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

analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

algebraic structure of tt equations for calabi yau sigma models the tt equations define a flat connection on the moduli spaces of 2d mathcal n 2 quantum field theories for conformal theories with c 3d which can be realized as nonlinear sigma models into calabi yau d folds this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions we show that the non holomorphic content of the tt equations restricted to the conformal directions in the cases d 1 2 3 is captured in terms of finitely many generators of special functions which close under derivatives the generators are understood as coordinates on a larger moduli space this space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric geometrically the freedom corresponds to a choice of forms on the target space respecting the hodge filtration and having a constant pairing linear combinations of vector fields on that space are identified with the generators of a lie algebra this lie algebra replaces the non holomorphic derivatives of tt and provides these with a finer and algebraic meaning for sigma models into lattice polarized k3 manifolds the differential ring of special functions on the moduli space is constructed extending known structures for d 1 and 3 the generators of the differential rings of special functions are given by quasi modular forms for d 1 and their generalizations in d 2 3 some explicit examples are worked out including the case of the mirror of the quartic in mathbbm p 3 where due to further algebraic constraints the differential ring coincides with quasi modular forms

counting bps states on the enriques calabi yau we study topological string amplitudes for the fhsv model using various techniques this model has a type ii realization involving a calabi yau threefold with enriques fibres which we call the enriques calabi yau by applying heterotic type iia duality we compute the topological amplitudes in the fibre to all genera it turns out that there are two different ways to do the computation that lead to topological couplings with different bps content one of them gives the standard d0 d2 counting amplitudes and from the other one we obtain information about bound states of d0 d4 d2 branes on the enriques fibre we also study the model using mirror symmetry and the holomorphic anomaly equations we verify in this way the heterotic results for the d0 d2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms and up to genus three this model turns out to be much simpler than the generic b model and might be exactly solvable

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

a criterion for the integrality of the taylor coefficients of mirror maps in several variables we give a necessary and sufficient condition for the integrality of the taylor coefficients at the origin of formal power series qi z ziexp gi z f z with z z1 zd and where f z and gi z log zi f z i 1 d are particular solutions of certain a systems of differential equations this criterion is based on the analytical properties of landau s function which is classically associated with sequences of factorial ratios and it generalizes the criterion in the case of one variable presented in e delaygue critere pour l integralite des coefficients de taylor des applications miroir j reine angew math 662 2012 205 252 one of the techniques used to prove this criterion is a generalization of a version of a theorem of dwork on formal congruences between formal series proved by krattenthaler and rivoal in c krattenthaler t rivoal multivariate p adic formal congruences and integrality of taylor coefficients of mirror maps in l di vizio t rivoal eds theories galoisiennes et arithmetiques des equations differentielles in seminaire et congres vol 27 soc math france paris 2011 pp 279 307

solitary shock waves and other travelling waves in a general compressible hyperelastic rod in the literature it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods recently it has been shown that this conjecture is true one might guess that when compressibility is taken into account such a wave which is both a solitary wave and a shock wave can still arise one of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description it is difficult to directly tackle the fully nonlinear rod equations here by using a non dimensionalization process and the reductive perturbation technique we derive a new type of nonlinear dispersive equation as the model equation we then focus on the travelling wave solutions of this new equation as a result we obtain a system of ordinary differential equations an important feature of this system is that there is a vertical singular line in the phase plane which leads to the appearance of shock waves by considering the equilibrium points and their relative positions to the singular line we are able to determine all qualitatively different phase planes those paths in phase planes which represent physically acceptable solutions are discussed one by one it turns out that there is a variety of travelling waves including solitary shock waves solitary waves periodic shock waves etc analytical expressions for all these waves are obtained a new phenomenon is also found a solitary wave can suddenly change into a periodic wave with finite period in dynamical systems this represents a homoclinic orbit suddenly changing into a closed orbit to the authors9 knowledge such a bifurcation has not been found in any other dynamical systems

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

linear differential equations and multiple zeta values iii zeta 3 we consider the hypergeometric equation 1 t t t g x3g 0 whose unique analytic solution 1 t x 1 o t near t 0 for t 1 becomes a generating function for multiple zeta values 1 1 x f3 x 1 3 x3 3 3 x6 we apply the so called wkb method to study solutions of the hypergeometric equation for large x and we calculate corresponding stokes matrices we prove that the function f3 x near x is also expressed via wkb type functions which subject to some stokes phenomenon this implies that f3 x satisfies a sixth order linear differential equation with irregular singularity at infinity

the transcendental part of the regulator map for k_1 on a mirror family of k3 surfaces uller stach abstract we compute the transcendental part of the normal function corresponding to the deligne class of a cycle in k1 of a mirror family of quartic k 3 surfaces the re sulting multivalued function does not satisfy the hypergeometric differential equation of the periods and we conclude that the cycle is indecomposable for most points in the mirror family the occurring inhomogenous picard fuchs equations are related to painlev e vi type differential equations 1 the regulator map and picard fuchs equations in this paper we study the first nonclassical higher k group k 1 x for a smooth com plex projective surface x it was conjectured by h esnault around 1995 that certain elements in this group can be detected in the transcendental part of the deligne coho mology group h 3

arithmetic harmonic analysis on character and quiver varieties ii we study connections between the topology of generic character varieties of fundamental groups of punctured riemann surfaces macdonald polynomials quiver representations hilbert schemes on c x x c x modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups c 2012 elsevier inc all rights reserved

ray nonsingularity of cycle tree matrices abstract ray nonsingular rns matrices are a generalization of sign nonsingular sns matrices from the real field to the complex field a matrix with positive diagonals is called a cycle tree matrix if the adjacency structure of all the cycles in its associated digraph is a tree in this paper the ray nonsingularity of cycle tree matrices is studied and a general method is given to recognize ray nonsingular cycle tree matrices

bifurcation of travelling wave solutions in generalized phi four equation by using the method of planar dynamical systems to a nonlinear variant of the phi four equation the existence of smooth and nonsmooth solitary wave so called peakon and valleyon and infinite many periodic wave solutions is shown under different regions of parametric spaces various sufficient conditions to guarantee the existence of above solutions are given the formulas to compute the travelling waves are also educed

rational conformal field theories and complex multiplication we study the geometric interpretation of two dimensional rational conformal field theories corresponding to sigma models on calabi yau manifolds we perform a detailed study of rcft s corresponding to the t2 target and identify the cardy branes with geometric branes the t2 s leading to rcft s admit complex multiplication which characterizes cardy branes as specific d0 branes we propose a condition for the conformal sigma model to be rcft for arbitrary calabi yau n folds which agrees with the known cases together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories and sometimes appear to be finite in number for calabi yau n folds for n 2 rcft s on k3 may be dense we speculate about the meaning of these special points in the moduli spaces of calabi yau n folds in connection with freezing geometric moduli

the douady space of a complex surface abstract we prove that a standard realization of the direct image complex via the so called douady barlet hilbert chow in the algebraic case morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi isomorphism of complexes this is a more precise form of a special case of the decomposition theorems of beilinson bernstein deligne gabber and m saito of which the proof we present is independent in addition it is elementary and transparent and does not use either perverse sheaves the methods of positive characteristic nor saito s theory of mixed hodge modules the proof hinges on the special case of the bi disk in the complex affine plane where we make explicit use of a construction of nakajima s and of the corresponding representation theoretic interpretation foreseen by vafa and witten some consequences of the decomposition theorem the gottsche formula holds for complex surfaces interpretation of the rational cohomologies of douady spaces as a kind of fock space new proofs of results of briancon and ellingsrud and stromme on punctual hilbert schemes and computation of the mixed hodge structure of the douady spaces in the kahler case we also derive a natural connection with equivariant k theory for which in the case of algebraic surfaces bezrukavnikov and ginzburg have proposed a different approach

ray patterns of matrices and nonsingularity a complex matrix a is ray nonsingular if det x 0 a f 0 for every matrix x with positive entries a sufficient condition for ray nonsingularity is that the origin is not in the relative interior of the convex hull of the signed transversal products of a the concept of an isolated set of transversals is defined and used to obtain a neces

bifurcations of travelling wave solutions in a model of the hydrogen bonded systems by using the theory of bifucations of dynamical systems to a model of the hydrogen bonded systems the existence of solitary wave kink and anti kink wave solutions and uncountably infinite many smooth and non smooth periodic wave solutions is obtained under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given in some simple parametric conditions exact explicit and implicit solution formulas are listed

bifurcations of travelling wave solutions for modified nonlinear dispersive phi four equation abstract by using the bifurcation theory of dynamical systems to modified nonlinear dispersive phi four equation we analysis all bifurcations and phase portraits in the parametric space the existence of solitary wave solutions and uncountably infinite many smooth and non smooth periodic wave solutions is obtained under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given some explicit exact solution formulas are acquired for some special cases

