on stable nonconstant stationary solutions and mesoscopic patterns for fitzhugh nagumo equations in higher dimensions abstract fitzhugh nagumo equation has been studied extensively in the field of mathematical biology it has the mechanism of lateral inhibition which seems to play a big role in the pattern formation of plankton distribution we consider fitzhugh nagumo equation in high dimension and show the existence of stable nonconstant stationary solutions which have fine structures on a mesoscopic scale we construct spatially periodic stationary solutions moreover we compute the singular limit energy which suggests that the transition from planar structure to droplet pattern can occur when parameters change

wavefront propagation for reaction diffusion systems of pde m i freidlin has introduced probabilistic techniques to study propagation for systems of reaction diffusion pde the motivating idea is that should a reaction diffusion system possess only a single unstable and a single stable equilibrium then the solution u of the system will presumably tend to switch for large times from near the former to near the latter state this paper brings to bear purely pde techniques to this problem especially the theory of viscosity solutions on hamilton jacobi equations due to crandall lions

duality gaps in nonconvex stochastic optimization we consider multistage stochastic optimization models containing nonconvex constraints e g due to logical or integrality requirements we study three variants of lagrangian relaxations and of the corresponding decomposition schemes namely scenario nodal and geographical decomposition based on convex equivalents for the lagrangian duals we compare the duality gaps for these decomposition schemes the first main result states that scenario decomposition provides a smaller or equal duality gap than nodal decomposition the second group of results concerns large stochastic optimization models with loosely coupled components the results provide conditions implying relations between the duality gaps of geographical decomposition and the duality gaps for scenario and nodal decomposition respectively

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

anisotropic singular perturbations the vectorial case we obtain the l 1 limit of the sequence where e e is the family of anisotropic perturbations of the nonconvex functional of vector valued functions the proof relies on the blow up argument introduced by fonseca and muller

coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction diffusion system abstract we establish the existence of standing waves with one pulse multiple spikes and transition layers in the nonlinear reaction diffusion system u t f u w u xx w t 2 g u w w xx x r where 0 is a small parameter f u w 0 has three branches u h 1 w u h 2 w and u h 3 w on an interval of w and g u w 0 intersects u h 1 w once and u h 3 w at most once we use a different method to prove the existence of a single pulse the existence of multiple spikes and transition layers is a new result the new method is a topological shooting plan consisting of various shooting arguments the main technical advance of this plan is that it provides a setup for proving the existence of not only a single pulse but also multiple spikes and transition layers the proof of the existence of spikes and transition layers involves more delicate estimates in order to assure the existence of certain types of solutions necessary in the topological shooting arguments a two dimensional topological shooting principle and another elementary topological principle are applied in the shooting arguments

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

some mathematical aspects of the micro phase separation in diblock copolymers abstract a free energy functional of nonlocal type is considered that was originally introduced to describe the micro phase separation of diblock copolymer a mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology the scaling law is equivalent to finding a nice rescaling in order to have a well defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer and the associated stationary problem becomes a morphology equation that governs the configuration of final patterns although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nature a possible rigorous approach is presented to stability and morphological selection problems for them

three dimensional static analysis of multi layered piezoelectric hollow spheres via the state space method a general non axisymmetric exact analysis of the statics of a laminated piezoelectric hollow sphere is presented in the paper by using a state space method to select a proper set of state variables three displacement functions and two stress functions are introduced it is found that the basic equations of a spherically isotropic piezoelectric medium are eventually turned to two separated state equations with constant coefficients the solutions of which are then obtained by virtue of matrix theory the continuity conditions at each interface are then used to derive two relationships between respective boundary variables at the inner and outer spherical surfaces no matter how many layers the sphere contains the orders of the final solving equations remain unaltered

criteria of spontaneous interconversions of traveling and static arbitrary dimensional dissipative structures abstract the general criteria of spontaneous transformation of traveling dissipative structures dss autowaves into static dss are derived on the basis of translational symmetry of the problem these criteria determine the system parameters in particular the excitation level a am when the velocity of autowaves of arbitrary dimension and shape becomes zero it is demonstrated that the criteria coincide with the instability condition of static dss with respect to the aperiodic increase of a certain fluctuation describing a small translation of the dss along arbitrary axis i e with the condition of spontaneous transformation of static dss into traveling ones it is shown that the velocity of autowaves depends on parameter a near am v a m a 1 2 the models of fitzhugh nagumo type are analyzed and it is shown that the particular conditions obtained earlier follow from the general criteria derived here some scenarios of interconversion of traveling and static dss are discussed

analysis of microstructure of a non convex functional with penalization term we study the properties of the minimizers of a singularly perturbed non convex functional with a penalization term fe v u i 12e2 ivxx2dx 12 iw vx dx 2e2 i v u 2dx where i 0 1 0 0 are parameters u is a given function and w p p2 1 2 we analyze the phenomenon of microstructure formation and show the dependence on and u

a variational approach for standing waves of fitzhugh nagumo type systems abstract we study the existence of radially symmetric solutions of fitzhugh nagumo type elliptic systems in r n n 2 u g u v in r n d v v u in r n u x v x 0 0 as x we utilize a truncation technique and apply minimax arguments to the corresponding strongly indefinite functional i u v 1 2 r n u 2 d v 2 d x r n g u 2 v 2 u v d x defined on h r 1 r n h r 1 r n to find positive and possibly sign changing solutions of in particular we overcome difficulty related to palais smale condition via our new scaling argument when g 1 0 1 2 we improve the existence result of reinecke sweers 23

interior peak solutions for an elliptic system of fitzhugh nagumo type abstract we consider the dirichlet problem for an elliptic system of fitzhugh nagumo type we prove that the problem has a solution with a sharp peak inside the domain

stochastic dual dynamic programming applied to nonconvex hydrothermal models abstract in this paper we apply stochastic dual dynamic programming decomposition to a nonconvex multistage stochastic hydrothermal model where the nonlinear water head effects on production and the nonlinear dependence between the reservoir head and the reservoir volume are modeled the nonconvex constraints that represent the production function of a hydro plant are approximated by mccormick envelopes these constraints are split into smaller regions and the mccormick envelopes are used for each region we use binary variables for this disjunctive programming approach and solve the problem with a decomposition method we resort to a variant of the l shaped method for solving the mip subproblem with binary variables at any stage inside the stochastic dual dynamic programming algorithm a realistic large scale case study is presented

a geometric study of duality gaps with applications lagrangian relaxation is often an efficient tool to solve large scale optimization problems even nonconvex however it introduces a duality gap which should be small for the method to be really efficient here we make a geometric study of the duality gap given a nonconvex problem we formulate in a first part a convex problem having the same dual this formulation involves a convexification in the product of the three spaces containing respectively the variables the objective and the constraints we apply our results to several relaxation schemes especially one called lagrangean decomposition in the combinatorial optimization community or operator splitting elsewhere we also study a specific application highly nonlinear the unit commitment problem

a proximal bundle method with approximate subgradient linearizations we give a proximal bundle method for minimizing a convex function f over a closed convex set it only requires evaluating f and its subgradients with an accuracy epsilon 0 which is fixed but possibly unknown it asymptotically finds points that are epsilon optimal when applied to lagrangian relaxation it allows for epsilon accurate solutions of lagrangian subproblems and finds epsilon optimal solutions of convex programs

demand allocation with latency cost functions we address the exact resolution of a mixed integer non linear programming model where resources can be activated in order to satisfy a demand a covering constraint while minimizing total cost for each resource there is a fixed activation cost and a variable cost expressed by means of latency functions we prove that this problem is mathcal n mathcal p hard even for linear latency functions a branch and bound algorithm is devised having two important features first a dual bound equal to that obtained by continuous relaxation can be computed very efficiently at each node of the enumeration tree second to break symmetries resulting in improved efficiency the branching scheme is n ary instead of binary these features lead to a successful comparison against two popular commercial and open source solvers cplex and bonmin

large amplitude stationary waves in an excitable lateral inhibitory medium pattern formation in a system of reaction diffusion equations which exhibit both lateral inhibition and excitability is investigated the system studied is derived from the fitzhugh nagumo nerve conduction model the existence of solitary and periodic stationary wave solutions is proved numerical computations show that some of these waves are stable as solutions of the partial differential equations further arguments are given which indicate that these patterns arise as a result of excitability of the system and not as a result of a turing mechanism

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

stationary wave solutions of a system of reaction diffusion equations derived from the fitzhugh nagumo equations we consider an extension of the fitzhugh nagumo model namely the system u t d 1 u xx f u w qquad w t d 2 w xx varepsilon u gamma w where varepsilon 0 gamma 0 d 1 0 d 2 0 and f u is cubic we allow gamma to be large which implies that there are three constant solutions we show that over an appropriate range of parameters the system has time independent pulse solutions and an infinite number of periodic solutions depending on the particular choice of parameters we show that the pulse solution leads to either the first constant solution or the third constant solution

an application of the modular function in nonlocal variational problems using the modular function and its invariance under the action of a modular group and an heuristic reduction of a mathematical model we present a mathematical account of a hexagonal pattern selection observed in di block copolymer melts

on the bifurcation curve for an elliptic system of fitzhugh nagumo type abstract we study a system of partial differential equations derived from the fitzhugh nagumo model in one dimension solutions are required to satisfy zero dirichlet boundary conditions on the interval 1 1 estimates are given to describe bounds on the range of parameters over which solutions exist numerical computations provide the global bifurcation diagram for families of symmetric and asymmetric solutions in the two dimensional case we use numerical methods for zero dirichlet boundary conditions on the square domain 1 1 1 1 numerical computations are given both for symmetric and asymmetric and for stable and unstable solutions

homogenization of linear and nonlinear transport equations we study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields two types of homogenized equations are derived for general n dimensional linear and nonlinear problems we derive homogenized equations by introducing additional independent variables to represent the small scales for the two dimensional linear transport equations we derive effective equations for the averaged quantities such equations take the form of either a degenerate non local diffusion equation with memory or a higher order hyperbolic equation to study the nonlinear transport equations we introduce the concept of two scale young measure and extend diperna s method to prove that it reduces to a family of dirac measures

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

a new approach to variational problems with multiple scales we introduce a new concept the young measure on micropatterns to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter e this allows one to extract in the limit e 0 the relevant information at the macroscopic scale as well as the coarsest microscopic scale say e and to eliminate all finer scales to achieve this we consider rescaled functions rx t x s e t viewed as maps of the macroscopic variable s with values in a suitable function space the limiting problem can then be formulated as a variational problem on the young measures generated by rex as an illustration we study a one dimensional model that describes the competition between formation of microstructure and highest gradient regularization we show that the unique minimizer of the limit problem is a young measure supported on sawtooth functions with a given period 2001 john wiley sons inc

standing waves in the fitzhugh nagumo system and a problem in combinatorial geometry we show that there is a close relation between standing wave solutions for the fitzhugh nagumo system

on a primal proximal heuristic in discrete optimization lagrangian relaxation is useful to bound the optimal value of a given optimization problem and also to obtain relaxed solutions to obtain primal solutions it is conceivable to use a convexification procedure suggested by d p bertsekas in 1979 based on the proximal algorithm in the primal space the present paper studies the theory assessing the approach in the framework of combinatorial optimization our results indicate that very little can be expected in theory even though fairly good practical results have been obtained for the unit commitment problem

domain branching in uniaxial ferromagnets a scaling law for the minimum energy we address the branching of magnetic domains in a uniaxial ferromagnet our thesis is that branching is required by energy minimization to show this we consider the nonlocal nonconvex variational problem of micromagnetics we identify the scaling law of the minimum energy by proving a rigorous lower bound which matches the already known upper bound we further show that any domain pattern achieving this scaling law must have average width of order l 2 3 where l is the length of the magnet in the easy direction finally we argue that branching is required by considering the constrained variational problem in which branching is prohibited and the domain structure is invariant in the easy direction its scaling law is different

traveling waves for the fitzhugh nagumo system on an infinite channel abstract we are concerned with the traveling wave solutions for the fitzhugh nagumo system on an infinite channel based on a variational formulation in which a non local term depends on a parameter the speed of a traveling wave can be selected out furthermore to show the existence of a traveling wave solution with such a speed we seek a minimizer subject to a constraint in the way of solving the variational problem we apply a truncation technique to the nonlocal term to obtain a minimizer located in a bounded invariant region

propagation phenomena in a bistable reaction diffusion system consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical biochemical systems these equations are of the fitzhugh nagumo type and contain three parameters for certain ranges of the parameters the system exhibits two stable singular points a singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves a complete rigorous description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity numerical results of solutions to these equations are also presented these calculations illustrate how waves are generated from initial data how they interact during collisions and how they are influenced by local disturbances and boundary conditions in one example a righltward moving

multipeak solutions for an elliptic system of fitzhugh nagumo type we consider the dirichlet problem for an elliptic system of fitzhugh nagumo type we prove that the problem has solutions with arbitrary many sharp peaks near the boundary but not on the boundary

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

free axisymmetric vibration of transversely isotropic piezoelectric circular plates abstract based on three dimensional elastic theory of piezoelectric materials the axisymmetric state space formulation of piezoelectric laminated circular plates is derived finite hankel transforms are used and the boundary variables in free terms are replaced for two kinds of boundary conditions to obtain ordinary differential equations with constant coefficients regarding the axisymmetric free vibration problem two exact solutions for two different boundary conditions are found discarding piezoelectric effect the exact solutions for transversely isotropic circular laminates are also obtained through the same procedure numerical examples are given and compared with those of finite element method fem

nonanticipative duality relaxations and formulations for chance constrained stochastic programs we propose two new lagrangian dual problems for chance constrained stochastic programs based on relaxing nonanticipativity constraints we compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed integer programming mip formulation for a given dual solution the associated lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem we also derive two new primal mip formulations and demonstrate that for chance constrained linear programs the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds we propose a new heuristic method and two new exact algorithms based on these duals and formulations the first exact algorithm applies to chance constrained binary programs and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems the second exact method is a branch and cut algorithm for solving either of the primal formulations our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently and the gaps between the best dual bounds and the heuristic solutions are small

divide to conquer decomposition methods for energy optimization modern electricity systems provide a plethora of challenging issues in optimization the increasing penetration of low carbon renewable sources of energy introduces uncertainty in problems traditionally modeled in a deterministic setting the liberalization of the electricity sector brought the need of designing sound markets ensuring capacity investments while properly reflecting strategic interactions in all these problems hedging risk possibly in a dynamic manner is also a concern the fact of representing uncertainty and or competition of different companies in a multi settlement power market considerably increases the number of variables and constraints for this reason usually a trade off needs to be found between modeling and numerical tractability the more details are brought into the model the harder becomes the optimization problem for structured optimization and generalized equilibrium problems we explore some variants of solution methods based on lagrangian relaxation and on benders decomposition throughout we keep as a leading thread the actual practical value of such techniques in terms of their efficiency to solve energy related problems

closed form solution for free vibration of piezoelectric coupled annular plates using levinson plate theory abstract free vibration analysis of annular moderately thick plates integrated with piezoelectric layers is investigated in this study for different combinations of soft simply supported hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the levinson plate theory lpt the distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the maxwell static electricity equation is approximately satisfied the differential equations of motion are solved analytically for various boundary conditions of the plate in this study the closed form solution for characteristic equations displacement components of the plate and electric potential are derived for the first time in the literature to demonstrate the accuracy of the present solution comparison studies is first carried out with the available data in the literature and then natural frequencies of the piezoelectric coupled annular plate are presented for different thickness radius ratios inner outer radius ratios thickness of piezoelectric material of piezoelectric and boundary conditions present analytical model provides design reference for piezoelectric material application such as sensors actuators and ultrasonic motors

the exact elasto electric field of a rotating piezoceramic spherical shell with a functionally graded property a displacement separation technique is employed to simplify the basic equations of a piezoceramic body with radial inhomogeneity it is shown that the controlling equations are finally reduced to an uncoupled second order ordinary differential equation and a coupled system of three second order ordinary differential equations solutions to these differential equations are given for the case that material constants are of power functions of the radial coordinate the static analysis of a steadily rotating spherical shell is then presented

standing pulse solutions to fitzhugh nagumo equations reaction diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences a localized pattern is a stable non constant stationary solution usually located far away from neighborhoods of bifurcation induced by turing s instability in the study of fitzhugh nagumo equations we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial this amounts to seeking a homoclinic orbit for a corresponding hamiltonian system and we utilize a variational formulation which involves a nonlocal term such a functional is referred to as helmholtz free energy in modeling microphase separation in diblock copolymers while its global minimizer does not exist in our setting of dealing with standing pulse the homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions in contrast with the known results for positive standing pulses found in the literature a new technique is attempted by seeking a standing pulse solution with a sign change

lagrangian decomposition of block separable mixed integer all quadratic programs the purpose of this paper is threefold first we propose splitting schemes for reformulating non separable problems as block separable problems second we show that the lagrangian dual of a block separable mixed integer all quadratic program miqqp can be formulated as an eigenvalue optimization problem keeping the block separable structure finally we report numerical results on solving the eigenvalue optimization problem by a proximal bundle algorithm applying lagrangian decomposition the results indicate that appropriate block separable reformulations of miqqps could accelerate the running time of dual solution algorithms considerably

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

convergent lagrangian and contour cut method for nonlinear integer programming with a quadratic objective function in this paper we present an efficient exact solution method for solving nonlinear separable integer programming problems with a quadratic objective function the proposed method combines the lagrangian dual method with a duality reduction scheme using contour cut at each iteration of the algorithm lower and upper bounds of the problem are determined by the lagrangian dual search to eliminate the duality gap a novel cut and partition scheme is derived by exploring the special structure of the quadratic contour the method finds an exact solution of the problem in a finite number of iterations computational results are reported for problems with up to 2000 integer variables comparison results with other methods are also presented

bundle methods for sum functions with easy components applications to multicommodity network design we propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum function where some of the components are easy that is they are lagrangian functions of explicitly known compact convex programs this corresponds to a stabilized partial dantzig wolfe decomposition where suitably modified representations of the easy convex subproblems are inserted in the master problem as an alternative to iteratively inner approximating them by extreme points thus providing the algorithm with exact information about a part of the dual objective function the resulting master problems are potentially larger and less well structured than the standard ones ruling out the available specialized techniques and requiring the use of general purpose solvers for their solution this strongly favors piecewise linear stabilizing terms as opposed to the more usual quadratic ones which in turn may have an adverse effect on the convergence speed of the algorithm so that the overall performance may depend on appropriate tuning of all these aspects yet very good computational results are obtained in at least one relevant application the computation of tight lower bounds for fixed charge multicommodity min cost flow problems

a primal proximal heuristic applied to the french unit commitment problem this paper is devoted to the numerical resolution of unit commitment problems with emphasis on the french model optimizing the daily production of electricity the solution process has two phases first a lagrangian relaxation solves the dual to find a lower bound it also gives a primal relaxed solution we then propose to use the latter in the second phase for a heuristic resolution based on a primal proximal algorithm this second step comes as an alternative to an earlier approach based on augmented lagrangian i e a dual proximal algorithm we illustrate the method with some real life numerical results a companion paper is devoted to a theoretical study of the heuristic in the second phase

a general approach to lower semicontinuity and lower closure in optimal control theory a self contained approach to lower semicontinuity and lower closure evolves from an extension of relaxed control theory which is based on a central relative weak compactness criterion called tightness and relaxation in all but one variable two lower closure results for outer integral functionals with variable abstract time domain are developed the first of these has a convexity condition for the integrand and generalizes all similar results in the literature the second lower closure result is of a new kind among other things it implies a quite general version of fatou s lemma in several dimensions