the zeros of derivatives of entire functions and the polya wiman conjecture an entire function f x which assumes only real values on the real axis is said to be a real entire function thus if f is a real entire function then its maclaurin coefficients are all real and consequently the zeros of f are symmetrically located with respect to the real axis one of the intuitive principles underlying the theory of distribution of zeros of successive derivatives of real entire functions was formulated by polya p4 in his celebrated survey article entitled on the zeros of the derivatives of a function and its analytic character in this work p4 p 181 polya states the real axis seems to exert an influence on the complex zeros of f n z it seems to attract these zeros when the order is less than 2 and it seems to repel them when the order is greater than 2 indeed the first confirmation of this principle was made by alander a2 see also p6lya p2 who in 1930 showed that the polya wiman conjecture is true if the order of f x is less than 2 3 in wil and wi2 wiman established the validity of this conjecture for functions f x of order at most 1 while in 1937 polya p3 proved it for functions of order less than 4 3 a general survey of these results and related conjectures may be found in p2 and p4 for more recent surveys of this and related areas of research see boas b2 and prather pr it seems that no progress has been made on the p6lya wiman conjecture since the publication of polya s 4 3 theorem in 1937 our proof of the conjecture is self contained and does not rely on the results cited above in section 2 we will 1 introduce some definitions and notations 2 state some of the properties of functions in the laguerre p6lya class and 3 prove the polya wiman conjecture theorem 3 in the proof of theorem 3 we

total nonnegativity of finite hurwitz matrices and root location of polynomials abstract in 1970 b a asner jr proved that for a real quasi stable polynomial i e a polynomial whose zeros lie in the closed left half plane of the complex plane its finite hurwitz matrix is totally nonnegative i e all its minors are nonnegative and that the converse statement is not true in this work we explain this phenomenon in detail and provide necessary and sufficient conditions for a real polynomial to have a totally nonnegative finite hurwitz matrix

hermite biehler routh hurwitz and total positivity simple proofs of the hermite biehler and routh hurwitz theorems are presented the total nonnegativity of the hurwitz matrix of a stable real polynomial follows as an immediate corollary

proof of a conjecture concerning the hadamard powers of inverse m matrices abstract the main result of this paper is the following if a aij is an inverse m matrix a r a ij r denotes the rth hadamard power of a then a r is again an inverse m matrix for any real number r 1 this settles a conjecture proposed by wang et al b y wang x p zhang f z zhang on the hadamard product of inverse m matrices linear algebra appl 305 2000 23 31 affirmatively naturally it shows that the conjecture of neumann m neumann a conjecture concerning the hadamard product of inverses of m matrices linear algebra appl 285 1998 277 290 is also valid

a conjecture concerning the hadamard product of inverses of m matrices abstract we conjecture that for an n n matrix a which is an inverse of an m matrix the hadamard product a a is also an inverse of an m matrix we have checked this conjecture without failure on many many examples but here we show that for quite a few well known classes of inverses of m matrices the conjecture is true it is known that the more general conjecture that when a and b are n n inverses of m matrices then a b is also an inverse of an m matrix is false however here too we are able to display some classes of inverses of m matrices which are closed under taking hadamard products

hadamard products of stable polynomials are stable a real polynomial is asymptotically stable when all of its zeros lie in the open left half of the complex plane we show that the hadamard coefficient wise product of two stable polynomials is again stable improving upon some known results via the associated hurwitz matrices we find another example of a class of totally nonnegative matrices which is closed under hadamard multiplication

total nonnegativity of matrices related to polynomial roots and poles of rational functions abstract in this paper totally nonnegative positive matrices are considered which are matrices having all their minors nonnegative positive the almost totally positive matrices form a class between the totally nonnegative matrices and the totally positive ones an efficient determinantal test based on the cauchon algorithm for checking a given matrix for falling in one of these three classes of matrices is applied to matrices which are related to roots of polynomials and poles of rational functions specifically the hankel matrix associated with the laurent series at infinity of a rational function and matrices of hurwitz type associated with polynomials in both cases it is concluded from properties of one or two finite sections of the infinite matrix that the infinite matrix itself has these or related properties then the results are applied to derive a sufficient condition for the hurwitz stability of an interval family of polynomials finally interval problems for a subclass of the rational functions viz r functions are investigated these problems include invariance of exclusively positive poles and exclusively negative roots in the presence of variation of the coefficients of the polynomials within given intervals

f vectors of barycentric subdivisions for a simplicial complex or more generally boolean cell complex we study the behavior of the f and h vector under barycentric subdivision we show that if has a non negative h vector then the h polynomial of its barycentric subdivision has only simple and real zeros as a consequence this implies a strong version of the charney davis conjecture for spheres that are the subdivision of a boolean cell complex or the subdivision of the boundary complex of a simple polytope for a general d 1 dimensional simplicial complex the h polynomial of its n th iterated subdivision shows convergent behavior more precisely we show that among the zeros of this h polynomial there is one converging to infinity and the other d 1 converge to a set of d 1 real numbers which only depends on d

