decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

decay rates of inverses of banded m matrices that are near to toeplitz matrices the decaying behavior of inverses of positive definite band matrices is analysed for m matrices that are in some sense close to toeplitz matrices estimates based on the factorization are derived that are better than existing ones in particular for nonsymmetric matrices some examples are given

monotone convex sequences and cholesky decomposition of symmetric toeplitz matrices this paper studies off diagonal decay in symmetric toeplitz matrices it is shown that if the generating sequence of the matrix is monotone positive and convex then the monotonicity and positivity are maintained through triangular decomposition the work is motivated by recent results on explicit bounds for inverses of triangular matrices

the inverse positivity of perturbed tridiagonal m matrices abstract a well known property of an m matrix m is that the inverse is element wise non negative which we write as m 1 0 in this paper we consider element wise perturbations of non symmetric tridiagonal m matrices and obtain sufficient bounds on the perturbations so that the non negative inverse persists these bounds improve the bounds recently given by kennedy and haynes inverse positivity of perturbed tridiagonal m matrices linear algebra appl 430 2009 2312 2323 in particular when perturbing the second diagonals elements l l 2 and l l 2 of m these sufficient bounds are shown to be the actual maximum allowable perturbations numerical examples are given to demonstrate the effectiveness of our estimates

a quadratic system with two parallel straight line isoclines abstract in this paper a quadratic system with two parallel straight line isoclines is considered this system corresponds to the system of class ii in the classification of ye yanqian ye yanqian et al theory of limit cycles transl math monogr vol 66 american mathematical society providence ri 1986 using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles we prove that the maximum number of limit cycles in a quadratic system with two parallel straight line isoclines and two finite singular points is equal to two besides we obtain the same result in a different way applying the wintner perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed in v a gaiko global bifurcation theory and hilbert s sixteenth problem kluwer boston 2003

determination of bed elevation in the enhanced lattice boltzmann method for the shallow water equations the bed slope in the shallow water equations reflects the bed topography it is not a flow variable and cannot be determined in the solution to the flow equations an immovable nonflat bed affects a flow as a force term but the flow has no effect on it only when the bed term is correctly represented in a numerical method can it generate an accurate solution in the enhanced lattice boltzmann method for the shallow water equations elabswe using an individual chapman enskog analysis it is found that such a correct representation can be achieved by retaining c 2 in which c is the coefficient for bed elevation in the lattice boltzmann equation and is that for the water depth in the local equilibrium distribution function the finding has been validated through simulations of a water at rest in a dish shaped lake a wind induced shallow flow in the same lake and a steady flow over a two dimensional bed hump

persistently positive inverses of perturbed m matrices a well known property of an m matrix a is that the inverse is element wise non negative which we write as a 1 0 in this paper we consider perturbations of m matrices and obtain bounds on the perturbations so that the non negative inverse persists the bounds are written in terms of decay estimates which characterize the decay along rows of the elements of the inverse matrix we obtain results for diagonal and rank 1 perturbations of symmetric tridiagonal m matrices and rank 1 perturbations of non symmetric tridiagonal m matrices

on the inverses of general tridiagonal matrices in this work the sign distribution for all inverse elements of general tridiagonal h matrices is presented in addition some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained based on the sign distribution these bounds greatly improve some well known results due to ostrowski 1952 23 shivakumar and ji 1996 26 nabben 1999 21 22 and recently given by peluso and politi 2001 24 peluso and popolizio 2008 25 and so forth it is also stated that the inverse of a general tridiagonal matrix may be described by 2n 2 parameters k k 2n and k k 1n 1 instead of 2n 2 ones as given by el mikkawy 2004 3 el mikkawy and karawia 2006 4 and huang and mccoll 1997 10 according to these results a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented which improves some recent results finally several applications to the preconditioning technology the numerical solution of differential equations and the birth death processes together with numerical tests are given

global qualitative analysis of a quartic ecological model abstract in this paper we complete the global qualitative analysis of a quartic ecological model in particular studying global bifurcations of singular points and limit cycles we prove that the corresponding dynamical system has at most two limit cycles

limit cycles of quadratic systems abstract in this paper the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving hilbert s sixteenth problem in this special case of polynomial systems is suggested using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is 3 1 v a gaiko global bifurcation theory and hilbert s sixteenth problem kluwer boston 2003 besides applying the wintner perko termination principle for multiple limit cycles to our canonical system we prove in a different way that a quadratic system has at most three limit cycles around a singular point focus and give another proof of the same conjecture

study of the 1d lattice boltzmann shallow water equation and its coupling to build a canal network the d1q3 lattice boltzmann lb shallow water equation is analyzed in detail and compared with other numerical schemes analytical results are derived and used to discuss the accuracy and stability of the model we show how such d1q3 lb models for canal reaches may be easily coupled with various hydraulic interconnection structures to build models of complex irrigation networks

estimates for the inverse of tridiagonal matrices arising in boundary value problems by considering tridiagonal matrices as three term recurrence relations with dirichlet boundary conditions one can formulate their inverses in terms of green s functions this analysis is applied to three point difference schemes for 1 d problems and five point difference schemes for 2 d problems we derive either an explicit inverse of the jacobian or a sharp estimate for both uniform and nonuniform grids

enhancement of the labswe for shallow water flows in the lattice boltzmann method for the shallow water equations labswe the force term involves the first order derivative related to a bed slope which has to be determined using the centred scheme for an accurate solution however such calculation contradicts the spirit of only simple arithmetic calculations required in the lattice boltzmann method this paper shows how to remove this drawback from the labswe by incorporating the bed level into the lattice boltzmann equation in a consistent manner with the lattice boltzmann approach three numerical tests have been presented to verify the proposed method

