conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

fuzzy shortest paths abstract shortest paths are one of the simplest and most widely used concepts in nonfuzzy networks fuzzy shortest paths also have a variety of applications but are hampered by the possibility that the fuzzy shortest length may not correspond to an actual path in the network in this paper we present new models based on fuzzy shortest paths that circumvent this problem and we also give a general algorithm based on dynamic programming to solve the new models in addition we analyze the fuzzy shortest path algorithms in terms of submodular functions the analysis concepts developed for fuzzy shortest paths are also discussed in terms of general fuzzy mathematical programming

positive clustered layered solutions for the gierer meinhardt system abstract we consider the stationary gierer meinhardt system in a ball of r n e 2 u u u p v q 0 in v v u m v s 0 in u v 0 and u v 0 on where b r is a ball of r n n 2 with radius r e 0 is a small parameter and p q m s satisfy the following condition p 1 q 0 m 1 s 0 q m p 1 1 s 1 assume 0 p 1 q a if n 2 and 0 p 1 q 1 if n 3 where a 1 whose numerical value is a 1 06119 we prove that there exists a unique r a 0 such that for r r a r corresponds to r n case and for any fixed integer k 1 this system has at least one sometimes two radially symmetric positive solution u e k v e k which concentrate at k spheres j 1 k x r e j where r e 1 r e 2 r e k are such that r 0 r e 1 e log 1 e r e j 1 r e j e log 1 e j 2 k where r 0 r is a root of some function m r r this generalizes the results in w m ni j wei on positive solutions concentrating on spheres for the gierer meinhardt system j differential equations 221 2006 158 189 where a special case k 1 and n 2 n 1 p 1 q 1 was considered

on the existence of positive solutions for semilinear elliptic equations with neumann boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded lipschitz domainsd d d 3 having compact boundary with nonlinear neumann boundary conditions on the boundary ofd for this we use an implicit probabilistic representation schauder s fixed point theorem and a recently proved sobolev inequality forw 1 2 d special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics

on the application of fuzzy sets theory to the optimal flood control problem arising in water resources systems abstract we first survey the application of fuzzy sets theory to various problems occurring in water resources systems the problem of optimal flood control planning by an appropriate integration of structural and non structural measures with the objective of optimizing the flood damage reduction due to recurrent floods is modeled via fuzzy sets methodologies a two level optimization model appropriate for planning decisions first on a regional or local level and then on a national level is proposed a third phase involving coordination is appended in particular a fuzzy optimization model involving an adroit combination of fuzzy dynamic programming type reasoning and a branch and bound type search procedure is offered as a more utilitarian approach than its crisp equivalent the performance of the algorithm is illustrated with an example

stable and finite morse index solutions on ⁿ or on bounded domains with small diffusion in this paper we study bounded solutions of u f u on r n where n 2 and sometimes n 3 and show that for most f s the weakly stable and finite morse index solutions are quite simple we then use this to obtain a very good understanding of the stable and bounded morse index solutions of e 2 u f u on with dirichlet or neumann boundary conditions for small e

solutions of systems of elliptic differential equations on circular domains abstract we study global bifurcation of weak solutions of systems of elliptic differential equations considered on so 2 invariant domains we formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones as the main tool we use the degree for so 2 equivariant gradient maps defined by the second author in rybicki nonlinear anal tma 23 1 1994 83

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

a class of singular gierer meinhardt systems of elliptic boundary value problems abstract we establish existence results for singular gierer meinhardt elliptic systems with zero dirichlet boundary conditions with the different range of growth rates of the source term we use the schauder fixed point theorem and upper lower solution methods to obtain the existence results we also study boundary estimates for certain singularities

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

analysis on a diffusive sis epidemic model with saturated incidence rate and linear source in a heterogeneous environment abstract in this paper we are concerned with a reaction diffusion sis epidemic model with saturated incidence rate and linear source in spatially heterogeneous environments we derive the uniform bounds of parabolic system and study the extinction and persistence of the infectious disease in terms of the basic reproduction number the existence of the endemic equilibrium ee is established when the basic reproduction number is greater than one we further investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium our study shows that the linear source can enhance persistence of infectious disease and large saturation can help speed up the elimination of disease

the stability of spike solutions to the one dimensional gierer meinhardt model abstract the stability properties of an n spike equilibrium solution to a simplified form of the gierer meinhardt activator inhibitor model in a one dimensional domain is studied asymptotically in the limit of small activator diffusivity e the equilibrium solution consists of a sequence of spikes of equal height the two classes of eigenvalues that must be considered are the o 1 eigenvalues and the o e2 eigenvalues which are referred to as the large and small eigenvalues respectively the spike pattern is stable when the parameters in the gierer meinhardt model are such that both sets of eigenvalues lie in the left half plane for a certain range of these parameters and for n 2 and e 0 it is shown the o 1 eigenvalues are in the left half plane only when d d n where d n is another critical value of d which satisfies d n n thus when n 2 and e 1 the spike pattern is stable only when d n an explicit formula for d n is given for the special case n 1 it is shown that a one spike equilibrium solution is stable when d d1 e an asymptotic formula for d1 e is given finally the dynamics of a one spike solution is studied by deriving a differential equation for the trajectory of the center of the spike

the probabilistic solution of the third boundary value problem for second order elliptic equations using standard reflected brownian motion srbm and martingales we define in the spirit of stroock and varadhan see s v the probabilistic solution of the boundary value problem begin gathered frac 1 2 delta u qu 0 in d hfill frac partial u partial n cu f on partial d hfill end gathered whered is a bounded domain withc3 boundary andn is the inward unit oormal vector on d the assumptions forq c andf are quite general

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

bounds for blow up time in a semilinear parabolic problem with viscoelastic term abstract we consider a semilinear parabolic equation with viscoelastic term u t u 0 t g t s u x s d s u p 2 u by the means of differential inequality technique we obtain a lower bound for blow up time of the solution if blow up occurs at the same time we establish a new blow up criterion and give an upper bound for blow up time of the solution under some conditions on p g and u 0

nodal sets for groundstates of schrodinger operators with zero magnetic field in non simply connected domains we investigate nodal sets of magnetic schrodinger operators with zero magnetic field acting on a non simply connected domain in 2 for the case of circulation 1 2 of the magnetic vector potential around each hole in the region we obtain a characterisation of the nodal set and use this to obtain bounds on the multiplicity of the groundstate for the case of one hole and a fixed electric potential we show that the first eigenvalue takes its highest value for circulation 1 2

optimal multiple objective resource allocation using hybrid particle swarm optimization and adaptive resource bounds technique the multiple objective resource allocation problem morap seeks for an allocation of resource to a number of activities such that a set of objectives are optimized simultaneously and the resource constraints are satisfied morap has many applications such as resource distribution project budgeting software testing health care resource allocation etc this paper addresses the nonlinear morap with integer decision variable constraint to guarantee that all the resource constraints are satisfied we devise an adaptive resource bound technique to construct feasible solutions the proposed method employs the particle swarm optimization pso paradigm and presents a hybrid execution plan which embeds a hill climbing heuristic into the pso for expediting the convergence to cope with the optimization problem with multiple objectives we evaluate the candidate solutions based on dominance relationship and a score function experimental results manifest that the hybrid pso derives solution sets which are very close to the exact pareto sets the proposed method also outperforms several representatives of the state of the art algorithms on a simulation data set of the morap

a dynamic approach to multiple objective resource allocation problem a new approach using dynamic programming is developed for solving the multiple objective resource allocation problem there are two key issues being addressed in this approach the first one is to develop a methodology of fuzzy evaluation and fuzzy optimization for multiple objective systems the procedure of getting the marginal evaluation for each objective and aggregating them synthetically into a global evaluation is presented in this paper the second one is to design a dynamic optimization algorithm by incorporating the method of fuzzy evaluation and fuzzy optimization with the conventional dynamic programming technique a characteristic feature of the approach presented is that various objectives are synthetically considered by the fuzzy systematic technique instead of the frequently employed weighted average method numeric examples are also given to clarify the developed approach and to demonstrate its effectiveness

a fuzzy dynamic approach to the multicriterion resource allocation problem abstract in this paper a new approach using fuzzy dynamic programming is proposed for solving the multiple criteria resource allocation problems it has been concluded that solutions obtained by this approach are always efficient hence an optimal compromise solution can be introduced a characteristic feature of the presented approach is that the objective function scalarization can be taken into account by using fuzzy set notations a numerical example is given to clarify the developed approach

varying total population enhances disease persistence qualitative analysis on a diffusive sis epidemic model abstract this paper performs qualitative analysis on an sis epidemic reaction diffusion system with a linear source in spatially heterogeneous environment the main feature of our model lies in that its total population number varies compared to its counterpart proposed by allen et al 2 the uniform bounds of solutions are derived based on which the threshold dynamics in terms of the basic reproduction number is established and the global stability of the unique endemic equilibrium is discussed when spatial environment is homogeneous in particular the asymptotic profile of endemic equilibria is determined if the diffusion rate of the susceptible or infected population is small or large the theoretical results show that a varying total population can enhance persistence of infectious disease and therefore the disease becomes more threatening and harder to control

blow up phenomena for some nonlinear parabolic problems abstract we consider the blow up of solutions of equations of the form u t div u 2 grad u f u by means of a differential inequality technique a lower bound for blow up time is determined if blow up does occur as well as a criterion for blow up and conditions which ensure that blow up cannot occur

monotonicity of the period function for u u up 0 with p r and p 1 abstract for p r and p 1 we consider the second order ordinary differential equation u u u p 0 which has a one parameter family of periodic solutions satisfying u 0 u 0 u 1 0 with u 1 0 1 we prove that the so called period function t u 1 which represents the period of the periodic solution is strictly decreasing on 0 1 this result is also applied to completely determine the global bifurcation diagram of interior single peak solutions in the neumann boundary value problem of u u u p 0 in 1 1 with u 1 0 where is a control parameter

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

identification of space time distributed parameters in the gierer meinhardt reaction diffusion system we consider parameter identification for the classic gierer meinhardt reaction diffusion system the original gierer meinhardt model a gierer and h meinhardt kybernetik 12 1972 pp 30 39 was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology in our paper the parameters are extended in time and space and used as distributed control variables the methodology employs pde constrained optimization in the context of image driven spatiotemporal pattern formation we prove the existence of optimal solutions derive an optimality system and determine optimal solutions the results of numerical experiments in two dimensions are presented using the finite element method which illustrates the convergence of a variable step gradient algorithm for finding the optimal parameters of the system a practical target function is constructed for the optimal control algorithm corresponding to the actual image of a marine angel

fuzzy approach to multilevel knapsack problems this study proposes a fuzzy approach for solving the multiobjective and multilevel knapsack problems kps the problem was first formulated as a multilevel programming problem with multiple decision makers dms then the degree of satisfaction of each dm was established and represented by their individual membership functions the recursive formulation of dynamic programming was used to solve the decisions of the interrelated stages the overall satisfaction of the decision was obtained through this stage wise operation on the hierarchical structure capacity allocation was developed and a step by step solution procedure was illustrated a detailed comparison between multiobjective and multilevel kps was also carried out finally the possible use of turnpike theorem in kps was scrutinized in the fuzzy domain

solving semilinear partial differential equations with probabilistic potential theory techniques of probabilistic potential theory are applied to solve lu f u u where u is a signed measure f a possibly discontinuous function and l a second order elliptic or parabolic operator on rj or more generally the infinitesi mal generator of a markov process a1so formulated are sufficient conditions guaranteeing existence of a solution to a countably infinite system of such equations

a particle swarm optimization approach to the nonlinear resource allocation problem the resource allocation problem seeks to find an optimal allocation of a limited amount of resource to a number of activities for optimizing the objective under the resource constraint most existing methods use mathematical programming techniques but they may fail to derive exact solutions for large sized problems with reasonable time an alternative is to use meta heuristic algorithms for obtaining approximate solutions this paper presents a particle swarm optimization pso algorithm for conquering the nonlinear resource allocation problem to ensure the resource constraint is satisfied we propose adaptive resource bounds for guiding the search the experimental results manifest that the proposed method is more effective and efficient than a genetic algorithm the convergence behavior of the proposed method is analyzed by observing the variations of particle entropy finally a worst case analysis is conducted to provide a reliable performance guarantee

fuzzy dynamic programming main developments and applications abstract we survey major developments and applications of fuzzy dynamic programming which is here advocated as a promising attempt at making dynamic programming models more realistic by a relaxation of often artificial assumptions of precision as to the constraints goals states and their transitions termination time etc issues related to numerical efficiency are considered applications in r d planning health care and medicine socioeconomic regional development energy systems water and environmental systems chemical engineering etc are surveyed

lower bounds for blow up time in parabolic problems under dirichlet conditions we consider an initial boundary value problem for the semilinear heat equation whose solution may blow up in finite time we use a differential inequality technique to determine a lower bound on blow up time if blow up occurs a second method based on a comparison principle is also presented

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

existence and regularity for a class of integrodifferential equations of parabolic type the method we use rests on the inversion of the formal laplace transform of r and allows one to analyze the regularity of the solution u problem 1 1 has been studied by several authors using various methods see for instance l 18 211 the laplace transform approach presented here has been used in s 7 9 14 153 in our previous papers s 7 93 we have in fact analyzed a more general situation while here we take full 36 0022 247x 85 3 00

singular gierer meinhardt systems of elliptic boundary value problems abstract we establish existence results for singular gierer meinhardt elliptic systems with zero dirichlet boundary conditions gierer meinhardt systems are model problems for pattern formations of spatial tissue structures of morphogenesis the mathematical difficulties are that the system becomes singular near the boundary and it is non quasimonotone we show the existence of positive solutions for the activator inhibitor model with common sources

on a conjecture of lin ni for a semilinear neumann problem let q be a bounded domain in rn n 3 and a 0 we consider au au u n 2 n 2 in q2 u 0 in q au av 0 on i2 and show that for a sufficiently small the minimal energy solutions are only constants

exact multiplicity of positive solutions for a class of semilinear problems we establish an exact multiplicity result of positive solutions of u f u 0inbn u 0on bn wherebnis the unit ball fsatisfiesf changes sign only once and asymptotically sublinear or linear the nonlinearities which we are concerned with here includef u u u b c u for 0 2b candf u up uqfor 1 p q precise global bifurcation diagrams are obtained

non existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry abstract we give two sufficient conditions for a branch consisting of non trivial solutions of an abstract equation in a banach space not to have a secondary bifurcation point when the equation has a certain symmetry when the nonlinearity f is of allen cahn type for instance f u u u 3 we apply these results to an unbounded branch consisting of non radially symmetric solutions of the neumann problem on a disk d r 2 u f u 0 in d u 0 on d and emanating from the second eigenvalue we show that the maximal continuum containing this branch is homeomorphic to r s 1 and that its closure is homeomorphic to r 2

solutions on a torus for a semilinear equation we are interested in the positive doubly periodic solutions which are even in each variable of a stationary nonlinear schrodinger equation in 2 with a small parameter for any pair of periods 2 a 2 b we construct a branch of solutions that concentrate uniformly to the ground state solution of the equation

lower bounds for blow up time in a nonlinear parabolic problem abstract for a parabolic problem with a gradient nonlinearity which was introduced by chipot and weissler m chipot f b weissler some blow up results for a nonlinear parabolic problem with a gradient term siam j math anal 20 1989 886 907 see also b kawohl l a peletier observations on blow up and dead cores for nonlinear parabolic equations math z 202 1989 207 217 the question of blow up is investigated specifically if the solution blows up a lower bound for the time of blow up is derived

global existence and finite time blow up of solutions of a gierer meinhardt system abstract we are concerned with the gierer meinhardt system with zero neumann boundary condition u t d 1 u a 1 u u p v q 1 x x t 0 v t d 2 v a 2 v u r v s 2 x x t 0 u x 0 u 0 x v x 0 v 0 x x where p 1 s 1 q r d 1 d 2 a 1 a 2 are positive constants 1 2 u 0 v 0 are nonnegative smooth functions r d d 1 is a bounded smooth domain we obtain new sufficient conditions for global existence and finite time blow up of solutions especially in the critical exponent cases p 1 r and q r p 1 s 1

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

asymptotic profiles of the positive steady state for an sis epidemic reaction diffusion model part i abstract to capture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease allen et al in l j s allen b m bolker y lou a l nevai asymptotic profiles of the steady states for an sis epidemic reaction diffusion model discrete contin dyn syst ser a 21 1 2008 1 20 proposed a spatial sis susceptible infected susceptible reaction diffusion model and studied the existence uniqueness and particularly the asymptotic behavior of the endemic equilibrium as the diffusion rate of the susceptible individuals goes to zero in the case where a so called low risk subhabitat is created in this work we shall provide further understanding of the impacts of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease which leads us to determine the asymptotic behaviors of the endemic equilibrium when the diffusion rate of either the susceptible or infected population approaches to infinity or zero in the remaining cases consequently our results reveal that in order to eliminate the infected population at least in low risk area it is necessary that one will have to create a low risk subhabitat and reduce at least one of the diffusion rates to zero in this case our results also show that different strategies of controlling the diffusion rates of individuals may lead to very different spatial distributions of the population moreover once the spatial environment is modified to include a low risk subhabitat the optimal strategy of eradicating the epidemic disease is to restrict the diffusion rate of the susceptible individuals rather than that of the infected ones

dynamic programming for fuzzy systems with fuzzy environment abstract a dynamic programming functional equation for a multi stage decision problem with fuzzy dynamics and environment is formulated and solved by a process of fuzzy interpolation this is an extension of the bellman zadeh model

on the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and dirichlet boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains d subset bbb r n n ge 3 with dirichlet boundary conditions we impose mild conditions on the domains and lower order nonlinear coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by kato class functions our approach uses an implicit probabilistic representation schauder s fixed point theorem and new a priori estimates for solutions of the corresponding linear elliptic equations in the course of deriving these a priori estimates we show that the green functions for operators of the form 1 over 2 delta b cdot nabla on d are comparable when one modifies the drift term b on a compact subset of d this generalizes a previous result of ancona 2 obtained under an l p condition on b to a kato condition on b 2

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

boundary bubbling solutions for a supercritical neumann problem with mixed nonlinearities abstract in this paper we investigate the existence of boundary bubbling solutions for a supercritical neumann problem with mixed nonlinearities under a suitable assumption on the mean curvature of the boundary the behavior of the solutions is as a tower of k bubbles when the exponent goes to sobolev critical exponent and the blow up point is a critical point of the mean curvature

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

probabilistic approach to the neumann problem the basic problem considered in this paper is to solve the following neumann boundary value problem probabilistically where we assume that q is in a certain functional class to be specified below and is a bounded measurable function on the boundary we give a martingale formulation of the neumann problem and show that this formulation is essentially equivalent to the classical formulation the paper culminates in an explicit formula for the solution of this problem in terms of reflecting brownian motion and its boundary local time

on the global existence and finite time blow up of shadow systems abstract shadow systems are often used to approximate reaction diffusion systems when one of the diffusion rates is large in this paper we study the global existence and blow up phenomena for shadow systems our results show that even for these fundamental aspects there are serious discrepancies between the dynamics of the reaction diffusion systems and that of their corresponding shadow systems

an eigenvalue problem arising from spiky steady states of a minimal chemotaxis model abstract asymptotic expansion of singular integrals is presented for the study of the existence and uniqueness of spiky steady states of keller segel s minimal chemotaxis model a tool is developed to find asymptotic expansion of eigenpairs of a nonlocal singular eigenvalue problem arising from the study of stability of the spiky steady state new insights are given to simplify significantly the proofs of the main results in 1

on the neumann problem for some semilinear elliptic equations and systems of activator inhibitor type on deduit des estimations a priori pour des solutions positives du probleme de neumann pour des systemes elliptiques semilineaires ainsi que pour des equations isolees semilineaires reliees a ces systemes