closure properties of certain positivity classes of matrices under various algebraic operations abstract we present a table indicating whether or not each of five positivity classes of matrices positive definite hermitian matrices m matrices inverse m matrices totally positive matrices and inverse totally positive matrices is closed under each of seven algebraic operations conventional multiplication addition powers extraction of roots hadamard multiplication the hadamard product of one element and the inverse of another and lu factorization three of these 35 facts seem not to have been previously known and key examples and references are given

finite bandwidth kramers kronig relations for acoustic group velocity and attenuation derivative applied to encapsulated microbubble suspensions kramers kronig kk analyses of experimental data are complicated by the conflict between the inherently bandlimited data and the requirement of kk integrals for a complete infinite spectrum of input information for data exhibiting localized extrema kk relations can provide accurate transforms over finite bandwidths due to the local weighting properties of the kk kernel recently acoustic kk relations have been derived for the determination of the group velocity cg and the derivative of the attenuation coefficient components of the derivative of the acoustic complex wave number these relations are applicable to bandlimited data exhibiting resonant features without extrapolation or unmeasured parameters in contrast to twice subtracted finite bandwidth kk predictions for phase velocity and attenuation coefficient components of the undifferentiated wave number these more recently derived relations for cg and provide stricter tests of causal consistency because the resulting shapes are inv

the generalized touchard polynomials revisited we discuss the generalized touchard polynomials introduced recently by dattoli et al as well as their extension to negative order introduced by the authors with operational methods the connection to generalized stirling and bell numbers is elucidated and analogs to burchnall s identity are derived a recursion relation for the generalized touchard polynomials is established and it is shown that one can interpret some of the resulting formulas as binomial theorems for particular noncommuting variables we suggest to generalize the generalized touchard polynomials still further and introduce so called comtet touchard functions which are associated to the powers of an arbitrary derivation

brief paper necessary conditions for hadamard factorizations of hurwitz polynomials a property of hurwitz polynomials is related with the hadamard product garloff and wagner proved that hadamard products of hurwitz polynomials are hurwitz polynomials and garloff and shrinivasan shown that there are hurwitz polynomials of degree 4 which do not have a hadamard factorization into two hurwitz polynomials of the same degree 4 in this paper we give necessary conditions for an even degree hurwitz polynomial to have a hadamard factorization into two even degree hurwitz polynomials such conditions are given in terms of the coefficients of the given polynomial alone furthermore we show that if an odd degree hurwitz polynomial has a hadamard factorization then a system of nonlinear inequalities has at least one solution

complex zero strip decreasing operators abstract let z be a function in the laguerre polya class write z e z 2 1 z where 0 and where 1 z is a real entire function of genus 0 or 1 let f z be any real entire function of the form f z e z 2 f 1 z where 0 and f 1 z is a real entire function of genus 0 or 1 having all of its zeros in the strip s r z c r im z r where r 0 if 1 4 the linear differential operator d f z where d denotes differentiation is known to converge to a real entire function whose zeros also belong the strip s r we describe several necessary and sufficient conditions on z such that all zeros of d f z belong to a smaller strip s r 1 z c r 1 im z r 1 where 0 r 1 r and r 1 depends on z but is independent of f z we call a linear operator having this property a complex zero strip decreasing operator or czsdo we examine several relevant examples in certain cases we give explicit upper and lower bounds for r and we state several conjectures and open problems regarding complex zero strip decreasing operators

a logic of probability with application to the foundations of statistics a simple example shows that the classical theory of probability implies more than one can deduce via kolmogorov s calculus of probability developing dawid s ideas i propose a new calculus of probability which is free from this drawback this calculus naturally leads to a new interpretation of probability i argue that attempts to create a general empirical theory of probability should be abandoned and we should content ourselves with the logic of probability establishing relations between probabilistic theories and observations my approach to the logic of probability is based on a variant of ville s principle of the excluded gambling strategy

finite bandwidth effects on the causal prediction of ultrasonic attenuation of the power law form kramers kronig k k relations exist as a consequence of causality placing nonlocal constraints on the relationship between dispersion and absorption the finite bandwidth method of applying these relations is examined where the k k integrals are restricted to the spectrum of the experimental data these finite bandwidth k k relations are known to work with resonant type data and here are applied to dispersion data consistent with a power law attenuation coefficient exponent from 1 to 2 bandwidth restricted forms of the zero and once subtracted k k relations are used to determine the attenuation coefficient from phase velocity analytically it is shown that these transforms produce the proper power law form of the attenuation coefficient as a stand alone term summed with artifacts that are dependent on the integration limits calculations are performed to demonstrate how these finite bandwidth artifacts affect the k k predictions under a variety of conditions the predictions are studied in a local context as a function of subtraction frequency bandwidth and power law exponent the k k predictions of the power law exponent within various decades of the spectrum are also examined in general the agreement between finite bandwidth k k predictions and exact values grows as the power law exponent approaches 1 and with increasing bandwidth

on the extension of some total positivity inequalities abstract some inequalities recently obtained for totally positive matrices are extended to more general classes of matrices among these larger classes of matrices sign consistent sign regular and matrices with signed bidiagonal decomposition are considered the class of matrices with a signed bidiagonal decomposition contains the nonsingular totally positive matrices and their inverses

causal theories and data for acoustic attenuation obeying a frequency power law this study compares causal theories based on kramers kronig relations fractional calculus and on those derived from new time domain causal relationships to diverse data all these theories are based on the assumptions that the functional form of the attenuation persists beyond the measurement range and that attenuation is much smaller than the wave number the data for lossy media with attenuation having a power law frequency dependence with an exponent y include cases for both liquids and solids ranging from acoustic to ultrasound frequencies data are in closer correspondence with the new theory which predicts decreasing dispersion as the power exponent y approaches zero or an even integer experimental results and supporting evidence show that the classical case of frequency squared attenuation is dispersionless an approximate nearly local kramers kronig theory is in agreement with the time causal theory when the exponent is close to one but deviates for other values the comprehensive time causal theory is shown to be equivalent to two other theories derived from exact kramers kronig relations and from fractional calculus and it covers the y odd integer cases which are missing or incomplete in these approaches attenuation dispersion relations are presented in two forms one for a general frequency range and another for a finite range it is demonstrated that complete velocity dispersion within a signal bandwidth can be predicted from knowledge of the attenuation data and velocity at a single frequency including the velocity at either zero frequency y 1 and or at a high frequency limit 0 y 1 in addition attenuation data can be predicted from velocity data

a unified presentation of the gamma type functions occurring in diffraction theory and associated probability distributions in view of their importance and usefulness in diffraction theory and probability distributions several generalizations of the gamma type functions were investigated extensively in recent years the main object of this paper is to present a systematic study of a unification and generalization of these gamma type functions which is defined here by means of the fox wright hypergeometric function email protected q a probability density function associated with the generalized gamma type function investigated in this paper together with several other related results in the theory of probability and statistics are also considered

a family of summation formulas on the fox wright function abstract by applying the triple form of gould hsu inversions to the pfaff saalschutz theorem a family of summation formulas on the fox wright function which implies several known 3 f 2 4 3 series identities due to chen and chu 2011 20 and numerous new hypergeometric series identities is established

the s eulerian polynomials have only real roots we study the roots of generalized eulerian polynomials via a novel approach we interpret eulerian polynomials as the generating polynomials of a statistic over inversion sequences inversion sequences also known as lehmer codes or subexcedant functions were recently generalized by savage and schuster to arbitrary sequences s of positive integers which they called s inversion sequences our object of study is the generating polynomial of the ascent statistic over the set of s inversion sequences of length n since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations we call this generalized polynomial the s eulerian polynomial the main result of this paper is that for any sequence s of positive integers the s eulerian polynomial has only real roots this result is first shown to generalize several existing results about the real rootedness of various eulerian polynomials we then show that it can be used to settle a conjecture of brenti that eulerian polynomials for all finite coxeter groups have only real roots and partially settle a conjecture of dilks petersen stembridge on type b affine eulerian polynomials it is then extended to several q analogs we show that the macmahon carlitz q eulerian polynomial has only real roots whenever q is a positive real number confirming a conjecture of chow and gessel the same holds true for the hyperoctahedral group and the wreath product groups confirming further conjectures of chow and gessel and chow and mansour respectively our results have interesting geometric consequences as well

the riemann hypothesis and the turan inequalities a solution is given to a fifty eight year old open problem of g polya involving the taylor coefficients of the riemann function 1 introduction the purpose of this paper is to solve a fifty eight year old problem of polya p p 16 related to the riemann hypothesis this problem may be described as follows starting with riemann s definition of his function cf titchmarsh t p 16 in a slightly different notation i e

inverse m matrices ii this is an update of the 1981 survey by the first author in the meantime a considerable amount has been learned about the very special structure of the important class of inverse m matrices developments since the earlier survey are emphasized but we have tried to be somewhat complete and some results have not previously been published some proofs are given where appropriate and references are given for others after some elementary preliminaries results are grouped by certain natural categories

zero loci of skew growth functions for dual artin monoids abstract we show that the skew growth function of a dual artin monoid of finite type p has exactly rank p l simple real zeros on the interval 0 1 the proofs for types a l and b l are based on an unexpected fact that the skew growth functions up to a trivial factor are expressed by jacobi polynomials due to a rodrigues type formula in the theory of orthogonal polynomials the skew growth functions for type d l also satisfy rodrigues type formulae but the relation with jacobi polynomials is not straightforward and the proof is intricate we show that the smallest root converges to zero as the rank l tends to infinity

orthogonal bases that lead to symmetric nonnegative matrices abstract in a paper dating back to 1983 soules constructs from a positive vector x an orthogonal matrix r which has the property that for any nonnegative diagonal matrix with nonincreasing diagonal entries the matrix r rt has all its entries nonnegative independently fiedler in 1988 showed that any symmetric irreducible nonsingular matrix whose powers are all m matrices and hence an mma matrix in the language of friedland hershkowitz and schneider must have an orthogonal matrix of eigenvectors r which has similar properties to those of r here for a given positive n vector x we investigate the structure of all orthogonal matrices r for which for any nonnegative diagonal matrix as above the matrices r rt are nonnegative up to a permutation of its columns each such r corresponds to a binary tree whose vertices are subsets of the set 1 2 n with the property that each vertex has either no successor or exactly two disjoint successors for such orthogonal matrices r and such nonsingular diagonal matrices we show that the set of matrices of the form r rt and the set of inverse mma matrices i e matrices whose inverses are mma matrices coincide using this result we establish a relation between strictly ultrametric matrices and inverse mma matrices finally we show that the qr factorization of r rt for certain such r s has a special sign pattern

on the number of nonreal zeros of real entire functions and the fourier polya conjecture this paper is concerned with a general theorem on the number of nonreal zeros of transcendental functions j fourier formulated the theorem in his work analyse des equations determineesin 1831 but he did not give a proof roughly speaking the theorem states that if a real entire function f x can be expressed as a product of linear factors then we can count the nonreal zeros of f x by observing the behavior of the derivatives of f x on the real axis alone as we shall see in the sequel this theorem completely justifies his former argument by which he tried to prove that the function j0 2 x has only real zeros it seems that no complete proof of the theorem is known and no general theorem has been published that justifies the argument later in 1930 g polya published a paper entitled some problems connected with fourier s work on transcendental equations p3 in this paper polya conjectured two hypothetical theorems that are closely related to fourier s unproved theorem in fact he conjectured three but he proved that two of them are equivalent to each other the first hypothetical theorem is a modernized formulation of the theorem and it justifies fourier s argument completely the second conjecture was proved in 1990 but it is impossible to justify the argument using the conjecture alone in the present paper we prove polya s formulation of the theorem his first conjecture as well as its extensions give a very simple and direct proof of the second conjecture mentioned above and exhibit some applications of the results in particular we completely justify fourier s argument by our general theorems acknowledgments professor fefferman has encouraged and helped us to publish this paper the authors truly thank him for this the authors also thank professor csordas for his kind interest in the results and his valuable suggestions on the proof of the polya wiman conjecture 1 historical introduction in this section we briefly explain our results as well as their background a real entire function is an entire function that assumes only real values on the real axis fourier s unproven theorem asserts that we can know the number of nonreal zeros of such real entire functions by counting their critical points which are defined as follows let f x be a real analytic function defined in an open

inclusion intervals of singular values and applications abstract in this paper we improve the brauer type inclusion interval and present a modified brauer type theorem for singular values numerical examples show that our estimation for singular values is more precise than those corresponding results in recent literature moreover we also consider the marginal singular value on the brauer type inclusion interval some applications of our results are presented

total positivity of a kind of generalized toric bernstein basis abstract the normalized totally positive bases are widely used in many fields based on the generalized vandermonde determinant the normalized total positivity of a kind of generalized toric bernstein basis is proved which is defined on a set of real points by this result then the progressive iterative approximation property of the generalized toric bezier curve is obtained and some examples are provided to illustrate this property

accurate inverses of nekrasov z matrices abstract we present a parametrization of a nekrasov z matrix that allows us to compute its inverse with high relative accuracy numerical examples illustrating the accuracy of the method are included

majorization and multiplier sequences we present applications of matrix methods to the analytic theory of polynomials we first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials then we use matrix methods to establish a new log majorization result on roots of polynomials the theory of multiplier sequences gives the common link between the applications

intervals of totally nonnegative matrices totally nonnegative matrices i e matrices having all their minors nonnegative and matrix intervals with respect to the checkerboard ordering are considered it is proven that if the two bound matrices of such a matrix interval are nonsingular and totally nonnegative and in addition all their zero minors are identical then all matrices from this interval are also nonsingular and totally nonnegative with identical zero minors

generalizations of m matrices which may not have a nonnegative inverse abstract generalizations of m matrices are studied including the new class of gm matrices the matrices studied are of the form si b with b having the perron frobenius property but not necessarily being nonnegative results for these classes of matrices are shown which are analogous to those known for m matrices also various splittings of a gm matrix are studied along with conditions for their convergence

touchard like polynomials and generalized stirling numbers abstract the theory of touchard polynomials is generalized using a method based on the definition of exponential operators which extend the notion of the shift operator the proposed technique along with the use of the relevant operational formalism allows the straightforward derivation of properties of this family of polynomials and their relationship to different forms of stirling numbers

