weighted shift matrices unitary equivalence reducibility and numerical ranges abstract an n by n n 3 weighted shift matrix a is one of the form 0 a 1 0 a n 1 a n 0 where the a j s called the weights of a are complex numbers assume that all a j s are nonzero and b is an n by n weighted shift matrix with weights b 1 b n we show that b is unitarily equivalent to a if and only if b 1 b n a 1 a n and for some fixed k 1 k n b j a k j a n j a j for all j next we show that a is reducible if and only if a j j 1 n is periodic that is for some fixed k 1 k n 2 n is divisible by k and a j a k j for all j 1 j n k finally we prove that a and b have the same numerical range if and only if a 1 a n b 1 b n and s r a 1 2 a n 2 s r b 1 2 b n 2 for all 1 r n 2 where s r s are the circularly symmetric functions

bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

scaled generalized bernstein polynomials and containment regions for the zeros of polynomials let p n x email protected k 0 email protected kb k n x a b b n 0 where b k n x a b a x k b x n k k 0 1 n a email protected c a email protected a generalized scaled bernstein polynomial extending a result of winkler a resultant matrix for scaled bernstein polynomials linear algebra appl 319 2000 179 191 a companion matrix for p n is given the application of some matrix methods to this companion matrix yields regions for the zeros of p n

companion matrices reducibility numerical ranges and similarity to contractions abstract in this paper we study some unitary equivalence properties of the companion matrices we obtain a criterion for a companion matrix to be reducible and show that the numerical range of a companion matrix is a circular disc centered at the origin if and only if the matrix equals the nilpotent jordan block however the more general assertion that a companion matrix is determined by its numerical range turns out to be false we also determine for an n n matrix a with eigenvalues in the open unit disc the defect index of a contraction to which a is similar

containment regions for zeros of polynomials from numerical ranges of companion matrices containment regions for the zeros of a monic polynomial are given with the aid of results for containment regions for the numerical range of certain bordered diagonal matrices which are applied to different types of companion matrices of the polynomial

blow up solutions for a class of nonlinear parabolic equations with dirichlet boundary conditions in the present paper the blow up of smooth local solutions for a class of nonlinear parabolic equations u t a u u f x u q t q u 2 with dirichlet boundary conditions are studied by constructing an auxiliary function and using hopf s maximum principles on it the sufficient conditions for blow up solutions are obtained and the upper bound of blow up time is given under some suitable assumptions on a f and initial date the obtained results are applied to some examples in which a and f are exponential functions

higher rank numerical ranges and compression problems we consider higher rank versions of the standard numerical range for matrices a central motivation for this investigation comes from quantum error correction we develop the basic structure theory for the higher rank numerical ranges and give a complete description in the hermitian case we also consider associated projection compression problems

blow up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions abstract by constructing different auxiliary functions and using hopf s maximum principle the sufficient conditions for the blow up and global solutions are presented for nonlinear parabolic equation u t a u b x c t u f x u q t with different kinds of boundary conditions the upper bounds of the blow up time and the upper estimates of global solutions are provided finally some examples are presented as the application of the obtained results

a bibliography on roots of polynomials abstract a categorized bibliography on roots of polynomials is presented covering the period from the earliest times until late 1990 the actual bibliography is on a computer diskette accompanying this issue together with programs designed to print parts of the main work such as entries belonging to a certain category

singular points of the algebraic curves of symmetric hyperbolic forms abstract we investigate singular points of the algebraic curve defined by a hyperbolic form satisfying some symmetric properties we show that all singular points of such a curve are real nodes

efficient evaluation of the fourier transform over products of slater type orbitals on different centers using the shift operator technique a compact formula for the fourier transform of a product of two slater type orbitals located on different atomic centers is derived the result is valid for arbitrary quantum numbers and was found to be numerically stable for a wide range of geometrical parameters and momenta details of the implementation are presented together with benchmark data for representative integrals we also discuss the assets and drawbacks of alternative algorithms available and analyze the numerical efficiency of the new scheme

global existence and blow up problems for reaction diffusion model with multiple nonlinearities abstract in this paper we study the following reaction diffusion model u t a u b x c t u g x t f u in d 0 t u n h x t r u on d 0 t u x 0 u 0 x 0 in d where d is a bounded domain in r n with smooth boundary d n 2 this paper deals with interactions among three kinds of nonlinear mechanisms nonlinear reaction nonlinear convection and nonlinear boundary flux the existence theorems of blow up positive solutions upper bound of blow up time upper estimates of blow up rate existence theorems of global positive solutions and upper estimates of global positive solutions are given under suitable assumptions on a b c f g h r and initial data u 0 x

an updated supplementary bibliography on roots of polynomials in 1 theauthorhaspresentedabibliographyonrootsofpolynomials coveringmostpublishedworksonthistopicfromthebeginninguntilapproximatelymid 1990 see 1 forsomeverybriefremarksonthehistoryandapplicationsofthesubject later in 2 wehavepresentedasupple mentarybibliography thiscontainedreferencestoworksnotincludedintheoriginalbibliography eitherbecausetheywerenotyetpublished orbecausetheyhadnotyetbeendiscoveredbytheauthor nowwepresentafurthersupplementarybibliography containingworkspublishedordiscoveredsincethepublicationof 2 atthetimeofpublicationof 2 thematerialsin 1 2 werecombinedintoone leandmadeavailableviatheworldwideweb thenewmaterialhasnowbeenincorporatedintothatsame le instructionsforaccessareasfollows 1 thebibliographycanbeaccessedfromthehomepageofthejournal of computational and ap plied mathematics http www elsevier nl locate cam byactivatingthehyperlinkbibliographyonrootsofpolynomials 2 thebibliographyisdividedinto32categories frombracketing methods tocontinuation eachofwhichcanbebrowsedandperusedbyactivatingtheappropriateentryinthecategorieslisting 3 a simple searchengineisalsoavailable byenteringthewordyouarelookingforinthetextwindowandpressingthesubmitbutton youwillseeallbibliographicreferencescontainingyoursearchword booleanconnectivesarenot supported youcanlimitthescopeofthesearchenginebycheckingoneormorecategoriesbefore submittingthesearch

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

a class of degenerate diffusion equations with mixed boundary conditions abstract this paper is concerned with a class of degenerate diffusion equations subject to mixed boundary conditions under some structure conditions we discuss the blow up property of local solutions and estimate the bounds of blow up time

on the numerical range of tridiagonal operators it is shown that the numerical range of a tridiagonal operator with 0 main diagonal is symmetric with respect to the origin some related specialized results under stronger hypotheses are included in particular if the order of a matrix is 3 then its numerical range is an elliptical disk and is explicitly described

the diameter and width of numerical ranges abstract we characterize the diameter and width of the numerical range present an algorithm for computing the diameter and width of the numerical range and formulate the diameters of the numerical ranges of unitary bordering matrices for lower dimensions we also determine the boundaries of the numerical ranges of certain nilpotent toeplitz matrices to be curves of constant width

zero dilation index of a finite matrix abstract for an n by n complex matrix a we define its zero dilation index d a as the largest size of a zero matrix which can be dilated to a this is the same as the maximum k 1 for which 0 is in the rank k numerical range of a using a result of li and sze we show that if d a 2 n 3 then under unitary similarity a has the zero matrix of size 3 d a 2 n as a direct summand it complements the known fact that if d a n 2 then 0 is an eigenvalue of a we then use it to give a complete characterization of n by n matrices a with d a n 1 namely a satisfies this condition if and only if it is unitarily similar to b 0 n 3 where b is a 3 by 3 matrix whose numerical range w b is an elliptic disc and whose eigenvalue other than the two foci of w b is 0 we also determine the value of d a for any normal matrix a and any weighted permutation matrix a with zero diagonals

