kac s question planar isospectral pairs and involutions in projective space ii classification of generalized projective isospectral data in am math monthly 73 1 23 1966 kac asked his famous question can one hear the shape of a drum which was eventually answered negatively in gordon et al 1992 invent math 110 1 22 by construction of planar isospectral pairs giraud 2005 j phys a math gen 38 l477 83 observed that most of the known examples can be generated from solutions of a certain equation which involves a set of involutions of an n dimensional projective space over some finite field he then generated all possible solutions for n 2 when the involutions fix the same number of points in thas 2006 j phys a math gen 39 l385 8 we showed that no other examples arise for any other dimension still assuming that the involutions fix the same number of points in this paper we study the problem for involutions not necessarily fixing the same number of points and solve the problem completely

dynamical spectral rigidity among mathbb z _2 symmetric strictly convex domains close to a circle appendix b coauthored with h hezari

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

geodesics and commensurability classes of arithmetic hyperbolic 3 manifolds this sharpens 10 where it was shown that the complex length spectrum of m determines its commensurability class suppose m is an arithmetic hyperbolic 3 manifold which is not commensurable to m theorem 1 1 implies ql m 6 ql m though by example 2 1 below it is possible that one of ql m or ql m contains the other by the length formulas recalled in 2 1 and 2 2 each element of ql m ql m is a rational multiple of the logarithm of a real algebraic number as noted by prasad and rapinchuk in 9 the gelfond schneider theorem 1 implies that a ratio of such logarithms is transcendental if it is irrational thus if ql m ql m then is transcendental for all non zero ql m recently prasad and rapinchuk have shown in 9 that if m is an arithmetic hyperbolic manifold of even dimension then ql m and the commensurability class of m determine one another in addition they have shown that this is not always true for arithmetic hyperbolic 5 manifolds however they have announced a proof that for all locally symmetric spaces associated to a specified absolutely simple lie group there are only finitely many commensurability classes of arithmetic lattices giving rise to a given rational length spectrum it is known see 4 pp 415 417 that for closed hyperbolic manifolds the spectrum of the laplace beltrami operator action on l2 m counting multiplicities determines the set of lengths of closed geodesics on m without counting multiplicities hence theorem 1 1 implies

isospectral graphs with identical nodal counts according to a recent conjecture isospectral objects have different nodal count sequences gnutzmann et al 2005 j phys a math gen 38 8921 33 we study generalized laplacians on discrete graphs and use them to construct the first non trivial counterexamples to this conjecture in addition these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

an analogue of the prime number theorem for closed orbits of axiom a flows for an axiom a flow restricted to a basic set we extend the zeta function to an open set containing w s h where h is the topological entropy this enables us to give an asymptotic formula for the number of closed orbits by adapting the wiener ikehara proof of the prime number theorem

twisted perron frobenius theorem and l functions abstract a theorem of perron frobenius type and its twisted version are established in a setting of topological graphs the applications include a partial extension of selberg s results on his zeta functions and a result by parry and pollicott on meromorphic continuations of dynamical zeta functions to certain l functions associated to a dynamical system of anosov type

many hard examples for resolution for every choice of positive integers c and k such that k 3 and c 2 k 0 7 there is a positive number e such that with probability tending to 1 as n tends to a randomly chosen family of cn clauses of size k over n variables is unsatisfiable but every resolution proof of its unsatisfiability must generate at least 1 e n clauses

calcul du spectre d une nilvariete de rang deux et applications on calcule en utilisant la theorie des orbites de kirillov le spectre d une nilvariete compacte de rang deux puis on utilise ce calcul pour etudier et caracteriser les deformations isospectrales de ces varietes

the covering spectrum of a compact length space we define a new spectrum for compact length spaces and riemannian manifolds called the covering spectrum which roughly measures the size of the one dimensional holes in the space more specifically the covering spectrum is a set of real numbers 0 which identify the distinct covers of the space we investigate the relationship between this covering spectrum the length spectrum the marked length spectrum and the laplace spectrum we analyze the behavior of the covering spectrum under gromov hausdorff convergence and study its gap phenomenon

kac s question planar isospectral pairs and involutions in projective space in a paper published in am math mon 1966 73 1 23 kac asked his famous question can one hear the shape of a drum gordon et al answered this question negatively by constructing planar isospectral pairs in their paper published in invent math 1992 110 1 22 only a finite number of pairs have been constructed till now further in j phys a math gen 2005 38 l477 83 giraud showed that most of the known examples can be generated from solutions of a certain equation which involves certain involutions of an n dimensional projective space over some finite field he then generated all possible solutions when n 2 in this letter we handle all dimensions and show that no other examples arise

subgroups of finite soluble groups inducing the same permutation character in this paper there are found necessary and sufficient conditions that a pair of solvable finite groups say g and k must satisfy for the existence of a solvable finite group l containing two isomorphic copies of g and h inducing the same permutation character also a construction of l is given as an iterated wreath product with respect to their actions on their natural modules of finite one dimensional affine groups

coarse non amenability and covers with small eigenvalues given a closed riemannian manifold m and a virtual epimorphism pi 1 m twoheadrightarrow mathbb f 2 of the fundamental group onto a free group of rank 2 we construct a tower of finite sheeted regular covers left m n right n 0 infty of m such that 1 mn 0 as n this is the first example of such a tower which is not obtainable up to uniform quasi isometry or even up to uniform coarse equivalence by the previously known methods where 1 m is supposed to surject onto an amenable group

counting isospectral manifolds abstract given a simple lie group h of real rank at least 2 we show that the maximum cardinality of a set of isospectral non isometric h locally symmetric spaces of volume at most x grows at least as fast as x c log x log log x 2 where c c h is a positive constant in contrast with the real rank 1 case this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of belolipetsky and lubotzky 2 our proof uses sunada s method results of 2 and some deep results from number theory we also discuss an open number theoretical problem which would imply an even faster growth estimate

cospectral graphs gm switching and regular rational orthogonal matrices of level p abstract the gm switching invented by godsil and mckay is a powerful method to construct cospectral graphs in its simplest version the method can be roughly described as follows let a g be the adjacency matrix of a graph g on n vertices let q diag r i n 2 m be a block diagonal matrix where r 1 m j i 2 m is an orthogonal matrix of order 2m j is the all one matrix and i k denotes the identity matrix of order k if g satisfies certain conditions then q t a g q is an adjacency matrix of a graph g which is cospectral with g with cospectral complements in this paper we consider the problem of replacing the above r with another rational orthogonal matrix u of order 2p p an odd prime we characterize all graphs g admitting a gm switching corresponding to q diag u i n 2 p i e such that q t a g q is an adjacency matrix of a graph g and hence giving a counterpart of the gm switching method for generating cospectral graphs with cospectral complements which can be used to construct pairs of non isomorphic cospectral graphs that are not obtainable by the original gm switching method next we consider the problem of how to determine whether a given graph has a non trivial gm switching associated with some u a simple algebraic condition is presented which partially answers this question

generalized pulse spectrum technique the generalized pulse spectrum technique gpst is a versatile and efficient iterative numerical algorithm for solving multiparameter inverse problems of a system of nonlinear partial differential equations here it is used to determine bulk modulus shear modulus and density variation from seismic data numerical simulations of simple but nontrivial examples are carried out on coarse uniform grids and adaptive grids to test and to study the general characteristics of gpst without the real measurements seismograms gpst does give reasonably good results it can be made more efficient by implementing parallelism automatic adaptive grids and methods of domain decomposition into the numerical algorithm

equivalence between isospectrality and isolength spectrality for a certain class of planar billiard domains isospectrality of the planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to isolength spectrality of the corresponding domains although an explicit and exact trace formula such as poisson s summation formula or selberg s trace formula is not known to exist for such planar domains equivalence between isospectrality and isolength spectrality in a certain setting can be proved by employing the matrix representation of transplantation of eigenfunctions as an application of the equivalence transplantable pairs of domains which are all isospectral pair of planar domains and therefore counter examples of kac s question can one hear the shape of a drum are numerically enumerated and it is found that at least up to the domain composed of 13 building blocks transplantable pairs coincide with those constructed by the method due to sunada

a widely convergent generalized pulse spectrum technique for the inversion of two dimensional acoustic wave equation an algorithm of the generalized pulse spectrum technique gpst is used for the inverse problems of two dimensional acoustic wave equation and a skill of saving computational cost is designed in choosing regularizing parameters to overcome the defect of the local convergence of the gpst a widely convergent gpst wcgpst is constructed numerical simulations are carried out to test the feasibility and to study the general characteristics of the wcgpst it is found that the wcgpst is not only as robust as the standard gpst but also possessing widely convergent region

integral gassman equivalence of algebraic and hyperbolic manifolds in this paper we construct arbitrarily large families of smooth projective varieties and closed riemannian manifolds that share many algebraic and analytic invariants for instance every non arithmetic closed hyperbolic 3 manifold admits arbitrarily large collections of non isometric finite covers which are strongly isospectral length isospectral and have isomorphic integral cohomology where the isomorphisms commute with restriction and co restriction we can also construct arbitrarily large collections of pairwise non isomorphic smooth projective surfaces where these isomorphisms in cohomology are natural with respect to hodge structure or as galois modules in particular the projective varieties have isomorphic picard and albanese varieties and they also have isomorphic effective chow motives our construction employs an integral refinement of the gassman sunada construction that has recently been utilized by d prasad one application of our work shows the non injectivity of the map from the grothendieck group of varieties over overline mathbf q to the grothendieck group of the category of effective chow motives we also answer a question of d prasad

finite geometries and diffractive orbits in isospectral billiards several examples of pairs of isospectral planar domains have been produced in the two dimensional euclidean space by various methods we show that all these examples rely on the symmetry between points and blocks in finite projective spaces from the properties of these spaces one can derive a relation between green s functions as well as a relation between diffractive orbits in isospectral billiards

line graphs and the transplantation method abstract we study isospectrality for mixed dirichlet neumann boundary conditions and extend the previously derived graph theoretic formulation of the transplantation method led by the theory of brownian motion we introduce vertex colored and edge colored line graphs that give rise to block diagonal transplantation matrices in particular we rephrase the transplantation method in terms of representations of free semigroups and provide a method for generating adjacency cospectral weighted directed graphs

can one hear the shape of a drum revisited a famous inverse problem posed by m kac can one hear the shape of a drum is concerned with isospectrality of drums or planer billiards and the first counter example was constructed by gordon webb and wolpert 1992 invent math 110 1 here we present pieces of numerical evidence showing that one can distinguish isospectral drums by measuring the scattering poles of exterior neumann problems this is based on the observation that the fredholm determinant appearing in the boundary element method admits a factorization into interior and exterior parts

isospectral cayley graphs of some finite simple groups we apply spectral analysis of quotients of the bruhat tits buildings of type ad 1 to construct isospectral non isomorphic cayley graphs of the finite simple groups psld fq for every d 5 d 6 6 and prime power q 2

isospectral drums in involution graphs and euclidean ti domains the widely investigated question can one hear the shape of a drum which kac posed in his published lecture kac 1966 am math mon 73 1 23 was eventually answered negatively in gordon et al 1992 invent math 110 1 22 by construction of isospectral pairs in up to present all known planar counter examples are constructed by a certain tiling method and in this communication we call such examples isospectral euclidean ti domains from counter examples of this type one can construct a pair of finite involution graphs in this communication we address the question as to how the isospectrality of the domains for the laplacian influences the cospectrality of the involution graphs

discrete homotopies and the fundamental group abstract we generalize and strengthen the theorem of gromov that the fundamental group of any compact riemannian manifold of diameter at most d has a set of generators g 1 g k of length at most 2 d and relators of the form g i g m g j in particular we obtain an explicit bound for the number k of generators in terms of the number of short loops at every point and the number of balls required to cover a given semi locally simply connected geodesic space as a corollary we obtain a fundamental group finiteness theorem new even for riemannian manifolds that replaces the curvature and volume conditions of anderson and the 1 systole bound of shen wei by more general geometric hypothesis implied by these conditions this theorem in turn is a special case of a theorem for arbitrary compact geodesic spaces proved using the method of discrete homotopies introduced by the first author and v n berestovskii central to the proof is the notion of homotopy critical spectrum introduced in this paper as a natural consequence of discrete homotopy methods this spectrum is closely related to the sormani wei covering spectrum which is a subset of the classical length spectrum studied by de verdiere and duistermaat guillemin it is completely determined including multiplicity by special closed geodesics called essential circles

a widely convergent generalized pulse spectrum technique for the coefficient inverse problem of differential equations abstract a new algorithm of the generalized pulse spectrum technique gpst is constructed for the solution of coefficient inverse problems of differential equations and a skill of saving computational cost is designed in choosing regularizing parameters to overcome the defect of the local convergence of the gpst a widely convergent gpst is illustrated numerical simulations are carried out for solving the inverse problem of 1 d seismic prospection to test the feasibility without real measurement data the computational results show the method s effectiveness

counting and effective rigidity in algebra and geometry the purpose of this article is to produce effective versions of some rigidity results in algebra and geometry on the geometric side we focus on the spectrum of primitive geodesic lengths resp complex lengths for arithmetic hyperbolic 2 manifolds resp 3 manifolds by work of reid this spectrum determines the commensurability class of the 2 manifold resp 3 manifold we establish effective versions of these rigidity results by ensuring that for two incommensurable arithmetic manifolds of bounded volume the length sets resp the complex length sets must disagree for a length that can be explicitly bounded as a function of volume we also prove an effective version of a similar rigidity result established by the second author with reid on a surface analog of the length spectrum for hyperbolic 3 manifolds these effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras to prove these effective rigidity results we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

a strategy for nonlinear elastic inversion of seismic reflection data the problem of interpretation of seismic reflection data can be posed with sufficient generality using the concepts of inverse theory in its roughest formulation the inverse problem consists of obtaining the earth model for which the predicted data best fit the observed data if an adequate forward model is used this best model will give the best images of the earth s interior three parameters are needed for describing a perfectly elastic isotropic earth the density x and the lame parameters x and x or the density x and the p wave and s wave velocities x and x the choice of parameters is not neutral in the sense that although theoretically equivalent if they are not adequately chosen the numerical algorithms in the inversion can be inefficient in the long spatial wavelengths of the model adequate parameters are the p wave and s wave velocities while in the short spatial wavelengths p wave impedance s wave impedance and density are adequate the problem of inversion o

decomposition of tautologies into regular formulas and strong completeness of connection graph resolution this paper addresses and answers a fundamental question about resolution informally what is gained with respect to the search for a proof by performing a single resolution step it is first shown that any unsatisfiable formula may be decomposed into regular formulas provable in linear time by resolution a relevant resolution step strictly reduces at least one of the formulas in the decomposition while an irrelevant one does not contribute to the proof in any way the relevance of this insight into the nature of resolution and of the unsatisfiability problem for the development of proof strategies and for complexity considerations are briefly discussed the decomposition also provides a technique for establishing completeness proofs for refinements of resolution as a first application connection graph resolution is shown to be strongly complete this settles a problem that remained open for two decades despite many proff attempts the result is relevant for theorem proving because without strong completeness a connection graph resolution prover might run into an infinite loop even on the ground level

psln q as operator group of isospectral drums in a paper by m kac 1966 am math mon 73 1 23 kac asked his famous question can one hear the shape of a drum which was answered negatively in gordon et al 1992 invent math 110 1 22 by construction of planar isospectral pairs in buser et al 1994 int math res not 9 it is observed that all operator groups associated with the known counter examples are isomorphic to one of psl3 2 psl3 3 psl4 2 and psl3 4 we show that if d1 d2 is a pair of non congruent planar isospectral domains constructed from unfolding a polygonal base tile and with associated operator group psln q then n q belongs to this very restricted list

the intractability of resolution abstract we prove that for infinitely many disjunctive normal form propositional calculus tautologies the length of the shortest resolution proof of cannot be bounded by any polynomial of the length of the tautologies we use were introduced by cook and reckhow 1979 and encode the pigeonhole principle extended resolution can furnish polynomial length proofs of these formulas

efficiency improvement of gpst inversion algorithm abstract to demonstrate the possibility of improving the efficiency of the generalized pulse spectrum technique gpst inversion algorithm by implementing special re structuralization and high level parallelism into the system of discretized fredholm integral equations of the first kind a simple two parameter inverse problem of two dimensional linear evolution equation is considered numerical simulations are carried out to test the feasibility and to study the general characteristics of the improved gpst without real measurement data it is found that the improved gpst is not only as robust as the standard gpst but also possessing the speedup very close to the estimated one by performing the computational complexity analysis based upon flo count

length spectra and degeneration of flat metrics in this paper we consider flat metrics semi translation structures on surfaces of finite type there are two main results the first is a complete description of when a set of simple closed curves is spectrally rigid that is when the length vector determines a metric among the class of flat metrics secondly we give an embedding into the space of geodesic currents and use this to obtain a compactification for the space of flat metrics the geometric interpretation is that flat metrics degenerate to mixed structures on the surface part flat metric and part measured foliation

hard examples for resolution exponential lower bounds are proved for the length of resolution refutations of sets of disjunctions constructed from expander graphs using the method of tseitin since these sets of clauses encode biconditionals they have short polynomial length refutations in a standard axiomatic formulation of propositional calculus

diffractive orbits in isospectral billiards isospectral domains are non isometric regions of space for which the spectra of the laplace beltrami operator coincide in the two dimensional euclidean space instances of such domains have been given it has been proved for these examples that the length spectrum that is the set of the lengths of all periodic trajectories coincides as well however there is no one to one correspondence between the diffractive trajectories it will be shown here how the diffractive contributions to the green functions match nevertheless in a one to three correspondence

transplantation and jacobians of sunada isospectral riemann surfaces abstract we consider relations among the jacobians of isospectral compact riemann surfaces constructed using sunada s theorem we use a simple algebraic formulation of transplantation of holomorphic 1 forms and singular 1 cycles to obtain two main results first we obtain a geometric proof of a result of prasad and rajan that sunada isospectral riemann surfaces have isogenous jacobians second we determine a relationship weaker than isogeny that holds among the jacobians of sunada isospectral riemann surfaces when the jacobians extra structure as principally polarized abelian varieties is taken into account we also show all sunada isospectral manifolds have isomorphic real cohomology algebras finally we exhibit transplantation of cycles explicitly in a concrete example of a pair of isospectral riemann surfaces constructed by brooks and tse

free energy as a geometric invariant the free energy plays a fundamental role in statistical and condensed matter physics a related notion of free energy plays an important role in the study of hyperbolic dynamical systems in this paper we introduce and study a natural notion of free energy for surfaces with variable negative curvature this geometric free energy encodes a new type of marked length spectrum of closed geodesics which lies between the well known marked length spectrum marked by the corresponding element of the fundamental group and the unmarked length spectrum we prove that the free energy parametrizes the boundary of the domain of convergence of a poincare series which also encodes this spectrum we also show that this new length spectrum or equivalently the geometric free energy is not an isometry invariant in the final section we use tools from multifractal analysis to effect a fine asymptotic comparison of word length and geodesic length of closed geodesics we hope that our geometric understanding of free energy will provide new insight into this fundamental physical and dynamical quantity

inverse problems for elastic plates with variable flexural rigidity abstract the pulse spectrum technique pst an iterative numerical algorithm is extended and further developed to solve inverse problems in the dynamic structural identification and the dynamic structural design based on the two dimensional elastic plate model with variable flexural rigidity numerical simulations on several simple examples are carried out to test the feasibility and to study the general characteristics of pst without the real measurement data for the case of dynamic structural identification or with the prescribed design data for the case of dynamic structural design it is found that pst does give satisfactory results and also fares very well in regard to the given four practical criteria for evaluating numerical methods

comment on resolving isospectral drums by counting nodal domains in gnutzmann et al 2005 j phys a math gen 38 8921 33 the authors studied the 4 parameter family of isospectral flat 4 tori t a b c d discovered by conway and sloane with a particular method of counting nodal domains they were able to distinguish these tori numerically by computing the corresponding nodal sequences relative to a few explicit tuples a b c d in this note we confirm the expectation expressed in gnutzmann et al 2005 j phys a math gen 38 8921 33 by proving analytically that their nodal count distinguishes any 4 tuple of distinct positive real numbers