arithmetic properties of mirror map and quantum coupling we study some arithmetic properties of the mirror maps and the quantum yukawa couplings for some 1 parameter deformations of calabi yau manifolds first we use the schwarzian differential equation which we derived previously to characterize the mirror map in each case for algebraic k3 surfaces we solve the equation in terms of thej function by deriving explicit modular relations we prove that some k3 mirror maps are algebraic over the genus zero function fieldq j this leads to a uniform proof that those mirror maps have integral fourier coefficients regarding the maps as riemann mappings we prove that they are genus zero functions by virtue of the conway norton conjecture proved by borcherds using frenkel lepowsky meurman s moonshine module we find that these maps are actually the reciprocals of the thompson series for certain conjugacy classes in the griess fischer group this also gives as an immediate consequence a second proof that those mirror maps are integral we thus conjecture a surprising connection between k3 mirror maps and the thompson series for threefolds we construct a formal nonlinear ode for the quantum coupling reduced modp under the mirror hypothesis and an integrality assumption we derive modp congurences for the fourier coefficients for the quintics we deduce at least for 5 d that the degreed instanton numbersnd are divisible by 53 a fact first conjectured by clemens

phase transitions in dynamical random graphs we study a large time limit of a markov process whose states are finite graphs the number of the vertices is described by a supercritical branching process and the dynamics of edges is determined by the rates of appending and deleting we find a phase transition in our model similar to the one in the random graph model g n p we derive a formula for the line of critical parameters which separates two different phases one is where the size of the largest component is proportional to the size of the entire graph and another one where the size of the largest component is at most logarithmic with respect to the size of the entire graph in the supercritical phase we find the asymptotics for the size of the largest component

on globally nilpotent differential equations abstract in a previous work of the authors a middle convolution operation on the category of fuchsian differential systems was introduced in this note we show that the middle convolution of fuchsian systems preserves the property of global nilpotence this leads to a globally nilpotent fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems

generalized hypergeometric functions and rational curves on calabi yau complete intersections in toric varieties we formulate general conjectures about the relationship between the a model connection on the cohomology of ad dimensional calabi yau complete intersectionv ofr hypersurfacesv1 vr in a toric varietyp and the system of differential operators annihilating the special generalized hypergeometric series 0 constructed from the fan using this generalized hypergeometric series we propose conjectural mirrorsv ofv and the canonicalq coordinates on the moduli spaces of calabi yau manifolds

lattice green functions and calabi yau differential equations by making the connection between four dimensional lattice green functions lgfs and picard fuchs ordinary differential equations of calabi yau manifolds we have given explicit forms for the coefficients of the four dimensional lgfs on the simple cubic and body centred cubic lattices in terms of finite sums of products of binomial coefficients and have shown that the corresponding four dimensional face centred cubic lgf satisfies a fourth order ode of degree 7 which we have found but not solved the fact that the picard fuchs equations that appear here are similar to those that appear in string theory perhaps makes the first connection between these two apparently unrelated topics

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

random graphs with hidden color we propose and investigate a unifying class of sparse random graph models based on a hidden coloring of edge vertex incidences extending an existing approach random graphs with a given degree distribution in a way that admits a nontrivial correlation structure in the resulting graphs the approach unifies a number of existing random graph ensembles within a common general formalism and allows for the analytic calculation of observable graph characteristics in particular generating function techniques are used to derive the size distribution of connected components clusters as well as the location of the percolation threshold where a giant component appears

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

monstrous moonshine from orbifolds we consider the monster module of frenkel lepowsky and meurman as az2 orbifold of a bosonic string compactified by the leech lattice we show that the main conway and norton monstrous moonshine properties stating that the thompson series for each monster group conjugacy class has a modular invariance group of genus zero follow from an orbifold construction based on an orbifold group composed of monster group elements it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each thompson series it is also shown that the power map formula of conway and norton follows from the same vacuum structure finally we demonstrate the validity of the vacuum conjectures for sectors twisted by leech lattice automorphisms in many cases

spectral statistics for the dirac operator on graphs we determine conditions for the quantization of graphs using the dirac operator for both two and four component spinors according to the bohigas giannoni schmit conjecture for such systems with time reversal symmetry the energy level statistics are expected in the semiclassical limit to correspond to those of random matrices from the gaussian symplectic ensemble this is confirmed by numerical investigation the scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor we derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

semiclassical form factor for chaotic systems with spin we study the properties of the two point spectral form factor for classically chaotic systems with spin in the semiclassical limit with a suitable semiclassical trace formula as our principal tool to this end we introduce a regularized form factor and discuss the limit in which the so called diagonal approximation can be recovered the incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics provided this skew product is mixing we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory

beyond the heisenberg time semiclassical treatment of spectral correlations in chaotic systems with spin 1 2 the two point correlation function of chaotic systems with spin 1 2 is evaluated using periodic orbits the spectral form factor for all times thus becomes accessible equivalence with the predictions of random matrix theory for the gaussian symplectic ensemble is demonstrated a duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established

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

picard fuchs uniformization and modularity of the mirror map arithmetic properties of mirror symmetry type iia iib string duality are studied we give criteria for the mirror map q series of certain families of calabi yau manifolds to be automorphic functions for families of elliptic curves and lattice polarized k3 surfaces with surjective period mappings global torelli theorems allow one to present these criteria in terms of the ramification behavior of natural algebraic invariants the functional and generalized functional invariants respectively in particular when applied to one parameter families of rank 19 lattice polarized k3 surfaces our criterion demystifies the mirror moonshine phenomenon of lian and yau and highlights its non monstrous nature the lack of global torelli theorems and presence of instanton corrections makes calabi yau threefold families more complicated via the constraints of special geometry the picard fuchs equations for one parameter families of calabi yau threefolds imply a differential equation criterion for automorphicity of the mirror map in terms of the yukawa coupling in the absence of instanton corrections the projective periods map to a twisted cubic space curve a hierarchy of algebraic instanton corrections correlated with the differential galois group of the picard fuchs equation is proposed

modular solutions to equations of generalized halphen type solutions to a class of differential systems that generalize the halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group these functions uniformize riemann surfaces of genus zero and have q series with integral coefficients rational maps between these functions are derived implying subgroup relations between their automorphism groups as well as symmetrization maps between the associated differential systems

algebraic hypergeometric transformations of modular origin it is shown that ramanujan s cubic transformation of the gauss hypergeometric function 2 f 1 arises from a relation between modular curves namely the covering of x 0 3 by x 0 9 in general when 2 n 7 the n fold cover of x 0 n by x 0 n 2 gives rise to an algebraic hypergeometric transformation the n 2 3 4 transformations are arithmetic geometric mean iterations but the n 5 6 7 transformations are new in the final two cases the change of variables is not parametrized by rational functions since x 0 6 x 0 7 are of genus 1 since their quotients x 0 6 x 0 7 under the fricke involution an atkin lehner involution are of genus 0 the parametrization is by two valued algebraic functions the resulting hypergeometric transformations are closely related to the two valued modular equations of fricke and h cohn

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

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

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

on the integrality of the taylor coefficients of mirror maps we continue our study begun in on the integrality of the taylor coefficients of mirror maps duke math j to appear of the fine integrality properties of the taylor coefficients of the series q z z exp g z f z where f z and g z log z f z are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at z 0 more precisely we address the question of finding the largest integer v such that the taylor coefficients of z 1q z 1 v are still integers in particular we determine the dwork kontsevich sequence un n 1 where un is the largest integer such that qn z n is a series with integer coefficients where qn z exp gn z fn z fn z m 0 nm z m and gn z m 1 hnm hm nm z m with hn denoting the n th harmonic number conditional on the conjecture that there are no prime number p and integer n such that the p adic valuation of hn 1 is strictly greater than 3

simple strong glass forming models mean field solution with activation we introduce simple models inspired by previous models for froths and covalent glasses with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems these models are also a generalization of backgammon or urn models to a non constant number of particles where entropic barriers are replaced by energy barriers allowing for the existence of activated processes we formulate a mean field version of the models which keeps most of the features of the finite dimensional ones and solve analytically the out of equilibrium dynamics in the low temperature regime where activation plays an essential role

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

monodromy of picard fuchs differential equations for calabi yau threefolds abstract in this paper we are concerned with the monodromy of picardfuchs differential equations associated with one parameter families of calabiyau threefolds our results show that in the hypergeometric cases the matrix representations of monodromy relative to the frobenius bases can be expressed in terms of the geometric invariants of the underlying calabi yau threefolds this phenomenon is also verified numerically for other families of calabi yau threefolds in the paper furthermore we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of sp 4 z of finite index and whose levels are related to the geometric invariants of the calabi yau threefolds

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

on differential equations satisfied by modular forms we use the theory of modular functions to give a new proof of a result of p f stiller which asserts that if t is a non constant meromorphic modular function of weight 0 and f is a meromorphic modular form of weight k with respect to a discrete subgroup of sl 2 commensurable with sl 2 then f as a function of t satisfies a k 1 st order linear differential equation with algebraic functions of t as coefficients furthermore we show that the schwarzian differential equation for the modular function t can be extracted from any given k 1 st order linear differential equation of this type one advantage of our approach is that every coefficient in the differential equations can be relatively easily determined

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

fricke s two valued modular equations the modular equation of order b is a polynomial relation between j z and j z b which has astronomically large coefficients even for small values of b fricke showed that a two valued relation exists for 37 small values of b this relation would have much smaller coefficients and would also be convenient for finding singular moduli although fricke produced no two valued relations explicitly no doubt because of the tedious amount of algebraic manipulation they are now found by use of macsyma for 31 cases ranging from b 2 to 49 fricke provided the equations necessary to generate the relations with two corrections required the remaining six cases of order 39 41 47 50 59 71 require an extension of fricke s methods using the discriminant function theta functions and power series approximations

leading off diagonal approximation for the spectral form factor for uniformly hyperbolic systems we consider the semiclassical approximation to the spectral form factor k for two dimensional uniformly hyperbolic systems with time reversal symmetry and derive the first off diagonal correction for small the result agrees with the 2 term of the form factor for the goe random matrix ensemble

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

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

semiclassical approach to parametric spectral correlation with spin 1 2 the spectral correlation of a chaotic system with spin 1 2 is universally described by the gse gaussian symplectic ensemble of random matrices in the semiclassical limit in semiclassical theory the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere in this paper instead of the uniform distribution we introduce brownian motion on a sphere to yield the parametric motion of the energy levels as a result the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the gse universality class moreover by starting the brownian motion from a point distribution on the sphere we gradually increase the effect of the spin and calculate the form factor describing the transition from the gaussian orthogonal ensemble class to the gse class

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

intermediate statistics for a system with symplectic symmetry the dirac rose graph we study the spectral statistics of the dirac operator on a rose shaped graph a graph with a single vertex and all bonds connected at both ends to the vertex we formulate a secular equation that generically determines the eigenvalues of the dirac rose graph which is seen to generalize the secular equation for a star graph with neumann boundary conditions we derive approximations to the spectral pair correlation function at large and small values of spectral spacings in the limit as the number of bonds approaches infinity and compare these predictions with results of numerical calculations our results represent the first example of intermediate statistics from the symplectic symmetry class

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

the semiclassical origin of the logarithmic singularity in the symplectic form factor sieber and richter achieved a breakthrough towards a proof of the universality of spectral fluctuations of chaotic quantum systems conjectured by bohigas giannoni and schmidt by calculating semiclassically the first term beyond the diagonal approximation of the orthogonal form factor in this letter the semiclassical origin of the logarithmic singularity of the symplectic form factor is deduced perturbatively by combining this result with the contribution that arises due to the spin this approach stands in contrast to the duality approach introduced by bogomolny and keating which is essentially non perturbative and where the structure around the heisenberg time is related to the structure for very small time which can be deduced using the diagonal approximation

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

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

properties of random graphs with hidden color we investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub coloring with or without the restriction to nondegenerate graphs the calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated cluster size statistics are derived with generating function techniques yielding a well defined percolation threshold explicit rules are derived for the enumeration of small subgraphs duality and redundancy is discussed and subclasses corresponding to commonly studied models are identified

automorphisms of algebraic varieties and yang baxter equations it is shown that commuting transfer matrix models in statistical mechanics are parametrized by algebraic varieties having a set of automorphisms deduced from the so called inversion relation in general this set of automorphisms is infinite this shows that for algebraic varieties of dimension 1 the models are parametrized by algebraic curves of genus 0 or 1

clustering determines the dynamics of complex contagions in multiplex networks we present the mathematical analysis of generalized complex contagions in a class of clustered multiplex networks the model is intended to understand spread of influence or any other spreading process implying a threshold dynamics in setups of interconnected networks with significant clustering the contagion is assumed to be general enough to account for a content dependent linear threshold model where each link type has a different weight for spreading influence that may depend on the content e g product rumor political view that is being spread using the generating functions formalism we determine the conditions probability and expected size of the emergent global cascades this analysis provides a generalization of previous approaches and is especially useful in problems related to spreading and percolation the results present nontrivial dependencies between the clustering coefficient of the networks and its average degree in particular several phase transitions are shown to occur depending on these descriptors generally speaking our findings reveal that increasing clustering decreases the probability of having global cascades and their size however this tendency changes with the average degree there exists a certain average degree from which on clustering favors the probability and size of the contagion by comparing the dynamics of complex contagions over multiplex networks and their monoplex projections we demonstrate that ignoring link types and aggregating network layers may lead to inaccurate conclusions about contagion dynamics particularly when the correlation of degrees between layers is high

spectral correlations of individual quantum graphs we investigate the spectral properties of chaotic quantum graphs we demonstrate that the energy average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear model action this proves that spectral correlations of individual quantum graphs behave according to the predictions of wigner dyson random matrix theory we explore the stability of the universal random matrix behavior with regard to perturbations and discuss the crossover between different types of symmetries

analysis of complex contagions in random multiplex networks we study the diffusion of influence in random multiplex networks where links can be of r different types and for a given content e g rumor product or political view each link type is associated with a content dependent parameter ci in 0 that measures the relative bias type i links have in spreading this content in this setting we propose a linear threshold model of contagion where nodes switch state if their perceived proportion of active neighbors exceeds a threshold namely a node connected to mi active neighbors and ki mi inactive neighbors via type i links will turn active if cimi ciki exceeds its threshold under this model we obtain the condition probability and expected size of global spreading events our results extend the existing work on complex contagions in several directions by i providing solutions for coupled random networks whose vertices are neither identical nor disjoint ii highlighting the effect of content on the dynamics of complex contagions and iii showing that content dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature

on reducing the heun equation to the hypergeometric equation abstract the reductions of the heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified heun to hypergeometric reductions are similar to classical hypergeometric identities but the conditions for the existence of a reduction involve features of the heun equation that the hypergeometric equation does not possess namely its cross ratio and accessory parameters the reductions include quadratic and cubic transformations which may be performed only if the singular points of the heun equation form a harmonic or an equianharmonic quadruple respectively and several higher degree transformations this result corrects and extends a theorem in a previous paper which found only the quadratic transformations siam j math anal 10 3 1979 655

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 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