three friendly walkers more than 15 years ago guttmann and voge 2002 j stat plan inference 101 107 introduced a model of friendly walkers since then it has remained unsolved in this paper we provide the exact solution to a closely allied model which essentially only differs in the boundary conditions the exact solution is expressed in terms of the reciprocal of the generating function for vicious walkers which is a d finite function however ratios of d finite functions are inherently not d finite and in this case we prove that the friendly walkers generating function is the solution to a non linear differential equation with polynomial coefficients it is in other words d algebraic we find using numerically exact calculations a conjectured expression for the generating function of the original model as a ratio of a d finite function and the generating function for vicious walkers we obtain an expression for this d finite function in terms of a hypergeometric function with a rational pullback and its first and second derivatives

path integral method for dna denaturation the statistical physics of homogeneous dna is investigated by the imaginary time path integral formalism the base pair stretchings are described by an ensemble of paths selected through a macroscopic constraint the fulfillment of the second law of thermodynamics the number of paths contributing to the partition function strongly increases around and above a specific temperature t c ensuremath ast whereas the fraction of unbound base pairs grows continuously around and above t c ensuremath ast the latter is identified with the denaturation temperature thus the separation of the two complementary strands appears as a highly cooperative phenomenon displaying a smooth crossover versus t the thermodynamical properties have been computed in a large temperature range by varying the size of the path ensemble at the lower bound of the range no significant physical dependence on the system size has been envisaged the entropy grows continuously versus t while the specific heat displays a remarkable peak at t c ensuremath ast the location of the peak versus t varies with the stiffness of the anharmonic stacking interaction along the strand the presented results suggest that denaturation in homogeneous dna has the features of a second order phase transition the method accounts for the cooperative behavior of a very large number of degrees of freedom while the computation time is kept within a reasonable limit

pinning and wetting transition for 1 1 dimensional fields with laplacian interaction we consider a random field 1 n as a model for a linear chain attracted to the defect line 0 that is the x axis the free law of the field is specified by the density exp iv i with respect to the lebesgue measure on n where is the discrete laplacian and we allow for a very large class of potentials v the interaction with the defect line is introduced by giving the field a reward 0 each time it touches the x axis we call this model the pinning model we consider a second model the wetting model in which in addition to the pinning reward the field is also constrained to stay nonnegative we show that both models undergo a phase transition as the intensity of the pinning reward varies both in the pinning a p and in the wetting a w case there exists a critical value ca such that when ca the field touches the defect line a positive fraction of times localization while this does not happen for ca delocalization the two critical values are nontrivial and distinct 0 cp cw and they are the only nonanalyticity points of the respective free energies for the pinning model the transition is of second order hence the field at cp is delocalized on the other hand the transition in the wetting model is of first order and for cw the field is localized the core of our approach is a markov renewal theory description of the field

probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

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

poland scheraga models and the dna denaturation transition poland scheraga models were introduced to describe the dna denaturation transition we give a rigorous and refined discussion of a family of these models we derive possible scaling functions in the neighborhood of the phase transition point and review common examples we introduce a self avoiding poland scheraga model displaying a first order phase transition in two and three dimensions we also discuss exactly solvable directed examples this complements recent suggestions as to how the poland scheraga class might be extended in order to display a first order transition which is observed experimentally

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 algebraic solutions of lame s differential equation in two previous papers 1 21 we reconsidered and improved klein s method for establishing whether a given second order linear differential equation with rational function coefficients over an algebraic curve has a full set of algebraic solutions that procedure being rather complicated we were interested in checking whether when applied to classical examples it would lead to nontrivial new results the hypergeometric equation being in that respect well understood lame s equation see 0 1 was the simplest type of differential equation for which we could hope to get interesting information by our method before explaining in detail the content of this paper it may be useful to the reader if we recall a few facts proved in 1 21 of which constant use will be made throughout this paper let c be a nonsingular algebraic curve over the complex field c with function field k and let l be a second order linear differential operator on c that is

exact scaling behavior of partially convex vesicles we solve analytically for the perimeter area generating functions for two models of vesicles while from the solution of the first model staircase polygons one can easily extract the asymptotic scaling behavior the exact solution of the second column convex polygons is difficult to analyze this leads us to apply a recently developed method for deriving the scaling behavior indirectly utilizing a set of nonlinear differential equations one result of this work is a nontrivial confirmation of the scaling universality hypothesis

on the cubic lattice green functions wheatstone physics laboratory king s college university of london strand london wc2r 2ls u k it is proved that k k 4 eta frac 1 2 1 eta frac 1 2 k k where eta is a complex variable which lies in a certain region scr r 2 of the eta plane and k k pm are complete elliptic integrals of the first kind with moduli k pm which are given by k pm 2 equiv k pm 2 eta frac 1 2 pm frac 1 4 eta 4 eta frac 1 2 frac 1 4 2 eta 1 eta frac 1 2 this basic result is then used to express the face centred cubic and simple cubic lattice green functions at the origin in terms of the square of a complete elliptic integral of the first kind several new identities involving the heun function f a b alpha beta gamma delta eta are also derived next it is shown that the three cubic lattice green functions all have parametric representations which involve the green function for the two dimensional honeycomb lattice finally the results are applied to a variety of problems in lattice statistics in particular a new simplified formula for the generating function of staircase polygons on a four dimensional hypercubic lattice is derived