expansions of chromatic polynomials and log concavity in this paper we present several results and open problems about log concavity properties of sequences associated with graph colorings five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros combinatorial and log concavity properties are studied four of these polynomials have never been considered before in the literature and some yield new expansions for the chromatic polynomial

a property concerning the hadamard powers of inverse m matrices abstract the main result of this paper is the following if a a ij is an inverse m matrix a k a ij k denotes the k th hadamard power of a then a k is an inverse m matrix for any positive integer k 1 in the case k 2 this settles a conjecture of neumann linear algebra appl 285 1998 278 affirmatively

hadamard powers and totally positive matrices considered are continuous positive hadamard powers of entry wise positive nonnegative matrices those that are eventually in the sense of all hadamard powers beyond some point totally positive totally nonnegative doubly nonnegative and doubly positive are characterized for example for matrices with at least four rows and four columns hadamard powers greater than one of totally positive matrices need not be totally positive but they are eventually totally positive

contiguous relations for the fox wright function abstract contiguous relations are a powerful tool for dealing with hypergeometric functions in this paper four different types of contiguous relations for the fox wright function are given we demonstrate the usefulness of our formulas by showing how they can be used to recover the famous hagen rothe convolution

the hadamard core of the totally nonnegative matrices abstract an m by n matrix a is called totally nonnegative if every minor of a is nonnegative the hadamard product of two matrices is simply their entry wise product this paper introduces the subclass of totally nonnegative matrices whose hadamard product with any totally nonnegative matrix is again totally nonnegative many properties concerning this class are discussed including a complete characterization for min m n a characterization of the zero nonzero patterns for which all totally nonnegative matrices lie in this class and connections to oppenheim s inequality

inner totally positive matrices abstract suppose a m n is a staircase matrix with row and column staircase sequences 1 2 n 1 2 n a minor a det a with 1 k 1 k is said to be an inner minor of a if i i i i for i 1 2 k a is said to be inner totally positiv e itp if every inner minor of a is positive we prove that a is itp if a 0 for all inner minors with q k n 0 k 1 2 n also a m n is itp iff it is totally non negative and its extreme inner minors are positive we show that an itp matrix may be reduced by similarity transformations to an itp band matrix and may alternatively be filled in by similarity transformations to become a tp matrix in both the reduction and the filling in the matrix is itp at each stage the analysis is applied to some inverse eigenvalue problems for band matrices

arrangements of equal minors in the positive grassmannian abstract we discuss arrangements of equal minors of totally positive matrices more precisely we investigate the structure of equalities and inequalities between the minors we show that arrangements of equal minors of largest value are in bijection with sorted sets which earlier appeared in the context of alcoved polytopes and grobner bases maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex and the number of such arrangements equals the eulerian number on the other hand we prove in many cases that arrangements of equal minors of smallest value are weakly separated sets weakly separated sets originally introduced by leclerc and zelevinsky are closely related to the positive grassmannian and the associated cluster algebra however we also construct examples of arrangements of smallest minors which are not weakly separated using chain reactions of mutations of plabic graphs

similarity to totally positive matrices and accurate computations abstract matrices similar to nonsingular totally positive matrices are analyzed in the particular case of sbd matrices new characterizations are presented for more general classes of matrices methods to assure accurate computations are analyzed

on nonsingular sign regular matrices abstract an m n matrix a is sign regular if for each k 1 k min m n all k k submatrices of a have determinant with the same nonstrict sign the zero pattern of nonsingular sign regular matrices is analyzed it is proved that the number of zero entries which can appear in a nonsingular sign regular matrix depends on its signature a matrix is totally nonpositive if all its minors are nonpositive a test for recognizing nonsingular totally nonpositive matrices is also provided

totally nonnegative cells and matrix poisson varieties we describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices that is the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by a postnikov in order to do this we relate the totally nonnegative cells to torus orbits of symplectic leaves of the poisson varieties of complex matrices in particular we describe the minors that vanish on a torus orbit of symplectic leaves we prove that such families of minors are exactly the admissible families and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices

multivariate polya schur classification problems in the weyl algebra a multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts we classify all linear partial differential operators in the weyl algebra a n that preserve stability an important tool that we develop in the process is the higher dimensional generalization of polya schur s notion of multiplier sequence we characterize all multivariate multiplier sequences as well as those of finite order next we establish a multivariate extension of the cauchy poincare interlacing theorem and prove a natural analog of the lax conjecture for real stable polynomials in two variables using the latter we describe all operators in a 1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in a n preserves stability if and only if its fischer fock adjoint does these are powerful multivariate extensions of the classical hermite poulain jensen theorem polya s curve theorem and schur malo szegocomposition theorems examples and applications to strict stability preservers are also discussed

on the applicability of kramers kronig relations for ultrasonic attenuation obeying a frequency power law in the recent literature concern has been raised regarding the validity of kramers kronig relations for media with ultrasonic attenuation obeying a frequency power law it is demonstrated however that the kramers kronig dispersion relations for application to these types of media are available the developed dispersion relations are compared with measurements on several liquids and agreement is found to better than 1 m s over the experimentally available bandwidth a discussion regarding the validity of these dispersion relations in particular how the dispersion relations relate to the so called paley wiener conditions forms the conclusion

simplified expressions of the subtracted kramers kronig relations using the expanded forms applied to ultrasonic power law systems the kramers kronig kk relations are a large class of integral transformations that exploit the broad principle of simple causality in order to link the physical properties of matter and materials in applications to the complex valued wavenumber for acoustic propagation the method of subtractions is used to form convergent integral relations between the phase velocity and the attenuation coefficient when the method of subtractions is applied in the usual manner the integrands in the relations become unnecessarily complicated in this work an expanded form of the subtracted relations is presented which is essentially a truncated taylor series expansion of the hilbert transforms the implementation of the relations only requires the explicit evaluation of two simply expressed integrals involving the hilbert transform kernel these two integrals determine the values of the other terms in the subtracted relations demonstrating the computational efficiency of the technique the method is illustrated analytically through its application to power law attenuation coefficients and its associated dispersion which are observed in a wide variety of materials this approach explicitly shows the central role of the hilbert transform kernel in the kk relations which can become obscured in other formulations

inverse m matrix inequalities and generalized ultrametric matrices abstract we use weighted directed graphs to introduce a class of nonnegative matrices which under a simple condition are inverse m matrices we call our class the generalized ultrametic matrices since it contains the class of symmetric ultrametric matrices and some unsymmetric matrices we show that a generalized ultrametric matrix is the inverse of a row and column diagonally dominant m matrix if and only if it contains no zero row and no two of its rows are identical this theorem generalizes the known result that a symmetric strictly ultrametric matrix is the inverse of a strictly diagonally dominant m matrix we also present inequalities and conditions for equality among the entries of the inverse of a row diagonally dominant m matrix some of these inequalities and conditions for equality generalize results of stieltjes on inverses of symmetric diagonally dominant m matrices

the approximation of a totally positive band matrix by a strictly banded totally positive one every nonsingular totally positive m banded matrix is shown to be the product of m totally positive one banded matrices and therefore the limit of strictly m banded totally positive matrices this result is then extended to bi infinite m banded totally positive matrices with linearly independent rows and columns in the process such matrices are shown to possess at least one diagonal whose principal sections are all nonzero as a consequence such matrices are seen to be approximable by strictly m banded totally positive ones

an analytic approach to the asymptotic variance of trie statistics and related structures we develop analytic tools for the asymptotics of general trie statistics which are particularly advantageous for clarifying the asymptotic variance many concrete examples are discussed for which new fourier expansions are given the tools are also useful for other splitting processes with an underlying binomial distribution we specially highlight philippe flajolet s contribution in the analysis of these random structures

on the hadamard product of inverse m matrices abstract we investigate the hadamard product of inverse m matrices and present two classes of inverse m matrices that are closed under the hadamard multiplication in the end we give some inequalities on the fan product of m matrices and schur complements

total positivity of sums hadamard products and hadamard powers results and counterexamples abstract we show that for hankel matrices total nonnegativity resp total positivity of order r is preserved by sum hadamard product and hadamard power with real exponent t r 2 we give examples to show that our results are sharp relative to matrix size and structure general symmetric or hankel some of these examples also resolve the hadamard critical exponent problem for totally positive and totally nonnegative matrices

on linear transformations preserving the polya frequency property we prove that certain linear operators preserve the polya frequency property and real rootedness and apply our results to settle some conjectures and open problems in combinatorics proposed by bona brenti and reiner welker

on inverse m matrices abstract we generalize a previous result of lewin and neumann and present a characterization of inverse m matrices the approach is mainly graph theoretical

the fox wright functions and laguerre multiplier sequences abstract linear operators which 1 preserve the reality of zeros of polynomials having only real zeros and 2 map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain fox wright functions and generalized mittag leffler functions the paper includes several open problems and questions

structured matrices continued fractions and root localization of polynomials we give a detailed account of various connections between several classes of objects hankel hurwitz toeplitz vandermonde and other structured matrices stietjes and jacobi type continued fractions cauchy indices moment problems total positivity and root localization of univariate polynomials along with a survey of many classical facts we provide a number of new results

on sufficient conditions for the total positivity and for the multiple positivity of matrices abstract the following theorem is proved theorem suppose m a i j be a k k matrix with positive entries and a i j a i 1 j 1 4 cos 2 k 1 a i j 1 a i 1 j 1 i k 1 1 j k 1 then det m 0 the constant 4 cos 2 k 1 in this theorem is sharp a few other results concerning totally positive and multiply positive matrices are obtained

the gauss lucas theorem and jensen polynomials a characterization is given of the sequences fyj o vvith the property that for any complex polynomial z 1akzk and convex region kcontaining the origin and the zeros of the zeros of 2 yi 0 for all n k