a necessary and sufficient condition for global existence for a degenerate parabolic boundary value problem consider the degenerate parabolic boundary value problemut u f u on 0 in which is a bounded domain inrnand thec 0 functionsfand are nonnegative and nondecreasing with s f s 0 ifs 0 and 0 0 assume homogeneous neumann boundary conditions and an initial condition that is nonnegative nontrivial and continuous on because the function is not sufficiently nice to allow this problem to have a classical solution we consider generalized solutions in a manner similar to that of benilan crandall and sacks appl math optim 17 1988 203 224 we show that this initial boundary value problem has such a nonnegative generalized solution if and only ifi 0 ds 1 f s

numerical ranges of weighted shift matrices abstract we show that if a is an n by n n 3 matrix of the form 0 a 1 0 a n 1 a n 0 then the boundary of its numerical range contains a line segment if and only if the a j s are nonzero and the numerical ranges of the n 1 by n 1 principal submatrices of a are all equal for n 3 this is the case if and only if a 1 a 2 a 3 0 in which case w a the numerical range of a is the equilateral triangular region with vertices the three cubic roots of a 1 a 2 a 3 for n 4 the condition becomes a 1 a 3 0 and a 2 a 4 0 in which case w a is the convex hull of two degenerate or otherwise ellipses

global existence and blowup solutions for quasilinear parabolic equations the authors discuss the quasilinear parabolic equation ut g u u h u u f u with u 0 u x 0 x i ff g and h are polynomials with proper degrees and proper coefficients they show that the blowup property only depends on the first eigenvalue of in with dirichlet boundary condition for a special case they obtain a sharp result

numerical ranges of row stochastic matrices abstract in this paper we consider properties of the numerical range of an n by n row stochastic matrix a it is shown that the numerical radius of a satisfies 1 w a 1 n 2 and moreover w a 1 resp w a 1 n 2 if and only if a is doubly stochastic resp a 0 1 1 0 j th for some j 1 j n a complete characterization of the a s for which the zero matrix of size n 1 can be dilated to a is also given finally for each n 2 we determine the smallest rectangular region in the complex plane whose sides are parallel to the x and y axis and which contains the numerical ranges of all n by n row stochastic matrices

numerical ranges of weighted shift matrices with periodic weights abstract let a be an n by n n 2 matrix of the form 0 a 1 0 a n 1 a n 0 we show that if the a j s are nonzero and their moduli are periodic then the boundary of its numerical range contains a line segment we also prove that w a contains a noncircular elliptic arc if and only if the a j s are nonzero n is even a 1 a 3 a n 1 a 2 a 4 a n and a 1 a 2 finally we give a criterion for a to be reducible and completely characterize the numerical ranges of such matrices

the fourier transform method and the sd approach for the analytical and numerical treatment of multicenter overlap like quantum similarity integrals molecular overlap like quantum similarity measurements imply the evaluation of overlap integrals of two molecular electronic densities related by dirac delta function when the electronic densities are expanded over atomic orbitals using the usual lcao mo approach linear combination of atomic orbitals overlap like quantum similarity integrals could be expressed in terms of four center overlap integrals it is shown that by introducing the fourier transform of delta dirac function in the integrals and using the fourier transform approach combined with the so called b functions one can obtain analytic expressions of the integrals under consideration these analytic expressions involve highly oscillatory semi infinite spherical bessel functions which are the principal source of severe numerical and computational difficulties in this work we present a highly efficient algorithm for a fast and accurate numerical evaluation of these multicenter overlap like quantum similarity integrals over slater type functions this algorithm is based on the s d approach due to safouhi recurrence formulae are used for a better control of the degree of accuracy and for a better stability of the algorithm the numerical result section shows the efficiency of our algorithm compared with the alternatives using the one center two range expansion method which led to very complicated analytic expressions the epsilon algorithm and the nonlinear d transformation

determinantal representations of hyperbolic forms via weighted shift matrices we characterize hyperbolic ternary forms of degrees 3 4 and 5 that admit determinantal representations via cyclic weighted shift matrices

on the numerical range of some weighted shift matrices and operators abstract in this paper we compute and compare the numerical radii of certain weighted shift matrices also we compute the numerical radius of a weighted shift operator on the hardy space h 2 the purpose of this paper is to develop results in 7

singular points of cyclic weighted shift matrices abstract let s be an n by n cyclic weighted shift matrix and f s t x y det t i x s y s be a ternary form associated with s we investigate the number of singular points of the curve f s t x y 0 and show that the number of singular points of f s t x y 0 associated with a cyclic weighted shift matrix whose weights are neither 1 periodic nor 2 periodic is less than or equal to n n 3 2 furthermore we verify the upper bound n n 3 2 is sharp for 4 n 7

diagonals and numerical ranges of weighted shift matrices for any n by n matrix a we consider the maximum number k k a forwhichthereisak by kcompressionofawithallitsdiagonal entries in the boundary w a of the numerical range w a of a for any such compression we give a standard model under unitary equivalence fora this is then applied to determine the value ofk a for a of size 3 in terms of the shape of w a w hena is a matrix of the form

extremely compact formulas for the fourier transform of a product of two centre slater type orbitals a compact formula for the fourier transform of a product of slater type orbitals on different centres is derived the integral is reduced to a finite one dimensional integration over non oscillatory hypergeometric functions of type the formula is valid for all quantum numbers and does not involve the reduced bessel functions that are usually used to evaluate these integrals reduced formulas are calculated for some special directions in the reciprocal space also some useful identities for the fourier transforms of a product of slater type orbitals with correlated sets of parameters are obtained in order to illustrate simple and efficient use of the presented results we have applied them to graphene

blow up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition this paper is devoted to a nonlinear parabolic equation subject to nonlocal boundary condition and with nonlocal reaction source some criteria on this problem which determine whether the solutions blow up in a finite time or exist globally are given moreover the blow up rate estimates are also obtained

a supplementary bibliography on roots of polynomials in 1 this author presented a bibliography on roots of polynomials covering most published works on this topic from the beginnings until approximately mid 1990 see 1 for some very brief remarks on the history and applications of the subject now we present a supplementary bibliography this contains references to works not included in the original bibliography either because they were not yet published or because they had not yet been discovered by the author formerly the bibliography was presented on a diskette included with the journal issue together with a c program which facilitated access to the material e g certain categories could be selected now however the material is being made available via the world wide web instructions for access are as follows 1 the bibliography can be accessed from the homepage of the journal of computational and applied mathematics

blow up solutions and global solutions for a class of quasilinear parabolic equations with robin boundary conditions the type of problem under consideration isu email protected email protected a u b x u g x t f u indx 0 t email protected email protected x t u 0 email protected dx 0 t u x 0 u 0 x 0 ind where d is a smooth bounded domain of r n by constructing an auxiliary function and using hopf s maximum principles on it existence theorems of blow up solutions upper bound of blow up time upper estimates of blow up rate existence theorems of global solutions and upper estimates of global solutions are given under suitable assumptions on a b f g s and initial data u 0 x the obtained results are applied to some examples in which a b f g and s are power functions or exponential functions

the fourier transform of a two center product of exponential type functions and its efficient evaluation we analyze the analytical and numerical properties of the hitherto known formulas of the fourier transform of a two center product of exponentially declining functions exponential type functions etfs which are derived with the help of the fourier convolution theorem and feynman s identity in detail we consider b functions which are a special class of etfs having advantageous properties under fourier transformation other etfs orbitals can be expressed in terms of b functions by linear combinations in our derivations we use the properties of the differential operator ylm specifying a solid harmonic whose argument is the nabla operator r instead of the vector r in order to generate multicenter integrals over nonscalar functions from integrals over scalar functions applying the generating differential operator ylm we obtain a recently derived new formula for the fourier transform of a two center product of b functions in a much more straightforward manner furthermore we present an efficient procedure to compute this new formula which is valid for arbitrary quantum numbers and exponential parameters and report various numerical test values