spaces of matrices of constant rank and uniform vector bundles abstract we consider the problem of determining l r a the maximal dimension of a subspace of a a matrices of rank r we first review in the language of vector bundles the known results then using known facts on uniform bundles we prove some new results and make a conjecture finally we determine l r a for every r 1 r a when a 10 showing that our conjecture holds true in this range

a mathematical model of adomian polynomials the adomian decomposition method is reviewed and a mathematical model of adomian polynomials is introduced to show the convenience of this model for computer programming the mathematica package is used to generate the adomian polynomials for any order

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

a new approach to the wave equation an application of the decomposition method abstract in the present paper the inhomogeneous wave equation is solved quickly easily and elegantly by adomian s decomposition method

an application of the adomian decomposition method to the transient behavior of a model biochemical reaction abstract an approximate analytical solution for the transient phase of the michaelis menten reaction is derived using the adomian decomposition method the analytical solution which is given in the form of a power series is found to be highly accurate in predicting the behavior of the reaction in the very early stages to accelerate the convergence of the power series solution and extend its region of applicability throughout the entire transient phase we have used a the method of pade approximants and b the iterated shanks transformation both the pade approximant and the shanks transformation are shown to converge rapidly throughout and beyond the transient period and yield very accurate results a comparison of the various analytical approximations and a direct numerical solution of the nonlinear initial value problem is also presented

an algorithm for solving boundary value problems in this paper a numerical algorithm based on the decomposition technique is presented for solving a class of nonlinear boundary value problems the method is implemented for well known examples including troesch s and bratu s problems which have been extensively studied the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions

on the dimension of spaces of linear transformations satisfying rank conditions abstract let l c n c m be the space from c n to c m let l k n m be the maximal dimension of a linear subspace v of l c n c m such that every nonzero map in v has rank k in this paper we develop some techniques for estimating l from above

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

a new iterative method to compute nonlinear equations the aim of this paper is to construct a new efficient iterative method to solve nonlinear equations the new method is based on the proposals of abbasbandy on improving the order of accuracy of newton raphson method s abbasbandy improving newton raphson method for nonlinear equations by modified adomian decomposition method applied mathematics and computation 145 2003 887 893 and on the proposals of babolian and biazar on improving the order of accuracy of adomian s decomposition method e babolian j biazar on the order of convergence of adomian method applied mathematics and computation 130 2002 383 387 the convergence of the new scheme is proved and at least the cubic order of convergence is established several examples are presented and compared to other methods showing the accuracy and fast convergence of this new method also it is shown in this paper that the modified adomian s method developed by babolian and biazar to solve nonlinear equations e babolian j biazar solution of nonlinear equations by modified adomian decomposition method applied mathematics and computation 132 2002 167 172 should be slightly modified due to the fact that convergence of adomian s method does not ensure convergence of the modified method an example illustrates this fact which unlike what is claimed by the authors does not converge with their method but with a simple different choice of the zero component becomes convergent

revisit on partial solutions in the adomian decomposition method solving heat and wave equations the adomian decomposition method is considered in application to heat and wave equations the so called partial solution technique is used it is shown that the fundamental equation of the method is well defined only for certain types of boundary conditions in cases involving inhomogeneous boundary conditions improper results may be obtained by former method this paper presents a further insight into partial solutions in the decomposition method and the resolution of such cases

a new approach to the heat equation an application of the decomposition method abstract the author s decomposition method adomian stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press bellman and adomian partial differential equations methods for their treatment and solution reidel dordrecht 1984 adomian j comput appl math 2 1984 225 230 shown elsewhere to provide computable accurate approximate convergent solutions to nonlinear stochastic operator equations is applied to the heat equation

the decomposition approach to inverse heat conduction adomian s decomposition approach is employed for solving some inverse boundary value problems in heat conduction further the mollification method is applied to deal with noisy input data and obtain a stable approximate solution

linear spaces of real matrices of constant rank abstract the largest possible dimensions of linear spaces of real n n matrices of constant rank n 1 or n 2 are determined using topological k theory and expressed in terms of hurwitz radon numbers

