approximating bayesian belief networks by arc removal i propose a general framework for approximating bayesian belief networks through model simplification by arc removal given an upper bound on the absolute error allowed on the prior and posterior probability distributions of the approximated network a subset of arcs is removed thereby speeding up probabilistic inference

the monotonicity of the period function for planar hamiltonian vector fields if the hamiltonian system with hamiltonian h x y 12y2 v x has a center at the origin it is shown that the period function for the family of periodic trajectories surrounding the center is monotone when the function v v 2 is convex this theorem is applied to determine existence and uniqueness of solutions for the neumann boundary value problem x a x 2 bx c x 0 x 1 0

global continuation of branches of nondegenerate solutions on etudie les solutions des problemes aux valeurs limites de la forme u t u yd y f u 0 y l 0 l a u u y y 0 r a u u y y l ou l a et r a sont de la forme l a u u y au y 1 a u r a u u y au y 1 a u 0 a 1 le terme non lineaire f est de la forme f u u 1 u u 0 1 2

population models with diffusion strong allee effect and nonlinear boundary conditions abstract we consider a population model with diffusion a strong allee effect per capita growth function and constant yield harvesting in particular we focus our study on a population living in a patch r n with n 1 that satisfies a certain nonlinear boundary condition we establish our existence results by the method of sub supersolutions

on s shaped bifurcation curves where sz c r n 2 1 is bounded with smooth boundary aq readers interested in the derivation of the problem are referred to the work by bebernen l particularly section 2 relevant work on the analysis of this problem has been done by brown et al 2 shivaji 3 61 hastings and mcleod 7 sattinger 8 parks 9 and tam lo clearly u 0 is a solution of 1 l when 2 0 it follows from the contraction mapping theorem that l l always has a solution for a sufficiently small moreover let

bounds for the loss in probability of correct classification under model based approximation in many pattern recognition classification problem the true class conditional model and class probabilities are approximated for reasons of reducing complexity and or of statistical estimation the approximated classifier is expected to have worse performance here measured by the probability of correct classification we present an analysis valid in general and easily computable formulas for estimating the degradation in probability of correct classification when compared to the optimal classifier an example of an approximation is the naive bayes classifier we show that the performance of the naive bayes depends on the degree of functional dependence between the features and labels we provide a sufficient condition for zero loss of performance too

a correction for a paper by j smoller and a wasserman abstract in j differential equations 39 1981 269 290 j smoller and a wasserman presented some rich ideas and results on the bifurcation of steady state solutions of a reaction diffusion equation in one space variable however the proof of their theorem 2 1 is incorrect the case a 0 and to correct it we find one additional hypothesis is necessary

bifurcation studies in reaction diffusion ii and estimated the critical lengths of the domain at which bifurcation occurs in the cases b 0 a and 1 also we were able to characterise some of the non constant steady state solutions which appear beyond the points of instability by a steady state solution of the form u x

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

rigorous analysis and estimates of s shaped bifurcation curves in a combustion problem with general arrhenius reaction rate laws we study the bifurcation of solutions of the combustion problem with general arrhenius reaction rate laws d 2 u d x 2 1 e u m e u 1 e u 0 1 x 1 u 1 u 1 0 for this two point boundary value problem we give a bifurcation analysis in the physically important ranges 0 m 1 and m 0 it is proved that the bifurcation curve is s shaped in certain parameter ranges and a lower bound is given for the frank kamenetskii transition value

quenching for solutions of a plasma type equation here 0 l the so called slow diffusion case the analysis will be carried out in the manner of 5 in which levine gave a complete qualitative study of problem i with m 1 the plan of our paper is as follows in the next section we present the comparison theorem and local existence of solutions for i in the third section we characterize the set of stationary solutions of i finally in the last section we establish stability and quenching results for problem i

