integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity this article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices under polynomial growth assumptions we prove that the energy functionals f varepsilon stored in the deformation of an varepsilon scaling of a stochastic lattice converge to a continuous energy functional when varepsilon goes to zero in particular the limiting energy functional is of integral type and deterministic if the lattice is ergodic we also generalize to systems and nonlinear settings well known results on stochastic homogenization of discrete elliptic equations as an application of the main result we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics we finally address some mechanical properties of the limiting models such as frame invariance isotropy and natural states

optimality of the triangular lattice for a particle system with wasserstein interaction we prove strong crystallization results in two dimensions for an energy that arises in the theory of block copolymers the energy is defined on sets of points and their weights or equivalently on the set of atomic measures it consists of two terms the first term is the sum of the square root of the weights and the second is the quadratic optimal transport cost between the atomic measure and the lebesgue measure

discrete breathers in a two dimensional fermi pasta ulam lattice using asymptotic methods we investigate whether discrete breathers are supported by a two dimensional fermi pasta ulam lattice a scalar one component two dimensional fermi pasta ulam lattice is shown to model the charge stored within an electrical transmission lattice a third order multiple scale analysis in the semi discrete limit fails since at this order the lattice equations reduce to the 2 1 dimensional cubic nonlinear schrodinger nls equation which does not support stable soliton solutions for the breather envelope we therefore extend the analysis to higher order and find a generalised 2 1 dimensional nls equation which incorporates higher order dispersive and nonlinear terms as perturbations we find an ellipticity criterion for the wave numbers of the carrier wave numerical simulations suggest that both stationary and moving breathers are supported by the system calculations of the energy show the expected threshold behaviour whereby the energy of breathers does em not go to zero with the amplitude we find that the energy threshold is maximised by stationary breathers and becomes arbitrarily small as the boundary of the domain of ellipticity is approached

harnack inequalities and discrete continuous error estimates for a chain of atoms with two body interactions we consider deformations in r 3 of an infinite linear chain of atoms where each atom interacts with all others through a two body potential we compute the effect of an external force applied to the chain at equilibrium the positions of the particles satisfy an euler lagrange equation for large classes of potentials we prove that every solution is well approximated by the solution of a continuous model we establish an error estimate between the discrete and the continuous solution based on a harnack lemma of indepen dent interest finally we apply our results to some lennard jones potentials

higher dimensional coulomb gases and renormalized energy functionals we consider a classical system of n charged particles in an external confining potential in any dimension d larger than 2 the particles interact via pairwise repulsive coulomb forces and the coupling parameter scales like the inverse of n mean field scaling by a suitable splitting of the hamiltonian we extract the next to leading order term in the ground state energy beyond the mean field limit we show that this next order term which characterizes the fluctuations of the system is governed by a new renormalized energy functional providing a way to compute the total coulomb energy of a jellium i e an infinite set of point charges screened by a uniform neutralizing background in any dimension the renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale as in onsager s lemma we obtain consequences for the statistical mechanics of the coulomb gas next to leading order asymptotic expansion of the free energy or partition function characterizations of the gibbs measures estimates on the local charge fluctuations and factorization estimates for reduced densities this extends results of sandier and serfaty to dimension higher than two by an alternative approach

combined breathing kink modes in the fpu lattice abstract combined breather kink modes have been observed in several systems here we show that in the fpu lattice small amplitude breathing kink modes are three soliton solutions of an associated mkdv equation since this is integrable the modes can be constructed using the backlund transform as well as finding explicit solutions we consider the stability of the combined mode using variational techniques and illustrate stability for some parameter values and instability for others

regularity of local minimizers of the interaction energy via obstacle problems the repulsion strength at the origin for repulsive attractive potentials determines the regularity of local minimizers of the interaction energy in this paper we show that if this repulsion is like newtonian or more singular than newtonian but still locally integrable then the local minimizers must be locally bounded densities and even continuous for more singular than newtonian repulsion we prove this and some other regularity results by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems

counterintuitive ground states in soft core models it is well known that statistical mechanics systems exhibit subtle behavior in high dimensions in this paper we show that certain natural soft core models such as the gaussian core model have unexpectedly complex ground states even in relatively low dimensions specifically we disprove a conjecture of torquato and stillinger who predicted that dilute ground states of the gaussian core model in dimensions 2 through 8 would be bravais lattices we show that in dimensions 5 and 7 there are in fact lower energy non bravais lattices the nearest three dimensional analog is the hexagonal close packing but it has higher energy than the face centered cubic lattice we believe these phenomena are in fact quite widespread and we relate them to decorrelation in high dimensions