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

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

energy based balance control approach to the ball and beam system this article investigates the set point balance control problem of the ball and beam system acting on the ball which is an underactuated system the control law is designed based on the total mechanical energy and the passivity properties of the system according to the desired set point the zero equilibrium point and non zero equilibrium points are studied respectively for the zero case it is proved that a single pd feedback controller is sufficient to bring the state to zero from any initial condition on condition that the control parameters satisfy an inequality for the non zero case the control problem is much more complicated unlike the previous energy based control laws a new form of lyapunov function candidate is constructed a complete analysis of the convergence of the energy and the dynamics is given and the characteristics of the closed loop system with the proposed feedback control law are illustrated moreover it is proved that with the parameter choice rules proposed in this article the ball and beam system will eventually converge exactly to the desired non zero equilibrium point furthermore since the length of the beam is not unlimited the trajectory of the ball should be restricted within a limited range the balance control laws are modified to avoid the ball running beyond the joint range limitation for the zero case and the non zero case respectively simulation results show that the control laws proposed in this article are effective for the set point balance control problem of the ball and beam system

lattice green s functions for kagome diced and hyperkagome lattices the lattice green s functions lgfs and density of states dos for noninteracting models on three related lattices are presented the dos and lgfs at the origin for the kagome and diced lattices are rederived furthermore from the form obtained for the dos of the hyperkagome lattice it seems highly suggestive that a closed form expression can be obtained if a particular complicated yet highly symmetrical integral can be solved the results have been checked numerically the continuum of energy states in the hyperkagome lattice and the appearance of a zero in the kagome lattice may be explicitly seen from the formulas obtained

box integrals by a box integral we mean here an expectation where email protected runs over the unit n cube with email protected and s fixed explicitly 0 email protected 0 1 r 1 q 1 2 r n q n 2 s 2dr 1 dr n the study of box integrals leads one naturally into several disparate fields of analysis while previous studies have focused upon symbolic evaluation and asymptotic analysis of special cases notably s 1 we work herein more generally in interdisciplinary fashion developing results such as 1 analytic continuation in complex s 2 relevant combinatorial identities 3 rapidly converging series 4 statistical inferences 5 connections to mathematical physics and 6 extreme precision quadrature techniques appropriate for these integrals these intuitions and results open up avenues of experimental mathematics with a view to new conjectures and theorems on integrals of this type

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

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

universal properties of group actions on locally compact spaces abstract we study universal properties of locally compact g spaces for countable infinite groups g in particular we consider open invariant subsets of the g space g and their minimal closed invariant subspaces these are locally compact free g spaces and the latter are also minimal we examine the properties of these g spaces with emphasis on their universal properties as an example of our results we use combinatorial methods to show that each countable infinite group admits a free minimal action on the locally compact non compact cantor set

lattice green function at 0 for the 4d hypercubic lattice the generating function for recurrent polya walks on the four dimensional hypercubic lattice is expressed as a kampe de feriet function various properties of the associated walks are enumerated

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

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

symmetry and the chazy equation there are three different actions of the unimodular lie group sl 2 c on a two dimensional space in every case we show how an ordinary differential equation admitting sl 2 as a symmetry group can be reduced in order by three and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation a particular example is the chazy equation whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation the reduction method leads to an alternative formula in terms of solutions to the lame equation resulting in a surprising transformation between the lame and hypergeometric equations finally we discuss the painleve analysis of the singularities of solutions to the chazy equation

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

parallel integer relation detection techniques and applications let x1 x2 xn be a vector of real numbers an integer relation algorithm is a computational scheme to find the n integers ak if they exist such that a1x1 a2x2 anxn 0 in the past few years integer relation algorithms have been utilized to discover new results in mathematics and physics existing programs for this purpose require very large amounts of computer time due in part to the requirement for multiprecision arithmetic yet are poorly suited for parallel processing this paper presents a new integer relation algorithm designed for parallel computer systems but as a bonus it also gives superior results on single processor systems single and multi level implementations of this algorithm are described together with performance results on a parallel computer system several applications of these programs are discussed including some new results in number theory quantum field theory and chaos theory

