finitized conformal spectrum of the ising model on the cylinder and torus abstract the spectrum of the critical ising model on a lattice with cylindrical and toroidal boundary conditions is calculated by commuting transfer matrix methods using a simple truncation procedure we obtain the natural finitizations of the conformal spectra recently proposed by melzer these finitizations imply polynomial identities which in the large lattice limit give rise to the rogers ramanujan identities for the c 1 2 virasoro characters

correspondence between the xxz model in roots of unity and the one dimensional quantum ising chain with different boundary conditions we consider the integrable xxz model with special open boundary conditions that renders its hamiltonian su 2 q symmetric and the one dimensional quantum ising model with four different boundary conditions we show that for each boundary condition the ising quantum chain is given exactly by the minimal model of integrable lattice theory lm 3 4 this theory is obtained as the result of the quantum group reduction of the xxz model at anisotropy q q 1 2 2 1 2 2 with a number of sites that depends on the type of imposed boundary condition

λ difference sequence spaces of fuzzy numbers in this paper using the difference operator of order m and an orlicz function we introduce and examine some classes of sequences of fuzzy numbers we give the relations between the strongly cesaro type convergence and statistical convergence in these spaces furthermore we study some of their properties like completeness solidity symmetricity etc we also give some inclusion relations related to these classes

limit structures and many valued topologies abstract this paper presents a canonical construction of limit spaces from many valued topological spaces in particular there exists a limit structure on l0 0 1 b which characterizes pointwise almost everywhere convergence

conformal invariance and the critical behavior of the triplet xy quantum chain we introduce and study the critical properties of the tripletxy quantum chain this system is described in terms of three spin interactions and is the generalization of the standardxy quantum chain we show that this model with periodic boundaries has a local gauge invariance and can be described by the composition of two triplet ising models with general toroidal boundary conditions from this composition the phase diagram as well the conformai anomaly and critical exponents are determined by exploring their relations with the mass gap amplitudes predicted by conformai invariance

finitized conformal spectra of the ising model on the klein bottle and mobius strip we study the conformal spectra of the critical square lattice ising model on the klein bottle and mobius strip using yang baxter techniques and the solution of functional equations in particular we obtain expressions for the finitized conformal partition functions in terms of finitized virasoro characters this demonstrates that yang baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies the results rely on certain properties of the eigenvalues which are confirmed numerically

invariant distributions and x ray transform for anosov flows for anosov flows preserving a smooth measure on a closed manifold m we define a natural self adjoint operator which maps into the space of flow invariant distributions in r 0 h s m we describe relations to the livsic theorem and recover regularity properties of cohomological equations using this operator for anosov geodesic flows on the unit tangent bundle m sm of a compact manifold m we apply this theory to study x ray transform on symmetric tensors on m in particular we prove existence of flow invariant distributions on sm with prescribed push forward on m and a similar version for tensors this allows us to show injectivity of the x ray transform on an anosov surface any divergence free symmetric tensor on m which integrates to 0 along all closed geodesics is zero

bulk and surface critical behavior of the three dimensional ising model and conformal invariance using a continuous cluster monte carlo algorithm we investigate the critical three dimensional ising model in its anisotropic limit from the ratio of the magnetic correlations in the strong and the weak coupling directions we determine the length ratio relating the isotropic ising model and the anisotropic limit on this basis we simulate the critical ising model on a spherocylinder s2 x r1 i e a curved geometry obtained from a conformal mapping of the infinite space r3 from correlation lengths along the spherocylinder combined with the prediction of conformal invariance we estimate the magnetic and thermal scaling dimensions as x h 0 5182 6 and x t 1 419 7 respectively the behavior of the binder cumulant is also determined in the limit of an infinitely long spherocylinder next free boundary conditions are imposed on the equators of the spherocylinder and thus the geometry s1 x s x r1 is obtained the surface magnetic scaling dimension is estimated as x s h 1 263 5 the consistency of the aforementioned estimations and existing results confirms that the three dimensional ising model is conformally invariant further the precision of these results reveals that as in two dimensions conformal mappings provide a powerful tool to investigate critical phenomena with the continuous cluster algorithm we also perform simulations of systems inside a conventional solid cylinder the surface magnetic correlation length differs within the estimated error margin by a factor pi 2 from that along a half spherocylinder s1 x s x r1 with the same radius