approximations in hypergroups and fuzzy hypergroups the first part of this paper represents a continuation of the study of approximations in a hypergroup we analyse the lower and upper approximations of a subset in a hypergroup with respect to some operators particularly the completeness operator in the last part we introduce and analyse the lower and upper approximation of a subset in a fuzzy hypergroup

applications of interval valued fuzzy n ary polygroups with respect to t norms t conorms using the triangular norm t conorm s in the context of n ary polygroups we introduce the concept of an interval valued anti fuzzy n ary subpolygroup with respect to t s respectively a necessary and sufficient condition for an interval valued fuzzy subset in order to be an interval valued anti fuzzy n ary subpolygroup is established and some important results are presented this study is useful for the most computational part of fuzzy control which is the defuzzification

fuzzy hyperrings in this paper we introduce and analyze the fuzzy hyperring notion we consider and study the subfuzzy structures of a such notion and homomorphisms between fuzzy hyperrings finally fundamental relations on fuzzy hyperrings are analyzed in all the paper we follow a connection between hyperrings and fuzzy hyperrings

stationary breather modes of generalized nonlinear klein gordon lattices this paper studies the form of stationary breather modes in discrete generalized nonlinear klein gordon equations with symmetric and non symmetric potential energy functions in the case of static breathers the discrete nature of the spatial dimension has a much more subtle effect on the breather than the moving breather mode this effect is analysed using a variety of approximating partial differential equations whose solutions are found by using an extended multiple scales asymptotic approach to reduce the equation to a nonlinear schrodinger equation at leading order and more complex equations at higher order where secularity conditions are required to fully specify the solution as well as the much studied discrete sine gordon equation the methods are demonstrated on a discrete nonlinear klein gordon equation with second neighbour interactions and non symmetric on site potential new partial differential equations which approximate these lattice systems are also proposed and analysed

the energy of some microscopic stochastic lattices we introduce a notion of energy for some microscopic stochastic lattices such lattices are broad generalizations of simple periodic lattices for which the question of the definition of an energy was examined in a series of previous works 14 18 note that slightly more general deterministic geometries were also considered in 6 these lattices are involved in the modelling of materials whose microscopic structure is a perturbation in a sense made precise in the article of the periodic structure of a perfect crystal the modelling considered here is either a classical modelling where the sites of the lattice are occupied by ball like atomic systems that interact by pair potentials or a quantum modelling where the sites are occupied by nuclei equipped with an electronic structure spread all over the ambient space the corresponding energies for the infinite stochastic lattices are derived consistently with truncated systems of finite size by application of a thermodynamic limit process subsequent works 7 8 will be devoted to the macroscopic limits of the energies of such microscopic lattices thereby extending to a stochastic context the results of 4 5 such convergences in a stochastic setting in dimension 1 have been studied in 21 22 we will also study in 8 some variants and extensions of the stationary setting presented here

hydrodynamic limit for an evolutional model of two dimensional young diagrams we construct dynamics of two dimensional young diagrams which are naturally associated with their grandcanonical ensembles by allowing the creation and annihilation of unit squares located at the boundary of the diagrams the grandcanonical ensembles which were introduced by vershik 17 are uniform measures under conditioning on their size or equivalently area we then show that as the averaged size of the diagrams diverges the corresponding height variable converges to a solution of a certain non linear partial differential equation under a proper hydrodynamic scaling furthermore the stationary solution of the limit equation is identified with the so called vershik curve we discuss both uniform and restricted uniform statistics for the young diagrams

fuzzy h v ideals in γ h v rings in this paper by means of a kind of new idea we consider the q fuzzy h v ideals of a c h v ring the concept of c h v rings which is a generalization of c rings and h v rings is introduced the notion of q fuzzy h v ideals of a c h v ring is provided and some related properties are investigated the concept of implication based fuzzy h v ideals of a c h v ring is studied in particular the implication operators in the lukasiewicz system of continuous valued logic are discussed finally the homomorphism of q fuzzy h v ideals of a c h v ring is explored

