stability and isolation phenomena for yang mills fields in this article a series of results concerning yang mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods one of our main results is to prove that any weakly stable yang mills field overs4 with groupg su2 su3 oru2 is either self dual or anti self dual the analogous statement for so4 bundles is also proved the other main series of results concerns gap phenomena for yang mills fields as a consequence of the non linearity of the yang mills equations we can give explicitc0 neighbourhoods of the minimal yang mills fields which contain no other yang mills fields in this part of the study the nature of the groupg does not matter neither is the dimension of the base manifold constrained to be four

density of states potts zeros and fisher zeros of the q state potts model for continuous q the q state potts model can be extended to noninteger and even complex q by expressing the partition function in the fortuin kasteleyn f k representation in the f k representation the partition function z q a is a polynomial in q and v a ensuremath 1 a e ensuremath beta j and the coefficients of this polynomial ensuremath phi b c are the number of graphs on the lattice consisting of b bonds and c connected clusters we introduce the random cluster transfer matrix to compute ensuremath phi b c exactly on finite square lattices with several types of boundary conditions given the f k representation of the partition function we begin by studying the critical potts model z mathrm cp z q a c q where a c q 1 sqrt q we find a set of zeros in the complex w sqrt q plane that map to or close to the beraha numbers for real positive q we also identify q c l the value of q for a lattice of width l above which the locus of zeros in the complex p v sqrt q plane lies on the unit circle by finite size scaling we find that 1 q c l ensuremath rightarrow 0 as stackrel ensuremath rightarrow l ensuremath infty we then study zeros of the antiferromagnetic af potts model in the complex q plane and determine q c a the largest value of q for a fixed value of a below which there is af order we find excellent agreement with baxter s conjecture q c mathrm af a 1 ensuremath a a 3 we also investigate the locus of zeros of the ferromagnetic potts model in the complex q plane and confirm that q c mathrm fm a a ensuremath 1 2 we show that the edge singularity in the complex q plane approaches q c as q c l ensuremath sim q c al ensuremath y q and determine the scaling exponent y q for several values of q finally by finite size scaling of the fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y t as a function of q in the range 2l ql 3 using data for lattices of size 3l ll 8 we find that y t is a smooth function of q and is well fitted by y t 1 au bu 2 c du where u ensuremath 2 ensuremath pi mathrm cos ensuremath 1 sqrt q 2 for q 3 we find y t ensuremath simeq 0 6 however if we include lattices up to l 12 we find y t ensuremath simeq 0 50 8 in rough agreement with a recent result of ferreira and sokal j stat phys 96 461 1999

the discrete zak transform application to time frequency analysis and synthesis of nonstationary signals algorithms to compute coefficients of the finite double sum expansion of time varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented the algorithms are based on the computation of the discrete zak transform dzt fast algorithms to compute dzt are presented modifications to the algorithms which increase robustness are given

the undersampled discrete gabor transform conventional studies on discrete gabor transforms have generally been confined to the cases of critical sampling and oversampling in which the gabor families span the whole signal space in this paper we investigate undersampled discrete gabor transforms for an undersampled gabor triple g a b i e a spl middot b n we show that the associated generalized dual gabor window gdgw function is the same as the one associated with the oversampled g n b n a except for the constant factor ab n computations of undersampled gabor transforms are made possible by applying the methods algorithms developed in oversampled settings the undersampled gdgw is determined then we are able to obtain the best approximation of a signal x by linear combinations of vectors in the gabor family

dual gabor frames theory and computational aspects we consider a general method for constructing dual gabor elements different from the canonical dual our approach is based on combining two gabor frames such that the generated frame type operator s sub g spl gamma is nonsingular we provide necessary and sufficient conditions on the gabor window functions g and spl gamma such that s sub g spl gamma is nonsingular for rational oversampling considering both the continuous time and the discrete time settings in contrast to the frame operator the operator s sub g spl gamma is in general not positive therefore all results in gabor analysis that are based on the positivity of the frame operator cannot be applied directly the advantage of the proposed characterization is that the algebraic system for computing the gabor dual elements preserves the high structure of usual gabor frames leading to computationally efficient algorithms in particular we consider examples in which both the condition number and the computational complexity in computing the proposed dual gabor elements decrease in comparison to the canonical dual gabor elements

complex temperature phase diagrams for the q state potts model on self dual families of graphs and the nature of the q limit exact calculations of the potts model partition function z g q v have been presented for arbitrary q and temperature like variable v on self dual strip graphs g of the square lattice with fixed width l y and arbitrarily great length lx with two types of boundary conditions letting lx infinity the resultant free energy and complex temperature phase diagram have been computed including the locus b where the free energy is nonanalytic results are analyzed for widths l y 1 2 3 these results have been used to study the approach to the large q limit of b

exact potts model partition functions for strips of the triangular lattice we present exact calculations of the potts model partition function z g q v for arbitrary q and temperature like variable v on n vertex strip graphs g of the triangular lattice for a variety of transverse widths equal to l vertices and for arbitrarily great length equal to m vertices with free longitudinal boundary conditions and free and periodic transverse boundary conditions these partition functions have the form z g q v sum j 1 n z g lambda c z g j z g j m 1 we give general formulas for n z g j and its specialization to v 1 for arbitrary l the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed it is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the potts model on the infinite triangular lattice considering the full generalization to arbitrary complex q and v we determine the singular locus mathcal b arising as the accumulation set of partition function zeros as m in the q plane for fixed v and in the v plane for fixed q explicit results for partition functions are given in the text for l 3 free and l 3 4 cylindrical and plots of partition function zeros and their asymptotic accumulation sets are given for l up to 5 a new estimate for the phase transition temperature of the q 3 potts antiferromagnet on the 2d triangular lattice is given

exact potts model partition functions on strips of the honeycomb lattice abstract we present exact calculations of the partition function of the q state potts model on i open ii cyclic and iii mobius strips of the honeycomb brick lattice of width l y 2 and arbitrarily great length in the infinite length limit the thermodynamic properties are discussed the continuous locus of singularities of the free energy is determined in the q plane for fixed temperature and in the complex temperature plane for fixed q values we also give exact calculations of the zero temperature partition function chromatic polynomial and w q the exponent of the ground state entropy for the potts antiferromagnet for honeycomb strips of type iv l y 3 cyclic v l y 3 mobius vi l y 4 cylindrical and vii l y 4 open in the infinite length limit we calculate w q and determine the continuous locus of points where it is nonanalytic we show that our exact calculation of the entropy for the l y 4 strip with cylindrical boundary conditions provides an extremely accurate approximation to a few parts in 10 5 for moderate q values to the entropy for the full 2d honeycomb lattice where the latter is determined by monte carlo measurements since no exact analytic form is known

transfer matrices for the partition function of the potts model on cyclic and mobius lattice strips abstract we present a method for calculating transfer matrices for the q state potts model partition functions z g q v for arbitrary q and temperature variable v on cyclic and mobius strip graphs g of the square sq triangular tri and honeycomb hc lattices of width l y vertices and of arbitrarily great length l x vertices for the cyclic case we express the partition function as z l y l x q v d 0 l y c d tr t z l y d m where denotes lattice type c d are specified polynomials of degree d in q t z l y d is the transfer matrix in the degree d subspace and m l x l x 2 for sq tri hc respectively an analogous formula is given for mobius strips we exhibit a method for calculating t z l y d for arbitrary l y explicit results for arbitrary l y are given for t z l y d with d l y and l y 1 in particular we find very simple formulas the determinant det t z l y d and trace tr t z l y corresponding results are given for the equivalent tutte polynomials for these lattice strips and illustrative examples are included we also present formulas for self dual cyclic strips of the square lattice