a lattice boltzmann model for the shallow water equations a lattice boltzmann model for the shallow water equations labswe is developed it solves the equations with the source terms such as bed slope bed friction the model has been verified by solving both steady and unsteady flow problems excellent agreement is obtained between numerical predictions and analytical solutions the results show that the labswe is a very competitive method for computational hydraulics or hydrodynamics in terms of computational efficiency and accuracy

block diagonal dominance of matrices revisited bounds for the norms of inverses and eigenvalue inclusion sets abstract we generalize the bounds on the inverses of diagonally dominant matrices obtained in 16 from scalar to block tridiagonal matrices our derivations are based on a generalization of the classical condition of block diagonal dominance of matrices given by feingold and varga in 11 based on this generalization which was recently presented in 3 we also derive a variant of the gershgorin circle theorem for general block matrices which can provide tighter spectral inclusion regions than those obtained by feingold and varga

two sided bounds on the inverses of diagonally dominant tridiagonal matrices abstract we establish upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices these bounds improve the bounds recently given by shivakumar and ji moreover we show how to improve our bounds iteratively for an n n m matrix this iterative refinement yields the exact inverse after n 1 steps

improved bounds for the inverses of diagonally dominant tridiagonal matrices abstract we obtain new bounds for the entries of the inverse of a diagonally dominant tridiagonal matrix which improve the best previous ones due to h b li et al we apply our bounds to the tridiagonal matrices arising in the second order finite difference discretization of certain boundary value problems of parabolic type establishing asymptotically optimal bounds

the surface gradient method for the treatment of source terms in the shallow water equations a novel scheme has been developed for data reconstruction within a godunov type method for solving the shallow water equations with source terms in contrast to conventional data reconstruction methods based on conservative variables the water surface level is chosen as the basis for data reconstruction this provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a riemann solver the main advantages are 1 a simple centered discretization is used for the source terms 2 the scheme is no more complicated than the conventional method for the homogeneous terms 3 small perturbations in the water surface elevation can be accurately predicted and 4 the method is generally suitable for both steady and unsteady shallow water problems the accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems excellent agreement has been obtained between the numerical predictions and analytical solutions the results indicate that the new scheme is accurate simple efficient and robust

semi implicit finite difference methods for the two dimensional shallow water equation abstract in this paper a semi implicit finite difference method for the 2 dimensional shallow water equations is derived and discussed a characteristic analysis of the governing equations is carried out first in order to determine those terms to be discretized implicitly so that the stability of the method will not depend upon the celerity such terms are the gradient of the water surface elevation in the momentum equations and the velocity divergence in the continuity equation the convective terms are discretized explicitly the simpler explicit discretization for the convective terms is the upwind discretization which is conditionally stable and introduces some artificial viscosity it is shown that the stability restriction is eliminated and the artificial viscosity is reduced when an eulerian lagrangian approach with large time steps is used to discretize the convective terms this method at each time step requires the solution of a linear symmetric 5 diagonal system such a system is diagonally dominant with positive elements on the main diagonal and negative ones elsewhere thus existence and uniqueness of the numerical solution is assured the resulting algorithm is mass conservative and fully vectorizable for an efficient implementation on modern vector computers the performance of this method is further improved when used in combination with an adi technique which results in two sets of simpler linear 3 diagonal systems and maintains all the properties described above

inverse positivity of perturbed tridiagonal m matrices a well known property of an m matrix m is that the inverse is element wise non negative which we write as m 1 0 in this paper we consider element wise perturbations of non symmetric tridiagonal m matrices and obtain bounds on the perturbations so that the non negative inverse persists sufficient bounds are written in terms of decay estimates which characterize the decay of the elements of the inverse of the unperturbed matrix results for general symmetric matrices and symmetric toeplitz matrices are obtained as special cases and compared with known results

upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices abstract we give easily computable upper and lower bounds for the inverse elements of finite tridiagonal diagonally dominant matrices and we improve the well known upper bounds due to ostrowski the results are extended to infinite systems the theory is also applied to some earlier results and to the evaluation of bessel functions and mathieu functions by using their recurrence relations

some improvements for two sided bounds on the inverse of diagonally dominant tridiagonal matrices abstract in this paper we obtain lower and upper bounds for the entries of the inverses of diagonally dominant tridiagonal matrices first of all we derive the bounds for off diagonal elements of the inverse as a function of the diagonal ones then we improve the two sided bounds for the diagonal entries obtaining sharper lower and upper bounds for all the elements of the inverse

multiple limit cycle bifurcations of the fitzhugh nagumo neuronal model abstract in this paper we complete the global qualitative analysis of the well known fitzhugh nagumo neuronal model in particular studying global limit cycle bifurcations and applying the wintner perko termination principle for multiple limit cycles we prove that the corresponding dynamical system has at most two limit cycles

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

nonhydrodynamic modes and a priori construction of shallow water lattice boltzmann equations lattice boltzmann equations for the isothermal navier stokes equations have been constructed systematically using a truncated moment expansion of the equilibrium distribution function from continuum kinetic theory applied to the shallow water equations with its different equation of state the same approach yields discrete equilibria that are subject to a grid scale computational instability different and stable equilibria were previously constructed by salmon j marine res 57 503 1999 the two sets of equilibria differ through a nonhydrodynamic or ghost mode that has no direct effect on the hydrodynamic behavior derived in the slowly varying limit however salmon s equilibria eliminate a coupling between hydrodynamic and ghost modes one that leads to instability with a growth rate increasing with wave number previous work has usually assumed that truncated moment expansions lead to stable schemes such instabilities have implications for lattice boltzmann equations that simulate other nonideal equations of state or that simulate fully compressible nonisothermal fluids using additional particles

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets