correction induced by irrelevant operators in the correlators of the two dimensional ising model in a magnetic field we investigate the presence of irrelevant operators in the two dimensional ising model perturbed by a magnetic field by studying the corrections induced by these operators in the spin spin correlator of the model to this end we perform a set of high precision simulations for the correlator both along the axes and along the diagonal of the lattice by comparing the numerical results with the predictions of a perturbative expansion around the critical point we find unambiguous evidence of the presence of such irrelevant operators it turns out that among the irrelevant operators the one which gives the largest correction is the spin 4 operator t 2 2 which accounts for the breaking of the rotational invariance due to the lattice this result agrees with what was already known for the correlator evaluated exactly at the critical point and also with recent results obtained in the case of the thermal perturbation of the model

equational specifications complete term rewriting systems and computable and semicomputable algebras we classify the computable and semicomputable algebras in terms of finite equational initial algebra specifications and their properties as term term rewriting systems such as completeness further results on properties of these specifications such as on their size and orthogonality are provided which show that our main results are the best possible

cooperative received signal strength based sensor localization with unknown transmit powers cooperative localization also known as sensor network localization using received signal strength rss measurements when the source transmit powers are different and unknown is investigated previous studies were based on the assumption that the transmit powers of source nodes are the same and perfectly known which is not practical in this paper the source transmit powers are considered as nuisance parameters and estimated along with the source locations the corresponding cramer rao lower bound crlb of the problem is derived to find the maximum likelihood ml estimator it is necessary to solve a nonlinear and nonconvex optimization problem which is computationally complex to avoid the difficulty in solving the ml estimator we derive a novel semidefinite programming sdp relaxation technique by converting the ml minimization problem into a convex problem which can be solved efficiently the algorithm requires only an estimate of the path loss exponent ple we initially assume that perfect knowledge of the ple is available but we then examine the effect of imperfect knowledge of the ple on the proposed sdp algorithm the complexity analyses of the proposed algorithms are also studied in detail computer simulations showing the remarkable performance of the proposed sdp algorithm are presented

high precision estimate of g4 in the 2d ising model we compute the renormalized four point coupling in the 2d ising model using transfer matrix techniques we greatly reduce the systematic uncertainties which usually affect this type of calculation by using the exact knowledge of several terms in the scaling function of the free energy our final result is g4 14 697 35 3

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility 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

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

bias reduction for an explicit solution of source localization using tdoa this paper proposes two methods to reduce the bias of the well known algebraic explicit solution chan and ho a simple and efficient estimator for hyperbolic location ieee trans signal process vol 42 pp 1905 1915 aug 1994 for source localization using tdoa bias of a source location estimate is significant when the measurement noise is large and the geolocation geometry is poor bias also dominates performance when multiple times of independent measurements are available such as in uwb localization or in target tracking the paper starts by deriving the bias of the source location estimate from chan and ho the bias is found to be considerably larger than that of the maximum likelihood estimator two methods called biassub and biasred are developed to reduce the bias the biassub method subtracts the expected bias from the solution of chan and ho s work where the expected bias is approximated by the theoretical bias using the estimated source location and noisy data measurements the biasred method augments the equation error formulation and imposes a constraint to improve the source location estimate the biassub method requires the exact knowledge of the noise covariance matrix and biasred only needs the structure of it analysis shows that both methods reduce the bias considerably and achieve the crlb performance for distant source when the noise is gaussian and small the biassub method can nearly eliminate the bias and the biasred method is able to lower the bias to the same level as the maximum likelihood estimator the biasred method is extended for tdoa and fdoa positioning simulations corroborate the performance of the proposed methods

on the singularity structure of the 2d ising model susceptibility some simplifications of the integrals 2n 1 derived by wu et al 1976 13 316 that contribute to the zero field susceptibility of the 2d square lattice ising model are reported in particular several alternate expressions for the integrands in 2n 1 are determined which greatly facilitate both the generation of high temperature series and analytical analysis one can show that as series 2n 1 22n s 2 4n n 1 1 o s where s is the high temperature variable sin 2k with k the conventional normalized inverse temperature analysis of the integrals near symmetry points of the integrands shows that 2n 1 s is singular on the unit circle at sk exp ik where 2cos k cos 2 k 2n 1 cos 2 2n 1 n k n the singularities k 0 excepted are logarithmic branch points of order 2n n 1 1ln with 1 s sk there is numerical evidence from series that these van hove points in addition to the known points at s 1 and i exhaust the singularities on the unit circle barring cancellation from extra unobserved singularities one can conclude that s 1 is a natural boundary for the susceptibility

low temperature series expansions for the spin 1 ising model the finite lattice method of series expansion has been used to extend low temperature series for the partition function order parameter and susceptibility of the spin 1 ising model on the square lattice a new formalism is described which uses two distinct transfer matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order subsequent analysis of the series clearly confirms that the spin 1 model has the same dominant critical exponents as the spin 1 2 ising model accurate estimates for both the critical temperature and non physical singularities are obtained in addition evidence for a non analytic confluent correction with exponent delta 1 1 1 or 0 1 is found

ising field theory in a magnetic field analytic properties of the free energy we study the analytic properties of the scaling function associated with the 2d ising model free energy in the critical domain t tc h 0 the analysis is based on numerical data obtained through the truncated free fermion space approach we determine the discontinuities across the yang lee and langer branch cuts we confirm the standard analyticity assumptions and propose extended analyticity roughly speaking the latter states that the yang lee branching point is the nearest singularity under langer s branch cut we support the extended analyticity by evaluating numerically the associated extended dispersion relation

the rational numbers as an abstract data type we give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations a consequence of this specification is that 0 1 e 0 an interesting equation consistent with the ring axioms and many properties of division the existence of an equational specification of the rationals without hidden functions was an open question we also give an axiomatic examination of the divisibility operator from which some interesting new axioms emerge along with equational specifications of algebras of rationals including one with the modulus function finally we state some open problems including does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting systemq

magnetic responses in the presence of a partial magnetic field on the rectangular ising lattice abstract we study the two dimensional ising model for the square lattice in the presence of a magnetic field which applies on a part of spins specifically we consider the situation in which only the spins on a row or a diagonal line can interact with the magnetic field the low temperature series expansions of the reduced partition functions for the square lattice with a magnetic field on a row are presented for both ferromagnetic and antiferromagnetic spin spin couplings the low and high temperature series expansions of the zero field susceptibility for the above lattice are performed and the critical amplitudes are estimated the critical exponent r of the susceptibility for a ferromagnet is confirmed to be 3 4 the spontaneous magnetization per field applied site of the same system is found to be the same as the ordinary spontaneous magnetization per site with an uniform magnetic field the exact partition functions are also obtained for the super exchange model applying both on a boundary row and on the central row of the rectangular ising lattice

form factor expansion of the row and diagonal correlation functions of the two dimensional ising model we derive and prove exponential and form factor expansions of the row correlation function 0 0 0 n and the diagonal correlation function 0 0 n n of the two dimensional ising model

ising model susceptibility the fuchsian differential equation for χ 4 and its factorization properties we give the fuchsian linear differential equation satisfied by x 4 the four particle contribution to the susceptibility of the isotropic square lattice ising model this fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to x 4 the corresponding order ten linear differentia operator exhibits a large set of factorization properties among these factorizations one is highly remarkable it corresponds to the fact that the two particle contribution x 2 is actually a solution of this order ten linear differential operator this result together with a similar one for the order seven differential operator corresponding to the three particle contribution x 3 leads us to a conjecture on the structure of all the n particle contributions x n

irrelevant operators in the two dimensional ising model by using conformal field theory we classify the possible irrelevant operators for the ising model with nearest neighbour interactions on the square and triangular lattices we analyse the existing results for the free energy and its derivatives and for the correlation length showing that they are in agreement with the conformal field theory predictions moreover these results imply that the nonlinear scaling field of the t operator where t is the energy momentum tensor vanishes at the critical point several other peculiar cancellations are explained in terms of a number of general conjectures we show that all existing results on the square and triangular lattices are consistent with the assumption that only nonzero spin operators are present

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

factorization of the ising model form factors we present a general method for analytically factorizing the n fold form factor integrals f n n n t for the correlation functions of the ising model on the diagonal in terms of the hypergeometric functions 2f1 1 2 n 1 2 n 1 t which appear in the form factor f 1 n n t new quadratic recursion and quartic identities are obtained for the form factors for n 2 3 for n 2 3 4 explicit results are given for the form factors these factorizations are proved for all n for n 2 3 these results yield the emergence of palindromic polynomials canonically associated with elliptic curves as a consequence understanding the form factors amounts to describing and understanding an infinite set of palindromic polynomials canonically associated with elliptic curves from an analytical viewpoint the relation of these palindromic polynomials with hypergeometric functions associated with elliptic curves is made very explicitly and from a differential algebra viewpoint this corresponds to the emergence of direct sums of differential operators homomorphic to symmetric powers of a second order operator associated with elliptic curve

on the singularities in the susceptibility expansion for the two dimensional ising model for temperatures below the critical temperature the magnetic susceptibility for the two dimensional isotropic ising model can be expressed in terms of an infinite series of multiple integrals with respect to a parameter related to temperature and the interaction constant the integrals may be extended to functions analytic outside the unit circle in a groundbreaking paper nickel j phys a 32 3889 3906 1999 identified a class of singularities of these integrals on the unit circle in this note we show that there are no other singularities on the unit circle

lattice green functions the seven dimensional face centred cubic lattice we present a recursive method to generate the expansion of the lattice green function of the d dimensional face centred cubic fcc lattice we produce a long series for d 7 then we show and recall that in order to obtain the linear differential equation annihilating such a long power series the most economic way amounts to producing the non minimal order differential equations we use the method to obtain the minimal order linear differential equation of the lattice green function of the seven dimensional fcc lattice we give some properties of this irreducible order eleven differential equation we show that the differential galois group of the corresponding operator is included in this order eleven operator is non trivially homomorphic to its adjoint and we give a decomposition of this order eleven operator in terms of four order one self adjoint operators and one order seven self adjoint operator furthermore using the landau conditions on the integral we forward the regular singularities of the differential equation of the d dimensional lattice and show that they are all rational numbers we evaluate the return probability in random walks in the seven dimensional fcc lattice we show that the return probability in the d dimensional fcc lattice decreases as d 2 as the dimension d goes to infinity

addendum to on the singularity structure of the 2d ising model susceptibility a remarkable product formula first derived by palmer and tracy 1981 adv appl math 2 329 for the integrand of the two dimensional ising model susceptibility expansion coefficients 2n for temperatures t less than the critical t c is shown to apply equally for 2n 1 for t t c and agrees with formulae derived by yamada 1984 prog theor phys 71 1416 this new representation simplifies the derivation of the results in the original paper of this title 1999 j phys a math gen 32 3889 to the extent that the leading series behaviour and the singularity structure can be deduced almost by inspection the derivation of series is also simplified and i show using extended series and knowledge of the singularity structure that there is now unambiguous evidence for correction to scaling terms in the susceptibility beyond those inferred from a nonlinear scaling field analysis

the diagonal ising susceptibility we use the recently derived form factor expansions of the diagonal two point correlation function of the square ising model to study the susceptibility for a magnetic field applied only to one diagonal of the square lattice for the isotropic ising model we exactly evaluate the one and two particle contributions 1 d and 2 d of the corresponding susceptibility and obtain linear differential equations for the three and four particle contributions as well as the five particle contribution 5 d but only modulo a given prime we use these exact linear differential equations to show that not only the russian doll structure but also the direct sum structure on the linear differential operators for the n particle contributions n d are quite directly inherited from the direct sum structure on the form factors f n we show that the nth particle contributions n d have their singularities at roots of unity these singularities become dense on the unit circle sinh 2ev ktsinh 2eh kt 1 as n

rss based sensor localization in the presence of unknown channel parameters this correspondence studies the received signal strength based localization problem when the transmit power or path loss exponent is unknown the corresponding maximum likelihood estimator mle poses a difficult nonconvex optimization problem to avoid the difficulty in solving the mle we use suitable approximations and formulate the localization problem as a general trust region subproblem which can be solved exactly under mild conditions simulation results show a promising performance for the proposed methods which also have reasonable complexities compared to existing approaches

singularities of n fold integrals of the ising class and the theory of elliptic curves we introduce some multiple integrals that are expected to have the same singularities as the singularities of the n particle contributions n to the susceptibility of the square lattice ising model we find the fuchsian linear differential equation satisfied by these multiple integrals for n 1 2 3 4 and only modulo some primes for n 5 and 6 thus providing a large set of possible new singularities of n we discuss the singularity structure for these multiple integrals by solving the landau conditions we find that the singularities of the associated odes identify up to n 6 with the leading pinch landau singularities the second remarkable obtained feature is that the singularities of the odes associated with the multiple integrals reduce to the singularities of the odes associated with a finite number of one dimensional integrals among the singularities found we underline the fact that the quadratic polynomial condition 1 3w 4w2 0 that occurs in the linear differential equation of 3 actually corresponds to a remarkable property of selected elliptic curves namely the occurrence of complex multiplication the interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group namely isogenies of elliptic curves is sketched most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations suggesting an interpretation in terms of motivic mathematical structures beyond the theory of elliptic curves

high order fuchsian equations for the square lattice ising model χ 6 this paper deals with the six particle contribution to the magnetic susceptibility of the square lattice ising model we have generated modulo a prime series coefficients for the length of the series is sufficient to produce the corresponding fuchsian linear differential equation modulo a prime we obtain the fuchsian linear differential equation that annihilates the depleted series the factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper the depleted differential operator is shown to have a structure similar to the corresponding operator for it splits into factors of smaller orders with the left most factor of order 6 being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral e the right most factor has a direct sum structure and using series calculated modulo several primes all the factors in the direct sum have been reconstructed in exact arithmetics

exact and approximate solutions of source localization problems we consider least squares ls approaches for locating a radiating source from range measurements which we call r ls or from range difference measurements rd ls collected using an array of passive sensors we also consider ls approaches based on squared range observations sr ls and based on squared range difference measurements srd ls despite the fact that the resulting optimization problems are nonconvex we provide exact solution procedures for efficiently computing the sr ls and srd ls estimates numerical simulations suggest that the exact sr ls and srd ls estimates outperform existing approximations of the sr ls and srd ls solutions as well as approximations of the r ls and rd ls solutions which are based on a semidefinite relaxation

least squares algorithms for time of arrival based mobile location localization of mobile phones is of considerable interest in wireless communications in this correspondence two algorithms are developed for accurate mobile location using the time of arrival measurements of the signal from the mobile station received at three or more base stations the first algorithm is an unconstrained least squares ls estimator that has implementation simplicity the second algorithm solves a nonconvex constrained weighted least squares cwls problem for improving estimation accuracy it is shown that the cwls estimator yields better performance than the ls method and achieves both the crame spl acute r rao lower bound and the optimal circular error probability at sufficiently high signal to noise ratio conditions

robust differential received signal strength based localization source localization based on signal strength measurements has become very popular due to its practical simplicity however the severe nonlinearity and non convexity make the related optimization problem mathematically difficult to solve especially when the transmit power or the path loss exponent ple is unknown moreover even if the ple is known but not perfectly estimated or the anchor location information is not accurate the constructed data model will become uncertain making the problem again hard to solve this paper particularly focuses on differential received signal strength drss based localization with model uncertainties in case of unknown transmit power and ple a new whitened model for drss based localization with unknown transmit powers is first presented and investigated when assuming the ple is known we introduce two estimators based on an exact data model an advanced best linear unbiased estimator a blue and a lagrangian estimator le and then we present a robust semidefinite programming sdp based estimator rsdpe which can cope with model uncertainties imperfect ple and inaccurate anchor location information the three proposed estimators have their own advantages from different perspectives the a blue has the lowest complexity the le holds the best accuracy for a small measurement noise and the rsdpe yields the best performance under a large measurement noise and possesses a very good robustness against model uncertainties finally we propose a robust sdp based block coordinate descent estimator rsdp bcde to deal with a completely unknown ple and its performance converges to that of the rsdpe using a perfectly known ple

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

improved methods of estimating critical indices via fractional calculus efficiencies of certain methods for the determination of critical indices from power series expansions are shown to be considerably improved by a suitable implementation of fractional differentiation in the context of the ratio method rm kinship of the modified strategy with the ad hoc shifted rm is established and the advantages are demonstrated further in the course of the estimation of critical points significant betterment of convergence properties of diagonal pade approximants is observed on several occasions by invoking this concept test calculations are performed on i various ising spin 1 2 lattice models for susceptibility series attended with a ferromagnetic phase transition ii complex model situations involving confluent and antiferromagnetic singularities and iii the chain generating functions for self avoiding walks on triangular square and simple cubic lattices

exact finite size scaling corrections to the critical two dimensional ising model on a torus ii triangular and hexagonal lattices we compute the finite size corrections to the free energy internal energy and specific heat of the critical two dimensional spin 1 2 ising model on triangular and hexagonal lattices wrapped on a torus we find the general form of the finite size corrections to these quantities as well as explicit formulae for the first coefficients of each expansion we analyse the implications of these findings for the renormalization group description of the model

concrete models of computation for topological algebras a concrete model of computation for a topological algebra is based on a representation of the algebra made from functions on the natural numbers the functions computable in a concrete model are computable in the representation in the classical sense of the chruch turing thesis moreover the functions turn out to be continuous in the topology of the algebra in this paper we consider different concrete models for computing in topological algebras and prove their mutual equivalence in certain commonly occurring circumstances for topological algebras the concrete models we use are effective representation by algebraic domains stoltenberg hansen and tucker effective representation by continuous domains edelat effective representation by type two recursion on baire space weihrauch and for metric and normed algebras we use effective metric spaces moschovakis and computability structures pour el and richards the result are evidence that computability theory for topological algebras is a stable theory independent of the specific models of computation just as classical computability theory for discrete algebras is stable

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

landau singularities and singularities of holonomic integrals of the ising class we consider families of multiple and simple integrals of the ising class and the linear ordinary differential equations with polynomial coefficients they are solutions of we compare the full set of singularities given by the roots of the head polynomial of these linear odes and the subset of singularities occurring in the integrals with the singularities obtained from the landau conditions for these ising class integrals we show that the landau conditions can be worked out either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral the singular behaviour of these integrals is obtained in the self dual variable w s 2 1 s2 with s sinh 2k where k j kt is the usual ising model coupling constant switching to the variable s we show that the singularities of the analytic continuation of series expansions of these integrals actually break the kramers wannier duality we revisit the singular behaviour zenine et al 2005 j phys a math gen 38 9439 74 of the third contribution to the magnetic susceptibility of ising model 3 at the points 1 3w 4w2 0 and show that 3 s is not singular at the corresponding points inside the unit circle s 1 while its analytical continuation in the variable s is actually singular at the corresponding points 2 s s2 0 outside the unit circle s 1

integrals of the ising class from an experimental mathematical perspective we analyze ising class integrals our experimental results involvedextreme precision multidimensional quadrature on intricate integrands thus highly parallel computation was required

diagonal ising susceptibility elliptic integrals modular forms and calabi yau equations we give the exact expressions of the partial susceptibilities 3 d and 4 d for the diagonal susceptibility of the ising model in terms of modular forms and calabi yau odes and more specifically 3f2 1 3 2 3 3 2 1 1 z and 4f3 1 2 1 2 1 2 1 2 1 1 1 z hypergeometric functions by solving the connection problems we analytically compute the behavior at all finite singular points for 3 d and 4 d we also give new results for 5 d we see in particular the emergence of a remarkable order 6 operator which is such that its symmetric square has a rational solution these new exact results indicate that the linear differential operators occurring in the n fold integrals of the ising model are not only derived from geometry globally nilpotent but actually correspond to special geometry homomorphic to their formal adjoint this raises the question of seeing if these special geometry ising operators are special ones reducing in fact systematically to selected k balanced q 1fq hypergeometric functions or correspond to the more general solutions of calabi yau equations

on abstract data types presented by multiequations equational presentation of abstract data types is generalized to presentation by multiequations i e exclusive or s of equations in order to capture parametric data types such as array or set multiinitial algebra semantics for such data types is introduced classes of algebras described by multiequations are characterized

non analyticity of the callan symanzik β function of two dimensional o n models we discuss the analytic properties of the callan symanzik function g associated with the zero momentum four point coupling g in the two dimensional 4 model with o n symmetry using renormalization group arguments we derive the asymptotic behaviour of g at the fixed point g we argue that g g o g g 1 7 for n 1 and g g o 1 log g g for n 3 our claim is supported by an explicit calculation in the ising lattice model and by a 1 n calculation for the two dimensional 4 theory we discuss how these non analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g

comment on the fuchsian differential equation of the square lattice ising model χ 3 susceptibility the new singularities in 3 found by zenine et al 2004 j phys a math gen 37 9651 are pinch type singularities analogous to those found in feynman graphs in field theory a general formula for the possible location of all such singularities for arbitrary n is given

integrability versus non integrability hard hexagons and hard squares compared in this paper we compare the integrable hard hexagon model with the nonintegrable hard squares model by means of partition function roots and transfer matrix eigenvalues we consider partition functions for toroidal cylindrical and free free boundary conditions up to sizes 40 40 and transfer matrices up to 30 sites for all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis the density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest and equal moduli and exhibits much greater structure than the corresponding density of hard hexagons we also study the special point z 1 of hard squares where all eigenvalues have unit modulus and we give several conjectures for the value at z 1 of the partition functions

linear least squares approach for accurate received signal strength based source localization a conventional approach for passive source localization is to utilize signal strength measurements of the emitted source received at an array of spatially separated sensors the received signal strength rss information can be converted to distance estimates for constructing a set of circular equations from which the target position is determined nevertheless a major challenge in this approach lies in the shadow fading effect which corresponds to multiplicative measurement errors by utilizing the mean and variance of the squared distance estimates we devise two linear least squares lls estimators for rss based positioning in this paper the first one is a best linear unbiased estimator while the second is its improved version by exploiting the known relation between the parameter estimates the variances of the position estimates are derived and confirmed by computer simulations in particular it is proved that the performance of the improved lls estimator achieves cramer rao lower bound at sufficiently small noise conditions

square lattice ising model susceptibility connection matrices and singular behaviour of χ 3 and χ 4 we present a simple but efficient way to calculate connection matrices between sets of independent local solutions defined at two neighbouring singular points of fuchsian differential equations of quite large orders such as those found for the third and fourth contribution 3 and 4 to the magnetic susceptibility of the square lattice ising model we deduce all the critical behaviours of the solutions 3 and 4 as well as the asymptotic behaviour of the coefficients in the corresponding series expansions we confirm that the newly found quadratic singularities of the fuchsian ode associated with 3 are not singularities of the particular solution 3 itself we use the previous connection matrices to get the exact expressions of all the monodromy matrices of the fuchsian differential equation for 3 and 4 expressed in the same basis of solutions these monodromy matrices are the generators of the differential galois group of the fuchsian differential equations for 3 and 4 whose analysis is just sketched here as far as the physics implications of the solutions are concerned we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibility at high temperature respectively low temperature and the first two terms 1 and 3 respectively 2 and 4

holonomy of the ising model form factors we study the ising model two point diagonal correlation function c n n by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of wu mccoy tracy and barouch we extend this expansion weighting by powers of a variable the j particle contributions f j n n in the form factor expansion the corresponding extension of the two point diagonal correlation function c n n is shown for arbitrary to be a solution of the sigma form of the painleve vi equation introduced by jimbo and miwa in their isomonodromic approach to the ising model fuchsian linear differential equations for the form factors f j n n are obtained for j 9 and shown to have both a russian doll nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral e from this we show that each f j n n is unexpectedly simple being expressed polynomially in terms of the elliptic integrals e and k in contrast we exhibit some mathematical objects built from these form factors f j n n which break the direct sum of symmetric powers decomposition with its associated polynomial expressions first we show that the scaling limit of these differential operators and form factors breaks the direct sum structure but not the russian doll structure secondly we show that the previous extension of two point diagonal correlation functions c n n is for singled out values cos m n m n integers also solutions of fuchsian linear differential equations these solutions of painleve vi are not polynomial in e and k but are actually algebraic functions being associated with modular curves

initial and final algebra semantics for data type specifications two characterization theorems we prove that those data types which may be defined by conditional equation specifications and final algebra semantics are exactly the cosemicomputable data types those data types which are effectively computable but whose inequality relations are recursively enumerable and we characterize the computable data types as those data types which may be specified by conditional equation specifications using both initial algebra semantics and final algebra semantics numerical bounds for the number of auxiliary functions and conditional equations required are included in both theorems

exact finite size scaling corrections to the critical two dimensional ising model on a torus we analyse the finite size corrections to the energy and specific heat of the critical two dimensional spin ising model on a torus we extend the analysis of ferdinand and fisher to compute the correction of order l 3 to the energy and the corrections of order l 2 and l 3 to the specific heat we also obtain general results on the form of the finite size corrections to these quantities only integer powers of l 1 occur unmodified by logarithms except of course for the leading log l term in the specific heat and the energy expansion contains only odd powers of l 1 in the specific heat expansion any power of l 1 can appear but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion

square lattice ising model tilde chi 5 ode in exact arithmetic we obtain in exact arithmetic the order 24 linear differential operator l24 and the right hand side e 5 of the inhomogeneous equation l24 5 e 5 where is a linear combination of n particle contributions to the susceptibility of the square lattice ising model in bostan et al 2009 j phys a math theor 42 275209 the operator l24 modulo a prime was shown to factorize into here we prove that no further factorization of the order 12 operator is possible we use the exact ode to obtain the behaviour of at the ferromagnetic critical point and to obtain a limited number of analytic continuations of beyond the principal disc defined by its high temperature series contrary to a speculation in boukraa et al 2008 j phys a math theor 41 455202 we find that is singular at w 1 2 on an infinite number of branches

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

meadows and the equational specification of division the rational real and complex numbers with their standard operations including division are partial algebras specified by the axiomatic concept of a field since the class of fields cannot be defined by equations the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational real and complex numbers we study a new axiomatic concept for number systems with division that uses only equations a meadow is a commutative ring with a total inverse operator satisfying two equations which imply 0 1 0 all fields and products of fields can be viewed as meadows after reviewing alternate axioms for inverse we start the development of a theory of meadows we give a general representation theorem for meadows and find as a corollary that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields we also prove representation results for meadows of finite characteristic

on tree coalgebras and coalgebra presentations for deterministic systems expressed as coalgebras over polynomial functors every tree t an element of the final coalgebra turns out to represent a new coalgebra at the universal property of this family of coalgebras resembling freeness is that for every state s of every system s there exists a unique coalgebra homomorphism from a unique at which takes the root of t to s consequently the tree coalgebras are finitely presentable and form a strong generator thus these categories of coalgebras are locally finitely presentable in particular every system is a filtered colimit of finitely presentable systems in contrast for transition systems expressed as coalgebras over the finite power set functor we show that there are systems which fail to be filtered colimits of finitely presentable finite ones surprisingly if is an uncountable cardinal then presentation is always well behaved whenever an endofunctor f preserves filtered colimits i e is accessible then presentable coalgebras are precisely those whose underlying objects are presentable consequently every f coalgebra is a filtered colimit of presentable coalgebras thus coalg f is a locally presentable category this holds for all endofunctors of accessible categories with colimits of chains corollary a set functor is bounded at in the sense of kawahara and mori iff it is accessible

experimental mathematics on the magnetic susceptibility of the square lattice ising model we calculate very long low and high temperature series for the susceptibility of the square lattice ising model as well as very long series for the five particle contribution 5 and six particle contribution 6 these calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 cpu hours on computer clusters the series for low and high temperature regimes 5 and 6 are now extended to 2000 terms in addition for 5 10 000 terms of the series are calculated modulo a single prime and have been used to find the linear ode satisfied by 5 modulo a prime a diff pade analysis of the 2000 terms series for 5 and 6 confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n particle components of the susceptibility up to a small set of additional singularities the exponents at all the singularities of the fuchsian linear ode of 5 and the as yet unknown ode of 6 are given they are all rational numbers we find the presence of singularities at w 1 2 for the linear ode of 5 and w2 1 8 for the ode of 6 which are not singularities of the physical 5 and 6 that is to say the series solutions of the ode s which are analytic at w 0 furthermore analysis of the long series for 5 and 6 combined with the corresponding long series for the full susceptibility yields previously conjectured singularities in some n n 7 the exponents at all these singularities are also seen to be rational numbers we also present a mechanism of resummation of the logarithmic singularities of the n leading to the known power law critical behaviour occurring in the full and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility

the fuchsian differential equation of the square lattice ising model χ 3 susceptibility using an expansion method in the variables xi that appear in n 1 dimensional integrals representing the n particle contribution to the ising square lattice model susceptibility we generate a long series of coefficients for the three particle contribution 3 using an n4 polynomial time algorithm we give the fuchsian differential equation of order 7 for 3 that reproduces all the terms of our long series an analysis of the properties of this fuchsian differential equation is performed

final algebras cosemicomputable algebras and degrees of unsolvability this paper studies some computability notions for abstract data types and in particular compares cosemicomputable many sorted algebras with a notion of finality to model minimal state realizations of abstract software machines given a finite many sorted signature and a set v of visible sorts for every algebra a with co r e behavior and nontrivial computable v behavior there is a finite signature extension of without new sorts and a finite set e of equations such that a is isomorphic to a reduct of the final e algebra relative to v this uses a theorem due to bergstra and tucker 3 if a is computable then a is also isomorphic to the reduct of the initial e algebra we also prove some results on congruences of finitely generated free algebras we show that for every finite signature there are either countably many congruences on the free algebra or else there is a continuum of such congruences there are several necessary and sufficient conditions which separate these two cases we introduce the notion of the turing degree of a minimal algebra using the results above prove that there is a fixed one sorted signature such that for every r e degree d there is a finite set e of equations such the initial e algebra has degree d there is a two sorted signature 0 and a single visible sort such that for every r e degree d there is a finite set e of equations such that the initial e v algebra is computable and the final e v algebra is cosemicomputable and has degree d

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

the completeness of the algebraic specification methods for computable data types the following fundamental theorem about the adequacy of the algebraic specification methods for data abstractions is proved let a be a data type with n subtypes then a is computable if and only if a possesses an equational specification involving at most 3 n 1 hidden operators and 2 n 1 axioms which defines it under initial and final algebra semantics simultaneously

a simple and efficient estimator for hyperbolic location an effective technique in locating a source based on intersections of hyperbolic curves defined by the time differences of arrival of a signal received at a number of sensors is proposed the approach is noniterative and gives an explicit solution it is an approximate realization of the maximum likelihood estimator and is shown to attain the cramer rao lower bound near the small error region comparisons of performance with existing techniques of beamformer spherical interpolation divide and conquer and iterative taylor series methods are made the proposed technique performs significantly better than spherical interpolation and has a higher noise threshold than divide and conquer before performance breaks away from the cramer rao lower bound it provides an explicit solution form that is not available in the beamforming and taylor series methods computational complexity is comparable to spherical interpolation but substantially less than the taylor series method

complex temperature singularities of the susceptibility in the d 2 ising model i square lattice we investigate the complex temperature singularities of the susceptibility of the 2d ising model on a square lattice from an analysis of low temperature series expansions we find evidence that as one approaches the point u us 1 where u e 4k from within the complex extensions of the fm or afm phases the susceptibility has a divergent singularity of the form x approximately as 1 u gamma s with exponent gamma s 3 2 the critical amplitude as is calculated other critical exponents are found to be alpha s sigma s 0 and beta s 1 4 so that the scaling relation alpha s 2 beta s gamma s is satisfied however using exact results for beta s on the square triangular and honeycomb lattices we show that universality is violated at this singularity beta s is lattice dependent finally from an analysis of spin spin correlation functions we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u 1 from within the symmetric phase this is confirmed by an explicit study of high temperature series expansions

the ising model from elliptic curves to modular forms and calabi yau equations we show that almost all the linear differential operators factors obtained in the analysis of the n particle contributions s of the susceptibility of the ising model for n 6 are linear differential operators associated with elliptic curves beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral e the second and third order differential operators z2 f2 f3 can actually be interpreted as modular forms of the elliptic curve of the ising model a last order 4 globally nilpotent linear differential operator is not reducible to this elliptic curve modular form scheme this operator is shown to actually correspond to a natural generalization of this elliptic curve modular form scheme with the emergence of a calabi yau equation corresponding to a selected 4f3 hypergeometric function this hypergeometric function can also be seen as a hadamard product of the complete elliptic integral k with a remarkably simple algebraic pull back square root extension the corresponding calabi yau fourth order differential operator having a symplectic differential galois group the mirror maps and higher order schwarzian odes associated with this calabi yau ode present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms in particular an exact isogenies representation of the generators of the renormalization group extending the modular group to a symmetry group