kolmogorov sinai entropy of many body hamiltonian systems the kolmogorov sinai ks entropy is a central measure of complexity and chaos its calculation for many body systems is an interesting and important challenge in this paper the evaluation is formulated by considering n dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem this approach makes explicit a duality relation that is exactly analogous to one found in a generalized anderson tight binding model and leads to a formally exact expression for the finite time ks entropy within this formalism there is a hierarchy of approximations the final one being a diagonal approximation that only makes use of instantaneous hessians of the potential to find the ks entropy by way of a nontrivial illustration the ks entropy of n identically coupled kicked rotors standard maps is investigated the validity of the various approximations with kicking strength particle number and time are elucidated an analytic formula for the ks entropy within the diagonal approximation is derived and its range of validity is also explored

efficient discrete approximations of quantum gates quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set it has been shown that this is possible for almost all choices of base sets and furthermore that the number of gates required for precision e is only polynomial in log 1 e here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1 e a result which matches the lower bound from counting volume up to constant factor

on the spectral gap for finitely generated subgroups of su 2 we prove the spectral gap property for free subgroups of su 2 generated by elements satisfying a noncommutative diophantine property in particular for free subgroups generated by elements with algebraic entries

identities and exponential bounds for transfer matrices this paper is about analytic properties of single transfer matrices originating from general block tridiagonal or banded matrices such matrices occur in various applications in physics and numerical analysis the eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity if the block tridiagonal matrix is invertible it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain and the other half have an upper bound that is exponentially small this is a consequence of a theorem by demko moss and smith on the decay of matrix elements of the inverse of banded matrices this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

a generalisation to birkhoff von neumann theorem abstract the classic birkhoff von neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices in this paper we study a generalisation of this theorem in the type ii 1 setting namely we replace a doubly stochastic matrix with a collection of measure preserving partial isomorphisms of the unit interval with similar properties we show that a weaker version of this theorem still holds

local spectral gap in simple lie groups and applications we introduce a novel notion of local spectral gap for general possibly infinite measure preserving actions we establish local spectral gap for the left translation action gamma curvearrowright g whenever gamma is a dense subgroup generated by algebraic elements of an arbitrary connected simple lie group g this extends to the non compact setting works of bourgain and gamburd invent math 171 83 121 2008 j eur math soc jems 14 1455 1511 2012 and benoist and de saxce invent math 205 337 361 2016 we present several applications to the banach ruziewicz problem orbit equivalence rigidity continuous and monotone expanders and bounded random walks on g in particular we prove that up to a multiplicative constant the haar measure is the unique gamma invariant finitely additive measure defined on all bounded measurable subsets of g

a fast algorithm for solving a toeplitz system of equations a fast algorithm for the solution of a toeplitz system of equations is presented the algorithm requires order n log n 2 computations where n is the number of equations for banded toeplitz matrices the order of computations is reduced to only n log n m log m 2 where 2m is the maximum number of nonzero principal subdiagonals of the toeplitz matrix the algorithm is in essence a fast implementation of the trench algorithm in reverse thus the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast hd half divisor algorithm to compute the first row of the inverse matrix the desired solution is then obtained directly from the first row by applying fast fourier transform techniques in order n log n computations finally the extension of the algorithm to block toeplitz matrices is also presented

exact synthesis of single qubit unitaries over clifford cyclotomic gate sets we generalize an efficient exact synthesis algorithm for single qubit unitaries over the clifford t gate set which was presented by kliuchnikov maslov and mosca quantum inf comput 13 7 8 607 630 2013 their algorithm takes as input an exactly synthesizable single qubit unitary one which can be expressed without error as a product of clifford and t gates and outputs a sequence of gates which implements it the algorithm is optimal in the sense that the length of the sequence measured by the number of t gates is smallest possible in this paper for each positive even integer n we consider the clifford cyclotomic gate set consisting of the clifford group plus a z rotation by n we present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of n z rotations for the clifford t case n 4 the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring z e i n 1 2 we prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100

the levinson recurrence and fast algorithms for solving toeplitz systems of linear equations this work brings together classical polynomial theory as it relates to the levinson recurrence for a hermitian toeplitz operator and matrix theory as it relates to the class of hermitian centro hermitian matrices a new computationally efficient alternative is presented to the levinson recurrence on either the hermitian or skew hermitian polynomial spaces this approach also leads to an entirely new algorithm for solving systems of linear equations when the coefficient matrix is hermitian toeplitz or real symmetric toeplitz analysis of the computational complexity of the algorithms presented is also performed and it is shown that these algorithms lead to significant improvements in the computational complexity as compared to the previously best known recursive algorithms they also provide further insight into the mathematical properties of the structurally rich toeplitz matrices

maximal determinant over a certain class of matrices and its application to d optimality of designs it is known that some optimality criteria of experimental designs are functionals of the eigenvalues of their information matrices in this context we study the problem of maximizing the determinant of it p pt 2 over the class of t by t permutation matrices and the determinant of it p pt 2 5 over the class of t by t permutation matrices with zero diagonal derangement matrices the results are used to characterize d optimal complete block designs under an interference model

determinants of block tridiagonal matrices an identity is proven that evaluates the determinant of a block tridiagonal matrix with or without corners as the determinant of the associated transfer matrix or a submatrix of it

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

comments on a note on a three term recurrence for a tridiagonal matrix abstract in el mikkawy a note on a three term recurrence for a tridiagonal matrix appl math comput 139 2003 503 511 the author obtained a two term recurrence for computing the determinant of a tridiagonal matrix in this note we show that this is an special case of a recurrence for block tridiagonal matrix

transfer matrices non hermitian hamiltonians and resolvents some spectral identities i consider the n step transfer matrix t for a general block hamiltonian with eigenvalue equation where and are matrices and provide its explicit representation in terms of blocks of the resolvent of the hamiltonian matrix for the system of length n with boundary conditions i then introduce the related hamiltonian for the case and and provide an exact relation between the trace of its resolvent and together with an identity of thouless type connecting tr with the hamiltonian eigenvalues for the results are then extended to by showing that it is itself a transfer matrix besides being of mathematical interest the identities should be useful for an analytical approach in the study of spectral properties of a physically relevant class of transfer matrices

transfer matrices and tridiagonal block hamiltonians with periodic and scattering boundary conditions general hamiltonian matrices with tridiagonal block structure and the associated transfer matrices are investigated in the cases of periodic and scattering boundary conditions they arise from tight binding models with finite range hopping in one or more dimensions of space in the presence of a aharonov bohm flux or in multichannel scattering an identity relating the characteristic equation of the periodic hamiltonian with that of the transfer matrix is found allowing a detailed analysis of the bands a velocity matrix is defined with properties relevant for the band structure or for the channel structure in the scattering problem

super golden gates for pu 2 abstract to each of the symmetry groups of the platonic solids we adjoin a carefully designed involution yielding topological generators of pu 2 which have optimal covering properties as well as efficient navigation these are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups the generators give super efficient 1 qubit quantum gates and are natural building blocks for the design of universal quantum gates

on the derivation of the semiclassical approximation to the quantum propagator in order to rigorously derive the amplitude factor of the semiclassical approximation to the quantum propagator we extend an existing method originally devised to evaluate gaussian path integral expressions using a result which relates the determinant of symmetric block tridiagonal matrices to the determinants of their blocks two difference equations are obtained the first one allows to establish the connection of the amplitude factor to jacobi s accessory equations in the continuous time limit while the second one leads to an additional factor which however contributes to the final result only in exceptional cases in order to demonstrate the wide applicability of these difference equations we treat explicitly the case where the time sliced lagrangian is written in generalized coordinates for which a general derivation has so far been unavailable

on biorthogonality of hermitian and skew hermitian szego levinson polynomials a system of polynomials biorthogonal with respect to a given weight function which is not necessarily even is constructed from a corresponding system of szego or levinson polynomials two term coupled recurrences are presented for the biorthogonal polynomial system and it is shown that these recurrences reduce to a pair of three term uncoupled recurrences in the case that the unit circle weight function is even several observations relative to recent linear prediction algorithms utilizing hermitian and skew hermitian levinson polynomials are discussed

subconvex bounds for hecke maass forms on compact arithmetic quotients of semisimple lie groups let h be a semisimple algebraic group k a maximal compact subgroup of g h mathbb r and gamma subset h mathbb q a congruence arithmetic subgroup in this paper we generalize existing subconvex bounds for hecke maass forms on the locally symmetric space gamma backslash g k to corresponding bounds on the arithmetic quotient gamma backslash g for cocompact lattices using the spectral function of an elliptic operator the bounds obtained extend known subconvex bounds for automorphic forms to non trivial k types yielding such bounds for new classes of automorphic representations they constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature we also obtain new subconvex bounds for holomorphic modular forms in the weight aspect

finite precision arithmetic and the split schur algorithms the split schur algorithms of p delsarte and y genin 1987 represent methods of computing reflection coefficients that are computationally more efficient in terms of multiplications than the conventional schur algorithm by a constant factor the authors investigate the use of fixed point binary arithmetic with quantization due to rounding in the implementation of the symmetric and antisymmetric split schur algorithms it is shown through a combination of analysis and simulation that the errors in the reflection coefficient estimates due to quantization are large when the input signal is either a narrowband high pass signal or a narrowband low pass signal

quadratic forms in unitary operators abstract let u 1 u n be unitary matrices on l 2 denote by the matrix a defined by a i i j j k 1 n u k i j u k i j acting as a bounded operator on l 2 n n in other words a is the sum of the kronecker products of u k with their complex conjugates we show the following sharp inequality as an application we show that the natural representation of u n n 1 acting on l 2 of the unit sphere in c n and restricted to mean zero functions satisfies for any choice 1 n in u n the lower bound this extends a result due to lubotzky phillips and sarnak who proved this with so 3 in the place of u n

computationally efficient reduced polynomial based algorithms for hermitian toeplitz matrices in this work the mathematical structure is analyzed that is associated with the fast order recursive algorithms for computing the reflection coefficients and the levinson polynomial associated with a hermitian positive definite toeplitz matrix a new form of three term recurrence relation is derived anda computationally efficient alternative to the classical levinson durbin algorithm is obtained the computational complexity of the new algorithm is the same as those of the split algorithms described in the recent literature the new algorithm also provides further insight into the mathematical properties of the structurally rich toeplitz matrices

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

spherical designs discrepancy and numerical integration a spherical design is a point configuration on the sphere which yields exact equal weight quadrature formulae for polynomials up to a given de gree until now only very specific constructions for spherical designs are known we establish connections to spherical cap discrepancy and show some general discrepancy bounds furthermore we reformulate the problem of construct ing designs as an optimization problem and develop an algorithm for finding practical designs

spectral duality and distribution of exponents for transfer matrices of block tridiagonal hamiltonians i consider a general block tridiagonal matrix and the corresponding transfer matrix by allowing for a complex bloch parameter in the boundary conditions the two matrices are related by a spectral duality as a consequence i derive some analytic properties of the exponents of the transfer matrix in terms of the eigenvalues of the non hermitian block matrix some of them are the single matrix analogues of results holding for lyapunov exponents of an ensemble of block matrices which occur in models of transport the counting function of exponents is related to winding numbers of eigenvalues i discuss some implications of duality for the distribution real bands and complex arcs and the dynamics of eigenvalues

trigonometric approximation and a general form of the erdos turan inequality there exists a positive function psi t on t 0 with fast decay infinity such that for every measurable set omega in the euclidean space and r 0 there exist entire functions a x and b x of exponential type r satisfying a x chi omega x b x and b x a x psi r dist x boundary omega this leads to erdos turan estimates for discrepancy of point set distributions in the multi dimensional torus analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds

on a two term recurrence for the determinant of a general matrix recently a two term recurrence for computing the determinant of a tridiagonal matrix has been found by el mikkawy m el mikkawy a note on a three term recurrence for a tridiagonal matrix appl math comput 139 2003 503 511 then the result has been extended to a block tridiagonal matrix by salkuyeh d k salkuyeh comments on a note on a three term recurrence for a tridiagonal matrix appl math comput 176 2006 442 444 in this note we show that the relation can be obtained for a general matrix and that as a by product we obtain a generalization of the detgtri algorithm m el mikkawy a fast algorithm for evaluating nth order tri diagonal determinants j comput appl math 166 2004 581 584

improving formulas for the eigenvalues of finite block toeplitz tridiagonal matrices abstract after a short overview improvements based on the kronecker product are proposed for the eigenvalues of n n block toeplitz tridiagonal block tt matrices with k k matrix entries common in applications some extensions of the spectral properties of the toeplitz tridiagonal matrices are pointed out the eigenvalues of diagonalizable symmetric and skew symmetric block tt matrices are studied besides if certain matrix square root is well defined it is proved that every block tt matrix with commuting matrix entries is isospectral to a related symmetric block tt one further insight about the eigenvalues of hierarchical hermitian block tt matrices of use in the solution of pdes is also achieved

split algorithms for hermitian toeplitz matrices with arbitrary rank profile abstract split algorithms for toeplitz matrices exploit besides the toeplitz structure additional symmetry properties to reduce the number of operations in this paper split levinson and schur algorithms for hermitian toeplitz matrices are presented that work in contrast to previous algorithms without additional conditions like strong nonsingularity the main contribution is the generalization of the split levinson type algorithms of b krishna h krishna and h krishna s morgera to general nonsingular hermitian toeplitz matrices furthermore a schur type counterpart of this algorithm is presented that is also new in the strongly nonsingular case some auxiliary considerations concerning the kernel structure of hermitian toeplitz matrices might be of independent interest

best possible rates of distribution of dense lattice orbits in homogeneous spaces this paper establishes upper and lower bounds on the speed of approximation in a wide range of natural diophantine approximation problems the upper and lower bounds coincide in many cases giving rise to optimal results in diophantine approximation which were inaccessible previously our approach proceeds by establishing more generally upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup in a connected lie or algebraic group g acting on suitable homogeneous spaces g h the upper bound is derived using a quantitative duality principle for homogeneous spaces reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on h and acting on g in particular the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of h in the automorphic representation on l 2 g we show that the rate is best possible when the representation in question is tempered and show that the latter condition holds in a wide range of examples

monotonicity of zeros for a class of polynomials including hypergeometric polynomials we study the monotonicity of zeros in connection with perturbed recurrence coefficients of polynomials satisfying certain three term recurrence relations of frobenius type these recurrence relations are the key ingredient for the tridiagonal approach developed by delsarte and genin to solve the standard linear prediction problem as a particular case we consider the askey para orthogonal polynomials on the unit circle 2 f 1 n a b i 2 a 1 z a b r extending a recent result about the monotonicity of their zeros with respect to the parameter b finally the consequences of our results in the theory of orthogonal polynomials on the real line are discussed

distribution of eigenvalues in non hermitian anderson models we develop a theory which describes the behavior of eigenvalues of a class of one dimensional random non hermitian operators introduced recently by hatano and nelson we prove that the eigenvalues are distributed along a curve in the complex plane an equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a reference hermitian disordered system the generic properties of the eigenvalue distribution are discussed

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

the central extension of the reflection equations and an analog of miki s formula two different types of centrally extended quantum reflection algebras are introduced realizations in terms of the elements of the central extension of the yang baxter algebra are exhibited a co action map is identified for the special case of a realization in terms of elements satisfying the zamolodchikov faddeev algebra a boundary analog of miki s formula is also proposed providing a free field realization of q onsager currents

euclid algorithm orthogonal polynomials and generalized routh hurwitz algorithm this paper is concerned with the relations between the euclid algorithm the theory of orthogonal polynomials and the problem of locating the zeros of a complex polynomial with respect to the imaginary axis in particular a simple generalized routh hurwitz algorithm is proposed which allows one to determine in any situation the numbers of zeros of an arbitrary complex polynomial in the right half plane on the imaginary axis and hence in the left half plane moreover it turns out that this algorithm yields as a side result a well defined factorization of the considered polynomial furthermore as a straightforward consequence of the adopted approach two presumably original algorithms are put into light which involve linear arithmetic operations only a polynomial nonnegativity test on the real axis and a characterization of positive real functions

semi classical limit for random walks let g u be a discrete symmetric random walk on a compact lie group g with step distribution and let t be the associated transition operator on l2 g the irreducibles vp of the left regular representation of g on l2 g are finite dimensional invariant subspaces for t and the spectrum of t is the union of the sub spectra a t 1vp on the irreducibles which consist of real eigenvalues apl ap dim vp our main result is an asymptotic expansion for the spectral measures dim vp m dim vp j along rays of representations in a positive weyl chamber t i e for sequences of representations kp k c n with k oo as a corollary we obtain some estimates on the spectral radius of the random walk we also analyse the fine structure of the spectrum for certain random walks on u n for which tm is essentially a direct sum of harper operators

fast triangular factorization of the sum of quasi toeplitz and quasi hankel matrices abstract the literature contains several recent fast algorithms for the triangular factorization of strongly regular toeplitz plus hankel matrices in this paper we study the rather more general sum of quasi toeplitz and quasi hankel matrices both hermitian and non hermitian quasi toeplitz and quasi hankel matrices are those that are congruent to toeplitz and hankel matrices in a special sense the derivation is based on the concept of displacement structure and its intimate relation to the schur reduction procedure for triangular factorization various special cases covering displacement ranks from two to eight are considered several other problems e g factorization of the inverse matrix solution of exact or overdetermined linear systems can be reduced to the direct factorization problem

comparison between some stability criteria of discrete time filters routh s algorithm is written in the form of polynomial recursion and transposed for testing stability in the wide sense of discrete time linear systems the algorithm obtained is a three term recursion of symmetric polynomials this recursion easily gives both the coefficients appearing in the continued fraction expansion associated to a given polynomial in the z domain and a computation scheme for the bilinear transformation of polynomials finally this recursion is compared to other algorithms already known for testing stability

toeplitz properties of the block matrices encountered in the processing of spatiotemporal signals various signal processing applications such as system identification fir finite impulse response wiener filtering linear prediction and spectral estimation are known to lead to the solution of toeplitz systems for the case of one dimensional signals and to toeplitz block toeplitz systems for the case of two dimensional signals the authors investigate such systems for the case of l dimensional signals with l 2 and emphasize their block and persymmetrical nature such important characteristics allow the use of fast algorithms for signals of any dimension the case of three dimensional signals is treated in detail not only because it is of special interest for video applications but also because the reasoning used for this case can be straightforwardly applied to higher dimension signals

shrinkage estimation of common breaks in panel data models via adaptive group fused lasso abstract in this paper we consider estimation and inference of common breaks in panel data models via adaptive group fused lasso we consider two approaches penalized least squares pls for first differenced models without endogenous regressors and penalized gmm pgmm for first differenced models with endogeneity we show that with probability tending to one both methods can correctly determine the unknown number of breaks and estimate the common break dates consistently we establish the asymptotic distributions of the lasso estimators of the regression coefficients and their post lasso versions we also propose and validate a data driven method to determine the tuning parameter used in the lasso procedure monte carlo simulations demonstrate that both the pls and pgmm estimation methods work well in finite samples we apply our pgmm method to study the effect of foreign direct investment fdi on economic growth using a panel of 88 countries and regions from 1973 to 2012 and find multiple breaks in the model

a note on a three term recurrence for a tridiagonal matrix in this article the author shows that under certain conditions a three term recurrence for a tridiagonal matrix becomes a two term recurrence using this new recurrence the possibility of the lu factorization of any tridiagonal matrix is now easy to investigate the positive definiteness of any real symmetric tridiagonal matrix is now easy to check an algorithm for solving any linear system with positive definite tridiagonal matrix is given some numerical examples are given

invariant measures for algebraic actions zariski dense subgroups and kazhdan s property t let k be any locally compact non discrete field we show that finite invariant measures for k algebraic actions are obtained only via actions of compact groups this extends both borel s density and fixed point theorems over local fields for semisimple solvable groups resp we then prove that for k algebraic actions finitely additive finite invariant measures are obtained only via actions of amenable groups this gives a new criterion for zariski density of subgroups and is shown to have representation theoretic applications the main one is to kazhdan s property t for algebraic groups which we investigate and strengthen

on the splitting of classical algorithms in linear prediction theory the levinson algorithm has been shown recently to be redundant in complexity it can be broken down into two simpler algorithms either of these split levinson algorithms needs only to be processed in the present paper it is shown that the schur algorithm the lattice algorithm and the normalized lattice algorithm for computing reflection coefficients can be broken down in a similar manner the number of multiplications in each of the resulting split algorithms is reduced with respect to the classical version by a factor lying between 1 5 and 2

levinson type algorithms for polynomial fitting and for cholesky and q factors of hankel and vandermonde matrices this paper presents levinson 1947 type algorithms for i polynomial fitting ii obtaining a q decomposition of vandermonde matrices and a cholesky factorization of hankel matrices iii obtaining the inverse of hankel matrices the algorithm for the least squares solution of hankel systems of equations requires 3n sup 2 9n 3 multiply and divide operation mdo the algorithm for obtaining an orthogonal representation of an m spl times n vandermonde matrix x and computing the cholesky factors f of hankel matrices requires 5mn n sup 2 2n 3m mdo and the algorithm for generating the inverse of hankel matrices requires 3 n sup 2 n 2 2 mdo our algorithms have been tested by means of fitting of polynomials of various orders and fortran versions of all subroutines are provided in the appendix