orthogonal designs with maximal rates orthogonal designs have been used as space time block codes for wireless communications with multiple transmit antennas which can achieve full transmit diversity and have a very simple decoupled maximum likelihood decoding algorithm the rate of an orthogonal design is defined as the ratio of the number of transmitted information symbols in a block of channel uses to the length of the given block which reflects the bandwidth efficiency of the employed space time block code constructed from the orthogonal design this paper focuses on the analysis and synthesis of orthogonal designs with the maximum possible rates which may be real or complex and square or rectangular matrices we first provide several representations of orthogonal designs and their characterizations in terms of hurwitz radon matrix equations next we observe that the real orthogonal designs square or rectangular and the complex square orthogonal designs with maximal rates have been well understood from the existing results in the mathematics literature which can be dated back to 1890s however unfortunately it is not the case for the complex rectangular orthogonal designs with rates as high as possible we then construct a class of complex orthogonal designs for any number of transmit antennas the proposed complex orthogonal designs for the number of transmit antennas n 2m 1 and 2m have the same rate m 1 and 2m where m is any natural number finally we demonstrate that for n 2m 1 and 2m with any given natural number m the value m 1 and 2m is the maximum possible rate that the complex orthogonal designs square or rectangular with n transmit antennas can achieve

probabilistic estimates for the two dimensional stochastic navier stokes equations we consider the navier stokes equation on a two dimensional torus with a random force white noise in time and analytic in space for arbitrary reynolds number r we prove probabilistic estimates for the long time behavior of the solutions that imply bounds for the dissipation scale and energy spectrum as r

an algorithm for solving a nonlinear integro differential equation in this paper an algorithm based on adomian s decomposition method is developed to approximate the solution of the nonlinear integro differential equationu t x t email protected t 0a email protected email protected email protected u x x t email protected f x t 0

experimentation with two step adomian decomposition method to solve evolution models the two step adomian decomposition method tsadm is derived by separate the standard adomian decomposition method into two steps although the tsadm needs only a slight variation from the standard adomian decomposition method experimentation shows that the tsadm is also promising for systems of inhomogeneous differential and integral equations hyperbolic partial differential equations and singular initial value problems

ergodicity of 2d navier stokes equations with random forcing and large viscosity the stochastically forced two dimensional incompressable navier stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough this result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions that is to say the system has a trivial random attractor along the way results controling the expectation and averaging time of the energy and enstrophy are given

linear spaces of matrices of constant rank and instanton bundles abstract we present a new method to study 4 dimensional linear spaces of skew symmetric matrices of constant co rank 2 based on rank 2 vector bundles on p 3 and derived category tools the method allows one to prove the existence of new examples of size 10 10 and 14 14 via instanton bundles of charge 2 and 4 respectively and it provides an explanation for what used to be the only known example westwick 1996 25 we also give an algorithm to construct explicitly a matrix of size 14 of this type

the inverse of a totally positive biinfinite band matrix it is shown that a bounded bi infinite banded totally positive matrix a is boundedly invertible iff there is one and only one bounded sequence mapped by a to the sequence the argument shows that such a matrix has a main diagonal i e the inverse of a is the bounded pointwise limit of inverses of finite sections of a principal with respect to a particular diagonal hence 4 j 7 or its inverse is again totally positive 0 introduction this paper is a further step in a continuing effort to understand certain linear spline approximation schemes convergence of such processes is intimately tied to their stability i e to their boundedness as maps on c say use of the b spline basis shows this question to be equivalent to bounding the inverse of certain totally positive band matrices the calculation of bounds on the inverse of a given matrix is in general a difficult task it is hoped that the present investigation into the consequences of bandedness and total positivity for the structure of the inverse may ultimately prove helpful in obtaining such bounds the results in this paper were obtained in the study of a conjecture due to c a micchelli 7 in connection with his work on the specific approximation scheme of interpolation at a strictly increasing point sequence t by elements of sm t i e by splines of some order m with some knot sequence t micchelli became convinced that every bounded function has one and only one bounded spline interpolant iff the particular function which satisfies t all has a bounded spline interpolant in sm if ni ni mt denotes the corresponding b spline basis for sm then micchelli s conjecture can be phrased thus the matrix

uniqueness of the invariant measure for a stochastic pde driven by degenerate noise we consider the stochastic ginzburg landau equation in a bounded domain we assume the stochastic forcing acts only on high spatial frequencies the low lying frequencies are then only connected to this forcing through the non linear cubic term of the ginzburg landau equation under these assumptions we show that the stochastic pde has a unique invariant measure the techniques of proof combine a controllability argument for thelow lying frequencies with an infinite dimensional version of the malliavin calculus to show positivity and regularity of the invariant measure this then implies the uniqueness of that measure

linear transformations which preserve or decrease rank abstract we describe some results concerning a linear transformation on a space v of matrices which is rank preserving or rank nonincreasing on a certain subset of v

a computational algebraic investigation of the decomposition method for time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat or wave equations is investigated numerically using computational algebraic techniques

the restrictions and improvement of the adomian decomposition method the adomian decomposition method is a creative and effective method for exactly solving functional equations of various kinds in this paper the principle of the decomposition method is described and its restrictions are discussed then an improvement is proposed which overcomes the restrictions and complements the ability of the adomian decomposition method some concrete examples are also studied to illustrate the validity of the improvement

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

the berry esseen theorem for strongly mixing harris recurrent markov chains let 0 1 be a stationary harris recurrent markov chain with state space e and let f e ir xi f i it is known that the sequence xi i 0 is strongly mixing i e n 0 where n are the strong or rosenblatt mixing coefficients if n decreases at a sufficiently fast rate and f is suitable then a central limit holds for the partial sums sum limits i 0 n x i the present paper gives conditions in order that the convergence rates are o n 1 2

numerical study of modified adomian s method applied to burgers equation the application of adomian s decomposition method to partial differential equations when the exact solution is not reached demands the use of truncated series but the solution s series may have small convergence radius and the truncated series may be inaccurate in many regions in order to enlarge the convergence domain of the truncated series pade approximants pas to the adomian s series solution have been tested and applied to partial and ordinary differential equations with good results in this paper pas both in x and t directions applied to the truncated series solution given by adomian s decomposition technique for burgers equation are tested numerical and graphical illustrations show that this technique can improve the accuracy and enlarge the domain of convergence of the solution it is also shown in this paper that the application of adomian s method to the ordinary differential equations set arising from the discretization of the spatial derivatives by finite differences the so called method of lines may reduce the convergence domain of the solution s series

exponential mixing properties of stochastic pdes through asymptotic coupling we consider parabolic stochastic partial differential equations driven by white noise in time we prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions these conditions amount essentially to the fact that the equation transmits the noise to all its determining modes several examples are investigated including some where the noise does not act on every determining mode directly

law of large numbers and central limit theorem for randomly forced pde s we consider a class of dissipative pde s perturbed by an external random force under the condition that the distribution of perturbation is sufficiently non degenerate a strong law of large numbers slln and a central limit theorem clt for solutions are established and the corresponding rates of convergence are estimated it is also shown that the estimates obtained are close to being optimal the proofs are based on the property of exponential mixing for the problem in question and some abstract slln and clt for mixing type markov processes

some limiting properties of randomly forced two dimensional navier stokes equations we consider random perturbations of two dimensional navier stokes equations under some natural conditions on random forces we study asymptotic properties of solutions and stationary measures

the decomposition method for forward and backward time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat wave and beam equations is investigated for both forward direct and backward inverse problems it is shown that for forward problems the convergence is faster than for backward problems

a coupling approach to randomly forced nonlinear pde s ii we consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure the result obtained applies in particular to the 2d navier stokes system and multidimensional complex ginzburg landau equation with random kick force

adomian s method of decomposition critical review and examples of failure abstract adomian s method of decomposition is considered in application to initial boundary value problems for the one space dimensional spatially homogeneous heat conduction equation it is shown that the fundamental equation of the method is well defined only for certain restricted types of boundary conditions within the class of such boundary conditions examples are given such that the fundamental equation fails to have a unique solution and such that the sequence produced by iteration of this equation is divergent the latter is a counterexample to a published assertion of convergence

spaces of real matrices of fixed small rank in this paper we investigate the maximal dimension for k spaces of real matrices for small values of k we also give some bounds for the dimension of spaces of real matrices of high rank

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

convergence of adomian s decomposition method periodic temperatures abstract the convergence of adomian s decomposition method 1 as applied to periodic temperature fields in heat conductors is investigated

application of the adomian decomposition method for two dimensional parabolic equation subject to nonstandard boundary specifications in this work we are concerned with the solution of a nonlocal boundary value problem an approach is presented for solving the two dimensional parabolic partial differential equation subject to integral boundary specifications the main objective is to propose an alternative method of solution one not based on finite difference methods or finite element schemes or spectral techniques the aim of the present paper is to investigate the application of the adomian decomposition method for solving the two dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions the adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques the introduction of this idea as will be discussed not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume the solutions will be handle more easily quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions in this approach the solution is found in the form of a convergent power series with easily computed components the adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization to give a clear overview of the methodology we have selected illustrative example

on two parameter families of symmetric matrices abstract lef f be a homogeneous polynomial mapping of degree d from c 2 into the space of complex n n symmetric matrices it is shown that if rank f x k for all 0 x c 2 then either k is even or d is even

an algorithm for solving bond pricing problem the nonlinear bond pricing problem has been extensively studied in the literature since an analytical solution is not readily available we will seek to find an approximate solution we will present a decomposition method due to adomian and show how to obtain a reasonable numerical solution to the bond pricing problem

a new modification of adomian decomposition method for solving a kind of evolution equation in this paper we propose a new modification of adomian decomposition method for solving a kind of evolution equation the modification will improve the convergence of the series solution the validity of the modified technique is verified through illustrative examples

on the dimension of some real spaces of bounded rank matrices abstract given n n let x be either the set of hermitian or real n n matrices of rank at least n 1 if n is even we give a sharp estimate on the maximal dimension of a real vector space v x 0 the results are obtained via k theory by studying a bundle map induced by the adjunction of matrices

a technique for approximate solutions to schrodinger like equations in the present paper the schro dinger like equations are solved easily quickly and elegantly using adomian s decomposition method the technique has been illustrated by some examples which are of physical interest

the decomposition method for cauchy advection diffusion problems in this paper the solution of cauchy problems for the advection diffusion equation is obtained using the decomposition method in the case when the flow velocity is constant an analytical solution can be derived whilst for variable flow velocity symbolic numerical computations need to be performed

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

ergodic theory of infinite dimensional systems with applications to dissipative parabolic pdes we consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general nonlinear dissipative parabolic pdes results on invariant measures and their exponential mixing properties are proved and applications to 2d navier stokes systems are included

solving nonlocal initial boundary value problems for the lotka von foerster model in this paper we present a new approach to solve nonlocal initial boundary value problems for the lotka von foerster model subject to initial and nonlocal boundary conditions of integral type we first transform the given nonlocal initial boundary value problem into an equivalent local initial boundary value problem then we apply a modified adomian decomposition method which permits convenient resolution of these problems also a new method is developed to solve this nonlocal model which reduces it to an equivalent local initial boundary value problem for a system of ordinary differential equations

spaces of matrices of fixed rank ii abstract when min m n k 1 the exact value of l k m n the maximum dimension of all possible linear spaces of rank k matrices of order m n is known the situation when min l k m n k 2 is not clear partial results are obtained for l k k 2 k 2 in this paper

exponential mixing of the 2d stochastic navier stokes dynamics we consider the navier stokes equation on a two dimensional torus with a random force which is white noise in time and excites only a finite number of modes the number of excited modes depends on the viscosity v and grows like v 3 when v goes to zero we prove that this markov process has a unique invariant measure and is exponentially mixing in time

a note on spaces of symmetric matrices we calculate the maximal dimension of linear spaces of symmetric and hermitian matrices with given high rank generalizing a well known result of adams et al

solving nonlocal initial boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the adomian decomposition method in this paper we present a new approach to solve nonlocal initial boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type we first transform the given nonlocal initial boundary value problems of integral type for the linear and nonlinear parabolic and hyperbolic partial differential equations into local dirichlet initial boundary value problems and then use a relatively new modified adomian decomposition method adm furthermore we investigate the fourier adomian method which also does not require any a priori assumptions on the solution for the solution of nonlocal initial boundary value problems combined with our new approach several examples are presented to demonstrate the efficiency of the adm