nucleation and growth in one dimension we study statistical properties of the kolmogorov avrami johnson mehl nucleation and growth model in one dimension we obtain exact results for the gap density as well as the island distribution when all nucleation events occur simultaneously we show that the island distribution has discontinuous derivatives on the rays mathit x mathit n t nt n 1 2 3 we introduce an accelerated growth mechanism with growth rate increasing linearly with the island size we solve for the interisland gap density and show that the system reaches complete coverage in a finite time and that the near critical behavior of the system is robust i e it is insensitive to details such as the nucleation mechanism textcopyright 1996 the american physical society

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

isospectral domains with mixed boundary conditions we construct a series of examples of planar isospectral domains with mixed dirichlet neumann boundary conditions this is a modification of a classical problem proposed by m kac

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

isospectral deformations i riemannian structures on two step nilspaces on montre comment construire des exemples de deformations isospectrales de metriques de riemann sur des varietes sur lesquelles des recouvrements de certains groupes de lie nilpotents agissent

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

eigenmodes of isospectral drums recently it was proved that there exist nonisometric planar regions that have identical laplace spectra that is one cannot hear the shape of a drum the simplest isospectral regions known are bounded by polygons with reentrant corners while the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenvalues themselves furthermore standard numerical methods for computing the eigenvalues such as adaptive finite elements are highly inefficient physical experiments have been performed to measure the spectra but the accuracy and flexibility of this method are limited we describe an algorithm due to descloux and tolley comput methods appl mech engrg 39 1983 pp 37 53 that blends singular finite elements with domain decomposition and show that with a modification that doubles its accuracy this algorithm can be used to compute efficiently the eigenvalues for polygonal regions we present results accurate to 12 digits for the most famous pair of isospectral drums as well as results for another pair

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

on isospectral periodic potentials on a discrete lattice i on considere une version discretisee et generalisee de l equation de schrodinger et on etudie le probleme aux valeurs propres

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

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

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

on ergodic one dimensional cellular automata we show that all onto cellular automata defined on the binary sequence space are invariant with respect to the haar measure and that an extensive class of such maps including many nonlinear ones are strongly mixing with respect to the haar measure

the isospectral fruits of representation theory quantum graphs and drums we present a method which enables one to construct isospectral objects such as quantum graphs and drums one aspect of the method is based on representation theory arguments which are shown and proved the complementary part concerns techniques of assembly which are both stated generally and demonstrated for this purpose quantum graphs are grist to the mill we develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects we discuss the theoretical implications which include sunada s theorem of isospectrality sunada 1985 ann math 121 169 arising as a particular case of this method a gallery of new isospectral examples is presented and some known examples are shown to result from our theory

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

ergodicity transitivity and regularity for linear cellular automata over z m 1 we study the dynamical behavior of d dimensional linear cellular automata over zm we provide an easy to check necessary and sufficient condition for a d dimensional linear cellular automata over zm to be ergodic and topologically transitive as a byproduct we get that for linear cellular automata ergodicity is equivalent to topological transitivity finally we prove that for 1 dimensional linear cellular automata over zm regularity denseness of periodic orbits is equivalent to surjectivity

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

continuous families of isospectral metrics on simply connected manifolds we construct continuous families of riemannian metrics on certain simply connected manifolds with the property that the resulting riemannian manifolds are pairwise isospectral for the laplace operator acting on functions these are the first examples of simply connected riemannian manifolds without boundary which are isospectral but not isometric for example we construct continuous isospectral families of metrics on the product of spheres s4 x s3 x s3 the metrics considered are not locally homogeneous for a big class of such families the set of critical values of the scalar curvature function changes during the deformation moreover the manifolds are in general not isospectral for the laplace operator acting on 1 forms

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

spectral problems with mixed dirichlet neumann boundary conditions isospectrality and beyond consider a bounded domain with the dirichlet condition on a part of the boundary and the neumann condition on its complement does the spectrum of the laplacian determine uniquely which condition is imposed on which part we present some results conjectures and problems related to this variation on the isospectral theme

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

small isospectral and nonisometric orbifolds of dimension 2 and 3 revisiting a construction due to vigneras we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic fuchsian and kleinian groups that are laplace isospectral in fact representation equivalent but nonisometric

ergodic endomorphisms of compact abelian groups we show that for a surjective endomorphism of a compact abelian group ergodicity is equivalent to a condition which impliesr mixing for allr 1 and we characterize such maps algebraically this is then used in proving the ergodicity of an extensive class of endomorphisms of the binary sequence space as a simple corollary it is found that one dimensional linear cellular automata and the accumulator automata arer mixing for allr 1

reconstructing dna replication kinetics from small dna fragments in higher organisms dna replicates simultaneously from many origins recent in vitro experiments have yielded large amounts of data on the state of replication of dna fragments from measurements of the time dependence of the average size of replicated and nonreplicated domains one can estimate the rate of initiation of dna replication origins as well as the average rate at which dna bases are copied one problem in making such estimates is that in the experiments the dna is broken up into small fragments whose finite size can bias downward the measured averages here we present a systematic way of accounting for this bias by deriving theoretical relationships between the original domain length distributions and fragment domain length distributions we also derive unbiased average domain length estimators that yield accurate results even in cases where the replicated or nonreplicated domains are larger than the average dna fragment then we apply these estimators to previously obtained experimental data to extract improved estimates of replication kinetics parameters

nucleation and growth in one dimension i the generalized kolmogorov johnson mehl avrami model motivated by a recent application of the kolmogorov johnson mehl avrami skjmad model to the study of dna replication we consider the one dimensional s1dd version of this model we generalize previous work to the case where the nucleation rate is an arbitrary function istd and obtain analytical results for the timedependent distributions of various quantities ssuch as the island distributiond we also present improved computer simulation algorithms to study the 1d kjma model the analytical results and simulations are in excellent agreement

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

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

mean field theory of polynuclear surface growth we study statistical properties of a continuum model of polynuclear surface growth on an infinite substrate we develop a self consistent mean field theory which is solved to deduce the growth velocity and the extremal behaviour of the coverage numerical simulations show that this theory gives an improved approximation for the coverage compared to the previous linear recursion relations approach furthermore these two approximations provide useful upper and lower bounds for a number of characteristics including the coverage growth velocity and the width exponent

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

emulation and subshifts of finite type in cellular automata abstract the dynamics of the class of nearest neighbour automata on binary sequence space z z 2 is studied the surjective mappings are identified and these are all strongly mixing except for the identity and parity maps those automata which act like emulate either the identity shift or inverse shift maps are identified and the shift emulators are shown to be subshifts of finite type on their limits sets

determinants of laplacians and isopolar metrics on surfaces of infinite area we construct a determinant of the laplacian for infinite area surfaces which are hyperbolic near infinity and without cusps in the case of a convex co compact hyperbolic metric the determinant can be related to the selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the laplacian in the hyperbolic near infinity case the determinant is analyzed through the zeta regularized relative determinant for a conformal metric perturbation we establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics these results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances

spectral decomposition of square tiled surfaces we consider n copies of a square s0 and define selfadjoint extensions of the euclidean laplacian acting on left mathcal c 0 infty s 0 right n by choosing some boundary conditions that are parametrized by two unitary matrices h and v acting on mathbb c n denoting by sp n h v the spectrum of such an operator we derive conditions on h and v so that the following spectral decomposition holds rm sp delta n h v bigcup 1 leq i leq k rm sp delta n i h i v i quad rm with sum 1 k n i n if h and v are permutation matrices this gives a spectral decomposition of the spectrum of the square tiled surface defined by the corresponding permutations we apply this to derive examples related to isospectrality and to high multiplicity

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

kinetics of magnetization switching in a 1 d system size distribution of unswitched domains abstract we construct the evolution equation of the size distribution function of the unswitched domains of a model of a 1 d system in a switching field and solve it as an initial value problem as the time goes on the distribution approaches to the fixed point which is the exponential distribution this feature is unaffected by the finiteness of the critical radius of nucleation

quasi regular representations of discrete groups and associated c algebras let g be a countable group we introduce several equivalence relations on the set sub g of subgroups of g defined by properties of the quasi regular representations g h associated to h sub g and compare them to the relation of g conjugacy of subgroups we define a class sub sg g of subgroups these are subgroups with a certain spectral gap property and show that they are rigid in the sense that the equivalence class of h sub sg g for any one of the above equivalence relations coincides with the g conjugacy class of h next we introduce a second class sub w par g of subgroups these are subgroups which are weakly parabolic in some sense and we establish results concerning the ideal structure of the c

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

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

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

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

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