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

noncore square matrices miscellany a noncore square matrix is one whose unique core nilpotent decomposition does not exclusively consist of the core part if the core part is idempotent the matrix itself is called quasiidempotent this paper makes a brief survey of these and other noncore matrices and their properties

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

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

on embedding between 2d meshes of the same size mesh is one of the most commonly used interconnection networks and therefore embedding between different meshes becomes a basic embedding problem not only does an efficient embedding between meshes allow one mesh connected computing system to efficiently simulate another but it also provides a useful tool for solving other embedding problems the authors study how to embed an s sub 1 spl times t sub 1 mesh into an s sub 2 spl times t sub 2 mesh where s sub i spl les t sub i i 1 2 s sub 1 spl les t sub 1 s sub 2 t sub 2 such that the minimum dilation and congestion can be achieved first they present a lower bound on the dilations and congestions of such embeddings for different cases then they propose an embedding with dilation s sub 1 s sub 2 2 and congestion s sub 1 s sub 2 4 for the case s sub 2 spl ges s sub 2 both of which almost match the lower bound s sub 1 s sub 2 finally for the case s sub 1 s sub 2 we present an embedding which has a dilation less than or equal to 2 spl radic s sub 1

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

the perimeter generating function of punctured staircase polygons using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of punctured staircase polygons staircase polygons with a single internal staircase hole we find that all the terms in the generating function can be reproduced from a linear fuchsian differential equation of order 8 we perform an analysis of the properties of the differential equation

transcendence of formal power series with rational coefficients we give algebraic proofs of transcendence over q x of formal power series with rational coefficients by using inter alia reduction modulo prime numbers and the christol theorem applications to generating series of languages and combinatorial objects are given

partial orders based on core nilpotent decomposition abstract in this paper we study partial orders in terms of the core nilpotent decomposition we derive some characterizations of the c n partial ordering create a new partial ordering the g drazin partial ordering which is a generalization of c n partial ordering and give some properties and characterizations of the g drazin partial ordering

optimal routing in double loop networks in this paper we study the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps we provide algorithms specifically tailored for weighted undirected and directed circulant graphs with two jumps which compute the shortest path our method only requires o logn arithmetic operations and the total bit complexity is o log 2nloglognlogloglogn where n is the number of the graph s vertices this elementary and efficient shortest path algorithm has been derived from the closest vector problem cvp of lattices in dimension two and with an 1 norm

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

on a new generalized inverse for matrices of an arbitrary index the purpose of this paper is to introduce a new generalized inverse called dmp inverse associated with a square complex matrix using its drazin and moore penrose inverses dmp inverse extends the notion of core inverse introduced by baksalary and trenkler for matrices of index at most 1 in baksalary and trenkler 2010 1 to matrices of an arbitrary index dmp inverses are analyzed from both algebraic as well as geometrical approaches establishing the equivalence between them

differential algebra on lattice green functions and calabi yau operators we revisit miscellaneous linear differential operators mostly associated with lattice green functions in arbitrary dimensions but also calabi yau operators and order 7 operators corresponding to exceptional differential galois groups we show that these irreducible operators are not only globally nilpotent but are such that they are homomorphic to their formal adjoints considering these operators or sometimes equivalent operators we show that they are also such that either their symmetric square or their exterior square have a rational solution this is a general result an irreducible linear differential operator homomorphic to its formal adjoint is necessarily such that either its symmetric square or its exterior square has a rational solution and this situation corresponds to the occurrence of a special differential galois group we thus define the notion of being special geometry for a linear differential operator if it is irreducible globally nilpotent and such that it is homomorphic to its formal adjoint since many derived from geometry n fold integrals periods occurring in physics are seen to be diagonals of rational functions we address several examples of minimal order operators annihilating diagonals of rational functions and remark that they also seem to be systematically associated with irreducible factors homomorphic to their adjoint