general structural results for potts model partition functions on lattice strips abstract we present a set of general results on structural features of the q state potts model partition function z g q v for arbitrary q and temperature boltzmann variable v for various lattice strips of arbitrarily great width ly vertices and length lx vertices including i cyclic and mobius strips of the square and triangular lattices and ii self dual cyclic strips of the square lattice we also present an exact solution for the chromatic polynomial for the cyclic and mobius strips of the square lattice with width ly 5 the greatest width for which an exact solution has been obtained so far for these families in the lx limit we calculate the ground state degeneracy per site w q and determine the boundary b across which w q is singular in the complex q plane

almost crystallographic groups as quotients of artin braid groups abstract let n k 3 in this paper we analyse the quotient group b n k p n of the artin braid group b n by the subgroup k p n belonging to the lower central series of the artin pure braid group p n we prove that it is an almost crystallographic group we then focus more specifically on the case k 3 if n 5 and if n is such that gcd 6 1 we show that b n 3 p n possesses an element of order if and only if s n does and we prove that there is a one to one correspondence between the conjugacy classes of elements of order in b n 3 p n with those of elements of order in the symmetric group s n we also exhibit a presentation for the almost crystallographic group b n 3 p n finally we obtain some 4 dimensional almost bieberbach subgroups of b 3 3 p 3 we explain how to construct almost bieberbach subgroups of b 4 3 p 4 and b 3 4 p 3 and we exhibit explicit elements of order 5 in b 5 3 p 5

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

non negative pinching moduli spaces and bundles with infinitely many souls we show that in each dimension n 10 there exist infinite sequences of homotopy equivalent but mutually non homeomorphic closed simply connected riemannian n manifolds with 0 sec 1 positive ricci curvature and uniformly bounded diameter we also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non homeomorphic finally we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec 0 have infinitely many connected components

robustness of oversampled gabor transient detectors a comparison of energy and known location detectors this correspondence provides a context for previous studies on the use of linear time frequency transforms for transient signal detection in the case where signal component locations are assumed to be known the robustness of a known location detector kld based on the oversampled gabor transform is compared with that of the corresponding energy detector in the gabor domain as expected the kld was sensitive to location mismatch unexpectedly it was extremely sensitive to location mismatch

exact potts model partition functions on wider arbitrary length strips of the square lattice abstract we present exact calculations of the partition function of the q state potts model for general q and temperature on strips of the square lattice of width l y 3 vertices and arbitrary length l x with periodic longitudinal boundary conditions of the following types i fbc y pbc x cyclic ii fbc y tpbc x mobius iii pbc y pbc x toroidal and iv pbc y tpbc x klein bottle where fbc and t pbc refer to free and twisted periodic boundary conditions results for the l y 2 torus and klein bottle strips are also included in the infinite length limit the thermodynamic properties are discussed and some general results are given for low temperature behavior on strips of arbitrarily great width we determine the submanifold in the c 2 space of q and temperature where the free energy is singular for these strips our calculations are also used to compute certain quantities of graph theoretic interest

partition function zeros of a restricted potts model on lattice strips and effects of boundary conditions we calculate exactly the partition function z g q v of the q state potts model with temperature like boltzmann variable v for strip graphs g of the square and triangular lattices of various widths ly and arbitrarily great lengths lx with a variety of boundary conditions and with q and v restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two dimensional lattices from these calculations in the limit lx we determine the continuous accumulation loci of the partition function zeros in the v and q planes strips of the honeycomb lattice are also considered we discuss some general features of these loci

on the weyl heisenberg frames generated by simple functions let x n 0 cn e x n with cn n 0 l1 and let a 1 0 a 1 be a weyl heisenberg system e2 imx x na m n z we show that if e 0 1 and some modulo extension of e then a 1 is a frame for each 0 a 1 for certain a respectively if and only if the analytic function h z n 0 cnzn has no zero on the unit circle z z 1 these results extend the case of casazza and kalton 2002 6 that x i 1k 0 1 x ni and a 1 which brought together the frame theory and the function theory on the closed unit disk our techniques of proofs are based on the zak transform and the distribution of fractional parts of na n z