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

exact structure of positive solutions for some p laplacian equations this paper establishes the exact multiplicities and properties of positive solutions for some second order differential equations involving p laplacian operator

on positive solutions of some pairs of differential equations this paper is a sequel to the author s paper 6 which studied positive solutions of a pair of differential equations introduced by conway gardner and smoller 4 their equations were equations for population problems in biology their significance is discussed further in 141 the purpose of the present paper is to obtain further results for the conwat gardner smoller equations and to show how the methods in 6 can be modified to decide exactly when the classical predator prey system as in blat and brown 2 has a strictly positive solution we also show how the methods can be applied to a number of other problems in particular our results answer some questions in conway s secondly we show how the asymptotic methods developed in 6 can be used to study the existence and uniqueness of strictly positive solutions of competing species problems our results suggest that this is a much more complicated problem than the predator prey problem we do obtain a condition which is except for a few special cases necessary and sufficient for the existence of a strictly positive solution however the condition is complicated and rather implicit thirdly we prove the uniqueness of the strictly positive solution in the conway smoller system if d is small and n 1 that is the case of ordinary differential equations this partially answers a question in 4 our proof is a local result and can be used in other situations for example it can be used to show that in certain cases of asocial nonlinearities where our notation follows 4 sect 4 there are exactly two strictly positive solutions for small d we also show how our method applies to more general predator prey problems in section 1 we find necessary and sufficient conditions for the existence of strictly positive solutions of predator prey systems in section 2 we study competing species models by iteration and asymptotic methods finally in section 3 we prove our uniqueness results in an appendix we briefly discuss how the methods of section 3 apply to more general models

exact number of solutions for a class of two point boundary value problems with one dimensional p laplacian abstract in this paper exact number of solutions are obtained for the one dimensional p laplacian in a class of two point boundary value problems the interesting point is that the nonlinearity f is general form f u g u h u meanwhile some properties of the solutions are given in details the arguments are based upon a quadrature method

eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance the abstract boundary value problem lu gu f u dom l h is considered here h is used to denote a real separable hilbert space l a closed symmetric linear operator and g a nonlinear operator assumed to be lipschitz continuous and strongly monotone in addition l is assumed to have a complete set of eigenfunctions in h and is allowed to have an infinite dimensional null space the existence of unique solutions depending continuously on f is established by a constructive approach galerkin approximations are considered and error estimates are given as an application of the main result the existence of time periodic weak solutions of the n dimensional wave equation is shown

geometric methods for two point nonlinear boundary value problems on considere une classe d equations differentielles de la forme x f x g x avec des conditions aux limites soit de dirichlet soit de neumann on etudie l existence de la solution triviale

multiple solutions of some semilinear elliptic equations in slender cylindrical domains we analyze the semilinear problem ul ul 0 in 0 2l ul 0 at 0 2l where rn 1 is a bounded smooth domain and is a given nonlinearity in the limit l the solutions of that problem are related to those of the n 1 dimensional problem v v 0 in v 0 at in particular necessary and sufficient conditions on a given solution of the latter v are given for the former to have a solution ul such that ul x1 x2 xn v x2 xn as l in most part of the cylinder 0 2l related bifurcation problems and some generalizations are also considered

a finite element method for computing the bifurcation function for semilinear elliptic bvps abstract the method of alternative problems can be used to show that a semilinear elliptic boundary value problem lu g x u 0 with gu x u bounded below is equivalent to a finite dimensional problem f c 0 r d c r d in the sense that their solution sets which are not necessarily singletons are in a one to one correspondence this correspondence is based on a map from low frequency to high frequency fourier components of solutions a numerical method is presented for approximating and hence also solutions of the bvp the method uses finite element approximations and avoids the use of eigenfunction expansions existence uniqueness and error estimates for the approximations of and solutions u are derived

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary abstract the structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined in particular a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary our existence results are established via the well known sub super solution method

comparison theorems for radial solutions of semilinear elliptic equations on obtient des resultats sur les solutions radiales des equations elliptiques de la forme u f u 0

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation abstract in this work we study the period function t of solutions to the conservative equation x t f x t 0 we present conditions on f that imply the monotonicity and convexity of t as a consequence we obtain the criterium established by c chicone and find conditions easier to apply we also get a condition obtained by cima gasull and manosas about monotonicity and following some of their calculations present results on the period function of hamiltonian systems where h x y f x n 1 y n using the monotonicity of t we count the homogeneous solutions to the semilinear elliptic equation u u 1 in two dimensions

on the return time function around monodromic polycycles abstract in this paper we study the period function of centers of planar polynomial differential systems with a convenient compactification of the phase portrait the boundary of the period annulus of the center has two connected components the center itself and a polycycle we are interested in the behaviour of the period function near the polycycle the desingularization of its critical points gives rise to a new polycycle monodromic as well with hyperbolic saddles or saddle nodes at the vertices in this paper we compute the first terms in the asymptotic development of the time function around any orbitally linearizable saddle that may come from this desingularization process in addition we use these developments to study the bifurcation diagram of the period function of the dehomogenized loud s centers more generally the tools developed here can be used to study the return time function around a monodromic polycycle this work is a continuation of the results in p mardesic d marin j villadelprat on the time function of the dulac map for families of meromorphic vector fields nonlinearity 16 2003 855 881 p mardesic d marin j villadelprat the period function of reversible quadratic centers j differential equations 224 2006 120 171

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

the period function of the generalized lotka volterra centers abstract the present paper deals with the period function of the quadratic centers in the literature different terminologies are used to classify these centers but essentially there are four families hamiltonian reversible q 3 r codimension four q 4 and generalized lotka volterra systems q 3 lv chicone c chicone review in mathscinet ref 94h 58072 conjectured that the reversible centers have at most two critical periods and that the centers of the three other families have a monotonic period function with regard to the second part of this conjecture only the monotonicity of the hamiltonian and q 4 families w a coppel l gavrilov the period function of a hamiltonian quadratic system differential integral equations 6 1993 1357 1365 y zhao the monotonicity of period function for codimension four quadratic system q 4 j differential equations 185 2002 370 387 has been proved concerning the q 3 lv family no substantial progress has been made since the middle 80s when several authors showed independently the monotonicity of the classical lotka volterra centers f rothe the periods of the volterra lokta system j reine angew math 355 1985 129 138 r schaaf global behaviour of solution branches for some neumann problems depending on one or several parameters j reine angew math 346 1984 1 31 j waldvogel the period in the lotka volterra system is monotonic j math anal appl 114 1986 178 184 by means of the first period constant one can easily conclude that the period function of the centers in the q 3 lv family is monotone increasing near the inner boundary of its period annulus i e the center itself thus according to chicone s conjecture it should be also monotone increasing near the outer boundary which in the poincare disc is a polycycle in this paper we show that this is true in addition we prove that except for a zero measure subset of the parameter plane there is no bifurcation of critical periods from the outer boundary finally we show that the period function is globally i e in the whole period annulus monotone increasing in two other cases different from the classical one

diffusive effects on dispersion in excitable media perturbation and numerical methods are used to study traveling and standing wave solutions for a system of reaction diffusion equations modeled after excitable media and the effect of variation of the diffusion coefficients it is shown that traveling waves and standing waves can coexist and that as a certain diffusion coefficient is varied a branch of traveling waves can bifurcate from a branch of standing waves providing a transition between only traveling solutions in one limit and only standing waves in the other limit key words traveling waves standing waves bifurcation dispersion reaction diffusion equations singular perturbation

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

maximal and minimal eigenvalues and their associated nonlinear equations the spectral theory of uniformly elliptic operators a under perturbations v giving rise to operators of the form hv a v x on a bounded or unbounded region such as schrodinger operators are considered suppose that v p is constrained but v is otherwise unspecified the theory of the potentials v that maximize or minimize the eigenvalues of hv is presented the optimizing potentials are typically determined by equations of the form u w x u cu u the optimization of eigenvalues also turns out to be related to the determination of the best constants in sobolev s inequality and in its one dimensional simplification to a classical oscillator problem with instanton properties

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

existence uniqueness and nondegeneracy of positive solutions of semilinear elliptic equations we study positive solutions of the dirichlet problem a u x f u x 0 x d u x o x e ad where d is an n ball we find necessary and sufficient conditions for solutions to be nondegenerate we also give some new existence and uniqueness theorems in this paper we study positive solutions of the dirichlet problem au x f u x o x f2

a numerical study of spontaneous bifurcation abstract in the study of nonlinear boundary value problems it has been observed that bifurcations may take place which are not local to the origin in function space in such problems the analysis of the bifurcation and the computation of solutions is complicated by the nonlocal nature of the bifurcation and the multiplicity of solutions in this work a model problem exhibiting this phenomena which has been of recent interest is numerically studied the method of alternative problems is employed to provide a complete numerical description of the bifurcation occurring in the model problem

comparative analysis of extrinsic and intrinsic cohesive models of dynamic fracture a comparative analysis of intrinsic and extrinsic cohesive models has been performed for the case of spontaneous and steady state dynamic crack propagation spontaneous crack propagation was simulated using a spectral form of the elastodynamic boundary integral equation while steady state solutions were obtained by numerically integrating the governing cauchy singular integral equation spontaneous crack propagation results showed that intrinsic models are less numerically stable than the extrinsic ones under steady state propagation conditions some intrinsic cohesive models lead to unrealistic results as the crack opening velocity becomes negative at the cohesive zone tip by imposing a positive crack opening acceleration at the cohesive zone tip the envelope of the required minimum initial strength has been calculated

representation and asymptotic formulas for 1 dimensional linearized eigenvalue problems with dirichlet boundary condition abstract in this article nontrivial solutions to a nonlinear dirichlet boundary value problem with a small parameter and the corresponding linearized eigenvalue problems are investigated in a typical case of nonlinearity asymptotic formulas of all eigenvalues and asymptotic profiles of all eigenfunctions are proved proofs are based on the method of representation equation and asymptotic results in the corresponding neumann problem obtained in the recent papers by the author and s yotsutani

bifurcation and stability of stationary solutions of the fitz hugh nagumo equations the fitz hugh nagumo equations have been of some interest to both mathematicians and theoretical biologists for several years the reason for this stems from the fact that they can be considered as a simpler model for the celebrated hodgkin huxley equations in that they exhibit many of the features of this latter system indeed mathematicians have studied them because their structure is different from the usually encountered equations in physics and they therefore admit solutions with less familiar properties homoclinic travelling waves threshold effects etc the equations can be written as

a singularity theory approach to a semilinear boundary value problem where a b 3 are real parameters with 0 a b 1 this problem arises in population genetics with the unknown function u corresponding to the relative frequency of a gene the parameter a is given by a combination of the death rates of the various genotypes involved b gives the frequency of the gene on the boundary of the region and 1 is the reciprocal of the diffusion rate of the gene see aronson and weinberger 1 for further details solutions of 1 1 correspond to possible steady states of the gene populations we are interested mainly in how the bifurcation diagram changes as a and b vary and we investigate this question by exploiting singularity theory for bifurcation problems as developed by golubitsky and schaeffer c51 equation 1 1 is a simple example of a semilinear elliptic boundary value problem and a standard method of investigating the bifurcation of

exact multiplicity results for a p laplacian problem with concave convex concave nonlinearities we study the exact number of positive solutions of a two point dirichlet boundary value problem involving the p laplacian operator we consider the case p 2 as well as the case p 1 when the nonlinearity f satisfies f 0 0 and has two distinct simple positive zeros and such that f changes sign exactly twice on 0 note that we may allow that f changes sign more than twice on 0 some interesting examples of quartic polynomials are given in particular for f u u2 u 1 u 2 we study the evolution of the bifurcation curves of the p laplacian problem as p increases from 1 to infinity and hence are able to determine the exact multiplicity of positive solutions for each p 1

time maps and exact multiplicity results for one dimensional prescribed mean curvature equations ii abstract we investigate various properties of time maps for one dimensional prescribed mean curvature equations using these properties we obtain some exact multiplicity results of positive solutions and sign changing solutions as it turned out these quasilinear problems show many different phenomena from semilinear problems our methods are based on a detailed analysis of time maps

exact multiplicity results for boundary value problems with nonlinearities generalizing cubic using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type the first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u the second one deals with a class of nonlinearities with explicit x dependence

diffusive logistic equation with non linear boundary conditions abstract we analyze the solutions of a population model with diffusion and logistic growth in particular we focus our study on a population living in a patch r n with n 1 that satisfies a certain non linear boundary condition and on its survival when constant yield harvesting is introduced we establish our existence results by the method of sub super solutions

spatial convergence of crack nucleation using a cohesive finite element model on a pinwheel based mesh we consider the use of initially rigid cohesive interface models in a two dimensional dynamic finite element solution of a fracture process our focus is on convergence of finite element solutions to a solution of the undiscretized medium as the mesh spacing x and therefore time step t tends to zero we propose the use of pinwheel meshes which possess the isoperimetric property that for any curve c in the computational domain there is an approximation to c using mesh edges that tends to c including a correct representation of its length as the grid size tends to zero we suggest that the isoperimetric property is a necessary condition for any possible spatial convergence proof in the general case that the crack path is not known in advance conversely we establish that if the pinwheel mesh is used the discrete interface first activated in the finite element model will converge to the initial crack in the undiscretized medium finally we carry out a mesh refinement experiment to check convergence of both nucleation and propagation our results indicate that the crack path computed in the pinwheel mesh is more stable as the mesh is refined compared to other types of meshes copyright 2006 john wiley sons ltd

global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications we study the global bifurcation and exact multiplicity of positive solutions of u x lambda f epsilon u 0 1 f epsilon u epsilon u 3 sigma u 2 tau u rho where lambda epsilon 0 are two bifurcation parameters and sigma rho 0 tau 0 are constants by developing some new time map techniques we prove the global bifurcation of bifurcation curves for varying epsilon 0 more precisely we prove that for any sigma rho 0 tau 0 there exists epsilon 0 such that on the lambda parallel to u parallel to infinity plane the bifurcation curve is s shaped for 0 epsilon we also prove the global bifurcation of bifurcation curves for varying lambda 0 thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda we give an application to prove a long standing conjecture for global bifurcation of positive solutions for the problem u x lambda epsilon u 3 u 2 u 1 0 1 u 1 u 1 0 which was studied by crandall and rabinowitz arch rational mech anal 52 1973 p 177 in addition we give an application to prove a conjecture of smoller and wasserman j differential equations 39 1981 p 283 lines 2 3 on the maximum number of positive solutions of a positone problem

on the evolution and qualitative behaviors of bifurcation curves for a boundary value problem abstract we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for u x u 1 sin u u p 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and p 1 is an evolution parameter on the u plane we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p 2 it is a monotone curve for p 2 it has at least two turning points for 1 p p 1 2 cos 1 sin 1 1 2 cos 1 sin 1 1 629 and it has infinitely many turning points for p 1 hence we are able to determine the exact number of positive solutions in particular we give complete descriptions of the structure of bifurcation curves when p 2 we also give some numerical simulations of bifurcation curves for p 1

on substantive research hypotheses conditional independence graphs and graphical chain models summary graphs consisting of points and lines or arrows as connections between selected pairs of points are used to formulate hypotheses about relations between variables points stand for variables connections represent associations when a missing connection is interpreted as a conditional independence the graph characterizes a conditional independence structure as well statistical models called graphical chain models correspond to special types of graphs which are interpreted in this fashion examples are used to illustrate how conditional independences are reflected in summary statistics derived from the models and how the graphs help to identify analogies and equivalences between different models graphical chain models are shown to provide a unifying concept for many statistical techniques that in the past have proven to be useful in analyses of data they also provide tools for new types of analysis

mini buckets a general scheme for bounded inference this article presents a class of approximation algorithms that extend the idea of bounded complexity inference inspired by successful constraint propagation algorithms to probabilistic inference and combinatorial optimization the idea is to bound the dimensionality of dependencies created by inference algorithms this yields a parameterized scheme called mini buckets that offers adjustable trade off between accuracy and efficiency the mini bucket approach to optimization problems such as finding the most probable explanation mpe in bayesian networks generates both an approximate solution and bounds on the solution quality we present empirical results demonstrating successful performance of the proposed approximation scheme for the mpe task both on randomly generated problems and on realistic domains such as medical diagnosis and probabilistic decoding

low dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem abstract a nonlinear boundary value problem exhibiting a nonlocal bifurcation in its solution set is considered it is shown that for some values of the problem parameters there is only the zero solution while for other values of these parameters two nontrivial solutions also appear these nontrivial solutions do not lie in small neighborhoods of the zero function so that the bifurcation analysis is not local to the origin the method of reduction to alternative finite dimensional problems is applied to give a numerical description of the bifurcation and in particular to describe a surface of bifurcating solutions in addition the solution set of the 1 dimensional galerkin approximation is also shown to have the same qualitative nature as the solution set of the full problem

global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity abstract we study the global bifurcation and exact multiplicity of positive solutions of u x f e u 0 1 x 1 u 1 u 1 0 f e u e u 3 u 2 u where e 0 are two bifurcation parameters and 0 0 are constants we prove the global bifurcation of bifurcation curves for varying e 0 more precisely there exists e 0 such that on the u plane the bifurcation curve is s shaped for 0 e e and is monotone increasing for e e thus we are able to determine the exact number of positive solutions by the values of e and our results extend those of hung and wang k c hung s h wang global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications trans amer math soc in press from 0 to

verification of bifurcation diagrams for polynomial like equations the results of our recent paper p korman y li t ouyang computing the location and the direction of bifurcation math res lett 12 2005 933 944 appear to be sufficient to justify computer generated bifurcation diagram for any autonomous two point dirichlet problem here we apply our results to polynomial like nonlinearities

bifurcation diagrams of population models with nonlinear diffusion we develop analytical and numerical tools for the equilibrium solutions of a class of reaction diffusion models with nonlinear diffusion rates such equations arise from population biology and material sciences we obtain global bifurcation diagrams for various nonlinear diffusion functions and several growth rate functions

quasi static crack propagation in plane and plate structures using set valued traction separation laws we introduce a numerical technique to model set valued traction separation laws in plate bending and also plane crack propagation problems by using of recent developments in thin kirchhoff love shell models and the extended finite element method a complete and accurate algorithm for the cohesive law is presented and is used to determine the crack path the cohesive law includes softening and unloading to origin adhesion and contact pure debonding and contact are obtained as particular degenerate cases a smooth root finding algorithm based on the trust region method is adopted a step driven algorithm is described with a smoothed law which can be made arbitrarily close to the exact non smooth law in the examples shown the results were found to be step size insensitive and accurate in addition the method provides the crack advance law extracted from the cohesive law and the absence of stress singularity at the tip copyright 2007 john wiley sons ltd

stabilization of solutions of nonlinear and degenerate evolution equations on etudie le comportement asymptotique des solutions bornees des equations de la forme u t u f x u dans r u x o u 0 x dans u x t u 1 x t dans r ou est un domaine borne de r n avec de classe c 2 0 1