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

on the convergence of rows of vector pade approximants we state and prove a de montessus like theorem for vector valued meromorphic functions using vector pade approximants which are defined in the context of clifford algebras the proof is based on that used by saff for the scalar case the theorem may be regarded as an answer to a general problem posed by graves morris and saff

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

a companion matrix approach to the study of zeros and critical points of a polynomial in this paper we introduce a new type of companion matrices namely d companion matrices by using these d companion matrices we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial in fact this new approach will allow us to prove quite a number of new as well as known results on this topic for example we prove some results on the majorization of the critical points of a polynomial by its zeros in particular we give a different proof of a recent result of gerhard schmeisser on this topic the same method allows us to prove a higher order schoenberg type conjecture proposed by m g de bruin and a sharma

on a schoenberg type conjecture abstract for an arbitrary polynomial p n z z n a 1 z n 1 a 2 z n 2 1 n a n 1 n z z j with the sum of all zeros equal to zero a 1 1 n z j 0 the quadratic mean radius is defined by r p n 1 n 1 n z j 2 1 2 and the quartic mean radius by s p n 1 n 1 n z j 4 1 4 this paper studies a schoenberg type conjecture using the quartic mean radius in the following form n 4 n 1 s p n 4 2 n 1 r p n 4 s p n 4 with equality if and only if the zeros all lie on a straight line through the origin in the complex plane

a note on matrix valued rational interpolants in this paper by defining a kind of transformation from matrix to vector we succeed in transferring some results on vector valued rational interpolants to those corresponding to the matrix valued rational interpolants moreover it is pointed out through a numerical example that the statement of the so called uniqueness theorem in 4 is incorrect and what is more the proof is also wrong a new uniqueness theorem is given

clifford algebras and vector valued rational forms i we demonstrate how the use of clifford algebras in the theory of vector valued rational forms leads to practical recurrence relations which do not involve representations of the algebras

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

relationship between the zeros of two polynomials abstract in this paper we shall follow a companion matrix approach to study the relationship between zeros of a wide range of pairs of complex polynomials for example a polynomial and its polar derivative or sz nagy s generalized derivative we shall introduce some new companion matrices and obtain a generalization of the weinstein aronszajn formula which will then be used to prove some inequalities similar to sendov conjecture and schoenberg conjecture and to study the distribution of equilibrium points of logarithmic potentials for finitely many discrete charges our method can also be used to produce in an easy and systematic way a lot of identities relating the sums of powers of zeros of a polynomial to that of the other polynomial

an extension of a row convergence theorem for vector pade approximants abstract a row convergence theorem of de montessus type is established for vector pade approximants to a vector valued meromorphic function f while in a previous paper the authors established such a theorem for the case when f has simple poles the essential feature of the present paper is to treat the situation when f has multiple poles

bivariate composite vector valued rational interpolation in this paper we point out that bivariate vector valued rational interpolants bvri have much to do with the vector grid to be interpolated when a vector grid is well defined one can directly design an algorithm to compute the bvri however the algorithm no longer works if a vector grid is ill defined taking the policy of divide and conquer we define a kind of bivariate composite vector valued rational interpolant and establish the corresponding algorithm a numerical example shows our algorithm still works even if a vector grid is ill defined

row convergence theorems for generalised inverse vector valued pade approximants abstract the approximants mentioned in the title are related to vector valued continued fractions and the vector algorithm devised by wynn in 1963 here we establish a unitary invariance property of these approximants and describe how the classical 1 dimensional pade approximants can be obtained as a special case the main results of the paper consist of de montessus de ballore type convergence theorems for row sequences having fixed denominator degree of vector valued approximants to meromorphic vector functions

minimal vector pade approximation abstract we describe the minimal vector pade approximation problem which consists in finding pade approximants with a common denominator for a number of series these approximants are minimal in the sense that for a given order of approximation and a given discrepancy in numerator and denominator degrees the degree of the rational approximant is minimal properties and solution methods are derived from an associated partial realization problem

adaptive osculatory rational interpolation for image processing image interpolation is a common problem in image applications although many interpolation algorithms have been proposed in the literature these methods suffer from the effects of imperfect reconstruction to some degree most often these effects manifest themselves as jagged contours or blurred edges in the image this paper presents a method for preserving the contours or edges based on adaptive osculatory rational interpolation kernel function which is built up by approximating the ideal interpolating kernel function by continued fractions it is a more accurate approximation for the ideal interpolation in space domain or frequency domain than by other linear polynomial interpolation kernel functions simulation results are also presented to demonstrate the superior performance of image magnification

a sleszyňski pringsheim theorem for vector valued continued fractions and its optimal error bounds the first purpose of this work is to prove a convergence theorem for vector valued continued fractions which is an exact extension of a well known theorem of sleszynski pringsheim and our proof is based on the truncation error analysis for such continued fractions the second purpose is to give refined error bounds for such continued fractions these bounds are also improvements of the scalar results and the numerical example given explains the effect of our results

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

scale invariances and lamperti transformations for stochastic processes scale invariant processes and hereafter processes with broken versions of this symmetry are studied by means of the lamperti transformation a one to one transformation linking stationary and self similar processes a general overview of the use of the transformation and of the stationary generators it builds is given for modelling and analysis of scale invariance we put an emphasis on generalizations to non strictly scale invariant situations the examples of discrete scale invariance and finite size scale invariance are developed by means of the lamperti transformation framework and some specific examples of processes with these generalized symmetries are given

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

problems and progress in vector pade approximation abstract some open problems in vector pade approximation are stated and some recent de montessus type theorems governing convergence of rows of the vector pade table are contrasted weshow how these results indicate when it is more appropriate to use generalised inverse vector pade approximants or their hybridised form we show how a straightforward analogue of the berlekamp massey algorithm may be used to calculate generalised inverse vector pade approximants this algorithm is applied to the derivation of a new low order hybrid vector approximant related results include the case of a row convergence theorem for a complex valued power series using a clifford inverse instead of the moore penrose inverse

a neville like method via continued fractions as we know the classical neville s algorithm is an effective method used to solve the interpolation problem by polynomials in this paper we adopt the idea of the neville s algorithm to construct a kind of blending rational interpolants via continued fractions for a given set of support points there are many ways to build up the interpolation schemes by which we mean that there are many choices to make to determine the initial interpolants on subsets of support points and then update them step by step to form a solution to the full interpolation problem numerical examples are given to show the advantage of our method and a multivariate analogy is also discussed

on the theory and computation of nonperfect pade hermite approximants abstract for a vector of k 1 power series we introduce two new types of rational approximations the weak pade hermite form and the weak pade hermite fraction a recurrence relation is then presented which computes pade hermite forms along with their weak counterparts along a sequence of perfect points in the pade hermite table the recurrence relation results in a fast algorithm for calculating a pade hermite approximant of any given type when the vector of power series is perfect the algorithm is shown to calculate a pade hermite form of type n0 nk in o kn 2 operations where n n0 nk this complexity is the same as that of other fast algorithms the new algorithm also succeeds in the nonperfect case usually with only a moderate increase in cost

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

determinantal inequalities for diagonally signed matrices and an application to gram cauchy matrices abstract the condition that a hermitian matrix is diagonally signed complementary has recently been shown to guarantee that its signature is invariant with respect to hadamard products with gram matrices in this paper we establish inequalities for the determinants of these diagonally signed matrices that are analogs of well known inequalities for positive definite matrices because hermitian cauchy matrices and their confluent forms are diagonally signed we can then infer from the new inequalities the existence in general of inverses of the confluent forms of hermitian gram cauchy matrices

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

weak log majorization mahler measure and polynomial inequalities abstract we explore the use of the weak log majorization order in the analytic theory of polynomials we examine the relationship between weak log majorization and mahler measure we also improve the weak log majorization form of the de bruijn springer mahler inequality

extrapolation methods for vector sequences an analogue of aitken s 2 method suitable for vector sequences is proposed aspects of the numerical performance of the vector algorithm based on using the moore penrose inverse are investigated the fact that the denominator polynomial associated with a vector pade approximant is the square of its equivalent in the scalar case is shown to be a source of approximation error in cases where the convergence of the vector sequence is dominated by real eigenvalues a hybrid form of the vector pade approximant having a denominator polynomial of minimal degree is proposed and its effectiveness is demonstrated on several standard examples

a new method for computing the matrix exponential operation based on vector valued rational approximations in this paper a new method for computing the action of the matrix exponential on a vector e a tb where a is a complex matrix and t is a positive real number is proposed our approach is based on vector valued rational approximation where the approximants are determined by the denominator polynomials whose coefficients are obtained by solving an inexpensive linear least squares problem no matrix multiplications or divisions but matrix vector products are required in the whole process a technique of scaling and recurrence enables our method to be more effective when the problem is for fixed a b and many values of t we also give a backward error analysis in exact arithmetic for the truncation errors to derive our new algorithm preliminary numerical results illustrate that the new algorithm performs well

a new structure for analyzing discrete scale invariant processes covariance and spectra improving the efficiency of discrete scale invariant dsi sequence we consider some flexible sampling of a continuous time dsi process on positive real line with scale greater than one this sampling has the advantage to have a corresponding multi dimensional self similar process this enables us to obtain spectral representation of such sampled dsi process and corresponding spectral density matrix by imposing wide sense markov property on the dsi process we show that the covariance function and the spectral density matrix are characterized by variances and covariances of adjacent samples in the first scale interval finally we present an example as simple brownian motion and provide its simulations to clarify this study we also study the performance of this structure on the s p500 indices for some special period too

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

vector rational interpolation algorithms of burlisch stoer neville form we present algorithms of the bulirsch stoer neville form for calculating the value of the vector rational interpolant the derivation parallels that of the bulirsch stoer algorithm for the usual scalar rational interpolation algorithm of stoer and bulirsch using the clifford algebra foundations for vector rational interpolants of roberts

a new approach to the rational interpolation problem the vector case we generalize our earlier results on rational interpolation which were given in van barel and bultheel this journal 1990 for the scalar case and in bultheel and van barel 1990 for the vector case when all the interpolation points coincide to the case of vector data given at arbitrary points that may coincide or not this is the vector valued newton pade problem we give a recursive algorithm which has the important advantage over other algorithms that we do not need a reordering of the given interpolation data to overcome a singularity in the interpolation table not even in the nonnormal vector case it also generates all the information needed to give all solutions of the problem

differentiators and the geometry of polynomials in 1959 davis introduced the concept of a differentiator of an operator on a finite dimensional hilbert space we prove that every such operator possesses a differentiator we also use the theory of differentiators to solve several problems in the geometry of polynomials for instance we answer in the affirmative a twenty year old unsolved conjecture of schoenberg a related conjecture of katsoprinakis and a fifty year old unsolved conjecture of de bruijn and springer

circulants and critical points of polynomials abstract we prove that for any circulant matrix c of size n n with monic characteristic polynomial p z the spectrum of its n 1 n 1 submatrix c n 1 consisting of the first n 1 rows and columns of c consists of all critical points of p z using this fact we provide a simple proof for the schoenberg conjecture recently proved by r pereira and s malamud we also prove full generalization of a higher order schoenberg type conjecture proposed by m de bruin and a sharma and recently proved by w s cheung and t w ng in its original form i e for polynomials whose mass centre of roots equals zero in this particular case our inequality is stronger than it was conjectured by de bruin and sharma some schmeisser s like results on majorization of critical point of polynomials are also obtained

block based thiele like blending rational interpolation thiele type continued fraction interpolation may be the favoured nonlinear interpolation in the sense that it is constructed by means of the inverse differences which can be calculated recursively and produce useful intermediate results however thiele s interpolation is in fact a point based interpolation by which we mean that a new interpolating continued fraction with one more degree of its numerator or denominator polynomial is obtained by adding a tail to the current one or in other words this can be reshaped simply by adding a new support point to the current set of support points once at a time in this paper we introduce so called block based inverse differences to extend the point based thiele type interpolation to the block based thiele like blending rational interpolation the construction process may be outlined as follows first of all divide the original set of support points into some subsets blocks then construct each block by using whatever interpolation means linear or rational and finally assemble these blocks by thiele s method to shape the whole interpolation scheme clearly many flexible rational interpolation schemes can be obtained in this case including the classical thiele s continued fraction interpolation as its special case as an extension a bivariate analogy is also discussed and numerical examples are given to show the effectiveness of our method

weak convergence of positive self similar markov processes and overshoots of levy processes using lamperti s relationship between levy processes and positive self similar markov processes pssmp we study the weak convergence of the law x of a pssmp starting at x 0 in the skorohod space of cadlag paths when x tends to 0 to do so we first give conditions which allow us to construct a cadlag markov process x 0 starting from 0 which stays positive and verifies the scaling property then we establish necessary and sufficient conditions for the laws x to converge weakly to the law of x 0 as x goes to 0 in particular this answers a question raised by lamperti z wahrsch verw gebiete 22 1972 205 225 about the feller property for pssmp at x 0

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

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

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

bounds and majorization relations for the critical points of polynomials abstract we apply several matrix inequalities to the derivative companion matrices of complex polynomials to establish new bounds and majorization relations for the critical points of these polynomials in terms of their zeros

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

solution of integral equations using generalised inverse function valued pade approximants i abstract neumann series solutions for the linear integral equation x 1 1 0 1 x y y d y converge for small values of we use generalised inverse function valued pade approximants to accelerate the convergence of the neumann series and to estimate the singular value of these estimates are found to compare very favourably with those from the fredholm determinant method and the pade approximant method

on the realizability of the critical points of a realizable list abstract the nonnegative inverse eigenvalue problem niep is to characterize the spectra of entrywise nonnegative matrices a finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix monov conjectured that the kth moments of the list of critical points of a realizable list are nonnegative johnson further conjectured that the list of critical points must be realizable in this work johnson s conjecture and consequently monov s conjecture is established for a variety of important cases including ciarlet spectra sule manova spectra spectra realizable via companion matrices and spectra realizable via similarity by a complex hadamard matrix additionally we prove a result on differentiators and trace vectors and use it to provide an alternative proof of a result due to malamud and a generalization of a result due to kushel and tyaglov on circulant matrices implications for further research are discussed

vector valued rational interpolants i it is shown explicitly that the samelson inverse may be used to define vector valued thiele type rational interpolants and the equivalence with claessens algorithm is established for the special case of vector valued pade approximants the general form of mccleod s theorem follows as a corollary

the epsilon algorithm and related topics the epsilon algorithm is recommended as the best all purpose acceleration method for slowly converging sequences it exploits the numerical precision of the data to extrapolate the sequence to its limit we explain its connections with pade approximation and continued fractions which underpin its theoretical base then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector valued sequences and some related topics in this paper we consider the class of methods based on using generalised inverses of vectors and the formulation specifically includes the complex case wherever possible

convergence theorems for rows of differential and algebraic hermite pade approximations abstract the authors investigate the asymptotic behaviour of hermite pade polynomials of latin type which are formed from differential or algebraic expressions involving a meromorphic function the degrees of the denominator polynomials are chosen to correspond to rows of the hermite pade table for special choices of denominator degrees these asymptotic are used to prove de montessus type theorems for hermite pade approximants formed by solving differential or algebraic equations

generalized inverse matrix pade approximation on the basis of scalar products a new type of generalized matrix inverse is used to define the generalized inverse matrix pade approximants gmpa gmpa is introduced on the basis of scalar product of matrices with the form of matrix numerator and scalar denominator it is different from the existing matrix pade approximants in that it does not need multiplication of matrices in the construction process some algebraic properties are discussed the representations of gmpa are provided with the following three forms i the explicit determinantal formulas for the denominator scalar polynomials and the numerator matrix polynomials ii e algorithm expression iii thiele type continued fraction expression the equivalence relations above three representations are proposed

spectral analysis of multi dimensional self similar markov processes in this paper we consider a discrete scale invariant dsi process x t t r with scale l 1 we consider a fixed number of observations in every scale say t and acquire our samples at discrete points k k w where is obtained by the equality l t and w 0 1 we thus provide a discrete time scale invariant dt si process x with the parameter space k k w we find the spectral representation of the covariance function of such a dt si process by providing the harmonic like representation of multi dimensional self similar processes spectral density functions of them are presented we assume that the process x t t r is also markov in the wide sense and provide a discrete time scale invariant markov dt sim process with the above scheme of sampling we present an example of the dt sim process simple brownian motion by the above sampling scheme and verify our results finally we find the spectral density matrix of such a dt sim process and show that its associated t dimensional self similar markov process is fully specified by rhj 1 rjh 0 j 0 1 t 1 where rhj is the covariance function of jth and j th observations of the process

a worpitzky theorem for vector valued continued fractions generalizations of worpitzky s convergence theorem for ordinary continued fractions are proved for vector valued continued fractions defined by the samelson inverse for vectors our convergence proof is constructive and thus yields more refined truncation error estimates this work shows that some of our results are exact generalizations or improvements of the scalar ones

is every matrix similar to a polynomial in a companion matrix abstract given a field f an integer n 1 and a matrix a m n f are there polynomials f g f x with f monic of degree n such that a is similar to g c f where c f is the companion matrix of f for infinite fields the answer is easily seen to positive so we concentrate on finite fields in this case we give an affirmative answer provided f n 2 moreover for any finite field f with f m we construct a matrix a m m 3 f that is not similar to any matrix of the form g c f of use above but also of independent interest is a constructive procedure to determine the similarity type of any given matrix g c f purely in terms of f and g without resorting to polynomial roots in f or in any extension thereof this in turn yields an algorithm that given g and the invariant factors of any a returns the elementary divisors of g a it is a rational procedure as opposed to the classical method that uses the jordan decomposition of a to find that of g a finally extending prior results by the authors we show that for an integrally closed ring r with field of fractions f and companion matrices c d the subalgebra r c d of m n r is a free r module of rank n n m n 1 where m is the degree of gcd f g f x and a presentation for r c d is given in terms of c and d a counterexample is furnished to show that r c d need not be a free r module if r is not integrally closed the preceding information is used to study m n r and others as r x modules

inverse spectral problem for normal matrices and the gauss lucas theorem we establish an analog of the cauchy poincare interlacing theorem for normal matrices in terms of majorization and we provide a solution to the corresponding inverse spectral problem using this solution we generalize and extend the gauss lucas theorem and prove the old conjecture of de bruijn springer on the location of the roots of a complex polynomial and its derivative and an analog of rolle s theorem conjectured by schoenberg

confluent hermitian cauchy matrices are diagonally signed abstract it is shown that cauchy matrices admit a confluent extension in much the same way that confluent vandermonde matrices or cauchy alternants have confluent forms cauchy matrices formed from distinct elements have been proved recently to be diagonally signed and this is equivalent to showing that they are complementary matrices we show here that this property extends to the confluent case too

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

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

the simplified topological varepsilon algorithms for accelerating sequences in a vector space when a sequence of numbers is slowly converging it can be transformed into a new sequence which under some assumptions converges faster to the same limit one of the best known sequence transformations is the shanks transformation which can be recursively implemented by the varepsilon algorithm of wynn this transformation and this algorithm have been extended in two different ways to a sequence of elements of a topological vector space e in this paper we present new algorithms for implementing these topological shanks transformations they no longer require the manipulation of elements of the algebraic dual space e of e nor do they use the duality product inside the rules of the algorithms they need the storage of fewer elements of e and the stability is improved they also allow us to prove convergence and acceleration results for some types of sequences various applications involving sequences of vectors or matrices show the interest of the new algorithms

the construction of λµ b spline curves and its application to rotational surfaces in order to provide more flexible approaches for designers the b spline curves are constructed as a generalization of the traditional cubic uniform b spline curves possessing multiple local shape control parameters b spline curves not only inherit the properties of cubic uniform b spline curves but also exhibit better performance when adjusting its local shapes through two local shape control parameters particularly to adjust and control the shapes of rotational surfaces more elegantly the b spline rotational surfaces with two local shape parameters are presented and utilized a rotational surface is produced by combining b spline with a transfinite vector valued rational interpolation function further the properties of rotational surfaces as well as its applications in rotational surface designs are explored finally the modeling examples are supplied to illustrate the proposed method in admitting the easy control of the shape of a surface which suggest the much wider applications to the pattern design system of apparel cad cam

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