discrete zak transforms polyphase transforms and applications we consider three different versions of the zak 1967 transform zt for discrete time signals namely the discrete time zt the polyphase transform and a cyclic discrete zt in particular we show that the extension of the discrete time zt to the complex z plane results in the polyphase transform an important and well known concept in multirate signal processing and filter bank theory we discuss fundamental properties relations and transform pairs of the three discrete zt versions and we summarize applications of these transforms in particular the discrete time zt and the cyclic discrete zt are important for discrete time gabor 1946 expansion weyl heisenberg frame theory since they diagonalize the weyl heisenberg frame operator for critical sampling and integer oversampling the polyphase representation plays a fundamental role in the theory of filter banks especially dft filter banks simulation results are presented to demonstrate the application of the discrete zt to the efficient calculation of dual gabor windows tight gabor windows and frame bounds

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

zeros of jones polynomials for families of knots and links abstract we calculate jones polynomials v l t for several families of alternating knots and links by computing the tutte polynomials t g x y for the associated graphs g and then obtaining v l t as a special case of the tutte polynomial for each of these families we determine the zeros of the jones polynomial including the accumulation set in the limit of infinitely many crossings a discussion is also given of the calculation of jones polynomials for non alternating links

a simple proof of geodesical completeness for compact space times of zero curvature any riemannian metric on a compact manifold is geodesically complete by contrast it is widely known that incomplete compact lorentzian manifolds exist in this article it will be proven that a compact manifold with a smooth lorentz metric must be complete if the metric is flat everywhere this question of completeness which has significant implications for the classification of compact lorentzian space forms has remained unresolved until quite recently when it was finally established by carriere as a special case of an even more general result here a simple alternative proof of this special case of carriere s theorem is given the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space time

flow polynomials and their asymptotic limits for lattice strip graphs we present exact calculations of flow polynomials f g q for lattice strips of various fixed widths l y 4 and arbitrarily great lengths l x with several different boundary conditions square honeycomb and triangular lattice strips are considered we introduce the notion of flows per face fl in the infinite length limit we study the zeros of f g q in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite length limit for the various families of strips the function fl is nonanalytic on this locus the loci are found to be noncompact for many strip graphs with periodic or twisted periodic longitudinal boundary conditions and compact for strips with free longitudinal boundary conditions we also find the interesting feature that aside from the trivial case l y 1 the maximal point q cf where crosses the real axis is universal on cyclic and mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf 3 for the infinite square lattice

efficient algorithms for discrete gabor transforms on a nonseparable lattice the discrete gabor transform dgt is the most commonly used transform for signal analysis and synthesis using a linear frequency scale it turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup most of the literature focuses on separable subgroups in this paper we will survey existing methods for a generalization to arbitrary groups as well as present an improvement on existing methods comparisons are made with respect to the computational complexity and the running time of optimized implementations in the c programming language the new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download by summarizing general background information on the state of the art this article can also be seen as a research survey sharing with the readers experience in the numerical work in gabor analysis

discrete multiwindow gabor type transforms the discrete finite gabor scheme is generalized by incorporating multiwindows two approaches are presented for the analysis of the multiwindow scheme the signal domain approach and the zak transform domain approach issues related to undersampling critical sampling and oversampling are considered the analysis is based on the concept of frames and on generalized moore penrose inverses the approach based on representing the frame operator as a matrix valued function is far less demanding from a computational complexity viewpoint than a straightforward matrix algebra in various operations such as the computation of the dual frame dft based algorithms including complexity analysis are presented for the calculation of the expansion coefficients and for the reconstruction of the signal in both signal and transform domains the scheme is further generalized and incorporates kernels other than the complex exponential representations other than those based on the dual frame and nonrectangular sampling of the combined space are considered as well an example that illustrates the advantages of the multiwindow scheme over the single window scheme is presented

