solutions of the yang baxter equation for isotropic quantum spin chains the author considers solutions of the yang baxter equation such that the logarithmic derivative of the transfer matrix yields a quantum spin hamiltonian which is isotropic in spin space i e su 2 invariant four such solutions are known for each value of the spin s for s 1 2 they degenerate into the same solution and for s 1 they only give three different solutions for s or 6 he shows that these are the only solutions which are su 2 invariant except for s 3 when there is a fifth solution

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

a sketched finite element method for elliptic models abstract we consider a sketched implementation of the finite element method for elliptic partial differential equations on high dimensional models motivated by applications in real time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low dimensional subspace and sketching the reduced equations using randomised sampling we show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete laplacian operator can achieve nearly optimal performance and a significant speedup we derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability and an upper bound for the distance between the sketched and the high dimensional solutions our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction

iterative reweighted algorithms for matrix rank minimization the problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning and is known to be np hard a tractable relaxation for this problem is nuclear norm or trace norm minimization which is guaranteed to find the minimum rank matrix under suitable assumptions in this paper we propose a family of iterative reweighted least squares algorithms irls p with 0 p 1 as a computationally efficient way to improve over the performance of nuclear norm minimization the algorithms can be viewed as locally minimizing certain smooth approximations to the rank function when p 1 we give theoretical guarantees similar to those for nuclear norm minimization that is recovery of low rank matrices under certain assumptions on the operator defining the constraints for p 1 irls p shows better empirical performance in terms of recovering low rank matrices than nuclear norm minimization we provide an efficient implementation for irls p and also present a related family of algorithms sirls p these algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem as well as on the movielens movie recommendation data set

finite size corrections and numerical calculations for long spin 1 2 heisenberg chains in the critical region leading and next to leading order finite size corrections to the ground and first excited states are calculated for the spin 1 2 anisotropic heisenberg model in the critical region the analytic results are compared to numerical data obtained for chains up to a length of n 1024 it is found that near the isotropic point the asymptotic region where the results obtained for n to infinity are applicable sets in at very large n values and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms even at large n 100 chain lengths

ground state energy and mass gap of a generalized quantum spin ladder we show that a two leg ladder hamiltonian introduced recently by albeverio and fei can be made to satisfy the hecke algebra as a result we have found an equivalent representation of the eigenspectrum in terms of the spin antiferromagnetic xxz chain at 5 3 the values of thermodynamic quantities such as the ground state energy and mass gap follow from the known xxz results

a class of interaction round a face models and its equivalence with an ice type model a new model called the temperley lieb interactions model is introduced in two dimensional lattice statistics on a square lattice the temperley lieb equivalence of this model to the six vertex self dual potts critical hard hexagons and critical nonintersecting string models is established a graphical equivalence of this model to the six vertex model generalizes this equivalence to noncritical cases of the above models the order parameters of a specialization of this model are studied

free fermion checkerboard and z invariant lattice models in statistical mechanics it is shown that the two dimensional free fermion model is equivalent to a checkerboard ising model which is a special case of the general z invariant ising model expressions are given for the partition function and local correlations in terms of those of the regular square lattice ising model corresponding results are given for the self dual potts model and the application of the methods to the three dimensional zamolodchikov model is discussed the paper ends with a discussion of the critical and disorder surfaces of the checkerboard potts model

conformal anomaly and critical exponents of the spin 1 takhtajan babujian model the authors consider the critical behaviour of the spin 1 hamiltonian of the takhtajan babujian model the bethe ansatz equations for this antiferromagnet in a finite chain are investigated analytically and numerically the conformal anomaly and critical exponents are obtained by exploiting their relations with the eigenspectrum of the finite system the results strongly support the conjecture that the wess zumino witten non linear sigma model with topological charge k 2 is the underlying field theory for this statistical mechanics model

the weighted dual functionals for the univariate bernstein basis abstract we find an explicit formula for the weighted dual functions of the bernstein polynomials with respect to the jacobi weight function using the usual inner product in the hilbert space l 2 0 1 we define the weighted dual functionals of the bernstein polynomials which are used to find the coefficients in the least squares approximation

nonintersecting string model and graphical approach equivalence with a potts model using a graphical method we establish the exact equivalence of the partition function of aq state nonintersecting string nis model on an arbitrary planar even valenced lattice with that of a q2 state potts model on a related lattice the nis model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types and the resulting potts model generally has multispin interactions for the square and kagome lattices this leads to the equivalence of a staggered nis model with potts models with anisotropic pair interactions indicating that these nis models have a first order transition forq 2 for the triangular lattice the nis model turns out to be the five vertex model of wu and lin and it relates to a potts model with two and three site interactions the most general model we discuss is an oriented nis model which contains the six vertex model and the nis models of stroganov and schultz as special cases

multicoloured temperley lieb lattice models the ground state using the inversion relation method we calculate the ground state energy for the lattice integrable models based on baxterization of multicoloured generalization of temperley lieb algebras the simplest vertex model is analysed and found to have a mass gap

construction of dual b spline functions using new properties of dual functions we show that the method of constructing the dual bases recently presented in p wozny construction of dual bases journal of computational and applied mathematics 245 2013 75 85 can be significantly simplified next we propose a simple algorithm of constructing the dual b spline functions which can be useful in solving some approximation problems related to numerical analysis or computer graphics

quantum line operators from lax pairs motivated by the realization of the yang baxter equation of 2d integrable models in the 4d gauge theory of costello witten yamazaki cwy we study the embedding of integrable 2d toda field models inside this construction this is done by using the lax formulation of 2d integrable systems and by thinking of the standard lax pair l in terms of components of cwy gauge connection propagating along particular directions in the gauge bundle we also use the results of the cwy theory to build quantum line operators for 2d toda models and compute the one loop contribution of two intersecting lines exchanging one gluon other features such as local symmetries and comments on extension of our method to other 2d integrable models are also discussed we also comment on some basic points that still need a refinement before discussing about a fully consistent embedding of lax pairs into cwy theory

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

anisotropic correlation length in the eight vertex model we investigate the directional dependence or anisotropy of the correlation length of the eight vertex model by a method which introduces the shift operator into the usual transfer matrix argument for a given x 0 x 1 there are two cases with respect to a parameter q in the case 0 q x3 the anisotropic correlation length acl is independent of q noting that the eight vertex model factors into two ising lattices in the q x4 limit we can directly relate the acl to that of the square lattice ising model when x3 q x2 the acl depends on q using an algebraic curve we show that the acl in this case is connected with that of the ising model on a 4 8 or union jack lattice we suggest that a wide class of models have essentially the same acl as the eight vertex model has

the dual bases for the bezier said wang type generalized ball polynomial bases and their applications abstract a unifying representation for the existing generalized ball bases and the bernstein bases are given then the dual bases for the bezier said wang type generalized bases bswgb for short are presented the marsden identity and the mutual transformation formulas between bezier curve and bezier said wang type generalized curve bswgb curve are also given these results are very useful for the applications of bswgb curves and their popularization in cagd numerical examples are also given to show the effectiveness of our methods

the operator content of the exactly integrable su n magnets the operator content of the exactly integrable models associated with the su n lie algebras with periodic boundaries are studied the authors results were obtained from a numerical and analytical analysis of the associated bethe ansatz equations the resulting spectra indicate that the ferromagnetic and antiferromagnetic models have different operator content for n 2 being realizations of distinct modular invariant solutions

finite size scaling for correlations of quantum spin chains at criticality we study the finite size scaling behavior of two point correlation functions of translationally invariant many body systems at criticality we propose an efficient method for calculating the two point correlation functions in the thermodynamic limit from numerical data of finite systems our method is most effective when applied to a two dimensional classical system which possesses a conformal invariance by using this method with numerical data obtained from exact diagonalizations and monte carlo simulations we study the spin spin correlations of the quantum spin 1 2 and 3 2 antifierromagnetic chains in particular the logarithmic corrections to power law decay of the correlation of the spin 1 2 isotropic heisenberg antiferromagnetic chain are studied thoroughly we clarify the cause of the discrepancy in previous calculations for the logarithmic corrections our result strongly supports the field theoretic prediction based on the mappings to the wess zumino witten nonlinear model or the sine gordon model we also treat logarithmic corrections and crossover phenomena in the spin spin correlation of the spin 3 2 isotropic heisenberg antiferromagnetic chain our results are consistent with the affleck haldane prediction that the correlation of the spin 3 2 chain exhibits a crossover to the same asymptotic behavior as in the spin 1 2 chain

critical behaviors of mutual information in the one dimensional spin 1 bilinear biquadratic model the critical behaviors of the mutual information in the one dimensional spin 1 bilinear biquadratic model are studied it is found that all the quantum phase transition points including the order disorder ones and the infinite order ones in the system can be quantified from the singularity analysis of the mutual information of two sites with the longest distance because the mutual information captures the correlation properties about the critical fluctuations more interestingly the special structure of the wavefunction and the special symmetric property of the system can be indicated by the entropy correlation behaviors the finite size scaling laws of the mutual information and its derivative are also studied

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

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

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

simple algorithms for computing the bezier coefficients of the constrained dual bernstein polynomials abstract simple recurrence relations for the bezier coefficients of the dual bernstein polynomials are given using the recurrence relations we propose fast methods for computing these coefficients the complexity of the proposed algorithms is o n where n is the degree of the considered dual bernstein polynomial basis

conditioning of finite element equations with arbitrary anisotropic meshes this is the published version also available here http dx doi org 10 1090 s0025 5718 2014 02822 6 first published in math comput in 2014 published by the american mathematical society

fa yueh wu s contributions in physics abstract we sketch a subset of professor f y wu s contributions in lattice statistical mechanics solid state physics graph theory enumerative combinatorics and other domains of physics and mathematics we will recall some of f y wu s most important and well known classic results and we will also sketch some of his most recent researches dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics

iterative hessian sketch fast and accurate solution approximation for constrained least squares we study randomized sketching methods for approximately solving least squares problem with a general convex constraint the quality of a least squares approximation can be assessed in different ways either in terms of the value of the quadratic objective function cost approximation or in terms of some distance measure between the approximate minimizer and the true minimizer solution approximation focusing on the latter criterion our first main result provides a general lower bound on any randomized method that sketches both the data matrix and vector in a least squares problem as a surprising consequence the most widely used least squares sketch is sub optimal for solution approximation we then present a new method known as the iterative hessian sketch and show that it can be used to obtain approximations to the original least squares problem using a projection dimension proportional to the statistical complexity of the least squares minimizer and a logarithmic number of iterations we illustrate our general theory with simulations for both unconstrained and constrained versions of least squares including l1 regularization and nuclear norm constraints we also numerically demonstrate the practicality of our approach in a real face expression classification experiment

diagonalization of the transfer matrix of a nonintersecting string model abstract in this paper we obtain the eigenvectors of the transfer matrix of a nonintersecting string model using the bethe ansatz the result is in agreement with baxter s conjecture of weak equivalence of the present model with the six vertex model

the lowest excitations in the spin s xxx magnet and conformal invariance the bethe ansatz equations for the integrable spin s isotropic heisenberg antiferromagnet are solved numerically at a finite number of sites n the vacuum and the lowest singlet and triplet solutions are presented for s 1 2 2 up to sn 240 through extrapolation of the finite size data the central charge and anomalous dimensions of the scaling operators for the underlying conformal field theory are calculated and found to agree with the expected theoretical values the status of the bethe string hypothesis about the structure of the solutions is discussed based on the obtained computer data up to s 9 2

the spectra of q state vertex models and related antiferromagnetic quantum spin chains some exactly solvable q state vertex models are investigated the author employs inversion relations to determine directly the spectra of the transfer matrices in the thermodynamic limit by avoiding the more cumbersome bethe ansatz the results are applied to related quantum spin chains of which two families are su q and so q invariant respectively various quantities are calculated e g energy momentum excitations and the correlation length for q 3 the su q invariant chain is the pure biquadratic spin 1 hamiltonian which turns out to be non critical the ground state energy the gap and the correlation length are given

projected equation methods for approximate solution of large linear systems we consider linear systems of equations and solution approximations derived by projection on a low dimensional subspace we propose stochastic iterative algorithms based on simulation which converge to the approximate solution and are suitable for very large dimensional problems the algorithms are extensions of recent approximate dynamic programming methods known as temporal difference methods which solve a projected form of bellman s equation by using simulation based approximations to this equation or by using a projected value iteration method

construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes abstract given a sequence of finite element spaces which form a de rham sequence we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de rham sequence the dual representations also need to satisfy the de rham sequence on the domain boundary the matrix which converts primal representations to dual representations the hodge matrix is the mass or gram matrix it will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients degrees of freedom of both representations this leads to very sparse system matrices even for high order methods the differential operators for dual representations will be defined vector operations grad curl and div for primal and dual representations are both topological and do not depend on the metric i e the size and shape of the mesh or the order of the numerical method derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations as illustration of the use of dual representations the method will be applied to i a mixed formulation for the poisson problem in 3d ii it will be shown that this approach allows one to preserve the equivalence between dirichlet and neumann problems in the finite dimensional setting and iii the method will be applied to the approximation of grad div eigenvalue problem on affine and non affine meshes

dual basis functions for the ns power basis and their applications considering the outstanding properties of the ns power basis which was given by sanchez reyes in 1997 the dual basis functions for the ns power basis are presented in this paper marsden identity and the transformation formulae from bezier curves uniform b spline curves to ns power curves are also given these results are useful for the applications of ns power curves and their popularization in computer aided geometric design

the completely packed o n loop model on the square lattice we present an investigation of the completely packed o n loop model on the square lattice by means of the transfer matrix method and finite size scaling we investigate the model for a number of n values covering a wide range this model is known to be equivalent with the q state potts model with q n 2 but here we also investigate the range n 2 the system is in an ordered phase where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence furthermore we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six vertex model this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

on the solution of some vertex models using factorizable s matrices the intimate connection between factorizables matrices and some vertex models in two dimensions to be reviewed here is exploited to show that the knowledge of thes matrix not only allows us to define a solvable vertex modela la zamolodchikov but often to write down the free energy by inspection the prototype for discussion is baxter s eight vertex model generated by zamolodchikov s z4 s matrix the method is then applied to a hitherto unsolved 19 vertex model based on the isospin 1s matrix of zamolidchikov and fateev and agreement is checked to fourth order in a perturbation series the possibility of molding other problems like theq state potts model into this framework is considered

fast approximation of matrix coherence and statistical leverage the statistical leverage scores of a matrix a are the squared row norms of the matrix containing its top left singular vectors and the coherence is the largest leverage score these quantities are of interest in recently popular problems such as matrix completion and nystrom based low rank matrix approximation as well as in large scale statistical data analysis applications more generally moreover they are of interest since they define the key structural nonuniformity that must be dealt with in developing fast randomized matrix algorithms our main result is a randomized algorithm that takes as input an arbitrary n d matrix a with n d and that returns as output relative error approximations to all n of the statistical leverage scores the proposed algorithm runs under assumptions on the precise values of n and d in o nd logn time as opposed to the o nd2 time required by the naive algorithm that involves computing an orthogonal basis for the range of a our analysis may be viewed in terms of computing a relative error approximation to an underconstrained least squares approximation problem or relatedly it may be viewed as an application of johnson lindenstrauss type ideas several practically important extensions of our basic result are also described including the approximation of so called cross leverage scores the extension of these ideas to matrices with n d and the extension to streaming environments

conformal invariance and the xxz chain with arbitrary spin abstract the critical properties of the anistropic heisenberg chain or xxz model with arbitrary spins s are studied the eigenspectrum of these models for finite chain length is calculated by solving their associated bethe ansatz equations the operator content with an even and odd number of sites is obtained by exploring the consequences of conformal invariance the results indicate the existence of critical lines governed by conformal theories with central charge c 3s 1 s and c 1 with critical exponents varying continuously with the anisotropy like in the 8 vertex model

finite size corrections for ground states of the xxz heisenberg chain the method of de vega and woynarovitch 1985 is used to calculate finite size corrections to the ground state energy in different sectors for the xxz heisenberg chain finite size scaling amplitudes and correction to scaling exponents in the critical region are derived using conformal invariance a scaling dimension x pi gamma 2 pi is extracted corresponding to the electric field operator in the 8 vertex model this confirms a conjecture of baxter and kelland 1974 finite size scaling properties near the kosterlitz thouless critical point delta 1 are discussed

algebraic bethe ansatz for 19 vertex models with reflection conditions in this work we solve the 19 vertex models with the use of algebraic bethe ansatz for diagonal reflection matrices sklyanin k matrices the eigenvectors eigenvalues and bethe equations are given in a general form quantum spin chains of spin one derived from the 19 vertex models were also discussed

graphical approach to the nonintersecting string model star triangle equation inversion relation and exact solution abstract a general q state nonintersecting string model for q 2 is studied and solved using a graphical approach this model of which special cases were first introduced by strogonov and schultz has recently drawn further attention because of its relation to potts models in its simplest version a uniform separable nonintersecting string model on a quadratic lattice it corresponds to a q2 state potts model with potts spins on every other lattice face we first formulate the related star triangle equation which we solve in terms of a line variable parametrization such a parametrization of a solution of the star triangle equation leads to a simple and direct graphical derivation of an inversion relation for the partition function our graphical analysis also shows that the inversion relation holds if certain boundary effects can be neglected as we shall give a special finite lattice from which the inversion relation can be read off immediately this relation is next solved under appropriate analyticity assumptions again using a simple and direct approach and the result is applied to obtain exact solutions for specific string models

bezier representation of the constrained dual bernstein polynomials abstract explicit formulae for the bezier coefficients of the constrained dual bernstein basis polynomials are derived in terms of the hahn orthogonal polynomials using difference properties of the latter polynomials efficient recursive scheme is obtained to compute these coefficients applications of this result to some problems of cagd is discussed

the o n model on the triangular lattice to find critical points of o n models on the triangular lattice we apply two methods first we investigate the yang baxter equations on the triangular lattice we find only solvable points directly related to those for the square lattice second we construct intersections with the potts model this yields eight branches of critical points parametrized by n we establish the equivalence of these branches with the known critical points of the o n model on the square lattice transfer matrix calculations are performed to obtain the conformal anomaly and the thermal exponent of these branches these results include a numerical analysis of a q 3 potts tricritical point we find analytic relations between potts and o n models as well as among o n models with different values of n and among potts models with different values of q

solvable spin 1 models in one dimension we look for solvable spin 1 su 3 models in the seven parameter manifold of translation and reflection invariant hamiltonians with nearest neighbour couplings we prove that only discrete solutions of the reshetikhin criterion exist therefore the situation here differs greatly from the corresponding spin 1 2 su 2 case where reshetikhin s criterion is satisfied for any model of the xyz type

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

a dual functional to the univariate b spline in this paper dual functionals with local supports to the univariate b splines are constructed by symmetrization which are a linear combination of function values

geometric aspects of quantum spin states a number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system s equilibrium states methods of general applicability are introduced in the context of the su 2s 1 invariant quantum spin s chains with the interaction p o wherep o is the projection onto the singlet state of a pair of nearest neighbor spins the phenomena discussed here include the absence of neel order the possibility of dimerization conditions for the existence of a spectral gap and a dichotomy analogous to one found by affleck and lieb stating that the systems exhibit either slow decay of correlations or translation symmetry breaking our representation elucidates the relation evidence for which was found earlier of the p o spin s systems with the potts and the fortuin kasteleyn random cluster models in one more dimension the method reveals the geometric aspects of the listed phenomena and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin e g within such structure the dichotomy is implied by a topological argument and the alternatives correspond to whether or not the clusters are of finite mean length

stable finite elements for problems with wild coefficients we consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain a linear finite element method is used it is shown that the standard method for solving the resulting linear equations in finite precision arithmetic can give an arbitrarily inaccurate answer because of ill conditioning in the stiffness matrix a new method for solving the linear equations is proposed this method is based on a mixed formulation and gives a numerically accurate answer independent of the variation in the coefficients the numerical error in the soution of the linear system for the new method is shown to depend on the aspect ratio of the triangulation

spin swapping operator as an entanglement witness for quantum heisenberg spin s systems using the su n representation of the group theory we derive the general form of the spin swapping operator for the quantum heisenberg spin s systems we further prove that such a spin swapping operator is equal to the spin singlet pairing operator under the partial transposition for su 2 invariant states it is shown that the expectation value of the spin swapping operator and its generalizations the permutations can be used as an entanglement witness especially for the formulation of observable conditions of entanglement

constrained multi degree reduction of triangular bezier surfaces using dual bernstein polynomials we propose a novel approach to the problem of multi degree reduction of bezier triangular patches with prescribed boundary control points we observe that the solution can be given in terms of bivariate dual discrete bernstein polynomials the algorithm is very efficient thanks to using the recursive properties of these polynomials the complexity of the method is o n 2m 2 n and m being the degrees of the input and output bezier surfaces respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined triangular bezier surfaces the result is a composite surface of global c r continuity with a prescribed order r some illustrative examples are given

fractional strings hypothesis and non simple laced integrable models the author discusses the ground state and some elementary excitations of the bp non simple laced integrable models the zeros of the associated bethe ansatz equations are characterized by a special distribution in the complex plane which is different from the usual string hypothesis the introduction of this new solution makes it possible to calculate exactly various properties in the thermodynamic limit the finite size corrections to the eigenspectrum strongly suggest that the bp wess zumino witten novikov models are the underlying conformal field theory for these non simple laced models

weighted dual functions for bernstein basis satisfying boundary constraints in this paper we consider the issue of dual functions for the bernstein basis which satisfy boundary conditions the jacobi weight function with the usual inner product in the hilbert space are used some examples of the transformation matrices are given some figures for the weighted dual functions of the bernstein basis with respect to the jacobi weight function satisfying boundary conditions are plotted we discuss special cases of the jacobi weight function as the legendre weight function and the chebyshev weight functions of the first second and third kinds

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

construction of dual bases let b n b 0 b 1 b n n 0 1 n n n be the sets of linearly independent functions we give a simple method of construction the dual functions d n d 0 n d 1 n d n n 0 n n satisfying the following conditions spand n spanb n and d i j 0 i j n n where d i i 1 d i j 0 for i j and is a given inner product the proposed algorithm allows us to construct all the sets of the dual functions d 0 d 1 d n in the time o n 3 where n is a natural number four illustrative examples presenting the possible applications of obtained results are given

multi degree reduction of tensor product bezier surfaces with general boundary constraints abstract we propose an efficient approach to the problem of multi degree reduction of rectangular bezier patches with prescribed boundary control points we observe that the solution can be given in terms of constrained bivariate dual bernstein polynomials the complexity of the method is o mn 1 n 2 with m min m1 m2 where n1 n2 and m1 m2 is the degree of the input and output bezier surface respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined rectangular bezier surfaces the result is a composite surface of global cr continuity with a prescribed r 0 in the detailed discussion we restrict ourselves to r 0 1 which is the most important case in practical application some illustrative examples are given

connections between two variable bernstein and jacobi polynomials on the triangle connection coefficients between the two variable bernstein and jacobi polynomial families on the triangle are given explicitly as evaluations of two variable hahn polynomials dual two variable bernstein polynomials are introduced explicit formula in terms of two variable jacobi polynomials and a recurrence relation are given

dual bases for wang bezier basis and their applications this paper presents the dual bases for wang bezier curves with a position parameter l which include bezier curve wang ball curve and some intermediate curves the marsden identity and the transformation formulas from bezier curve to wang bezier curve are also given these results are useful for the application of wang bezier curve and their popularization in computer aided geometric design

solving ill conditioned and singular linear systems a tutorial on regularization it is shown that the basic regularization procedures for finding meaningful approximate solutions of ill conditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course apart from rewriting many known results in a more elementary form we also derive a new two parameter family of merit functions for the determination of the regularization parameter the traditional merit functions from generalized cross validation gcv and generalized maximum likelihood gml are recovered as special cases

conformal invariance and critical exponents of the takhtajan babujian models the authors are concerned with the critical properties of antiferromagnetic takhtajan babujian models with spin s 1 3 2 and 2 the leading eigenenergies of this hamiltonian in a finite chain are calculated by investigating numerically and analytically the bethe ansatz equations for the finite system the critical exponents and the conformal anomaly are obtained from their relations with the eigenspectrum of the finite hamiltonian the appearance of logarithmic corrections produces poor estimates however a combination of analytical and numerical methods produces very good estimates their results strongly support the conjecture that the wess zumino witten novikov non linear sigma models with topological charge k 2s are the underlying field theories for these spin s statistical mechanics models

finite size corrections for spin s heisenberg chains and conformal properties the author compares recent results on finite size corrections for the isotropic xxx heisenberg chain and its integrable generalisations with former numerical and analytic calculations for the ground state and its lowest excitations leading and non leading terms are considered the analysis confirms the predictions of conformal invariance and sheds some new light on the fine structure of finite size corrections

on spin systems related to the temperley lieb algebra the spectrum of the transfer matrices constructed from the spectral parameter dependent temperley lieb r matrix is found using functional relations identical to those of the spin 1 2 xxz magnet

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

the weighted dual functions for wang bezier type generalized ball bases and their applications by introducing the inner product matrix of two vector functions and using conversion matrix explicit formulas for the dual basis functions of wang bezier type generalized ball bases wbgb with respect to the jacobi weight function are given the dual basis functions with and without boundary constraints are also considered as a result the paper includes the weighted dual basis functions of bernstein basis wang ball basis and some intermediate bases dual functionals for wbgb and the least square approximation polynomials are also obtained

generalized t splines and vmcr t meshes abstract the paper considers the extension of the t spline approach to the generalized b splines gb splines a relevant class of non polynomial splines the generalized t splines gt splines are based both on the framework of classical polynomial t splines and on the trigonometric gt splines tgt splines a particular case of gt splines our study of gt splines introduces a class of t meshes named vmcr t meshes for which both the corresponding gt splines and the corresponding polynomial t splines are linearly independent a practical characterization can be given for a sub class of vmcr t meshes which we refer to as weakly dual compatible t meshes which properly includes the class of dual compatible equivalently analysis suitable t meshes for an arbitrary polynomial order

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

simple derivation of the baxter model free energy starting with some observations due to a b zamolodchikov a short cut to the determination of the baxter eight vertex model free energy is developed the method involves factorizable s matrices in 1 1 dimensions which are briefly reviewed if the method generalizes we may associate with each known s matrix a baxter like model with a known free energy

equilibrium crystal shape of hard squares with diagonal attractions the author exactly calculates the anisotropic interfacial tension of a square lattice gas model with nearest neighbour exclusion and next nearest neighbour attractions this is achieved by a method which introduces the shift operator into the standard transfer matrix argument from the calculated anisotropic interfacial tension the equilibrium crystal shape ecs is derived by the use of the wulff construction the ecs is a closed curve in the x y plane represented as cosh lambda x y kbt a3cosh lambda x y kbt a4 2 0 with a scale factor lambda and constants a3 a4 he argues that this shape is a universal one which appears as the ecss of a wide class of models