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

an approximation of the analytic solution of the shock wave equation in this article we discuss the analytic solution of the fully developed shock waves the adomian decomposition method is used to solve the shock wave equation which describes the flow of gases unlike the various numerical techniques which are usually valid for short period of time the solution of the presented equation is analytic for 0 t also the results presented here indicate that the method is reliable accurate and converges very rapidly

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

donaldson invariants of product ruled surfaces and two dimensional gauge theories using the u plane integral of moore and witten we derive a simple expression for the donaldson invariants of product ruled surfaces g s 2 where g is a riemann surface of genus g this expression generalizes a theorem of morgan and szabo for g 1 to any genus g we give two applications of our results 1 we derive thaddeus formulae for the intersection pairings on the moduli space of rank two stable bundles over a riemann surface 2 we derive the eigenvalue spectrum of the fukaya floer cohomology of g s 1

galerkin method for the numerical solution of the rlw equation using quintic b splines the regularized long wave equation rlw is solved numerically by using the quintic b spline galerkin finite element method the same method is applied to the time split rlw equation comparison is made with both analytical solutions and some previous results propagation of solitary waves interaction of two solitons are studied

a bifurcation method for solving multiple positive solutions to the boundary value problem of the henon equation on a unit disk three algorithms based on the bifurcation theory are proposed to compute the o 2 symmetric positive solutions to the boundary value problem of the henon equation on the unit disk taking l in the henon equation as a bifurcation parameter the symmetry breaking bifurcation point on the branch of the o 2 symmetric positive solutions is found via the extended systems finally other symmetric positive solutions are computed by the branch switching method based on the lyapunov schmidt reduction

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

on the boundaries of the determinantal regions of ray pattern matrices abstract in c a eschenbach f j hall z li from real to complex sign pattern matrices bull austral math soc 57 1998 159 172 eschenbach et al proposed the problem concerning whether the boundaries of the complex determinantal regions s a are always on the axes in the complex plane in jia yu shao hai ying shan the determinantal regions of complex sign pattern matrices and ray pattern matrices linear algebra appl 395 2005 211 228 an affirmative answer to this problem was obtained in this paper we generalize this result from complex determinantal regions s a to ray determinantal regions r a let t a be the set of the nonzero terms in the determinantal expansion of the matrix a then we show that the boundary of the ray determinantal region r a is always a subset of the union of all those rays starting at the origin and passing through some one element of the set t a we also define a so called canonical form a whose entries are all on the axes of a complex matrix a and show that s a r a for all complex square matrices a then the affirmative answer of the above problem will be a direct consequence of this and the result on the boundaries of the ray determinantal regions this result s a r a also shows that the study of the complex determinantal regions can be turned to the study of the ray determinantal regions

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

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

donaldson invariants for non simply connected manifolds we study coulomb branch u plane integrals for supersymmetric su 2 so 3 yang mills theory on 4 manifolds x of b 1 x 0 b 2 x 1 using wall crossing arguments we derive expressions for the donaldson invariants for manifolds with b 1 x 0 b 2 x 0 explicit expressions for where f g is a riemann surface of genus g are obtained using kronecker s double series identity the result might be useful in future studies of quantum cohomology

the sixth painleve transcendent and uniformization of algebraic curves abstract we exhibit a remarkable connection between sixth equation of painleve list and infinite families of explicitly uniformizable algebraic curves fuchsian equations congruences for group transformations differential calculus of functions and differentials on corresponding riemann surfaces abelian integrals analytic connections generalizations of chazy s equations and other attributes of uniformization can be obtained for these curves as byproducts of the theory we establish relations between picard hitchin s curves hyperelliptic curves punctured tori heun s equations and the famous differential equation which apery used to prove the irrationality of riemann s 3

inverse closed ray nonsingular cones of matrices abstract a description is given of those ray patterns which will be called inverse closed ray nonsingular of complex matrices that contain invertible matrices only and are closed under inversion here two n n matrices are said to belong to the same ray pattern if in each position either the entries of both matrices are zeros or both entries are nonzero and their quotient is positive possible jordan forms of matrices in the inverse closed ray nonsingular ray patterns are characterized

nilpotent singular points and compactons abstract in this paper dynamical system theory is applied to several types of fully nonlinear wave equations these equations can be reduced to planar polynomial differential systems by transformation of variables we treat these polynomial differential systems by phase space analytical technique the results of our study demonstrate that there exist close connection between nilpotent singular points and compactons moreover we find some new elliptic function compactons instead of well known trigonometric function compactons by analyzing nilpotent points two new compactons induced by singular elliptic are also obtained

ray nonsingularity of cycle chain matrices abstract ray nonsingular rns matrices are a generalization of sign nonsingular sns matrices from the real field to complex field the problem of how to recognize ray nonsingular matrices is still open in this paper we give an algorithm to recognize the ray nonsingularity for the so called cycle chain matrices whose associated digraphs are of certain simple structure furthermore the cycle chain matrices that are not ray nonsingular are classified into two classes according to whether or not they can be majorized by some rns cycle chain matrices and the case when they can is characterized

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

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

on the algebras of bps states we define an algebra on the space of bps states in theories with extended supersymmetry we show that the algebra of perturbative bps states in toroidal compactification of the heterotic string is closely related to a generalized kac moody algebra we use d brane theory to compare the formulation of rr charged bps algebras in type ii compactification with the requirements of string string duality and find that the rr charged bps states should be regarded as cohomology classes on moduli spaces of coherent sheaves the equivalence of the algebra of bps states in heterotic iia dual pairs elucidates certain results and conjectures of nakajima and gritsenko nikulin on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras in particular to any calabi yau 3 fold there are two canonically associated algebras exchanged by mirror symmetry

the topological vertex we construct a cubic field theory which provides all genus amplitudes of the topological a model for all non compact toric calabi yau threefolds the topology of a given feynman diagram encodes the topology of a fixed calabi yau with schwinger parameters playing the role of kahler classes of the threefold we interpret this result as an operatorial computation of the amplitudes in the b model mirror which is the quantum kodaira spencer theory the only degree of freedom of this theory is an unconventional chiral scalar on a riemann surface in this setup we identify the b branes on the mirror riemann surface as fermions related to the chiral boson by bosonization

the decomposability of the matrix with two determinantal regional components abstract the ray of a complex number a is either 0 or a a depending on whether a is 0 or nonzero the ray pattern of a complex matrix a denoted by ray a is the matrix obtained by replacing each entry of a with its ray the determinantal region of a square matrix a denoted by r a is the set of the determinants of all the complex matrices with the same ray pattern as a a connected component of the set r a 0 is called a determinantal regional component of a the number of determinantal regional components of r a is denoted by n r a it was proved in shao et al jia yu shao yue liu ling zhi ren the inverse problems of the determinantal regions of ray pattern and complex sign pattern matrices linear algebra appl 416 2006 835 843 that n r a 2 for any complex square matrix a when n r a 2 the two determinantal regional components are either two opposite open rays or two opposite open sectors with the angle no more than in this paper we prove that any square matrix a with n r a 2 is partly decomposable if one of its determinantal regional components is an open sector with the angle less than as a main graph theoretical technique we also discuss a property of strongly connected digraphs

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

compact and noncompact physical structures for the zk bbm equation a variety of exact solutions for the 2 1 dimensional zk bbm equation are developed by means of the tanh method and the sine cosine methods generalized forms of the zk bbm equation are studied the tanh and the sine cosine methods are reliable to derive solutions of distinct physical structures compactons solitons solitary patterns and periodic solutions

exact traveling wave solutions of the zakharov kuznetsov benjamin bona mahony equation abstract bifurcation method of dynamical systems is employed to investigate traveling wave solutions in the 2 1 dimensional zakharov kuznetsov benjamin bona mahony equation under some parameter conditions exact solitary wave solutions and kink wave solutions are obtained

the determinantal regions of complex sign pattern matrices and ray pattern matrices abstract this paper studies the determinantal regions sa of complex sign pattern matrices and the determinantal regions ra of ray pattern matrices we give an affirmative answer to a problem proposed in c a eschenbach f j hall z li from real to complex sign pattern matrices bull australian math soc 57 1998 159 172 concerning the boundary points of the regions sa we also prove that the region sa 0 and ra 0 is an open set in the complex plane c except the cases that sa or ra is entirely contained in a line through the origin we determine all the 21 possible regions sa of complex sign nonsingular matrices a and all the possible regions ra of ray nonsingular matrices a we also study some further properties of sa and ra when a is not necessarily complex sign nonsingular or ray nonsingular

solutions of compact and noncompact structures for nonlinear klein gordon type equation the focusing and the defocusing branches of nonlinear klein gordon type equation kg n n are considered a framework is implemented to show that the first model exhibits compactons solitons that do not have exponential tails whereas the second model demonstrates solitary patterns solutions the two variants of kg n n equation will be examined in one and higher dimensions general formulas will be developed to present a fairly complete understanding of the solutions of compact and noncompact structures

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

gromov witten pairs correspondence for the quintic 3 fold we use the gromov witten pairs descendent correspondence for toric 3 folds and degeneration arguments to establish the gw p correspondence for several compact calabi yau 3 folds including all cy complete intersections in products of projective spaces a crucial aspect of the proof is the study of the gw p correspondence for descendents in relative geometries projective bundles over surfaces relative to a section play a special role the gw p correspondence for calabi yau complete intersections provides a structure result for the gromov witten invariants in a fixed curve class after change of variables the gromov witten series is a rational function in the variable q eiu invariant under q q 1

periodic wave solutions and their limits for the zk bbm equation abstract in this paper we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the zk bbm equation some explicit periodic wave solutions are obtained these solutions contain smooth periodic wave solutions and periodic blow up solutions their limits contain solitary wave solutions periodic wave solutions kink wave solutions unbounded solutions and blow up wave solutions

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

qualitative behavior and exact travelling wave solutions of the zhiber shabat equation in this paper the qualitative behavior and exact travelling wave solutions of the zhiber shabat equation are studied by using qualitative theory of polynomial differential system the phase portraits of system are given under different parametric conditions some exact travelling wave solutions of the zhiber shabat equation are obtained the results presented in this paper improve the previous results

the inverse problems of the determinantal regions of ray pattern and complex sign pattern matrices abstract we study the inverse problems for the determinantal regions r a of the ray pattern matrices and the determinantal regions s a of the complex sign pattern matrices we determine all but eight possible exceptions in two equivalent classes the determinantal regions s a when a ranges over all complex square matrices we also determine all the possible determinantal regions r a except those regions which are the union of 0 and an open sector with an angle greater than we also answer several questions proposed in j y shao h y shan the determinantal regions of complex sign pattern matrices and ray pattern matrices linear algebra appl 395 2005 211 228 concerning the number of the connected components of the set r a 0 and of the set s a 0

bifurcations of traveling wave solutions for zhiber shabat equation abstract in this paper we investigate the dynamical behavior of traveling wave solutions in the zhiber shabat equation by using the bifurcation theory and the method of phase portraits analysis as a result we obtain the conditions under which smooth and non smooth traveling wave solutions exist and give some exact explicit solutions for some special cases

topological field theory and rational curves we analyze the quantum field theory corresponding to a string propagating on a calabi yau threefold this theory naturally leads to the consideration of witten s topological non linear model and the structure of rational curves on the calabi yau manifold we study in detail the case of the world sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by candelas de la ossa green and parkes

a forbidden structure of ray nonsingular matrices abstract a complex square matrix is called a ray nonsingular matrix rns matrix if its ray pattern implies that it is nonsingular in this paper a necessary condition for rns matrices is provided by showing that if a i a d is ray nonsingular then the arc weighted digraph d contains no forbidden cycle chains

bifurcations of travelling wave solutions for the generalized double sinh gordon equation in this paper the generalized double sinh gordon equation is studied the existence of periodic wave solitary wave kink and anti kink wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given some exact explicit parametric representations of the above waves are determined

theta functions and hodge numbers of moduli spaces of sheaves on rational surfaces we compute generating functions for the hodge numbers of the moduli spaces of h stable rank 2 sheaves on a rational surface s in terms of theta functions for indefinite lattices if h lies in the closure of the ample cone and has self intersection 0 it follows that the generating functions are jacobi forms in particular the generating functions for the euler numbers can be expressed in terms of modular forms and their transformation behaviour is compatible with the predictions of s duality we also express the generating functions for the signatures in terms of modular forms it turns out that these generating functions are also with respect to another developing parameter the generating function for the donaldson invariants of s evaluated on all powers of the point class

on the elliptic genus of generalised kummer varieties borisov and libgober 2 recently proved a conjecture of dijkgraaf moore verlinde and verlinde see 6 on the elliptic genus of a hilbert scheme of points on a surface we show how their result can be used together with our work on complex genera of generalised kummer varieties 17 to deduce the following formula conjectured by kawai and yoshioka 15 on the elliptic genus of a generalised kummer variety a n of dimension 2 n 1 here is the weak jacobi form of weight 1 and index and v n is the hecke operator sending jacobi forms of index r to jacobi forms of index nr see 7