some results on discrete gabor transforms for finite periodic sequences in this correspondence using linear system theory and frame theory we address computation methods for analysis and synthesis sequences of the discrete gabor transform dgt where all sequences are periodic with the same period l and show that the minimum energy solution to the wexler and raz 1990 biorthogonal condition and the solution to the frame operator are the same via linear system theory and frame theory

matrix characterization for gfdm low complexity mmse receivers and optimal filters in this paper a new matrix based characterization of generalized frequency division multiplexing gfdm transmitter matrices is proposed as opposed to traditional vector based characterization with prototype filters the characterization facilitates deriving properties of gfdm transmitter matrices including conditions for gfdm matrices being nonsingular and unitary respectively using the new characterization the necessary and sufficient conditions for the existence of a form of low complexity implementation for a minimum mean square error mmse receiver are derived such an implementation exists under multipath channels if the gfdm transmitter matrix is selected to be unitary for cases where this implementation does not exist a low complexity suboptimal mmse receiver is proposed with its performance approximating that of an mmse receiver the new characterization also enables derivations of optimal prototype filters in terms of minimizing receiver mean square error mse they are found to correspond to the use of unitary gfdm matrices under many scenarios the use of such optimal filters in gfdm systems does not cause the problem of noise enhancement thereby demonstrating the same mse performance as orthogonal frequency division multiplexing moreover we find that gfdm matrices with a size of power of two are verified to exist in the class of unitary gfdm matrices finally while the out of band oob radiation performance of systems using a unitary gfdm matrix is not optimal in general it is shown that the oob radiation can be satisfactorily low if parameters in the new characterization are carefully chosen

scalar curvature and hammocks scalar curvature is the simplest generalization of gaussian curvature to higher dimensions however there are many questions open with regard to its relation to other geometric quantities and topology here we will prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curvature namely the one sided affinity for curvature decreasing deformations the first one is concerned with some prescribed decrease of the scalar curvature scal g of some riemannian metric g on a given manifoldmn of dimension 3 we denote the e neighborhood of some set u with respect to g by ue

rational time frequency super gabor frames and their duals abstract this paper addresses general super gabor systems with rational time frequency product lattices a zak transform matrix method is developed for such super gabor systems we characterize complete super gabor systems and super gabor frames given a super gabor frame we obtain an explicit expression of its canonical dual and a parametrization of all its super gabor duals and prove that the canonical dual is the norm minimal one among all super gabor duals we also prove that a super gabor frame has a unique super gabor dual if and only if it is a riesz basis

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

gabor transform of time varying systems exact representation and approximation approximate modeling of discrete time linear time varying systems is studied based on a representation of linear systems in the gabor time frequency space the time varying system is assumed to be given in input output or kernel representation this representation has previously received attention because of its applicability in frozen time analysis and design of optimal control for time varying systems but requires a large number of coefficients motivated by its application to signal analysis the gabor transform is considered as a tool for the representation and approximation of linear time variant systems in order to show the main results the class of systems considered is restricted to the one usually considered in the frozen time approach an example is included to illustrate the potential application of the technique

potts model partition functions for self dual families of strip graphs abstract we consider the q state potts model on families of self dual strip graphs g d of the square lattice of width l y and arbitrarily great length l x with periodic longitudinal boundary conditions the general partition function z and the t 0 antiferromagnetic special case p chromatic polynomial have the respective forms j 1 n f l y c f l y j f l y j l x with f z p for arbitrary l y we determine i the general coefficient c f l y j in terms of chebyshev polynomials ii the number n f l y d of terms with each type of coefficient and iii the total number of terms n f l y we point out interesting connections between the n z l y d and temperley lieb algebras and between the n f l y and enumerations of directed lattice animals exact calculations of p are presented for 2 l y 4 in the limit of infinite length we calculate the ground state degeneracy per site exponent of the ground state entropy w q generalizing q from z to c we determine the continuous locus b in the complex q plane where w q is singular we find the interesting result that for all l y values considered the maximal point at which b crosses the real q axis denoted q c is the same and is equal to the value for the infinite square lattice q c 3 this is the first family of strip graphs of which we are aware that exhibits this type of universality of q c