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

fuzzy h v groups in this paper we introduce the concept of fuzzy subhypergroup of a hypergroup as well as the fuzzy hv group we prove some results in this respect we also consider the fundamental relation defined on an hv group h and for a fuzzy subset of h we define the fuzzy subset of h and prove a fundamental theorem concerning the group h

fuzzy hypervector spaces based on fuzzy singletons we introduce and study fuzzy hypervector spaces based on fuzzy singletons in this regard by considering the notion of fuzzy singletons we characterize a fuzzy hypervector space fuzzily spanned by a fuzzy subset then we use these results to introduce the concept of fuzzy freeness of a fuzzy subset m of a hypervector space v and finally we characterize it in terms of linear independence in the usual sense

a new view of fuzzy hypernear rings in this survey article we first introduce the concept of quasi coincidence of a fuzzy interval value with an interval valued fuzzy set this concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we consider the interval valued q fuzzy sub hypernear rings hyperideals of a hypernear ring and hence a generalization of a fuzzy sub near ring ideal is given some related properties of fuzzy hypernear rings are described finally we consider the concept of implication based interval valued fuzzy sub hypernear rings hyperideals in a hypernear ring in particular the implication operators in lukasiewicz system of continuous valued logic are discussed

froth like minimizers of a non local free energy functional with competing interactions we investigate the ground and low energy states of a one dimensional non local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions in particular the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one the competition between these two effects is expected to lead to the spontaneous emergence of a regular alternation of long intervals on which the spin profile is magnetized either up or down with an oscillation scale intermediate between the range of the ferromagnetic and that of the antiferromagnetic interaction in this sense the optimal or quasi optimal profiles are froth like if seen on the scale of the antiferromagnetic potential they look neutral but if seen at the microscope they actually consist of big bubbles of two different phases alternating among each other in this paper we prove the validity of this picture we compute the oscillation scale of the quasi optimal profiles and we quantify their distance in norm from a reference periodic profile the proof consists of two main steps we first coarse grain the system on a scale intermediate between the range of the ferromagnetic potential and the expected optimal oscillation scale in this way we reduce the original functional to an effective sharp interface one next we study the latter by reflection positivity methods which require as a key ingredient the exact locality of the short range term our proof has the conceptual interest of combining coarse graining with reflection positivity methods an idea that is presumably useful in much more general contexts than the one studied here

a proof of crystallization in two dimensions many materials have a crystalline phase at low temperatures the simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type where y x 2 is the position of particle x and v r is the pair interaction energy of two particles which are placed at distance r due to the mermin wagner theorem it can t be expected that at finite temperature this system exhibits long range ordering we focus on the zero temperature case and show rigorously that under suitable assumptions on the potential v which are compatible with the growth behavior of the lennard jones potential the ground state energy per particle converges to an explicit constant e where e is the minimum of a simple function on 0 furthermore if suitable dirichlet or periodic boundary conditions are used then the minimizers form a triangular lattice to the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the euclidean symmetry group consisting of rotations and translations

the lower and upper approximations in a hypergroup this paper presents a relationship between rough sets and hypergroup theory we analyze the lower and upper approximations of a subset with respect to an invertible subhypergroup and we consider some particular situations moreover the notion of a rough subhypergroup is introduced finally fuzzy rough subhypergroups are introduced and characterized

ground states of the 2d sticky disc model fine properties and n 3 4 law for the deviation from the asymptotic wulff shape we investigate ground state configurations for a general finite number n of parti cles of the heitmann radin sticky disc pair potential model in two dimensions exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic wulff shape which is a regular hexagon there are arbitrarily large n with ground state configura tions deviating from the nearest regular hexagon by a number of n 3 4 particles we also prove that for any n and any ground state configuration this deviation is bounded above by n 3 4 as a consequence we obtain an exact scaling law for the fluctuations about the asymptotic wulff shape in particular our results give a sharp rate of convergence to the limiting wulff shape

high order breather solutions to a discrete nonlinear klein gordon model abstract a general discrete nonlinear klein gordon model is studied with regards to the movability of breathers approximations to moving breather solutions are found by a semi discrete multiple scale perturbation expansion to 3rd order the higher order effects of discreteness have a remarkable connection to nonlinear fiber optics effects such as higher order dispersion self steepening of the pulse edge and self frequency shift are found modifying both the shape velocity and frequency of the solution here only their combined influence is studied discreteness is found to trap breathers with initial amplitudes higher than a critical value which depends on the internal state of the breather this shows that no strict definition of the peierls nabarro potential can be given for breathers furthermore we find that the envelope velocity of the breather cannot exceed the maximum group velocity which converges to zero as the degree of discreteness is increased thus no moving breathers exist in a highly discrete system

redefined fuzzy hv submodules and many valued implications this paper considers a relationship between fuzzy sets and algebraic hyperstructures a generalization of ideas presented by davvaz b davvaz fuzzy h v submodules fuzzy sets syst 117 2001 477 484 and bhakat and das s k bhakat p das email protected q fuzzy subgroup fuzzy sets syst 80 1996 359 368 is presented in fact the main purpose of this paper is to introduce and study a new sort of fuzzy h v submodule of an h v module called email protected q fuzzy h v submodules these fuzzy h v submodules are characterized by their level h v submodules the concept of a fuzzy h v submodule with thresholds is introduced and some interesting properties are investigated

fuzzy krasner m n hyperrings recently krasner m n hyperrings were introduced and analyzed by mirvakili and davvaz krasner m n hyperrings are a suitable generalization of krasner hyperrings this paper concerns a relationship between fuzzy sets and hyperstructure theory it is a continuation of the ideas presented by davvaz et al 2009 b davvaz p corsini v leoreanu fotea fuzzy n ary subpolygroups computers mathematics with applications 57 2009 141 152 the aim of the paper is to introduce the notion of a fuzzy hyperideal of a krasner m n hyperring and to extend the fuzzy results to krasner m n hyperrings

on probabilistic n ary hypergroups abstract the t fuzzy n ary subhypergroups of an n ary hypergroup are defined by using triangular norms and some related properties are hence obtained in particular we consider the probabilistic version of n ary hypergroups by using random sets and show that the fuzzy n ary hypergroups defined by triangular norms are consequences of some probabilistic n ary hypergroups under certain conditions some results on n ary hypergroups recently given by davvaz and corsini are extended

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

asymptotic analysis of combined breather kink modes in a fermi pasta ulam chain we find approximations to travelling breather solutions of the one dimensional fermi pasta ulam fpu lattice both bright breather and dark breather solutions are found we find that the existence of localised bright solutions depends upon the coefficients of cubic and quartic terms of the potential energy generalising an earlier inequality derived by james cr acad sci paris 332 581 2001 we use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear schr o dinger equation at leading order and hence construct an asymptotic form for the breather we show that in the absence of a cubic potential energy term the lattice supports combined breathing kink waveforms the amplitude of breathing kinks can be arbitrarily small as opposed to traditional monotone kinks which have a nonzero minimum amplitude in such systems we also present numerical simulations of the lattice verifying the shape and velocity of the travelling waveforms and confirming the long lived nature of all such modes

fuzzy hypermodules the fuzzy hypermodule notion is introduced and studied subfuzzy structures of such a notion as well as homomorphism between fuzzy hypermodules and fuzzy quotient hypermodules are analyzed

construction of n ary f polygroups by an n ary f hyperoperation on a non empty set a we mean a function from a a to the set of all non zero fuzzy subsets of a where a appears n times using this concept we introduce the notion of n ary f polygroups which is a generalization of ideas presented by zahedi hasankhani and davvaz et al some basic results are obtained

phase transitions for dilute particle systems with lennard jones potential we consider a classical dilute particle system in a large box with pair interaction given by a lennard jones type potential the inverse temperature is picked proportionally to the logarithm of the particle density we identify the free energy per particle in terms of a variational formula and show that this formula exhibits a cascade of phase transitions as the temperature parameter ranges from zero to infinity loosely speaking the particle system separates into spatially distant components in such a way that within each phase all components are of the same size which is the larger the lower the temperature the main tool in our proof is a new large deviation principle for sparse point configurations

generalized fuzzy hyperideals of hyperrings the concept of quasi coincidence of a fuzzy interval value with an interval valued fuzzy set is considered in fact this is a generalized concept of the quasi coincidence of a fuzzy point with a fuzzy set by using this new idea the notion of interval valued a b fuzzy hyperideals in a hyperring is introduced and consequently a generalization of a fuzzy hyperideal is defined in this paper we study the related properties of the fuzzy hyperideals and in particular the interval valued q fuzzy hyperideals in a hyperring will be investigated moreover we also consider the concept of implication based interval valued fuzzy hyperideals in a hyperring some recent results obtained by b davvaz and j m zhan on hyperstructures are extended and consolidated

the lower and upper approximations in a quotient hypermodule with respect to fuzzy sets this paper provides a continuation of ideas presented by davvaz and mahdavipour b davvaz m mahdavipour roughness in modules inform sci 176 2006 3658 3674 the notion of hypermodule is a generalization of the notion of module in this paper we consider the quotient hypermodule m a and interpret the lower and upper approximations as subsets of the quotient hypermodule m a then we introduce the concept of quotient rough sub hypermodule also using the concept of fuzzy sets we introduce and discuss the concept of fuzzy rough hypermodules and then we obtain the relation between fuzzy rough sub hypermodules and level rough sets this relation is characterized as a necessary and sufficient condition

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

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

fuzzy hypergroups based on fuzzy relations based on fuzzy reasoning in fuzzy logic this paper studies a fuzzy hyperoperation and a fuzzy hypergroupoid associated with a fuzzy relation a sufficient and necessary condition for such a fuzzy hypergroupoid being a fuzzy hypergroup is given and the properties of the fuzzy hypergroups associated with fuzzy relations are investigated furthermore the definition of normal fuzzy hypergroups is put forward and it is shown that the category nfhg of normal fuzzy hypergroups satisfies all the axioms of topos except for the subobject classifier axiom

atomic to continuum passage for nanotubes a discrete saint venant principle and error estimates we consider general infinite nanotubes of atoms in mathbb r 3 where each atom interacts with all the others through a two body potential at the equilibrium the positions of the atoms satisfy a euler lagrange equation when there are no exterior forces and for a suitable geometry a particular family of nanotubes is the set of perfect nanotubes at the equilibrium when exterior forces are applied on the nanotube we compare the nanotube to nanotubes of the previous family in part i of the paper this quantitative comparison is formulated in our first main result as a discrete saint venant principle as a corollary we also give a liouville classification result our saint venant principle can be derived for a large class of potentials including the lennard jones potential when the perfect nanotubes at the equilibrium are stable the approach is designed to be applicable to nanotubes that can have general shapes like for instance carbon nanotubes or dna under the oversimplified assumption that all the atoms are identical in part ii of the paper we derive from our saint venant principle a macroscopic mechanical model for general nanotubes we prove that every solution is well approximated by the solution of a continuum model involving stretching and twisting but no bending we establish error estimates between the discrete and the continuous solution more precisely we give two error estimates one at the microscopic level and one at the macroscopic level

hexagonal patterns in a simplified model for block copolymers in this paper we study a new model for patterns in two dimensions inspired by diblock copolymer melts with a dominant phase the model is simple enough to be amenable not only to numerics but also to analysis yet sophisticated enough to reproduce hexagonally packed structures that resemble the cylinder patterns observed in block copolymer experiments starting from a sharp interface continuum model a nonlocal energy functional involving a wasserstein cost we derive the new model using gamma convergence in a limit where the volume fraction of one phase tends to zero the limit energy is defined on atomic measures in three dimensions the atoms represent small spherical blobs of the minority phase and in two dimensions they represent thin cylinders of the minority phase we then study local minimizers of the limit energy numerical minimization is performed in two dimensions by recasting the problem as a computational geometry problem involving power diagrams the numerical results suggest that the sma

a uniqueness result for minimizers of the 1d log gas renormalized energy abstract in 21 sandier and serfaty studied the one dimensional log gas model in particular they gave a crystallization result by showing that the one dimensional lattice z is a minimizer for the so called renormalized energy which they obtained as a limit of the n particle log gas hamiltonian for n however this minimizer is not unique among infinite point configurations for example local perturbations of z leave the renormalized energy unchanged in this paper we establish that uniqueness holds at the level of stationary point processes the only minimizer being given by averaging z over a choice of the origin in 0 1 this is proved by showing a quantitative estimate on the two point correlation function of a process in terms of its renormalized energy

fuzzy subgroups and anti fuzzy subgroups abstract we introduce the concept of anti fuzzy subgroups of groups lower level subset of a fuzzy subset lower level subgroups and prove some results on these we show that a fuzzy subset of a group is a fuzzy subgroup iff the complement of this fuzzy subset is anti fuzzy subgroup we fuzzify lower level subsets and prove that if a fuzzy set is an anti fuzzy subgroup then the fuzzifications of its lower level subsets are also anti fuzzy subgroups

new integral estimates for deformations in terms of their nonlinear strains if u is a bi lipschitzian deformation of a bounded lipschitz domain in ln n 2 we show that the lp norm p 1 p n of a certain nonlinear strain function e u associated with u dominates the distance in lq q np n p if p n from u to a suitably chosen rigid motion of n this work extends that of f john who proved corresponding estimates for p 1 under the hypothesis that u has uniformly small strain we also obtain a bound for the oscillation of du in l2 these estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u in 3 the integral int limits omega e u 2d 3 is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber thus the case n 3 p 2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work

equivalence of ensembles under inhomogeneous conditioning and its applications to random young diagrams we prove the equivalence of ensembles or a realization of the local equilibrium for bernoulli measures on mathbb z conditioned on two conserved quantities under the situation that one of them is spatially inhomogeneous for the proof we extend the classical local limit theorem for a sum of bernoulli independent sequences to those multiplied by linearly growing weights the motivation comes from the study of random young diagrams and their evolutional models which were originally suggested by herbert spohn we discuss the relation between our result and the so called vershik curve which appears in a scaling limit for height functions of two dimensional young diagrams we also discuss a related random dynamics

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

bounds on the geometric complexity of optimal centroidal voronoi tesselations in 3d gersho s conjecture in 3d asserts the asymptotic periodicity and structure of the optimal centroidal voronoi tessellation this relatively simple crystallization problem remains to date open we prove bounds on the geometric complexity of optimal centroidal voronoi tessellations as the number of generators tends to infinity combined with an approach of gruber in 2d these bounds reduce the resolution of the 3d gersho s conjecture to a finite albeit very large computation of an explicit convex problem in finitely many variables

existence of compactly supported global minimisers for the interaction energy the existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses the main assumption on the potential is that it is catastrophic or not h stable which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics the proof is based on a uniform control on the local mass around each point of the support of a global minimiser together with an estimate on the size of the gaps it may have the class of potentials for which we prove the existence of global minimisers includes power law potentials and for some range of parameters morse potentials widely used in applications we also show that the support of local minimisers is compact under suitable assumptions

ground states and formal duality relations in the gaussian core model we study dimensional trends in ground states for soft matter systems specifically using a high dimensional version of parrinello rahman dynamics we investigate the behavior of the gaussian core model in up to eight dimensions the results include unexpected geometric structures with surprising anisotropy as well as formal duality relations these duality relations suggest that the gaussian core model possesses unexplored symmetries and they have implications for a broad range of soft core potentials

a microscopic justification of the wulff construction we report results about a rigorous microscopic justification of the wulff construction for the two dimensional ising model at low temperatures and under periodic boundary conditions the idea of the proof is sketched

discrete breathers in a two dimensional hexagonal fermi pasta ulam lattice we consider a two dimensional fermi pasta ulam fpu lattice with hexagonal symmetry using asymptotic methods based on small amplitude ansatz at third order we obtain a eduction to a cubic nonlinear schr o dinger equation nls for the breather envelope however this does not support stable soliton solutions so we pursue a higher order analysis yielding a generalised nls which includes known stabilising terms we present numerical results which suggest that long lived stationary and moving breathers are supported by the lattice we find breather solutions which move in an arbitrary direction an ellipticity criterion for the wavenumbers of the carrier wave symptotic estimates for the breather energy and a minimum threshold energy below which breathers cannot be found this energy threshold is maximised for stationary breathers and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed several of the results obtained are similar to those obtained for the square fpu lattice butt wattis em j phys a bf 39 4955 2006 though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics and the isotropy of the generalised nls equation

ground states and hyperuniformity of the hierarchical coulomb gas in all dimensions stochastic point processes with coulomb interactions arise in various natural examples of statistical mechanics random matrices and optimization problems often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties that is the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i i d random points a well known conjecture from physics appearing in the works of jancovici lebowitz manificat martin and yalcin see jancovici et al in j stat phys 72 3 4 773 787 1993 lebowitz in phys rev a 27 1491 1494 1983 martin and yalcin in j stat phys 22 4 435 463 1980 states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i i d random points in a recent beautiful work chatterjee in rigidity of the three dimensional hierarchical coulomb gas 2017 arxiv preprint arxiv 1708 01965 to appear in probability theory and related fields chatterjee gave the first proof of such a result in dimension three for a coulomb type system known as the hierarchical coulomb gas inspired by dyson s hierarchical model of the ising ferromagnet dyson in phys rev 92 1331 1338 1953 dyson in commun math phys 12 2 91 107 1969 however the case of dimensions greater than three had remained open in this paper we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three for the hierarchical model using similar methods we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two a key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems and could be of independent interest

from the ginzburg landau model to vortex lattice problems we introduce a coulombian renormalized energy w which is a logarithmic type of interaction between points in the plane computed by a renormalization we prove various of its properties such as the existence of minimizers and show in particular using results from number theory that among lattice configurations the triangular lattice is the unique minimizer its minimization in general remains open

fuzzy hv submodules abstract in this paper we define the concept of fuzzy h v submodule of an h v module and study a few results in this respect we also consider the fundamental relation e defined on an h v module m and prove a fundamental theorem concerning the module m e

fuzzy vector spaces and fuzzy cosets abstract a fuzzy subspace is characterized by its level subspaces the algebraic nature of fuzzy subspaces under homomorphisms is studied certain conditions are obtained under which a given fuzzy subspace can or cannot be realized as a union of two fuzzy subspaces such that none is contained in the other the construction of a fuzzy subspace generated by a given fuzzy subset of a vector space is given the set of all the fuzzy cosets of a fuzzy subspace is shown to be a vector space and the dimension of the same is obtained a fuzzy quotient space is defined and an analogue of a consequence of the fundamental theorem of homomorphisms is obtained

approximations to solitary waves on lattices ii quasi continuum methods for fast and slow waves a number of highly accurate analytic approximations to solitary waves on lattices are detailed and the results of numerical tests presented two types of approximation are considered the first set of approximations are generalizations of the continuum approximation based around various pade expansions of the operator these are convergent to the exact solution in the limit of wavespeed c to 1 or a dense lattice the author also considers ways of constructing approximations to fast solitary waves from a continuum type theory the two types are closely related in both cases significant improvements are made on existing approximations a symplectic hamiltonian integration scheme is used to perform computer simulations a new method of measuring the accuracy of a predicted waveform is also used on new and old approximations alike to compare the various methods considered

aggregation patterns from nonlocal interactions discrete stochastic and continuum modeling conservation equations governed by a nonlocal interaction potential generate aggregates from an initial uniform distribution of particles we address the evolution and formation of these aggregating steady states when the interaction potential has both attractive and repulsive singularities currently no existence theory for such potentials is available we develop and compare two complementary solution methods a continuous pseudoinverse method and a discrete stochastic lattice approach and formally show a connection between the two interesting aggregation patterns involving multiple peaks for a simple doubly singular attractive repulsive potential are determined for a swarming morse potential characteristic slow fast dynamics in the scaled inverse energy is observed in the evolution to steady state in both the continuous and discrete approaches the discrete approach is found to be remarkably robust to modifications in movement rules related to the potential function the comparable evolution dynamics and steady states of the discrete model with the continuum model suggest that the discrete stochastic approach is a promising way of probing aggregation patterns arising from two and three dimensional nonlocal interaction conservation equations

existence of ground states of nonlocal interaction energies we investigate which nonlocal interaction energies have a ground state global minimizer we consider this question over the space of probability measures and establish a sharp condition for the existence of ground states we show that this condition is closely related to the notion of stability i e h stability of pairwise interaction potentials our approach uses the direct method of the calculus of variations

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

fuzzy n ary subpolygroups recently n ary hypergroups were introduced and analyzed by davvaz and vougiouklis n ary polygroups are suitable generalizations of polygroups and a special case of n ary hypergroups the aim of this paper is to introduce the notion of a fuzzy n ary subpolygroup of an n ary polygroup and to extend the fuzzy results of fundamental equivalence relations to n ary polygroups

on the crystallization of 2d hexagonal lattices it is a fundamental problem to understand why solids form crystals at zero temperature and how atomic interaction determines the particular crystal structure that a material selects in this paper we focus on the zero temperature case and consider a class of atomic potentials v v 2 v 3 where v 2 is a pair potential of lennard jones type and v 3 is a three body potential of stillinger weber type for this class of potentials we prove that the ground state energy per particle converges to a finite value as the number of particles tends to infinity this value is given by the corresponding value for a optimal hexagonal lattice optimized with respect to the lattice spacing furthermore under suitable periodic or dirichlet boundary condition we show that the minimizers do form a hexagonal lattice

fuzzy hyperrings hv rings based on fuzzy universal sets the aim of this paper is to define and study the concept of a fuzzy hyperring which depends on the concept of a fuzzy universal set a correspondence relation between a fuzzy hyperring based on a fuzzy universal set and both ordinary and fuzzy hyperideal in the sense of davvaz is obtained