conformal invariance and finite one dimensional quantum chains based on previous work of cardy 1984 the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions the method is illustrated in the case of the three and four state potts models where the anomalous dimensions of the para fermionic operators are found

conformal invariance and critical dynamics the author discusses the behaviour of dynamic correlation functions under conformal transformations in particular the exact form of the dynamic scaling function at the critical point for an arbitrary isotropic two dimensional model is obtained with either conserved or nonconserved order parameter

a kind of helicoidal surfaces in 3 dimensional minkowski space abstract in this paper we constructed a helicoidal surface with a light like axis with prescribed mean curvature or gauss curvature given by smooth function in 3 dimensional minkowski space e 1 3 and solved an open problem left by beneki kaimakamis and papantoniou in j math anal appl 275 2002 586 614

wavevector dependent susceptibility in z invariant pentagrid ising model we study the q dependent susceptibility q of a z invariant ferromagnetic ising model on a penrose tiling as first introduced by korepin using de bruijn s pentagrid for the rapidity lines the pair correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers its fourier transform q is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins reducing this calculation to linear programming since the lattice is quasiperiodic we find that q is aperiodic and has everywhere dense peaks which are not all visible at very low or high temperatures more and more peaks become visible as the correlation length increases that is as the temperature approaches the critical temperature

integrable dynamics of toda type on square and triangular lattices in a recent paper we constructed an integrable generalization of the toda law on the square lattice we construct other examples of integrable dynamics of toda type on the square lattice as well as on the triangular lattice as nonlinear symmetries of the discrete laplace equations on square and triangular lattices we also construct the ensuremath tau function formulations and darboux b acklund transformations of these dynamics

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

on classification of n 2 supersymmetric theories we find a relation between the spectrum of solitons of massiven 2 quantum field theories ind 2 and the scaling dimensions of chiral fields at the conformal point the condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetricn 2 conformal theories and their massive deformations in terms of a suitable generalization of dynkin diagrams which coincides with the a d e dynkin diagrams for minimal models the landau ginzburg theories are a proper subset of this classification in the particular case of lg theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of picard lefschetz theory

rigidity of self shrinkers and translating solitons of mean curvature flows abstract in this paper we prove that any complete m dimensional spacelike self shrinkers in pseudo euclidean spaces r n m n must be affine planes and there exists no complete m dimensional spacelike translating soliton in r n m n these results are proved by using a new omori yau maximal principle we also derive a rigidity theorem of self shrinking hypersurfaces in euclidean space with gauss image lies in a regular ball

surface critical behaviour of the smoothly inhomogeneous planar ising model the pfaffian method a semi infinite nearest neighbour square ising lattice is investigated whose couplings j l at a distance l from the boundary differ from homogeneity by an amount j l j infinity approximately a l on the basis of the pfaffian method the critical behaviour at the surface of this system is investigated the exponents eta11 nu11 beta1 gamma11 and delta11 all display rich non universal behaviour as a function of the amplitude a for a below a critical value there is a spontaneous surface magnetisation when the bulk l infinity is critical and an asymmetry between the exponents on either side of the critical point

complex wishart matrices and conductance in mesoscopic systems exact results it is noted that the hypothesis of independent random complex elements for the off diagonal blocks say m2 and m3 of the transfer matrix describing conductance in a mesoscopic wire allows the eigenvalue distribution of the matrix product m3 m3 or m2 m2 to be computed exactly exact expressions in terms of double wronskian and toeplitz determinants are derived for the distribution of the smallest and second smallest eigenvalue of this and similar random matrix products

quantitative index bounds for translators via topology we obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form the estimate involves the dimension of the space of weighted square integrable f harmonic 1 forms by the adaptation to the weighted setting of li tam theory developed in previous works this yields estimates in terms of the number of ends of the hypersurface when this is contained in a upper halfspace with respect to the translating direction when there exists a point where all principal curvatures are distinct we estimate the nullity of the stability operator this permits to obtain quantitative estimates on the stability index via the topology of translators with bounded norm of the second fundamental form which are either two dimensional or in higher dimension have finite topological type and are contained in a upper halfspace

multi spin correlation functions for the z invariant ising model abstract continuing our work 1 where a explicit formula for the two point functions of the two dimensional z invariant ising model were found i obtain here different results for the higher correlation functions and several consistency checks are done

third order phase transition in random tilings we consider the domino tilings of an aztec diamond with a cut off corner of macroscopic square shape and given size and address the bulk properties of tilings as the size is varied we observe that the free energy exhibits a third order phase transition when the cut off square increasing in size reaches the arctic ellipse the phase separation curve of the original unmodified aztec diamond we obtain this result by studying the thermodynamic limit of a certain nonlocal correlation function of the underlying six vertex model with domain wall boundary conditions the so called emptiness formation probability efp we consider efp in two different representations as a function for toda chains and as a random matrix model integral the latter has a discrete measure and a linear potential with hard walls the observed phase transition shares properties with both gross witten wadia and douglas kazakov phase transitions

exact expressions for row correlation functions in the isotropic d 2 ising model we present exact explicit expressions for the row spin spin correlation functions 00 n0 in the isotropicd 2 ising model in terms of elliptic integrals forn 5 we also give a general structural formula for 00 n0

the inversion relation method for some two dimensional exactly solved models in lattice statistics the partition functions per site of several two dimensional models notably the eight vertex self dual potts and hard hexagon models can be easily obtained by using an inversion relation for local transfer matrices together with symmetry and analyticity properties this technique is discussed the analyticity properties compared and some equivalences and nonequivalences pointed out in particular the critical hard hexagon model is found to have the same as the self dualq state potts model withq 3 5 2 2 618 the temperley lieb equivalence between the potts and six vertex models is found to fail in certain nonphysical antiferromagnetic cases

finite size effects at 2d ising critical points via conformal mapping using recent results by cardy 1984 based on conformal invariance of critical correlation functions the authors calculate universal results for scattering functions s k susceptibilities correlation lengths and specific heat correction terms for finite ising systems in two dimensions with circular elliptical and rectangular shapes and free boundary conditions results show the effects of critical order on these quantities for a circle s k decays as 1 k2 eta for a large range of intermediate k values with an apparent exponent eta 0 09 the probable influence of end edge and domain wall effects in the rectangular geometry is discussed application of the results to experimental systems and other theoretical models is discussed

further exact solutions of the eight vertex sos model and generalizations of the rogers ramanujan identities the restricted eight vertex solid on solid sos model is an exactly solvable class of two dimensional lattice models to each sitei of the lattice there is associated an integer heightli restricted to the range 1 li r 1 the boltzmann weights of the model are expressed in terms of elliptic functions of period 2k and involve a parameter in an earlier paper we considered the case k r here we generalize those considerations to the case sk r s an integer relatively prime tor we are again led to generalizations of the rogers ramanujan identities

critical correlations in a z invariant inhomogeneous ising model abstract we study the critical spin spin correlation function in the z invariant inhomogeneous ising model which includes the rectangular triangular honeycomb and checkerboard lattices as special cases using quadratic difference equations we derive various explicit representations for the correlation function confirming and simplifying also a miraculous result of baxter

the susceptibility of the square lattice ising model new developments we have made substantial advances in elucidating the properties of the susceptibility of the square lattice ising model we discuss its analyticity properties certain closed form expressions for subsets of the coefficients and give an algorithm of complexity o n6 to determine its first n coefficients as a result we have generated and analyzed series with more than 300 terms in both the high and low temperature regime we quantify the effect of irrelevant variables to the scaling amplitude functions in particular we find and quantify the breakdown of simple scaling in the absence of irrelevant scaling fields arising first at order t tc 9 4 though high low temperature symmetry is still preserved at terms of order t tc 17 4 and beyond this symmetry is no longer present the short distance terms are shown to have the form t tc p log t tc q with p q2 conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments

the critical z invariant ising model via dimers locality property we study a large class of critical two dimensional ising models namely critical z invariant ising models fisher j math phys 7 1776 1781 1966 introduced a correspondence between the ising model and the dimer model on a decorated graph thus setting dimer techniques as a powerful tool for understanding the ising model in this paper we give a full description of the dimer model corresponding to the critical z invariant ising model consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph our main result is an explicit local formula for the inverse kasteleyn matrix in the spirit of kenyon invent math 150 2 409 439 2002 as a contour integral of the discrete exponential function of mercat discrete period matrices and related topics 2002 and kenyon invent math 150 2 409 439 2002 multiplied by a local function using results of boutillier and de tiliere prob theor rel fields 147 3 4 379 413 2010 and techniques of de tiliere prob th rel fields 137 3 4 487 518 2007 and kenyon invent math 150 2 409 439 2002 this yields an explicit local formula for a natural gibbs measure and a local formula for the free energy as a corollary we recover baxter s formula for the free energy of the critical z invariant ising model baxter in exactly solved models in statistical mechanics academic press london 1982 and thus a new proof of it the latter is equal up to a constant to the logarithm of the normalized determinant of the laplacian obtained in kenyon invent math 150 2 409 439 2002

critical scaling for monodromy fields the large scale asymptotics of the correlations for a family of two dimensional lattice field theories is calculated at the critical temperature

the early history of the integrable chiral potts model and the odd even problem in the first part of this paper i shall discuss the round about way of how the integrable chiral potts model was discovered about 30 years ago as there should be more higher genus models to be discovered this might be of interest in the second part i shall discuss some quantum group aspects especially issues of odd versus even n related to the serre relations conjecture in our quantum loop subalgebra paper of 5 years ago and how we can make good use of coproducts also borrowing ideas of drinfeld jimbo deguchi fabricius mccoy and nishino

wavevector dependent susceptibility in quasiperiodic ising models using the various functional relations for correlation functions in planar ising models new results are obtained for the correlation functions and the q dependent susceptibility for ising models on a quadratic lattice with quasiperiodic coupling constants the effects are clearest if the interactions are both attractive and repulsive according to a quasiperiodic pattern in particular an exact scaling limit result for the two point correlation function of the z invariant inhomogeneous ising model is presented and the q dependent susceptibility is calculated for some cases where the coupling constants vary according to fibonacci rules it is found that the ferromagnetic case differs drastically from the case with both ferro and antiferromagnetic bonds in the mixed case the peaks of the q dependent susceptibility are everywhere dense for temperature t both above or below the critical temperature tc but due to overlap only a finite number of peaks is visible this number of visible peaks decreases as t moves away from tc in the ferromagnetic case there is typically only one single peak at q 0 in spite of the aperiodicity present in the lattice these results provide evidence that in real systems even if the atoms arrange themselves aperiodically there will be no dramatic difference in the diffraction pattern unless the pair correlation function has clear aperiodic oscillations the number of oscillations per correlation length determines the number of visible peaks

helicoidal surfaces under the cubic screw motion in minkowski 3 space abstract in this paper we construct helicoidal surfaces under the cubic screw motion with prescribed mean or gauss curvature in minkowski 3 space e 1 3 we also find explicitly the relation between the mean curvature and gauss curvature of them furthermore we discuss helicoidal surfaces under the cubic screw motion with h 2 k and prove that these surfaces have equal constant principal curvatures

translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in minkowski space abstract in this paper we study the existence uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in minkowski space we also study the asymptotic behavior and the strict convexity of general solitons of such flows

entire spacelike translating solitons in minkowski space abstract in this paper we study entire spacelike translating solitons in minkowski space by constructing convex spacelike solutions to 1 3 in bounded convex domains we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity

space like loxodromes on rotational surfaces in minkowski 3 space abstract we find all space like loxodromes on rotational surfaces which have space like meridians or time like meridians respectively by using a relevant lorentzian angle in minkowski 3 space to understand loxodromes better we draw some pictures of them via mathematica computer program

exactly solved models on planar graphs with vertices in z 3 it is shown how exactly solved edge interaction models on the square lattice may be extended onto more general planar graphs with edges connecting a subset of next nearest neighbour vertices of mathbb z 3 this is done by using local deformations of the square lattice that arise through the use of the star triangle relation similar to baxter s z invariance property these local deformations leave the partition function invariant up to some simple factors coming from the star triangle relation the deformations used here extend the usual formulation of z invariance by requiring the introduction of oriented rapidity lines which form directed closed paths in the rapidity graph of the model the quasi classical limit is also considered in which case the deformations imply a classical z invariance property as well as a related local closure relation for the action functional of a system of classical discrete laplace equations

q dependent susceptibilities in ferromagnetic quasiperiodic z invariant ising models we study the q dependent susceptibility q of a series of quasiperiodic ising models on the square lattice several different kinds of aperiodic sequences of couplings are studied including the fibonacci and silver mean sequences some identities and theorems are generalized and simpler derivations are presented we find that the q dependent susceptibilities are periodic with the commensurate peaks of q located at the same positions as for the regular ising models hence incommensurate everywhere dense peaks can only occur in cases with mixed ferromagnetic antiferromagnetic interactions or if the underlying lattice is aperiodic for mixed interaction models the positions of the peaks depend strongly on the aperiodic sequence chosen

mean curvatures and gauss maps of a pair of isometric helicoidal and rotation surfaces in minkowski 3 space it is proved that in minkowski 3 space a csm helicoidal surface i e a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface by distinguishing a csm helicoidal surface as three cases that is the case of type i the case of type ii with negative and positive pitch the relations are discussed between the mean curvatures or gauss maps of a pair of isometric helicoidal and rotation surface a csm helicoidal surface of case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures resp gauss maps if and only if they are minimal but each pair of isometric csm helicoidal surface of case 3 and rotation surface with spacelike axis have different gauss maps

random fixed points for several classes of 1 ball contractive and 1 set contractive random maps a general random fixed point theorem for continuous random operators is proved as applications a number of random fixed points theorems for various classes of 1 set and 1 ball contractive random operators e g operators of contractive type with compact or completely continuous perturbations operators of semicontractive type etc are derived our results unify and extend most of the known random fixed points theorems 1991 ams subject classification 47h10 60h25

rotation hypersurfaces in spaces of constant curvature rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed a local characterization of such hypersurfaces with dimensions greater than two is given in terms of principal curvatures some special cases of rotation hypersurfaces with constant mean curvature in hyperbolic space are studied in particular it is shown that the well known conjugation between the helicoid and the catenoid in euclidean three space extends naturally to hyperbolic three space h3 in the latter case catenoids are of three different types and the explicit correspondence is given it is also shown that there exists a family of simply connected complete embedded nontotally geodesic stable minimal surfaces in h3

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

isometries between minimal helicoidal surfaces and rotation surfaces in minkowski space in minkowski 3 space a helicoidal surface is a generalization of rotation surface which is somewhat similar except the translation part therefore a natural question arises whether an isometry between these surfaces exists it is proved that in minkowski 3 space a minimal helicoidal surface with gauss curvature k has an isometric minimal rotation surface if and only if k 0 especially we show that a timelike right helicoid does not have an isometric minimal rotation surface

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

correlation functions for the heisenberg xxz antiferromagnet the general method for calculation of correlation functions in integrable quantum models has been given in papers 1 2 the correlation function of the third components of local spins for the heisenberg one dimensionalxxz antiferromagnet is calculated in this paper the answer is a series which gives in particular an improved version of the usual perturbative expansion in the anisotropy parameter the remarkable property of the series obtained is that the long distance asymptotics of the correlator is given already by the first term the arguments are given in favour of the convergence of the series

determinant formula for the six vertex model the partition function of a six vertex model with domain wall boundary conditions is considered on the finite lattice the authors show that the partition function satisfies a recursive relation they solve the recursion relation by a determinant formula this gives a determinant representation for the partition function they use the quantum inverse scattering method qism

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

critical behavior and associated conformal algebra of the z 3 potts model the conformal algebra for operators of thez 3 model at the phase transition point is built critical exponents are found in this approach as solutions of simple algebraic equations which are consistency conditions of the algebra multipoint correlation functions obey linear differential equations some solutions are given for the four point correlation functions of thez 3 model at the phase transition point

a dual resonance model solves the yang baxter equation the duality of dual resonance models is shown to imply that the four point string correlation function solves the yang baxter equation a reduction of transfer matrices to symmetry is described by a restriction of the kp function to toda molecules

the use of backlund transformations in obtaining n soliton solutions in wronskian form the wronskian formulation of the n soliton solutions of various nonlinear evolution equations modified korteweg de vries with both zero and non zero asymptotic conditions sine gordon korteweg de vries kadomtsev petviashvili and boussinesq equations are obtained by inductive use of backlund transformations in hirota s 1980 bilinear notation

helicoidal surfaces in three dimensional minkowski space abstract in this paper we solve the problem of finding explicitly the helicoidal surfaces of the minkowski space r 1 3 with prescribed gaussian or mean curvature given by smooth functions we distinguish three kinds of helicoidal surfaces corresponding to the space like time like or light like axes of revolution and give some geometric meanings of the helicoidal surfaces of the space like type we also define certain solinoid tubular surfaces of type i around a hyperbolic helix in r 1 3 and study some of their geometric properties

selected non holonomic functions in lattice statistical mechanics and enumerative combinatorics we recall that the full susceptibility series of the ising model modulo powers of the prime reduce to algebraic functions we also recall the nonlinear polynomial differential equation obtained by tutte for the generating function of the q coloured rooted triangulations by vertices which is known to have algebraic solutions for all the numbers of the form the holonomic status of being unclear we focus on the analysis of the case showing that the corresponding series is quite certainly non holonomic along the line of a previous work on the susceptibility of the ising model we consider this series modulo the first eight primes and show that this probably non holonomic function reduces modulo these primes to algebraic functions we conjecture that this probably non holonomic function reduces to algebraic functions modulo almost every prime or power of prime numbers this raises the question of whether such remarkable non holonomic functions can be seen as a ratio of diagonals of rational functions or even algebraic functions of diagonals of rational functions

on lorentzian surfaces with h2 k in minkowski 3 space abstract a well known classical result classifies the surfaces in 3 space r 3 with the mean curvatures h and the gauss curvatures k satisfying h 2 k as pieces of planes or two spheres corresponding results are given here for lorentzian surfaces in minkowski 3 space this work is done by proving that all the lorentzian surfaces with h 2 k in minkowski 3 space are null scrolls locally and can be classified into three cases two of which have no counterparts in euclidean space

exact solution of the lock step model of vicious walkers the lock step model of vicious walkers on a one dimensional lattice allows each walker at the tick of a clock to move either one lattice site to the left or one lattice site to the right the only restriction is that no two walkers may arrive at the same lattice site or pass one another in periodic boundary conditions the partition function and correlation function for this model are calculated exactly taking the continuum limit gives an exactly solvable model of vicious walkers undergoing brownian motion

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

new results for the correlation functions of the ising model and the transverse ising chain in this paper we show how an infinite system of coupled toda type nonlinear differential equations derived by one of us can be used efficiently to calculate the time dependent pair correlations in the ising chain in a transverse field the results are seen to match extremely well long large time asymptotic expansions newly derived here for our initial conditions we use new long asymptotic expansions for the equal time pair correlation functions of the transverse ising chain extending an old result of t t wu for the 2d ising model using this one can also study the equal time wavevector dependent correlation function of the quantum chain a k a the q dependent diagonal susceptibility in the 2d ising model in great detail with very little computational effort

rotationally symmetric spacelike translating solitons for the mean curvature flow in minkowski space abstract the rotationally symmetric spacelike translating solitons for the mcf in the minkowski space are considered in this paper to be specific the complete classification for such translating solitons according to their rotation axes which are timelike spacelike and lightlike is provided in particular the explicit parametrization of the rotationally symmetric spacelike hypersurface of the lightlike rotation axis with the prescribed mean curvature is obtained

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

susceptibility calculations in periodic and quasiperiodic planar ising models new results are presented for the wavevector dependent susceptibility of z invariant periodic and quasiperiodic ising models in the scaling limit generalizing old results of tracy and mccoy for the square lattice explicit results are worked out for the two leading singular terms of the susceptibility on four regular isotropic lattices the methods used provide a proof of the extended lattice lattice scaling hypothesis for the class of models under consideration

quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

rational mappings arborescent iterations and the symmetries of integrability we describe a class of nonlinear birational representations of groups generated by a finite number of involutions these groups are symmetries of the yang baxter equations and their higher dimensional generalizations they provide discrete dynamical systems with a variety of behaviors from chaotic to integrable according to the number of invariants of the representation

scaling functions in the square ising model we show and give the linear differential operators of order for the integrals which appear in the two point correlation scaling function of ising model the integrals are given in expansion around in the basis of the formal solutions of with transcendental combination coefficients we find that the expression is a solution of the painleve vi equation in the scaling limit combinations of the analytic at solutions of sum to we show that the expression is the scaling limit of the correlation function and the differential galois groups of the factors occurring in the operators are given

a symmetric generalization of linear backlund transformation associated with the hirota bilinear difference equation the hirota bilinear difference equation is generalized to discrete space of arbitrary dimension solutions to the nonlinear difference equations can be obtained via backlund transformation of the corresponding linear problems

new fixed point theorems for 1 set contractive operators in banach spaces abstract w v petryshyn remark on condensing and k set contractive mappings j math anal appl 39 1972 717 741 and r d nussbaum degree theory for local condensing maps j math anal appl 37 1972 741 766 defined the topological degree of 1 set contractive fields and studied fixed point theorems of 1 set contractive operators by virtue of the potential tool following them g z li the fixed point index and the fixed point theorems of 1 set contraction mappings proc amer math soc 104 1988 1163 1170 introduced the concept of semi closed 1 set contractive operators and studied the fixed point theorems for such a class of operators in this paper we continue to study semi closed 1 set contractive operators a and investigate the boundary conditions under which the topological degrees of 1 set contractive fields deg i a p are equal to 1 correspondingly we can obtain some new fixed point theorems for 1 set contractive operators which improve and extend many famous theorems such as the leray schauder theorem rothe s theorem altman s theorem petryshyn s theorem etc on the other hand this class of 1 set contractive operators includes strict set contractive operators condensing operators semi contractive operators and others see g z li the fixed point index and the fixed point theorems of 1 set contraction mappings proc amer math soc 104 1988 1163 1170 l s liu approximation theorems and fixed point theorems for various class of 1 set contractive mappings in banach spaces acta math sinica 17 2001 103 112 so the results in this paper remain valid for the above mentioned operators in addition our conclusions and methods are different from the ones in many recent works see l s liu approximation theorems and fixed point theorems for various class of 1 set contractive mappings in banach spaces acta math sinica 17 2001 103 112 n shahzad s latif random fixed points for several classes of 1 ball contractive and 1 set contractive random maps j math anal appl 237 1999 83 92 t c lin random approximations and random fixed point theorems for continuous 1 set contractive random maps proc amer math soc 123 1995 1167 1176

extended z invariance for integrable vector and face models and multi component integrable quad equations in a previous paper kels in j phys a 50 49 495202 2017 the author has established an extension of the z invariance property for integrable edge interaction models of statistical mechanics that satisfy the star triangle relation str form of the yang baxter equation ybe in the present paper an analogous extended z invariance property is shown to also hold for integrable vector models and interaction round a face irf models of statistical mechanics respectively as for the previous case of the str the z invariance property is shown through the use of local cubic type deformations of a 2 dimensional surface associated to the models which allow an extension of the models onto a subset of next nearest neighbour vertices of mathbb z 3 while leaving the partition functions invariant these deformations are permitted as a consequence of the respective ybe s satisfied by the models the quasi classical limit is also considered and it is shown that an analogous z invariance property holds for the variational formulation of classical discrete laplace equations which arise in this limit from this limit new integrable 3d consistent multi component quad equations are proposed which are constructed from a degeneration of the equations of motion for irf boltzmann weights

isomonodromic deformation theory and the next to diagonal correlations of the anisotropic square lattice ising model in 1980 jimbo and miwa evaluated the diagonal two point correlation function of the square lattice ising model as a function of the sixth painleve system by constructing an associated isomonodromic system within their theory of holonomic quantum fields more recently an alternative isomonodromy theory was constructed based on bi orthogonal polynomials on the unit circle with regular semi classical weights for which the diagonal ising correlations arise as the leading coefficient of the polynomials specialized appropriately here we demonstrate that the next to diagonal correlations of the anisotropic ising model are evaluated as one of the elements of this isomonodromic system or essentially as the cauchy hilbert transform of one of the bi orthogonal polynomials

rigidity of complete spacelike translating solitons in pseudo euclidean space abstract in this paper we investigate the parametric version and non parametric version of rigidity theorem of spacelike translating solitons in pseudo euclidean space r n m n firstly we classify m dimensional complete spacelike translating solitons in r n m n and prove the only complete spacelike translating solitons in r n m n are the spacelike m planes secondly we generalize the rigidity theorem of entire spacelike lagrangian translating solitons to spacelike translating solitons with general codimensions

the quantum inverse scattering method approach to correlation functions the inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with ther matrix of xxx type these models include the one dimensional bose gas and the heisenberg xxx model the algebraic questions of the problem are considered

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

the ising susceptibility scaling function we have dramatically extended the zero field susceptibility series at both high and low temperature of the ising model on the triangular and honeycomb lattices and used these data and newly available further terms for the square lattice to calculate a number of terms in the scaling function expansion around both the ferromagnetic and for the square and honeycomb lattices the antiferromagnetic critical point

continued fraction as a discrete nonlinear transform the connection between a taylor series and a continued fraction involves a nonlinear relation between the taylor coefficients an and the continued fraction coefficients bn in many instances it turns out that this nonlinear relation transforms a complicated sequence an into a very simple one bn this simplification is illustrated in the context of graph combinatorics

the backlund transformation equations for the ultradiscrete kp equation the backlund transformation for the ultradiscrete kp equation is proposed an algorithm to eliminate variables from the ultradiscrete linear equations is proposed the consistency condition for the backlund transformation equations is obtained via the algorithm

helicoidal surfaces with h2 k in minkowski 3 space in this paper we prove that any helicoidal surface with h2 k in minkowski 3 space is locally a generalized umbilical surface i e a surface with the shape operator having the minimal polynomial x a 2 where a is constant we also give the representation formulas for these helicoidal surfaces

null scrolls as solutions of a sigma model the two dimensional o 2 1 nonlinear sigma model with boundary is considered we calibrate the size of its space of field configurations by exhibiting new and wide classes of solutions we first construct solutions by evolving under a certain group of transformations free elastic curves in any surface either riemannian or lorentzian of constant curvature furthermore we show that any null scroll can provide a solution of this sigma model this surprising phenomenon which obviously has no euclidean counterpart guarantees the existence of an ample class of solutions which are generated by null or lightlike curves evolving through null ruling flows

minimum selections and fixed points of set valued operators in banach spaces with some uniform convexity a new continuity theorem of minimum selection is presented for a continuous set valued operator from a topological space into a banach space with some uniform convexity as applications some problems concerning minimum right inverses for linear operators and minimum fixed points for condensing set valued nonlinear operators are discussed also the existence of minimum solutions for an integral inclusion is proved

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