a cubic spline method for the solution of a linear fourth order two point boundary value problem abstract a cubic spline method is described for the numerical solution of a two point boundary value problem involving a fourth order linear differential equation this spline method is shown to be closely related to a known fourth order finite difference scheme

winding of planar brownian curves the authors compute the joint probability for a closed brownian curve to wind n times around a prescribed point and to enclose a given algebraic area an estimate from below of the arithmetic area is obtained

the monadic second order logic of graphs ix machines and their behaviours abstract we establish that every monadic second order property of the behaviour of a machine transition systems and tree automata are typical examples of machines is a monadic second order property of the machine itself in this way we clarify the distinction between dynamic properties of machines i e properties of their behaviours and their static properties i e properties of the graphs or relational structures representing them it is important for program verification that the dynamic properties that one wants to verify can be formulated statically in the simplest possible way as a corollary of our main result we also obtain that the monadic theory of an algebraic tree is decidable

some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

computations by fly automata beyond monadic second order logic parameterized algorithms based on automata infinite automata computing their transitions problems described in monadic second order logic counting and other problems not expressible in monadic second order logic the validity of a monadic second order ms expressible property can be checked in linear time on graphs of bounded tree width or clique width given with appropriate decompositions this result is proved by constructing from the ms sentence expressing the property and an integer that bounds the tree width or clique width of the input graph a finite automaton intended to run bottom up on the algebraic term representing a decomposition of the input graph as we cannot construct practically the transition tables of these automata because they are huge we use fly automata whose states and transitions are computed on the fly only when needed for a particular input furthermore we allow infinite sets of states and we equip automata with output functions thus they can check properties that are not ms expressible and compute values for example the number of p colorings of a graph we obtain xp and fpt graph algorithms parameterized by tree width or clique width we show how to construct easily such algorithms by combining predefined automata for basic functions and properties these combinations reflect the structure of the ms formula that specifies the property to check or the function to compute

counting modulo quantifiers on finite structures we give a combinatorial method for proving elementary equivalence in first order logic fo with counting modulo n quantifiers dn inexpressibility results for fo dn with built in linear order are also considered for instance the class of linear orders of length divisible by n 1 cannot be expressed in fo dn using this result we prove that comparing cardinalities or connectivity of ordered graphs are not definable in fo dn we also show that the height of complete n ary trees cannot be expressed in fo dn with linear order interpreting the predicate y nx as a complete n ary tree we show that the predicate y px cannot be defined in fo dn with linear order whenever p has a prime factor that does not divide n this solves the problem raised by niwi ski and stolboushkin lics 93 we also discuss a connection between our results and the well known open problem in circuit complexity theory whether acc nc1

non gaussian random walks the authors present an explicit expression for the probability distribution for the position of a continuous time random walker in an arbitrary number of dimensions when the interjump density has a long time tail in contrast to earlier results which require numerical inversion of a fourier integral they replace this numerical procedure by one that relies on the method of steepest descents their results are applied to diffusion on a comb and on a percolation cluster generated on a cayley tree at criticality and are confirmed numerically

spectral determinants and zeta functions of schrodinger operators on metric graphs a derivation of the spectral determinant of the schrodinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of kostrykin and schrader to formulate the spectral determinant we first derive the spectral zeta function of the schrodinger operator using an appropriate secular equation the result obtained for the spectral determinant is along the lines of the recent conjecture texier 2010 j phys a math theor 43 425203

high order accuracy nonpolynomial spline solutions for 2μth order two point boundary value problems sixth order accurate method based on quintic nonpolynomial spline function for the numerical solution of 2 th order two point bvps where is a positive integer is presented by transforming the problem into a system of second order two point bvps the nonpolynomial spline function is used to drive some consistency relations for computing approximation to the solution of this problem the proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost convergence analysis of the proposed method is discussed numerical examples are included to illustrate the practical usefulness of our method

structural properties of context free sets of graphs generated by vertex replacement abstract we establish that a vertex replacement set of graphs i e a set of graphs generated by a c ednce or equivalently by a separated handle rewriting graph grammar is hyperedge replacement i e is generated by a hyperedge replacement graph grammar iff its graphs do not contain arbitrary large complete bipartite graphs kn n as subgraphs another equivalent condition is that its graphs have a number of edges that is linearly bounded in terms of the number of vertices these properties are decidable by means of an appropriate extension of the theorem by parikh that characterizes the commutative images of context free languages we extend these results to hypergraphs