fuzzy smoothness and sequences of fuzzy smooth functions abstract this paper is a continuation of the work of kalina tatra mountains math publ 14 1998 153 159 it further investigates fuzzy smoothness of fuzzy continuous but crisp functions finally sequences of fuzzy smooth functions and their sums are considered

fuzzy sets and sheaves part ii this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete w valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

comparison of fuzzy partitions based on their a cuts abstract this paper proposes a measure of degree of equivalence between two fuzzy partitions a method which allows one to find values of 0 1 at which two or more partitions are considered to be equivalent is introduced partial order of fuzzy partitions with respect to their sharpness is discussed a procedure for constructing a lower bound and an upper bound of a family of fuzzy partitions is suggested this procedure can be used in aggregation of fuzzy partitions

fermionic solution of the andrews baxter forrester model i unification of tba and ctm methods the problem of computing the one dimensional configuration sums of the abf model in regime iii is mapped onto the problem of evaluating the grandcanonical partition function of a gas of charged particles obeying certain fermionic exclusion rules we thus obtain a newfermionic method to compute the local height probabilities of the model combined with the originalbosonic approach of andrews baxter and forrester we obtain a new proof of some of melzer s polynomial identities in the infinite limit these identities yield rogers ramanujan type identities for the virasoro characters l 1 r l r q as conjectured by the stony brook group as a result of our work the corner transfer matrix and thermodynamic bethe ansatz approaches to solvable lattice models are unified

on the ideal convergence of sequences of fuzzy numbers in this paper we introduce and study the concepts of i convergence i convergence and i cauchy sequence for sequences of fuzzy numbers where i denotes the ideal of subsets of n the set of positive integers

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

multicriticality in a self dual potts model the authors explore the self dual plane of a two dimensional potts model with vacancies and four spin interactions using finite size scaling of transfer matrix results since the transfer matrix construction is based on the random cluster representation of the potts model its application is not restricted to an integer number q of potts states for q or 4 the self dual plane contains a critical region belonging to the ordinary potts universality class and a region of first order transition points the two regions are separated by a line of which a range consists of potts tricritical points while another range is of a first order nature the point separating these two ranges is for 0 q or 9 4 identified with the multicritical point of which an exact solution has recently been given by nienhuis et al 1979

on the asymptotics of dimers on tori we study asymptotics of the dimer model on large toric graphs let mathbb l be a weighted mathbb z 2 periodic planar graph and let mathbb z 2 e be a large index sublattice of mathbb z 2 for mathbb l bipartite we show that the dimer partition function textsf textit z e on the quotient mathbb l mathbb z 2 e has the asymptotic expansion begin aligned textsf textit z exp a mathbf f 0 textsf fsc o 1 end aligned where a is the area of mathbb l mathbb z 2 e mathbf f 0 is the free energy density in the bulk and textsf fsc is a finite size correction term depending only on the conformal shape of the domain together with some parity type information assuming a conjectural condition on the zero locus of the dimer characteristic polynomial we show that an analogous expansion holds for mathbb l non bipartite the functional form of the finite size correction differs between the two classes but is universal within each class our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double dimer models

fuzzification of the theory of continuous functions abstract there are classical results in mathematics which are incomplete an example is given by the well known result of the classical mathematical analysis stating that a continuous function defined on a closed interval is bounded students study this theorem at the beginning of their courses of higher mathematics but if we ask whether the converse is true we reveal that the answer is negative the criterion of boundedness may be found only in terms of fuzzy set theory the main result of this paper theorem 2 states that a function defined on a closed interval is bounded if and only if is fuzzy continuous the local form of this result theorem 1 is also proved in the third section of the paper in order to do this in the second section of the paper after the introduction the main concepts and constructions of the theory of fuzzy continuous functions are introduced and discussed the third section is devoted to the problems of boundedness in the fourth section operations on sets of fuzzy continuous functions are investigated

fuzzy conditional convergence and nearness relations this paper is concerned with fuzziness in basic mathematical structures such as sequences and series formalization of this fuzziness gives such concepts as conditional fuzzy limit conditional fuzzy convergence nearness limit and nearness convergence properties of these constructions and relations between them are obtained it is discussed how results and constructions of this paper are related to some principle problems of numerical computation

lens rigidity for manifolds with hyperbolic trapped set for a riemannian manifold m g with strictly convex boundary m the lens data consists in the set of lengths of geodesics with endpoints on m together with their endpoints x x m m and tangent exit vectors v v t x m t x m we show deformation lens rigidity for manifolds with hyperbolic trapped set and no conjugate points a class which contains all manifolds with negative curvature and strictly convex boundary including those with non trivial topology and trapped geodesics for the same class of manifolds in dimension 2 we prove that the set of endpoints and exit vectors of geodesics ie the scattering data determines the riemann surface up to conformal diffeomorphism

fermionic solution of the andrews baxter forrester model ii proof of melzer s polynomial identities we compute the one dimensional configuration sums of the abf model using the fermionic technique introduced in part i of this paper combined with the results of andrews baxter and forrester we prove polynomial identities for finitizations of the virasoro characters chi b a r 1 r q as conjectured by melzer in the thermodynamic limit these identities reproduce rogers ramanujan type identities for the unitary minimal virasoro characters conjectured by the stony brook group we also present a list of additional virasoro character identities which follow from our proof of melzer s identities and application of bailey s lemma

notes on fuzzy complex analysis several fuzzy complex analysis problems are discussed pointing out some errors in literature and some weaknesses of introduced concepts a modified approach to fuzzy complex analysis is proposed and its impact is discussed

fuzzy sets and sheaves part ii sheaf theoretic foundations of fuzzy set theory with applications to algebra and topology this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

invariant distributions beurling transforms and tensor tomography in higher dimensions in the recent articles paternain et al j differ geom 98 147 181 2014 invent math 193 229 247 2013 a number of tensor tomography results were proved on two dimensional manifolds the purpose of this paper is to extend some of these methods to manifolds of any dimension a central concept is the surjectivity of the adjoint of the geodesic ray transform or equivalently the existence of certain distributions that are invariant under the geodesic flow we prove that on any anosov manifold one can find invariant distributions with controlled first fourier coefficients the proof is based on subelliptic type estimates and a pestov identity we present an alternative construction valid on manifolds with nonpositive curvature based on the fact that a natural beurling transform on such manifolds turns out to be essentially a contraction finally we obtain uniqueness results in tensor tomography both on simple and anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified jacobi equation

nearness derivatives and fuzzy differentiability we introduce a fuzzy differentiability and a type of generalized derivatives nearness derivatives based on the notion of nearness operation using this type of differentiability we show that some basic statements of mathematical analysis remain valid for a wider class of functions we state generalized versions of rolle and lagrange mean value theorems for the case of non differentiable and even non continuous functions in terms of the fuzzy theory

on the restudy of fuzzy complex analysis part ii the continuity and differentiation of fuzzy complex functions we consider functions mapping complex numbers into fuzzy complex numbers or mapping fuzzy complex numbers into complex numbers we study the continuity and differentiation of such functions

similarity based fuzzy limits in this paper the concept of similarity based fuzzy limits in metric spaces is introduced and it is shown that it is an application of limit maps of topologies on 0 1 valued sets introduced by hohle and sostak furthermore it is observed that the similarity based setting introduced is a more general approach to fuzzy limits than the one proposed by burgin

lattice integrals of motion of the ising model on the cylinder abstract we consider the 2d critical ising model with spatially periodic boundary conditions it is shown that for a suitable reparameterization of the known boltzmann weights the transfer matrix becomes a polynomial in the variable csc 4 u u being the spectral parameter the coefficients of this polynomial are decomposed on the periodic temperley lieb algebra by introducing a lattice version of the local integrals of motion

local state probabilities for an infinite sequence of solvable lattice models the authors present a new infinite sequence of solvable lattice models they contrast strikingly with the eight vertex solid on solid models and admit extra degrees of freedom for the local fluctuation variables the exact one point functions are obtained the result is neatly described in terms of theta function identities using their modular invariance critical behaviour is studied and exponents evaluated

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

interaction round a face models with fixed boundary conditions the abf fusion hierarchy we use boundary weights and reflection equations to obtain families of commuting double row transfer matrices for interaction round a face models with fixed boundary conditions in particular we consider the fusion hierarchy of the andrews baxter forrester abf models for which we obtain diagonal elliptic solutions to the reflection equations and find that the double row transfer matrices satisfy functional equations with the same form as in the case of periodic boundary conditions

theory of fuzzy limits theory of limits is the base of the classical mathematical analysis the calculus in a similar way theory of fuzzy limits presented in this paper is the base of neoclassical analysis which is a new branch of fuzzy mathematics and extends possibilities provided by classical analysis the concepts of a fuzzy limit of a sequence and a fuzzy limit of a function are introduced and studied

integrable and conformal boundary conditions for widehat s ell 2 a d e lattice models and unitary minimal conformal field theories integrable boundary conditions are studied for critical a d e and general graph based lattice models of statistical mechanics in particular using techniques associated with the temperley lieb algebra and fusion a set of boundary boltzmann weights which satisfies the boundary yang baxter equation is obtained for each boundary condition when appropriately specialized these boundary weights each of which depends on three spins decompose into more natural two spin edge weights the specialized boundary conditions for the a d e cases are naturally in one to one correspondence with the conformal boundary conditions of widehat s ell 2 unitary minimal conformal field theories supported by this and further evidence we conclude that in the continuum scaling limit the integrable boundary conditions provide realizations of the complete set of conformal boundary conditions in the corresponding field theories

neoclassical analysis fuzzy continuity and convergence abstract the neoclassical analysis is a field in which fuzzy continuous functions are investigated for this purpose new measures of continuity and discontinuity or defects of continuity are introduced and studied based on such measures classes of fuzzy continuous functions are defined and their properties are obtained the class of fuzzy continuous functions may be considered as a fuzzy set of continuous functions its support consists of all functions on some topological space x into a metric space y and its membership function is the corresponding continuity measure such an expansion provides a possibility to complete some important classical results connections between boundedness and fuzzy continuity are investigated criteria of boundedness and local boundedness are obtained for functions on eucleadean spaces besides such new concepts as fuzzy convergence and fuzzy uniform convergence are introduced and investigated their properties and connections with fuzzy continuous functions are explicated some results which are obtained here are similar to the results of classical mathematical analysis while others differ essentially from those that are proved in classical mathematics in the first case classical results are consequences of the corresponding results of neoclassical analysis

possibly exact fractal dimensions from conformal invariance based on an equivalence between the q state potts model and conformally invariant field theories the author speculates that all fractal dimensionalities of two dimensional percolation are given by the formula d 100 x2 48 where x is an integer the same formula should also apply to self avoiding walks and lattice animals this approach is extended to cluster weighted percolation his conjecture seems to agree with previous exact and numerical results except for the backbone exponent and possibly for lattice animals

finite size scaling and conformal invariance in a self dual quantum z 5 model considers the critical behaviour of a particular set of onedimensional self dual models with z n symmetry n or 5 the critical indices are evaluated using standard finite size scaling and by exploiting their relations with the mass gap amplitudes predicted by conformal invariance the results strongly suggest that the recently introduced z n quantum field theory is the underlying field theory for these statistical mechanics models

lattice integrals of motion of the ising model on the strip we consider the 2d critical ising model on a strip with fixed boundary conditions it is shown that for a suitable reparametrization of the known boltzmann weights the transfer matrix becomes a polynomial in the variable csc 4u being u the spectral parameter the coefficients of this polynomial are decomposed on the fixed boundaries temperley lieb algebra by introducing a lattice version of the local integrals of motion

the equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets in this paper with the help of the convergence of sequences of fuzzy numbers with respect to the lebesgue measure we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric the sendograph metric and the l p metric we prove that these convergences are equivalent under proper conditions in addition by applying our result we give a new characterization of compact sets in fuzzy number space with the sendograph metric

general approach to continuity measures the neoclassical analysis is a field where an approach to investigation of different kinds of fuzzy continuous functions is suggested in this paper we consider such property as continuity local and global of functions and investigate what does it mean that a function is continuous only to some extent

correlation functions of twist operators applied to single self avoiding loops the o n spin model in two dimensions may equivalently be formulated as a loop model and then mapped to a height model which is conjectured to flow under the renormalization group to a conformal field theory cft at the critical point the order n terms in the partition function and correlation functions describe single self avoiding loops we investigate the ensemble of these self avoiding loops using twist operators which count loops which wind non trivially around them with a factor 1 these turn out to have level 2 null states and hence their correlators satisfy a set of partial differential equations we show that partly connected parts of the four point function count the expected number of loops which separate one pair of points from the other pair and find an explicit expression for this we argue that the differential equation satisfied by these expectation values should have an interpretation in terms of a stochastic schramm loewner evolution sle process with 6 the two point function in a simply connected domain satisfies a closely related set of equations we solve these and hence calculate the expected number of single loops which separate both points from the boundary

partition functions loop measure and versions of sle we discuss the partition function view of the schramm loewner evolution after reviewing a number of known results in the framework of brownian loop measures and scaling rules for partition functions we give some speculation about multiply connected domains

topological aspects of non convergent sequences a comment on burgin s concept of fuzzy limits abstract it is well known that metric spaces can be provided with the structure of 0 1 valued sets on this background this paper explains an application of topologies on 0 1 valued sets to metric spaces and shows that every bounded sequence has always local limit points represented by singletons of the underlying 0 1 valued set

phase structure of the three matrix chain model coupled to the magnetic field the author investigates the critical properties of the three matrix chain model corresponding to the blume emery griffiths model in a homogeneous magnetic field on a fluctuating lattice the third order critical line and the tricritical point disappear after the magnetic field is switched on it is however found that another critical line exists as a result of the naive perturbative expansion and the string susceptibility exponent is 1 3 on the line it is also shown that the critical exponent of the magnetic moment beta is 1 6 which coincides with the kpz solution

functions defined on a set of z numbers abstract a wide class of problems in science and engineering requires mathematical formalization one of the important aspects in this regard is that we should account for various types of uncertainties which are simultaneously present in real world problems and result from imprecise and partially reliable information this mandates a need for a theoretical basis of uncertain functional dependencies the theories of interval valued functions functions of random variables and fuzzy functions are well developed the main disadvantage of these theories is that each of them can deal with only a single type of uncertainty in order to ground the formal basis for dealing with real world information zadeh suggested the concept of a z number as an ordered pair z a b of fuzzy numbers used to describe a value of a random variable x where a is a fuzzy constraint on values of x and b is a fuzzy value of probability measure of a unfortunately there is no study on functions of z numbers in existence in this paper for the first time we propose a general methodology for a construction of such functions on the basis of the extension principle applied to z numbers the proposed methodology is helpful in restricting the increase of uncertainty in computation the values of z valued functions some properties of functions of z numbers are considered we construct typical and commonly encountered functions such as addition of z numbers minimum of z numbers square root of z numbers and others a spectrum of examples is provided to illustrate the proposed approach

conformal invariance and the operator content of the xxz model with arbitrary spin the authors are concerned with the critical properties of the anisotropic heisenberg chain or xxz model with arbitrary integer or half integer spin the eigenspectra of these hamiltonians with periodic boundaries are calculated for finite chains by solving numerically their associated bethe ansatz equations the resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified for lattices with an even and odd number of sites the results for spin 1 and spin 3 2 indicate that the conformal anomaly c for these models in the gapless regime has the value c 3s 1 s independent of the anisotropy and the exponents vary continuously with the anisotropy as in the eight vertex model the operator content of these models indicate that the underlying field theory governing these critical spin s models is described by composite fields formed by the product of gaussian and z n fields with n 2s finally some of the irrelevant operators which produce the leading finite size corrections of the eigenenergies are identified

fractal dimensions of potts clusters potts clusters are connected sets of nearest neighbour sites for which the potts variable is in the same state at criticality there exists a fractal cluster using arguments from the renormalization group conformal invariance and numerical simulations the author determines the bulk surface hull and red bond dimension of these clusters as a function of the number of potts states

the 1 1 do 2 model a finite lattice analysis finite size methods and the theory of conformal invariance are applied to the 1 1 do 2 or x y planar model the critical point is found to be xc approximately 2 0 the value sigma 0 501 or 0 005 is obtained for the index governing the critical divergence of the correlation length and the conformal anomaly is confirmed to be c 1 the kosterlitz thouless features of a low temperature phase with zero spontaneous magnetisation but algebraic decay of correlations together with a standard high temperature phase are observed the critical exponent eta is determined as a function of coupling constant and compared with theoretical expectations

the baxter revolution i review the revolutionary impact rodney baxter has had on statistical mechanics beginning with his solution of the 8 vertex model in 1971 and the invention of corner transfer matrices in 1976 to the creation of the rsos models in 1984 and his continuing current work on the chiral potts model