sendograph metric and relatively compact sets of fuzzy sets let k y be the family of all fuzzy subsets of an arbitrary metric space y which are upper semicontinuous normal with nonempty compact support the sendograph distance h between fuzzy sets is the hausdorff distance of their reduced hypographs called sendographs fan fuzzy sets and systems 143 2004 471 477 characterized compact sets of fuzzy numbers convex fuzzy subsets of the real line r with respect to sendograph metric and declared open the problem of finding a criterion for convex fuzzy subsets of r n in this paper relatively compact subsets of k y h are characterized see theorem 8 and consequently it is shown that fan s theorem holds for arbitrary metric spaces without any convexity assumption in the proofs a variational convergence called c convergence introduced by de giorgi franzoni atti accad naz lincei rend cl sc mat fis natur 58 8 1975 842 850 rend sem mat brescia 3 1979 63 101 and author s previous results are fundamental no isometrical embedding theorem is used co et al ann univ ferrara sez vii sc mat 44 1998 27 39 greco and moschen nonlinear anal tma to appear no isometrical embedding theorem is used

semilinear copulas a family of copulas called semilinear is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square this family contains some other known families of copulas e g cuadras auge frechet and has a nice statistical interpretation several construction methods are provided especially concerning aggregation of semilinear copulas and a special form of ordinal sum construction is introduced some results about related families of quasi copulas and semicopulas are hence given

a metric space of subcopulas an approach via hausdorff distance abstract in this work we define a distance function on the set of bivariate subcopulas to generate a compact metric space moreover the copula space equipped with the uniform distance is essentially a metric subspace of this subcopula space we also characterize the convergence in this space and provide the interrelationship with the convergence of the distribution functions

critical percolation and the minimal spanning tree in slabs the minimal spanning forest on d is known to consist of a single tree for d 2 and is conjectured to consist of infinitely many trees for large d in this paper we prove that there is a single tree for quasi planar graphs such as 2 0 k d 2 our method relies on generalizations of the gluing lemma of duminil copin sidoravicius and tassion a related result is that critical bernoulli percolation on a slab satisfies the box crossing property its proof is based on a new russo seymour welsh type theorem for quasi planar graphs thus at criticality the probability of an open path from 0 of diameter n decays polynomially in n this strengthens the result of duminil copin et al where the absence of an infinite cluster at criticality was first established 2017 wiley periodicals inc

connection probabilities and rsw type bounds for the two dimensional fk ising model we prove russo seymour welsh type uniform bounds on crossing probabilities for the fk ising fk percolation with cluster weight q 2 model at criticality independent of the boundary conditions our proof relies mainly on smirnov s fermionic observable for the fk ising model 24 which allows us to get precise estimates on boundary connection probabilities we stay in a discrete setting in particular we do not make use of any continuum limit and our result can be used to derive directly several noteworthy properties including some new ones among which are the fact that there is no infinite cluster at criticality tightness properties for the interfaces and the existence of several critical exponents in particular the half plane one arm exponent such crossing bounds are also instrumental for important applications such as constructing the scaling limit of the ising spin field 6 and deriving polynomial bounds for the mixing time of the glauber dynamics at criticality 17

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

some results on the convergence of quasi copulas it is shown that pointwise convergence of a sequence a n n n of copulas to a copula a is equivalent 1 to the convergence of the corresponding endographs and 2 to the convergence of the corresponding upper or lower a levels for all but at most countably many a in 0 1 all with respect to the hausdorff metric examples are given that show that the countably many exceptions in 2 cannot be omitted it is furthermore shown that the main results also hold on the bigger class of quasi copulas

absence of infinite cluster for critical bernoulli percolation on slabs we prove that for bernoulli percolation on a graph 2 0 k k 0 there is no infinite cluster at criticality almost surely the proof extends to finite range bernoulli percolation models on 2 that are invariant under 2 rotation and reflection 2016 wiley periodicals inc

a way to define the phase distribution for a single mode quantum field the idea that the phase of single mode field may be correctly defined as a phase difference between the state considered and a highly excited coherent state treated as the reference phase state although present in discussions about the quantum phase problem have not been directly concretized and its consequences fully understood in the present work we succeed in finding an effective mathematical procedure which corresponds exactly to this idea and so derive using the results related to phase difference available in the literature the phase distribution for a single mode field we discuss the obtained results

an interesting property of the husimi function and its implications we prove that after the transformation q p to lambda q lambda p where 0 lambda 1 every husimi function with appropriate renormalization remains in the class of husimi distributions we discuss some implications of this fact on the problem of the phase distribution of a quantum state we show that wigner and p functions do not have this property

ising field theory quadratic difference equations for the n point green s functions on the lattice for the two dimensional ising model at arbitrary temperature a system of quadratic difference equations is presented involving the correlations of n order variables and that of n ensuremath 2 order and two disorder variables with suitable boundary conditions these equations specify the correlations

quantum states which behave classically we give a definition of quantum states which behave classically based on minimal number of physically reasonable requirements we prove that an infinite unique class of states exists which satisfies this definition and we show that every state from this class may be generated in the unique way departing from some corresponding strictly quantum state we discuss some implications of the obtained results

a hitchhiker s guide to quasi copulas abstract this paper presents a review of the concept of an n quasi copula starting from the characterization of functions that can be derived from operations on random variables we recall several other characterizations and properties that have been proven in the literature highlighting the differences between the bivariate case and the higher dimensional case additionally we recall several applications of n quasi copulas such as their role in the study of bounds on sets of n copulas we also recall various classes of n quasi copulas that have been studied in the literature finally we present some questions that have not yet been answered in the literature

extensions of subcopulas abstract in view of sklar s theorem the probability distribution function of every not necessarily continuous random vector can be uniquely decomposed in terms of the marginal distributions of its components and a suitable subcopula the study of such latter functions is therefore of interest for understanding the dependence information of non continuous variables here we investigate some analytical properties of the class of subcopulas including compactness with respect to a novel metric approximations and baire category results moreover under a suitable assumption we describe all possible extensions from a subcopula to a copula in any dimension

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

crossing probabilities for voronoi percolation we prove that the standard russo seymour welsh theory is valid for voronoi percolation this implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio this result has many consequences such as the polynomial decay of the one arm event at criticality

the scaling limit of the energy correlations in non integrable ising models we obtain an explicit expression for the multipoint energy correlations of a non solvable two dimensional ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory which generalizes the one originally used by schultz mattis and lieb for the nearest neighbor ising model the interacting model is then analyzed by a multiscale method first proposed by pinson and spencer if the lattice spacing is finite then the correlations cannot be computed in closed form rather they are expressed in terms of infinite convergent power series in in the scaling limit these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable ising model up to a finite renormalization of the parameters explicit bounds on the speed of convergence to the scaling limit are derived

characterization of the sendograph convergence of fuzzy sets by means of their lp and levelwise convergence one particular way to define a metric on the family of all d dimensional fuzzy sets with non empty convex compact a cuts is the sendograph metric which measures the hausdorff distance of the sendographs of fuzzy sets as first result it will be proved that if a sequence of fuzzy sets converges to a limit fuzzy set with respect to the sendograph metric then it also converges with respect to the l p metric and that the converse is true if in addition the supports of the sequence converge to the support of the limit additionally convergence with respect to the sendograph metric will be shown to be equivalent to almost everywhere convergence of the a cuts of the sequence to the a cut of the limit plus the convergence of the supports of the sequence to the support of the limit already proved in the one dimensional setting by fan as second and stronger result it will be proved that analogous convergence interrelations also hold for the bigger class of fuzzy sets not necessarily having compact support if the hausdorff distance is replaced by a metrization of the fell topology this in particular implies that the two characterizations proved in the first step do not depend on the choice of the metric used in the definition of the hausdorff metric in addition it is demonstrated that the characterization of weakly compact sets with respect to the sendograph metric as stated by greco is a straightforward consequence of the abovementioned equivalences and an alternative definition of fuzzy random variables that is equivalent to the common definition is given

the relation between the scaling of husimi functions and the linear phase insensitive amplification of the corresponding quantum states and its implications using the description of the linear phase insensitive amplification of a quantum state available in the literature we prove that to every state produced from the initial state by such amplification there corresponds a phase space distribution where is the husimi function of the initial state the scaling parameter satisfies the relation and decreases with time while as we have shown earlier the scaled function is again a husimi distribution we prove using these facts that if the definition of the phase insensitive amplification is physically correct then the distribution of the phase of a state is unambiguously given as the corresponding marginal distribution obtained from its husimi distribution represented in polar coordinates

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

quasi copulas and signed measures we study the relationship between multivariate quasi copulas and measures that they may or may not induce on 0 1 n we first study the mass distribution of the pointwise best possible lower bound for the set of n quasi copulas for n 3 as a consequence we show that not every n quasi copula induces a signed measure on 0 1 n

kac ward formula and its extension to order disorder correlators through a graph zeta function a streamlined derivation of the kac ward formula for the planar ising model s partition function is presented and applied in relating the kernel of the kac ward matrices inverse with the correlation functions of the ising model s order disorder correlation functions a shortcut for both is facilitated by the bowen lanford graph zeta function relation the kac ward relation is also extended here to produce a family of non planar interactions on mathbb z 2 for which the partition function and the order disorder correlators are solvable at special values of the coupling parameters temperature

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

emergent planarity in two dimensional ising models with finite range interactions the known pfaffian structure of the boundary spin correlations and more generally order disorder correlation functions is given a new explanation through simple topological considerations within the model s random current representation this perspective is then employed in the proof that the pfaffian structure of boundary correlations emerges asymptotically at criticality in ising models on mathbb z 2 with finite range interactions the analysis is enabled by new results on the stochastic geometry of the corresponding random currents the proven statement establishes an aspect of universality seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases