on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

effect of boundary constraint on the frequency response of moderately thick doubly curved cross ply panels using mixed fourier solution functions abstract a hitherto unavailable analytical solution to the boundary value problem of free vibration response of shear flexible cross ply laminated doubly curved panels is presented the laminated shell theory formulation is based on the first order shear deformation theory fsdt including rotatory and surface parallel inertias the governing equations of the panel are defined by five highly coupled partial differential equations in five unknowns three displacements and two rotations the assumed solution functions for the eigen boundary value problem are selected in terms of mixed type double fourier series extensive numerical results that are presented in this study include 1 convergence characteristics of computed natural frequencies and 2 effects of length to thickness ratio radius to length ratio lamination sequence boundary constraint and shell geometry on the normalized natural frequencies of interest also numerically investigated is the highly complex interaction among bending stretching type coupling effect membrane action due to shell curvature surface parallel end constraints or lack thereof and the effects of transverse shear deformation rotatory inertias and surface parallel inertias

distance statistics in quadrangulations with no multiple edges and the geometry of minbus we present a detailed calculation of the distance dependent two point function for quadrangulations with no multiple edges various discrete observables measuring this two point function are computed and analyzed in the limit of large maps for large distances and in the scaling regime we recover the same universal scaling function as for general quadrangulations we then explore the geometry of lsquo minimal neck baby universes rsquo minbus which are the outgrowths to be removed from a general quadrangulation to transform it into a quadrangulation with no multiple edges the lsquo mother universe rsquo we give a number of distance dependent characterizations of minbus such as the two point function inside a minbu or the law for the distance from a random point to the mother universe

anomalous diffusion of random walk on random planar maps we prove that the simple random walk on the uniform infinite planar triangulation uipt typically travels graph distance at most n 1 4 o n 1 in n units of time together with the complementary lower bound proven by gwynne and miller 2017 this shows that the typical graph distance displacement of the walk after n steps is n 1 4 o n 1 as conjectured by benjamini and curien geom funct anal 2 2 501 531 2013 arxiv 1202 5454 more generally we show that the simple random walks on a certain family of random planar maps in the gamma liouville quantum gravity lqg universality class for gamma in 0 2 including spanning tree weighted maps bipolar oriented maps and mated crt maps typically travels graph distance n 1 d gamma o n 1 in n units of time where d gamma is the growth exponent for the volume of a metric ball on the map which was shown to exist and depend only on gamma by ding and gwynne commun math phys 374 1877 1934 2018 arxiv 1807 01072 since d gamma 2 this shows that the simple random walk on each of these maps is subdiffusive our proofs are based on an embedding of the random planar maps under consideration into mathbb c wherein graph distance balls can be compared to euclidean balls modulo subpolynomial errors this embedding arises from a coupling of the given random planar map with a mated crt map together with the relationship of the latter map to sle decorated lqg

a class of contractions in fuzzy metric spaces using the notion of geometrically convergent t norms a fixed point theorem in fuzzy metric spaces in the sense of kramosil and michalek for a class of contractions larger than the class of e l contraction mappings has been proved

subtree prune and regraft a reversible real tree valued markov process we use dirichlet form methods to construct and analyze a reversible markov process the stationary distribution of which is the brownian continuum random tree this process is inspired by the subtree prune and regraft spr markov chains that appear in phylogenetic analysis a key technical ingredient in this work is the use of a novel gromov hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure also the investigation of the dirichlet form hinges on a new path decomposition of the brownian excursion

invariance principles for random bipartite planar maps it is conjectured in the physics literature that properly rescaled random planar maps when conditioned to have a large number of faces should converge to a limiting surface whose law does not depend up to scaling factors on details of the class of maps that are sampled previous works on the topic starting with chassaing schaeffer have shown that the radius of a random quadrangulation with n faces converges in distribution once rescaled by n 1 4 to the diameter of the brownian snake up to a scaling constant using a bijection due to bouttier di francesco guitter between bipartite planar maps and a family of labeled trees we show the corresponding invariance principle for a class of random maps that follow a boltzmann distribution the radius of such maps conditioned to have n faces or n vertices and under a criticality assumption converges in distribution once rescaled by n 1 4 to a scaled version of the diameter of the brownian snake convergence results for the so called profile of maps are also provided as well as convergence of rescaled bipartite maps to the brownian map as introduced by marckert mokkadem the proofs of these results rely on a new invariance principle for two type spatial galton watson trees

on the roles of complementary and admissible boundary constraints in fourier solutions to the boundary value problems of completely coupled r th order pdes abstract a heretofore unavailable double fourier series based approach for obtaining non separable solution to a system of completely coupled linear r th order partial differential equations with constant coefficients and subjected to general completely coupled boundary conditions has been presented the method has been successfully implemented to solve a class of hitherto unsolved boundary value problems pertaining to free and forced vibrations of arbitrarily laminated anisotropic doubly curved thin panels of rectangular planform with arbitrarily prescribed both symmetric and asymmetric with respect to the panel centerlines admissible boundary conditions and subjected to general transverse loading existing solutions such as those due to navier or levy are based on the well known method of separation of variables such solutions represent particular solutions whenever the method of separation of variables work and when these particular solution functions fortuitously satisfy the boundary conditions for derivation of the complementary solution the complementary boundary constraints are introduced through boundary discontinuities of some of the particular solution functions and their partial derivatives such discontinuities form sets of measure zero various cases of lamination geometry and dynamic response forced and free vibrations of a class of thin anisotropic laminated shells curved panels have been shown to follow from the above six sets of boundary conditions are used to illustrate the present method for the derivation of complementary solutions navier type solutions whenever available form special cases of the present general solution

design of composite laminates for optimum frequency response in this study natural frequency response of symmetrically laminated composite plates was optimized an analytical model accounting for bending twisting effects was used to determine the laminate natural frequency two different problems fundamental frequency maximization and frequency separation maximization were considered fiber orientation angles were chosen as design variables because of the existence of numerous local optimums a global search algorithm a variant of simulated annealing was utilized to find the optimal designs results were obtained for different plate aspect ratios effects of the number of design variables and the range of values they may take on the optimal frequency were investigated problems in which fiber angles showed uncertainty were considered optimal frequency response of laminates subjected to static loads was also investigated

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

stochastic de giorgi iteration and regularity of stochastic partial differential equations under general conditions we devise a stochastic version of de giorgi iteration scheme for semilinear stochastic parabolic partial differential equation of the form tu div a u f t x u gi t x u w ti with progressively measurable diffusion coefficients we use the scheme to show that the solution of the equation is almost surely holder continuous in both space and time variables

local convergence of large random triangulations coupled with an ising model we prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size n decorated by an ising configuration with a weight proportional to the energy of this configuration to do so we establish the algebraicity and the asymptotic behaviour of the partition functions of triangulations with spins for any boundary condition in particular we show that these partition functions all have the same phase transition at the same critical temperature some properties of the limiting object called the infinite ising planar triangulation are derived including the recurrence of the simple random walk at the critical temperature

a new class of fuzzy contractive mappings and fixed point theorems abstract the main aim of this work is to unify different classes of fuzzy contractive mappings by introducing a new class of fuzzy contractive mappings called fuzzy z contractive mappings for this new class of mappings suitable conditions are framed to ensure the existence of fixed point in m complete fuzzy metric spaces in the sense of george and veeramani a comprehensive set of examples are presented to support the claim

sensitivity of the response of moderately thick cross ply doubly curved panels to lamination and boundary constraint i theory abstract hitherto unavailable analytical solutions to the boundary value problem of moderately thick general cross ply laminated doubly curved panels of rectangular planform subjected to various boundary conditions are presented the five highly coupled second order linear partial differential equations that characterize the deformation of such laminates are solved in part i of the paper using a recently developed double fourier series based approach together with the ss1 ss2 and ss4 types of simply supported and c4 type of clamped boundary conditions prescribed at all four edges the issues of derivation of the linear algebraic equations arising from these boundary conditions together with an efficient method of solving the complete system of linear algebraic equations convergence characteristics and other numerical results are addressed in the accompanying part ii of this investigation

on contractions in probabilistic and fuzzy metric spaces in this paper by means of weakening conditions of the gauge function a new fixed point theorem for probabilistic contraction in menger probabilistic metric spaces with a t norm of h type is established this theorem improves and generalizes the recent results of iric 2010 4 jachymski 2010 12 and xiao et al 2013 25 by using the theorem we also obtain some important corollaries and the fixed point theorems for contraction in fuzzy metric spaces

the scaling limit of random simple triangulations and random simple quadrangulations let mnmn be a simple triangulation of the sphere s2s2 drawn uniformly at random from all such triangulations with nn vertices endow mnmn with the uniform probability measure on its vertices after rescaling graph distance by 3 4n 1 4 3 4n 1 4 the resulting random measured metric space converges in distribution in the gromov hausdorff prokhorov sense to the brownian map in proving the preceding fact we introduce a labelling function for the vertices of mnmn under this labelling distances to a distinguished point are essentially given by vertex labels with an error given by the winding number of an associated closed loop in the map we establish similar results for simple quadrangulations

fourier solution to higher order theory based laminated shell boundary value problem a complete i e particular as well as complementary fourier solution to the boundary value problem of static response under transverse load of a general cross ply thick doubly curved panel of rectangular planform is presented a boundary discontinuous double fourier series approach is used to solve a system of five partial differential equations generated by a higher order shear deformation theory based shell analysis with the ss2 type of simply supported boundary condition prescribed at all four edges unlike the conventional navier and levy type approaches which can only provide particular solutions the present method is general enough to provide the complete particular as well as the complementary solution for any arbitrary combination of admissible boundary conditions with almost equal ease the numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of cross ply spherical panels and also by comparison with the available first order shear deformation theory and classical lamination theory based analytical solutions hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to lamination boundary constraint and thickness and curvature effects as well as their interactions

compact brownian surfaces i brownian disks we show that under certain natural assumptions large random plane bipartite maps with a boundary converge after rescaling to a one parameter family mathrm bd l 0 l infty of random metric spaces homeomorphic to the closed unit disk of mathbb r 2 the space mathrm bd l being called the brownian disk of perimeter l and unit area these results can be seen as an extension of the convergence of uniform plane quadrangulations to the brownian map which intuitively corresponds to the limit case where l 0 similar results are obtained for maps following a boltzmann distribution in which the perimeter is fixed but the area is random

the topological structure of scaling limits of large planar maps we discuss scaling limits of large bipartite planar maps if p 2 is a fixed integer we consider for every integer n 2 a random planar map mn which is uniformly distributed over the set of all rooted 2p angulations with n faces then at least along a suitable subsequence the metric space consisting of the set of vertices of mn equipped with the graph distance rescaled by the factor n 1 4 converges in distribution as n towards a limiting random compact metric space in the sense of the gromov hausdorff distance we prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence and that this space can be obtained as the quotient of the continuum random tree for an equivalence relation which is defined from brownian labels attached to the vertices we also verify that the hausdorff dimension of the limit is almost surely equal to 4

stability of geodesics in the brownian map the brownian map is a random geodesic metric space arising as the scaling limit of random planar maps we strengthen the so called confluence of geodesics phenomenon observed at the root of the map and with this reveal several properties of its rich geodesic structure our main result is the continuity of the cut locus at typical points a small shift from such a point results in a small local modification to the cut locus moreover the cut locus is uniformly stable in the sense that any two cut loci coincide outside a closed nowhere dense set of zero measure we obtain similar stability results for the set of points inside geodesics to a fixed point furthermore we show that the set of points inside geodesics of the map is of first baire category hence most points in the brownian map are endpoints finally we classify the types of geodesic networks which are dense for each k 1 2 3 4 6 9 k 1 2 3 4 6 9 there is a dense set of pairs of points which are joined by networks of exactly kk geodesics and of a specific topological form we find the hausdorff dimension of the set of pairs joined by each type of network all other geodesic networks are nowhere dense

limit of normalized quadrangulations the brownian map consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces in this paper we show that when n goes to qn suitably normalized converges weakly in a certain sense to a random limit object which is continuous and compact and that we name the brownian map the same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths a metric space of rooted resp pointed abstract maps that contains the model of discrete rooted resp pointed quadrangulations and the model of the brownian map is defined the weak convergences hold in these metric spaces

static analysis of moderately thick finite antisymmetric angle ply cylindrical panels and shells abstract the analytical exact in the limit or strong or differential form of solutions to the bench mark problems of i axisymmetric angle ply circular cylindrical panels of rectangular planform and ii circumferentially complete circular cylindrical shells subjected to transverse load and with ss2 type simply supported boundary conditions prescribed at the edges are presented the problems investigated which were hitherto thought to be incapable of admitting analytical solutions have been solved utilizing a recently developed novel boundary discontinuous double fourier series approach for three kinematic relations which are extensions of those due to sanders love and donnell to the first order shear deformation theory fsdt numerical results presented for two layer square antisymmetric angle ply panels which demonstrate good convergence and show the effects of fiber orientation and thickness on the static response of these panels should serve as baseline solutions in the context of fsdt for future comparison with various approximate weak forms of solutions with either local e g finite element methods or global supports e g raleigh ritz galerkin

a generalization of tardiff s fixed point theorem in probabilistic metric spaces and applications to random equations using the infinitely countable extension of triangular norms a generalization of tardiff s fixed point theorem in probabilistic metric spaces is proved as a consequence an application to random equations is obtained ained

multivalued generalizations of probabilistic contractions we generalize some well known fixed point theorems for probabilistic contractions to multivalued contractions the contraction maps are considered in generalized menger spaces

some questions in fuzzy metric spaces the george and veeramani s fuzzy metric defined by m x y t min x y t max x y t on 0 the set of non negative real numbers has shown some advantages in front of classical metrics in the process of filtering images in this paper we study from the mathematical point of view this fuzzy metric and other fuzzy metrics related to it as a consequence of this study we introduce throughout the paper some questions relative to fuzzy metrics also as another practical application we show that this fuzzy metric is useful for measuring perceptual colour differences between colour samples

uniqueness and universality of the brownian map we consider a random planar map mn which is uniformly distributed over the class of all rooted q angulations with n faces we let mn be the vertex set of mn which is equipped with the graph distance dgr both when q 4 is an even integer and when q 3 there exists a positive constant cq such that the rescaled metric spaces mn cqn 1 4dgr converge in distribution in the gromov hausdorff sense toward a universal limit called the brownian map the particular case of triangulations solves a question of schramm

on fixed point theorems of gregori and sapena using the theory of countable extension of a t norm we prove in this paper that the fixed point theorem of gregori and sapena holds under more general conditions

the brownian map is the scaling limit of uniform random plane quadrangulations we prove that uniform random quadrangulations of the sphere with n faces endowed with the usual graph distance and renormalized by n 1 4 converge as n in distribution for the gromov hausdorff topology to a limiting metric space we validate a conjecture by le gall by showing that the limit is up to a scale constant the so called brownian map which was introduced by marckert mokkadem and le gall as the most natural candidate for the scaling limit of many models of random plane maps the proof relies strongly on the concept of geodesic stars in the map which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere

random planar lattices and integrated superbrownian excursion in this paper a surprising connection is described between a specific brand of random lattices namely planar quadrangulations and aldous integrated superbrownian excursion ise as a consequence the radius r n of a random quadrangulation with n faces is shown to converge up to scaling to the width r r l of the support of the one dimensional ise or precisely more generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ise shifted to set the minimum of its support in zero the first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half line reminiscent of cori and vauquelin s well labelled trees the second step relates these trees to embedded discrete trees in the sense of aldous via the conjugation of tree principle an analogue for trees of vervaat s construction of the brownian excursion from the bridge from probability theory we need a new result of independent interest the weak convergence of the encoding of a random embedded plane tree by two contour walks to the brownian snake description of ise our results suggest the existence of a continuum random map describing in term of ise the scaled limit of the dynamical triangulations considered in two dimensional pure quantum gravity

levy type boundary fourier analysis of thick cross ply panels with negative gaussian curvature a boundary discontinuous generalized double fourier series based solution to the problem of deformation of a finite dimensional general cross ply thick doubly curved panel with negative gaussian curvature and of rectangular planform modeled using a higher order shear deformation theory is presented this spectral method is used to solve a system of five highly coupled linear partial differential equations generated by the higher order shear deformation theory based laminated shell analysis with the simply supported of type 2 boundary condition prescribed on two opposite edges whereas the remaining two edges are subjected to the simply supported of type 3 constraint the hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to shell geometry negative or positive gaussian curvature lamination and thickness effects as well as their interactions

levy type boundary fourier analysis of thick clamped hyperbolic paraboloidal cross ply panels an analytical solution to the problem of deformation of a finite dimensional general cross ply thick doubly curved panel with negative gaussian curvature and of rectangular planform modeled using a higher order shear deformation theory is presented a solution methodology based on a boundary discontinuous generalized double fourier series approach is used to solve a system of five highly coupled linear partial differential equations generated by the higher order shear deformation theory based laminated shell analysis with the c4 type rigidly clamped boundary condition prescribed on two opposite edges whereas the remaining two edges are subjected to the ss3 type constraint hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to applied loading shell geometry negative or positive gaussian curvature edge clamping constraints and lamination and thickness effects as well as their possible interactions comparison with the

on the existence and the uniqueness of fixed points of sehgal contractions we discuss the problem of existence and uniqueness of fixed points for fuzzy contraction maps in fuzzy menger spaces and quasi pseudo menger spaces

on fixed point theorems in fuzzy metric spaces we give fixed point theorems for complete fuzzy metric spaces in the sense of george and veeramani and also for kramosil and michalek s fuzzy metric spaces which are complete in grabiec s sense

subordination of trees and the brownian map we discuss subordination of random compact mathbb r trees we focus on the case of the brownian tree where the subordination function is given by the past maximum process of brownian motion indexed by the tree in that particular case the subordinate tree is identified as a stable levy tree with index 3 2 as a more precise alternative formulation we show that the maximum process of the brownian snake is a time change of the height process coding the levy tree we then apply our results to properties of the brownian map in particular we recover in a more precise form a recent result of miller and sheffield identifying the metric net associated with the brownian map

geodesics in large planar maps and in the brownian map we study geodesics in the random metric space called the brownian map which appears as the scaling limit of large planar maps in particular we completely describe geodesics starting from the distinguished point called the root and we characterize the set s of all points that are connected to the root by more than one geodesic the set s is dense in the brownian map and homeomorphic to a non compact real tree furthermore for every x in s the number of distinct geodesics from x to the root is equal to the number of connected components of s x in particular points of the brownian map can be connected to the root by at most three distinct geodesics our results have applications to the behavior of geodesics in large planar maps