time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

wavelet based iterative methods for a class of 2d partial integro differential equations abstract in this paper an iterative method based on quasilinearization is presented to solve a class of two dimensional partial integro differential equations that arise in nuclear reactor models and population models two different approaches based on haar and legendre wavelets are studied to develop numerical methods in the first approach time domain is approximated with the help of forward finite difference approach in the second approach both time as well as space domains are approximated by wavelets appropriate examples are solved using these methods and the obtained results are compared with the methods available in the recent literature

infinitesimally rigid polyhedra i statics of frameworks from the time of cauchy mathematicians have studied the motions of convex polyhedra with the faces held rigid while changes are allowed in the dihedral angles in the 1940s alexandrov proved that even with additional vertices along the natural edges and with an arbitrary triangulation of the natural faces on these vertices such polyhedra are infinitesimally rigid in this paper the dual and equivalent concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex and other polyhedra the static techniques introduced provide a new simplified proof of alexandrov s theorem as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces including frameworks with vertices interior to the faces the static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron nonconvex toroidal etc the techniques are also applied to derive the static rigidity of tensegrity frameworks with cables and struts in place of bars and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks finally as an exercise to give an additional perspective to the results in 3 space detailed analogues of alexandrov s theorem are presented for convex 4 polytopes built as bar and joint frameworks in 4 space

monotone iterates for solving nonlinear integro parabolic equations of volterra type the paper deals with numerical solving of nonlinear integro parabolic problems based on the method of upper and lower solutions a monotone iterative method is constructed existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given construction of initial upper and lower solutions is discussed numerical experiments are presented

zero temperature limit for interacting brownian particles i motion of a single body we consider a system of interacting brownian particles in r d with a pairwise potential which is radially symmetric of finite range and attains a unique minimum when the distance of two particles becomes a 0 the asymptotic behavior of the system is studied under the zero temperature limit from both microscopic and macroscopic aspects if the system is rigidly crystallized namely if the particles are rigidly arranged in an equal distance a the crystallization is kept under the evolution in macroscopic time scale then assuming that the crystal has a definite limit shape under a macroscopic spatial scaling the translational and rotational motions of such shape are characterized

context free complexity of finite languages abstract in this paper the investigation of the theory of grammatical complexity as started by bucher 1981 and bucher culik ii maurer and wotschke 1981 is continued the basic question we are concerned with is the following given some finite language l what is the smallest number of context free productions needed to generate l the so called context free complexity of l we strengthen some of the results given by bucher et al 1981 the main result being a necessary condition for certain sequences of finite languages to be of sublinear complexity

the multiplicative complexity of quadratic boolean forms abstract let the multiplicative complexity l f of a boolean function f be the minimal number of gates that are sufficient to evaluate f by circuits over the basis 1 we characterize the multiplicative complexity l f of quadratic boolean forms f it is half the rank of an associated matrix two quadratic boolean forms f g have the same complexity l f l g iff they are isomorphic by a linear isomorphism we characterize computational independence of two quadratic boolean forms f 1 f 2 in the sense that l f 1 f 2 l f 1 l f 2

generating all permutations by context free grammars in chomsky normal form let ln be the finite language of all n strings that are permutations of n different symbols n 1 we consider context free grammars gn in chomsky normal form that generate ln in particular we study a few families gn n 1 satisfying l gn ln for n 1 with respect to their descriptional complexity i e we determine the number of nonterminal symbols and the number of production rules of gn as functions of n

a note on a problem in the theory of grammatical complexity abstract in this note we answer an open question in the theory of grammatical complexity we show that if l is an infinite context free language then l n the set of words in l of length less than or equal to n can be generated by o l n 2 3 context free productions and if l is an infinite linear language then l n can be generated by o l n linear productions we also show that these bounds are the best possible

generating all permutations by context free grammars in greibach normal form we consider context free grammars g n in greibach normal form and particularly in greibach m form m 1 2 which generates the finite language l n of all n strings that are permutations of n different symbols n 1 these grammars are investigated with respect to their descriptional complexity i e we determine the number of nonterminal symbols and the number of production rules of g n as functions of n as in the case of chomsky normal form these descriptional complexity measures grow faster than any polynomial function

standard realizations of crystal lattices via harmonic maps an eells sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals it is thus observed that the mimimum principle frames symmetry of crystals

block monotone iterative method for semilinear parabolic equations with nonlinear boundary conditions two block monotone iterative schemes called jacobi and gauss seidel monotone iterations are presented for numerical solutions of a class of semilinear parabolic equations under nonlinear boundary conditions by the finite difference method these iteration schemes extend the method for semilinear elliptic boundary value problems to parabolic equations including a comparison result between them it is shown that by using an upper solution and a lower solution as initial iterations each of the iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system some error estimates and a convergence theorem are given and various sufficient conditions for the construction of upper and lower solutions are obtained numerical results are presented for some physical model problems including some problems with known continuous solutions and two problems with l shaped and trapezoidal domains

numerical methods for nonlinear integro parabolic equations of fredholm type abstract this paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction diffusion convection equations under either linear or nonlinear boundary conditions the discrete approximation of the problem is based on the finite difference method and the computation of the finite difference solution is by the method of upper and lower solutions three types of quasi monotone reaction functions are considered and for each type a monotone iterative scheme is obtained each of these iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system this monotone convergence leads to an existence uniqueness theorem as well as a computational algorithm for the computation of the solution an error estimate between the computed approximations and the true finite difference solution is obtained for each iterative scheme these error estimates are given in terms of the strength of the reaction function and the effect of diffusion convection and are independent of the true solution applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm

numerical solving nonlinear integro parabolic equations by the monotone weighted average method the paper deals with numerical solving nonlinear integro parabolic problems of volterra type based on the weighted average method a monotone iterative method is presented construction of initial upper and lower solutions is given existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given numerical experiments are presented

on the compressibility of finite languages and formal proofs abstract we consider the minimal number of productions needed for a grammar to cover a finite language l as the grammatical complexity of l we study this measure for several types of word and tree grammars and show that it is closely connected to well known measures for the complexity of formal proofs in first order predicate logic we construct an incompressible sequence of finite word languages and transfer this and several other results about the complexity of word and tree languages to formal proofs

concise description of finite languages abstract this paper is a contribution to the theory of grammatical complexity in particular to the following basic question consider some language l and grammars of some type x what is the smallest number of productions of a type x grammar required to generate l this complexity measure the so called x complexity of l has been investigated before we study the more basic case when the languages l considered are finite a case which has been neglected so far we obtain a number of results and insights suggesting that such study is of importance in addition to a number of expected but not necessarily easy to prove results that with type x grammars more productions are necessary than with some other type y grammars even if types x and y define the same family of languages we show that if the limit of certain sequences of finite languages is of type x1 then the type x complexity of each of the finite languages involved must be low

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations