tool. (National Center for Education Statistics, 2016)How AI Helps:AI proactively notiﬁes educators and school resource oﬃcers when something unusual happens like a door is propped open or a banned vehicle drives on campus.Challenge: As workplaces reopen during the COVID-19pandemic, businesses are working to remain compliant with social distancing and mask recommendations.How AI Helps:AI actively monitors social distancing and mask wearing efforts within a facility recognizing and notifying facility managers of high violation zones and times requiring corrective measures.Challenge:Among people with dementia who wander,at least 50 percent could suffer serious injury or die if they remain missing for more than 24 hours. (The Alzheimer’s Foundation)How AI Helps:AI can help identify those who cannot identify themselves by matching against a database.Challenge: 30% of an oﬃcer’s time is spent on administrative tasks with 400,000 incident reports created annually in a large city (2018 Motorola Solutions Law Enforcement Survey Report,LAPD CompStat Division)How AI Helps:AI can transcribe a spoken narrative recorded by the oﬃcer – rather than typing up a report in their vehicle or back at the precinct – and automatically populate it in the incident record, where the original audio ﬁle is kept as evidence.Challenge:8% of high school students in 2019 reported being in a physical ﬁght on school property at least once in the previous year. (Centers for Disease Control and Prevention 1991-2019High School Youth Risk Behavior Survey Data.)How AI Helps:AI can detect congregating groups in video that indicate a ﬁght or other disturbance is imminent, and alerts security to defuse the situation.Challenge:$61.7B in 2019 retail losses from theft,fraud and shrink (National Retail Federation, July2020)How AI Helps:AI can monitor video to help minimize shrink and theft by notifying security personnel when people are present at off hours or in the wrong location.。

组织行为学罗宾斯英文版PPT整理

Chapter1 Introduction to organizational behavior✓Organizational Behavior:The systematic study of the actions and attitudes that people exhibit within organizations✓Systematic Study of Determinants of Employee Performance:➢Actions or Behaviors：Productivity, Absenteeism, Turnover , Organizational citizenship➢Attitudes– Job Satisfaction: a. Possible link between satisfaction and productivityb.Satisfaction appears to be negatively related to absenteeism andproductivityc.Humanistic responsibility to provide employees with challenging,intrinsically rewarding, and satisfying job✓Organization: a. Consciously coordinated social unitb. Composed of two or more peoplec. Functions to achieve a common goal or set of goalsd. Formal roles define and shape the behavior of its members✓OB Encompasses Behavior in Diverse Organizations: Manufacturing:Service firms Schools Hospitals Churches Military units Charitable organizations Local, state, and federal government agencies✓Contributing Disciplines(Level of Analysis):➢Micro (individual): Psychology➢Macro (group processes and organization) : Sociology, Social Psychology, Anthropology, Political Science✓Toward an OB discipline P4 1.1✓Goals of Organizational Behavior: explanation, prediction, control✓Challenges and Opportunities: a.Increasing age of typical workerb.More women and minorities in the workplacec.Requirements to meet global competitiond.Severed loyalty bonds between employees and employers ✓What is Quality Management?➢Intense focus on customer→Outsiders -- purchasers of products and services→Insiders -- interact with and serve others in the organization➢Concern for continual improvement→Commitment to never be satisfied→“Very good” is not good enough→Quality can always be improved➢Improvement in quality of everything the organization does“Quality” applies not only to the final product, but to→How organization handles deliveries→How rapidly it responds to complaints→How politely the phones are answered➢Accurate measurement→Uses statistical techniques to measure every critical performance variable in operations➢Empowerment of employees→Involves people on the line in the improvement process→Teams are widely used as empowerment vehicles for finding and solving problems ✓ A Managerial Perspective：a. Improving People Skills b. Managing Work Force Diversityc. Responding to Globalizationd. Empowering Peoplee.Stimulating Innovation andChange f. Coping with “Temporariness” g. Helping Employee Balance Work-Life Conflicts h. Declining Employee Loyalty i. Improving Ethical Behavior✓Levels of OB Analysis: Individual Level Group Level Organization System Level Chapter2 Job Attitudes✓What the fundamental values of the organizational development can be found in the general manager’s approach to management? Respect, Support, Trust, Innovation ✓What contribution to the organization can be found in those values?A good work environment will be benefit to employees’ self-realization and theestablishment of team and learning organization.✓Attitude:Attitudes are evaluative statements or judgments concerning objects, people, or events. They reflect how we feel about something. When I say I like my job, I am expressing my attitude about work.✓Three components of Attitudes : Cognitive, Affective, Behavioral✓What are the Major Job Attitudes?➢Job Satisfaction: A positive feeling about the job resulting from an evaluation of its characteristics➢Job Involvement: Degree of psychological identification with the job where perceived performance is important to self-worth. High level of both job involvement andpsychological employment are positively related to organizational citizenship and jobperformance. High job involvement is also related to reduced absences and lowerresignation rates.➢Psychological Empowerment (PE): a. Belief in the degree of influence over the job, competence, job meaningfulness, and autonomy. b. Good leaders empower theiremployees by involving them in decisions, making them feel their work is important,and giving them discretion to do their own thing. c. Higher level of Job Involvement andPE are positively related to Organizational citizenship and job performance.✓other Major Job Attitudes:➢Organizational Commitment: Identifying with a particular organization and its goals and wishes to remain a member.✧The three forms of OC:Affective – emotional attachment to organization (e.g. pro-environmental firms)Continuance Commitment – economic value of staying with an org (e.g. high salary)Normative -moral or ethical obligations with employers (e.g. personal promise) There appears to be a positive relationship between organizational commitment andjob productivity.---has strong relation to performance, especially for new employees.---In general, affective commitment is most likely to relate to organizational outcomes such as performance and turnover.➢Perceived Organizational Support (POS)a.Degree to which employees believe the organization values their contributionand cares about their well-being.b.People perceive OS is higher when rewards are fair, employees are involved indecision-making, and supervisors are seen as supportive.c.High POS is related to higher OB outcomes (performance).➢Employee Engagementa. The degree of an individual’s involvement with, satisfaction with, and enthusiasm for the job.b. Engaged employees are passionate about their work and company.c. According to researches, they contribute high customer satisfaction, highprofits, and lower level turnover and accidents.✓Is there cognitive dissonance?--Your friends or relatives won’t disagree with you because of the close relation.--People do seek consistency among their attitudes and between their attitudes and their behavior. (E.g. I don’t marry her because love her.)✓The relationship between attitudes and behavior:a.Important attitudes reflect our fundamental values, self-interest, or identification withindividuals or groups we value. These attitudes tend to show a strong relationship to our behavior.b.The more you talk about your attitude on a subject, the more likely you are toremember it, and the more likely it to shape your behavior. (e.g. changing a job)c.Discrepancies between attitudes and behavior tend to occur when social pressures tobehave in certain ways hold exceptional power.d.The attitude-behavior relationship is likely to be much stronger if an attitude refers tosomething with which we have direct personal experience.✓the closer the match between attitude and behavior, the stronger the relationship Chapter3 Moods, Emotions and Organizational Behavior✓Why Were Emotions Excluded from OB Study?➢Myth of rationality – emotions were the antithesis of rationality and should not be seen in the workplace➢Belief that emotions of any kind are disruptive in the workplace✓Emotional Terminology:➢affect: A generic term that encompasses a broad range of feelings that people experience➢emotion: Intense feelings that are directed at someone or somethingShort termed and action-oriented.➢Mood: Feelings that tend to be less intense and longer-lasting than emotions and often lack a contextual stimulusP27 3.1✓The Basic Emotions:➢positive emotions→positive affect: The mood dimension consisting of positive emotions such as excitement, self-assurance, and cheerfulness at the high end with boredom,sluggishness, and tiredness at the low end.→negative affect: At zero input, when no stimulus is provided, most people experience a mildly positive mood. In fact, positive moods tend to be morecommon than negative ones.➢negative emotions➢negative affect: The mood dimension consisting of nervousness, stress, and anxiety at the high end with relaxation, tranquility, and poise at the low end.✓The Functions of Emotions:➢Emotions and Rationality: Emotions are critical to rational thought: they help in understanding the world around us.➢Evolutionary Psychology : Theory that emotions serve an evolutionary purpose: helps in survival of the gene pool. The theory is not universally accepted✓Sources of Emotions and Moods:➢Personality➢Day of the week and time of the day: More positive interactions will likely occur mid-day and later in the week➢Weather: no impact according to the research➢Stress: Increased stress worsens moods➢Social Activities: Physical, informal, and epicurean activities increase positive mood ➢Sleep: Lack of sleep increases negative emotions and impairs decision making➢Exercise: Mildly enhances positive mood➢Age: Older people experience negative emotions less frequently➢Gender: Women show greater emotional expression, experience emotions more intensely and display more frequent expressions of emotions. Could be due tosocialization✓Emotional Labor: An employee’s expression of organizationally desired emotions during interpersonal transactions at workEmotional dissonance is when an employee has to project one emotion while simultaneously feeling anotherFelt vs. Displayed Emotions:➢Felt Emotions: the individual’s actual emotions➢Displayed Emotions: the learned emotions that the organization requires workers to show and considers appropriate in a given job→Surface Acting is hiding one’s true emotions→Deep Acting is trying to change one’s feelings based on display rules ✓Emotional Intelligence:A person’s ability to:1)Be self-aware (to recognize his or her own emotions as experienced), 2)Detectemotions in others, and 3)Manage emotional cues and information.Moderately associated with high job performanceEmotional Intelligence on Trial➢The case for: a. Intuitive appeal – it makes sense b. EI predicts criteria that matter –positively correlated to high job performance c. Study suggests that EI isneurologically based➢The case against: a. EI is too vague a concept b. EI can’t be measured c. EI is so closely related to intelligence and personality that it is not unique when thosefactors are controlled✓OB Applications of Emotions and Moods➢Selection – Employers should consider EI a factor in hiring for jobs that demand a high degree of social interaction➢Decision Making – Positive emotions can increase problem-solving skills and help us understand and analyze new information➢Creativity – Positive moods and feedback may increase creativity✓More OB Applications of Emotions and Moods➢Motivation – Promoting positive moods may give a more motivated workforce➢Leadership – Emotions help convey messages more effectively➢Negotiation – Emotions may impair negotiator performance➢Customer Service – Customers “catch” emotions from employees, called emotional contagion✓Even More OB Applications of Emotions and Moods➢Job Attitudes – Emotions at work get carried home but rarely carry over to the next day ➢Deviant Workplace Behaviors – Those who feel negative emotions are more likely to engage in deviant behavior at work✓How Can Managers Influence Moods?➢Use humor to lighten the moment➢Give small tokens of appreciation➢Stay in a good mood themselves – lead by example➢Hire positive people✓Does the degree to which people experience emotions vary across cultures?Do people’s interpretations of emotions vary across cultures?Do the norms for the expressions of emotions differ across cultures?“YES” to all of the above!Chapter 5 Perception and Decision-making✓Perception：The process by which individuals organize and interpret their sensory impressions in order to give meaning to their environment✓Factors influencing perception：➢The perceiver：Attitudes，Motives，Interests，Experience，expectations➢The target：Novelty，Motion，Sound，Size，Background，proximity➢The Situation：Time，Work setting，Social setting✓Attribution Theory：trying to explain the ways in which we judge people differently, depending on the meaning we attribute to a given behavior.✓The three determining factors of attribution theory：➢Distinctiveness➢Consensus➢Consistency→Fundamental attribution error:1. When making judgments about the behavior of other people, we tend tounderestimate the influence of external factors and overestimate the influence ofinternal or personal factors2.Individuals and organizations tend to attribute their own successes to internal factors such as ability or effort, while putting the blame for failure on external factors such as bad luck or unproductive workers.3. Individuals whose intellectural and interpersonal abilieties are weakest are mostlikely to overestimate their performance and abilty.✓The Link Between Perception and Individual Decision making：Who makes decisions? What decisions to make?All the decisions are closed related to perceptions. (data collection and analysis)✓The Six Steps of Rational Decision-making model：➢Define the problem➢Identify the decision criterria➢Allocate weithgts to teh criteria➢Develop the alternatives➢Evaluate teh alternatives➢Select the best alternative➢Example:bicycle parking problem➢Bounded Rationality➢Intuitive decision making✓Common Biases and Erorrs in Decision Making：anchoring bias, confirmation bias, availabe bias, escalation of commitment, risk aversion, hindsight bias✓Organizatioal Constraints on Decision making: performance evaluation, reward systems, formal regulations, system-imposed time constraints, historical precdidents✓Three Ethical Decision Criteria:➢Utilitarianism（providing the greatest benefits for the greatest number功利主义，实用主义）➢Rights(respecting and protecting the basic rights of individuals,eg.right to privacy, free speech ,and due process)➢Justice(imposing and enforceing rules afaily and impartially to ensure justice or an equitalbe distribution of benefits and costs.) Comment on the three choices.✓Three-component Model of Creativity:➢Expertise(abilities, knowledge, proficiencies, and similar expertise )➢Creative thinking skills(personality ——creativity, the ability to use analogies, and the talent ot see the familiar in a different light)➢Intrinsic task motivation (interesting , involving , exciting, satisfying,persionally challengfing jobs, etc.)Chapter8 Groups✓Group: Two or more individuals, interacting and interdependent, who come together to achieve particular objectives. Groups can be either formal or informal, and further subclassified into command, task, interest, or friendship categories.✓Four Types of Groups:Command group, Task group, Interest group, Friendship group✓Why People Join Groups: (benefits)➢Security Reduce the insecurity of “standing alone”; feel stronger, fewer self-doubts, and more resistant to threats➢Status Inclusion in a group viewed by outsiders as important; provides recognition and status➢Self-esteem Provides feelings of self-worth to group members, in addition to conveying status to outsiders➢Affiliation Fulfills social needs. Enjoys regular interaction; can be primary source for fulfilling need for affiliation➢Power What cannot be achieved individually often becomes possible; power in numbers➢Goal achievement Some tasks require more than one person; need to pool talents, knowledge, or power to complete the job. In such instances, management may rely onthe use of a formal group✓Basic Group Concepts:➢Roles→Role research conclusions: a.People play multiple roles b.People learn roles from stimuli around them c.People can shift roles rapidly when the situation demandsd.People experience major role conflict between roles➢Norms: Acceptable standards of behavior within a group that are adopted and shared by the group’s members→The Hawthorne Studies→Conformity and the Asch Studies➢Cohesiveness: The degree to which members of the group are attracted to each other and motivated to stay in the group→Relationship of Cohesiveness to Productivity→Managers Can Encourage Cohesiveness: a.Make the group smaller b.Encourage agreement on group goals c.Increase the time spent together d.Increase thestatus and perceived difficulty of group membership→More Ways Managers Can Encourage Cohesiveness: a.Stimulate competition with other groups b.Give rewards to the group rather than members c.Physicallyisolate the group➢Size→How Size Effects a Group: a.Smaller groups are faster at completing tasks rge groups are consistently better at problem solving c.Social loafing - tendency toexpend less effort in a group than as an individual d.Increases in group size areinversely related to individual performance➢Composition: Diversity increases effectiveness due to the variety of viewpoints.Diversity promotes conflict, which stimulates creativity, which leads to improveddecision making. May take more time to work smoothly. May lead to turnover ➢Status: A prestige grading, position, or rank within the group. It may be formally imposed by the group, or informally acquired through characteristics such aseducation, age, gender, skill, or experience→Effects of High Status: a.Resist conformity or receive more freedom b.Do not need or care about social rewards c.Members must believe status hierarchy isequitable d.Inequities produce corrective behaviors and conflict✓Individual versus Group Decision Making:➢Individual: More efficient, Speed, No meetings, No discussion, Clear accountability, Consistent values➢Group: More effective, More information and knowledge, Diversity of views, Higher-quality decisions, Increased acceptance✓Symptoms of Group Think: a.Group members rationalize any resistance to their assumptionsb.Members pressure any doubters to support the alternative favored by the majorityc.Doubters keep silent about misgivings(doubts) and minimize their importanced.Groupinterprets members’ silence as a “yes” vote for the majorityVariables Influencing Group Think: Group’s cohesiveness, Leader’s behavior, Insulation from outsiders, Time pressures, Failure to follow methodical decision-making procedures✓GroupShift: A special case of groupthink. The decision of the group reflects the dominant decision-making norm that develops during the group discussion, whether shift is toward greater caution or more risk depends on the dominant prediscussion norm.✓Selecting the Best Decision-Making Technique:➢Brainstorming➢Nominal group technique➢Electronic meetingsChapter9 Teams✓Reasons for Team Popularity: a.Outperform on tasks requiring multiple skills, judgment, and experience b.Better utilization of employee talents c.More flexible and responsive to changing events d.Facilitate employee participation in operating decisions e.Effective in democratizing the organization and increasing employee motivation✓Work Group: A group who interacts primarily to share information and to make decisions to help one another perform within each member’s area of responsibility. Individuals work alone, not collectively, on a task. Performance is the summation of all of the group member’s individual contributions.✓Work Team:Generates positive synergy through coordinated effort. Their individual efforts result in a level of performance that is greater than the sum of those individual inputs.✓Comparing Work Groups and Work Teams P123 9.1✓Four Types of Teams P124 9.2➢Problem-Solving Teams: a.Share ideas or offer suggestions on how work processes and methods can be improved. b.Rarely given authority to unilaterally implement any oftheir suggested actions c.Typically composed of 5-12 hourly employees from thesame departmentExample: Quality Circles➢Self-Managed Work Teams: a.Collectively control pace of work b.Determine work assignments anize breaks d.Collectively choose inspection procedurese.Select their own members and evaluate each other’s performancef.Generallycomposed of 10-15 people➢Cross-Functional Teams: a.Members from diverse areas within and between organizations b.Exchange information c.Develop new ideas and solve problemsd.Coordinate complex projects f.Development is time-consuming due to complexity anddiversityExamples: Task Force and Committees➢Virtual Teams: Computer technology ties physically dispersed members together to achieve a common goal→Differentiating factors from other teams: Absence of para-verbal and non-verbal cues, Limited social context, Ability to overcome time and space constraints✓Creating Effective Teams:Effectiveness of teams is defined by:➢Objective measures of the team’s productivity➢Manager’s ratings of team performance➢Aggregate measures of member satisfactionA Team Effectiveness Model P126 9.3✓Turning Individuals into Team Players: To perform well as team members, individuals must be able to 1)Communicate openly and honestly 2)Confront differences and resolve conflicts 3)Sublimate personal goals for the good of the team✓The Challenge in Shaping Team Players:➢Greatest where... a.The national culture is highly individualistic b.Introduced into organizations that historically value c.individual achievement➢Less demanding... a.Where employees have strong collectivist values, such as Japan or Mexico b.In new organizations that use teams as their initial form for structuringwork✓Shaping Team Players:➢Selection: Ensure that candidates can fulfill their team roles in addition to having the technical skills required for the job➢Training: Provide workshops in problem-solving, communication, negotiation, conflict-management, coaching, and group-development skills➢Rewards: Rework reward systems to encourage cooperative efforts rather than competitive onesChapter 10 Communication✓Functions of Communication➢Control - both formal and informal➢Motivation - clarification and feedback➢Emotional expression - fulfillment of social needs➢Information - facilitating decision making✓The Communication ProcessSource, Encoding, Channel, decoding, Receiver✓Downward Communication：Assign goals，Provide job instructions，Inform employees of policies and procedures，Point out problems that need attention，Offer feedback about performance，Letters and email from leaders to members of the team✓Upward Communication：Provide feedback to higher-ups，Inform them of progress toward goals，Relay current problems，Keep managers aware of how employees feel，Ideas on how things can be improved✓Lateral Communication：Save time and facilitate coordination，Formally sanctioned or informally created，Enhance efficient and accurate transfer of information，Can create dysfunctional conflicts when formal vertical channels are breached✓Oral Communication：➢Advantage: Speed , Feedback➢Disadvantage: Potential for distorted message, Content at destination is different from the original✓Written Communication:➢Advantage: Provide a tangible and verifiable record, Can be stored for an indefinite period of time, Physically available for later reference, Well thought-out, logical, andclear➢Disadvantage: Time consuming, Lack of feedback, No guarantee how reader will interpret it✓Non-verbal Communication:➢Kinesics - Gestures, facial configurations, and other movements of the body➢Body movement -Body language adds to, and often complicates, verbal communication➢Intonations - Change the meaning of the message➢Facial expression -Characteristics that would never be communicated if you read a transcript of what is said➢Physical distance - Proper spacing is largely dependent cultural norms✓Formal Small-Group Networks P140 10.3✓Small-Group Networks and Effectiveness Criteria p140 10.4✓The Grapevine:Not controlled by management, Perceived as being more believable and reliable, Largely used to serve self-interest, Appear in response to situations: Important to us, Where there is ambiguity, Under conditions that arouse anxiety✓Computer-Aided Communication: Electronic mail (e-mail), Intranet and Extranet links, Videoconferencing✓Barriers to Effective Communication: Filtering, Selective Perception, Information Overload, Gender Styles, Emotions, Language✓ A Cultural Guide: Assume differences until similarity is proved, Emphasize description rather than interpretation or evaluation, Practice empathy, Treat your interpretation as a working hypothesis✓Improving Feedback Skills: 1. Focus on specific behaviors 2. Keep feedback impersonal 3.Keep feedback goal oriented 4. Make feedback well timed 5. Ensure understanding 6. Direct negative feedback toward behavior that is controllable by the recipient✓Improving Active Listening Skills: 1. Make eye contact 2. Exhibit affirmative head nods and appropriate facial expressions 3. Avoid distracting actions or gestures 4. Ask questions 5.Paraphrase 6. Avoid interrupting the speaker 7. Do not over talkChspter11 Leadership✓Leadership: Ability to influence a group toward the achievement of goals. The source of influence may be formal, provided by managerial rank in an organization. Non-sanctioned leadership(不具制裁力的领导) is the ability to influence that arises from outside of the formal structure of the organization.✓Trait Theories: Assumes that leaders are born, Characteristics that differentiate leaders from non-leaders, Personality traits in leaders that non-leaders do not possess, Characteristics of individuals who meet the definition of leader, Provides the basis of selecting the right person for leadership✓Traits Consistently Associated with Leadership:Drive and ambition, Desire to lead and influence others, Honesty and integrity, Self-confidence, Intelligence, In-depth technical knowledge✓Traits Alone Do Not Explain Leadership: Ignore situational factors. Leaders must take “the right actions”“The right actions” differ by situation✓Behavioral Theories: Assumes people can be trained to lead Researched the behaviors of specific leaders. Critical behavioral determinants of leadership. Specific behaviors identify leaders. Provides the basis of design for training programs✓Ohio State Studies:Sought to identify independent dimensions of leader behavior.Developed two categories of leadership behavior.:→Initiating structure - attempts to organize work, work relationships, and goals.→Consideration - concern for followers’ comfort, well-being, status, and satisfaction ✓University of Michigan Studies: Locate behavioral characteristics of leaders that appear related to measures of performance effectivenessTwo dimensions:→Employee-oriented - emphasize interpersonal relations→Production-oriented - emphasize the technical or task aspects of the job✓Limitations of Behavioral Theories:Did not identify consistent relationships between leadership behavior and group performance. Missing consideration of the situational factors that influence success and failure. Could not clarify situational factors✓Contingency Theories:➢Fiedler Leadership Model -Proper match of leader’s style of interacting with subordinates➢Path-Goal Model -Leader assists followers in attaining goals and ensures goals are compatible with overall objectives➢Leader-Participation Model - Leader behavior must adjust to reflect the task structure ✓Least-Preferred Co-Worker (LPC) Questionnaire: Individual’s basic leadership style is a key factor in leadership success. Assumed that individual leadership style is fixed,。

Scattering of a Klein-Gordon particle by a Hulth'en potential

a r X i v :0710.2575v 1[q u a n t -p h ] 13 O c t 2007Scattering of a Klein-Gordon particle by a Hulth´e n potentialJian You Guo,1,∗Xiang Zheng Fang,1and Chuan Mei Xie 11School of physics and material science,Anhui university,Hefei 230039,P.R.ChinaThe Klein-Gordon equation in the presence of a spatially one-dimensional Hulth´e n potential is solved exactly and the scattering solutions are obtained in terms of hypergeometric functions.The transmission coeﬃcient is derived by the matching conditions on the wavefunctions and the condition for the existence of transmission resonances are investigated.It is shown how the zero-reﬂection condition depends on the shape of the potential.PACS numbers:03.65.Nk,03.65.PmThe study of low-momentum scattering in the Schr¨o inger equation in one-dimensional even potentials shows that,as momentum goes to zero,the reﬂection coeﬃcient goes to unity unless the potential V (x )supports a zero-energy resonance[1].In this case the transmission coeﬃcient goes to unity,becoming a transmission resonance[2].Recently,this result has been generalized to the Dirac equation[3],showing that transmission resonances at k =0in the Dirac equation take place for a potential barrier V =V (x )when the corresponding potential well V =−V (x )supports a supercritical state.This conclusion is demonstrated in both special examples as square potential and Gaussian potential,where the phenomenon of transmission resonance is exhibited clearly in Dirac spinors in the appropriate shapes and strengths of the potentials.Except for the both special examples,the transmission resonance is also investigated in the realistic physical system.In Ref.[4],a key potential in nuclear physics is introduced,and the scattering and bound states are obtained by solving the Dirac equation in the presence of Woods-Saxon potential,which has been extensively discussed in the literature[5,6,7,8,9].The transmission resonance is shown appearing at the spinor wave solutions with a functional dependence on the shape and strength of the potential.The presence of transmission resonance in relativistic scalar wave equations in the potential is also investigated by solving the one-dimensional Klein-Gordon equation.The phenomenon of resonance appearing in Dirac equation is reproduced at the one-dimensional scalar wave solutions with a functional dependence on the shape and strength of the potential similar to those obtained for the Dirac equation[10].Due to the transmission resonance appearing in the realistic physical system for not only Dirac particle but also Klein Gordon particle as illustrated in the Woods-Saxon potential,it is indispensable to check the existence of the phenomenon in some other ﬁelds.Considering that the Hulth´e n potential[11]is an important realistic model,it has been widely used in a number of areas such as nuclear and particle physics,atomic physics,condensed matter and chemical physics[12,13,14,15].Hence,to discuss the scattering problem for a relativistic particle moving in the potential is signiﬁcant,which may provide more knowledge on the transmission resonance.Recently,there have been a great deal of works to be put to the Hulth´e n potential in order to obtain the bound and scattering solutions in the case of relativity and non relativity[16].However,the transmission resonance is not still checked for particle moving in the potential in the relativistic case.In this paper,we will derive the scattering solution of the Klein-Gordon equation in the presence of the general Hulth´e n potential,and show the phenomenon of transmission resonance as well as its relation to the parameters of the potential.Following Ref.[10],one-dimensional Klein-Gordon equation,minimally coupled to a vector potential A µ,is written asηαβ(∂α+ieA α)(∂β+ieA β)φ+φ=0,(1)where the metric ηαβ=diag(1,-1).For simplicity,the natural units =c =m =1are adopted,and Eq.(1)is simpliﬁed into the following formd 2φ(x )e −ax −q+Θ(x )V 0∗Electronicaddress:jianyou@where all the parameters V0,a,and q are real and positive.To remove oﬀthe divergence of Hulth´e n potential,q<1 is required.If q=−1is taken,the Hulth´e n potential turns into a Woods-Saxon potential.Θ(x)is the Heaviside step function.The form of the Hulth´e n potential is shown in Fig.1and2at diﬀerent values of parameters.From Fig.1and2one readily notices that for a given value of the potential strength parameter V0,as q increases, the height of potential barrier increases.When q−→1,the height of potential barrier goes to inﬁnity.Similarly,the potential becomes more diﬀusible with the decreasing of the diﬀuseness parameter a.In order to obtain the scattering solutions for x<0with E2>1,we solve the diﬀerential equationd2φ(x)e−ax−q 2−1 φ(x)=0.(4)On making the substitution y=qe ax,Eq.(4)becomesa2y2d2φdy+ E−V01−y 2−1 φ(x)=0.(5)In order to derive the solution of Eq.(5),we putφ=yµ(1−y)λf,then Eq.(5)reduces to the hypergeometric equationy(1−y)f′′+[1+2µ−(2µ+2λ+1)y]f′− λ(1+2µ)+2EV0E2−1,λ±=12 µ2+λ2−λ−2EV0dx2+ E−V0dz2+a2zdφq(1−z) 2−1φ(x)=0.(11)Putφ=zµ(1−z)λg,Eq.(11)reduces to the hypergeometric equationz(1−z)g′′+[1+2µ−(2µ+2λ+1)z]g′− λ(1+2µ)+2EV02m φdφ∗dx .(16)The current as x−→−∞can be decomposed as j L=j in−j reﬂwhere j in is the incident current and j reﬂis the reﬂected one.Analogously we have that,on the right side,as x−→∞the current is j R=j trans,where j trans is the transmitted ing the reﬂected j reﬂand transmitted j trans currents,we have that the reﬂection coeﬃcient R,and the transmission coeﬃcient T can be expressed in terms of the coeﬃcients A,B,and D asR=j reﬂ|A|2,(17)T=j trans|A|2.(18)Obviously,R and T are not independent;they are related via the unitarity conditionR+T=1.(19)In order to obtain R and T we proceed to equate at x=0the rightφR and leftφLwave functions and theirﬁrstderivatives.From the matching condition we derive a system of equations governing the dependence of coeﬃcients A and B on D that can be solved numerically.The calculated transmission coeﬃcient T varying with the energy E is displayed in Figs.3-6at the diﬀerent values of the parameters in the Hulth´e n potential.From Figs.3-6,one can see that the transmission resonance appears in all the Hulth´e n potential considered here.But the intensity and width of resonance as well as the condition for the existence of resonance are diﬀerent,and they depend on the shape of the pared Fig.3with Fig.4,it can be seen that the width of resonance decreases as the decreasing of diﬀuseness a,which is similar to that of Woods-Saxon potential as shown in Figs.3and5in Ref.[10].The same dependence can also be observed from Figs.5and6. Compared Fig.3with Fig.5,one canﬁnd that the condition for the existence of transmission resonance does also relate to the parameter q.As q decreases,the height of potential barrier increases,the widths of the transmission resonance increases.The conclusion can also be obtained by comparing Fig.4with Fig.6.In order to obtain more knowledge on the dependence of transmission resonance on the shapes of the potential,the transmission coeﬃcient T varying with the strength of potential V0is plotted in Figs.7and8.Beside of the phenomenon of transmission resonance,similar to the Fig.3and4,the width of resonance decreasing as the decreasing of diﬀuseness a is disclosed.All these show the transmission resonances in Hulth´e n potential for Klein-Gordon particle possess the same rich structure with that we observe in Woods-Saxon potential.AcknowledgmentsThis work was partly supported by the National Natural Science Foundation of China under Grant No.10475001and 10675001,the Program for New Century Excellent Talents in University of China under Grant No.NCET-05-0558,the Program for Excellent Talents in Anhui Province University,and the Education Committee Foundation of Anhui Province under Grant No.2006KJ259B[1]R.Newton,Scattering Theory of Waves and Particles (Springer-Verlag,Berlin,1982).[2]D.Bohm,Quantum Mechanics (Prentice-Hall,Englewood Cliﬀs,NJ,1951).[3]N.Dombey,P.Kennedy,and A.Calogeracos,Phys.Rev.Lett.85,1787(2000).[4]P.Kennedy,J.Phys.A 35,689(2002).[5]J.Y.Guo,X.Z.Fang,and F.X.Xu,Phys.Rev.A 66,062105(2002).[6]V.Petrillo and D.Janner,Phys.Rev.A 67,012110(2003).[7]G.Chen,Phys.Scr.69,257(2004).[8]J.Y.Guo,J.Meng,and F.X.Xu,Chin.Phys.Lett.20,602(2003).[9]A.D.Alhaidari,Phys.Rev.Lett.87,210405(2001);88,189901(E)(2002).[10]C.Rojas and V.M.Villalba,Phys.Rev.A 71,052101(2005).[11]L.Hulth´e n,Ark.Mat.Astron.Fys.A 28,5(1942).[12]Y.P.Varshni,Phys.Rev.A 41,4682(1990).[13]M.Jameelt,J.Phys.A:Math.Gen.19,1967(1986).[14]R.Barnana and R.Rajkumar R,J.Phys.A:Math.Gen.20,3051(1987).[15]L.H.Richard,J.Phys.A:Math.Gen.25,1373(1992).[16]C.-Y.Chen,D.-S.Sun,F.-L.Lu,Phys.Lett.A,(2007)(in press),and the Reﬀerences there.[17]W.-C.Qiang,R.-S.Zhou,Y.Gao,Phys.Lett.A,(2007)(in press).42024X510152025303540VFIG.1:Hulth´e n potential for a=1.0and q=0.9with V 0=4,of which the peak of barrier reaches 40.0.42024X12345678VFIG.2:Hulth´e n potential for a=0.5and q=0.5with V 0=4,of which the peak of barrier reaches 8.0.246810E0.20.40.6T FIG.3:The transmission coeﬃcient for the relativistic Hulth´e n potential barrier.The plot illustrate T for varying energy E with V 0=4,a =1,and q =0.9.246810E0.20.40.60.81T FIG.4:Similar to Fig.3,but with V 0=4,a =0.5,and q =0.9.246810E0.20.40.60.81T FIG.5:Similar to Fig.3,but with V 0=4,a =1,and q =0.5.246810E0.20.40.6T FIG.6:Similar to Fig.3,but with V 0=4,a =0.5,and q =0.5.1234V00.20.40.60.81T FIG.7:The transmission coeﬃcient for the relativistic Hulth´e n potential barrier.The plot illustrate T for varying barrier height V 0with E =2,a =1,and q =0.9.1234V00.20.40.60.81T FIG.8:Similar to Fig.7,but with E =2,a =0.5,and q =0.9.。

Asymptotic behavior of discrete holomorphic maps z^c, log(z) and discrete Painleve transced

Abstract It is shown that discrete analogs of z and log(z ) have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painleve-II equations, asymptotics of these solutions providing the behaviour of discrete z c and log(z ) at inﬁnity.
This deﬁnition is motivated by the following properties: 1) it is M¨ obius invariant, 2) a smooth map f : D ⊂ C → C is conformal (holomorphic or antiholomorphic) if and only if
c
1
Introduction
In this paper we discuss a very interesting and rich object: circle patterns mimicking the holomorphic maps z c and log(z ). Discrete z 2 and log(z ) were guessed by Schramm and Kenyon (see [25]) as examples from the class of s of the square grid introduced by Schramm in [24]. This class opens a new page in the classical theory of circle packings, enjoying new golden age after Thurston’s idea [26] about approximating the Riemann mapping by circle packings: it turned out that these circle patterns are governed by discrete integrable equation (the stationary Hirota equation), thus providing one with the whole machinery of the integrable system theory ([11]). The striking analogy between circle patterns and holomorphic maps resulted in the development of discrete analytic function theory (for a good survey see[13]). Classical circle packings comprised of disjoint open disks were later generalized to circle patterns where the disks may overlap. Discrete versions of uniformization theorem, maximum principle, Schwarz’s lemma and rigidity properties and Dirichlet principle were established ([19],[16],[24]). Diﬀerent underlying combinatorics were considered: Schramm introduced square grid circle patterns, generalized by Bobenko and Hoﬀmann to hexagonal patterns with constant intersection angles in [8], hexagonal circle patterns with constant multi-ratios were studied by Bobenko, Hoﬀman and Suris in [7]. The diﬃcult question of convergence was settled by Rodin and Sullivan [22] for general circle packings, He and Schramm [15] showed that the convergence is C ∞ for hexagonal packings, the uniform convergence for square grid circle patterns was established by Schramm [24]. On the other hand not very many examples are known: for circle packings with the hexagonal combinatorics the only explicitly described examples are Doyle spirals, which are discrete analogues of exponential maps [12], and conformally symmetric packings, which are analogues 1

脉冲微分方程解的存在性和稳定性

中国地质大学（武汉）硕士学位论文脉冲微分方程解的存在性和稳定性姓名：***申请学位级别：硕士专业：应用数学指导教师：***20080501“’Ｏ）＋兄甜（ｆ）＝／（ｆ，“（，一ｒ）），ｆ≠０，ｆ∈，＝【ｏ，丁】，Ａｕ（ｔ』）＝Ｉｊ（“（ｆ伪，／＝１，…Ｐ，甜（ｏ）＝“（丁）＝Ｕｏ，“（，）＝ｏ，ｆ∈卜ｆ，ｏ）．甜７（，）＋名“（，）＝厂（，，甜（ｒ））＋Ｐ（，），ｔｅＪ＝［Ｏ，丁】，ｕ（ｏ）＝甜（丁）＝‰．（７）甜’（，）＋力甜（ｆ）＝厂（ｒ，“（，））＋Ｐ（，），，∈／＝【ｏ，ｒ】，Ａｕ（ｔｊ）＝‘＠ｑ）），／＝１，２，…Ｐ，ｕ（ｏ）＝ｕ（ｒ）－Ｕｏ．对于后面的五类问题，我们均可以得到其相应的反周期边值问题解的性质。

通过讨论，我们可以清晰地看到：所研究的脉冲周期边值问题解的存在性及稳定性的结论与脉冲条件密不可分。

关键词：周期边值问题，脉冲，时滞，存在性，稳定性ＴｈｅＥｘｉｓｔｅｎｃｅａｎｄＳｔａｂｉｌｉｔｙｏｆＩｍｐｕｌｓｉＶｅＤｉｆｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎｓＭａｓｔｅｒＣａｎｄｉｄａｔｅ：ＬｉＹ砒ｉｎｇＳｕｐｅｒｖｉｓｏｒ：ＬｉｕＡｍｐｉｎｇＩｍｐｕｌｓｉｖｅｄｉ舵ｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｃ锄ｂｅｓｕｃｃｅｓｓｆｕｌｌｙ吣ｅｄｆｏｒｍａｔｈｅｍａｔｉｃａｌｓｉｍｕｌａｔｉｏｎｉｎｔｌｌｅｏｒｅｔｉｃａｌｐｈｙｓｉｃｓ，ｃｈｅｍｉｓｔⅨｂｉｏｔｅｃｈｎｏｌｏ烈ｍｅｄｉｃｉｎｅ，ｐｏｐｕＩａｔｉｏｎｄｙｎ锄ｉｃｓ，ｏｐｔｉｍａｌｃｏｎｔｍｌ，锄ｄｉｎｏｔｈｅｒｐｎｏｃｅｓｓｅｓ鲫ｄｐｈｅｎｏｍｅｍｉｎｓｃｉｅｎｃｅ觚ｄｔｅｃｈｎｏｌｏｇｙ．ＴｈｅＳｔａｂｉｌｉｔｙｔｈｅｏ搿ｏｆｉｍｐｕｌｓｉＶｅｄｉｆｆ－ｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｈ嬲ｂｅｅｎｄｅｖｅｌｏｐｅｄｂｙａｌａ呼ｎ啪ｂｅｒｏｆｍａｔｌｌｅｍａｔｉｃｉａｎｓ，ａ１１ｄｔｈｅｉｒＳｔｕｄｉｅｓｈａｖｅａｔｔｒ暑Ｉｃｔｅｄｍｕｃｈａｔｔｅｎｔｉｏｎ．Ｔｈｅｖｈａｖｅｂｅｅｎｓｕｃｃｅｓｓｆｕｌｉｎｄｉ行ｅｒｅｎｔａｐｐｒｏａｃｈｅｓｂ舔ｅｄ０ｎＬｙ印ｕｎｏＶｄｉｒｅｃｔｍｅｔｈｏｄ锄ｄｃｏｍｐ耐ｓｏｎｔｅｃｌｌｌｌｉｑｕｅ．Ｔｈｉｓｍｅｔｈｏｄｈ∞ｇｅｎｅｒａｌｉｔ），ｆ．ｒｏｍｔｈｅｔ１１ｅｏｒｅｔｉｃａｌｓｔａｎｄｐｏｉｍ，ｂｕｔｉｔｉｓｎｏｔｃｏｎｖｅｎｉｅｎｔｆｏｒｐｒａｃｔｉｃａｌｍｅｓｏｍｅｔｉｍｅｓ．Ｔｈｅｍａｉｎｄｉ师ｃｕｌｔｙｌｉｅｓｉｎｃｏｎｓｔｍｃｔｉｎｇｔｈｅＬｙａｐｕｎｏＶｆｕｎｃｔｉｏｎａｌ．Ｉｎｔｈｉｓｐ印ｅｒ，ｗｅｅｍｐｌｏｙ廿ｌｅｉｔｅｒａｔｉＶｅｍｅｔｈｏｄｗｈｉｃｈｉｓＶｅｒｙｃｏｎｃｒｅｔｅｔ０ｏｂｔａｉｎｔｈｅｅ心ｓｔｅｎｃｅ锄ｄｓｔａｂｉｌｉ锣ｏｆｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙＶａｌｕｅｐｒｏｂｌｅｍｓｗ曲ｉｍｐｕＩｓｅｓ．Ｉｎｔｈｉｓ硎ｃｌｅ，ｗｅｍａｉｎｌｙｏｂｔａｉｌｌｔｈｅｅ虹ｓｔｅｎｃｅｓ锄ｄｓ劬ｉｌｉｔｉｅｓｏｆｓｏｌｕｔｉｏｎＳｏｆｐｅｒｉｏｄｉｃｂｏｕｎｄａ巧Ｖａｌｕｅｐｒｏｂｌｅｍｓ舔ｆｏｌｌｏｗｓ：（１）材’（，）＋兄“（，）＝厂（，，甜（，）），，∈，＝【ｏ，丁】，（２）（３）“（ｏ）＝“（丁）＝‰．甜’（，）＋五材（，）＝厂（，，“（，）），，∈，＝【ｏ，丁】，“（ｏ）＝一“（ｒ）＝‰．材’（ｆ）＋勉（ｆ）＝厂（ｆ，材（ｆ））＋Ｐ（ｆ），ｆ∈‘，＝【ｏ，丁】，甜（ｏ）＝“（ｒ）＝‰．研究生学位论文原创性声明我以诚信声明：本人呈交的硕士学位论文是在刘安平教授指导下开展研究工作所取得的研究成果。

Asymptotic behaviour of granular materials

Granular Matter(2012)14:759–774DOI10.1007/s10035-012-0372-xORIGINAL PAPERAsymptotic behaviour of granular materials David MašínReceived:8December2011/Published online:23September2012©Springer-Verlag2012Abstract The concept of the asymptotic behaviour of par-ticulate materials is described,including its enhancement by considering asymptotic states in extension.A3D discrete ele-ment model with elastic spherical particles and the granulom-etry of a real sand is set up.The numerical sample is stretched from different initial states,and the inﬂuence of the strain rate direction on theﬁnal state is studied within the stress ratio,void ratio and mean stress space.Asymptotic behav-iour is clearly observed,although the grains remain intact (no grain crushing is considered).The extension asymptotic states are observed,and the notion of a normal extension line is introduced.The extension asymptotic states coincide with the peak states observed in the shear tests with constant stress path direction in dense samples.Keywords Asymptotic behaviour·Critical state·Discrete element method·Particle crushing·Sand1IntroductionAsymptotic behaviour is one of the most striking features in the behaviour of granular materials.Speciﬁc asymptotic states have been known since the early studies of soil mechan-ics.Casagrande[4],Hvorslev[27]and Taylor[61]are among the pioneers who observed critical state behaviour of soils—a particular asymptotic state related to constant volume shear-ing.Schoﬁeld and Wroth[54]and Roscoe and Burland[52] combined the existence of critical states and compression asymptotic states(revealed in normal compression behav-iour)into a uniﬁed framework of critical state soil mechanics.D.Mašín(B)Faculty of Science,Charles University in Prague,Albertov6,12843Prague2,Czech Republice-mail:masin@natur.cuni.cz More generally,Gudehus et al.[21,25]understood asymp-totic states to be attractors in the behaviour of granular materi-als,which are independent of the initial state.They proposed that each direction of strain rate with a volume decrease is uniquely linked to a particular asymptotic stress ratio and a particular path in the mean stress versus void ratio plane (normal compression line).Gudehus[22]later argued that the asymptotic stress ratio should not be expected to remain con-stant in the course of loading because of grain crushing and changing granulometry.The asymptotic state has also been denoted as the state limit[24],or the swept-out-memory state [25].More recently,Gudehus[23]and Gudehus and Mašín [24]have extended the asymptotic state concept into the vol-ume increase(extension)regime,and identiﬁed theoretical limits to the asymptotic behaviour.The“extension asymp-totic states”have not yet been observed experimentally.Asymptotic behaviour has been the subject of experi-mental investigation;critical state and normal compression behaviour have been well documented.More generally,com-pression asymptotic states have been studied,and conﬁrmed, by Goldscheider[20]and Chu and Lo[9],who performed true and axisymmetric triaxial tests on sands with strain path control.They observed a unique relationship between the strain path direction and asymptotic stress ratio.Asymp-totic behaviour has also been studied by testingﬁne-grained soils(clays),most notably by Topolnicki et al.[62].They observed that the stress paths of tests starting at an arbitrary state became parallel to the asymptotic path in some cases. Further insight into the micro-mechanics of granular materi-als has been given,for example,in[19].Another means of investigating the asymptotic state is the discrete element method(DEM).Different authors typ-ically focused on speciﬁc asymptotic states;the existence of the critical state is conﬁrmed in[7,8,48,53,56,66,67]. Salot et al.[53]and Wang et al.[66]demonstrated that the760 D.Mašín value of the critical state friction anglecle shape.The inﬂuence of the particle shapetotic behaviour was also emphasized in[17,Luding and Alonso-Marroquín[36]observedcal state friction angle was constant forbehaviour,but showed a pressure dependenceinter-particle contacts.For further details ofinvestigation of the micromechanics ofthe reader is referred to[1,32,51]and theA number of authors argued that theasymptotic states is directly linked to[2,7,8,43–45,65].Using DEM simulations,normal compression lines was explained byin references[2,7,8,44,65].Cheng et al.[7,8]critical states and normal compression lines,parameter sets,creating a more completeular material behaviour.They argued thatis a cause of the asymptotic behaviour.The aim of this paper is to provide asation of the asymptotic behaviour ofAfter introducing the asymptotic statehensive DEM model was set ing thisical experiments were performed to revealstates.This included characteristics that havetigated before,such as the asymptotic behaviourIt is also discussed whether particle crushing isconsequence,of the asymptotic propertiesassemblies.2Asymptotic state frameworkAsymptotic state is deﬁned as that stateﬁciently long proportional stretching,i.e.constant direction of the strain rate.tation of asymptotic states has been proposed[23]and Gudehus and Mašín[24].In thisaxisymmetric stress and deformation states,rate tensor is fully characterised by axial˙ acomponents.Similarly,the stress tensor isstress)andσr(radial stress).It is assumedmaterial behaviour is governed only by itsvoid ratio e(deﬁned as the void volume overume).The strain rate direction may beangleψ˙ (see Fig.1a),and the stress obliquityby the angleψσ(Fig.1b).2.1Compression and constant volume asymptotic states According to the current understanding of the asymptotic behaviour of a granular assembly,proportional deformation (constantψ˙ )will ultimately lead to an asymptotic state char-acterised by a constantψσ.Not all stretching directions will,iting case)and−90◦≤ψ˙ ≤90◦.These directions ofψ˙ are represented in Fig.2a.Isotropic compression1ψ˙ =0◦is indicated with the index‘i’;limiting valuesψ˙ =±90◦1Note that the isotropic asymptotic state is deﬁned here byψ˙ =0◦; the corresponding asymptoticψσmay then differ from0◦in the case of anisotropic structure.in Fig.2b.In the special case of isochoric deformation(±c,critical state),the corresponding value ofψσis directly linked to the critical state friction angle by 2OC R is traditionally deﬁned as OC R=p c/p,where p c is the preconsolidation pressure.We prefer the deﬁnition(2),as no762 D.Mašínr e=e−e de c−e dOC R=p∗ep(2)where e d and e c are minimum and critical state void ratios at the current mean stress respectively.p∗e is the Hvorslev equivalent pressure,deﬁned as the mean stress at the isotropic normal compression line at the current void ratio.It is clear that the isotropic normal compression line is characterised by r e=r ei>1and OC R=1,and the critical state line by r e=1and OC R=OC R c>1.Each of the compressionasymptotic states can be attributed to a unique value of1≤r e≤r ei,or OC R c≥OC R≥1.Compression asymptotic states are predicted by constitu-tive models based on the critical state theory,such as the Mod-iﬁed Cam clay model[52],or hypoplastic models[39,40,64]. The models incorporate the notion of a state boundary sur-face(SBS),which is deﬁned as a boundary of all possible states in the stress versus void ratio space.For the purpose of the present discussion,we also deﬁne the asymptotic state boundary surface(ASBS)(also known as the swept-out-memory surface[41]),which is an envelope of all asymp-totic states.Critical state soil mechanics models consider the two surfaces to coincide.A constant void ratio cross-section through this surface in the compression regime is sketched in Fig.2d.In the principal stress versus void ratio space,the ASBS is a four-dimmensional object;this is shown in Fig.3 for the axisymmetric case.An important property of compression asymptotic states is that they may be reached by proportional compression from the stress-free state.2.2Extension asymptotic statesAsymptotic behaviour in compression and under constant volume is considered to be a well-proven property of a granu-lar assembly.However,the contrary is true regarding asymp-totic behaviour in extension.As suggested by Gudehus[23] (Chapters2and3)and Gudehus and Mašín[24],asymptotic states can also be reached after proportional stretching along extension(volume increase)paths.These asymptotic states have,to the author’s knowledge,not yet been observed exper-imentally;their existence is expected solely from theoretical arguments.The stretching directions that lead to extension asymptotic states are depicted in Fig.4a.Limiting values of ψ˙ andψσare denoted with indices‘d’(asymptoticσr=0) and‘−d’(asymptoticσa=0)[24].The maximum values of |ψσ|at the limiting states±d correspond to mobilised fric-tion angles equal to90◦.The relationship betweenψ˙ and ψσfor asymptotic extension paths is shown in Fig.4b.Footnote2continuedadditional assumptions about the quasi-elastic soil behaviour are needed for its quantiﬁcation.Fig.6Grain size distribution of Zbraslav sand from[16],and the reduced grain size distribution adopted in the simulationsFig.7Periodic cell used in the simulations,consisting of150,000 spherical particlesEach extension asymptotic state is also associated with its trace in the mean stress versus void ratio plane.In the following,we denote these traces as normal extension lines (adopting a parallel with the well-known notion of normal compression lines),seen in Fig.4c.Gudehus[23]suggested that the minimum void ratio e d represents a normal exten-sion line for states±d.Because proportional stretching along extension asymptotic states involves volume increase and mean stress decrease,these states cannot be approached from the stress-free state.Interestingly,although knowledge of the extension asymp-totic states is scarce,these states are predicted by virtually all constitutive models based on critical state soil mechan-ics(for example,[37,39,52,64]).The reason for this may be explained with the aid of Fig.4d,which shows a complete ASBS of a granular material,with the highlighted portion corresponding to the extension asymptotic states.The mod-els were developed to correctly represent the so-called peakAsymptotic behaviour of granular materials763 Fig.8Frequency–magnitudediagrams of relative overlapsΔu/(r1+r2)between particlesinψ˙ =0◦test.Δu is theoverlap and r1and r2are radii ofthe spheres in contactstates.Peak states are attained after loading with the stresspaths in aﬁxed direction(for example,a drained triaxial test,which is characterised by˙σr=0).It has been shown experi-mentally that such loading,from the initial state with r e<1(and OC R>OC R c),leads to a higher mobilised frictionangle than that which corresponds to±c states.The strain rateat the peak state is dilatant.To represent this behaviour,theSBS for r e<1extends into the region of|ψσ|>|ψσ(±c)|,with|ψ˙ |>|ψ˙ (±c)|,as depicted in Fig.4d.As mentioned earlier,ASBS in the critical state models isconsidered to coincide with the SBS.For this reason,consti-tutive models based on critical state soil mechanics implic-itly predict extension asymptotic states,although the model developers did not speciﬁcally aim to predict them.To demonstrate the asymptotic property of constitutive models,Fig.5shows the predicted compression(path“A”), constant volume(path“B”)and extension(path“C”)asymp-totic state using the critical state soil mechanics-based hypoplastic constitutive model of Mašín[39].This model is capable of predicting all three types of asymptotic states;the explicit mathematical formulation of the asymptotic states predicted by this model are given in[41].0.40.50.60.70.80.91 10 100 1000 10000ln(1+e)[-]p [kPa]cyclic isotropic testisotropic compression testFig.9Monotonous and cyclic(unloading–reloading)isotropic test results(ψ˙ =0◦in loading andψ˙ =180◦in unloading)3Discrete element model characteristicsIn order to model the granular assembly,the open-source3D discrete element software Yade[57]was used.This software764 D.MašínFig.10ψ˙ =90◦tests on normally consolidated samples.a Sketch of the stress paths expected from the theory,(b–d)results of DEM simulations. bψσversus mean stress diagram,c normal compression lines in the ln(1+e)versus log p plane,dψσversus ln(1+e)diagramutilises the DEM formulation by Cundall and Strack[12]. The algorithm involves two steps:First,based on constitu-tive laws,the interaction forces between the discrete ele-ments are computed.The elements are allowed to slightly interpenetrate each other,which actually represents the rel-ative deformation of the surface layers of the particles[11]. Second,Newton’s second law is applied to determine,for each element,the resulting acceleration,which is then time integrated toﬁnd its new position.This process is repeated until the simulation isﬁnished[55,15].Chen et al.[6]demon-strated that the Yade software yields comparable predictions to the commercial software package PFC[28].We present the results for a specimen consisting of elastic spherical particles.To eliminate the inﬂuence of the model boundaries,periodic boundary conditions have been adopted [46],so that the modelled unit cell(as well as all its parti-cles and all their properties)is surrounded by identical cells shifted along the cell edges[59].In order to resemble a real granular material,particles of different sizes have been considered following the grain-size-distribution curve of a real sand.Figure6shows grain size distribution of Zbraslav sand[16].For the purpose of the discrete element modelling, particles above1and below0.2mm were removed from the specimen(“reduced GSD”in Fig.6).The sample consisted of150,000spherical particles,generated using an algorithm to ensure that they followed the prescribed particle size dis-tribution,were randomly distributed,and were initially not in contact.The periodic cell was cubic,with the initial side length of31mm.The specimen was generated only once and used in all subsequent simulations;all the results thus repre-sent the response of an identical sample.It has been veriﬁed that practically identical results were obtained if new sample was generated in each simulation.The specimen in its initial state is depicted in Fig.7.The contact properties of the spherical particles were governed by a basic linear elastic perfectly plastic model without cohesion[12],which speciﬁes the contact normal stiffness k n,shear stiffness k s and friction angleϕ.Asymptotic behaviour of granular materials 765Fig.11ψ˙ =90◦tests on overconsolidated samples.a Sketch of the stress paths expected from the theory,(b –d )results of DEM simulations.b ψσversus mean stress diagram,c normal compression lines in the ln (1+e )versus log p plane,d ψσversus ln (1+e )diagramThese parameters are calculated from the particle proper-ties E (Young’s modulus),ν(Poisson’s ratio)and ϕ.In the present simulations,constant values E =500MPa,ν=0.3and ϕ=0.5Rad (coefﬁcient of friction μ=0.546)have been used.The rolling resistance (rotational spring)was not considered.The prescribed particle density was ρs =2650kg/m 3and acceleration due to gravity was zero.The inﬂuence of selected contact parameter values is clariﬁed in Sect.5.The periodic cell boundaries were subjected to a con-stant velocity gradient ∇v ,which means to constant value of the Euler stretching tensor D =(∂v i /∂x j +∂v j /∂x i )/2,as no rotations have been considered.Axisymmetric condi-tions were applied,such that D 22=D 33(subscripts 2and 3represent the horizontal directions,1the vertical direction).The angle ψ˙ can then be calculated by means of Fig.1a,where ˙ a =D 11and ˙r =D 22=D 33.Off-diagonal com-ponents of D were always zero.A constant magnitude of the stretching rate D =√=167s −1and step size Δt =1.2×10−7s was applied at all times.Local non-viscous damping [5,10],has been used,with a damping coefﬁcient χ=0.5.The inﬂuence of the selected step size and stretch-ing rate is discussed in more detail in Sect.5.The simulation results have been evaluated in terms of the Cauchy stress tensor σand void ratio e .The void ratio has been calculated from the current cell size and total par-ticle volume (particle overlaps have thus been neglected).The macroscopic stress σwas obtained from the inter-particle forces using the procedure outlined in [31].The angle ψσwas calculated using Fig.1b,with σa =σ11and σr =(σ22+σ33)/2(note that the slight difference between σ22and σ33has occured due to the initial random inhomo-geneities of the sample).Only positive values of the angle ψ˙ were considered in this study.An increase in the magnitude of particle overlaps decreases the accuracy and the physical relevance of the simulations.This overlap magnitude is quantiﬁed in Fig.8,where the distribution of the relative overlap magnitude is plotted forseveral stress levels along the ψ˙ =0◦asymptotic stress path.be an approximate limit of accuracy for the present simu-lations.At10MPa the relative overlaps reach10%,which renders the results unreliable.At stresses lower than100kPa the relative overlaps are small,lower than0.5%.Figure8 also demonstrates that an increase in the stress level is asso-ciated with a dramatic increase in the number of inter-particle contacts.4DEM simulation results4.1Compression and constant volume asymptotic states First,we investigate the compression and constant volume asymptotic states.Figure9shows results on a sample loaded uniformly along the pathψ˙ =0◦,together with the results ofa test where the loading direction was reversed several times toψ˙ =180◦.The uniformly loaded sample clearly deﬁnes an isotropic normal compression line,which is approxi-mately linear in the ln p versus ln(1+e)plane.The sample with the unloading-reloading cycles shows that the isotropic asymptotic state on the initial conditions.The samples were ﬁrst loaded along theψ˙ =0◦path until p=1,000kPa, then unloaded along theψ˙ =180◦path until different pre-scribed mean stresses were reached,andﬁnally sheared along ψ˙ =90◦path up to the asymptotic state.The same proce-dure was adopted in all subsequent simulations on overcon-solidated(OC)samples.All the samples thus had a similar initial void ratio(the differences were only caused by the rel-atively high unloading modulus of the assembly),but differ-ent relative void ratios r e(different overconsolidation ratios OC R).For comparison,Fig.11also shows the results on nor-mally consolidated samples.The asymptotic states coincide exactly with the states reached after normal compression, demonstrating that the isochoric asymptotic state is indepen-dent of the initial stresses and void ratios.It is also interesting to point out that the maximum value ofψσreached in the tests increases with increasing overconsolidation ratio.In other words,the peak friction angle increases with OC R,while the critical state friction angle remains unchanged.This is consistent with predictions of constitutive models based on critical state soil mechanics.Asymptotic behaviour of granular materials767Fig.13Constantσr tests on normally consolidated and overconsoli-dated samples.a Sketch of the stress paths expected from the theory, (b–d)results of DEM simulations.bψσversus mean stress diagram,c normal compression lines in the ln(1+e)versus log p plane,dψσversus ln(1+e)diagramIn the next set of simulations,we consider a shear testin which the stress path(rather than the strain path)direc-tion is ly,experiments with a constantσr(denoted as drained triaxial tests in soil mechanics terminol-ogy)have been simulated.The same stretching rate D11asin theψ˙ =90◦tests was imposed,and D22=D33were controlled in such a way thatσr remained constant.After suf-ﬁciently long shearing,the specimens reached a state withconstantσand constant e,therefore with constantψσandwithψ˙ =90◦(see Fig.12).Tests on both normally consol-idated and overconsolidated samples were considered.The asymptotic states reached by the NC and OC samples coin-cided for the givenσr(Fig.12).They,however,dependended on the radial stress.The results are compared with results of ψ˙ =90◦in Fig.13.Theﬁnal states are close to the results of the tests with theψ˙ =90◦strain path,but a slight deviation was observed.Note that the stress path at low stress obliq-uities deviated from the ideal constantσr path,caused by an imperfection in theσr control in the software.In the larger strain range and at asymptotic states,however,the results were not affected.The next series of numerical experiments involves loading along compression proportional paths with0◦<ψ˙ <90◦(Fig.14).The test results for the given directionψ˙ converge towards a unique asymptotic state.Similarly to the isochoric asymptotic state,ψσis not a constant depending primarily on ψ˙ ,but in addition it depends on the current mean stress level. In agreement with the framework from Sect.2,increasingψ˙ leads to a continuous increase of asymptoticψσ(for a given mean stress).In the volumetric space(Fig.14c),increasing ψ˙ leads to a continuous increase of the asymptotic OC R. The asymptotic value of OC R for the givenψ˙ depends on the void ratio e.4.2Extension asymptotic statesTo track the extension asymptotic states,the same proce-dure adopted in the case of compression states is used with110◦.For higher angles,asymptotic state would have been reached at extremely low stresses(below0.5kPa),where the results were scattered and unreliable.We thus could not conﬁrm the existence of the limit state denoted as±d in Fig.4a.4.3Summary of the simulation resultsFigure16attempts to summarise the asymptotic states obtained in the described discrete element simulations.While plotting Fig.16it was neccesary to descide whether the experiments are thus insufﬁcient to track the overall depen-dency of the asymptotic stress ratio on the mean stress(on void ratio).In these cases,portions of the test paths from Figs.14and15,which were reached by a number of tests with different consolidation stresses,were plotted in Fig.16. Since the same state was reached by different tests,these states are considered to properly represent the asymptotic response.Results from Fig.16allowed us to plot the shapes of the ASBS cross-sections for constant void ratio,which are shown in Fig.17in the plane of q versus p normalised by theof it.Figure18shows the normalised asymptotic state bound-ary surface at the void ratio ln(1+e)=0.58,plotted togetherwith results of constant volume(ψ˙ =90◦)tests at dif-ferent overconsolidation ly,the results of nor-mally consolidated test to p=500kPa and overconsolidatedtest unloaded to1kPa are plotted.Specimens in these twoexperiments had ln(1+e)very close to0.58(see Fig.11c).Figure18a shows that both the stress paths are bound bythe ASBS,indicating that the ASBS may well be consid-ered as an approximation of the state boundary surface.Figure18b,however,demonstrates that theψ˙ =90◦path does not follow the ASBSs in the course of loading.This is tion isﬁxed,whereas the direction ofψ˙ varies.In Fig.19, stress and void ratio states with speciﬁcψ˙ were extracted along constantσr test paths and compared with the asymp-totic behaviour obtained in constantψ˙ experiments.In gen-eral,reasonable agreement between the two data sets is obtained for the higher void ratios.In the lower void ratio range,more signiﬁcant deviations occur,particularly in the range of lowerψσvalues.The deviations may,however,be caused by non-negligible particle overlaps occuring at high stresses(Fig.8).Agreement between the two data sets would imply that at the ASBS the strain rate direction is indepen-dent of the stress rate direction.This assumption is implicit in elasto-plastic constitutive models.This was investigatedstates of an assemblyDEM simulations.compression lines in thediagramusing the DEM simulations by Tamagnini et al.[60],whodid not conﬁrm it.It is particularly interesting to concentrate on theψ˙ =95◦andψ˙ =100◦state occurring near the peak of the con-stantσr test on heavily overconsolidated soil(unloaded from 1,000kPa to2and5kPa),shown in Fig.19.This state plots close to the extension asymptotic state obtained byψ˙ =95◦andψ˙ =100◦stretching,respectively,thus supporting the structure of the critical state models,which consider that the peak states in drained triaxial test and extension asymptotic states coincide.5Discussion of the inﬂuence of the model characteristics 5.1Inter-particle contact parametersFirst,it has been veriﬁed that the actual contact parameter values do not qualitatively inﬂuence the results of the sim-ulations.Special attention has been given to the inﬂuence of the coefﬁcient of friction;Fig.20shows the response of theψ˙ =0◦test(a)andψ˙ =90◦test(b).Qualitatively similar results to those with frictional contact were obtainedgranular material and to the inter-particle friction.Even a soil withϕ=0Rad has a certain frictional resistance,which fur-ther increases with an increase in the contact friction angle (Fig.20b).5.2Integration step sizeOne of the possible approaches to estimate critical time-step size ensuring stable numerical integration is based on the characteristic duration of contact(response time)t c[14,34, 35,47,49,63].For undamped oscilator,it readst c=πω(3)Fig.21Dependency of the global friction coefﬁcientμ∗on the inertialnumber I for various values of restitution coefﬁcients and inter-particlefriction coefﬁcients(da Cruz et al.[13])whereω=√k n/m12is the eigenfrequency of the contact,k n is contact normal stiffness and m12=m1m2/(m1+m2)is the reduced mass of the two particles in contact.Inthe present case,t c for the contact of smallest particles ist c(min)=1.5×10−6s,and t c for the contact of largest par-ticles is t c(max)=7.4×10−6s.The time-step size adoptedin the present simulations wasΔt=1.2×10−7s,whichmeansΔt=t c(min)/12.4=t c(max)/61.9.To verify thetime-step size was sufﬁciently small,selected simulations0.0010.010.110.11 10 100 1000 10000I n e r t i a l n u m b e r [-]p [kPa]d=0.2 mm d=1 mmFig.22The dependency of the inertial number I on mean stress p in the present simulations,calculated for the smallest and largest particleswere repeated with ﬁve-times smaller Δt ;the results were identical.5.3Stretching rateThe results of discrete element simulations are in the dynamic regime inﬂuenced by the rate of shearing.Simulations of the simple plane shear tests by different authors [13,18,26,50]revealed that the dependency of the global coefﬁcient of friction μ∗on shear rate can be evaluated by means of a non-dimmenisonal variable I ,called inertial number [13,18].It is deﬁned asI =˙γd√P /ρs (4)where ˙γis the shear rate,d is particle diameter and P is normal stress.For low values of the inertial number (approx-imately I <10−2),the simulations are quasi-static and the observed friction coefﬁcient does not depend on the shear rate;for larger values of I ,however,the global coeﬁcient of friction μ∗increases with increasing I [13,18,26,50].This is demonstrated in Fig.21by Cruz et al.[13].Generalisation of the inertial number for general loading has been proposed by Jop et al.[29]I = D d2ρsp (5)where p is mean stress.Figure 22shows the values of I calculated for the smallest (d =0.2mm)and largest (d =1mm)particles within the system simulated in this paper.Because the simulations were performed at a constant stretching rate D ,the inertial number depends on the stress level and increases with decreasing mean stress.In the lower stress range,the value of I is well above the limitting value I =10−2.The dependency of I on mean stress is likely the cause of the observed dependency of the asymptotic ψσon mean stress level.Detailed investigation of this issue is out-side the scope of the present paper and will be addressed in the future work.6ConclusionsIn the paper,we investigated the existence of asymptotic states of granular assemblies and their characteristics.The concept was ﬁrst introduced;asymptotic stress ratio,reached after sufﬁciently long stretching with a constant strain rate direction,is considered in soil mechanics to be a constant.We further introduced extension asymptotic states,and the notion of normal extension lines.It was pointed out that the asymp-totic states were inherent in the critical state soil mechanics based constitutive models,although the existence of exten-sion states has not been investigated in detail in the past.We pointed out that many authors indicated that the asymptotic behaviour is a consequence of grain crushing.An extensive DEM study was undertaken.Although the particles were elastic (non-crushable),the asymptotic behaviour was clearly observed.Asymptotic behaviour thus appeared to be an inherent property of the granular assem-bly,caused primarily by particle rearrangement.The simula-tions indicated the existence of extension asymptotic states and normal extension lines.We could not,however,inves-tigate the behaviour up to limiting values of ψ˙ (±d );before reaching the asymptotic state,extremely low values of mean stresses were attained and then the results were scattered and unreliable.The asymptotic states were further compared with states along constant σr (drained)test paths.It was observed that the peak state in the drained test coincided with the extension asymptotic state.More generally,we could con-ﬁrm the primary property of the critical state models,which assumes coincidence of the state boundary and asymptotic state boundary surfaces.The asymptotic value of ψσfor the given ψ˙ was not constant,but depended on the mean stress.This fact was argued to be caused by the rate effects expressed in terms of the inertial number I .Acknowledgments The author would like to thank to Prof.Gerd Gudehus for valuable discussions on the subject,to Dr.Václav Šmi-lauer for the introduction to the discrete element software Yade and to an anonymous journal reviewer for his valuable comments on the man-uscript.Financial support by the research grants GACR P105/12/1705and TACR TA01031840is greatly appreciated.References1.Alonso-Marroquín,F.,Luding,S.,Herrmann,H.J.,Vardoulakis,I.:Role of anisotropy in the elastoplastic response of a polygonal packing.Phys.Rev.E 71,051304(2005)2.Ben-Nun,O.,Einav,I.,Tordesillas,A.:Force attractor in conﬁned comminution of granular materials.Phys.Rev.Lett.104,108,001–1/4(2010)3.Butterﬁeld,R.:A natural compression law for soils.Géotechnique 29(4),469–480(1979)4.Casagrande,A.:Characteristics of cohesionless soils affecting the stability of slopes and earth ﬁlls.J.Boston Soc.Civil Eng.23(Jan),257–276(1936)。

A Discontinuity Test of Endogeneity

A Discontinuity Test of Endogeneity∗
Maria Carolina Caetano
UC Berkeley, Department er 2009 (Last update: March 12, 2010)
PRELIMINARY AND INCOMPLETE, COMMENTS WELCOME
Abstract This paper develops a nonparametric test of endogeneity without the need of instrumental variables. The test ensues from the novel observation that the potentially endogenous variable x is often of a nature such that the distribution of the unobservable q conditional on x and covariates z is discontinuous in x at a known value in its range. This relationship arises, for example, when x is subject to corner solutions, default contracts, social norms or law imposed restrictions, and may be argued using both economic theory and empirical evidence. If also x has a continuous eﬀect on the dependent variable y , any discontinuity of y that is not accounted by the discontinuities in the covariates z is evidence that q and y are dependent conditional on z , i.e. it is evidence of the endogeneity of x. The analysis develops the test statistics and derives the asymptotic distribution for three versions of the test: linear, partially linear (nonparametric only on x but not on covariates) and non-parametric. Finally, the partially linear version of the test is applied to the estimation of the eﬀect of maternal smoking on birth weight and on the probability of low birth weight (LBW). For the most detailed speciﬁcation in the literature (Almond, Chay, and Lee (2005)), the test ﬁnds strong evidence of endogeneity in the case of birth weight, and very weak evidence in the case of the probability of LBW.

异质环境中西尼罗河病毒稳态问题解的存在唯一性

异质环境中西尼罗河病毒稳态问题解的存在唯一性葛静【摘要】利用最大模原理、上下解方法以及乘乘减积技巧探讨了异质环境中一类西尼罗河病毒稳态问题解的存在唯一性.【期刊名称】《淮阴师范学院学报（自然科学版）》【年(卷),期】2017(016)004【总页数】5页(P283-287)【关键词】西尼罗河病毒;异质环境;稳态解;基本再生数【作者】葛静【作者单位】淮阴师范学院数学科学学院,江苏淮安 223300【正文语种】中文【中图分类】O1750 引言考虑到蚊子和鸟都有空间扩散的特点,Lin等[1]研究了如下均质环境下西尼罗河病毒的自由边界问题(1)其中Vi表示染病蚊子的数量,Hi表示染病鸟类的数量,rv表示染病蚊子的新增率,βv 表示病毒从蚊子传染到鸟类的接触传播率,βh表示病毒从鸟类传染到蚊子的接触传播率,Dh,Dv分别表示鸟类和蚊子的扩散率,γh,dh分别表示鸟类的恢复率和自然死亡率,0<q=1描述病毒在蚊子中间的垂直传播率,分别表示蚊子和鸟类的总数,这里设为常数.早期传染病研究通常假设环境是均质的, 所研究的模型多数是仅与时间有关的常微分系统[2-3], 随着研究的不断深入人们渐渐意识到空间扩散和环境异质性对疾病传播有重要的影响[4-6]. 相对于蚊子而言, 鸟类的扩散受环境影响较为显著,仅仅考虑恢复率γh以及死亡率dh和空间x相关. 而且由于蚊子的活动范围相对鸟的活动范围要小得多,可以忽略不计, 本文主要考虑如下西尼罗河病毒模型:(2)首先,由最大模原理知Vi(x,t),Hi(x,t)对都为正的, 其中Tmax为问题(2)解的最大存在区间.再一次应用最大模原理得Vi(x,t),Hi(x,t)在上有界. 因此由经典抛物方程理论可知Tma x=∞, 即问题(2)存在唯一的全局古典解(Vi,Hi) .1 稳态问题解的存在唯一性考虑问题(2)相应的稳态问题(3)接下来,求其平衡解, 由式(3)第中的第1个方程得再将上式代入式(3)的第2个方程得-DhΔHi=βv--(γh(x)+dh(x))Hi=于是将方程简化为(4)由变分法可得模型(2)的基本再生数容易证明如下性质, 参见文[2,3,7].引理1 1-R0与λ1具有相同的符号,其中λ1为如下椭圆问题的主特征值这里ψ(x)>0, x∈Ω为λ1相应的特征函数.由上述R0的定义以及[4]中的引理可得如下结论.2 结论定理1 关于R0下列性质成立.(a) R0为正数, 且关于Dh是单调递减的;(b) 当Dh→0时,(c) 当Dh→∞时, R0→;(d) 存在使得当时, 有R0>1;而当时, 有R0<1.下面将依据阈值R0讨论问题(2)的稳态解的存在唯一性.定理2 如下论断成立.(i) 当R0≤1时, 问题(2)存在唯一的无病平衡点(ii) 当R0>1时, 问题(2)存在唯一的染病平衡点).证明考虑椭圆问题(4)易知问题(4)总存在一个平凡解, 即在上下面证明当R0≤1时,问题(4)的非负平衡点只有DFE. 事实上, 假设问题(4)存在另一个非负非平凡平衡点,则由强极值原理可知在Ω内都为正, 故从而(-DhΔ )<(-(γh(x)+dh(x))()2,对上述不等式两边在x∈Ω上积分得(-DhΔ )dx<-(γh(x)+dh(x))()2dx,而不等式右边故Dh| |2dx<-(γh(x)+dh(x))()2dx,所以有>1,这与条件R0≤1矛盾, 故非负平衡点只有DFE.下面证(ii), 当R0>1, 由引理1可知1-R0<0, 则如下特征问题总是存在唯一的正解ψ(x)(至多相差一个常数乘子)且‖ψ‖L∞(Ω)=1, λ1为问题(4)的主特征值.取则易验证是问题(4)的上解.取由上面的特征问题可知要保证εψ是问题(4)的下解, 只需即只要亦即ε≤,因为‖ψ‖L∞(Ω)=1, 故取从而与为问题(4)的一对有序上下解, 以此为初值构造迭代序列, 从而得到问题(4)在中的最大解, 最小解. 因此椭圆问题(4)至少存在1个正解, 将此正解记为取则为问题(2)的1个染病平衡点.最后用乘乘减积技巧证明染病平衡点的唯一性.设为问题(4)的2个不同的正解, 即在内成立, 运用极大值原理有可以选择ε充分小, 使在上成立. 令分别为问题(4)对应的最小解、最大解. 因为所以且由最大值原理可知在上成立.记g(Hi)=-(γh(x)+dh(x))-Hi,易知g′(Hi)=<0.在等式两边同乘得到(5)在等式两边同乘得到(6)式(5)～(6)得(7)且在Ω上对式(7)两端分别积分有又因为且g关于Hi严格递减, 所以从而得矛盾. 故问题(4)存在唯一的正解, 记为(x). 参考文献：[1] Lin Z G, Zhu H P. Spatial spreading model and dynamics of West Nilevirus in birds and mosquitoes with free boundary[J].J Math Biol,2017,75(6-7):1381-1409.[2] Jiang J F, Qiu Z P, Wu J H, et al. Threshold conditions for West Nile virus outbreaks[J].Bull Math Biol,2009,71(3),627-647.[3] Fan G H, Liu J L, Driessche P vanden, et al. The impact of maturation delay of mosquitoes on the transmission of West Nile virus[J].Math Biosci,2010,228(2):119-126.[4] Allen L J S, Bolker B M, Lou Y, et al. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model[J].Discrete Contin,Dyn Syst Ser A,2008,21(1):1-20.[5] Ge J, Kim K I, Lin Z G, et al. A SIS reaction-diffusion-advection model ina low-risk and high-risk domain[J].J Differential Equations,2015,259(10):5486-5509.[6] 林支桂. 张丹，宋燕，等.一类具有常数输入及饱和传染率的传染病模型[J].渤海大学学报(自然科学版)，2013，34(1)：8-15.[7] 林支桂. 数学生态学导引[M].北京: 科学出版社, 2013.。

Organizational Behavior_中国大学mooc课后章节答案期末考试题库2023年

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公共管理英语作文

Public management,as a field of study and practice,is essential for the efficient functioning of government entities and the delivery of public services.Here are some key points to consider when writing an essay on public management in English:1.Introduction to Public Management:Begin by defining public management and explaining its significance in the context of modern governance.Highlight the role of public management in ensuring accountability,transparency,and efficiency in public administration.2.Historical Context:Provide a brief overview of the evolution of public management, from traditional bureaucratic models to the adoption of new public management NPM principles that emphasize marketoriented reforms and performance measurement.3.Key Principles of Public Management:Discuss the fundamental principles that guide public management practices,such as efficiency,effectiveness,equity,and responsiveness to citizens needs.4.Public Management Theories:Explore various theories that influence public management,including the principalagent theory,transaction cost theory,and the public choice theory.Explain how these theories contribute to the understanding of public sector behavior and decisionmaking.5.Public Management Practices:Describe common practices in public management,such as performance budgeting,strategic planning,and the use of performance indicators. Discuss the challenges and benefits of implementing these practices.6.Stakeholder Engagement:Emphasize the importance of engaging with various stakeholders,including citizens,interest groups,and other government agencies,in the public management process.Explain how this engagement can lead to better policy outcomes and increased public trust.7.Ethical Considerations:Address the ethical dimensions of public management, including issues of corruption,conflicts of interest,and the need for ethical leadership in the public sector.parative Perspectives:Compare public management approaches across different countries or regions,highlighting the cultural,political,and economic factors that influence the effectiveness of public management systems.9.Case Studies:Include case studies to illustrate the application of public managementprinciples and practices in realworld scenarios.Analyze the successes and failures of these cases to draw lessons for future public management efforts.10.Challenges and Future Directions:Discuss current challenges facing public management,such as the impact of digital technology,globalization,and demographic changes.Consider potential future directions for the field,including the role of innovation,collaboration,and sustainability.11.Conclusion:Summarize the main points of your essay and reiterate the importance of effective public management for the wellbeing of society.Offer a final thought on the potential for public management to adapt and evolve in response to changing societal needs and expectations.12.References:Ensure that your essay includes a list of references to support your arguments and provide further reading for interested readers.When writing your essay,use clear and concise language,provide evidence to support your arguments,and engage with the topic critically.Remember to proofread your work for grammar,spelling,and punctuation errors to ensure a polished final product.。

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CS aij : aij
I (| x j xi |) N
MT and aij : aij
I (| x j xi |)
I (| x
k 1
N
,
(0.3)
k
xi |)
here, I (| x j xi |) ( j i) originally quantifies the interaction of agent j on the alignment of agent i and
80
Volume 6, Number 3, July 2016
International Journal of Applied Physics and Mathematics
N dxi aij ( x j xi ), dt j 1, j i
i, j
(0.1)
with initial conditions
I (r ) is decreasing on [0, ) such that I (0) 1 and lim I ( r ) 0 .
r
The classical model (1.1) shows that agents are assumed to be equal in terms of their inﬂuence on each other, but in reality, there always exists leader-follower relationship in aggregation (see, e.g., [2], [14]). In a corporation, for example, the boss plays a vital role in efficient teamwork for his decision-making exerts much more inﬂuence on his employees than that of employees exerted on themselves. Once the leader makes up his mind, a consensus opinion will always be achieved eventually. Therefore, we also consider an opinion dynamical system with a leader and assume that the N-th agent is the leader, then we can depict leader-follower relationship with a mathematical model as follows
1. Introduction
The consensus problems as one of the forms of self-organized systems have been widely observed from Physics [1], Chemistry [2], Biology [3]-[5], Computer science [6], [7], Human society, Psychology and Education [8]-[11] (in the emergence of language and spectator violence), Management science [12] and so on. Opinion consensus systems as one kind of consensus problems have attracted more and more researchers to study. There has been a lot of literature which focused on a natural question that how to form an opinion consensus by the interaction between individuals in the organization? Baum and Katz [13] considered the convergence rates in the law of large numbers and Couzin et al. [14] studied the effective leadership and decision making in animal groups on the move, but they did not model the consensus phenomenon. A natural problem is that is a system more quickly to reach an opinion consensus with a leader than that without a leader? What attracts us most is under which circumstances the given simple rule can lead agents’ opinions to a steady state, that is, whether agents’ opinions can reach a consensus by the proposed algorithm. Such problem carries critical applications for decision-making in social systems or ﬂocking behaviors of animal population. In this paper, inspired by [5], [8], [9], [10], [15], we study the celebrated opinion consensus problem which reﬂects opinion compromise of a certain event by different agents and give a simple proof to guarantee that an opinion dynamical system can reach a consensus. Moreover, we ﬁnd that the system with a leader reaches a consensus more slowly than that without a leader by simulations. For the purpose of the paper, we first give the mathematical formulation of the opinion dynamical system. Let = {1, 2, …, N} denote a group of N agents and xi denote the i-th personal opinion(or position). Then the classical opinion consensus problem can be written as follows in continuous time ( t )(see, e.g., [5], [6]),
College of Science, National University of Defense Technology, Changsha, Hunan, P. R. China. * Corresponding author. Email: wxiao_98@ Manuscript submitted April 1, 2016; accepted June 12, 2016. doi: 10.17706/ijapm.2016.6.3.80-87 Abstract: Consensus problems of an opinion dynamical system without a leader and with a leader under CS and MT inﬂuence functions are studied and a simple proof to guarantee that an opinion dynamical system can reach a consensus is also obtained in this paper. Moreover, results show that the system with a leader reaches a consensus more slowly than that without a leader by numerical simulations. Key words: Dini derivative, Lyapunov function, opinion consensus.
dxN dt dx i dt
0,
j 1, j i

N 1
(0.4)
aij ( x j (t ) xi (t )) aiN ( xN (t ) xi (t )),
where i 1, 2,..., N 1 and β > 0 expresses the impacts on every agent exerted by the leader N . In this paper, “consensus” refers to general phenomena where each agent's opinion reaches a same value ﬁnally. We will study opinion dynamics of the system (1.1) and (1.4). For the purpose, ﬁrstly, we will introduce the concept of reaching consensus. Definition 1.1. The system (1.1) or (1.4) is said to reach a consensus, if for any initial value there finally exists only one constant (opinion) c such that