on the fundamental groups of manifolds of positive sectional curvature let m be an n manifold with sectional curvature 0 6 k 1 the main result asserts that the fundamental group of m has a finite normal cyclic subgroup with index less than w n 8 oc a constant depending on n and 8

chromatic polynomials and representations of the symmetric group abstract the chromatic polynomials considered in this paper are associated with graphs constructed in the following way take n copies of a complete graph k b and for i 1 2 n join each vertex in the ith copy to the same vertex in the i 1 th copy taking n 1 1 by convention previous calculations for b 2 and b 3 suggest that the chromatic polynomial contains terms that occur in levels in the present paper the levels are explained by using a version of the sieve principle and it is shown that the terms at level l correspond to the irreducible representations of the symmetric group sym l in the case of the two linear representations the terms can be calculated explicitly by methods based on the theory of distance regular graphs for the nonlinear representations the calculations are more complicated an illustration is given in section 10 where the complete chromatic polynomial for the case b 4 is obtained

exact potts model partition functions for strips of the square lattice we present exact calculations of the potts model partition function z g q v for arbitrary q and temperature like variable v on n vertex square lattice strip graphs g for a variety of transverse widths l t and for arbitrarily great length l l with free longitudinal boundary conditions and free and periodic transverse boundary conditions these have the form z g q v sum j 1 n z g lambda c z g j lambda z g j l ell we give general formulas for n z g j and its specialization to v 1 for arbitrary l t for both types of boundary conditions as well as other general structural results on z the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed it is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the potts model on the infinite square lattice considering the full generalization to arbitrary complex q and v we determine the singular locus b arising as the accumulation set of partition function zeros as l l in the q plane for fixed v and in the v plane for fixed q

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

oversampled gabor representation for transient signals considers the gabor representation that uses a one sided exponential window for detection and analysis of transient signals earlier results on the critically sampled case are extended to the more practically useful oversampled case for oversampling by an integer factor the authors derive an explicit analytical expression for the dual window dual frame function required for computing the gabor representation based on this expression they develop an efficient procedure for computing the gabor coefficients finally they demonstrate the performance of the method by numerical examples

the generalized gabor scheme of image representation in biological and machine vision a scheme suitable for visual information representation in a combined frequency position space is investigated through image decomposition into a finite set of two dimensional gabor elementary functions gef the scheme is generalized to account for the position dependent gabor sampling rate oversampling logarithmic frequency scaling and phase quantization characteristic of the visual system comparison of reconstructed signal highlights the advantages of the generalized gabor scheme in coding typical bandlimited images it is shown that there exists a tradeoff between the number of frequency components used per position and the number of such clusters sampling rate utilized along the spatial coordinate

ends of finite volume nonpositively curved manifolds we study complete finite volume n manifolds m of bounded nonpositive sectional curvature a classical theorem of m gromov says that if such m has negative curvature then it is homeomorphic to the interior of a compact manifold with boundary and we denote this boundary partial m if n ge 3 we prove that the universal cover of the boundary widetilde partial m and also the pi 1m cover of the boundary partial widetilde m have vanishing n 2 dimensional homology for n 4 the first of these recovers a result of nguy tilde hat mathrm e n phan saying that each component of the boundary partial m is aspherical for any n ge 3 the second of these implies the vanishing of the first group cohomology group with group ring coefficients h 1 b pi 1m mathbb z pi 1m 0 a consequence is that pi 1m is freely indecomposable these results extend to manifolds m of bounded nonpositive curvature if we assume that m is homeomorphic to the interior of a compact manifold with boundary our approach is a form of homological collapse for ends of finite volume manifolds of bounded nonpositive curvature this paper is very much influenced by earlier yet still unpublished work of nguy tilde hat mathrm e n phan