asynchronous mappings and asynchronous cellular automata the aim of this paper is the study of asynchronous automata a special kind of automata which encode the independency relation between actions and which enable their concurrent execution these automata introduced by zielonka rairo inform theor appl 21 99 135 1987 constitute a natural extension of finite automata to the case of asynchronous parallelism their behaviour is described by trace languages subsets of partially commutative monoids the main result concerning this class of automata states that they accept exactly all recognizable trace languages in this paper we give new improved constructions of asynchronous automata in the final part of the paper we present a distributed system of messages with bounded time stamps based on asynchronous automata

solution of the system of fourth order boundary value problems using non polynomial spline technique non polynomial spline is used for the numerical solutions of a system of fourth order boundary value problems associated with obstacle unilateral and contact problems in addition to the boundary conditions corresponding to the second derivatives the boundary conditions corresponding to the first derivatives are also considered two examples one for each case of boundary conditions are considered for the numerical illustration of the method developed it has been observed that the method developed in this paper is better than the other collocation finite difference and spline methods

numerical solution of a system of fourth order boundary value problems using cubic non polynomial spline method abstract a cubic non polynomial spline technique is developed for the numerical solutions of a system of fourth order boundary value problems associated with obstacle unilateral and contact problems the end conditions consistent with the bvp are derived corresponding to the boundary conditions in terms of not only second derivatives but first derivatives as well the present method has less computational cost and gives better approximations than those produced by other collocation finite difference and spline methods the method developed is compared with those developed by khan et al a khan m a noor t aziz parametric quintic spline approach to the solution of a system of fourth order boundary value problems journal of optimization theory and applications 122 2 2004 309 322 al said and noor e a al said m a noor computational methods for fourth order obstacle boundary value problems communications in applied nonlinear analysis 2 1995 73 83 and siddiqi and ghazala s s siddiqi g akram solution of the system of fourth order boundary value problems using non polynomial spline technique applied mathematics and computation 185 2007 128 135 through different examples

the monadic second order logic of graphs x linear orderings graphs are finite and handled as relational structures we give some answers to the following general questions 1 1 for which classes of graphs c is it possible to specify a linear ordering of the set of vertices of each graph of c by fixed monadic second order formulas 2 2 for which classes of graphs c does there exist an extension l of monadic second order logic such that a subclass l of c is recognizable if and only if it is the class of graphs in c that satisfy a formula of l in this paper recognizability is understood in an algebraic sense relative to a finite set of graph operations and basic graphs that generate all graphs of c 3 3 for which classes of graphs c is it possible to construct in every graph of the class and by fixed formulas of a suitable extension of monadic second order logic its hierarchichal structure i e a finite term written with the operations and basic graphs of 2 that defines the considered graph applications concern dependency graphs of partially commutative words partial k paths cographs and graphs the modular decomposition of which uses prime graphs of bounded size

ζ regularized spectral determinants on metric graphs several general results for the spectral determinant of the schrodinger operator on metric graphs are reviewed then a simple derivation for the regularized spectral determinant is proposed based on the roth trace formula two types of boundary conditions are studied functions continuous at the vertices and functions whose derivative is continuous at the vertices the regularized spectral determinant of the schrodinger operator acting on functions with the most general boundary conditions is conjectured in conclusion the relation to the ihara bass and bartholdi formulae obtained for combinatorial graphs is also discussed

numerical solution of a system of fourth order boundary value problems using variational iteration method abstract in this paper the variational iteration method is used to solve a system of fourth order boundary value problems associated with obstacle unilateral and contact problems numerical solution obtained by the method is of high accuracy moreover the higher order derivatives of numerical solution can also approximate the higher order derivatives of exact solution well five examples compared with those considered by siddiqi and akram s s siddiqi g akram numerical solution of a system of fourth order boundary value problems using cubic non polynomial spline method applied mathematics and computation 190 2007 652 661 show that the method is more efficient

on the spectrum of the laplace operator of metric graphs attached at a vertex spectral determinant approach we consider a metric graph made of two graphs and attached at one point we derive a formula relating the spectral determinant of the laplace operator in terms of the spectral determinants of the two subgraphs the result is generalized to describe the attachment of n graphs the formulae are also valid for the spectral determinant of the schrodinger operator

theory of traces abstract the theory of traces originated by a mazurkiewicz in 1977 is an attempt to provide a mathematical description of the behavior of concurrent systems its aim is to reconcile the sequential nature of observations of the system behavior on the one hand and the nonsequential nature of causality between the actions of the system on the other hand one can see the theory of traces to be rooted in formal string language theory with the notion of partial commutativity playing the central role alternatively one can see the theory of traces to be rooted in the theory of labeled acyclic directed graphs or even in the theory of labeled partial orders this paper attempts to present a major portion of the theory of traces in a unified way however it is not a survey in the sense that a number of new notions are introduced and a number of new results are proved although traditionally most of the development in the theory of traces follows the string language theoretic line we try to demonstrate to the reader that the graph theoretic point of view may be more appropriate the paper essentially consists of two parts the first one sections 1 through 4 is concerned with the basic theory of traces the second one section 5 presents applications of the theory of traces to the theory of the behavior of concurrent systems where the basic system model we have chosen is the condition event system introduced by c a petri

the monadic second order logic of graphs vii graphs as relational structures abstract relational structures form a unique framework in which various types of graphs and hypergraphs can be formalized and studied we define operations on structures that are compatible with monadic second order logic and that are powerful enough to represent context free graph and hypergraph grammars of various types namely the so called hyperedge replacement c ednce and separated handle rewriting grammars several results concerning monadic second order properties of the generated sets are obtained in a uniform way we also give a logical characterization of the equational sets of structures that generalizes the ones obtained by engelfriet and courcelle for hyperedge replacement and c ednce sets of graphs

non polynomial spline approach to the solution of a system of third order boundary value problems we use a quartic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of third order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods a numerical example is given to illustrate the applicability and efficiency of the new method

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

an axiomatic definition of context free rewriting and its application to nlc graph grammars abstract an abstract notion of context free grammar is introduced it deals with abstract objects that can be words trees graphs or other combinatorial objects it is applied to nlc graph grammars introduced by rozenberg and janssens the monadic second order theory of a context free nlc set of graphs is decidable

the monadic second order logic of graphs xii planar graphs and planar maps we prove that we can specify by formulas of monadic second order logic the unique planar embedding of a 3 connected planar graph if the planar graph is not 3 connected but given with a linear order of its set of edges we can also define a planar embedding by monadic second order formulas we cannot do so in general without the ordering even for 2 connected planar graphs the planar embedding of a graph can be specified by a relational structure called a map which is a graph enriched with a circular ordering of the edges incident with each vertex this circular ordering called a rotation system represents a planar embedding of the neighbourhood of each vertex for each connected map one can define a linear order on its vertices by formulas of monadic second order logic hence we have for planar graphs some kind of equivalence between linear orderings and planar embeddings

a smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems methods of order two four and six based on exponential spline functions consisting of a polynomial part of degree three and an exponential part is developed to find approximation of linear and nonlinear fourth order two point boundary value problems it is shown that the free parameter k of the exponential part can be used to raise the order of accuracy of the new scheme convergence analysis of these methods is shown along with numerical examples each for linear and nonlinear are included to illustrate the practical usefulness of our methods

quantum oscillations in mesoscopic rings and anomalous diffusion we consider the weak localization correction to the conductance of a ring connected to a network we analyse the harmonics content of the al tshuler aronov spivak aas oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by aas the physical origin of this behaviour is the anomalous diffusion of brownian trajectories around the ring due to the diffusion in the wires we show that this problem is related to the anomalous diffusion along the skeleton of a comb we study in detail the winding properties of brownian curves around a ring connected to an arbitrary network our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales a general expression of this length is derived for arbitrary networks more specifically we consider the case of a ring connected to wires to a square network and to a bethe lattice

solution of the system of fourth order boundary value problems using cubic spline a system of fourth order boundary value problems associated with obstacle unilateral and contact problems is solved using cubic spline since the solution of system of fourth order bvp with boundary conditions in terms of values of function and first derivatives is not found in the literature so in this paper the boundary conditions are considered in terms of values of function and not only in first derivatives but in terms of second derivatives as well three examples are considered for the numerical illustration of the method developed it has been observed that the method developed in this paper is better than the other collocation finite difference and spline methods

random walk on the bethe lattice and hyperbolic brownian motion we give the exact solution to the problem of a random walk on the bethe lattice through a mapping on an asymmetric random walk on the half line we also study the continuous limit of this model and discuss in detail the relation between the random walk on the bethe lattice and brownian motion on a space of constant negative curvature

convergence of cubic spline approach to the solution of a system of boundary value problems abstract we use cubic spline to derive some consistency relations which are then used to develop a numerical method for the solution of a system of fourth order boundary value problems associated with obstacle unilateral and contact problems it is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations which are solved using some numerical method boundary formula of order o h 8 are formulated the most common approach for convergence analysis are using monotonicity of the coefficient matrix but here we study a new approach and give the convergence of prescribed method so that the matrix associated with the system of linear equations that arises is not required to be monotone numerical examples are given to show the applicability and efficiency of our method