Invariant and perceptually consistent texture mapping for content-based image retrieval
Gauge-Gravity Dualities, Dipoles and New Non-Kahler Manifolds
1. Introduction and Summary It is by now clear that the usual way to deal with flux compactifications is to replace the Calabi-Yau SU (n) holonomy condition with an SU (n) or SU (n−p) structure condition, with p being a specific integer [1], [2], [3]. This requires, for example, the existence of two globally defined spinors on the six-dimensional manifold which are everywhere parallel for the SU (3) structures and nowhere parallel for the SU (2) structure (here p = 1). These spinors appear in the supersymmetry transformations of the gravitino and dilatino fields and as a result the supersymmetry conditions drastically restrict the fluxes and the geometry. The restrictions on the geometry are seen in terms of the torsion classes which measure the departure from the Calabi-Yau condition. The SU (3) structure is much more tractable as there are only five torsion classes whereas for the SU (2) structure the torsion decomposes into ninety classes which makes the classification a formidable task [4]. An alternative route by which we retain the properties of SU (n) structures and their consequent classification in terms of torsion classes, yet do not explicitly consider the supersymmetry transformation, is to follow a U-duality map that appears directly from superstring compactifications. The beauty of this approach is that it gives solutions that are explicitly supersymmetric, satisfy the required equations of motion and fall under the classification of torsion classes for SU (n) structure, all in one smooth map. Such an approach was first elucidated in [5], and was later followed in various other works, for 1
希尔伯特的23个数学问题
BULLETIN(New Series)OF THEAMERICAN MATHEMATICAL SOCIETYVolume37,Number4,Pages407–436S0273-0979(00)00881-8Article electronically published on June26,2000MATHEMATICAL PROBLEMSDAVID HILBERTLecture delivered before the International Congress of Mathematicians at Paris in1900.Who of us would not be glad to lift the veil behind which the future lies hidden;to cast a glance at the next advances of our science and at the secrets of its development during future centuries?What particular goals will there be toward which the leading mathematical spirits of coming generations will strive?What new methods and new facts in the wide and richfield of mathematical thought will the new centuries disclose?History teaches the continuity of the development of science.We know that every age has its own problems,which the following age either solves or casts aside as profitless and replaces by new ones.If we would obtain an idea of the probable development of mathematical knowledge in the immediate future,we must let the unsettled questions pass before our minds and look over the problems which the science of to-day sets and whose solution we expect from the future.To such a review of problems the present day,lying at the meeting of the centuries,seems to me well adapted.For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.The deep significance of certain problems for the advance of mathematical science in general and the important rˆo le which they play in the work of the individual investigator are not to be denied.As long as a branch of science offers an abundance of problems,so long is it alive;a lack of problems foreshadows extinction or the cessation of independent development.Just as every human undertaking pursues certain objects,so also mathematical research requires its problems.It is by the solution of problems that the investigator tests the temper of his steel;hefinds new methods and new outlooks,and gains a wider and freer horizon.It is difficult and often impossible to judge the value of a problem correctly in advance;for thefinal award depends upon the grain which science obtains from the problem.Nevertheless we can ask whether there are general criteria which mark a good mathematical problem.An old French mathematician said:“A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to thefirst man whom you meet on the street.”This clearness and ease of comprehension,here insisted on for a mathematical theory,I should still more demand for a mathematical problem if it is to be perfect;for what is clear and easily comprehended attracts,the complicated repels us.Moreover a mathematical problem should be difficult in order to entice us,yet not completely inaccessible,lest it mock at our efforts.It should be to us a guide408DA VID HILBERTpost on the mazy paths to hidden truths,and ultimately a reminder of our pleasure in the successful solution.The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal.They knew the value of difficult problems.I remind you only of the“problem of the line of quickest descent,”proposed by John Bernoulli.Experience teaches,explains Bernoulli in the public announcement of this problem,that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems,and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne,Pascal, Fermat,Viviani and others and laying before the distinguished analysts of his time a problem by which,as a touchstone,they may test the value of their methods and measure their strength.The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.Fermat had asserted,as is well known,that the diophantine equationx n+y n=z n(x,y and z integers)is unsolvable—except in certain self-evident cases.The attempt to prove this impossibility offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science.For Kummer,incited by Fermat’s problem,was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circularfield into ideal prime factors—a law which to-day in its generalization to any algebraicfield by Dedekind and Kronecker,stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.To speak of a very different region of research,I remind you of the problem of three bodies.The fruitful methods and the far-reaching principles which Poincar´e has brought into celestial mechanics and which are to-day recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason,belonging to the region of abstract number theory,the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.But it often happens also that the same special problemfinds application in the most unlike branches of mathematical knowledge.So,for example,the problem of the shortest line plays a chief and historically important part in the foundations of geometry,in the theory of curved lines and surfaces,in mechanics and in the calculus of variations.And how convincingly has F.Klein,in his work on the icosahedron,pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry,in group theory,in the theory of equations and in that of linear differential equations.In order to throw light on the importance of certain problems,I may also refer to Weierstrass,who spoke of it as his happy fortune that he found at the outset of his scientific career a problem so important as Jacobi’s problem of inversion on which to work.MATHEMATICAL PROBLEMS409 Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely thefirst and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena.Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization,just as the child of to-day learns the application of these laws by empirical methods.The same is true of thefirst problems of geometry, the problems bequeathed us by antiquity,such as the duplication of the cube, the squaring of the circle;also the oldest problems in the theory of the solution of numerical equations,in the theory of curves and the differential and integral calculus,in the calculus of variations,the theory of Fourier series and the theory of potential—to say noting of the further abundance of problems properly belonging to mechanics,astronomy and physics.But,in the further development of a branch of mathematics,the human mind, encouraged by the success of its solutions,becomes conscious of its independence. It evolves from itself alone,often without appreciable influence from without,by means of logical combination,generalization,specialization,by separating and col-lecting ideas in fortunate ways,new and fruitful problems,and appears then itself as the real questioner.Thus arose the problem of prime numbers and the other problems of number theory,Galois’s theory of equations,the theory of algebraic invariants,the theory of abelian and automorphic functions;indeed almost all the nicer questions of modern arithmetic and function theory arise in this way.In the meantime,while the creative power of pure reason is at work,the outer world again comes into play,forces upon us new questions from actual experience, opens up new branches of mathematics,and while we seek to conquer these new fields of knowledge for the realm of pure thought,we oftenfind the answers to old unsolved problems and thus at the same time advance most successfully the old theories.And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions,methods and ideas of the various branches of his science,have their origin in this ever-recurring interplay between thought and experience.It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem.I should sayfirst of all,this:that it shall be possible to establish the correctness of the solution by means of afinite number of steps based upon afinite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated.This requirement of logical deduction by means of afinite number of processes is sim-ply the requirement of rigor in reasoning.Indeed the requirement of rigor,which has become proverbial in mathematics,corresponds to a universal philosophical necessity of our understanding;and,on the other hand,only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full effect.A new problem,especially when it comes from the world of outer experience,is like a young twig,which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem,the established achievements of our mathematical science.Besides it is an error to believe that rigor in the proof is the enemy of simplic-ity.On the contrary wefind it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended.The410DA VID HILBERTvery effort for rigor forces us tofind out simpler methods of proof.It also fre-quently leads the way to methods which are more capable of development than the old methods of less rigor.Thus the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of the more rigor-ous function-theoretical methods and the consistent introduction of transcendental devices.Further,the proof that the power series permits the application of the four elementary arithmetical operations a well as the term by term differentiation and integration,and the recognition of the utility of the power series depending upon this proof contributed materially to the simplification of all analysis,particularly of the theory of elimination and the theory of differential equations,and also of the existence proofs demanded in those theories.But the most striking example for my statement is the calculus of variations.The treatment of thefirst and second variations of definite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations.By the examples of the simple and double integral I will show briefly,at the close of my lecture,how this way leads at once to a surprising simplification of the calculus of variations.For in the demonstration of the necessary and sufficient criteria for the occurrence of a maximum and minimum,the calculation of the sec-ond variation and in part,indeed,the wearisome reasoning connected with thefirst variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the differential coefficients of the function vary but slightly.While insisting on rigor in the proof as a requirement for a perfect solution of a problem,I should like,on the other hand,to oppose the opinion that only the concepts of analysis,or even those of arithmetic alone,are susceptible of a fully rigorous treatment.This opinion,occasionally advocated by eminent men,I con-sider entirely erroneous.Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry,mechanics and physics,to a stoppage of theflow of new material from the outside world,and finally,indeed,as a last consequence,to the rejection of the ideas of the continuum and of the irrational number.But what an important nerve,vital to mathematical science,would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever,from the side of the theory of knowledge or in geometry,or from the theories of natural or physical science,mathematical ideas come up,the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms,that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.To new concepts correspond,necessarily,new signs.These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts.So the geometricalfigures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians.Who does not always use along with the double inequality a>b>c the picture of three points following one another on a straight line as the geometrical picture of the idea “between”?Who does not make use of drawings of segments and rectangles enclosed in one another,when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation?Who could dispense with thefigure of the triangle,the circle with its center,or with the crossMATHEMATICAL PROBLEMS411 of three perpendicular axes?Or who would give up the representation of the vector field,or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry,in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?The arithmetical symbols are written diagrams and the geometricalfigures are graphic formulas;and no mathematician could spare these graphic formulas,any more than in calculation the insertion and removal of parentheses or the use of other analytical signs.The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie thosefigures;and in order that these geometricalfigures may be incorporated in the general treasure of mathematical signs,there is necessary a rigorous axiomatic investigation of their conceptual content.Just as in adding two numbers,one must place the digits under each other in the right order,so that only the rules of calculation,i.e.,the axioms of arithmetic,determine the correct use of the digits,so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations.The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical,any more than in geometrical discussions.On the contrary we ap-ply,especially infirst attacking a problem,a rapid,unconscious,not absolutely sure combination,trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols,which we could dispense with as little in arithmetic as with the geometrical imagination in geometry.As an example of an arithmetical theory operating rigorously with geometrical ideas and signs,I may mention Minkowski’s work,Die Geometrie der Zahlen.1Some remarks upon the difficulties which mathematical problems may offer,and the means of surmounting them,may be in place here.If we do not succeed in solving a mathematical problem,the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. Afterfinding this standpoint,not only is this problem frequently more accessible to our investigation,but at the same time we come into possession of a method which is applicable also to related problems.The introduction of complex paths of integration by Cauchy and of the notion of the ideals in number theory by Kummer may serve as examples.This way forfinding general methods is certainly the most practicable and the most certain;for he who seeks for methods without having a definite problem in mind seeks for the most part in vain.In dealing with mathematical problems,specialization plays,as I believe,a still more important part than generalization.Perhaps in most cases where we seek in vain the answer to a question,the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved.All depends,then,onfinding out these easier problems,and on solving them by means of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important leers for overcoming mathematical difficulties and it seems to me that it is used almost always,though perhaps unconsciously.412DA VID HILBERTOccasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense,and for this reason do not succeed.The problem then arises:to show the impossibility of the solution under the given hypotheses,or in the sense contemplated.Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational.In later mathematics,the question as to the impossibility of certain solutions plays a pre¨e minent part,and we perceive in this way that old and difficult problems,such as the proof of the axiom of parallels,the squaring of the circle,or the solution of equations of thefifth degree by radicals havefinally found fully satisfactory and rigorous solutions,although in another sense than that originally intended.It is probably this important fact along with other philosophical reasons that gives rise to the conviction(which every mathematician shares,but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement,either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts.Take any definite unsolved problem,such as the question as to the irrationality of the Euler-Mascheroni constant C,or the existence of an infinite number of prime numbers of the form2n+1.However unapproachable these problems may seem to us and however helpless we stand before them,we have,nevertheless,thefirm conviction that their solution must follow by afinite number of purely logical processes.Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone,or is it possibly a general law inherent in the nature of the mind,that all questions which it asks must be answerable?For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility.I instance the problem of perpetual motion.After seeking in vain for the construction of a perpetual motion machine,the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible;2and this inverted question led to the discovery of the law of the conservation of energy,which,again,explained the impossibility of perpetual motion in the sense originally intended.This conviction of the solvability of every mathematical problem is a powerful incentive to the worker.We hear within us the perpetual call:There is the problem. Seek its solution.You canfind it by pure reason,for in mathematics there is no ignorabimus.The supply of problems in mathematics is inexhaustible,and as soon as one problem is solved numerous others come forth in its place.Permit me in the fol-lowing,tentatively as it were,to mention particular definite problems,drawn from various branches of mathematics,from the discussion of which an advancement of science may be expected.Let us look at the principles of analysis and geometry.The most suggestive and notable achievements of the last century in thisfield are,as it seems to me,the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor,and the discovery of non-euclidean geometry by Gauss,Bolyai,MATHEMATICAL PROBLEMS413 and Lobachevsky.I thereforefirst direct your attention to some problems belonging to thesefields.1.Cantor’s problem of the cardinal number of the continuumTwo systems,i.e.,two assemblages of ordinary real numbers or points,are said to be(according to Cantor)equivalent or of equal cardinal number,if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one definite number of the other.The inves-tigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless,in spite of the most strenuous efforts,no one has succeeded in proving.This is the theorem:Every system of infinitely many real numbers,i.e.,every assemblage of numbers (or points),is either equivalent to the assemblage of natural integers,1,2,3,...or to the assemblage of all real numbers and therefore to the continuum,that is,to the points of a line;as regards equivalence there are,therefore,only two assemblages of numbers,the countable assemblage and the continuum.From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage;the proof of this theorem would,therefore,form a new bridge between the countable assemblage and the continuum.Let me mention another very remarkable statement of Cantor’s which stands in the closest connection with the theorem mentioned and which,perhaps,offers the key to its proof.Any system of real numbers is said to be ordered,if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that,if a is before b and b is before c,then a always comes before c.The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger.But there are,as is easily seen,infinitely many other ways in which the numbers of a system may be arranged.If we think of a definite arrangement of numbers and select from them a particular system of these numbers,a so-called partial system or assemblage,this partial system will also prove to be ordered.Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way,that not only in the assemblage itself but also in every partial assemblage there exists afirst number.The system of integers1,2,3,...in their natural order is evidently a well ordered assemblage.On the other hand the system of all real numbers,i.e.,the continuum in its natural order,is evidently not well ordered.For,if we think of the points of a segment of a straight line,with its initial point excluded,as our partial assemblage,it will have nofirst element.The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have afirst element,i.e., whether the continuum cannot be considered as a well ordered assemblage—a ques-tion which Cantor thinks must be answered in the affirmative.It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor’s, perhaps by actually giving an arrangement of numbers such that in every partial system afirst number can be pointed out.414DA VID HILBERT2.The compatibility of the arithmetical axiomsWhen we are engaged in investigating the foundations of a science,we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science.The axioms so set up are at the same time the definitions of those elementary ideas;and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of afinite number of logical steps.Upon closer consideration the question arises:Whether,in any way,certain statements of single axioms depend upon one another,and whether the axioms may not therefore contain certain parts in common,which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms:To prove that they are not contradictory,that is,that afinite number of logical steps based upon them can never lead to contradictory results.In geometry,the proof of the compatibility of the axioms can be effected by constructing a suitablefield of numbers,such that analogous relations between the numbers of thisfield correspond to the geometrical axioms.Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of thisfield of numbers.In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms.The axioms of arithmetic are essentially nothing else than the known rules of calculation,with the addition of the axiom of continuity.I recently collected them3and in so doing replaced the axiom of continuity by two simpler axioms,namely,the well-known axiom of Archimedes,and a new axiom essentially as follows:that numbers form a system of things which is capable of no further extension,as long as all the other axioms hold(axiom of completeness).I am convinced that it must be possible tofind a direct proof for the compatibility of the arithmetical axioms,by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.To show the significance of the problem from another point of view,I add the following observation:If contradictory attributes be assigned to a concept,I say, that mathematically the concept does not exist.So,for example,a real number whose square is−1does not exist mathematically.But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of afinite number of logical processes,I say that the mathematical existence of the concept(for example,of a number or a function which satisfies certain conditions)is thereby proved.In the case before us,where we are concerned with the axioms of real numbers in arithmetic,the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum.Indeed,when the proof for the compatibility of the axioms shall be fully accomplished,the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless.The totality of real numbers,i.e., the continuum according to the point of view just indicated,is not the totality of。
tpo32三篇托福阅读TOEFL原文译文题目答案译文背景知识
tpo32三篇托福阅读TOEFL原文译文题目答案译文背景知识阅读-1 (2)原文 (2)译文 (5)题目 (7)答案 (16)背景知识 (16)阅读-2 (25)原文 (25)译文 (28)题目 (31)答案 (40)背景知识 (41)阅读-3 (49)原文 (49)译文 (53)题目 (55)答案 (63)背景知识 (64)阅读-1原文Plant Colonization①Colonization is one way in which plants can change the ecology of a site.Colonization is a process with two components:invasion and survival.The rate at which a site is colonized by plants depends on both the rate at which individual organisms(seeds,spores,immature or mature individuals)arrive at the site and their success at becoming established and surviving.Success in colonization depends to a great extent on there being a site available for colonization–a safe site where disturbance by fire or by cutting down of trees has either removed competing species or reduced levels of competition and other negative interactions to a level at which the invading species can become established.For a given rate of invasion,colonization of a moist,fertile site is likely to be much more rapid than that of a dry, infertile site because of poor survival on the latter.A fertile,plowed field is rapidly invaded by a large variety of weeds,whereas a neighboring construction site from which the soil has been compacted or removed to expose a coarse,infertile parent material may remain virtually free of vegetation for many months or even years despite receiving the same input of seeds as the plowed field.②Both the rate of invasion and the rate of extinction vary greatly among different plant species.Pioneer species-those that occur only in the earliest stages of colonization-tend to have high rates of invasion because they produce very large numbers of reproductive propagules(seeds,spores,and so on)and because they have an efficient means of dispersal(normally,wind).③If colonizers produce short-lived reproductive propagules,they must produce very large numbers unless they have an efficient means of dispersal to suitable new habitats.Many plants depend on wind for dispersal and produce abundant quantities of small,relatively short-lived seeds to compensate for the fact that wind is not always a reliable means If reaching the appropriate type of habitat.Alternative strategies have evolved in some plants,such as those that produce fewer but larger seeds that are dispersed to suitable sites by birds or small mammals or those that produce long-lived seeds.Many forest plants seem to exhibit the latter adaptation,and viable seeds of pioneer species can be found in large numbers on some forest floors. For example,as many as1,125viable seeds per square meter were found in a100-year-old Douglas fir/western hemlock forest in coastal British Columbia.Nearly all the seeds that had germinated from this seed bank were from pioneer species.The rapid colonization of such sites after disturbance is undoubtedly in part a reflection of the largeseed band on the forest floor.④An adaptation that is well developed in colonizing species is a high degree of variation in germination(the beginning of a seed’s growth). Seeds of a given species exhibit a wide range of germination dates, increasing the probability that at least some of the seeds will germinate during a period of favorable environmental conditions.This is particularly important for species that colonize an environment where there is no existing vegetation to ameliorate climatic extremes and in which there may be great climatic diversity.⑤Species succession in plant communities,i.e.,the temporal sequence of appearance and disappearance of species is dependent on events occurring at different stages in the life history of a species. Variation in rates of invasion and growth plays an important role in determining patterns of succession,especially secondary succession. The species that are first to colonize a site are those that produce abundant seed that is distributed successfully to new sites.Such species generally grow rapidly and quickly dominate new sites, excluding other species with lower invasion and growth rates.The first community that occupies a disturbed area therefore may be composed of specie with the highest rate of invasion,whereas the community of the subsequent stage may consist of plants with similar survival ratesbut lower invasion rates.译文植物定居①定居是植物改变一个地点生态环境的一种方式。
f(R) Theories Of Gravity
a r X i v :0805.1726v 2 [g r -q c ] 24 M a y 2008f (R )theories of gravityThomas P.SotiriouCenter for Fundamental Physics,University of Maryland,College Park,MD 20742-4111,USA ∗Valerio FaraoniPhysics Department,Bishop’s University,2600College St.,Sherbrooke,Qu`e bec,Canada J1M 1Z7†Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics,cosmology and astrophysics.Among numerous alternatives to Einstein’s theory of gravity,theories which include higher order curvature invariants,and specifically the particular class of f (R )theories,have a long history.In the last five years there has been a new stimulus for their study,leading to a number of interesting results.We review here f (R )theories of gravity in an attempt to comprehensively present their most important aspects and cover the largest possible portion of the relevant literature.All known formalisms are presented —metric,Palatini and metric-affine —and the following topics are discussed:motivation;actions,field equations and theoretical aspects;equivalence with other theories;cosmological aspects and constraints;viability criteria;astrophysical applications.ContentsI.Introduction1A.Historical1B.Contemporary Motivation 2C.f (R )theories as toy theories 3II.Actions and field equations4A.Metric formalism 5B.Palatini formalism6C.Metric-affine formalism81.Preliminaries 92.Field Equations10III.Equivalence with Brans–Dicke theory andclassification of theories 11A.Metric formalism 12B.Palatini formalism 13C.Classification13D.Why f (R )gravity then?14IV.Cosmological evolution and constraints15A.Background evolution151.Metric f (R )gravity 152.Palatini f (R )gravity 17B.Cosmological eras18C.Dynamics of cosmological perturbations19V.Other standard viability criteria 20A.Weak-field limit 201.The scalar degree of freedom 202.Weak-field limit in the metric formalism 223.Weak-field limit in the Palatini formalism 25B.Stability issues 271.Ricci stability in the metric formalism 272.Gauge-invariant stability of de Sitter space in themetric formalism 293.Ricci stability in the Palatini formalism 304.Ghost fields 30C.The Cauchy problem 312and,consequently,thefield equations with no apparent theoretical or experimental motivation is not very ap-pealing.However,the motivation was soon to come. Beginning in the1960’s,there appeared indications that complicating the gravitational action might indeed have its merits.GR is not renormalizable and,therefore, can not be conventionally quantized.In1962,Utiyama and De Witt showed that renormalization at one-loop de-mands that the Einstein–Hilbert action be supplemented by higher order curvature terms(Utiyama and DeWitt, 1962).Later on,Stelle showed that higher order ac-tions are indeed renormalizable(but not unitary)(Stelle, 1977).More recent results show that when quantum corrections or string theory are taken into account, the effective low energy gravitational action admits higher order curvature invariants(Birrell and Davies, 1982;Buchbinder et al.,1992;Vilkovisky,1992).Such considerations stimulated the interest of the scientific community in higher-order theories of gravity, i.e.,modifications of the Einstein–Hilbert action in order to include higher-order curvature invariants with respect to the Ricci scalar[see(Schmidt,2007)for a historical review and a list of references to early work].However, the relevance of such terms in the action was considered to be restricted to very strong gravity regimes and they were expected to be strongly suppressed by small couplings,as one would expect when simple effective field theory considerations are taken into account. Therefore,corrections to GR were considered to be important only at scales close to the Planck scale and, consequently,in the early universe or near black hole sin-gularities—and indeed there are relevant studies,such as the well-known curvature-driven inflation scenario (Starobinsky,1980)and attempts to avoid cosmo-logical and black hole singularities(Brandenberger, 1992,1993,1995;Brandenberger et al.,1993; Mukhanov and Brandenberger,1992;Shahid-Saless, 1990;Trodden et al.,1993).However,it was not expected that such corrections could affect the gravita-tional phenomenology at low energies,and consequently large scales such as,for instance,the late universe.B.Contemporary MotivationMore recently,new evidence coming from astrophysics and cosmology has revealed a quite unexpected picture of the universe.Our latest datasets coming from different sources,such as the Cosmic Microwave Background Radi-ation(CMBR)and supernovae surveys,seem to indicate that the energy budget of the universe is the following: 4%ordinary baryonic matter,20%dark matter and76% dark energy(Astier et al.,2006;Eisenstein et al.,2005; Riess et al.,2004;Spergel et al.,2007).The term dark matter refers to an unkown form of matter,which has the clustering properties of ordinary matter but has not yet been detected in the laboratory.The term dark en-ergy is reserved for an unknown form of energy which not only has not been detected directly,but also does not cluster as ordinary matter does.More rigorously, one could use the various energy conditions(Wald,1984) to distinguish dark matter and dark energy:Ordinary matter and dark matter satisfy the Strong Energy Condi-tion,whereas Dark Energy does not.Additionally,dark energy seems to resemble in high detail a cosmological constant.Due to its dominance over matter(ordinary and dark)at present times,the expansion of the universe seems to be an accelerated one,contrary to past expec-tations.1Note that this late time speed-up comes to be added to an early time accelerated epoch as predicted by the infla-tionary paradigm(Guth,1981;Kolb and Turner,1992; Linde,1990).The inflationary epoch is needed to ad-dress the so-called horizon,flatness and monopole prob-lems(Kolb and Turner,1992;Linde,1990;Misner,1968; Weinberg,1972),as well as to provide the mechanism that generates primordial inhomogeneities acting as seeds for the formation of large scale structures(Mukhanov, 2003).Recall also that,in between these two periods of acceleration,there should be a period of decelerated expansion,so that the more conventional cosmological eras of radiation domination and matter domination can take place.Indeed,there are stringent observational bounds on the abundances of light elements,such as deu-terium,helium and lithium,which require that Big Bang Nucleosynthesis(BBN),the production of nuclei other than hydrogen,takes place during radiation domination (Burles et al.,2001;Carroll and Kaplinghat,2002).On the other hand,a matter-dominated era is required for structure formation to take place.Puzzling observations do not stop here.Dark mat-ter does not only make its appearance in cosmological data but also in astrophysical observations.The“missing mass”question had already been posed in1933for galaxy clusters(Zwicky,1933)and in1959for individual galax-ies(Kahn and Woltjer,1959)and a satisfactoryfinal an-swer has been pending ever since(Bosma,1978;Ellis, 2002;Moore,2001;Persic et al.,1996;Rubin and Ford, 1970;Rubin et al.,1980).One,therefore,has to admit that our current picture of the evolution and the matter/energy content of the universe is at least surprising and definitely calls for an explanation.The simplest model which adequatelyfits the data creating this picture is the so called concordancea=−4πG3 model orΛCDM(Λ-Cold Dark Matter),supplemented bysome inflationary scenario,usually based on some scalarfield called inflaton.Besides not explaining the originof the inflaton or the nature of dark matter by itself,theΛCDM model is burdened with the well known cosmolog-ical constant problems(Carroll,2001a;Weinberg,1989):the magnitude problem,according to which the observedvalue of the cosmological constant is extravagantly smallto be attributed to the vacuum energy of matterfields,and the coincidence problem,which can be summed upin the question:since there is just an extremely shortperiod of time in the evolution of the universe in whichthe energy density of the cosmological constant is com-parable with that of matter,why is this happening todaythat we are present to observe it?These problems make theΛCDM model more of anempiricalfit to the data whose theoretical motivationcan be regarded as quite poor.Consequently,therehave been several attempts to either directly motivatethe presence of a cosmological constant or to proposedynamical alternatives to dark energy.Unfortunately,none of these attempts are problem-free.For instance,the so-called anthropic reasoning for the magnitude ofΛ(Barrow and Tipler,1986;Carter,1974),even whenplaced into thefirmer grounds through the idea of the“anthropic or string landscape”(Susskind,2003),stillmakes many physicists feel uncomfortable due to itsprobabilistic nature.On the other hand,simple sce-narios for dynamical dark energy,such as quintessence(Bahcall et al.,1999;Caldwell et al.,1998;Carroll,1998;Ostriker and Steinhardt,1995;Peebles and Ratra,1988;Ratra and Peebles,1988;Wang et al.,2000;Wetterich,1988)do not seem to be as well motivated theoreticallyas one would desire.2Another perspective towards resolving the issues de-scribed above,which might appear as more radical to some,is the following:gravity is by far the dominant in-teraction at cosmological scales and,therefore,it is the force governing the evolution of the universe.Could it be that our description of the gravitational interaction at the relevant scales is not sufficiently adequate and stands at the root of all or some of these problems?Should we con-sider modifying our theory of gravitation and if so,would this help in avoiding dark components and answering the cosmological and astrophysical riddles?It is rather pointless to argue whether such a perspec-tive would be better or worse than any of the other so-lutions already proposed.It is definitely a different way to address the same problems and,as long as these prob-lems do notfind a plausible,well accepted and simple,2κ d4x√2κ d4x√4 as a series expansion,i.e.,f(R)=...+α2R−2Λ+R+R2β3+...,(4)where theαi andβi coefficients have the appropriate dimensions,we see that the action includes a number of phenomenologically interesting terms.In brief,f(R) theories make excellent candidates for toy-theories—tools from which one gains some insight in such gravity mod-ifications.Second,there are serious reasons to believe that f(R)theories are unique among higher-order grav-ity theories,in the sense that they seem to be the only ones which can avoid the long known and fatal Ostro-gradski instability(Woodard,2007).The second question calling for an answer is related to a possible loophole that one may have already spotted in the motivation presented:How can high-energy modifi-cations of the gravitational action have anything to do with late-time cosmological phenomenology?Wouldn’t effectivefield theory considerations require that the co-efficients in eq.(4)be such,as to make any corrections to the standard Einstein–Hilbert term important only near the Planck scale?Conservatively thinking,the answer would be posi-tive.However,one also has to stress two other serious factors:first,there is a large ambiguity on how grav-ity really works at small scales or high energies.Indeed there are certain results already in the literature claiming that terms responsible for late time gravitational phe-nomenology might be predicted by some more funda-mental theory,such as string theory[see,for instance, (Nojiri and Odintsov,2003b)].On the other hand,one should not forget that the observationally measured value of the cosmological constant corresponds to some energy scale.Effectivefield theory or any other high-energy the-ory consideration has thus far failed to predict or explain it.Yet,it stands as an experimental fact and putting the number in the right context can be crucial in ex-plaining its value.Therefore,in any phenomenological approach,its seems inevitable that some parameter will appear to be unnaturally small atfirst(the mass of a scalar,a coefficient of some expansions,etc.according to the approach).The real question is whether this initial “unnaturalness”still has room to be explained.In other words,in all sincerity,the motivation for in-frared modifications of gravity in general and f(R)grav-ity in particular is,to some extent,hand-waving.How-ever,the importance of the issues leading to this motiva-tion and our inability tofind other,more straightforward and maybe better motivated,successful ways to address them combined with the significant room for specula-tion which our quantum gravity candidates leave,have triggered a,probably reasonable,increase of interest in modified gravity.To conclude,when all of the above is taken into ac-count,f(R)gravity should neither be over-nor under-estimated.It is an interesting and relatively simple alter-native to GR,from the study of which some useful con-clusions have been derived already.However,it is still a toy-theory,as already mentioned;an easy-to-handle deviation from Einstein’s theory mostly to be used in or-der to understand the principles and limitations of mod-ified gravity.Similar considerations apply to modifying gravity in general:we are probably far from concluding whether it is the answer to our problems at the moment. However,in some sense,such an approach is bound to be fruitful since,even if it only leads to the conclusion that GR is the only correct theory of gravitation,it will still have helped us to both understand GR better and secure our faith in it.II.ACTIONS AND FIELD EQUATIONSAs can be found in many textbooks—see,for ex-ample(Misner et al.,1973;Wald,1984)—there are actually two variational principles that one can apply to the Einstein–Hilbert action in order to derive Ein-stein’s equations:the standard metric variation and a less standard variation dubbed Palatini variation[even though it was Einstein and not Palatini who introduced it(Ferraris et al.,1982)].In the latter the metric and the connection are assumed to be independent variables and one varies the action with respect to both of them(we will see how this variation leads to Einstein’s equations shortly),under the important assumption that the mat-ter action does not depend on the connection.The choice of the variational principle is usually referred to as a for-malism,so one can use the terms metric(or second order) formalism and Palatini(orfirst order)formalism.How-ever,even though both variational principles lead to the samefield equation for an action whose Lagrangian is lin-ear in R,this is no longer true for a more general action. Therefore,it is intuitive that there will be two version of f(R)gravity,according to which variational principle or formalism is used.Indeed this is the case:f(R)gravity in the metric formalism is called metric f(R)gravity and f(R)gravity in the Palatini formalism is called Palatini f(R)gravity(Buchdahl,1970).Finally,there is actually even a third ver-sion of f(R)gravity:metric-affine f(R)gravity (Sotiriou and Liberati,2007a,b).This comes about if one uses the Palatini variation but abandons the assumption that the matter action is independent of the connection.Clearly,metric affine f(R)gravity is the most general of these theories and reduces to metric or Palatini f(R)gravity if further assumptions are made. In this section we will present the actions andfield equations of all three versions of f(R)gravity and point out their difference.We will also clarify the physical meaning behind the assumptions that discriminate them.For an introduction to metric f(R)gravity see also(Nojiri and Odintsov,2007a),for a shorter review of metric and Palatini f(R)gravity see (Capozziello and Francaviglia,2008)and for an ex-5 tensive analysis of all versions of f(R)gravity and otheralternative theories of gravity see(Sotiriou,2007b).A.Metric formalismBeginning from the action(3)and adding a matterterm S M,the total action for f(R)gravity takes the formS met=1−g f(R)+S M(gµν,ψ),(5)whereψcollectively denotes the matterfields.Variation with respect to the metric gives,after some manipula-tions and modulo surface termsf′(R)Rµν−1√δgµν,(7) a prime denotes differentiation with respect to the ar-gument,∇µis the covariant derivative associated withthe Levi-Civita connection of the metric,and ≡∇µ∇µ.Metric f(R)gravity wasfirst rigorously studied in(Buchdahl,1970).3It has to be stressed that there is a mathematical jump in deriving eq.(6)from the action(5)having to do with the surface terms that appear in the variation:as in the case of the Einstein–Hilbert action,the surface terms do not vanish just byfixing the metric on the boundary. For the Einstein–Hilbert action,however,these terms gather into a total variation of a quantity.Therefore, it is possible to add a total divergence to the action in order to“heal”it and arrive to a well-defined variational principle(this is the well known Gibbons–Hawking–York surface term(Gibbons and Hawking,1977;York,1972)). Unfortunately,the surface terms in the variation of the action(3)do not consist of a total variation of some quantity(the reader is urged to calculate the variation in order to verify this fact)and it is not possible to“heal”the action by just subtracting some surface term before performing the variation.The way out comes from the fact that the action in-cludes higher order derivatives of the metric and,there-fore,it should be possible tofix more degrees of freedom on the boundary than those of the metric itself.There is no unique prescription for such afixing in the literature so far.Note also that the choice offixing is not void of phys-ical meaning,since it will be relevant for the Hamiltonian formulation of the theory.However,thefield equations (6)would be unaffected by thefixing chosen and from a6 2006a).4Finally,let us note that it is possible to write thefieldequations in the form of Einstein equations with an ef-fective stress-energy tensor composed of curvature termsmoved to the right hand side.This approach is question-able in principle(the theory is not Einstein’s theory andit is artificial to force upon it an interpretation in termsof Einstein equations)but,in practice,it has been provedto be useful in scalar-tensor gravity.Specifically,eq.(6)can be written asGµν≡Rµν−1f′(R)+gµν[f(R)−Rf′(R)]f′(R)(10)orGµν=κκ f(R)−Rf′(R)4Energy-momentum complexes in the spherically symmetric case have been computed in(Multamaki et al.,2008).the independent connection.Note that the metric is not needed to obtain the latter from the former.For clarity of notation,we denote the Ricci tensor constructed with this independent connection as Rµνand the correspond-ing Ricci scalar5is R=gµνRµν.The action now takes the formS pal=1−g f(R)+S M(gµν,ψ).(13)GR will come about,as we will see shortly,when f(R)= R.Note that the matter action S M is assumed to de-pend only on the metric and the matterfields and not on the independent connection.This assumption is crucial for the derivation of Einstein’s equations from the linear version of the action(13)and is the main feature of the Palatini formalism.It has already been mentioned that this assumption has consequences for the physical meaning of the independent connection(Sotiriou,2006b,d;Sotiriou and Liberati, 2007b).Let us elaborate on this:recall that an affine connection usually defines parallel transport and the co-variant derivative.On the other hand,the matter action S M is supposed to be a generally covariant scalar which includes derivatives of the matterfields.Therefore,these derivatives ought to be covariant derivatives for a general matterfield.Exceptions exist,such as a scalarfield,for which a covariant and a partial derivative coincide,and the electromagneticfield,for which one can write a co-variant action without the use of the covariant derivative [it is the exterior derivative that is actually needed,see next section and(Sotiriou and Liberati,2007b)].How-ever,S M should include all possiblefields.Therefore, assuming that S M is independent of the connection can imply one of two things(Sotiriou,2006d):either we are restricting ourselves to specificfields,or we are implic-itly assuming that it is the Levi-Civita connection of the metric that actually defines parallel transport.Since the first option is implausibly limiting for a gravitational the-ory,we are left with the conclusion that the independent connectionΓλµνdoes not define parallel transport or the covariant derivative and the geometry is actually pseudo-Riemannian.The covariant derivative is actually defined by the Levi-Civita connection of the metric{λµν}.This also implies that Palatini f(R)gravity is a met-ric theory in the sense that it satisfies the metric pos-tulates(Will,1981).Let us clarify this:matter is mini-mally coupled to the metric and not coupled to any other fields.Once again,as in GR or metric f(R)gravity, one could use diffeomorphism invariance to show that the stress energy tensor is conserved by the covariant derivative defined with the Levi-Civita connection of the metric,i.e.,∇µTµν=0(but¯∇µTµν=0).This can7also be shown by using the field equations,which we will present shortly,in order to calculate the divergence of T µνwith respect to the Levi-Civita connection of the metric and show that it vanishes(Barracoet al.,1999;Koivisto,2006a).6Clearly then,Palatini f (R)gravityis ametrictheoryaccording to the definition of (Will,1981)(not to be confusedwiththeterm“metric”in“metricf(R)gravity”,which simply refers to the fact that one only varies the action with respect to the metric).Conven-tionally thinking,as a consequence of the covariant con-servation of the matter energy-momentum tensor,test particles should follow geodesics of the metric in Palatini f (R )gravity.This can be seen by considering a dust fluid with T µν=ρu µu νand projecting the conserva-tion equation ∇βT µβ=0onto the fluid four-velocity u β.Similarly,theories that satisfy the metric postulates are supposed to satisfy the Einstein Equivalence Principle as well (Will,1981).Unfortunately,things are more compli-cated here and,therefore,we set this issue aside for the moment.We will return to it and attempt to fully clarify it in Secs.VI.B and VI.C.2.For now,let us proceed with our discussion of the field equations.Varying the action (13)independently with respect to the metric and the connection,respectively,and using the formulaδR µν=¯∇λδΓλµν−¯∇νδΓλµλ.(14)yieldsf ′(R )R (µν)−1−gf ′(R )g µν+¯∇σ√−gf ′(R )g σµ=0,(17)which implies that we can bring the field equations intothe more economical formf ′(R )R (µν)−1−gf ′(R )g µν=0,(19)It is now easy to see how the Palatini formalism leads to GR when f (R )=R ;in this case f ′(R )=1and−h h µν=√7See,however,(Sotiriou,2007b)for further analysis of the f (R )action and how it can be derived from first principles in the two formalisms.8This calculation holds in four dimensions.When the num-ber of dimensions D is different from 4then,instead of us-ing eq.(22),the conformal metric h µνshould be introduced as h µν≡[f ′(R ]2/(D −2)g µνin order for eq.(23)to still hold.8Then,eq.(19)becomesthedefinitionofthe Levi-Civita connectionofhµνandcan besolved algebraicallyto giveΓλµν=hλσ(∂µhνσ+∂νhµσ−∂σhµν),(24)or,equivalently,in terms of gµν,Γλµν=121f′(R) ∇µ∇ν−12(f′(R))2(∇µf′(R))(∇µf′(R))+3f′Tµν−1f′ (28)+1212gµν(∇f′)2 .Notice that,assuming that we know the root of eq.(20),R=R(T),we have completely eliminated the indepen-dent connection from this equation.Therefore,we havesuccessfully reduced the number offield equations to oneand at the same time both sides of eq.(28)depend onlyon the metric and the matterfields.In a sense,the the-ory has been brought to the form of GR with a modifiedsource.We can now straightforwardly deduce the following:•When f(R)=R,the theory reduces to GR,asdiscussed previously.•For matterfields with T=0,due to eq.(21),R andconsequently f(R)and f′(R)are constants and thetheory reduces to GR with a cosmological constantand a modified coupling constant G/f′.If we de-note the value of R when T=0as R0,then thevalue of the cosmological constant is1f′(R0) =R02κ d4x√9Note that,apart from special cases such as a perfectfluid,Tµνand consequently T already includefirst derivatives of the matterfields,given that the matter action has such a dependence.Thisimplies that the right hand side of eq.(28)will include at leastsecond derivatives of the matterfields,and possibly up to thirdderivatives.9 1.PreliminariesBefore going further and derivingfield equations fromthis action certain issues need to be clarified.First,sincenow the matter action depends on the connection,weshould define a quantity representing the variation of S Mwith respect to the connection,which mimics the defi-nition of the stress-energy tensor.We call this quantitythe hypermomentum and is defined as(Hehl and Kerling,1978)∆µνλ≡−2−gδS M4Qνµν,(33)and the Cartan torsion tensorSλµν≡Γλ[µν],(34) which is the antisymmetric part of the connection.By allowing a non-vanishing Cartan torsion tensor we are allowing the theory to naturally include torsion.Even though this brings complications,it has been considered by some to be an advantage for a gravity theory since some matterfields,such as Diracfields,can be cou-pled to it in a way which might be considered more natural(Hehl et al.,1995):one might expect that at some intermediate or high energy regime,the spin of particles might interact with the geometry(in the same sense that macroscopic angular momentum interacts with geometry)and torsion can naturally arise.Theories with torsion have a long history,probably starting with the Einstein–Cartan(–Sciama–Kibble)theory(Cartan, 1922,1923,1924;Hehl et al.,1976;Kibble,1961;Sciama, 1964).In this theory,as well as in other theories with an independent connection,some part of the connection is still related to the metric(e.g.,the non-metricity is set to zero).In our case,the connection is left completely unconstrained and is to be determined by thefield equa-tions.Metric-affine gravity with the linear version of the action(30)was initially proposed in(Hehl and Kerling, 1978)and the generalization to f(R)actions was consid-ered in(Sotiriou and Liberati,2007a,b).Unfortunately,leaving the connection completely un-constrained comes with a complication.Let us consider the projective transformationΓλµν→Γλµν+δλµξν,(35) whereξνis an arbitrary covariant vectorfield.One can easily show that the Ricci tensor will correspondingly transform likeRµν→Rµν−2∂[µξν].(36) However,given that the metric is symmetric,this implies that the curvature scalar does not changeR→R,(37) i.e.,R is invariant under projective transformations. Hence the Einstein–Hilbert action or any other action built from a function of R,such as the one used here, is projective invariant in metric-affine gravity.However, the matter action is not generically projective invariant and this would be the cause of an inconsistency in the field equations.One could try to avoid this problem by generalizing the gravitational action in order to break projective in-variance.This can be done in several ways,such as allowing for the metric to be non-symmetric as well, adding higher order curvature invariants or terms in-cluding the Cartan torsion tensor[see(Sotiriou,2007b; Sotiriou and Liberati,2007b)for a more detailed discus-sion].However,if one wants to stay within the framework of f(R)gravity,which is our subject here,then there is only one way to cure this problem:to somehow constrain the connection.In fact,it is evident from eq.(35)that, if the connection were symmetric,projective invariance would be broken.However,one does not have to take such a drastic measure.To understand this issue further,we should re-examine the meaning of projective invariance.This is very similar to gauge invariance in electromagnetism(EM).It tells us that the correspondingfield,in this case the connections Γλµν,can be determined from thefield equations up to a projective transformation[eq.(35)].Breaking this invari-ance can therefore come byfixing some degrees of free-dom of thefield,similarly to gaugefixing.The number of degrees of freedom which we need tofix is obviously the number of the components of the four-vector used for the transformation,i.e.,simply four.In practice,this means that we should start by assuming that the con-nection is not the most general which one can construct, but satisfies some constraints.Since the degrees of freedom that we need tofix are four and seem to be related to the non-symmetric part of the connection,the most obvious prescription is to demand that Sµ=Sσσµbe equal to zero,which wasfirst suggested in(Sandberg,1975)for a linear ac-tion and shown to work also for an f(R)action in。
Review of Matrix Theory
Review of Matrix Theory
D. Bigatti† and L. Susskind
Stanford University
Abstract
In this article we present a self contained review of the principles of Matrix Theory including the basics of light cone quantization, the formulation of 11 dimensional M-Theory in terms of supersymmetric quantum mechanics, the origin of membranes and the rules of compactification on 1,2 and 3 tori. We emphasize the unusual origins of space time and gravitation which are very different than in conventional approaches to quantum gravity. Finally we discuss application of Matrix Theory to the quantum mechanics of Schwarzschild black holes. This work is based on lectures given by the second author at the Cargese ASI 1997 and at the Institute for Advanced Study in Princeton.
consistency regularization 出处 -回复
consistency regularization 出处-回复Consistency Regularization: An Overview and ApplicationsIntroductionConsistency regularization has emerged as a powerful technique in machine learning, specifically in the field of deep learning. It aims to improve the generalization and robustness of models by encouraging consistency in their predictions. This regularization technique has found applications in various domains, including image classification, natural language processing, and speech recognition. In this article, we will provide an overview of consistency regularization, discuss its theoretical foundations, and explore its applications in different areas.Theoretical FoundationsConsistency regularization is rooted in the principle of encouraging smoothness and stability in model predictions. The underlying assumption is that small changes in the input should not significantly alter the output of a well-trained model. This principle is particularly relevant in scenarios where the training data maycontain noisy or ambiguous samples.One of the commonly used methods for achieving consistency regularization is known as consistency training. In this approach, two different input transformations are applied to the same sample, creating two augmented versions. The model is then trained to produce consistent predictions for the transformed samples. Intuitively, this process encourages the model to focus on the underlying patterns in the data rather than being influenced by specific input variations.Consistency regularization can be formulated using several loss functions. One popular choice is the mean squared error (MSE) loss, which measures the discrepancy between predictions of the original input and transformed versions. Other approaches include cross-entropy loss and Kullback-Leibler divergence.Applications in Image ClassificationConsistency regularization has yielded promising results in image classification tasks. One notable application is semi-supervised learning, where the goal is to leverage a small amount of labeleddata with a larger set of unlabeled data. By applying consistent predictions to both labeled and unlabeled data, models can effectively learn from the unlabeled data and improve their performance on the labeled data. This approach has been shown to outperform traditional supervised learning methods in scenarios with limited labeled samples.Additionally, consistency regularization has been explored in the context of adversarial attacks. Adversarial attacks attempt to fool a model by introducing subtle perturbations to the input data. By training models with consistent predictions for both original and perturbed inputs, their robustness against such attacks can be significantly improved.Applications in Natural Language ProcessingConsistency regularization has also demonstrated promising results in natural language processing (NLP) tasks. In NLP, models often face the challenge of understanding and generating coherent sentences. By applying consistency regularization, models can be trained to produce consistent predictions for different representations of the same text. This encourages the model tofocus on the meaning and semantics of the text rather than being influenced by superficial variations, such as different word order or sentence structure.Furthermore, consistency regularization can be used in machine translation tasks, where the goal is to translate text from one language to another. By enforcing consistency between translations of the same source text, models can generate more accurate and consistent translations.Applications in Speech RecognitionSpeech recognition is another domain where consistency regularization has found applications. One of the key challenges in speech recognition is handling variations in pronunciation and speaking styles. By training models with consistent predictions for different acoustic representations of the same speech utterance, models can better capture the underlying patterns and improve their accuracy in recognizing speech in different conditions. This can lead to more robust and reliable speech recognition systems in real-world scenarios.ConclusionConsistency regularization has emerged as an effective technique for improving the generalization and robustness of models in various machine learning tasks. By encouraging consistency in predictions, models can better learn the underlying patterns in the data and generalize well to unseen examples. This regularization technique has been successfully applied in image classification, natural language processing, and speech recognition tasks, among others. As research in consistency regularization continues to advance, we can expect further developments and applications in the future.。
英语阅读-推理判断题
目 录
• Example analysis of reasoning and judgment questions
• Practice and improvement of reasoning and judgment questions
Understanding Texts
Inference questions promote deeper understanding of texts by encouraging readers to deduce meanings beyond the explicit information provided. True/False questions ensure that students comprehend the text correctly.
True/False Questions
These questions ask whether a statement is true or false based on a given passage. They test the reader's comprehension of the text and their ability to identify accurate information.
03
Use common sense
Apply your own knowledge and common sense to evaluate the credibility of the options.
Excluding interference items
Neutrino Mixing and Flavour Changing Processes
2
2.1
Patterns of supersymmetry breaking
Yukawa couplings and scalar masses
ห้องสมุดไป่ตู้
We consider the supersymmetric standard model with right-handed neutrinos, which is described by the superpotential 1 W = µH1 H2 + heij Eic Lj H1 + hν ij Nic Lj H2 + hrij Nic Njc R 2 c H1 + huij Qi Ujc H2 . +hdij Qi Dj
DESY 99-169 December 1999
arXiv:hep-ph/9912317v1 13 Dec 1999
NEUTRINO MIXING AND FLAVOUR CHANGING PROCESSES
Wilfried Buchm¨ uller∗ , David Delepine† , Laksana Tri Handoko‡ Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
mass matrices. Sec. 3 deals with quark flavour changing processes, and in Sec. 4 flavour changing radiative transitions are discussed, leading to the conlusions in Sec. 5.
Abstract We study the implications of a large νµ -ντ mixing angle on flavour changing transitions of quarks and leptons in supersymmetric extensions of the standard model. Two patterns of supersymmetry breaking are considered, models with modular invariance and the standard scenario of universal soft breaking terms at the GUT scale. The analysis is performed for two symmetry groups G ⊗ U (1)F , with G = SU (5) and G = SU (3)3 , where U (1)F is a family symmetry. Models with modular invariance are in agreement with observations only for restricted scalar 2 quark and gaugino masses, M q /m2 g ˜ ≃ 7/9 and m˜ b > 350 GeV. A characteristic feature of models with large tan β and radiatively induced flavour mixing is a large branching ratio for µ → eγ . For both symmetry groups and for the considered range of supersymmetry breaking mass parameters we find BR(µ → eγ ) > 10−14 .
翻译原文
AMS Classification: Primary: 60K35, 82C20, 82C22 1. INTRODUCTION Faggionato and Martinelli(3,4) have studied the hydrodynamic limit for a dynamics on lattice gas which depends on a collection of real bounded i.i.d. random variables α = (αx : x ∈ Zd ) corresponding to some external quenched disorder field. The hydrodynamic limit was obtained for a monocolour particles, an important byproduct of their study being the equivalence of ensembles shown in Lemma A.4.(4) The aim of our work is to give an equivalence of ensembles result with sharp bounds for multicolour particles. Our proof uses the projection on the monocolour system, and then the equivalence of ensembles result of the monocolour lattice gas.(3) After, we study the variation of the canonical Gibbs measures of the multicolour problem with respect to the monocolour one, showing that it does not add new terms. This is done by elementary computations. We point out that the
基于图像正则化的抗几何变换的感知哈希算法
收稿日期：3作者简介：孙锐（6），男，浙江余姚人，副研究员，博士，主要研究方向为多媒体安全，图像处理与理解。
2010年工程图学学报2010第2期J OURNAL OF ENG INEERING GRAPHICSNo.2基于图像正则化的抗几何变换的感知哈希算法孙锐1，闫晓星2，丁志中1（1.合肥工业大学计算机与信息学院，安徽合肥230009；2.合肥工业大学光电技术研究院，安徽合肥230009）摘要：图像哈希在内容认证、数据库搜索和水印等领域有广泛的应用。
该文提出的新的抗几何变换的感知哈希方法包括三个主要阶段：第一阶段通过图像正则化过程获得一个对任意仿射变换具有不变性的正则图像；第二阶段对随机选择的多个子图像进行小波变换产生一个包括图像主要特征的副图像；第三阶段采用奇异值分解捕获图像的局部感知成分并生成最终哈希。
仿真实验表明算法有效抵抗了几何变换、压缩等感知保持操作，内容篡改也被正确检测。
批量实验也证明算法有较好的稳健性和抗误分类能力。
关键词：计算机应用；感知哈希；图像正则化；几何变换；图像认证中图分类号：TP 391.41文献标识码：A文章编号：1003-0158(2010)02-0116-07Robust Perceptual Hashing Algorithm to Geometric Distor tionsBased on Image NormalizationSUN Rui 1,YAN Xiao-xing 2,DING Zhi-zhong 1(1.School of Computer and Information,Hefei Universi ty of Technology,Hefei Anhui 230009,China;2.Academy of Optoelectronic Technology,Hefei Uni versity of Technology,Hefei Anhui 230009,China )Abstr act:Image hashing functions have extensive applications in content authentication,database search and watermarking,etc.A novel perceptual hashing algorithm is developed which is robust to geometric distortions.The algorithm includes three stages:the first stage obtains the image that is invariant to any affine transforms through a normalization procedure;the second stage applies discrete wavelet transform to pseudo-randomly select sub-images to obtain secondary image including main features;the third stage uses singular value decomposing to capture local perceptual features and to get the final hash.The experiments show the method withstands perceptually preserved manipulations including geometric distortions,compression,and common signal processing operations.The content changes are also can be detected accurately.Thus the proposed method achieves perceptual robustness while avoiding misclassification.K ey wor ds:computer application;perceptual Hashing;image normalization;geometric distortions;image authentication2008-10-1197-图像哈希是图像内容或特征的一种简洁表示，即使两幅感知相近的图像在数值上有不同的表示，感知哈希函数也使它们产生相同或相近的哈希序列，这是感知哈希技术与传统密码学中的哈希的显著区别。
ò á êúD Y× × óò ú úD ò ìêì
C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c aPNAProbability, Networks and AlgorithmsProbability, Networks and AlgorithmsAn image retrieval system based on adaptive waveletliftingP.J. Oonincx, P.M. de ZeeuwR EPORT PNA-R0208 M ARCH 31, 2002CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organization for Scientific Research (NWO).CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.Probability, Networks and Algorithms (PNA)Software Engineering (SEN)Modelling, Analysis and Simulation (MAS)Information Systems (INS)Copyright © 2001, Stichting Centrum voor Wiskunde en InformaticaP.O. Box 94079, 1090 GB Amsterdam (NL)Kruislaan 413, 1098 SJ Amsterdam (NL)Telephone +31 20 592 9333Telefax +31 20 592 4199ISSN 1386-3711CWIP.O.Box94079,1090GB Amsterdam,The NetherlandsPatrick.Oonincx,Paul.de.Zeeuw@cwi.nl1.I NTRODUCTIONContent-based image retrieval(CBIR)is a widely used term to indicate the process of retrieving desired images from a large collection on the basis of features.The extraction process should be automatic(i.e. no human interference)and the features used for retrieval can be either primitive(color,shape,texture) or semantic(involving identity and meaning).In this paper we confine ourselves to grayscale images of objects against a background of texture.This class of images occurs for example in various databases created for the combat of crime:stolen objects[21],tyre tracks and shoe sole impressions[1].In this report we restrict ourselves to the following problem.Given an image of an object(a so-called query)we want to identify all images in a database which contain the same object irrespective of translation,rotation or re-sizing of the object,lighting conditions and the background texture.One of the most classical approaches to the problem of recognition of similar images is by the use of moment invariants[11].This method is based on calculating moments in both the-and-direction of the image density function up to a certain order.Hu[11]has shown that certain homogeneous polynomials of these moments can be used as statistical quantities that attain the same values for images that are of the same class,i.e.,that can be obtained by transforming one single original image(affine transforms and scaling).However,this method uses the fact that such images consists of a crisp object against a neutral background.If the background contains‘information’(noise,environment in a picture)the background should be the same for all images in one class and should also be obtained from one background using the same transformations.In general this will not be the case.The kind of databases we consider in this paper consists of classes of different objects pasted on different background textures.To deal with the problem of different backgrounds one may use somefiltering process as a preprocessing step.In Do et al.[7]the wavelet transform modulus maxima is used as such preprocessing step.To measure the(dis)similarity between images,moments of the set of maxima points are determined(per scale)and subsequently Hu’s invariants are computed.Thus,each image is indexed by a vector in the wavelet maxima moment space.By its construction,this vector predominantly represents shapes.In this report we propose to bring in adaptivity by using different waveletfilters for smooth and unsmooth parts of the image.Thefilters are used in the context of the(redundant)lifting scheme[18].The degree2of”smoothness”is determined by measuring the relative local variance(RLV),which indicates whether locally an image behaves smoothly or not.Near edges low order predictionfilters are activated which lead to large lifting detail coefficients along thin curves.At backgrounds of texture high order predictionfilters are activated which lead to negligible detail coefficients.Moments and subsequently moment invariants are computed with respect to these wavelet detail coefficients.With the computation of the detail coefficients a certain preprocessing is required to make the method robust for shifts over a non-integer number of gridpoints.Further we introduce the homogeneity condition which means that we demand a homogeneous change in the elements of a feature vector if the image seen as a density distribution is multiplied by a scalar.The report is organized as follows.In Sections2and3we discuss the lifting scheme and its adaptive version.Section4is devoted to the topic of affine invariances of the coefficients obtained from the lifting scheme.In Section5the method of moment invariants is recapitulated.The homogeneity condition is in-troduced which leads to a normalization.Furthermore,the mathematical consequences for the computation of moments of functions represented byfields of wavelet(detail)coefficients are investigated.Section6 discusses various aspects of thefinal retrieval algorithm,including possible metrics.Numerical results of the algorithm for a synthetic database are presented in Section7.Finally,some conclusions are drawn in Section8.2.T HE L IFTING S CHEMEThe lifting scheme as introduced by Sweldens in1997,see[18],is a method for constructing wavelet transforms that are not necessarily based on dilates and translates of one function.In fact the construction does not rely on the Fourier transform which makes it also suitable for functions on irregular grids.The transform also allows a fully in-place calculation,which means that no auxiliary memory is needed for the computations.The idea of lifting is based on splitting a given set of data into two subsets.In the one-dimensional case this can mean that starting with a signal the even and odd samples are collected into two new signals,i.e.,,where and,for all.The next step of the lifting scheme is to predict the value of given the sequence.This prediction uses a prediction operator acting on.The predicted value is subtracted from yielding a ‘detail’signal.An update of the odd samples is needed to avoid aliassing problems.This update is performed by adding to the sequence,with the update operator.The lifting procedure can also be seen as a2-bandfilter bank.This idea has been depicted in Figure1.The inverse lifting scheme can/.-,()*+/.-,()*+/.-,()*+/.-,()*+/.-,()*+Figure1:The lifting scheme:splitting,predicting,updating.immediately be found by undoing the prediction and update operators.In practice,this comes down in Figure1to simply changing each into a and vice versa.Compared to the traditional wavelet transform the sequence can be regarded as detail coefficients of the signal.The updated sequence can be regarded as the approximation of at a coarse ing again as input for the lifting scheme yields detail and approximation signals at lower resolution levels.We observe that every discrete wavelet transform can also be decomposed into a finite sequence of lifting steps[6].To understand the notion of vanishing moments in terms of the prediction and update operators,we com-pare the lifting scheme with a two-channelfilter bank with analysisfilters(lowpass)and(highpass) and synthesisfilters and.Such afilter bank has been depicted in Figure2.Traditionally we say that a389:;?>=<89:;?>=</.-,()*+89:;?>=<89:;?>=<Figure2:Classical2-band analysis/synthesisfilter bank.filter bank has primal and vanishing moments ifandwhere denotes the space of all polynomial sequences of order.Given thefilter operators and, the correspondingfilters and can be computed by(2.1)(2.2)where and denote thefilter sequences of the operators and respectively.In[12]Kovacevic and Sweldens showed that we can always use lifting schemes with primal vanishing moments and dual vanishing moments by taking for a Nevillefilter of order with a shift and for half the adjoint of a Nevillefilter of order and shift,see[17].Example2.1We take and.With these operators we getThefilter bank has only one vanishing moment.The lifting transform corresponds in this example to the Haar wavelet transform.Example2.2For more vanishing moments,i.e.,smoother approximation signals,we takeThese Nevillefilters give rise to a2-channelfilter bank with2primal and4dual vanishing moments. The lifting scheme can also be used for higher dimensional signals.For these signals the lifting scheme consists of channels,where denotes the absolute value of the determinant of the dilation matrix,that is used in the corresponding discrete wavelet transform.In each channel the signal is translated along one of the coset representatives from the unit cell of the corresponding lattice,see[12]. The signal in thefirst channel is then used for predicting the data in all other channels by using possible different prediction operators.Thereafter thefirst channel is updated using update operators on the other channels.Let us consider an image as a two-dimensional signal.An important example of the lifting scheme applied to such a signal is one that involves channels().We subdivide the lattice on which the signal has been defined into two sets on quincunx grids,see Figure3.This division is also called ”checkerboard”or”red-black”division.The pixels on the red spots()are used to predict the samples on the black spots(),while updating of the red spots is performed by using the detailed data on the black spots.An example of a lifting transform with second order prediction and updatefilters is given by4Figure3:A rectangular grid composed of two quincunx grids.order200000040000060008Table1:Quincunx Nevillefilter coefficientsThe algorithm using the quincunx lattice is also known as the red-black wavelet transform by Uytterhoeven and Bultheel,see[20].In general can be written as(2.3) with a subset of mod and,a set of coefficients in. In this case a general formula for reads(2.4)with depending on the number of required primal vanishing moments.For several elements in the coefficients attain the same values.Therefore we take these elements together in subsets of, i.e.,(2.5)Table1indicates the values of all,for different values of(2through8)when using quincunx Nevillefilters,see[12],which are thefilters we use in our approach.We observe that and so a44tapsfilter is used as prediction/update if the requiredfilter order is8.For an illustration of the Nevillefilter of order see Figure4.Here the numbers,correspond to the values of thefilter coefficients as given in and respectively at that position.The left-handfilter can be used to transform a signal defined on a quincunx grid into a signal defined on a rectangular grid,the right-hand filter is the degrees rotated version of the left-handfilter and can be used to transform a signal from a rectangular grid towards a quincunx grid.We observe that the quincunx lattice yields a non separable2D-wavelet transform,which is also sym-metric in both horizontal and vertical direction.Furthermore,we only need one prediction and one update operator for this2D-lifting scheme,which reduces the number of computations.The prediction and update operators for the quincunx lattice do also appear in schemes for other lattices, like the standard2D-separable lattice and the hexagonal lattice[12].The algorithm for the quincunx lattice can be extended in a rather straightforward way for these two other well-known lattices.5111122222222111122222222Figure 4:Neville filter of order :rectangular (left)and quincunx (right)Figure 5illustrates the possibility of the use of more than channels in the lifting scheme.Herechannels are employed,using a four-colour division of the 2D-lattice.It involves (interchange-able)prediction steps.Each of the subsets with colours ,and respectively,is predicted by application of a prediction filter on the subset with colour .Figure 5:Separable grid (four-colour division).3.A DAPTIVE L IFTINGWhen using the lifting scheme or a classical wavelet approach,the prediction/update filters or wavelet/scaling functions are chosen in a fixed fashion.Generally they can be chosen in such way that a signal is approximated with very high accuracy using only a limited number of coefficients.Discontinuities mostly give rise to large detail coefficients which is undesirable for applications like compression.For our purpose large detail coefficients near edges in an images are desirable,since they can be identified with the shape of objects we want to detect.However,they are undesirable if such large coefficients are related to the background of the image.This situation occurs if a small filter is used on a background of texture that contains irregularities locally.In this case a large smoothing filter gives rise to small coefficients for the background.These considerations lead to the idea of using different prediction filters for different parts of the signal.The signal itself should indicate (for example by means of local behavior information)whether a high or low order prediction filter should be used.Such an approach is commonly referred to as an adaptive approach.Many of these adaptive approaches have been described already thoroughly in the literature,e.g.[3,4,8,13,19].In this paper we follow the approach proposed by Baraniuk et al.in [2],called the space-adaptive approach.This approach follows the scheme as shown in Figure 6.After splitting all pixels of a given image into two complementary groups and (red/black),thepixels inare used to predict the values in .This is done by means of a prediction filter acting on ,i.e.,.In the adaptive lifting case this prediction filter depends on local information of the image pixels .Choices for may vary from high to low order filters,depending on the regularity of the image locally.For the update operator,we choose the update filter that corresponds to the prediction filter with lowest order from all possible to be chosen .The order of the update filter should be lower or equal to the order of the prediction filter as a condition to provide a perfect reconstruction filter bank.As with the classical wavelet filter bank approach,the order of the prediction filter equals the number of dual vanishing6/.-,()*+/.-,()*+Figure6:Generating coefficients via adaptive liftingmoments while the order of the updatefilter equals the number of primal vanishing moments,see[12].The above leads us to use a second order Nevillefilter for the update step and an th order Nevillefilter for the prediction step,where.In our application the reconstruction part of the lifting scheme is not needed.In[2],Baraniuk et al.choose to start the lifting scheme with an update operator followed by an adaptively chosen prediction operator.The reason for interchanging the prediction and update operator is that this solves stability and synchronization problems in lossy coding applications.We will not discuss this topic in further detail,but only mention that they took for thefilters of and the branch of the Cohen-Daubechies-Feauveau(CDF)filter family[5].The order of the predictionfilter was chosen to be,,or,depending on the local behavior of the signal.Thefilter orders of the CDFfilters in their paper correspond to thefilter orders of the Nevillefilters we are using in our approach.Relative local variance We propose a measure on which the decision operator in the2D adaptive lifting scheme can be based on,namely the relative local variance of an image.This relative local variance(RLV) of an image is given byrlv var(3.1) with(3.2) For the window size we take,since with this choice all that are used for the prediction of contribute to the RLV for,even for the8th order Nevillefilter.When the RLV is used at higher resolution levels wefirst have to down sample the image appropriately.Thefirst time the predictionfilter is applied(to the upper left pixel)we use the8th order Nevillefil-ter on the quincunx lattice as given in Table1.For all other subsequent pixels to be predicted,we first compute rlv.Then quantizing the values of the RLV yields a decisionmap indicating which predictionfilter should be used at which positions.Values above the highest quantizing level induce a 2nd order Nevillefilter,while values below the lowest quantizing levels induce an8th order Nevillefil-ter.For the quantizing levels we take multiples of the mean of the RLV.Test results have shown that rlv rlv rlv are quantizing levels that yield a good performance in our application.In Figure7we have depicted an image(left)and its decision map based on the RLV(right).4.A FFINE I NVARIANT L IFTINGAlthough both traditional wavelet analysis and the lifting scheme yield detail and approximation coeffi-cients that are localised in scale and space,they are both not translation invariant.This means that if a signal or image is translated along the grid,its lifting coefficients may not be just be given by a translation of the original coefficients.Moreover,in general the coefficients will attain values in the same range of the original values(after translation),but they will be totally different.7a) original image b) decision map (RLV)Figure7:An object on a wooden background and its rel.local variance(decision map):white=8th order, black=2nd order.For studying lifting coefficients of images a desirable property would also be invariance under reflections and rotations.However,for these two transformations we have to assumefirst that the values of the image on the grid points is not affected a rotation or reflection.In practice,this means that we only consider reflections in the horizontal,the vertical and the diagonal axis and rotations over multiples of.4.1Redundant LiftingFor the classical wavelet transform a solution for translation invariance is given by the redundant wavelet transform[15],which is a non-decimated wavelet(at all scales)transform.This means that one gets rid of the decimation step.As a consequence the data in all subbands have the same size as the size as the input data of the transform.Furthermore,at each scaling level,we have to use zero padding to thefilters in order to keep the multiresolution analysis consistent.Not only more memory is used by the redundant transform,also the computing complexity of the fast transform increases.For the non-decimated transform computing complexity is instead of for the fast wavelet transform.Whether the described redundant transform is also invariant under reflections and rotations depends strongly on thefilters(wavelets)themselves.Symmetry of thefilters is necessary to guarantee certain rotation and reflection invariances.This is a condition that is not satisfied by many well-known wavelet filters.The redundant wavelet transform can also be translated into a redundant lifting scheme.In one dimension this works out as follows.Instead of partitioning a signal into and we copy to both and.The next step of the lifting scheme is to predict by(4.1) The predictionfilter is the samefilter as used for the non-redundant case,however now it depends on the resolution level,since at each level zero padding is applied to.This holds also for the updatefilters .So,the update step reads(4.2)For higher dimensional signals we copy the data in all channels of the usedfilter bank. Next the-channel lifting scheme is applied on the data,using zero padding for thefilters at each resolu-8................ ........Figure8:Tree structure of the-channel lifting scheme.tion level.Remark,that for each lifting step in the redundant-channel lifting scheme we have to store at each scaling level times as much data as in the non-redundant scheme,see Figure8.We observe that in our approach Nevillefilters on a quincunx lattice are used.Due to their symmetry properties,see Table1,the redundant scheme does not only guarantee translation invariance,but also invariance under rotations over multiples of and reflections in the horizontal,vertical and diagonal axis is assured.Invariance under other rotations and reflections can not be guaranteed by any prediction and updatefilter pair,since the quincunx lattice is not invariant under these transformations.4.2An Attempt to Avoid Redundancy:Fixed Point LiftingAs we have seen the redundant scheme provides a way offinding detail and approximation coefficients that are invariant under translations,reflections and rotations,under which the lattice is also invariant.Due to its redundancy this scheme is stable in the sense that it treats all samples of a given signal in the same way.However redundancy also means additional computational costs and perhaps even worse additional memory to store the coefficients.Therefore we started searching for alternative schemes that are also invariant under the described class of affine transformation.Although we did not yet manage to come up with an efficient stable scheme,we would like to stretch the principal idea behind the building blocks of such approach.In the sequel we will only use the redundant lifting scheme as described in the preceding section.Before we start looking for possible alternative schemes we examine why the lifting scheme is not translation invariant.Assume we have a signal that is analysed with an-band lifting scheme. Then after one lifting step we have approximation data and detail data.Whether one sample,is determined to become either a sample of or a sample ofdepends only on its position on the lattice and the way we partition the lattice into groups.Of course, this partitioning is rather arbitrary.The more channels we use the higher the probability is that for afixed partitioning one sample that was determined to be used for predicting other samples,will become a sample of after translating.Following Figure8it is clear that any sample,can end up after lifting steps in ways,either in approximation data at level or in detail data at some level.The idea of the alternative scheme we propose here is to partition a signal not upon its position on the lattice but upon its structure.This means that for each individual signal we indicate afixed point for which we demand that it will end up in the approximation data after lifting steps.If this point can be chosen independent of its coordinates on the lattice,the lifting scheme based on this partitioning will then translation invariant.For higher dimensional signals we can also achieve invariance under the other discussed affine transformations,however then we have tofix more points,depending on the number ofchannels.In our approach the quincunx lattice is used and thereforefixing one approximation sample on scales will immediatelyfix the partitioning of all other samples on the quincunx lattice at scale. As a result thefixed point lifting scheme is invariant under translations,rotations and reflections that leave the quincunx lattice invariant.In the sequel of this chapter we will only discuss the lifting scheme for for the quincunx lattice.Although the proposedfixed point lifting scheme may seem to be a powerful tool for affine invariant lifting,it will be hard to deal with in practice.The problem we will have to face is how to choose afixed point in every image.In other words we have tofind a suitable decision operator that adds to every a unique,itsfixed point,i.e.,If we demand to depend only on and not on the lattice(coordinate free)it will be hard tofind such that is well defined.This independence of the coordinates is necessary for rotation invariances.However, this is not the only difficulty we have to face.Stability of the scheme is an other problem.If for some reason afixed point has been wrongly indicated,for example due to truncation errors,the whole scheme might collapse down.Although we cannot easily solve the problem of determining incorrectfixed points we can increase the stability of the scheme by not imposing that at each scale should be an index number of the coarse scale data after zero padding.Instead of this procedure we rather determine afixed point for both the original signal()and for the coarse scale data(at each scale.Then we impose that should be used for prediction in the th lifting step,for and with.Furthermore,stability may be increased by using decision operators that generate a set offixed points.However,since no stable method (uniform decision operator)is available yet,we will use the redundant lifting scheme in our approach and do not work out the idea offixed point lifting here at this moment.5.M OMENT I NVARIANTSAt the outset of this section we give a brief introduction into the theory of statistical invariants for imag-ing purposes,based on centralized moments.Traditionally,these features have been widely used in pat-tern recognition applications to recognize the geometrical shapes of different objects[11].Here,we will compute invariants with respect to the detail coefficients as produced by the wavelet lifting schemes of Sections2–4.We use invariants based on moments of the coefficients up to third order.We show how to construct a feature vector from the obtained wavelet coefficients at several scales It is followed by proposals for normalization of the moments to keep them in comparable range.5.1Introduction and recapitulationWe regard an image as a density distribution function,the Schwartz class.In order to obtain translation invariant statistics of such we use central moments of for our features.The order central moment of is given by(5.1) with the center of massand(5.2)Computing the centers of mass and of yieldsand bining this with(5.1)showsi.e.,the central moments are translation invariant.We also require that the features should be invariant under orthogonal transformations(rotations).For deriving these features we follow[11]using a method with homogeneous polynomials of order.These are given by(5.3) Now assume that the variables are obtained from other variables under some linear transformation,i.e.,then is an algebraic invariant of weight if(5.4) with the new coefficients obtained after transforming by.For orthogonal transformations we have and therefore is invariant under rotations ifParticularly we have from[11],that if is an algebraic invariant,then also the moments of order have the same invariant,i.e.,(5.5) From this equation2functions of second order can be derived that are invariant under rotations,see[11]. For we have the invariantsandIt was also shown that these two functions are also invariant under reflections,which can be a useful property for identifying reflected images.Since the way of deriving these invariants may seem a bit technical and artificial,we illustrate with straightforward calculus that and are indeed invariant under rotations.The invariance under reflections is left to the reader,since showing this follows the same calculations.We consider the rotated distribution functionand the corresponding invariants and,which are and but now based on moments calculated from.So what we have to show is that and.It follows from(5.1)and(5.2)that if and only ifwith and.Obviously this holds true,considering the trigonometric rule .To do the same for we also have to introduce and.Because we have to take products of integrals that define,we cannot use and in both integrals.As for we can now derive from(5.1)and(5.2)that if and only ifWe simplify the right-hand side term by term.Thefirst term,that is related to becomes The second term(related to)becomesAdding these two terms gives uswhich demonstrates that indeed also is invariant under rotations.Similar calculus shows that invariance under reflections also holds.From Equation(5.5)four functions of third order and one function of both second and third order can be derived that are invariant under both rotations and reflecting,namelyandwithThe last polynomial that is invariant under both rotations and reflections consists of both second and third order moments and is given bywith and as above.To these six invariants we can add a seventh one,which is only invariant under rotations and changes sign under reflections.It is given bySince we want to include reflections as well in our set of invariant transformations we will use instead of in our approach.From now on,we will identify with.We observe that all possible linear combinations of these invariants are invariant under proper orthogonal transformations and translations.Therefore we can call these seven invariants also invariant generators.。
Independence and concurrent separation logic
Logical Methods in Computer ScienceVol.4(1:6)2008,pp.1–68Abstract.A compositional Petri net-based semantics is given to a simple language al-lowing pointer manipulation and parallelism.The model is then applied to give a notionof validity to the judgements made by concurrent separation logic that emphasizes theprocess-environment duality inherent in such rely-guarantee reasoning.Soundness of therules of concurrent separation logic with respect to this definition of validity is shown.Theindependence information retained by the Petri net model is then exploited to characterizethe independence of parallel processes enforced by the logic.This is shown to permit arefinement operation capable of changing the granularity of atomic actions.1.IntroductionThe foundational work of Hoare on parallel programming[Hoa72]identified the fact that attributing an interleaved semantics to parallel languages is problematic.Three areas of difficulty were isolated,quoted directly:•That of defining a‘unit of action’.•That of implementing the interleaving on genuinely parallel hardware.•That of designing programs to control the fantastic number of combinations involved in arbitrary interleaving.The significance of these problems increases with developments in hardware,such as multiple-core processors,that allow primitive machine actions to occur at the same time.As Hoare went on to explain,a feature of concurrent systems in the physical world is that they are often spatially separated,operating on completely different resources and not interacting.When this is so,the systems are independent of each other,and therefore it is unnecessary to consider how they interact.This perspective can be extended by regarding computer processes as spatially separated if they operate on different memory locations. The problems above are resolved if the occurrence of non-independent parallel actions is prohibited except in rare cases where atomicity may be assumed,as might be enforced using the constructs proposed in[Dij68,Bri72].2J.HAYMAN AND G.WINSKELIndependence models for concurrency allow semantics to be given to parallel languages in a way that can tackle the problems associated with an interleaved semantics.The common core of independence models is that they record when actions are independent,and that independent actions can be run in either order or even concurrently with no consequence on their effect.This mitigates the increase in the state space since unnecessary interleavings of independent actions need not be considered(see e.g.[CGMP99]for applications to model checking).Independence models also permit easier notions of refinement which allow the assumed atomicity of actions to be changed.It is surprising that,to our knowledge,there has been no comprehensive study of the semantics of programming languages inside an independence model.Thefirst component of our work gives such a semantics in terms of a well-known independence model,namely Petri nets.Our model isolates the specification of the controlflow of programs from their effect on the shared state.It indicates what appears to be a general method(an alternative to Plotkin’s structural operational semantics)for giving a structural Petri net semantics to a variety of languages—see the Conclusion,Section7.The language that we consider is motivated by the emergence of concurrent separation logic[O’H07],the rules of which form a partial correctness judgement about the execution of pointer-manipulating concurrent programs.Reasoning about such programs has tradi-tionally proved difficult due to the problem of variable aliasing.For instance,Owicki and Gries’system for proving properties of parallel programs that do not manipulate pointers [OG76]essentially requires that the programs operate on disjoint collections of variables, thereby allowing judgements to be composed.In the presence of pointers,the same syntac-tic condition cannot be imposed to yield a sound logic since distinct variables may point to the same memory location,thereby allowing arbitrary interaction between the processes. To give a specific example,Owicki and Gries’system would allow a judgement of the form {x→0∧y→0}x:=1 y:=2{x→1∧y→2},indicating that the result of assigning1to the program variable x concurrently with assign-ing2to y from a state where x and y both initially hold value0is a state where x holds value1and y holds value2.The judgement is sound because the variables x and y are distinct.If pointers are introduced to the language,however,it is not sound to conclude that{[x]→0∧[y]→0}[x]:=1 [y]:=2{[x]→1∧[y]→2},which would indicate that assigning1to the location pointed to by x and2to the location pointed to by y yields a state in which x points to a location holding1and y points to a location holding2,since x and y may both point to the same location.At the core of separation logic[Rey00,IO01],initially presented for non-concurrent programs,is the separating conjunction,ϕ∗ψ,which asserts that the state in which processes execute may be split into two parts,one part satisfyingϕand the otherψ.The separating conjunction was used by O’Hearn to adapt Owicki and Gries’system to provide a rule for parallel composition suitable for pointer-manipulating programs[O’H07].As we shall see,the rule for parallel composition is informally understood by splitting the initial state into two parts,one owned by thefirst process and the other by the second. Ownership can be seen as a dynamic constraint on the interference to be assumed:parallel processes always own disjoint sets of locations and only ever act on locations that they own. As processes evolve,ownership of locations may be transferred using a system of invariants (an example is presented in Section4).A consequence of this notion of ownership is thatINDEPENDENCE AND CONCURRENT SEPARATION LOGIC∗3 the rules discriminate between the parallel composition of processes and their interleavedexpansion.For example,the logic does not allow the judgement{ℓ→0}[ℓ]:=1 [ℓ]:=1{ℓ→1},which informally means that the effect of two processes acting in parallel which both assign the value1to the memory locationℓfrom a state in whichℓholds0is to yield a state in whichℓholds1.However,if we adopt the usual rule for the nondeterministic sum of processes,the corresponding judgement is derivable for their interleaved expansion,([ℓ]:=1;[ℓ]:=1)+([ℓ]:=1;[ℓ]:=1).One would hope that the distinction that the logic makes between concurrent processes and their interleaved expansion is captured by the semantics;the Petri net model that we give does so directly.The rules of concurrent separation logic contain a good deal of subtlety,and so lacked a completely formal account until the pioneering proof of their soundness due to Brookes [Bro07].The proof that Brookes gives is based on a form of interleaved trace semantics.The presence of pointers within the model alongside the possibility that ownership of locations is transferred means,however,that the way in which processes are separated is absolutely non-trivial,which motivates strongly the study of the language within an independence model.We therefore give a proof of soundness using our net model and then characterize entirely semantically the independence of concurrent processes in Theorem5.4.It should be emphasized that the model that we present is different from Brookes’since it provides an explicit account of the intuitions behind ownership presented by O’Hearn. It involves taking the original semantics of the process and embellishing it to capture the semantics of the logic.The proof technique that we employ defines validity of assertions in a way that captures the rely-guarantee reasoning[Jon83]emanating from ownership in separation logic directly,and in a way that might be applied in other situations.In[Rey04],Reynolds argues that the separation of parallel processes arising from the logic allows store actions that were assumed to be atomic,in fact,to be implemented as composite actions(seen as a change in their granularity)with no effect on the validity of the judgement.Independence models are suited to modeling situations where actions are not atomic,a perspective advocated by Lamport and Pratt[Pra86,Lam86].We introduce a novel form of refinement,inspired by that of[vGG89],and show how this may be applied to address the issue of granularity using our characterization of the independence of processes arising from the logic.2.Terms and statesConcurrent separation logic is a logic for programs that operate on a heap.A heap is a structure recording the values held by memory locations that allows the existence of pointers as well as providing primitives for the allocation and deallocation of memory locations.A heap can be seen as afinite partial function from a set of locations Loc to a set of values Val:Heap def=Loc⇀fin ValWe will useℓto range over elements of Loc and v to range over elements of Val.As stated, a heap location can point to another location,so we require that Loc⊆Val.We shall say that a location is current(or allocated)in a heap if the heap is defined at that location.The4J.HAYMAN AND G.WINSKELprocedure of making a non-current location current is allocation,and the reverse procedure is called deallocation.If h is a heap and h(ℓ)=ℓ′,there is no implicit assumption that h(ℓ′) is defined.Consequently,heaps may contain dangling pointers.In addition to operating on a heap,the programs that we shall consider shall make use of critical regions[Dij68]protected by resources.The mutual exclusion property that they provide is that no two parallel processes may be inside critical regions protected by the same resource.We will write Res for the set of resources and use r to range over its elements. Critical regions are straightforwardly implemented by recording,for each resource,whether the resource is available or unavailable.A process may enter a critical region protected by r only if r is available;otherwise it is blocked and may not resume execution until the resource becomes available.The process makes r unavailable upon entering the critical region and makes r available again when it leaves the critical region.The language also has a primitive,resource w do t od,which says that the variable w represents a resource local to t.The syntax of the language that we will consider is presented in Figure1.The symbol αis used to range over heap actions,which are actions on the heap that might change the values held at locations but do not affect the domain of definition of the heap.That is, they neither allocate nor deallocate locations.We reserve the symbol b for boolean guards, which are heap actions that may proceed without changing the heap if the boolean b holds.Provision for allocation within our language is made via the alloc(ℓ)primitive for ℓ∈Loc,which makes a location current and setsℓto point at this location.For symmetry, dealloc(ℓ)makes the location pointed to byℓnon-current ifℓpoints to a current location. Writing a heap as the set of values that it holds for each allocated location,the effect of the command alloc(ℓ)on the heap{ℓ→0}might be to form a heap{ℓ→ℓ′,ℓ′→1}if the locationℓ′is chosen to be allocated and is assigned initial value1.The effect of the command dealloc(ℓ)on the heap{ℓ→ℓ′,ℓ′→1}would be to form the heap{ℓ→ℓ′}.The guarded sumα.t+α′.t′is a process that executes as t ifαtakes place or as t′ifα′takes place.We refer the reader to Section?for a brief justification for disallowing non-guarded sums.As mentioned earlier,critical regions are provided to control concurrency:the sub-process t inside with r do t od can only run when no other process is inside a critical region protected by r.The term resource w do t od has the resource variable w bound within t, asserting that a resource is to be chosen that is local to t and used for w.Consequently,in the process(resource w do with w do t1od od) (resource w do with w do t2od od)the sub-processes t1and t2may run concurrently since they must be protected by different resources,one local to the process on the left and the other local to the process on the right. To model this,we shall say that the construct resource w do t od binds the variable w within t,and the variable w is free in with w do t od.We write fv(t)for the free variables in t and say that a term closed if it contains no free resource variables;we shall restrict attention to such terms.We write[r/w]t for the term obtained by substituting r for free occurrences of the variable w within t.As standard,we will identify terms‘up to’the standard alpha-equivalence≡induced by renaming bound occurrences of variables.The notation res(t)is adopted to represent the resources occurring in t.The semantics of the term resource w do t od will involvefirst picking a‘fresh’resource r and then running[r/w]t.It will therefore be necessary to record during the execution ofINDEPENDENCE AND CONCURRENT SEPARATION LOGIC∗5Figure1:Syntax of termsprocesses which resources are current(i.e.not fresh)as well as which current resources are available(i.e.not held by any process).The way in which we shall formally model the state in which processes execute is motivated by the way in which we shall give the net semantics to closed terms.We begin6J.HAYMAN AND G.WINSKELby defining the following sets:D def=Loc×ValL def={curr(ℓ)|ℓ∈Loc}R def=ResN def={curr(r)|r∈Res}.A stateσis defined to be a tuple(D,L,R,N)where D⊆D represents the values held by locations in the heap;L⊆L represents the set of current,or allocated,locations of the heap;R⊆R represents the set of available resources;and N⊆N represents the set of current resources.The sets D,L,R and N are disjoint,so no ambiguity arises from writing,for example,(ℓ,v)∈σ.The interpretation of a state for the heap is that(ℓ,v)∈D ifℓholds value v and that curr(ℓ)∈L ifℓis current.For resources,r∈R if the resource r is available and curr(r)∈N if r is current.It is clear that only certain such tuples of subsets are sensible. In particular,the heap must be defined precisely on the set of current locations,and only current resources may be available.Definition2.1(Consistent state).The state(D,L,R,N)is consistent if we have:•the sets D,L,R and N are allfinite,•D is a partial function:for allℓ,v and v′,if(ℓ,v)∈D and(ℓ,v′)∈D then v=v′,•L represents the domain of D:L={curr(ℓ)|∃v:(ℓ,v)∈D},and•all available resources are current:R⊆{r|curr(r)∈N}.It is clear to see that the L component of any given consistent state may be inferred from the D component.It will,however,be useful to retain this information separately for when the net semantics is given.We shall call D⊆D a heap when it is afinite partial function from locations to values,and shall writeℓ→v for its elements rather than(ℓ,v). We shall frequently make use of the following definition of the domain of a heap D:dom(D)def={ℓ|∃v.(ℓ→v)∈D}.3.Process modelsThe definition of state that we have adopted permits a net semantics to be defined. Before doing so,we shall define how heap actions are to be interpreted and then give a transition semantics to closed terms.3.1.Actions.The earlier definition of state allows a very general form of heap action to be defined that forms a basis for both the transition and net semantics.We assume that we are given the semantics of primitive actionsαas A α comprising a set of heap pairs:A α ⊆Heap×Heap.We require that whenever(D1,D2)∈A α ,it is the case that D1and D2are(the graphs of)partial functions with the same domain.The interpretation is thatαcan proceed in heap D if there are(D1,D2)∈A α such that D has the same value as D1wherever D1is defined.The resulting heap is formed byINDEPENDENCE AND CONCURRENT SEPARATION LOGIC∗7 updating D to have the same value as D2wherever it is defined.It is significant that this definition allows us to infer precisely the set of locations upon which an action depends. The requirement on the domains of D1and D2ensures that actions preserve consistent markings(Lemma3.25).Example3.1(Assignment).For any two locationsℓandℓ′,let[ℓ]:=[ℓ′]represent the action that copies the value held at locationℓ′to locationℓ.Its semantics is as follows:A [ℓ]:=[ℓ′] def={({ℓ→v,ℓ′→v′},{ℓ→v′,ℓ′→v′})|v,v′∈Val}Following the informal account above of the semantics of actions,because in the semantics we have({ℓ0→0,ℓ1→1},{ℓ0→1,ℓ1→1})∈A [ℓ0]:=[ℓ1] ,the state{ℓ0→0,ℓ1→1,ℓ2→2}is updated by[ℓ0]:=[ℓ1]to{ℓ0→1,ℓ1→1,ℓ2→2}.Example3.2(Booleans).Boolean guards b are actions that wait until the boolean expres-sion holds and may then take place;they do not update the state.A selection of literals may be defined.For example:A [ℓ]=v def={({ℓ→v},{ℓ→v})}A [ℓ]=[ℓ′] def={({ℓ→v,ℓ′→v},{ℓ→v,ℓ′→v})|v∈Val}Thefirst gives the semantics of an action that proceeds only ifℓholds value v and the second gives the semantics of an action that proceeds only if the locationsℓandℓ′hold the same value.Since boolean actions shall not modify the heap,they shall possess the property that:if(D1,D2)∈A b then D1=D2.This is preserved by the operations defined below.For heaps D and D′,we use D↑D′to mean that D and D′are compatible as partial functions and D ↑D′otherwise,i.e.if they disagree on the values assigned to a common location.A true def={(∅,∅)}A false def=∅A b∧b′ def={({D∪D′},{D∪D′})|D↑D′and(D,D)∈A b and(D′,D′)∈A b′ } A b∨b′ def=A b ∪A b′A ¬b def={(D,D)|D is a⊆-minimal heap s.t.∀D′.(D′,D′)∈A b :D ↑D′}By insisting on minimality in the clause for¬b,we form an action that is defined at as few locations as possible to refute all grounds for b.3.2.Transition semantics.As an aid to understanding the net model,and in particular to give a model with respect to which we can prove its correspondence,a transition semantics for closed terms(terms such that fv(t)=∅)is given in Figure2.A formal relationship between the two semantics is presented in Theorem3.27.The transition semantics is given by means of labelled transition relations of the forms t,σ λ−→ t′,σ′ and t,σ λ−→σ′.As usual,thefirst form of transition indicates that t performs an action labelledλin stateσ8J.HAYMAN AND G.WINSKELto yield a resumption t′and a stateσ′.The second indicates that t in stateσperforms an action labelledλto terminate and yields a stateσ′.Labels follow the grammar λ::=act(D1,D2)heap action|alloc(ℓ,v,ℓ′,v′)heap allocation|dealloc(ℓ,ℓ′,v)heap disposal|decl(r)resource declaration|end(r)end of resource scope|acq(r)resource acquisition(critical region entry)|rel(r)resource release(critical region exit).In the transition semantics,we writeσ⊕σ′for the union of the components of two states where they are disjoint and impose the implicit side-condition that this is defined wherever it is used.For example,this implicit side-condition means,in the rule(Alloc),that for alloc(ℓ,v,ℓ′,v′)to occur we must have curr(ℓ′)∈σ,and henceℓ′was initially non-current. Similarly,the rule(Res)can only be applied to derive a transition labelled decl(r)if the resource r was not initially current.The syntax of terms is extended temporarily to include rel r and end r which are special terms used in the rules(Rel)and(End).These,respectively,are attached to the ends of terms protected by critical regions and the ends of terms in which a resource was declared.For conciseness,we do not give an error semantics to situations in which non-current locations or resources are used;instead,the process will become stuck.We show in Section 4.3that such situations are excluded by the logic.3.3.Petri nets.Petri nets,introduced by Petri in his1962thesis[Pet62],are a well-known model for concurrent computation.It is beyond the scope of the current article to provide a full account of the many variants of Petri net and their associated theories;we instead refer the reader to[BRR87]for a good account.Roughly,a Petri net can be thought of as a transition system where,instead of a transition occurring from a single global state,an occurrence of an event is imagined to affect only the conditions in its neighbourhood.Petri nets allow a derived notion of independence of events;two events are independent if their neighbourhoods of conditions do not intersect.We base our semantics on the following well-known variant of Petri net(cf.the‘basic’nets of[CW01]and[WN95]):Definition3.3(Petri net).A Petri net is afive-tuple,(B,E,•(−),(−)•,M0).The set B comprises the conditions of the net,the set E consists of the events of the net, and M0is the subset of B of marked conditions(the initial marking).The maps•(−),(−)•:E→P ow(B)are the precondition and postcondition maps,respectively.Petri nets have an appealing graphical representation,with:•circles to represent conditions,•bold lines to represent events,•arrows from conditions to events to represent the precondition map,INDEPENDENCE AND CONCURRENT SEPARATION LOGIC∗9α,(D,L,R,N) act(D1,D2)−→(D′,L,R,N)(Alloc): alloc(ℓ),σ⊕{ℓ→v} alloc(ℓ,v,ℓ′,v′)−→σ⊕{ℓ→ℓ′,ℓ′→v′,curr(ℓ′)} (Dealloc): dealloc(ℓ),σ⊕{ℓ→ℓ′,ℓ′→v′,curr(ℓ′)} dealloc(ℓ,ℓ′,v′)−→σ⊕{ℓ→ℓ′}(Seq): t1,σ λ−→ t′1,σ′t1;t2,σ λ−→ t2,σ′(Par-1): t1,σ λ−→ t′1,σ′t1 t2,σ λ−→ t1 t′2,σ′(Par′-1): t1,σ λ−→σ′t1 t2,σ λ−→ t1,σ′(Sum-1): α1,σ λ−→σ′α1.t1+α2.t2,σ λ−→ t2,σ′(While): b,σ λ−→σwhile b do t od,σ λ−→σ(With): with r do t od,σ⊕{r}) acq(r)−→ t;rel r,σ (Rel): rel r,σ rel(r)−→σ⊕{r}(Res): resource w do t od,σ decl(r)−→ [r/w]t;end r,σ⊕{r,curr(r)} (End): end r,σ⊕{r,curr(r)} end(r)−→σ10J.HAYMAN AND G.WINSKELSuch an event is said to have concession or to be enabled .The marking following the occurrence of e is obtained by removing the tokens from the preconditions of e and placing a token in every postcondition of e .We write M e −։M ′whereM ′=(M \•e )∪e •.If constraint (2)does not hold but constraint (1)does,so the preconditions are all marked (have a token inside)but following removal of the tokens from the preconditions there is a token in some postcondition,there is said to be contact in the marking and the event cannot fire.Consider the following example Petri net,with its transition system between markings derived according to the tokengame.{d,c,g }e 3K K K K K K K K K K {a,g }e 13.4.Overview of net semantics.Before giving the formal definition of the net semantics of closed terms,by means of an example we shall illustrate how our semantics shall be defined.First,we shall draw the semantics of an action toggle (ℓ,0,1)that toggles the value held at a location ℓbetween 0and1.terminal conditionsinitial conditions evolves tocontrol terminal conditions initial conditions state Notice that in the above net there are conditions to represent the shared state in which processes execute,including for example the values held at locations (we have only drawn conditions that are actually used by the net).There are also conditions to represent the control point of the process.The net pictured on the left is in its initial marking of control conditions and the net on the right is in its terminal marking of control conditions,indicating successful completion of the process following the toggle of the value;the marking of the net initially had the state condition ℓ→0marked and finished with the condition ℓ→1marked.There is an event present in the net for each way that the action could take place:one event for toggling the value from 0to 1and another event for toggling the value from 1to 0.Only the first event could occur in the initial marking of the net on the left,and no event can occur in the marking on the right since the control conditions are not appropriately marked.The parallel composition toggle (ℓ,0,1) toggle (ℓ,0,1)can be formed by taking two copies of the net toggle (ℓ,0,1)and forcing them to operate on disjoint sets of controlconditions.control state initial conditions terminal conditionsAn example run of this net would involve first the top event changing the value of ℓfrom 0to 1and then the bottom event changing ℓback from 1to 0.The resulting marking ofcontrol conditions would be equal to the terminal conditions of the net,so no event would have concession in this marking.The net representing the sequential composition(toggle(ℓ,0,1) toggle(ℓ,0,1));(toggle(ℓ,0,1) toggle(ℓ,0,1))is formed by a‘gluing’operation that joins the terminal conditions of one copy of the net for toggle(ℓ,0,1)to the initial conditions of another copy of the net for toggle(ℓ,0,1).(Inconditions.)this example net,for clarity we shall not show the state“gluing” structure.As outlined above,within the nets that we give for processes we distinguish two forms of condition,namely control conditions and state conditions.The markings of these sets of conditions determine the control point of the process and the state in which it is executing,respectively.When we give the net semantics,we will make use of the closure of the set of control conditions under various operations.Definition3.5(Conditions).Define the set of control conditions C,ranged over by c,to be the least set such that:•C contains distinguished elements i and t,standing for‘initial’and‘terminal’,respectively.•If c∈C then r:c∈C for all r∈Res and i:c∈C for all i∈{1,2},to distinguish processes working on different resources or arising from different subterms.•If c,c′∈C then(c,c′)∈C to allow the‘gluing’operation above.Define the set of state conditions S to be D∪L∪R∪N.A stateσ=(D,L,R,N)corresponds to the marking D∪L∪R∪N of state conditions in the obvious way.Similarly,if C is a marking of control conditions andσis a state,the pair (C,σ)corresponds to the marking C∪σ.We therefore use the notations interchangeably.The nets that we form shall be extensional in the sense that two events are equal if they have the same preconditions and the same postconditions.An event can therefore be regarded as a tuplee=(C,σ,C′,σ′)with preconditions•e def=C∪σand postconditions e•def=C′∪σ′.To obtain a concise notation for working with events,we write C e for the pre-control conditions of e:C e def=•e∩C.We likewise define notations e C,D e,L e etc.,and call these the components of e by virtue of the fact that it is sufficient to define an event through the definition of its components. The pre-state conditions of e are S e=D e∪L e∪R e∪N e,and we define e S similarly.Two markings of control conditions are of particular importance:those marked when the process starts executing and those marked when the process has terminated.We call these the initial control conditions I and terminal control conditions T,respectively.We shall call a net with a partition of its conditions into control and state with the subsets of control conditions I and T an embedded net.For an embedded net N,we write Ic(N)for I and Tc(N)for T,and we write Ev(N)for its set of events.Observe that no initial marking of state conditions is specified.The semantics of a closed term t shall be an embedded net,written N t .No confusion arises,so we shall write Ic(t)for Ic(N t ),and Tc(t)and Ev(t)for Tc(N t )and Ev(N t ), respectively.The nets formed shall always have the same sets of control and state conditions; the difference shall arise in the events present in the nets.It would be a trivial matter to restrict to the conditions that are actually used.As we give the semantics of closed terms,we will make use of several constructions on nets.For example,we wish the events of parallel processes to operate on disjoint sets of control conditions.This is conducted using a tagging operation on events.We define1:e to be the event e changed so thatC(1:e)def={1:c|c∈C e}(1:e)C def={1:c|c∈e C}but otherwise unchanged in its action on state conditions.We define the notations2:e and r:e where r∈Res similarly.The notations are extended pointwise to sets of events:1:E def={1:e|e∈E}.Another useful operation is what we call gluing two embedded nets together.For example,when forming the sequential composition of processes t1;t2,we want to enable the events of t2when t1has terminated.This is done by‘gluing’the two nets together at the terminal conditions of t1and the initial conditions of t2,having made them disjoint on control conditions using tagging.Wherever a terminal condition c of Tc(t1)occurs as a pre-or a postcondition of an event of t1,every element of the set{1:c}×(2:Ic(t2))would occur in its place.Similarly,the events of t2use the set of conditions(1:Tc(t1))×{2:c′} instead of an initial condition c′of Ic(t2).A variety of control properties that the nets we form possess(Lemma3.11),such as that all events have at least one pre-control condition, allows us to infer that it is impossible for an event of t2to occur before t1has terminated, and thereon it is impossible for t1to resume.An example follows shortly.Assume a set P⊆C×eful definitions to represent gluing are:P⊳C def={(c1,c2)|c1∈C and(c1,c2)∈P}∪{c1|c1∈C and∄c2.(c1,c2)∈P}P⊲C def={(c1,c2)|c2∈C and(c1,c2)∈P}∪{c2|c2∈C and∄c1.(c1,c2)∈P}Thefirst definition,P⊳C,indicates that an occurrence of c1in C is to be replaced by occurrences of(c1,c2)for every c2such that(c1,c2)occurs in P.The second definition, P⊲C,indicates that an occurrence of c2in C is to be replaced by occurrences of(c1,c2) for every c1such that(c1,c2)occurs in P.。
ProfileofKeithMoffatt
PROFILEProfile of Keith Moffatt Farooq AhmedScience WriterKeith Moffatt was born in1935and raised in Edinburgh,Scotland.He was four yearsold when his father left home to serve inthe Second World War.“At that age,youaccept things,”Moffatt recalls.“It was a time of great shortage.Everything in the UK wasrationed—food,clothes,fuel,even sweets!”The elder Moffatt,an accountant,hadintroduced his son to mathematical gamesand arithmetic puzzles,and,when he re-turned at the war’s end in1945,recreationalmathematics recommenced.The early tutelage had a lasting effect:Moffatt’s facility with numbers would trans-form into an aptitude for applied mathe-matics and its use in fluid mechanics andastrophysics.His contributions to our under-standing of magnetic and spin fields as wellas his advocacy for the study of mathematics around the world have earned him fellow-ships in the Royal Societies of both Londonand Edinburgh.A2005recipient of the RoyalSociety of London’s Hughes Medal,he servesas an emeritus professor of mathematicalphysics at the University of Cambridge.Hewas elected as a foreign associate of theNational Academy of Sciences in2008. Magnetic Fields in a Turbulent Medium Moffatt completed his undergraduate stud-ies in mathematics at the University ofEdinburgh.He left Scotland for TrinityCollege,Cambridge,in1957,where he cameunder the influence of George Batchelor,an Australian fluid dynamicist.Batchelor setMoffatt to work on his first research projectand played a decisive role in his career.“George was very sympathetic to students who came from outside Cambridge,as it had been his own career path,”Moffatt explains. In1959Batchelor founded the Depart-ment of Applied Mathematics and Theoreti-cal Physics at Cambridge—a department thatcounts in its ranks some of the world’s mostinfluential fluid dynamicists and theoreticalphysicists.He gave Moffatt a prepublication draft of his paper on how turbulence inter-acts with temperature or a contaminant that does not influence the turbulence(1).Two years later,Moffatt’s first paper in1961extended Batchelor’s ideas to the in-teraction of turbulence with a weak magnetic field and obtained the spectrum of the mag-netic fluctuations(2).The result would beverified experimentally nearly three decadeslater by French researchers,using liquidgallium and an applied magnetic field(3).“The application that I had most in mindat that time,”Moffatt notes,“was in astro-physics.”In the1950s,dynamo theory,whichis the study of how the earth,stars,and otherheavenly bodies generate and maintainmagnetic fields,was beset by conflictingideas.Furthermore,how these magneticfields interacted with the interstellar mediumand its turbulent ionized gases was poorlyunderstood.Knotted VorticesToward the end of the1960s,while teachinga course on magnetohydrodynamics,Moffattidentified an integral characteristic of fluidflow that he named“helicity”(4),building onthe observations of two scientists workingnearly a century apart.“In1869Lord Kelvin recognized the fro-zen property of vortex lines in fluid flows,and in1958Lodewijk Woltjer,an astro-physicist then at the University of Chicago,demonstrated the invariance of a puzzlingintegral quantity in the flow of a perfectlyconducting or Euler fluid.In struggling tofind a physical interpretation of Woltjer’sinvariant,it dawned on me that there mustbe an analogous result for the nonlinearEuler equations.”Helicity,as Moffatt explains,represents thedegree of linkage or entanglement in a fieldof vorticity,which can be thought of as thespin associated with a velocity field.Helicityis an invariant of the Euler equations,whichgovern ideal,nonviscous flows.This conceptestablished a bridge between classical fluidmechanics and topology,the study of shapesand their continuous deformation.Moffatt gives credit for the finding to Jean-Jacques Moreau,now an emeritus professorin Montpellier,France,who had found thesame invariant several years before his pub-lication(5).“As so often happens in oursubject,”Moffatt explains,“someone elseproved the concept in a paper that wascompletely buried.Moreau didn’t call ithelicity,but the result is there.He and Idiscovered it quite independently.”DynamosHelicity had immediate applications in dy-namo theory,and especially for the expla-nation of the earth’s magnetic field.Abouthalfway to the center of the earth,the mantlegives way to a liquid metal core,“a con-ducting fluid in random motion superposedon differential rotation,”Moffatt explains.This combination generates a magnetic field.He showed that dynamo action occurs evenin a weakly conducting fluid,provided thefluid domain is large enough and the turbu-lence has an average nonzero helicity(6).While on a sabbatical at the UniversitéPierre et Marie Curie in Paris,Moffatt wrotethe first monograph on dynamo theory,in-corporating helicity and its implications forplanetary and astrophysical dynamos(7).“Bythe late1970s,”he notes,“the basic principleswere well established,so the time was ripe forsuch a book,which is still a standard in-troductory text.”The publication led to anexplosion of activity in the field of dynamotheory,with three other books on the subjectproduced across the world within the nextfive years.A massive computational effortthroughout the1980s and1990s followed.Moffatt is currently working on an updatedversion of themonograph.Keith Moffatt.Photo by JillPaton-Walsh.This is a Profile of a recently elected member of the NationalAcademy of Sciences to accompany the member’s Inaugural Articleon page3663.3650–3652|PNAS|March11,2014|vol.111|/cgi/doi/10.1073/pnas.1400956111Nearly thirty years elapsed before a major experiment carried out at the International Thermonuclear Experimental Reactor site in Cadarache,France confirmed the basic ideas in Moffatt ’s book (8).In the experiment,a turbulent flow of liquid sodium driven by counter-rotating propellers produced dynamo action (9).More recently,physicists at the University of Chicago succeeded in creating knotted vortices in water using carefully fabricated hydrofoils (10).In his Inaugural Article,Moffatt recaps his inves-tigations into helicity and dynamo genera-tion.He remarks that “it ’s a good moment to talk about this work when theory meets experiment!”(11).Applying MathematicsAfter three years as professor of applied mathematics at the University of Bristol,Moffatt returned in 1980to Cambridge,where he has remained ever since as a pro-fessor of mathematical physics and a fellow of Trinity College.There,he focused on de-veloping the topological approach to fluid dynamics,leading to work on magnetic re-laxation,which has implications for ther-monuclear fusion devices.Similar research describes the evolution of the magnetic field in the solar corona (12,13).These efforts led Moffatt to consider what happens when a knotted magnetic flux tube relaxes to a minimum energy configuration while conserving its knot topology —a prob-lem closely related to the pure mathematical question:What is the smallest length of rope of a fixed diameter needed to tie a given knot?(14).“The theory of tight knots,”Moffatt explains,“has been considerably developed with applications in polymer physics and molecular biology ”(15).“The beauty of working in applied mathematics,”he continues,“is that your results some-times turn out to have application in fields far from your initial interests.”Newton InstituteIn 1991the British mathematician Sir Michael Atiyah and others founded the Isaac Newton Institute for Mathematical Sciences at Cambridge as a national visitor research center for the United Kingdom.Moffatt ran one of the first programs,on dynamo theory,and succeeded Atiyah as director of the institute for a five-year term beginning in 1996.Moffatt says that “the major problem for any director is fund-raising,and this was a major pre-occupation for me during these years.”While director,he became involved in the early planning of the African Institute forMathematical Sciences (AIMS),which was eventually founded in 2003by former Cambridge mathematical physicist Neil Turok.Located in a suburb of Cape Town,the institute trains graduate students from across the continent.“The concept has worked extremely well,”notes Moffatt,“and has now expanded through the AIMS Next Einstein Initiative to three other loca-tions across Africa in Senegal,Ghana,and Cameroon.”Mathematics of Toys and Soap FilmsIn recent years,Moffatt has worked with Japanese mathematician Yutaka Shimomura,Cambridge ’s Tadashi Tokieda,and others on mechanical toys that exhibit puzzling behaviors:the Euler disk,which has a per-ceived finite-time singularity (16),the spin-ning hard-boiled egg with a friction-driven instability (17),and the rattleback,with a chiral instability and asymmetric behavior (18).Moffatt says the real interest in these problems is that they illustrate behavior that appears in complex fluid situations like the helicity-driven dynamo,or in the friction-driven instability of pressure-driven flow in a 2D channel.With Raymond Goldstein and Adriana Pesci at Cambridge,Moffatt worked on the problem of topological jumps in soap films that span twisted wire loops.The researchers formed a one-sided Mobius-strip soap film on a doubled-over loop of wire and found that it jumps to a two-sided surface in mil-liseconds at a critical moment when the wire is slowly untwisted (19).With the aid of high-speed photography,they described the detailed mechanism of this jump,and in the process opened up a new branch of topological fluid dynamics.Topologists,Moffatt notes,have been studying minimal-surface area problems for more than a century,and “this new work provides welcome interaction between pure mathematics and fluid dynamics.”MusingsAlthough he still performs research,Moffatt retired from teaching in 2002.He recently built a personal Web site that chronicles his life and career in applied mathe-matics.The site details his early life in Edinburgh during the war,and catalogs major contributions,periods of sabbatical,and graduate students he has mentored.The Web site also serves as a repository for his poetry,much of it science themed.“I don ’t call myself a poet,”he says,“but when occasion demands or when the Muse inspires,I don ’t hesitate to put pen to paper!”His Royal Highness Prince Philip visits the Newton Institute,October 1992.Keith Moffatt (Left )demonstrates the production of a cusp singularity at a free surface.Sir Michael Atiyah,Founder Director of the Institute (Right ).Photo courtesy of MM Photographic.AhmedPNAS |March 11,2014|vol.111|no.10|3651P R O F I L E1Batchelor GK(1959)Small-scale variation of convected quantities like temperature in turbulent fluid.J Fluid Mech5(1):113–133.2Moffatt HK(1961)The amplification of a weak applied magnetic field by turbulence in fluids of moderate conductivity.J Fluid Mech 11(4):625–635.3Odier P,Pinton J-F,Fauve S(1998)Advection of a magnetic field by a turbulent swirling flow.Phys Rev E58:7397–7401.4Moffatt HK(1969)The degree of knottedness of tangled vortex lines.J Fluid Mech35(1):117–129.5Moreau J-J(1961)Constantes d’unîlot tourbillonnaire en régime permanent.CR Acad Sci Paris252:2810–2813.6Moffatt HK(1970)Turbulent dynamo action at low magnetic Reynolds number.J Fluid Mech41(2):435–452.7Moffatt HK(1978)Magnetic Field Generation in Electrically Conducting Fluids(Cambridge Univ Press,Cambridge,UK).8Monchaux R,et al.(2007)Generation of a magnetic field bydynamo action in a turbulent flow of liquid sodium.Phys Rev Lett98(4):044502–044505.9Monchaux R,et al.(2009)The von Kármán sodiumexperiment:Turbulent dynamical dynamos.Phys Fluids21(3):035108.10Kleckner D,Irvine WTM(2013)Creation and dynamics ofknotted vortices.Nat Phys9(4):253–258.11Moffatt HK(2014)Helicity and singular structures in fluiddynamics.Proc Natl Acad Sci USA111:3663–3670.12Moffatt HK(1985)Magnetostatic equilibria and analogousEuler flows of arbitrarily complex topology Part1.Fundamentals.J Fluid Mech159:359–378.13Moffatt HK(1986)Magnetostatic equilibria and analogous Eulerflows of arbitrarily complex topology.Part2.Stability considerations.J Fluid Mech166:359–378.14Moffatt HK(1990)The energy-spectrum of knots and links.Nature347(6291):367–369.15Stasiak A,Katrich V,Kauffman LH,eds(1998)Ideal Knots(World Scientific,Singapore).16Moffatt HK(2000)Euler’s disk and its finite-time singularity.Nature404(6780):833–834.17Moffatt HK,Shimomura Y(2002)Classical dynamics:Spinning eggs—a paradox resolved.Nature416(6879):385–386.18Moffatt HK,Tokieda T(2008)Celt reversals:A prototype ofchiral dynamics.Proc Roy Soc Edin A138(2):361–368.19Goldstein RE,Moffatt HK,Pesci AI,Ricca RL(2010)Soap-filmMöbius strip changes topology with a twist.Proc Natl Acad Sci USA107(51):21979–21984.3652|/cgi/doi/10.1073/pnas.1400956111Ahmed。
5-AnimalMetaphorinCognitiveLinguistics
Psychology Research, ISSN 2159-5542October 2011, Vol. 1, No. 4, 251-254Animal Metaphor in Cognitive LinguisticsMehri RouhiIslamic Azad University, Hamedan Branch, Iran Mohammad Rasekh Mahand Bu Ali University, Hamedan, IranThe phenomenon of AM (animal metaphor) can be discussed based on the class-inclusion model in cognitivelinguistics. In this article, we try to prove that this kind of metaphor accords more with this model than withcorrespondence model of Lakoff. It does not mean that the correspondence model is not valid in this regard, but weargue that depending on the nature of this kind of metaphor, class-inclusion model can explain some of itscharacteristics better than the other models. The correspondence model assumes that metaphors are essentiallyanalogical in character. Also, it suggests that mappings are one-to-one and structurally consistent. Invarianceprinciple of this model states that metaphorical mappings preserve the cognitive topology (that is, the image schemastructure) of the source domain, in a way consistent with the inherent structure of the target domain. But, theclass-inclusion model does not treat metaphors as analogies rather the source is treated as prototypical instantiationof a larger, newly created super-ordinate category, which is seen then as encompassing both source and targetdomains. This newly created category uses a prototypical member as an exemplar. We tried to compare these twomodels in explaining AM in Persian.Keyword s: AM (animal metaphor), correspondence model, class-inclusion modelIntroductionMetaphor has been studied for many years especially in cognitive linguistics. The importance of metaphor studies in cognitive linguistics maybe the result of the nature of it. If cognitive linguistics is the study of ways in which features of language reflect other aspects of human cognition, metaphors provide one of the clearest illustrations of this relationship (Grady, 2007).Koveceses (2002) defined metaphor as understanding one conceptual domain in terms of another conceptual domain. A conceptual domain is any coherent organization of experience. Thus, for example, we have coherently organized knowledge about journeys that we rely on in understanding life. This is the thing which has been studied in correspondence model.Correspondence model has been accepted by metaphor scholars from its introduction by Lakoff (1993). It has been used for describing metaphors, although some different versions of it were suggested.Some scholars tried to reform some of its ambiguities by adding different traits to it, as a result that the blending theory was introduced.But, none of these things have prevented it to fulfill its role as the best frame for describing metaphors, until Glucksberg and Keysar (1990) tried to show some of its shortcomings and weak points in explaining proper name metaphors. They introduced class-inclusion model as a substitute, then.Mehri Rouhi, English Department, Islamic Azad University.Mohammad Rasekh Mahand, Ph.D., associate professor, Department of Linguistics, Bu Ali University.ANIMAL METAPHOR IN COGNITIVE LINGUISTICS252The later acts more efficiently, at least in describing proper name metaphors. But, what are the characteristics of each model and why did the correspondence model force to withdraw at least in one front, regardless its worldwide acceptance? And what was the reason of current challenging situation that causes these two models to array troops against each other? Finally, we tried to explain why animal metaphors fail to be accommodated within a correspondence model as the case of proper name metaphor.Features of AM (Animal Metaphors)First, it is necessary to note that what counts as an AM is the use of an animal name as the source rather than the target. Let us flash back to our childhood stories; we may remember some sentences such as “Fox fired up/fox was fuming”. Here fox is target and fire is source, but if we say in Persian: “He is a fox”, then fox is source here and this sentence is an example of AM.Second, the animal’s name in an AM may be used either referentially (that is, it may be used simply as a label for an object) or predicatively (that is, it may be used as a description that an object may satisfy to varying degree or perhaps not satisfy at all), as it is illustrated in Example (1) and (2):Example (1) He is a lion;Example (2) He is a poor lion who has lost everything.In Example (1), the “lion” is used referentially. In the context, we can see that it represents all those traits which we have accepted for a lion (in folk model) and it attributed them to target. But in Example (2), using “poor” shows that the person fails to have those predicted features which we expected for a lion.Of course, it is worth to note that this animal’s name makes sense, only if we know those culturally accepted features for lion.Moreover, we should bear in mind those characteristics which conventionally stand for an animal and become fixed by repeated usage.Finally, aside from knowing the accepted characteristics, we should be aware of those irrelevant traits that must be ignored, in order to make a metaphor shaped. The main difference here is being “animal” and “human”. So, it is clear that metaphors are selective, highlighting particular aspects of the source and the target while hiding others (Lakoff, 1993).It is well worth considering that an animal name is used whether for a person who has the highlighted characteristics or lack it, while the “lion” can be used to admire a person because of his/her bravery or mock him/her for his cowardice.AM and Two Models of MetaphorThe correspondence model assumes that metaphors are essentially analogical in character; it means they possess the systematic pairing of relations and entities across the source and the target.In other words, there is a mapping between the source and the target which shows one to one correspondence. Lakoff (1993) suggested that particular correspondence between certain kinds of source and certain kinds of target already exist in our long-term semantic memory due to the sensory motor experiences we are exposed to as a consequence of our neurobiological make up. In this model, we always face some kinds of limitations, unless we may be surrounded by the large number of possible mapping. As a result, some particular kinds of mappings appear to be preferred over others. It refers to Lakoff’s (1993) “Invariance Principle” which emphasizes that metaphorical mappings should preserve the cognitive topology (that is, the image schema structure) of the sourceANIMAL METAPHOR IN COGNITIVE LINGUISTICS 253domain in a way consistent with the inherent structure of target domain. So, based on this principle, perseverance of structural relations is necessary in the course of mapping (Turner, 1991).In contrast to Lakoff’s model, the class-inclusion model does not treat metaphors as analogies. Rather, the source is treated as a prototypical instance of a larger, newly-created and super-ordinate category, which is then seen as encompassing both source and target domains (Glucksberg, 2001). Because it is a newly-created category, the super-ordinate category cannot be directly named, and hence, there is a need to use a prototypical member as an exemplar of such a category (Wee, 2004). Thus, according to the class-inclusion model, in understanding an expression, such as “He is a fox” in Persian, the speaker treats a fox as a prototypical member of the category of “individuals who are clever, cunning, etc.”. This is a super-ordinate category which allows both “fox” and “human being” to be seen as its members. This model does not assume that metaphors are necessarily analogical in nature. It may be the main difference between this model and correspondence model which is reflected in forming ad hoc super-ordinate category in comparison to systematic mapping of Lakoff’s model which is formed between the internal structures of the source and target domains. The ad hoc category is created in the mind of hearer/reader and source, and the target is introduced as category members. The source, as a prototypical member, simply helps in this process.The Role of Culture in AM Birth“He is a fox” in Persian and “He is an owl” in English are two metaphors with almost the same meaning (the first one is somehow negative and the second is positive, but both refer to cleverness). “Fox” is an animal that is well known because of its cleverness in Persian and “owl” is the bird whose main characteristic is its wisdom.When we speak about AM and face such a difference, we can see the traces of culture in forming this kind of metaphor. The culture’s role can be clearer when we see that the same animal is prototype of different features in various cultures. For example, “owl” is a sinister bird in Persian and is used to refer to sinister person (compare this with English). Of course, we may face some similarities in this kind of metaphor in different cultures. For example, “dog”⎯to the best of the author’s knowledge⎯is known for its loyalty in different cultures.We do not want to compare cultures in this regard, because this job is difficult and time-consuming. We just want to show the role of culture in forming AM by these simple examples.The other thing that worthes regarding is those characteristics which animals are famous for.Some AMs are formed based on physical appearance or feature of an animal (using elephant or chicken in Persian) and some of them are used because of those traits which folk models decide about (such as cleverness for fox and loyalty for dogs).Another point which was discussed earlier and worth to berepeated is the case of metaphors which are used to show the lack of some characteristics, that is, they are used to mock a person who does not have that feature, such as “You are chicken!”.Although it is clear, we note that some animals are used in this kind of metaphor for the sake of their positive features and for admiring someone who has that feature (e.g., lion), while the others are used because of their negative features to humiliate or mock someone (e.g., chicken).ConclusionsAlthough correspondence model is useful for describing different kinds of metaphor, it seems thatANIMAL METAPHOR IN COGNITIVE LINGUISTICS254class-inclusion is more efficient in explaining animal metaphor. It is more congruent with this kind of metaphor, because everything is ad hoc in AM forming and some mismatches of source and target characteristics are justified with this model. Also, the ignorance of some characteristics and highlighting the others are explained.ReferencesCampbell, J. D., & Albert, N. K. (2006). On reversing the topics and vehicles of the metaphor. Metaphor and Symbol, 21(1), 1-22. Croft, W. (1993). The role of domains in the interpretation of metaphors and metonymies. Cognitive Linguistics, 4(4), 335-370. Croft, W. (1995). Intonation units and grammatical structure. Linguistics, 33(5), 839-882.Croft, W., & Cruse, D. A. (2004). Cognitive linguistics. Cambridge: Cambridge University Press.Evan, V., & Green, M. (2006). Cognitive linguistics: An introduction. Edinburg: Edinburg University Press.Fauconnier, G., & Turner, M. (1998). Conceptual integration networks. Cognitive Science, 22(2), 133-187.Gibbs, R. (1990). Comprehending figurative referential descriptions. Journal of Experimental Psychology: Learning, Memory and Cognition, 16(1), 56-66.Glucksberg, S. (2001). Understanding figurative language: From metaphors to idioms. Oxford: Oxford University Press. Glucksberg, S., & Keysar, B. (1990). Understanding metaphorical comparisons: Beyond similarity. Psychological Review, 97, 3-18.Grady, E. J. (2007). Metaphor in Oxford handbook of cognitive linguistics (D. Geeraerts, & H. Cuyckens, Ed.). Oxford: Oxford University Press.Keysar, B. (1989). On the functional equivalence of literal and metaphorical interpretations in discourse. Journal of Memory and Language, 28, 375-385.King, B. (1989). The conceptual structure of emotional experience in Chinese (Doctoral dissertation, Ohio state University). Koveceses, Z. (2002). Metaphor: A practical introduction. Oxford: Oxford University Press.Koveceses, Z. (2005). Metaphor in culture. New York and Cambridge: Cambridge University Press.Lakoff, G. (1987). Women, fire, and dangerous things: What categories reveal about the mind. Chicago: University of Chicago Press.Lakoff, G. (1991). Cognitive versus generative linguistics: How commitments influence results. Language and Communication, 11(1 & 2), 53-62.Lakoff, G. (1993). The contemporary theory of metaphor. In A. Ortony (Ed.), Metaphor and thought. Cambridge: Cambridge University Press.Lakoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago: University of Chicago Press.Lakoff, G., & Turner, M. (1989). More than cool reason: A field guide to poetic metaphor. Chicago: University of Chicago Press. Lee, D. (2001). Cognitive linguistics: An Introduction. Oxford: Oxford University Press.Lobner, S. (2002). Understanding semantics. Arnold Publishers.Miall, D. (1982). Metaphor: Problems and perspectives. Harvester Press Limited.Turner, M. (1991). Reading minds: The study of English in the age of cognitive science. Princeton: Princeton University Press. Ungerer, F., & Schmid, H. J. (1996). An introduction to cognitive linguistics. Longman.Wee, L. (2004). Proper names and theory of metaphor. New York and Cambridge: Cambridge University Press.。
上课不应该迟到英语作文
Being punctual is a fundamental aspect of respect and discipline, especially in an educational setting. Heres a composition on why its important not to be late for class: The Importance of Punctuality in the ClassroomPunctuality is not just a matter of time management it is a reflection of ones character and respect for others. In the context of a classroom, arriving on time is crucial for several reasons.Firstly, punctuality sets a positive tone for the learning environment. When students arrive early or on time, it shows that they value the educational opportunity and are eager to learn. This attitude is contagious and can inspire a sense of enthusiasm among peers.Secondly, being late disrupts the flow of the class. Teachers often plan their lessons to start promptly, and a latecomer can disrupt the momentum of the lesson, causing a distraction not only for the teacher but also for the rest of the class. This can lead to a loss of valuable instructional time.Thirdly, punctuality is a skill that translates into professional life. Employers value employees who are reliable and consistent. Developing the habit of being on time for class can prepare students for the punctuality expected in the workplace.Moreover, being late can also affect a students academic performance. Missing the beginning of a class means missing out on important information that may be crucial for understanding the rest of the lesson. Over time, this can lead to gaps in knowledge and a weaker grasp of the subject matter.In addition, punctuality fosters a sense of responsibility. When students are accountable for their time, they learn to plan and prioritize effectively. This skill is essential for managing assignments, projects, and other responsibilities both in school and beyond. Lastly, respecting the time of others is a key aspect of social etiquette. In a classroom, this means being considerate of the teachers time and the time of fellow students who are waiting for the lesson to begin.In conclusion, punctuality is a virtue that enhances the educational experience for everyone involved. It demonstrates respect, fosters a positive learning atmosphere, and prepares students for future success. By cultivating the habit of being on time, studentscan set themselves up for a lifetime of effective time management and professional success.This composition emphasizes the multifaceted benefits of being punctual for class, from the immediate classroom environment to longterm personal development.。
Multivariable link invariants arising from Lie superalgebras of type I
MULTIVARIABLE LINK INVARIANTS ARISING FROM LIE SUPERALGEBRAS OF TYPE I
NATHAN GEER AND BERTRAND PATUREAU-MIRAND Abstract. In this paper we construct new links invariants from a type I basic Lie superalgebra g. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module, by non-trivial “fake quantum dimensions.” Using this, we get a multivariable link invariant associated to any one parameter family of irreducible g-modules.
Date : February 2, 2008.
1
2
NATHAN GEER AND BERTRANDe non-zero and lead to non-trivial link invariants. The first of these examples recover the hierarchy of invariants defined by Akutsu, Deguchi and Ohtsuki [1], using a regularize of the Markov trace and nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev’s quantum dilogarithm invariants of knot (see [16]). The second example, is the invariants defined in this paper. The definition of the “fake quantum dimensions” given in [9] is abstract where the analogous definition in this paper is given by explicit formulas. One can use general theory to show that these definitions are equivalent. The explicit formulas given in this paper are useful when one wants to compute the invariant or compare it to other invariants. • In the second subsequent paper the authors will use the explicit formulas for the “fake quantum dimensions” to define “fake superdimensions” of typical representations of the Lie superalgebra g. These “fake superdimensions” are non-zero and lead to a kind of supertrace on the category representations of g which is non-trivial and invariant. These statements are completely classical statements. However, the only proof we know of uses the quantum algebra and low-dimensional topology developed in this paper. • We will now discuss the final subsequent paper in relation with the multivariable invariants defined in this paper. In Section 3 we will show that for c ∈ Nr−1 the pair (g, c) gives rise to c a multivariable link invariant Mg . These invariants associate a variable to each component of the link. There are only a handful of such invariants including the multivariable Alexander polynomial and the ones defined in [1]. All of these invariants are related to the invariants defined in this paper. Let us now explain these relationships. First, in [8] we plan (0,...,0) on showing that the invariant Msl(m|1) specializes to the multivariable Alexander polynomial. Second, in order to define their link invariants the authors of [1] regularize the Markov trace. Although using different methods, the invariants of this paper have a similar regularization. In both case, the standard method using ribbon categories or the Markov trace is trivial. Moreover, both families of invariants are generalization of the multivariable Alexander polynomial. In [8], we plan on conjec(0,...,0) turing that the invariants Msl(m|1) , for m ∈ N, specialize to the hierarchy of invariants defined in [1]. (Note this specialization
拓扑理论
The topological approach to perceptual organizationLin ChenKey Laboratory of Cognitive Science,Graduate School and Institute of Biophysics,Chinese Academy of Sciences,Beijing,ChinaTo address the fundamental question of``what are the primitives of visual per-ception'',a theory of topological structure and functional hierarchy in visual perception has been proposed.This holds that the global nature of perceptual organization can be described in terms of topological invariants,global topological perception is prior to the perception of other featural properties,and the primitives of visual form perception are geometric invariants at different levels of structural stability.I n Part Iof this paper,Iwill illustrate why and how the topological approach to perceptual organization has been advanced.In Part II,I will provide empirical evidence supporting the early topological perception,while answering some commonly considered counteraccounts.In Part III,to complete the theory,I will apply the mathematics of tolerance spaces to describe global properties in discrete sets.In Part IV,I will further present experimental data to demonstrate the global-to-local functional hierarchy in form perception,which is stratified with respect to structural stability defined by Klein's Erlangen Program.Finally,in Part V,Iwill discuss relations of the global-to-local topological model to other theories:The topological approach reformulates both classical Gestalt holism and Gibson's direct perception of invariance,while providing a challenge to com-putational approaches to vision based on the local-to-global assumption.INTRODUCTIONA great divide:Local-to-global vs.global-to-localAs a Chinese proverb says:Everything is difficult at its very beginning. Historically,major schools of vision diverge in their answers to the question of ``Where visual processing begins?''(Pomerantz,1981)or``What are the pri-mitives of visual perception?''(Chen,1982).The question is so fundamental and also so controversial as to serve as a watershed,a Great Divide,separatingPlease address all correspondence to:Lin Chen,Key Laboratory of Cognitive Science,Graduate School and Institute of Biophysics,Chinese Academy of Sciences,15Datun Road,Beijing100101, China.Email:chenl@Supported by National Nature Science Foundation of China Grant(697900800);Ministry of Science and Technology of China Grant(1998030503);Chinese Academy of Sciences Grants (KGCX2-SW-101,KJCX1-07).This work was partly done during author's sabbaticals,at Institute of Medical Psychology,University of Munich,and at National Institute of Mental Health./journals/pp/13506285.html DOI:10.1080/13506280444000256554CHENtwo most basic and sharply contrasting lines of thinking in the study of perception.Early feature analysis:Local-to-global.On one side of the Great Divide,the early feature-analytic viewpoint holds that perceptual processing is from local to global:Objects are initially decomposed into separable properties and components,and only in subsequent processes are objects recognized,on the basis of the extracted features.The computational approach to vision by Marr (1982),representative of``early feature-analysis''viewpoint,claims that the primitives of visual-information representation are simple components of forms and their local geometric properties,such as,typically,line segments with slopes. Early holistic registration:Global-to-local.On the other side of the Great Divide,the viewpoint of early holistic registration claims that perceptual processing is from global to local:Wholes are coded prior to perceptual analysis of their separable properties or parts,as indicated by the conception of perceptual organization in Gestalt psychology.As we will see in the following discussion,with respect to the fundamental question of``Where to begin'',the core contribution of the Gestalt idea goes far beyond the notion that``Whole is more than the simple sum of it parts'';rather it is that``Holistic registration is prior to local analysis''.The idea of early feature analysis has gained wide acceptance,and dominates much of the current study of visual cognition.Intuitively,it seems to be natural and reasonable that visual processing begins with analysing simple components and their local geometric properties,typically as line segments with slopes,as they are readily to be considered physically simple and computationally easy. An underlying idea of Marr's computational system of vision was,in Marr's own words,``In the early stages of the analysis of an image,the representations used depend more on what it is possible to compute from an image than on what is ultimately desirable.''(Marr,1978).Nevertheless,a starting point of the present paper is that physically or computationally simple doesn't necessarily mean psychologically simple or perceptually primitive;therefore,the question of which variables are perceptual primitives is not a question answered primarily by logical reasoning or analysis of computational complexity but rather by empirical findings.Topological structure and functional hierarchyinform perceptionTo address the fundamental question of what are the primitives of visual percep-tion,based on a fairly large set of data on perceptual organization reviewed here,a theory of``early topological perception''has been proposed.This holds that:GLOBAL TO LOCAL TOPOLOGICAL PERCEPTION555A primitive and general function of the visual system is the perception of topo-logical properties.The time dependence of perceiving form properties is system-atically related to their structural stability under change,in a manner similar to Klein's hierarchy of geometries;in particular,topological perception(based on physical connectivity)is prior to the perception of other geometrical properties.The invariants at different geometrical levels are the primitives of visual form perception.These include,in a descending order of stability(from global to local), topological,projective,affine,and Euclidean geometrical invariants.The topological approach is based on one core idea and includes two main aspects.The core idea is that perceptual organization should be understood in the perspective of transformation and perception of invar-iance over transformation.The first aspect emphasizes the topological struc-ture in form perception,namely,that the global nature of perceptual organization can be described in terms of topological invariants.The sec-ond aspect further highlights early topological perception,namely,that topo-logical perception is prior to perception of local featural properties.The ``prior''has two strict meanings:First,it implies that global organizations, determined by topology,are the basis that perception of local geometrical properties depends on;and second,topological perception(based on physi-cal connectivity)occurs earlier than the perception of local geometrical properties.The hypothesis of early topological perception led to a major finding that the relative perceptual salience of different geometric properties is remarkably consistent with the hierarchy of geometries according to Klein's Erlangen Program(see Part II and IV),which stratifies geometries in terms of their relative stability over transformations.Based on the finding,a functional hier-archy in form perception has been established as a formal and systematic definition of``global-to-local''relations:A property is considered more global (or stable)the more general the transformation group is,under which this property remains invariant;relative to geometrical transformations,the topolo-gical transformation is the most general and hence topological properties are the most global.The framework of the topological structure and functional hierarchy high-lights a fundamental empirical prediction,namely a time dependence of per-ceiving form properties,in which visual processing is from global to local:The more global a form invariant is the earlier its perception occurs,with topological perception being the most global and occurring earliest.The framework further highlights a series of novel empirical predictions about long-standing issues related to the study of perceptual organization,and many Gestalt-type phe-nomena in form perception may be explained in this unified manner.They include the following examples:556CHEN.With respect to the relationship between different organizational factors, proximity is the most fundamental organizational factor(even in comparison with uniform connectivity;Palmer&Rock,1994)(see Part III),and there is a time course of processing different organizational principles:Proximity precedes similarity,and topological similarity precedes similarity of local geometric properties(see Part VI)..Early topological perception predicts the visual sensitivity to distinction made in topology.For example,two stimuli that are topologically different are more discriminable under a near-threshold condition than are other pairs of forms that are topologically equivalent despite their difference in local features(see Part II)..With respect to the question of whether motion perception precedes form perception or vice versa,topological discrimination should occur earlier than and determine motion perception(see Part II)..Configural superiority effects(Pomerantz,1981)demonstrated by configural relations between line segments,such as the``triangle±arrow pair'',may simply demonstrate the superiority effect for perception of holes over indi-vidual line segments(see Part IV)..With respect to``global precedence''(Navon,1977),according to the functional hierarchy,the performance advantage for compound letters is quite straightforward:Global precedence reflects the primacy of proximity in perceptual organization(see Part III)..If topological properties are primitive,illusory conjunctions(Treisman& Gelade,1980)of topological properties,such as holes,should sometimes take place(see Part II)..With respect to the definition of perceptual object,the topological approach ties a formal definition of``object''to invariance over topological trans-formation(see Part I).From this definition,it follows that as an object is moving along and a hole appears in it,this topological change would dis-turb object continuity,while changes of shape and colour wouldn't(Wolfe, personal communication).For example,in an MOT(multiple object track-ing)test(Pylyshyn&Storm,1988;vanMarle&Scholl,2003),attentive tracking processes would be impaired by objects changing topology by getting a hole,while it does not matter if they change local features and colours..With respect to its ecological function and functional anatomy,long-range apparent motion works by abstracting form invariants,and hence is asso-ciating with form perception and activates the ventral pathway in the two visual systems(Ungerleider&Mishkin,1982).Specifically,the fMRIacti-vation should be correlated with the form stability under change(see Parts II and IV)..From the perspective of biological evolution,if topological perception is indeed a fundamental property of vision,one might expect topologicalGLOBAL TO LOCAL TOPOLOGICAL PERCEPTION557 properties to be extracted by all visual systems,including the relativelysimple ones possessed by insects,such as bees(see Part II).In summary,the framework of topological structure and functional hierarchy in form perception provides a new analysis of the fundamental question,i.e., ``What are the primitives of visual perception?'',in which primitives of visual form perception are considered to be geometric invariants(as opposed to simple components of objects,such as line segments)at different levels of structural stability.In the following,I will illustrate why and how the topological approach to perceptual organization has been advanced(Part I);provide empirical evidence supporting the topological perception,while answering some commonly con-sidered counteraccounts(Part II);complete the theory of topological perception, using the mathematics of tolerance spaces that describe global properties in discrete sets(Part III);present experimental data to demonstrate the functional hierarchy in form perception,which is stratified with respect to structural sta-bility defined by Klein's Erlangen Program(Part IV);and finally,discuss relations of early topological perception to other theories,including Gestalt psychology,Gibsonian psychology,and the computational approach(Part V).PART I:WHY AND HOWÐA TOPOLOGICALAPPROACH TO PERCEPTUAL ORGANIZATIONA paradoxical problem of``where to put the master map'' Fundamental problems faced by the early feature-analysis approach are typically embodied in a paradoxical problem of``where to put the master map''as posed by the feature-integration theory of Treisman and co-workers(e.g.,Treisman& Gelade,1980).Feature-integration theory,consistent with the early feature-analysis approach,adopts a``two-stage model'':In the first,preattentive stage, primitive features,such as colours and orientations,are abstracted effortlessly and in parallel over the entire visual field,and registered in special modules of feature maps;and in the second,attentive stage,focal attention is required to recombine the separate features to form objects.A master map of locations plays a central role in feature binding by tying the separate feature maps together.Within the master map,a focal attention mechanism selects a filled location,binding the activated features linked to that location together to form a coherent object. Problems for feature-integration theory are,however,represented by the question of where exactly the master map of locations fits into the feature integration mechanism.In Treisman's own words,``I have hedged my bets on where to put the master map of locations by publishing two versions of the figure!In one of them,the location map received the output of the feature558CHENmodules(e.g.,Treisman,1986)and in other is placed at an earlier stage of analysis(e.g.,Treisman,1985;Treisman&Gormican,1988)''(Treisman,1988, pp.203±204).To place the master map of locations at an early stage of analysis,in Treisman's own words,``implies that different dimensions are initially conjoined in a single representation before being separately analysed''(Treisman,1988,pp. 203±204).This contradicts the basic position of early feature analysis.Placing the master map later,however,contradicts some behavioural data.One of the strengths of feature-integration theory is that it draws on a number of major pieces of counterintuitive evidence,including illusory conjunction and visual search, which appear to provide strong support for early feature analysis(e.g.,Treisman &Gelade,1980;Treisman&Gormican,1988).However,it is interesting to see that problems for this theory also arise here(e.g.,for a general review,see Humphreys&Bruce,1989).For example,despite the fact that line segments are commonly considered to be basic features,there is markedly little evidence for illusory conjunction where line segments are miscombined into letter-like objects,when letter shapes and line segments forming the letter shapes are used as stimulus forms(e.g.,Duncan,1984).In contrast,there is much stronger evidence that whole letter shapes migrate across words and produce illusory conjunctions of the entire letter shapes,rather than of line segments making up the letter shapes. These findings indicate that letter shapes,as combinations of line segments, behave psychologically as holistic objects,even though line segments are commonly considered to be basic features.Apparently attention,as it relates to feature binding,is not needed for holistic object perception.This suggests that before a stage of separate featural analysis,there must be a stage of early holistic perception in which objects like letters are coded as wholes.Treisman and co-workers,in their effort to explain these counterexamples, have augmented feature-integration theory with new strategies and new mechanisms of attention,such as``guided search''(e.g.,Wolfe,1994),``map suppression'',and dividing items into different depth planes.Nevertheless,these efforts are not completely successful(e.g.,Duncan&Humphreys,1989)but rather in fact illustrate that,despite the attractions of feature-integration theory, the paradoxical problem of``Where to put the master map''stems directly from the fundamental question of``Where visual processing begins''.Perceptual organization:To reverse back theinverted(upside-down)problem of feature binding Regardless of how an object is decomposed into properties and components,the decomposed features themselves are unlikely to be sufficient for achieving object recognition.Indeed,we do not normally perceive isolated features such as brightness,colours,and orientations free from an object,leading to the con-tention that there must be a further process of feature binding.This problem ofGLOBAL TO LOCAL TOPOLOGICAL PERCEPTION559 feature binding presents a central problem for current vision research in parti-cular,and for parallel and distributed modelling of cognition in general(e.g., MuÈller,Elliott,Herman,&Mecklinger,2001).However,despite the centrality of the issue for perceptual theory,it is questionable whether any breakthrough has been made after extensive efforts. Both spatial and temporal factors have been considered as cues for binding features together.But the principles for feature binding based on either space or time are neither always obeyed nor exclusive.Feature binding and perceptual organization appear to be very similar pro-blems(Duncan,1989)in the sense that both deal with similar processes,such as ``what goes with what'',and with similar concepts,such as belongingness and assignment.It turns out that,even though the early feature-analysis viewpoint emphasizes the fundamental importance of early parallel processing,the issue of perceptual organization remains indispensable.Yet,the concepts of``perceptual organization''and``feature binding''involve very different underlying issues, with the former rooted in the idea of early holistic registration and the latter originating from an assumption of early feature analysis.Thus,with respect to the fundamental question of``Where to begin'',perceptual organization and feature binding can be considered contrary concepts,going in opposite directions.In terms of our understanding of objects in the real word,there may be little disagreement that the real features of an object,whether geometrical or physical, exist together as a coherent whole belonging to one entity in the outside world. The question of how the perceptual system represents perceptual objects as fundamental units of conscious perceptual experience,however,has either given rise to much controversy when considered,or not been considered at all.But the truth remains that real features of a real object,at a given time,originally coexist together rather than being separated;a real object is an integral stimulus,a single thing.This truth is a fundamental property of a real-world object.No one doubts the direct perception of various featural properties such as brightness,colour, line orientation,and so on.Why,then,is only this fundamental property of ``belonging together as a whole''excluded from the membership of primitives in our perceptual world?The assumption that the visual system cannot directly perceive a real integral object has not yet been proved or disproved.Indeed,the continuing challenges to the issues of feature binding suggest that this question deserves closer scrutiny.From the perspective of early holistic registration,the feature-binding pro-blem is an ill-posed question:Not just a question of getting off on a wrong foot but even a question of``standing upside down''.In this sense,the feature-binding problem might be a wrong,inverted question.Kubovy and Pomerantz (1981)commented:``the main problems facing us today are quite similar to those faced by the Gestalt psychologists in the first half of this century''.After half a century,the study of visual perception appears,in some sense,to be back560CHENto square one.This situation leads us to wonder whether the problems of feature binding are due to difficulties in posing the fundamental question of``Where to begin''.Where does the above analysis leave us?It suggests that early holistic registration may provide a way to avoid the feature-binding problem by focusing on issues of perceptual organization.In other words,we may apply the concept of perceptual organization to reverse back the inverted(upside-down)question of feature binding.The topological approachDespite its deep and rational core in the idea of early holistic registration,the notion of perceptual organization has its own problems.In particular,like other Gestalt concepts,it has suffered from a lack of proper theoretical treatment. Gestalt evidence has often been criticized for being mainly phenomenological and relying mainly on conscious experience.Consequently,explanations from theories of perceptual organization usually rely on intuitive or mentalistic concepts that are somewhat vague and elusive.What is needed is a proper formal analysis of perceptual organization that goes beyond intuitive approa-ches,and provides a theoretical basis for describing or defining precisely the core concepts related to perceptual organization,e.g.,``global''vs.``local'', ``objects'',``grouping'',and others.Until the intuitive notions of these Gestalt-inspired concepts become properly and precisely defined,the proposed principles of perceptual organization cannot be entirely testable.Delimiting the concept of perceptual organizationTo give a precise description of the essence of perceptual organization,we first need to properly delimit the concept of perceptual organization.On the one hand,as Rock(1986)pointed out,``The concept of perceptual organization should not be defined so loosely as to be a synonym of perception'';on the other hand,this concept should not be so limited as to be unable to cover the great variety and the commonplace occurrence of perceptual organization.The fol-lowing definition of perceptual organization given by Rock(1986)is considered to define properly the very notion of perceptual organization:The meaning of organization here is the grouping of parts or regions of the field with one another,the``what goes with what''problem,and the differentiation of figure from ground.According to this definition,the study of perceptual organization is concerned with early stages of perceptual processing divorced from high-level cognition, and therefore such delimitation pitches our discussion at the right level to answer the basic question of``Where visual processing begins''.On the other hand,theGLOBAL TO LOCAL TOPOLOGICAL PERCEPTION561 concept of perceptual organization discussed in the present paper deals withgeneral processes,such as figure±ground differentiation,grouping,``what goes with what'',belonging and assignment(not particular processes,such as dif-ferentiating luminance flux,discriminating orientation,or recognizing a face), and with abstract things,such as objects,units,and wholes as well as their counterparts,such as items,elements,and parts(not concrete things,such as a line segments,a geometrical figure,a friend's face).These general processes and abstract things represent the variety and commonplace occurrence of per-ceptual organization.Figure1illustrates this.The percept of Figure1A may be described at a semantic level:Either a vase or two profiles face to face.On the other hand,it may be described in terms of the vocabulary of perceptual orga-nization:Two boundaries(units)grouped into one object(as the basis of the percept of a vase)or two boundaries(units)separated into two objects(as the basis of the percept of the two profiles face to face).It is the latter level,the level of perceptual organization,which our present research focuses on.Furthermore, as Figure1B demonstrates,perceptual organization may be perceived without semantic meaning.Here even though there is little semantic meaning involved in the stimulus,either the black parts are perceived to be an unified whole as a figure and the grey parts,another unified whole as background,or vice versa. Top-down processing of prior knowledge or expectation may influence per-ceptual organization,but it will avoid possible confusion if we consider per-ceptual organization and top-down processing of high-level cognition separately.This strengthens the rationale for defining the terminology for describing perceptual organization,emphasizing the primacy of perceptual organization.Major challenges to establishing a proper theoretical treatment on perceptual organization:Its commonplace,and its general and abstract characteristicsThe concept of perceptual organization reflects the most common fact that the phenomenal world contains objects separated from one another by space or background.Phenomena in perceptual organization are usually so common that they have not been looked on as an achievement of the perceptual system,and, thus,as something to be explained(Rock,1986).For example,tremendous efforts have been made to study how to detect line segments with orientation and location,but little attention has been paid to the question of how to perceive a line segment as a single object.While the study of face recognition has advanced considerably,the fundamental grouping question of``which eyes go with which noses,which noses with which mouths,and so forth''(Pomerantz,1981)has been almost completely ignored.One more example shows how commonplace characteristics of perceptual organization make it difficult to realize that there are problems requiringexplanation.In 1990,Rock and Palmer revealed two primary laws of perceptual organization:``Connectedness''and ``common region'',referring to the pow-erful tendency of the visual system to perceive any connected or enclosed regionas a single unit.The phenomena relating to the two laws are so common andself-evident that even classical Gestalt psychologists failed to realizethat an explanation was required for why elements that are either physically connected or enclosed by a closed curve are perceived as a single unit.As ourdiscussionABFigure 1.(A)An ambiguous figures of ``a vase vs.two faces'',showing competing organization.(B)An example of ambiguous figures,showing competing organization without involving semantic meaning.562CHENGLOBAL TO LOCAL TOPOLOGICAL PERCEPTION563 goes on,we will see that these two Gestalt laws closely approach the preciseformal(topological)description of the essence of Gestalt principles.Never-theless,they were neglected for more than a half a century!Besides the problem of being easily overlooked,one more major challenge to establishing a scientific framework for perceptual organization stems from the abstract and general characteristics of the concept.A theoretical explanation of perceptual organization,to possess explanatory power,must be built on even more general and abstract concepts than this vocabulary.The next question, therefore,is:What kinds of concept are more general and abstract than,for example,``what goes with what'',grouping,belongingness,wholes,and per-ceptual objects,and therefore,suitable for the formal analysis of organizational processes?It is not difficult to see that featural properties commonly used in the feature-analysis approach,such as orientation,distance,and size,cannot help out in dealing with the problems facing us in finding a formal explanation of perceptual organization.Topology provides a formal description of perceptual organization:Insight from invariants over shape-changing transformationsTopology has been considered a promising mathematical tool for providing a formal analysis of concepts and processing of perceptual organization(e.g., Chen,1982,2001).Topology is a branch of mathematics that aims at studying invariant properties and relationships under continuous and one-to-one trans-formations,termed topological transformations.The properties preserved under an arbitrary topological transformation are called topological properties.A topological framework of visual perception can be broad enough to encom-pass the variety of phenomena in perceptual organization,such as``what goes with what'',grouping,belonging,and parsing visual scenes into potential objects,and,on the other hand,precise enough to be free from intuitive approaches.Topology is often considered as one of the most abstract branches of mathematics.If the concepts of topology,their relevance and applicability to perceptual organization are difficult to contemplate in the abstract,an appeal to illustrative examples might be helpful.In the following,I will analyse in some depth three typical cases of perceptual organization to demonstrate why and how to advance the topological approach to perceptual organization.Question1:Figure and ground perceptionÐwhat attributes of stimuli determine the segregation of figure from background?Despite the common acceptance that figure±ground perception is fundamental and occurs at the early stage of perception,and despite the large body of empirical findings about。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
INV ARIANT AND PERCEPTUALLY CONSISTENT TEXTURE MAPPING FORCONTENT-BASED IMAGE RETRIEV ALHuizhong Long,Chee Wee Tan,and Wee Kheng LeowSchool of Computing,National University of Singapore,3Science Drive2,Singapore117543,SingaporeABSTRACTTexture is an important visual feature for content-based im-age retrieval.An ideal content-based retrieval system shouldcompare images in its database with the query in a mannerthat is consistent with human’s perception of visual simi-larity.Moreover,texture matching should be invariant totexture scale and orientation because the same texture canappear in the images in varying scales and orientations.Inpractice,however,texture similarity computed using com-putational texture features is not necessarily consistent withhuman’s perception.This paper presents a method of map-ping texture features into a texture space that is scale andorientation invariant,and at the same time,consistent withhuman’s perception.Test results show that this method achie-ves a better retrieval performance than methods that are notinvariant and not perceptually consistent.1.INTRODUCTIONTexture is an important visual feature for content-based im-age retrieval.An ideal content-based image retrieval system should compare images with the user-specified query in a manner that is consistent with human’s perception.More-over,texture matching should be invariant to texture scale and orientation because the same texture can appear in the images in varying scales and orientations.Computational texture features commonly used for im-age retrieval include the statistical features of Tamura et al.[13]used in IBM’s QBIC system,DFT in[12]and the Wold model[6],MRSAR in Wold model,Gabor features in[14], NeTra[9],and invariant texture space[5].Among these models,[5,12,14]are scale and orientation invariant.How-ever,our studies[8]show that texture similarity computed based on these methods are not necessarily consistent with human’s perception.On the other hand,Santini and Jain’s fuzzy features contrast model(FFC)[11]has the potential of accounting for various peculiarities of human’s percep-tual similarity.(1) where is the position of texture in PTS,and is the number of textures.Note that it is not practically viable to perform this mapping in the reverse order because this would require the measurement of human’s perception under various scales and orientations,which is too tedious to perform and too difficult to obtain reliable results.In the construction of the PTS,texture images are preprocessed to normalize texture intensity,contrast,scale,and orientation [7]so that texture similarity measured in PTS is not con-founded by these factors.2.1.Perceptual Texture SpaceSeveral previous works attempted to measure human’s per-ception of texture similarity by having human subjects rank textures along subjectively predefined properties such as di-rectionality,coarseness,busyness,and etc.[13,1]However, the subjects’interpretations of the meaning of these visual properties are expected to vary from one person to the other.To appear in Int.Conf.Image Processing2001.1Therefore,it is uncertain whether the individual ranking re-sults can be combined into group ranking results that rep-resent the perception of a typical person.Ranking along predefined properties may also bias subjects’judgments to-wards these properties rather than other potentially salient features[10].Moreover,recent psychological studies show that subjective texture properties cannot be reliably associ-ated with the dimensions of a perceptual texture space[3]. Therefore,instead of ranking textures according to prede-fined properties,we constructed a PTS using an experimen-tal procedure similar to those of[10,3].60subjects were asked to perform a free sorting task on60texture images according to their perceptual similar-ity.These60images consisted of50textures selected from the Brodatz album,and10other natural textures(such as foliage and roof tiles)not contained in the Brodatz album. Other textures in the Brodatz album were not selected be-cause they were either scaled versions of the50selected textures,or their contents(e.g.,flowers)were much stronger than their texture patterns.It is very difficult for the subjects to avoid classifying the images based on their contents.After obtaining the sorting results,pairwise distances between textures were computed using Donderi’s informa-tion measurement[2]:Table1.Mean squared errors of testing under various con-ditions.SVM0.2160.2450.00990.00740.0161.:new instance;normalized scale,orientation9samples were cropped from each of the60textureimages.For each texture,7samples were used fortraining,1for validation,and1for testing.2.:new type;normalized scale,orientationAll samples of48(80%)of the textures were used fortraining.All samples of6(10%)of the textures wereused for validation,and the rest for testing.3.:new instance;various scales,orientations56variations,with7scales and8orientations,wereproduced from each of the60textures.80%of thesevariations were randomly selected for training,10%for validation,and10%for testing.4.:new type;various scales,orientationsSimilar to condition2except that samples with vari-ous scales and orientations were used.5.:randomOf all the scale and orientation variations of the60textures,80%were randomly selected for training,10%for validation,and10%for testing.Ten-fold cross-validation tests were performed for all5con-ditions.Conditions2and4are the most stringent because their testing samples are perceptually different from their training samples.Their testing errors are expected to be larger than those of the other conditions.The last condi-tion mimics a practical situation in which random samples at random scales and orientations are available for training.To obtain the baseline results,Support Vector Machine (SVM)regression was applied to thefive conditions as well. For SVM regression,Gaussian kernels were chosen because our tests showed that they perform better than polynomial and sigmoid kernels.Various parameter values of Gaussian kernels were tested to obtain the best performance.Table1summarizes the testing results.Under the con-dition of normalized scale and orientation(conditions1and 2),the invariant network achieves a performance compara-ble to that of SVM regression.This is expected because both these methods are good at nonlinear regression.Also as expected,testing errors for condition2(new type)are larger than those for condition1(new instance).Under conditions3,4,and5,SVM regression does not perform as well as the invariant network because the net-work is designed to capture variations of scale and orienta-tion while the SVM is not.SVM’s testing errors for these three conditions are roughly the same as that for condition2 (new type,normalized scale and orientation).These results indicate that SVM treats textures with different scales and orientations as new texture types.On the other hand,the in-variant network performs texture mapping in a manner that is invariant to scale and orientation.The invariant network performs better in conditions3and4than in conditions1 and2because there are more training samples in the former conditions than in the latter.4.APPLICATION EXAMPLETo further assess the invariant network’s performance,tex-ture image retrieval tests were performed.In a similar test conducted by Ma and Manjunath[9],human subjects were asked to classify similar Brodatz textures into the same group.During retrieval,textures in the same group as the query are considered as relevant.In our tests,texture groupings were derived from-means clustering of the textures in PTS.Out of the60textures in PTS,20groups were derived,giving an average of3textures per group.Three types of retrieval tests were performed to assess different aspects of the invariant network:1.Assessment of the benefit of perceptual mapping.Texture images with normalized scale and orienta-tion were used.Invariant network’s performance wascompared with those of SVM regression and fuzzyfeatures contrast model(FFC)of Santini and Jain[11].Various parameter values of FFC were tested to ob-tain the best performance.In addition,scaled Eu-clidean distance was also tested because it was com-pared with FFC in[11].2.Assessment of the benefit of invariant mapping.Textures with various scales and orientations were used.Invariant network’s performance was compared withthat of SVM regression.3.Assessment of the combined benefit of invariant andperceptual mapping.Textures with various scales and orientations were used.Invariant network’s performance was compared withthose of SVM regression,FFC,and scaled Euclidean.In these tests,invariant network and SVM regression trained under the random condition in Section2.2were used since random condition is similar to practical application condi-tion.For all the tests,30textures were randomly selected as the query textures,and the average normalized precision and average normalized recall are reported in this paper.Table2summarizes the test results for retrieval condi-tion1.Both invariant network and SVM regression have 3Table2.Results of retrieving textures with normalized scale and orientation.norm.recall0.5890.6220.2910.143norm.precisioninvariant network0.959SVM regression0.833FFC0.650scaled Euclidean0.582been trained to map texture features to PTS.Therefore,their performances are better than those of FFC and scaled Eu-clidean.FFC has a better performance than does scaled Eu-clidean distance,indicating that it can indeed capture some aspects of human perception.Table3summarizes the test results for retrieval condi-tions2and3.Invariant network has the best performance because it can perform both invariant and perceptual map-pings well.Although SVM regression has been trained to perform perceptual mapping,it does not capture the varia-tions of texture scale and orientation.Therefore,it does not perform as well as the invariant network.FFC and scaled Euclidean distance have lower precision and recall than do invariant network and SVM because they do not perform invariant and perceptual mappings well.5.CONCLUSIONThis paper presented a neural network that is trained to opti-mize both invariant and perceptual mappings.Thefirst three layers of the network map the Gabor features of a texture to an invariant space,and the last three layers map the texture to the perceptual space,which is constructed using data ob-tained from psychological test of human’s perception.Ex-tensive tests were conducted to assess the performance of the invariant network.Test results show that the invariant network can perform invariant and perceptual mappings ac-curately,and invariant and perceptual mappings indeed im-prove the performance of texture image retrieval.6.REFERENCES[1]N.Abbadeni,D.Ziou,and putationalmeasures corresponding to perceptual textural fea-tures.In Proc.ICIP,2000.[2]rmation measurement of distinc-tiveness and similarity.Perception and Psychophysics,44(6):576–584,1988.[3]C.Heaps and S.Handel.Similarity and features ofnatural textures.J.Expt.Psycho.:Human Perceptionand Performance,25(2):299–320,1999.[4]Y.LeCun and Y.Bengio.Convolutional networks forimages,speech,and time series.In M.A.Arbib,ed-itor,The Handbook of Brain Theory and Neural Net-works,pages255–258.MIT Press,1995.[5]W.K.Leow and i.Scale and orientation-invariant texture matching for image retrieval.InM.K.Pietik¨a inen,editor,Texture Analysis in MachineVision.World Scientific,2000.[6]F.Liu and R.W.Picard.Periodicity,directionality,andrandomness:Wold features for image modeling andretrieval.IEEE Trans.PAMI,18(7):722–733,1996.[7]H.-Z.Long and W.K.Leow.Perceptual texture spacefor content-based image retrieval.In Proc.Int.Conf.on Multimedia Modeling(MMM2000),pages167–180,2000.[8]H.-Z.Long and W.K.Leow.Perceptual texture spaceimproves perceptual consistency of computational fea-tures.In Proc.IJCAI,2001.[9]W.Y.Ma and B.S.Manjunath.Texture features andlearning similarity.In Proc.CVPR,pages1160–1169,1996.[10]A.R.Rao and G.L.Lohse.Identifying high level fea-tures of texture perception.CVGIP:Graphical Modelsand Image Processing,55(3):218–233,1993.[11]S.Santini and R.Jain.Similarity measures.IEEETrans.PAMI,21(9):871–883,1999.[12]D.-G.Sim,H.-K.Kim,and D.-I.Oh.Translation,scale,and rotation invariant texture descriptor fortexture-based image retrieval.In Proc.ICIP,2000.[13]H.Tamura,S.Mori,and T.Yamawaki.Textural fea-tures corresponding to visual perception.IEEE Trans.SMC,8(6):460–47,1978.[14]A.Teuner,O.Pichler,J.E.Santos Conde,and B.J.Hosticka.Orientation-and scale-invariant recognitionof textures in multi-object scenes.In Proc.ICIP,pages174–177,1997.4。