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

algebraic solutions of the lame equation revisited a minor error in the necessary conditions for the algebraic form of the lam e equation to have a finite projective monodromy group and hence for it to have only algebraic solutions is pointed out see f baldassarri on algebraic solutions of lams differential equation j differential equations41 1 1981 44 58 it is shown that if the group is the octahedral group then the degree parameter of the equation may differ by from an integer this possibility was missed the omission affects a recent result on the monodromy of the weierstrass form of the lamequation see r c churchill two generator subgroups of and the hypergeometric riemann and lam e equations j symbolic computa tion 28 4 5 1999 521 545 the weierstrass form which is a differential equation on an elliptic curve may have after all an octahedral projective monodromy group

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

on an exactly solvable compact cluster growth model on a square lattice we consider a cluster growth model on a square lattice that is in the compact directed percolation universality class the model is exactly solvable as regards its static cluster properties and expressions are given for various quantities of interest such as the mean perimeter length and the mean cluster size the results also provide new information on area perimeter generating functions for a class of self avoiding polygons on square lattices

some cubic modular identities of ramanujan there is a beautiful cubic analogue of jacobi s fundamental theta function identity it is sum n m infty infty q n 2 nm m 2 sum n m infty infty n m q n nm m sum n m infty infty q n 1 3 n 1 3 m 1 3 m 1 3 here exp 2 i 3 in this note we provide an elementary proof of this identity and of a related identity due to ramanujan we also indicate how to discover and prove such identities symbolically

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

applications of fuchsian differential equations to free boundary problems in a couple of recent papers free boundary problems and a class of conformal mappings involving curvilinear quadrilaterals were analysed primarily using transform methods in both the free boundary and conformal mapping problems there is an underlying relation with fuchsian differential equations and an alternative conceptually simpler solution technique for these problems would use solutions of these fuchsian equations thus it was conjectured in those papers that a class of fuchsian differential equations commonly known as heun9s equation has in some special but relatively important cases degenerate solutions these involve hypergeometric functions this complementary solution method is in many cases more convenient than the transform approach the purpose of this paper is to explore the connections with the fuchsian equations directly and use the solutions to solve free boundary problems the specific examples treated here come from a quasi steady approximation to solidification problems and are not without interest in their own right there are few analytical solutions for solidification problems in geometries of practical interest and the solutions found here should be of use in that regard

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

nonlinear control via approximate input output linearization the ball and beam example approximate input output linearization of nonlinear systems which fail to have a well defined relative degree is studied for such systems a method for constructing approximate systems that are input output linearizable is provided the analysis presented is motivated through its application to a common undergraduate control laboratory experiment the ball and beam where it is shown to be more effective for trajectory tracking than the standard jacobian linearization

advances in the theory of box integrals box integrals expectations or over the unit n cube or n box have over three decades been occasionally given closed forms for isolated n s by employing experimental mathematics together with a new global analytic strategy we prove that for n le 4 dimensions the box integrals are for any integer s hypergeometrically closed in a sense we clarify herein for n 5 dimensions we show that a single unresolved integral we call k sub 5 stands in the way of such hyperclosure proofs we supply a compendium of exemplary closed forms that naturally arise algorithmically from this theory

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

explicitly symmetrical treatment of three body phase space we derive expressions for three body phase space that are explicitly symmetrical in the masses of the three particles we study geometrical properties of the variables involved in elliptic integrals and demonstrate that it is convenient to use the jacobian zeta function to express the results in four and six dimensions

two dimensional random walk and critical behavior of double strand dna the phase behavior in a double strand homopolynucleotide is considered the relative distance between the corresponding monomers of two polymer chains is modeled by the two dimensional random walk on the square lattice returns of the random walk to the origin describe the formation of base pairs between the complementary units we obtain a completely denatured state below the finite temperature tc it was shown that tc is the point of the infinite order phase transition the specific behavior of the order parameter free energy and correlation functions has been addressed

a large deviation perspective on ratio observables in reset processes robustness of rate functions we study large deviations of a ratio observable in discrete time reset processes the ratio takes the form of a current divided by the number of reset steps and as such it is not extensive in time a large deviation rate function can be derived for this observable via contraction from the joint probability density function of current and number of reset steps the ratio rate function is differentiable and we argue that its qualitative shape is robust i e it is generic for reset processes regardless of whether they have short or long range correlations we discuss similarities and differences with the rate function of the efficiency in stochastic thermodynamics

an exact solution of two friendly interacting directed walks near a sticky wall we find the exact solution of two interacting friendly directed walks modelling polymers on the square lattice these walks are confined to the quarter plane by a horizontal attractive surface to capture the effects of dna denaturation and adsorption we find the solution to the model s corresponding generating function by means of the obstinate kernel method specifically we apply this technique in two different instances to establish partial solutions for two simplified generating functions of the same underlying model that ignore either surface or shared contacts we then subsequently combine our two partial solutions to find the solution for the full generating function in terms of the two simpler variants this expression guides our analysis of the model where we find the system exhibits four phases and proceed to delineate the full phase diagram showing that all observed phase transitions are second order

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

partial stabilization of underactuated euler lagrange systems via a class of feedback transformations abstract a new class of non trivial feedback transformations which can be applied to any underactuated lagrangian system is proposed as shown these transformations play an important role in solving a stabilization problem with respect to a part of the actuated variables while the given additional constraints are imposed on the rest of the variables theoretical results are illustrated by solving the problem of swinging up the pendubot

the evaluation of integrals of bessel functions via g function identities a few transformations are presented for reducing certain cases of meijer s g function to a g function of lower order their applications to the integration of a product of bessel functions are given the algorithm has been implemented within mathematica 3 0

singularity analysis hadamard products and tree recurrences we present a toolbox for extracting asymptotic information on the coefficients of combinatorial generating functions this toolbox notably includes a treatment of the effect of hadamard products on singularities in the context of the complex tauberian technique known as singularity analysis as a consequence it becomes possible to unify the analysis of a number of divide and conquer algorithms or equivalently random tree models including several classical methods for sorting searching and dynamically managing equivalence relations

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

effective laguerre asymptotics it is known that the generalized laguerre polynomials can enjoy subexponential growth for large primary index in particular for certain fixed parameter pairs a z one has the large n asymptotic behavior l n a z sim c a z n a 2 1 4 e 2 sqrt nz we introduce a computationally motivated contour integral that allows efficient numerical laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior we present a fast algorithm for symbolic generation of the rather formidable expansion coefficients along the way we address the difficult problem of establishing effective i e rigorous and explicit error bounds on the general expansion a primary tool for these developments is an exp arc method giving a natural bridge between converging series and effective asymptotics

70 years of the watson integrals watson q j math oxford 10 266 276 1939 published in 1939 the evaluation of three integrals submitted to him which had arisen from a problem in physics van pepye in physica 5 465 482 1938 over the years these integrals have continued to occur in other aspects of physics such as random walk problems this article reviews these integrals and generalisations over the past 70 years

on kummer s twenty four solutions of the hypergeometric differential equation the p adic analyticity of the boyarsky matrix associated with the hypergeometric function f a b c x has been investigated in an earlier article the transformation of this matrix under translation of a b c by z3 was determined at that time this article gives the transformation of this matrix under the extended kummer group the p adic implications of quadratic and higher degree transformations remain open the object of this article is to explain the relation between kummer s twenty four solutions of this equation 1 ax i a 12 c a b ab y 0 and the p adic theory of this equation for simplicity of exposition let us assume that a b c satisfy the condition 2 a b c a c b all lie outside z the unique solution of 1 analytic at a 0 is given by the classical formula 3 f a b c e a s b s ms s o c s where a s f a s f a s e n s 0 the euler integral formula f a b c n 17 c jb f 4 bs c a f b r c b l a b c wa b c x b x 1 1 x cb1 l x a d valid for re c re b 0 a restriction which may be removed 7 p 60 equation 13 by changing the path of integration shows that we are concerned with the periods of a certain differential as is well known 5 6 11 p 107 12 the kummer list may be obtained by transformation of this integral representation indeed there received by the editors september 1 1982 and in revised form september 28 1983 1980 mathematics subject classification primary 12b40 34a30 1984 american mathematical society 0002 9947 84 1 00 25 per page

simple model for the dna denaturation transition we study pairs of interacting self avoiding walks mathit ensuremath omega 1 mathit ensuremath omega 2 on the 3d simple cubic lattice they have a common origin mathit ensuremath omega 0 1 mathit ensuremath omega 0 2 and are allowed to overlap only at the same monomer position along the chain mathit ensuremath omega i 1 ensuremath ne mathit ensuremath omega j 2 for i ensuremath ne j while mathit ensuremath omega i 1 mathit ensuremath omega i 2 is allowed the latter overlaps are indeed favored by an energetic gain ensuremath epsilon this is inspired by a model introduced long ago by poland and sheraga j chem phys 45 1464 1966 for the denaturation transition in dna where however self avoidance was not fully taken into account for both models there exists a temperature t m above which the entropic advantage to open up overcomes the energy gained by forming tightly bound two stranded structures numerical simulations of our model indicate that the transition is of first order the energy density is discontinuous but the analog of the surface tension vanishes and the scaling laws near the transition point are exactly those of a second order transition with crossover exponent ensuremath varphi 1 numerical and exact analytic results show that the transition is second order in modified models where the self avoidance is partially or completely neglected

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