high order differential approximants we introduce a new form of differential approximant for the summation of power series the method is a special type of pade hermite approximant it consists of a high order linear differential equation with polynomial coefficients that is satisfied approximately by the partial sum of the power series this method is able to reproduce the polylogarithmic functions exactly numerical evidence suggests that this is currently one of the best methods of singularity analysis for many problems

susceptibility of the checkerboard ising model the authors show that at the disorder variety the n point correlation functions of the checkerboard potts model have a simple causal structure and decay as simple exponentials for space like separations an exact expression for the susceptibility in the ising case is obtained the result is additional evidence in favour of the conjectured s4 symmetry of the checkerboard potts model in the presence of a finite field

automata and the susceptibility of the square lattice ising model modulo powers of primes we study the full susceptibility of the ising model modulo powers of primes we find exact functional equations for the full susceptibility modulo these primes revisiting some lesser known results on discrete finite automata we show that these results can be seen as a consequence of the fact that modulo 2 r one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2 r and consequently satisfy exact functional equations modulo 2 r we sketch a possible physical interpretation of these functional equations modulo 2 r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves one relevant example is the landen transformation which can be seen as an exact generator of the ising model renormalization group we underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries

a complementary survey on double loop networks we give a survey on double loop networks with emphasis on new development since the surveys in 1986 1991 and 1995

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

relationships between dmp relation and some partial orders let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h the equivalent definition of the dmp relation on b h using operator block matrices is introduced we present a new interpretation of this relation which allows to generalize many known results for matrices to general hilbert spaces regarding the dmp relation and its relationship with the star order and minus partial order also some properties of dmp relation defined with the help of idempotent operators are investigated

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

an efficient algorithm to find a double loop network that realizes a given l shape double loop networks have been widely studied as an architecture for local area networks it is well known that the minimum distance diagram of a double loop network yields an l shape given a positive integer n it is desirable to find a double loop network with its diameter being the minimum among all double loop networks with n nodes since the diameter of a double loop network can be easily computed from its l shape one method is to start with a desirable l shape and then find a double loop network to realize it this is a problem discussed by many authors f aguilo m a fiol an efficient algorithm to find optimal double loop networks discrete math 138 1995 15 29 r c chan c y chen z x hong a simple algorithm to find the steps of double loop networks discrete appl math 121 2002 61 72 c y chen f k hwang the minimum distance diagram of double loop networks ieee trans comput 49 2000 977 979 p esque f aguilo m a fiol double commutative step diagraphs with minimum diameters discrete math 114 1993 147 157 and it has been open for a long time whether this problem can be solved in o log n time in this paper we will provide a simple and efficient o log n time algorithm for solving this problem

on a generalized core inverse abstract the paper introduces the concept of a generalized core inverse of a matrix which extends the notion of the core inverse defined by baksalary and trenkler 1 while the original core inverse is restricted to matrices of index one the generalized core inverse exists for any square matrix several properties of the new concept are identified with the derivations based essentially on partitioned representations of matrices some of the features of the generalized core inverse coincide with those attributed to the core inverse but there are also such which characterize the core inverse only and not its generalization

recursive evaluation of space time lattice green s functions up to a multiplicative constant the lattice green s function lgf as defined in condensed matter physics and lattice statistical mechanics is equivalent to the z domain counterpart of the finite difference time domain green s function gf on a lattice expansion of a well known integral representation for the lgf on a dimensional hyper cubic lattice in powers of z 1 and application of the chu vandermonde identity results in 1 nested finite sum representations for discrete space time gfs due to severe numerical cancellations these nested finite sums are of little practical use for 2 the finite sum may be evaluated in closed form in terms of a generalized hypergeometric function for special lattice points that representation simplifies considerably while on the other hand the finite difference stencil may be used to derive single lattice point second order recurrence schemes for generating 2d discrete space time gf time sequences on the fly for arbitrary symbolic lattice points zeilberger s algorithm produces a third order recurrence operator with polynomial coefficients of the sixth degree the corresponding recurrence scheme constitutes the most efficient numerical method for the majority of lattice points in spite of the fact that for explicit numeric lattice points the associated third order recurrence operator is not the minimum recurrence operator as regards the asymptotic bounds for the possible solutions to the recurrence scheme perron s theorem precludes factorial or exponential growth along horizontal lattices directions rapid initial growth does occur but poses no problems in augmented dynamic range fixed precision arithmetic by analysing long distance wave propagation along a horizontal lattice direction we have concluded that the chirp up oscillations of the discrete space time gf are the root cause of grid dispersion anisotropy with each factor of ten increase in the lattice distance one would have to roughly double the pulse width of the source signature to keep pulse distortion at bay the gf time sequences can also be used for an efficient computation of discrete space frequency lgfs especially if one employs aitken s 2 process for the acceleration of the convergence of the consecutive partial sums

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

lattice green s functions of the higher dimensional face centered cubic lattices we study the face centered cubic fcc lattice in up to six dimensions in particular we are concerned with lattice green functions lgfs and return probabilities computer algebra techniques such as the method of creative telescoping are used for deriving an ode for a given lgf for the four and five dimensional fcc lattices we give rigorous proofs of the odes that were conjectured by guttmann and broadhurst additionally we find the ode of the lgf of the six dimensional fcc lattice a result that was not believed to be achievable with current computer hardware

painleve versus fuchs the sigma form of the painleve vi pvi equation contains four arbitrary parameters and generically the solutions can be said to be genuinely nonlinear because they do not satisfy linear differential equations of finite order however when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy fuchsian differential equations of finite order we study this phenomenon of fuchsian solutions to the painleve equation with a focus on the particular pvi equation which is satisfied by the diagonal correlation function c n n of the ising model we obtain fuchsian equations of order n i for c n n and show that the equation for c n n is equivalent to the nth symmetric power of the equation for the elliptic integral e we show that these fuchsian equations correspond to rational algebraic curves with an additional riccati structure and we show that the malmquist hamiltonian p q variables are rational functions in complete elliptic integrals fuchsian equations for off diagonal correlations c n m are given which extend our considerations to discrete generalizations of painleve

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

weighted g drazin inverses and a new pre order on rectangular matrices abstract this paper deals with weighted g drazin inverses which is a new class of matrices introduced to extend to the rectangular case g drazin inverses recently considered by wang and liu for square matrices first we define and characterize weighted g drazin inverses next we consider a new pre order defined on complex rectangular matrices based on weighted g drazin inverses finally we characterize this pre order and relate it to the minus partial order and to the weighted drazin pre order

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

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

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

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

the ising model and special geometries we show that the globally nilpotent g operators corresponding to the factors of the linear differential operators annihilating the multifold integrals n of the magnetic susceptibility of the ising model n 6 are homomorphic to their adjoint this property of being self adjoint up to operator homomorphisms is equivalent to the feature of their symmetric squares or their exterior squares having rational solutions the differential galois groups are in the special orthogonal or symplectic groups this self adjoint up to operator equivalence property means that the factor operators that we already know to be derived from geometry are special globally nilpotent operators they correspond to special geometries beyond the small order factor operators occurring in the linear differential operators associated with 5 and 6 and in particular those associated with modular forms we focus on the quite large order 12 and order 23 operators we show that the order 12 operator has an exterior square which annihilates a rational solution then its differential galois group is in the symplectic group the order 23 operator is shown to factorize into an order 2 operator and an order 21 operator the symmetric square of this order 21 operator has a rational solution its differential galois group is thus in the orthogonal group

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

universality considerations in vlsi circuits the problem of embedding the interconnection pattern of a circuit into a two dimensional surface of minimal area is discussed since even for some natural patterns graphs containing m connections may require m2 area in order to achieve compact embeddings restricted classes of graphs have to be considered for example arbitrary trees of bounded degree can be embedded in linear area without edges crossing over planar graphs can be embedded efficiently only if crossovers are allowed in the embedding

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

relations of multiple zeta values and their algebraic expression abstract we establish a new class of relations which we call the cyclic sum identities among the multiple zeta values k1 kl n1 nl 11 n1k1 nkkl these identities have an elementary proof and imply the sum theorem for multiple zeta values they also have a succinct statement in terms of cyclic derivations as introduced by rota sagan and stein in addition we discuss the expression of other relations of multiple zeta values via the shuffle and harmonic products on the underlying vector space h of the noncommutative polynomial ring q x y and also using an action of the hopf algebra of quasi symmetric functions on q x y

series analysis of the q state checkerboard potts model the low temperature expansion of the checkerboard potts model in a magnetic field is obtained up to order twelve from the so called disorder solutions various expansions and results known in the literature it is shown that this expansion drastically simplifies on the dual of the disorder variety the low temperature expansion of the magnetisation is seen to become equal to one up to order twelve when it is restricted to the dual of the disorder variety these results have to be seen as exact formal constraints on the analytic continuation of the low temperature expansion of the partition function per site similar simplifications occur for the susceptibility and higher derivatives with respect to the magnetic field these expansions are also analysed in the vicinity of these particular varieties

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

quantum spin chains and riemann zeta function with odd arguments the riemann zeta function is an important object of number theory we argue that it is related to the heisenberg spin 1 2 anti ferromagnet in the xxx spin chain we study the probability of formation of a ferromagnetic string in the anti ferromagnetic ground state in the thermodynamics limit we prove that for short strings the probability can be expressed in terms of the riemann zeta function with odd arguments

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

near embeddings of hypercubes into cayley graphs on the symmetric group simulations of hypercube networks by certain cayley graphs on the symmetric group are investigated let q k be the familiar k dimensional hypercube and let s n be the star network of dimension n defined as follows the vertices of s n are the elements of the symmetric group of degree n two vertices x and y being adjacent if xo 1 i y for some i that is xy is an edge if x and y are related by a transposition involving some fixed symbol which we take to be spl i bold 1 this network has nice symmetry properties and its degree and diameter are sublogarithmic as functions of the number of vertices making it compare favorably with the hypercube network these advantages of s n motivate the study of how well it can simulate other parallel computation networks in particular the hypercube the first step in such a simulation is the construction of a one to one map f q k spl rarr s n of dilation d for d small that is one wants a map f such that images of adjacent points are at most distance d apart in s n an alternative approach best applicable when one to one maps are difficult or impossible to find is the construction of a one to many map g of dilation d defined as follows for each point x spl isin q k there is an associated subset g x spl sube v s n such that for each edge xy in q k every x spl isin g x is at most distance d in s n from some y spl isin g y such one to many maps allow one to achieve the low interprocessor communication time desired in the usual one to one embedding underlying a simulation this is done by capturing the local structure of q k inside of s n via the one to many embedding when the global structure cannot be so captured our results are the following 1 there exist the following one to many embeddings a f q k spl rarr s 3k 1 with dilation f 1 b f q 11k 2 spl rarr s 13k 2 with dilation f 2 2 there exists a one to one embedding f q n2 sup n spl minus 1 spl rarr s 2 sup n with dilation f 3

column space equalities for orthogonal projectors in their paper y tian g p h styan rank equalities for idempotent and involutory matrices linear algebra appl 335 2001 101 117 tian and styan established several rank equalities involving a pair of idempotent matrices p and q subsequently these results are reinvestigated from the point of view of the following question provided that idempotent p q are hermitian which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces simultaneously some related results are established which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors

some theorems on the core inverse of matrices and the core partial ordering abstract this paper derives some further results on the core inverse of matrices and the core partial ordering first we treat two representations of core inverse obtained by baksalary and trenkler 2010 and derive maximal classes of matrices for which the representations are valid next we discuss several formulas on the core partial ordering presented by malik 2013 and extend them by deriving maximal classes of matrices for which the formulas remain true

ising n fold integrals as diagonals of rational functions and integrality of series expansions we show that the n fold integrals n of the magnetic susceptibility of the ising model as well as various other n fold integrals of the ising class or n fold integrals from enumerative combinatorics like lattice green functions correspond to a distinguished class of functions generalizing algebraic functions they are actually diagonals of rational functions as a consequence the power series expansions of the analytic at x 0 solutions of these linear differential equations derived from geometry are globally bounded which means that after just one rescaling of the expansion variable they can be cast into series expansions with integer coefficients we also give several results showing that the unique analytical solution of calabi yau odes and more generally picard fuchs linear odes with solutions of maximal weights are always diagonals of rational functions besides in a more enumerative combinatorics context generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions we finally address the question of the relations between the notion of integrality series with integer coefficients or more generally globally bounded series and the modularity of odes

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

the minimum distance diagram of double loop networks it is well known that the minimum distance diagram of a double loop network yields an l shape it is important to know for a given l shape whether there exists a double loop network realizing it necessary and sufficient conditions were given before but were said to be cumbersome they also need some clarification we give new conditions and a simple proof we also prove that all double loop networks realizing the same l shape are isomorphic as the existing proof was given in a spanish dissertation

optimal fault tolerant routing algorithm and fault tolerant diameter in directed double loop networks this paper addresses the reliability of directed double loop networks g n r s and studies the problems about optimal fault tolerant routing fault tolerant diameter etc in g n r s firstly we study the shapes of the l shaped tiles which are the minimum distance diagram of directed double loop networks we divide them into four types there are different parameters for each type according to the solutions to the congruence equation then we study the distributions of the optimal equivalent nodes for different destination nodes and present an optimal fault tolerant routing algorithm and a formula computing the fault tolerant diameter finally we prove that there is a lower bound for fault tolerant diameter and show that there may be lots of double optimal directed double loop networks in some infinite clusters of g n r s according to the proposed fault tolerant routing algorithm the reliability and transmission performance will be optimal when some faults occur in g n r s

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

embedding an arbitrary binary tree into the star graph binary tree structures have been very useful in solving divide and conquer type of problems embedding binary trees into another network the host network helps in designing solutions for the host network using the known solutions on binary trees embedding arbitrary binary trees into networks in particular into the hypercube has been addressed in the literature the latter was achieved with load 1 and constant dilation the n star graph is a recently introduced interconnection network for massively parallel systems it enjoys symmetry and fault tolerance properties that make it a viable alternative to the hypercube in this paper we address the problem of embedding arbitrary binary trees into the n star graph this work is the first to present such an embedding the tree has e n 1 1 or fewer vertices the embedding leads to load 1 and constant dilation for all values of n it therefore enables the star graph to efficiently simulate an arbitrary binary tree with only a constant factor of communication delay

embedding meshes and tori on double loop networks of the same size double loop networks are extensions of ring networks and are widely used in the design and implementation of local area networks and parallel processing architectures however embedding of other types of networks on double loop networks has not been well studied due to the topological complexity of double loop networks the traditional l shape designed to compute the diameter of double loop networks is not effective to solve the embedding problem we propose a novel tessellation approach to partition the geometric plane of double loop networks into a set of parallelogram tiles called p shape based on the characteristics of p shape we design a simple embedding scheme namely p shape embedding that embeds meshes and tori on double loop networks in a systematic way under p shape embedding we evaluate the embedding metrics of dilation average dilation and congestion which depend heavily on the parameters of p shape a main merit of p shape embedding is that a large fraction of embedded mesh torus edges have edge dilation 1 resulting in a low average dilation these are the first results to our knowledge for embedding meshes and tori on double loop networks which is of great significance due to the popularity of these architectures our p shape construction bridges between regular graphs and double loop networks and provides a powerful tool for studying double loop networks

the exact perimeter generating function for a model of punctured staircase polygons we have derived the perimeter generating function of a model of punctured staircase polygons in which the internal staircase polygon is rotated by a 90 angle with respect to the outer staircase polygon in one approach we calculated a long series expansion for the problem and found that all the terms in the generating function can be reproduced from a linear fuchsian differential equation of order 4 we then solved this ode and found a closed form expression for the generating function this is a highly unusual and most fortuitous result since odes of such high order very rarely permit a closed form solution in a second approach we proved the result for the generating function exactly using combinatorial arguments this latter solution allows many generalizations including models with other types of punctures and a model with any fixed number of nested rotated staircase punctures

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

discrete symmetry groups of vertex models in statistical mechanics we analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations they can be seen as generated by involutions corresponding respectively to two kinds of transformations onq q matrices the inversion of theq q matrix and an involutive permutation of the entries of the matrix we show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics this approach enables one to analyze two dimensionalq4 state vertex models as simply as three dimensional vertex models or higher dimensional vertex models various examples of birational symmetries of vertex models are analyzed a particular emphasis is devoted to a three dimensional vertex model the 64 state cubic vertex model which exhibits a polynomial growth of the complexity of the calculations a subcase of this general model is seen to yield integrable recursion relations we also concentrate on a specific two dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes yang baxter integrable it is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces or higher dimensional algebraic varieties

on a pair of vector spaces the paper provides an extensive investigation of a pair of vectors spaces with a particular attention paid to perpendicular subspaces several new properties of the pair are identified some of which enabled to derive known facts about subspaces by means of simpler proofs the approach utilized exploits partitioned representations of orthogonal projectors and is based on the fact that there is one to one correspondence between an orthogonal projector and the subspace onto which it projects

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

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

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

schlesinger transformations for painleve vi equation a method to obtain the schlesinger transformations for painleve vi equation is given the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential galois groups we first revisit an order six linear differential operator already introduced in a previous paper having a solution which is a diagonal of a rational function of three variables this linear differential operator is such that its exterior square has a rational solution indicating that it has a selected differential galois group and is actually homomorphic to its adjoint we obtain the two corresponding intertwiners giving this homomorphism to the adjoint we show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition already underlined in a previous paper in terms of order two self adjoint operators from these results we deduce a new form of decomposition of operators for this selected order six linear differential operator in terms of three order two self adjoint operators we generalize this decomposition to decomposition in terms of three self adjoint operators of arbitrary orders provided the three orders have the same parity we then generalize the previous decomposition to decompositions in terms of an arbitrary number of self adjoint operators of the same parity order this yields an infinite family of linear differential operators homomorphic to their adjoint and thus with a selected differential galois group we show that the equivalence of such operators with selected differential galois groups is compatible with these canonical decompositions the rational solutions of the symmetric or exterior squares of these selected operators are noticeably seen to depend only on the rightmost self adjoint operator in the decomposition these results and tools are applied on operators of large orders for instance it is seen that a large set of quite massive operators associated with reflexive 4 polytopes defining calabi yau three folds obtained recently by lairez correspond to a particular form of the decomposition detailed in this paper all the results of this paper can be seen as providing an algebraic characterization of linear differential operators with selected symplectic or orthogonal differential galois groups

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

core ep decomposition and its applications abstract in this paper we introduce a new decomposition called the core ep decomposition in the present paper for square matrices and give some of its applications by applying the decomposition we derive several characterizations of the core ep inverse introduce two orders the core ep order and the core minus partial order and characterize their properties we also give some properties of the decomposition

on a matrix decomposition of hartwig and spindelbock abstract the main task of the paper is to demonstrate that corollary 6 in r e hartwig k spindelbock matrices for which a and a commute linear and multilinear algebra 14 1984 241 256 provides a powerful tool to investigate square matrices with complex entries this aim is achieved on the one hand by obtaining several original results involving square matrices and on the other hand by reestablishing some of the facts already known in the literature often in extended and or generalized forms the particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings

on some new pre orders defined by weighted drazin inverses in this paper we investigate new binary relations defined on the set of rectangular complex matrices based on the weighted drazin inverse and give some characterizations of them these relations become pre orders and improve the results found by the authors in hernandez et al 2013 as well as extend those known for square matrices on the other hand some new weighted partial orders are also defined and characterized the advantages of these new relations compared to the ones considered in the mentioned paper are also pointed out

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