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

orthogonalization of circular stationary vector sequences and its application to the gabor decomposition certain vector sequences in hermitian or in hilbert spaces can be orthogonalized by a fourier transform in the finite dimensional case the discrete fourier transform dft accomplishes the orthogonalization the property of a vector sequence which allows the orthogonalization of the sequence by the dft called circular stationarity cs is discussed in this paper applying the dft to a given cs vector sequence results in an orthogonal vector sequence which has the same span as the original one in order to obtain coefficients of the decomposition of a vector upon a particular nonorthogonal cs vector sequence the decomposition is first found upon the equivalent dft orthogonalized one and then the required coefficients are found through the dft it is shown that the sequence of discrete gabor 1946 basis functions with periodic kernel and with a certain inner product on the space of n periodic discrete functions satisfies the cs condition the theory of decomposition upon cs vector sequences is then applied to the gabor basis functions to produce a fast algorithm for calculation of the gabor coefficients

a linear cost algorithm to compute the discrete gabor transform in this paper we propose an alternative efficient method to calculate the gabor coefficients of a signal given a synthesis window with a support of size much lesser than the length of the signal the algorithm uses the canonical dual of the window which does not need to be calculated beforehand and achieves a computational cost that is linear with the signal length in both analysis and synthesis this is done by exploiting the block structure of the matrices and using an ad hoc cholesky decomposition of the gabor frame matrix

t 0 partition functions for potts antiferromagnets on lattice strips with fully periodic boundary conditions abstract we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet equivalently the chromatic polynomial for families of arbitrarily long strip graphs of the square and triangular lattices with width ly 4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation i e of torus or klein bottle type these boundary conditions have the advantage of removing edge effects in the limit of infinite length we calculate the exponent of the entropy w q and determine the continuous locus b where it is singular we also give results for toroidal strips involving crossing subgraphs these make possible a unified treatment of torus and klein bottle boundary conditions and enable us to prove that for a given strip the locus b is the same for these boundary conditions

frame analysis of the discrete gabor scheme the properties of the discrete gabor scheme are considered in the context of oversampling the approach is based on the concept of frames and utilizes the piecewise finite zak transform pfzt the frame operator is represented as a matrix valued function in the pfzt domain and its properties are examined in relation to this function the frame bounds are calculated by means of the eigenvalues of the matrix valued function and the dual frame which is used in calculation of the expansion coefficients is expressed by means of the inverse matrix dft based algorithms for computation of the expansion coefficients and for the reconstruction of signals from these coefficients are generalized for the case of oversampling of the gabor space the algorithms are implemented in an example of representation of a nonstationary signal

the zak transform and decimated time frequency distributions in this paper the interrelation between the zak transform and the generalized discrete time frequency distribution gdtfd is examined starting with the discrete zak transform its definition is broadened to include arbitrary windows creating the windowed zak transform wzt the wzt is then combined with the spectrogram it is demonstrated that the spectrogram based upon the wzt called the zak spectrogram zs is a generalization of the standard spectrogram next building upon the idea of the weighted spectrogram the weighted zs is used to produce a new class of gdtfd called the decimated gdtfd d gdtfd the decimated gdtfd is similar to the gdtfd except it trades bandwidth for computational speed and requires significantly less storage in order to be implemented the reduction in discrete bandwidth is from 2 spl pi for the gdtfd to 2 spl pi m for the d gdtfd an important attribute of the dgdtfd is that it requires significantly less storage than the gdtfd the d gdtfd requires only 1 m sup 2 of the storage of the gdtfd an example using the binomial distribution is given to illustrate the connection between the d gdtfd and the gdtfd throughout the paper examples of multirate systems are given which could be used to implement the building blocks of the dgdtfd this makes the d gdtfd practical to implement using slower and less expensive elements

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly

generalized gabor expansions of discrete time signals in l sup 2 z via biorthogonal like sequences in this paper a biorthogonal like sequences bls theory and its application to the generalized gabor expansions equivalently the generalized short time fourier transform filterbank summation are presented a pair of bls s are defined to be two sequences satisfying a biorthogonal like condition blc which is a moment equation and equivalent to a linear difference equation we show that two collections in a hilbert space generated by a pair of bls s in the joint time frequency domain are complete either can be used as an analysis filter and the other can be used as a synthesis filter for a generalized gabor expansion of discrete time signals a sufficient and necessary condition on the existence of bls s based on the moment equation is presented which is simpler to use than frame theory given a filter generating a frame its bls s also generate frames the dual frame is one of them given a fir analysis synthesis filter there is a fir synthesis analysis filter if bls s exist the algorithm to compute fir analysis and synthesis filters based on the linear difference equation is presented in this paper which is simpler than frame operator

discrete weyl heisenberg transforms the concepts of von neumann lattices and tight frames are used for defining discrete quantum mechanical transforms in the phase plane these transforms are obtained by finite shifts la and mb in the coordinate x and momentum p respectively of the weyl heisenberg group and they are called the discrete weyl heisenberg transforms la mb here ab h n with h the planck constant l and m integers and n a positive integer a construction is carried out of la mb for a general weyl heisenberg set by using the kq representation in which a useful formula is established for the frame operator the construction is illustrated on an example of the ground state of a harmonic oscillator it is shown that any physical quantity can be described by the discrete weyl heisenberg transform connections are established between la mb the bargmann representation and the husimi distribution function

transfer matrices and partition function zeros for antiferromagnetic potts models i general theory and square lattice chromatic polynomial we study the chromatic polynomials zero temperature antiferromagnetic potts model partition functions p g q for m n rectangular subsets of the square lattice with m 8 free or periodic transverse boundary conditions and n arbitrary free longitudinal boundary conditions using a transfer matrix in the fortuin kasteleyn representation in particular we extract the limiting curves of partition function zeros when n which arise from the crossing in modulus of dominant eigenvalues beraha kahane weiss theorem we also provide evidence that the beraha numbers b2 b3 b4 b5 are limiting points of partition function zeros as n whenever the strip width m is 7 periodic transverse b c or 8 free transverse b c along the way we prove that a noninteger beraha number except perhaps b10 cannot be a chromatic root of any graph

structural properties of potts model partition functions and chromatic polynomials for lattice strips abstract the q state potts model partition function equivalent to the tutte polynomial for a lattice strip of fixed width l y and arbitrary length l x has the form z g q v j 1 n z g c z g j z g j l x where v is a temperature dependent variable the special case of the zero temperature antiferromagnet v 1 is the chromatic polynomial p g q using coloring and transfer matrix methods we give general formulas for c x g j 1 n x g c x g j for x z p on cyclic and mobius strip graphs of the square and triangular lattice combining these with a general expression for the unique coefficient c z g j of degree d in q c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we determine the number of z g j s with coefficient c d in z g q v for these cyclic strips of width l y to be n z l y d 2d 1 l y d 1 1 2l y l y d for 0 d l y and zero otherwise for both cyclic and mobius strips of these lattices the total number of distinct eigenvalues z g j is calculated to be n z l y 2l y l y results are also presented for the analogous numbers n p l y d and n p l y for p g q we find that n p l y 0 n p l y 1 1 m l y 1 motzkin number n z l y 0 c l y the catalan number and give an exact expression for n p l y our results for n z l y and n p l y apply for both the cyclic and mobius strips of both the square and triangular lattices we also point out the interesting relations n z l y 2 n da tri l y and n p l y 2 n da sq l y where n da n denotes the number of directed lattice animals on the lattice we find the asymptotic growths n z l y l y 1 2 4 l y and n p l y l y 1 2 3 l y as l y some general geometric identities for potts model partition functions are also presented

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics