a double bubble assembly as a new phase of a ternary inhibitory system a ternary inhibitory system is a three component system characterized by two properties growth and inhibition a deviation from homogeneity has a strong positive feedback on its further increase in the meantime a longer ranging confinement mechanism prevents unlimited spreading together they lead to a locally self enhancing and self organizing process the model considered here is a planar nonlocal geometric problem derived from the triblock copolymer theory an assembly of perturbed double bubbles is mathematically constructed as a stable stationary point of the free energy functional triple junction a phenomenon in which the three components meet at a single point is a key issue addressed in the construction coarsening an undesirable scenario of excessive growth is prevented by a lower bound on the long range interaction term in the free energy the proof involves several ideas perturbation of double bubbles in a restricted class use of internal variables to remove nonlinear constraints local minimization in a restricted class formulated as a nonlinear problem on a hilbert space and reduction to finite dimensional minimization this existence theorem predicts a new morphological phase of a double bubble assembly

compositional modeling in porous media using constant volume flash and flux computation without the need for phase identification abstract the paper deals with the numerical solution of a compositional model describing compressible two phase flow of a mixture composed of several components in porous media with species transfer between the phases the mathematical model is formulated by means of the extended darcy s laws for all phases components continuity equations constitutive relations and appropriate initial and boundary conditions the splitting of components among the phases is described using a new formulation of the local thermodynamic equilibrium which uses volume temperature and moles as specification variables the problem is solved numerically using a combination of the mixed hybrid finite element method for the total flux discretization and the finite volume method for the discretization of transport equations a new approach to numerical flux approximation is proposed which does not require the phase identification and determination of correspondence between the phases on adjacent elements the time discretization is carried out by the backward euler method the resulting large system of nonlinear algebraic equations is solved by the newton raphson iterative method we provide eight examples of different complexity to show reliability and robustness of our approach

the γ limit of the two dimensional ohta kawasaki energy droplet arrangement via the renormalized energy this is the second in a series of papers in which we derive a expansion for the two dimensional non local ginzburg landau energy with coulomb repulsion known as the ohta kawasaki model in connection with diblock copolymer systems in this model two phases appear which interact via a nonlocal coulomb type energy here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction thus creating small droplets of the minority phase in a sea of the majority phase in our previous paper we computed the limit of the leading order energy which yields the averaged behavior for almost minimizers namely that the density of droplets should be uniform here we go to the next order and derive a next order limit energy which is exactly the coulombian renormalized energy obtained by sandier and serfaty as a limiting interaction energy for vortices in the magnetic ginzburg landau model the derivation is based on the abstract scheme of sandier serfaty that serves to obtain lower bounds for 2 scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem thus without appealing to the euler lagrange equation we establish for all configurations which have almost minimal energy the asymptotic roundness and radius of the droplets and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense via a kind of equivalence the obtained results also yield an expansion of the minimal energy and a characterization of the zero super level sets of the minimizers for the original ohta kawasaki energy this leads to the expectation of seeing triangular lattices of droplets as energy minimizers

quantitative inequality for the eigenvalue of a schrodinger operator in the ball abstract the aim of this article is to prove a quantitative inequality for the first eigenvalue of a schrodinger operator in the ball more precisely we optimize the first eigenvalue v of the operator l v v with dirichlet boundary conditions with respect to the potential v under l 1 and l constraints on v the solution has been known to be the characteristic function of a centered ball but this article aims at proving a sharp growth rate of the following form if v is a minimizer then v v c v v l 1 2 for some c 0 the proof relies on two notions of derivatives for shape optimization parametric derivatives and shape derivatives we use parametric derivatives to handle radial competitors and shape derivatives to deal with normal deformation of the ball a dichotomy is then established to extend the result to all other potentials we develop a new method to handle radial distributions and a comparison principle to handle second order shape derivatives at the ball finally we add some remarks regarding the coercivity norm of the second order shape derivative in this context

on the first and second variations of a nonlocal isoperimetric problem we consider a nonlocal perturbation of an isoperimetric variational prob lem the problem may be viewed as a mathematical paradigm for the ubiquitous phenom enon of energy driven pattern formation associated with competing short and long range interactions in particular it arises as a g limit of a model for microphase separation of diblock copolymers in this article we establish precise conditions for criticality and stability i e we explicitly compute the first and second variations we also present some applications

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

exact solutions for models of evolving networks with addition and deletion of nodes there has been considerable recent interest in the properties of networks such as citation networks and the worldwide web that grow by the addition of vertices and a number of simple solvable models of network growth have been studied in the real world however many networks including the web not only add vertices but also lose them here we formulate models of the time evolution of such networks and give exact solutions for a number of cases of particular interest for the case of net growth and so called preferential attachment in which newly appearing vertices attach to previously existing ones in proportion to vertex degree we show that the resulting networks have power law degree distributions but with an exponent that diverges as the growth rate vanishes we conjecture that the low exponent values observed in real world networks are thus the result of vigorous growth in which the rate of addition of vertices far exceeds the rate of removal were growth to slow in the future for instance in a more mature future version of the web we would expect to see exponents increase potentially without bound

existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters abstract the existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved together with a partial regularity result

geometrically induced phase transitions in two dimensional dumbbell shaped domains abstract we continue the analysis started in 23 of a two dimensional non convex variational problem motivated by studies on magnetic domain walls trapped by thin necks the main focus is on the impact of extreme geometry on the structure of local minimizers representing the transition between two different constant phases we address here the case of general non symmetric dumbbell shaped domains with a small constriction and general multi well potentials our main results concern the existence and uniqueness of non constant local minimizers their full classification in the case of convex bulks and the complete description of their asymptotic behavior as the size of the constriction tends to zero

patterns and localized structures in a hybrid non equilibrium ising model we study a hybrid non equilibrium pattern formation model combining short scale ising model with a continuous slow or long scale inhibitor the computation combines alternates monte carlo algorithm with updating of the inhibitor field it is very fast and allows us to study the influence of various factors such as scale ratios coupling strength bias temperature level of noise anisotropy etc on the pattern formation and behavior of emerging non equilibrium structures

a non markovian sir network model with fixed infectious period and preventive rewiring abstract in the face of disease occurrence susceptible individuals tend to protect themselves by rewiring their links i e cutting off connection with infected person and switching to contact with healthy ones therefore the adaptive rewiring mechanism is considered in the network epidemic model moreover the infection periods for different diseases do not always follow exponential distributions in the process of disease transmission which may be of fixed length using the idea of age structure we establish the susceptible infected recovered epidemic model for each node and each link resulting in a delayed non markovian sir pairwise model with fixed infectious period and preventive rewiring then give the pairwise reproduction number r 0 p and provide the formula for the final epidemic size by rigorous analysis and tedious computation the simulation results show that adaptive rewiring inhibits the spread of disease and decreases the size of disease outbreak while the extension of the infectious period promotes disease transmission in addition the numerical simulation results are in good agreement with the stochastic simulation to our best knowledge the approach in building our model is novel the results may provide new insights into the study of the network disease transmission

sharp dimension free quantitative estimates for the gaussian isoperimetric inequality we provide a full quantitative version of the gaussian isoperimetric inequality the difference between the gaussian perimeter of a given set and a half space with the same mass controls the gap between the norms of the corresponding barycenters in particular it controls the gaussian measure of the symmetric difference between the set and the half space oriented so to have the barycenter in the same direction of the set our estimate is independent of the dimension sharp on the decay rate with respect to the gap and with optimal dependence on the mass

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

pattern propagation in nonlinear dissipative systems abstract we discuss the problem of pattern selection in situations where a stable nonuniform state of a nonlinear dissipative system propagates into an initially unstable homogeneous region our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known and we point out that these known properties are consistent with a marginal stability hypothesis that has been suggested in the theory of dendritic crystal growth we then describe a more general interpretation of the marginal stability hypothesis and finally present numerical evidence for its validity from three different pattern forming models

equilibrium configurations of epitaxially strained elastic films second order minimality conditions and qualitative properties of solutions we consider a variational model introduced in the physical literature to describe the epitaxial growth of an elastic film over a thick flat substrate when a lattice mismatch between the two materials is present we study quantitative and qualitative properties of equilibrium configurations that is of local and global minimizers of the free energy functional more precisely we determine analytically the critical threshold for the local minimality of the flat configuration and we also prove several results concerning its global minimality the non occurrence of singularities in non flat global minimizers is also addressed one of the main results of the paper is a new sufficient condition for local minimality which provides the first extension of the classical criteria based on the positivity of the second variation to the context of functionals with bulk and surface energies

general theory of instabilities for patterns with sharp interfaces in reaction diffusion systems an asymptotic method for finding instabilities of arbitrary d dimensional large amplitude patterns in a wide class of reaction diffusion systems is presented the complete stability analysis of two and three dimensional localized patterns is carried out it is shown that in the considered class of systems the criteria for different types of instabilities are universal the specific nonlinearities enter the criteria only via three numerical constants of order 1 the analysis performed explains the self organization scenarios observed in the recent experiments and numerical simulations of some concrete reaction diffusion systems textcopyright 1996 the american physical society

a double bubble in a ternary system with inhibitory long range interaction the free energy of a ternary system with a self organization property includes an interface energy and a longer ranging inhibitory interaction energy in a planar domain if the two energies are properly balanced and two of the three constituents make up an equal but small fraction the free energy admits a local minimizer that is shaped like a perturbed double bubble most difficulties in the proof of this result are related to the triple junction phenomenon that the three constituents of the ternary system meet at a point two techniques are developed to deal with the triple junction first one defines restricted classes of perturbed double bubbles each perturbed double bubble in a restricted class is obtained from a standard double bubble by a special perturbation the two triple junction points of the standard double bubble can only move along the line connecting them in opposite directions and by the same distance the second technique is the use of the so called internal variables these variables derive from the more geometric quantities that describe perturbed double bubbles in restricted classes the advantage of the internal variables is that they are only subject to linear constraints and perturbed double bubbles in a restricted class represented by internal variables are elements of a hilbert space a local minimizer of the free energy in each restricted class is found as a fixed point of a nonlinear equation by a contraction mapping argument the second variation at the fixed point within its restricted class is positive this perturbed double bubble satisfies three of the four equations for critical points of the free energy the unsolved equation is the 120 degree angle condition at triple junction points then perform another minimization among the local minimizers from all restricted classes a minimum of minimizers emerges and solves all the equations for critical points

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

faber krahn inequalities in sharp quantitative form the classical faber krahn inequality asserts that balls uniquely minimize the first eigenvalue of the dirichlet laplacian among sets with given volume in this paper we prove a sharp quantitative enhancement of this result thus confirming a conjecture by nadirashvili and bhattacharya weitsman more generally the result applies to every optimal poincare sobolev constant for the embeddings w 1 2 0 omega hookrightarrow l q omega

traveling spike autosolitons in the gray scott model abstract we developed singular perturbation techniques based on the strong separation of time and length scales to construct the solutions in the form of the traveling spike autosolitons self sustained solitary waves in the gray scott model of an autocatalytic reaction we found that when the inhibitor diffusion is sufficiently slow the ultrafast traveling spike autosolitons are realized in a wide range of parameters when the diffusion of the inhibitor is sufficiently fast the slower traveling spike autosolitons with the diffusion precursor are realized we asymptotically calculated the main parameters such as speed and amplitude of these autosolitons as well as the regions of their existence in the gray scott model we also showed that in certain parameter regions the traveling spike autosolitons coexist with static

the impact of the domain boundary on an inhibitory system existence and location of a stationary half disc the nonlocal geometric variational problem derived from the ohta kawasaki diblock copolymer theory is an inhibitory system with self organizing properties the free energy defined on subsets of a prescribed measure in a domain is a sum of a local perimeter functional and a nonlocal energy given by the green s function of poisson s equation on the domain with the neumann boundary condition the system has the property of preventing a disc from drifting towards the domain boundary this raises the question of whether a stationary set may have its interface touch the domain boundary it is proved that a small perturbed half disc exists as a stable stationary set where the circular part of its boundary is inside the domain as the interface and the almost flat part of its boundary coincides with part of the domain boundary the location of the half disc depends on two quantities the curvature of the domain boundary and a remnant of the green s function after one removes the fundamental solution and a reflection of the fundamental solution this reflection is defined with respect to any sufficiently smooth domain boundary it is an interesting new concept that generalizes the familiar notions of mirror image and circle inversion when the nonlocal energy is weighted less against the local energy the stationary half disc sits near a maximum of the curvature when the nonlocal energy is weighted more the half disc appears near a minimum of the remnant function there is also an intermediate case where the half disc is near a minimum of a combination of the curvature and the remnant function

the spectrum of the torus profile to a geometric variational problem with long range interaction abstract the profile problem for the ohta kawasaki diblock copolymer theory is a geometric variational problem the energy functional is defined on sets in r 3 of prescribed volume and the energy of an admissible set is its perimeter plus a long range interaction term related to the newtonian potential of the set this problem admits a solution called a torus profile that is a set enclosed by an approximate torus of the major radius 1 and the minor radius q the torus profile is both axially symmetric about the z axis and reflexively symmetric about the x y plane there is a way to set up the profile problem in a function space as a partial differential integro equation the linearized operator l of the problem at the torus profile is decomposed into a family of linear ordinary differential integro operators l m where the index m 0 1 2 is called a mode the spectrum of l is the union of the spectra of the l m s it is proved that for each m when q is sufficiently small l m is positive definite 0 is an eigenvalue for both l 0 and l 1 due to the translation and rotation invariance as q tends to 0 more and more l m s become positive definite however no matter how small q is there is always a mode m of which l m has a negative eigenvalue this mode grows to infinity like q 3 4 as q 0

symmetry of minimizers of a gaussian isoperimetric problem we study an isoperimetric problem described by a functional that consists of the standard gaussian perimeter and the norm of the barycenter the second term is in competition with the perimeter balancing the mass with respect to the origin and because of that the solution is not always the half space we characterize all the minimizers of this functional when the volume is close to one by proving that the minimizer is either the half space or the symmetric strip depending on the strength of the barycenter term as a corollary we obtain that the symmetric strip is the solution of the gaussian isoperimetric problem among symmetric sets when the volume is close to one as another corollary we obtain the optimal constant in the quantitative gaussian isoperimetric inequality

preferential urn model and nongrowing complex networks a preferential urn model which is based on the concept the rich get richer is proposed from a relationship between a nongrowing model for complex networks and the preferential urn model in regard to degree distributions it is revealed that a fitness parameter in the nongrowing model is interpreted as an inverse local temperature in the preferential urn model furthermore it is clarified that the preferential urn model with randomness generates a fat tailed occupation distribution the concept of the local temperature enables us to understand the fat tailed occupation distribution intuitively since the preferential urn model is a simple stochastic model it can be applied to research on not only the nongrowing complex networks but also many other fields such as econophysics and social sciences

existence and stability for a non local isoperimetric model of charged liquid drops we consider a variational problem related to the shape of charged liquid drops at equilibrium we show that this problem never admits local minimizers with respect to l1 perturbations preserving the volume however we prove that the ball is stable under small c1 1 perturbations when the charge is small enough

minimal lamellar structures in a periodic fitzhugh nagumo system abstract a singular limit of a fitzhugh nagumo system leads to a nonlocal geometric variational problem with periodic boundary conditions we study the stationary lamellar set and give a criterion to select out the one with the lowest energy such an optimal structure is called a minimal lamella while the empty set or the full torus is a global minimizer for appropriate parameter regimes the minimal lamellae beat both in other circumstances the concept of minimal lamella points out that a preferred 1d mesh size is universal

on periodic critical points and local minimizers of the ohta kawasaki functional abstract in this paper we collect some new observations about periodic critical points and local minimizers of a nonlocal isoperimetric problem arising in the modeling of diblock copolymers in the main result by means of a purely variational procedure we show that it is possible to construct locally minimizing periodic critical points whose shape resemble that of any given strictly stable constant mean curvature periodic hypersurface along the way we establish several auxiliary results of independent interest

fractional hardy lieb thirring and related inequalities for interacting systems we prove analogues of the lieb thirring and hardy lieb thirring inequalities for many body quantum systems with fractional kinetic operators and homogeneous interaction potentials where no anti symmetry on the wave functions is assumed these many body inequalities imply interesting one body interpolation inequalities and we show that the corresponding one and many body inequalities are actually equivalent in certain cases

on an isoperimetric problem with a competing nonlocal term ii the general case this paper is the continuation of a previous paper h knupfer and c b muratov comm pure appl math 66 2013 1129 1162 we investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance in this work we treat the case of a general space dimension we obtain basic existence results for minimizers with sufficiently small masses for certain ranges of the exponent in the kernel we also obtain nonexistence results for sufficiently large masses as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality the physically important special case of three space dimensions and coulombic repulsion is included in all the results mentioned above in particular our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter in all cases the minimal energy scales linearly with mass for large masses even if the infimum of energy cannot be attained 2014 wiley periodicals inc

remark on a nonlocal isoperimetric problem abstract we consider isoperimetric problem with a nonlocal repulsive term given by the newtonian potential we prove that regular critical sets of the functional are analytic this optimal regularity holds also for critical sets of the ohta kawasaki functional we also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set

isoperimetry and stability properties of balls with respect to nonlocal energies we obtain a sharp quantitative isoperimetric inequality for nonlocal s perimeters uniform with respect to s bounded away from 0 this allows us to address local and global minimality properties of balls with respect to the volume constrained minimization of a free energy consisting of a nonlocal s perimeter plus a non local repulsive interaction term in the particular case s 1 the s perimeter coincides with the classical perimeter and our results improve the ones of knuepfer and muratov comm pure appl math 66 7 1129 1162 2013 comm pure appl math 2014 concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term more precisely their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts

the γ limit of traveling waves in the fitzhugh nagumo system abstract patterns and waves are basic and important phenomena that govern the dynamics of physical and biological systems a common theme in investigating such systems is to identify the intrinsic factors responsible for such self organization the convergence is a well known technique applicable to variational formulations of concentration phenomena of stable patterns recently a geometric variational functional associated with the limit of standing waves of the fitzhugh nagumo system has been built this article studies the limit of traveling waves to the best of our knowledge this is the first attempt to expand the scope of applicability of convergence to cover non stationary problems

self organized scale free networks growth and preferential attachments have been coined as the two fundamental mechanisms responsible for the scale free feature in complex networks as characterized by an algebraic degree distribution there are situations particularly in biological networks where growth is absent or not important yet some of these networks still exhibit the scale free feature with a small degree exponent here we propose two classes of models to account for this phenomenon we show analytically and numerically that in the first model a spectrum of algebraic degree distributions with a small exponent can be generated the second model incorporates weights for nodes and it is able to generate robust scale free degree distribution with larger algebraic exponents our results imply that it is natural for a complex network to self organize itself into a scale free

schur complement systems in the mixed hybrid finite element approximation of the potential fluid flow problem the mixed hybrid finite element discretization of darcy s law and continuity equation describing the potential fluid flow problem in porous media leads to a symmetric indefinite linear system for the pressure and velocity vector components as a method of solution the reduction to three schur complement systems based on successive block elimination is considered the first and second schur complement matrices are formed eliminating the velocity and pressure variables respectively and the third schur complement matrix is obtained by elimination of a part of lagrange multipliers that come from the hybridization of a mixed method the structural properties of these consecutive schur complement matrices in terms of the discretization parameters are studied in detail based on these results the computational complexity of a direct solution method is estimated and compared to the computational cost of the iterative conjugate gradient method applied to schur complement systems it is shown that due to special block structure the spectral properties of successive schur complement matrices do not deteriorate and the approach based on the block elimination and subsequent iterative solution is well justified theoretical results are illustrated by numerical experiments

stability in shape optimization with second variation we are interested in the question of stability in the eld of shape optimization precisely we prove that under structural assumptions on the hessian of the considered shape functions the necessary second order minimality conditions imply that the shape hessian is coercive for a given norm moreover under an additional continuity condition for the second order derivatives we derive precise optimality results in the class of regular perturbations of a domain these conditions are quite general and are satised for the volume the perimeter the torsional rigidity and the rst dirichlet eigenvalue of the laplace operator as an application we provide non trivial examples of inequalities obtained in this way

stability of spot and ring solutions of the diblock copolymer equation the convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions we determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability in two dimensions a threshold exists for the stability of the spot solution it is stable if the sample size is small and unstable if the sample size is large the stability of the ring solutions is reduced to a family of finite dimensional eigenvalue problems in one study no two interface ring solutions are found by the convergence method if the sample is small a stable two interface ring solution exists if the sample size is increased it becomes unstable if the sample size is increased further

mixed hybrid finite element approximation of the potential fluid flow problem abstract in the paper a mixed hybrid approximation of the potential fluid flow problem based on prismatic discretization of the domain is presented trilateral prismatic elements with vertical faces and nonparallel bases suitable for the modelling of real geological circumstances are considered the set of linearly independent vector basis functions is defined and existence and uniqueness of the approximate solution from the resulting symmetric indefinite system are examined possible approaches to the solution of the discretized system are discussed

a variational method for second order shape derivatives we consider shape functionals obtained as minima on sobolev spaces of classical integrals having smooth and convex densities under mixed dirichlet neumann boundary conditions we propose a new approach for the computation of the second order shape derivative of such functionals yielding a general existence and representation theorem in particular we consider the p torsional rigidity functional for p geq 2

random models of scale free networks in this paper we study two models of network evolution and give more realistic descriptions of the local processes than barabasi albert model incorporating the addition of new nodes the addition of new links the rewiring and deleting of links to random networks model with single or double preferential attachment s we have proved that the two models are of scale free networks if the parameters are chosen properly and got the scaling exponent by the continuum theory

schur complement reduction in the mixed hybrid approximation of darcy s law rounding error analysis abstract mixed hybrid finite element approximation of the potential fluid flow problem leads to the solution of a large symmetric indefinite system for the velocity and potential head vector components such discretization gives rise to a very accurate approximation of the continuity equation in every element and for low order discretizations the structural properties of the discrete matrix blocks allow cheap block elimination of the positive definite diagonal block and subsequent reduction to the schur complement system for the pressure and lagrangian vector components this system is then frequently solved by the iterative conjugate gradient type method whereas this approach is well known considerably less attention has been paid to the numerical stability aspects of such transformation it was shown in 5 that block lu factorization can be unstable even when the system matrix is symmetric positive definite in this paper we examine this type of conditional stability for a particular application in the underground water flow modelling we show that the actual error of the computed approximate solution depends not only on the user defined tolerance in the conjugate gradient process but also on the spectral properties of the corresponding matrix blocks eliminated during the schur complement reduction it is often observed that although the backward error of the approximate solution in the iterative part is reduced to the level of machine accuracy the total residual norm after the back substitution process remains at certain accuracy level we give a bound for this maximal attainable accuracy and illustrate our theoretical results on a model example

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

exact solution for the time evolution of network rewiring models we consider the rewiring of a bipartite graph using a mixture of random and preferential attachment the full mean field equations for the degree distribution and its generating function are given the exact solution of these equations for all finite parameter values at any time is found in terms of standard functions it is demonstrated that these solutions are an excellent fit to numerical simulations of the model we discuss the relationship between our model and several others in the literature including examples of urn backgammon and balls in boxes models the watts and strogatz rewiring problem and some models of zero range processes our model is also equivalent to those used in various applications including cultural transmission family name and gene frequencies glasses and wealth distributions finally some voter models and an example of a minority game also show features described by our model

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

higher dimensional localized patterns in excitable media abstract the excitable reaction diffusion equation model of the form e u t e 2 2 u u v v t 2 v u v is considered when u is assumed to be of mckean s piecewise linear type the interfacial approach can be applied to the stability of various localized patterns in higher dimensional spaces it is shown that a band shaped localized pattern is destabilized into a zig zag mode or a varicose mode and that a disk shaped localized pattern is destabilized into static modes and when is small into an oscillatory mode like a breather motion numerical simulations are performed to confirm such destabilizations for more general nonlinear functions u

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

on an isoperimetric problem with power law potentials and external attraction abstract in this paper an isoperimetric problem with power law potentials and external attraction is considered using concentration compactness lemma for sets of finite perimeter which was introduced by frank and lieb we prove the existence of minimizers for the isoperimetric problem in the special case of quadratic attraction we also obtain that the ball is the unique minimizer of the isoperimetric problem

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

droplet minimizers of an isoperimetric problem with long range interactions we give a detailed description of the geometry of single droplet patterns in a non local isoperimetric problem in particular we focus on the sharp interface limit of the ohta kawasaki free energy for diblock copolymers regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long range interactions exploiting fine properties of the regularity theory for minimal surfaces we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns where u is the order parameter of a two phases system confined in r n and and m are two nonnegative numerical parameters the two terms in the energy mimic attractive short range and repulsive long range energies between the phases more precisely the first term is local it favors minimal interface area and drives the system toward a partition into few pure phases while the second term involving a coulomb like kernel g is non local and favor a fine mixing of the phases a detailed description of the energy is given in 1 the competition between these two terms is expected to induce the formation of highly regular mesoscopic patterns whose geometry strongly depends on the choice of the parameters and m e g spherical spots cylinders gyroids and lamellae 0 1 the ohta kawasaki functional for diblock copolymers the model we consider arises as a simplification of a ginzburg landau functional proposed by ohta and kawasaki in their pioneering paper 41 as a possible description of a diblock copolymers dbc system even though it is questionable whether such an energy actually describes dbcs see choksi and ren 15 muratov 39 and niethammer and oshita 40 nevertheless it is a first and mathematically non trivial attempt to capture some of the main features of these systems for such a reason it deserved over the last twenty years great attention from both the mathematical and the physical

nonexistence of a minimizer for thomas fermi dirac von weizsacker model 1 x y dx dy a nonexistence result similar to theorem 1 also holds theorem 2 there exists constant v0 0 such that the variational problem 4 inf v e does not have a minimizer if v v0 if the coulomb term is not present in 1 and 3 a standard modica mortela type result establishes a link between the two functional through gamma convergence when the volume constraint m is large this is no longer valid for the energy functionals 1 and 3 due to the coulomb term the connection between the two functionals is just on the formal level

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

stability of the static spike autosolitons in the gray scott model we performed an asymptotic linear stability analysis of the static spike autosolitons ass self sustained solitary inhomogeneous states in the gray scott model of an autocatalytic chemical reaction we found that in one dimension these ass destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough in higher dimensions the one dimensional static spike ass are always unstable with respect to corrugation and wriggling the higher dimensional radially symmetric static spike ass may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow

quantitative minimality of strictly stable extremal submanifolds in a flat neighbourhood abstract in this paper we extend the results of a strong minimax property of nondegenerate minimal submanifolds by white where it is proved that any smooth compact submanifold which is a strictly stable critical point for an elliptic parametric functional is the unique minimizer in a certain geodesic tubular neighbourhood we prove a similar result replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates our proof is based on the introduction of a penalized minimization problem in the spirit of a selection principle for the sharp quantitative isoperimetric inequality by cicalese and leonardi which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands

a selection principle for the sharp quantitative isoperimetric inequality we introduce a new variational method for the study of isoperimetric inequalities with quantitative terms the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter two notable applications are presented first we give a new proof of the sharp quantitative isoperimetric inequality in mathbb r n second we positively answer a conjecture by hall concerning the best constant for the quantitative isoperimetric inequality in mathbb r 2 in the small asymmetry regime

stability in the isoperimetric problem for convex or nearly spherical domains in ⁿ for convex bodies d in rn the deviation d from spherical shape is estimated from above in terms of the dimensionless isoperimetric deficiency a of d as follows d f a for a sufficiently small here f is an explicit elementary function vanishing continuously at 0 the estimate is sharp as regards the order of magnitude of f the dimensions n 2 and 3 present anomalies as to the form of f in the planar case n 2 the result is contained in an inequality due to t bonnesen a qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies d in rn in the sense that as d varies d p 0 for a a 0 the proof of the estimate d f a is based on a related estimate in the case of domains not necessarily convex that are supposed a priori to be nearly spherical in a certain sense

network evolution by different rewiring schemes many real world networks such as social networks are characterized by rearrangements of the links between nodes rewiring indeed very few natural networks are static in time and it is therefore important to study the properties of networks in which rewiring occurs in this paper two different rewiring schemes are formulated and compared using a general ordinary differential equation ode model the equilibrium distributions are analytically derived it is found that by uniformly choosing a node and a link connected to it rewiring from different ends of the link yields different equilibrium degree distributions rewiring from the neighbor generally produces more high degree nodes the equilibrium distributions of the ode model are compared with simulation results of the corresponding stochastic process for rewiring conditions are discussed under which our ode provides a good approximation for the mean of the corresponding stochastic process

on the phase diagram for microphase separation of diblock copolymers an approach via a nonlocal cahn hilliard functional we consider analytical and numerical aspects of the phase diagram for microphase separation of diblock copolymers our approach is variational and is based upon a density functional theory which entails minimization of a nonlocal cahn hilliard functional based upon two parameters which characterize the phase diagram we give a preliminary analysis of the phase plane that is we divide the plane into regions wherein a combination of analysis and numerics is used to describe minimizers in particular we identify a regime wherein the uniform disordered state is the unique global minimizer a regime wherein the constant state is linearly unstable and where numerical simulations are currently the only tool for characterizing the phase geometry and a regime of small volume fraction wherein we conjecture that small well separated approximately spherical objects are the unique global minimizer for this last regime we present an asymptotic analysis from the point of view of the energetics which will be comp

comments on scale free networks without growth in y b xie t zhou b h wang scale free networks without growth physica a 387 2008 1683 1688 a nongrowing scale free network model has been introduced which shows that the degree distribution of the model varies from the power law form to the poisson form as the free parameter increases and indicates that the growth may not be necessary for a scale free network structure to emerge however the model implicitly assumes that self loops and multiple links are allowed in the model and counted in the degree distribution in many real life networks such an assumption may not be reasonable we showed here that the degree distribution of the emergent network does not obey a power law form if self loops and multiple links are allowed in the model but not counted in the degree distribution we also observed the same result when self loops and multiple links are not allowed in the model furthermore we showed that the effect of self loops and multiple links on the degree distribution weakens as increases and even becomes negligible when is sufficiently large

study of island formation in epitaxially strained films on unbounded domains we consider a variational model related to the formation of islands in heteroepitaxial growth on unbounded domains we first derive the scaling regimes of the minimal energy in terms of the volume of the film and the amplitude of the crystallographic misfit for small volumes non existence of minimizers is then proven this corresponds to the experimentally observed wetting effect on the other hand we show the existence of minimizers for large volumes we finally study the asymptotic behavior of the optimal shapes