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

differential operators of infinite order and the distribution of zeros of entire functions whenever the right hand side of 1 2 represents an analytic function in a neighborhood of the origin when x is an entire function the operator d has been studied by several authors see for example 5 11 19 chapter ix 22 and 32 the conjecture of polya and wiman proved in 8 9 and 17 states that if f x l p then df x is in the laguerre polya class for all sufficiently large positive integers m in this paper we analyze the more general situation when d is replaced by the operator d in section 2 we consider real power series with zero linear term i e 1 0 in 1 1 and 0 2 0 and show that if f x is any real polynomial then d f x l p for all sufficiently large positive integers m theorem 2 4 if the linear term in 1 1 is nonzero

kramers kronig relations applied to finite bandwidth data from suspensions of encapsulated microbubbles in this work the kramers kronig k k relations are applied to experimental data of resonant nature by limiting the interval of integration to the measurement spectrum the data are from suspensions of encapsulated microbubbles albunex and have the characteristics of an ultrasonic notch filter the goal is to test the consistency of this dispersion and attenuation data with the kramers kronig relations in a strict manner without any parameters from outside the experimental bandwidth entering in to the calculations in the course of reaching the goal the artifacts associated with the truncation of the integrals are identified and it is shown how their impacts on the results can be minimized the problem is first approached analytically by performing the kramers kronig calculations over a restricted spectral band on a specific hilbert transform pair lorentzian curves the resulting closed form solutions illustrate the type of artifacts that can occur due to truncation and also show that accurate res

positive path product and inverse m matrices it is known that an inverse m matrix is strict path product but not necessarily vice versa for n 3 it is shown that any square positive matrix may be made strict path product by predictable additions to the diagonal and that any normalized strict path product matrix may be made inverse m by additions to the diagonal that are bounded in terms of the size of the matrix the latter has implications for pairs of inverse m matrices whose hadamard product is inverse m a determinantal inequality relating principal minors and certain associated almost principal minors is derived for normalized inverse m matrices

totally nonnegative moment matrices abstract the finite dimensional moment matrices of distributions which are either concentrated on the nonnegative half line or symmetric around zero are similar to totally nonnegative matrices an analogous result holds true for schur complements in these moment matrices as a consequence in the former case the nonzero eigenvalues of these matrices have multiplicity one and in the latter at most two this is a key result for finding e optimal polynomial regression designs in heiligers 1992 see also pukelsheim and studden 1993

polya schur master theorems for circular domains and their boundaries we characterize all linear operators on finite or infinite dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region omega subset of c for arbitrary

on the real rootedness of generalized touchard polynomials we consider the real rootedness of generalized touchard polynomials recently revisited by mansour and schork 2013 towards this end we first describe the normal form of the generalized touchard polynomials by which recurrence relations for the polynomial part are derived by using the recurrence relations we prove the real rootedness of the generalized touchard polynomials for the parameter m 1 k k 1 k n

the significance of jacob bernoulli s ars conjectandi for the philosophy of probability today abstract more than 300 years ago in a fertile period from 1684 to 1689 jacob bernoulli worked out a strategy for applying the mathematics of games of chance to the domain of probability that strategy came into public view in august 1705 eight years after jacob s death at the age of 50 when his masterpiece ars conjectandi was published unfortunately the message of ars conjectandi was not fully absorbed at the time of its publication and it has been obscured by various intellectual fashions during the past 300 years the theme of this article is that jacob s contributions to the philosophy of probability are still fresh in many ways after 300 years we still have something to learn from his way of seeing the problems and from his approach to solving them and the fog that obscured his ideas is clearing our intellectual situation today may permit us a deeper appreciation of jacob s message than any of our predecessors were able to achieve i will pursue this theme chronologically first i will review jacob s problem situation and how he dealt with it second i will look at the changing obstacles to the reception of his message over the last three centuries finally i will make a case for paying more attention today

theorems on m splittings of a singular m matrix which depend on graph structure abstract let a m ne r n n be a splitting we investigate the spectral properties of the iteration matrix m 1n by considering the relationships of the graphs of a m n and m 1n we call a splitting an m splitting if m is a nonsingular m matrix and n 0 for an m splitting of an irreducible z matrix a we prove that the circuit index of m 1n is the greatest common divisor of certain sets of integers associated with the circuits of a for m splittings of a reducible singular m matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting these results hold under somewhat weaker assumptions

the spectrum of the gegenbauer collocation approximation for the second derivative operator abstract it is proved that the eigenvalues of the gegenbauer collocation method for the second derivative operator with dirichlet boundary conditions are real negative and distinct for a range of the gegenbauer parameter this range includes the chebyshev collocation approximation the result is also extended to neumann and robin boundary conditions

suffridge s convolution theorem for polynomials and entire functions having only real zeros abstract we present a q extension of polya s and schur s characterization of multiplier sequences and a q extension and a converse of newton s inequalities

total positivity of hadamard products abstract the hadamard product of two totally positive toeplitz matrices m and n need not be totally positive when only finitely many diagonals of m and of n are non zero preservation of total positivity by hadamard product is essentially a theorem of malo here we establish another sufficient condition for the preservation of total positivity if both m and n are totally positive lower triangular toeplitz matrices such that the value on the nth diagonal is a polynomial function of n then the hadamard product m n is totally positive we use the characterization of the generating functions of polya frequency sequences given by aissen edrei schoenberg and whitney and in the course of the proof we extend the concept of sturm sequences and develop some other results on polynomials with only real roots which are of independent interest

imprecise probability trees bridging two theories of imprecise probability we give an overview of two approaches to probability theory where lower and upper probabilities rather than probabilities are used walley s behavioural theory of imprecise probabilities and shafer and vovk s game theoretic account of probability we show that the two theories are more closely related than would be suspected at first sight and we establish a correspondence between them that i has an interesting interpretation and ii allows us to freely import results from one theory into the other our approach leads to an account of probability trees and random processes in the framework of walley s theory we indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees and we prove an interesting and quite general version of the weak law of large numbers

h polynomials of zonotopes the ehrhart polynomial of a lattice polytope p encodes information about the number of integer lattice points in positive integral dilates of p the h polynomial of p is the numerator polynomial o

the role of the fox wright functions in fractional sub diffusion of distributed order the fundamental solution of the fractional diffusion equation of distributed order in time usually adopted for modelling sub diffusion processes is obtained based on its mellin barnes integral representation such solution is proved to be related via a laplace type integral to the fox wright functions a series expansion is also provided in order to point out the distribution of time scales related to the distribution of the fractional orders the results of the time fractional diffusion equation of a single order are also recalled and then re obtained from the general theory

new polya schoenberg type theorems abstract in this paper the theory of hadamard product multipliers is extended from the unit disk in the complex plane to arbitrary so called disk like domains i e such domains which are the union of disks or half planes all containing the origin in such a domain say we define the class r d of generalized prestarlike functions of order 1 and ask for hadamard multipliers g analytic at z 0 for which f r d implies g f r d we prove that such a multiplier necessarily has to be analytic in u v u v c in many cases we prove this for all proper disks containing the origin we actually find that r d is the precise description of the set of all such multipliers for these disks say the domains turn out to be bounded by the outer loops of certain limacons of pascal the parameter is related to the characteristic q 1 1 r s of the disk where r is the shortest distance of the origin to the boundary of that disk and s the largest large subclasses of r d are being explicitly determined for the case 0 i e d this result coincides with an old one by ruscheweyh and sheil small previously conjectured by g polya and i j schoenberg the notion of the characteristic of a disk containing the origin is then extended to general disk like domains and some multipliers are identified for those general classes r d the previously determined class of universally prestarlike functions defined in the slit domain c 1 is identified as the class of universal multipliers for r d in any disk like domain

some possible new directions for combinatorial matrix analysis using the facts that 1 much of the combinatorial matrix analysis to date has been concerned with properties of trees and 2 trees as digraphs have many important special properties we suggest several classes of matrices which can probably be successfully studied with the help of combinatorial methods these classes include the following a unipethic matrices a class derived by retaining the property that in the irreducible case the digraph defined in the usual way is strong and unipathic if it is a tree b bipartite matrices a class obtained by retaining the property that a tree is a bipartite digraph these matrices are closely related to 2 cyclic matrices but they are not the same class c matrices such that their digraph has at least one cycle of length p and no longer cycle these matrices are an obvious generalization of trees they are primarily of interest for small values of p relative to the size of the matrix d hessenberg matrices because their successive principal minors can be shown to always satisfy a difference equation and therefore a are one sort of generalization of the tridiagonal matrices outside of the class of trees e matrices which are such that the digraph is a k tree for k 1 in particular the 2 trees it is easy to show that k trees are a natural generalization of trees and they include the band matrices five diagonal matrices are for example 2 trees

inverting linear combinations of identity and generalized catalan matrices abstract we introduce the notion of the generalized catalan matrix as a kind of lower triangular toeplitz matrix whose nonzero elements involve the generalized catalan numbers inverse of the linear combination of the pascal matrix with the identity matrix is computed in aggarwala and lamoureux 2002 1 in this paper continuing this idea we invert various linear combinations of the generalized catalan matrix with the identity matrix a simple and efficient approach to invert the pascal matrix plus one in terms of the hadamard product of the pascal matrix and appropriate lower triangular toeplitz matrices is considered in yang and liu 2006 14 we derive representations for inverses of linear combinations of the generalized catalan matrix and the identity matrix in terms of the hadamard product which includes the generalized catalan matrix and appropriate lower triangular toeplitz matrix