asymptotic behavior of partial and false theta functions arising from jacobi forms and regularized characters motivated by recent developments in the representation theory of vertex algebras and conformal field theory we prove several asymptotic results for partial and false theta functions arising from jacobi forms as the modular variable tends to 0 along the imaginary axis and the elliptic variable z is unrestricted in the complex plane we observe that these functions exhibit stokes phenomenon the asymptotic behavior of these functions sharply differs depending on where the elliptic variable z is located within the complex plane we apply our results to study the asymptotic expansions of regularized characters and quantum dimensions of the 1 p singlet w algebra modules important in logarithmic conformal field theory this in particular recovers and extends several results from the work of t creutzig et al int math res not 2016 e print arxiv 1411 3282 pertaining to regularized quantum dimensions

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

false theta functions and the verlinde formula abstract we discover new analytic properties of classical partial and false theta functions and their potential applications to representation theory of w algebras and vertex algebras in general more precisely motivated by clues from conformal field theory first we are able to determine modular like transformation properties of regularized partial and false theta functions then after suitable identification of regularized partial false theta functions with the characters of atypical modules for the singlet vertex algebra w 2 2 p 1 we formulate a verlinde type formula for the fusion rules of irreducible w 2 2 p 1 modules

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

optimization over the efficient set abstract in this paper we study the problem of optimizing a linear function over the set of efficient solutions for a vector maximization problem this problem arises whenever a linear function is available which acts as a criterion for measuring the importance of or for discriminating among the efficient alternatives that are available using the concepts of an efficient direction and of a direction of recession we develop necessary and sufficient conditions for this problem to be unbounded we also present necessary and sufficient conditions for efficient and for arbitrary solutions of the underlying vector maximization problem to be optimal solutions for this problem some of these conditions are geometric in nature others algebraic the algebraic conditions suggest potential computational procedures for finding an optimal solution to the problem finally we give conditions under which the set of optimal solutions possesses certain special properties

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

a large k asymptotics of witten s invariant of seifert manifolds we calculate a largek asymptotic expansion of the exact surgery formula for witten ssu 2 invariant of some seifert manifolds the contributions of all flat connections are identified an agreement with the 1 loop formula is checked a contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series a 2 loop correction to the contribution of the trivial connection is found to be proportional to casson s invariant

finiteness of stationary configurations of the four vortex problem we show that the number of relative equilibria equilibria and rigidly translating configurations in the problem ot tour point vortices is hnite the proof is based on symbolic and exact integer computations which are carried out by computer we also provide upper bounds for these classes of stationary configurations

minimization of a quasi concave function over an efficient set the nonconvex programming problem of minimizing a quasi concave function over an efficient or weakly efficient set of a multiobjective linear program is studied a cutting plane algorithm which finds an approximate optimal solution in a finite number of steps is developed for the particular all linear case the algorithm performs better finding an optimal solution in a finite time and being more easily implemented

fermionic solution of the andrews baxter forrester model i unification of tba and ctm methods the problem of computing the one dimensional configuration sums of the abf model in regime iii is mapped onto the problem of evaluating the grandcanonical partition function of a gas of charged particles obeying certain fermionic exclusion rules we thus obtain a newfermionic method to compute the local height probabilities of the model combined with the originalbosonic approach of andrews baxter and forrester we obtain a new proof of some of melzer s polynomial identities in the infinite limit these identities yield rogers ramanujan type identities for the virasoro characters l 1 r l r q as conjectured by the stony brook group as a result of our work the corner transfer matrix and thermodynamic bethe ansatz approaches to solvable lattice models are unified

on q series identities for false theta series abstract we prove several infinite families of q series identities for false theta functions and related series these identities are motivated by considerations of characters of modules of vertex operator superalgebras and of quantum dilogarithms we also obtain closely related modular identities of the gollnitz gordon andrews type as a byproduct of our identities we establish several identities for the rogers dilogarithm function coming from multi q hypergeometric series with double poles

q series identities and values of certain l functions as usual define dedekind s eta function z by the infinite product eta z q 1 24 prod n 1 infty big 1 q n big quad big q e 2 pi i z text throughout big in a recent paper d zagier proved that note empty products equal 1 throughout sum n 0 infty big eta 24z q big 1 q 24 big big 1 q 48 big cdots big 1 q 24n big big eta 24z d q e q where the series d q and e q are defined by begin align d q frac 1 2 sum n 1 infty frac q 24n 1 q 24n frac 1 2 sum n 1 infty d n q 24n frac 1 2 q 24 2q 48 2q 72 3q 96 cdots quad e q frac 1 2 sum n 1 infty bigg frac 12 n bigg nq n 2 frac 1 2 q frac 5 2 q 25 frac 7 2 q 49 frac 11 2 q 121 cdots quad end align here d n denotes the number of positive divisors of n we obtain two infinite families of such identities and describe some consequences for l functions and partitions for example if 2 is the kronecker character for sqrt 2 these identities can be used to show that begin align 2e t 8 sum n 0 infty frac big 1 e 2t big big 1 e 4t big cdots big 1 e 2nt big big 1 e t big big 1 e 3t big cdots big 1 e 2n 1 t big hspace 110pt sum n 0 infty bigg frac 1 8 bigg n cdot l chi 2 2n 1 cdot frac t n n end align

equidistribution on the sphere a concept of generalized discrepancy which involves pseudodifferential operators to give a criterion of equidistributed pointsets is developed on the sphere a simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy with the help of this formula five kinds of point systems on the sphere namely lattices in polar coordinates transformed two dimensional sequences rotations on the sphere triangulations and sum of three squares sequence are investigated quantitative tests are done and the results are compared with one another our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems

residue formulas for the large k asymptotics of witten s invariants of seifert manifolds the case of su 2 we derive the largek asymptotics of the surgery formula forsu 2 witten s invariants of general seifert manifolds the contributions of connected components of the moduli space of flat connections are identified the contributions of irreducible connections are presented in the residue form this allows us to express them in terms of intersection numbers on their moduli spaces

quantum invariant for torus link and modular forms we consider an asymptotic expansion of kashaev s invariant or of the colored jones function for the torus link t 2 2m we shall give q series identity related to these invariants and show that the invariant is regarded as a limit of q being n th root of unity of the eichler integral of a modular form of weight 3 2 which is related to the character

the andrews gordon identities and q multinomial coefficients we prove polynomial boson fermion identities for the generating function of the number of partitions of n of the form n sum j 1 l 1 j f j with f 1 leq i 1 f l 1 leq i 1 and f j f j 1 leq k the bosonic side of the identities involves q deformations of the coefficients of x a in the expansion of 1 x cdots x k l a combinatorial interpretation for these q multinomial coefficients is given using durfee dissection partitions the fermionic side of the polynomial identities arises as the partition function of a one dimensional lattice gas of fermionic particles

representations of mock theta functions abstract motivated by the works of liu we provide a unified approach to find appell lerch series and hecke type series representations for mock theta functions we establish a number of parameterized identities with two parameters a and b specializing the choices of a b we not only give various known and new representations for the mock theta functions of orders 2 3 5 6 and 8 but also present many other interesting identities we find that some mock theta functions of different orders are related to each other in the sense that their representations can be deduced from the same a b parameterized identity furthermore we introduce the concept of false appell lerch series we then express the appell lerch series false appell lerch series and hecke type series in this paper using the building blocks m x q z and f a b c x y q introduced by hickerson and mortenson as well as m x q z and f a b c x y q introduced in this paper we also show the equivalences of our new representations for several mock theta functions and the known representations

proof of the flohr grabow koehn conjectures for characters of logarithmic conformal field theory in a recent paper flohr grabow and koehn conjectured that the characters of the logarithmic conformal field theory ck 1 admit fermionic representations labelled by the lie algebra dk in this paper we provide a simple analytic proof of this conjecture

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

truncated hecke rogers type series abstract the recent work of george andrews and mircea merca on the truncated version of euler s pentagonal number theorem has opened up a new study on truncated theta series since then several papers on the topic have followed the main purpose of this paper is to generalize the study to hecke rogers type double series which are associated with some interesting partition functions our proofs heavily rely on a formula from the work of zhi guo liu on the q partial differential equations and q series

fermionic solution of the andrews baxter forrester model ii proof of melzer s polynomial identities we compute the one dimensional configuration sums of the abf model using the fermionic technique introduced in part i of this paper combined with the results of andrews baxter and forrester we prove polynomial identities for finitizations of the virasoro characters chi b a r 1 r q as conjectured by melzer in the thermodynamic limit these identities reproduce rogers ramanujan type identities for the unitary minimal virasoro characters conjectured by the stony brook group we also present a list of additional virasoro character identities which follow from our proof of melzer s identities and application of bailey s lemma

quantum invariants modular forms and lattice points ii we study the su 2 witten reshetikhin turaev wrt invariant for the seifert fibered homology spheres with m exceptional fibers we show that the wrt invariant can be written in terms of differential of the eichler integrals of modular forms with weight 1 2 and 3 2 by use of nearly modular property of the eichler integrals we shall obtain asymptotic expansions of the wrt invariant in the large n limit we further reveal that the number of the gauge equivalent classes of flat connections which dominate the asymptotics of the wrt invariant in n is related to the number of integral lattice points inside the m dimensional tetrahedron

vortices and polynomials non uniqueness of the adler moser polynomials for the tkachenko equation stationary and translating relative equilibria of point vortices in the plane are studied it is shown that stationary equilibria of any system containing point vortices with arbitrary choice of circulations can be described with the help of the tkachenko equation it is also obtained that translating relative equilibria of point vortices with arbitrary circulations can be constructed using a generalization of the tkachenko equation roots of any pair of polynomials solving the tkachenko equation and the generalized tkachenko equation are proved to give positions of point vortices in stationary and translating relative equilibria accordingly these results are valid even if the polynomials in a pair have multiple or common roots it is obtained that the adler moser polynomial provides non unique polynomial solutions of the tkachenko equation it is shown that the generalized tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers

the leading root of the partial theta function abstract i study the leading root x 0 y of the partial theta function 0 x y n 0 x n y n n 1 2 considered as a formal power series i prove that all the coefficients of x 0 y are strictly positive indeed i prove the stronger results that all the coefficients of 1 x 0 y after the constant term 1 are strictly negative and all the coefficients of 1 x 0 y 2 after the constant term 1 are strictly negative except for the vanishing coefficient of y 3

minimum values over the efficient set in multiple objective decision making abstract in multiple objective decision making modm it is often helpful to provide the decision maker dm with bounds on the values of each of the objectives ideal solutions are relatively easy to calculate and provide upper bounds on the value of each objective over the set of efficient solutions ideal solutions also provide lower bounds on the value of each objective over the ideal set however the lower bounds over the set of efficient solutions can be strictly less than the ideal lower bounds but are in general more difficult to determine thus modm procedures which utilize the ideal lower bound may overlook elements of the set of efficient solutions this study explores the differences between the subset of the set of efficient solutions to a modm problem bounded by its ideal solutions and the complete efficient set

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

on the triplet vertex algebra w p abstract we study the triplet vertex operator algebra w p of central charge 1 6 p 1 2 p p 2 we show that w p is c 2 cofinite but irrational since it admits indecomposable and logarithmic modules furthermore we prove that w p is of finite representation type and we provide an explicit construction and classification of all irreducible w p modules and describe block decomposition of the category of ordinary w p modules all this is done through an extensive use of zhu s associative algebra together with explicit methods based on vertex operators and the theory of automorphic forms moreover we obtain an upper bound for dim a w p finally for p prime we completely describe the structure of a w p the methods of this paper are easily extendable to other w algebras and superalgebras

vortex lattice theory a particle interaction perspective recent experiments on the formation of vortex lattices in bose einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns ones that are free of discrete symmetries and can form in a random environment we give an overview of an n particle based hamiltonian theory which if formulated in terms of the o n 2 interparticle distances leads to the analysis of a nonnormal configuration matrix whose nullspace structure determines the existence or nonexistence of a lattice the singular value decomposition of this matrix leads to a method in which all lattice patterns and the associated particle strengths in principle can be classified and calculated by a random walk scheme which systematically uses the m smallest singular values as a ratchet mechanism to home in on lattices with m dimensional nullspaces where 0 m le n the resulting distribution of singular values encodes detailed geometric properties of the lattice and allows us to identify and calculate important quantitative measures associated with each lattice including its size as measured by the frobenius or 2 norms distance between the lattices hence lattice density robustness and shannon entropy as a quantitative measure of its level of disorder this article gives an overview of vortex lattice theory from 1957 to the present highlighting recent experiments in bose einstein condensate systems and formulating questions that can be addressed by understanding the singular value decomposition of the configuration matrix we discuss some of the computational challenges associated with producing large n lattices the subtleties associated with understanding and exploiting complicated hamiltonian energy surfaces in high dimensions and we highlight ten important directions for future research in this area

partial theta functions i beyond the lost notebook it is shown how many of the partial theta function identities in ramanujan s lost notebook can be generalized to infinite families of such identities key in our construction is the bailey lemma and a new generalization of the jacobi triple product identity by computing residues around the poles of our identities we find a surprising connection between partial theta function identities and garrett ismail stanton type extensions of multisum rogers ramanujan identities

on the quantum invariants for the spherical seifert manifolds we study the witten reshetikhin turaev su 2 invariant for the seifert manifolds s3 gamma where is a finite subgroup of su 2 we show that the wrt invariants can be written in terms of the eichler integral of modular forms with half integral weight and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the eichler integral we further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to

vertex algebras for s duality we define new deformable families of vertex operator algebras mathfrak a mathfrak g psi sigma associated to a large set of s duality operations in four dimensional supersymmetric gauge theory they are defined as algebras of protected operators for two dimensional supersymmetric junctions which interpolate between a dirichlet boundary condition and its s duality image the mathfrak a mathfrak g psi sigma vertex operator algebras are equipped with two mathfrak g affine vertex subalgebras whose levels are related by the s duality operation they compose accordingly under a natural convolution operation and can be used to define an action of the s duality operations on a certain space of vertex operator algebras equipped with a mathfrak g affine vertex subalgebra we give a self contained definition of the s duality action on that space of vertex operator algebras the space of conformal blocks in the derived sense i e chiral homology for mathfrak a mathfrak g psi sigma is expected to play an important role in a broad generalization of the quantum geometric langlands program namely we expect the s duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks this action should coincide with and generalize the usual quantum geometric langlands correspondence the strategy we use to define the mathfrak a mathfrak g psi sigma vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras

complex ginzburg landau equations and dynamics of vortices filaments and codimension 2 submanifolds here we study the asymptotic behavior of solutions to the complex ginzburg landau equations and their associated heat flows in arbitrary dimensions when the ginzburg landau parameter tends to infinity we prove that the energies of solutions in the flow are concentrated at vortices in two dimensions filaments in three dimensions and codimension 2 submanifolds in higher dimensions moreover we show the dynamical laws for the motion of these vortices filaments and codimension 2 submanifolds away from the energy concentration sets we use some measure theoretic arguments to show the strong convergence of solutions in both static and heat flow cases 1998 john wiley sons inc

hecke type double sums appell lerch sums and mock theta functions i by introducing a dual notion between partial theta functions and appell lerch sums we find and prove a formula which expresses hecke type double sums in terms of appell lerch sums not only does our formula prove classical hecke type sum identities such as those in work of kac and peterson on affine lie algebras and hecke modular forms but once we have the hecke type forms for ramanujan s mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities indeed we obtain a new proof of the mock theta conjectures our formula also applies to positive level string functions associated with admissible representations of the affine lie algebra a 1 1 of kac and wakimoto

vector valued higher depth quantum modular forms and higher mordell integrals abstract we introduce vector valued higher depth quantum modular forms and investigate examples coming from characters of representations of vertex algebras these are expressed as rank two false theta series generalizing unary false theta series studied from several standpoints we also discuss certain double integrals which may be viewed as the obstruction to modularity of the depth two quantum modular forms we then find explicit formulas for the double error integrals in the form reminiscent of the classical mordell integral

higher rank partial and false theta functions and representation theory abstract we study higher rank jacobi partial and false theta functions generalizations of the classical partial and false theta functions associated to positive definite rational lattices in particular we focus our attention on certain kostant s partial theta functions coming from ade root lattices which are then linked to representation theory of w algebras we derive modular transformation properties of regularized higher rank partial and false theta functions as well as kostant s version of these modulo conjectures in representation theory as an application we compute regularized quantum dimensions of atypical and typical modules of narrow logarithmic w algebras associated to rescaled root lattices this paper substantially generalize our previous work 19 pertaining to 1 p singlet w algebras the sl 2 case results in this paper are very general and are applicable in a variety of situations

rogers schur ramanujan type identities for the m p p minimal models of conformal field theory we present and prove rogers schur ramanujan bose fermi type identities for the virasoro characters of the minimal model m p p the proof uses the continued fraction decomposition of p p introduced by takahashi and suzuki for the study of the bethe s ansatz equations of the xxz model and gives a general method to construct polynomial generalizations of the fermionic form of the characters which satisfy the same recursion relations as the bosonic polynomials of forrester and baxter we use this method to get fermionic representations of the characters for many classes of r and s

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

the t coefficient method to partial theta function identities and ramanujan s 1ψ1 summation formula this paper is concerned with a uniform approach the t coefficient method to basic hypergeometric series by virtue of this method some new and elementary proofs for four partial theta function identities due to alladi and berkovich 2004 8 andrews and warnaar 2007 4 berkovich 5 and warnaar 2003 2 as well as for ramanujan s 1 1 summation formula are obtained

modular group representations and fusion in logarithmic conformal field theories and in the quantum group center the sl 2 representation on the center of the restricted quantum group open image in new window at the primitive 2pth root of unity is shown to be equivalent to the sl 2 representation on the extended characters of the logarithmic 1 p conformal field theory model the multiplicative jordan decomposition of the open image in new window ribbon element determines the decomposition of into a pointwise product of two commuting sl 2 representations one of which restricts to the grothendieck ring this restriction is equivalent to the sl 2 representation on the 1 p characters related to the fusion algebra via a nonsemisimple verlinde formula the grothendieck ring of open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the 1 p logarithmic conformal field theory model as a by product we derive q binomial identities implied by the fusion algebra realized in the center of open image in new window

three manifold invariants and their relation with the fundamental group we consider the 3 manifold invariant i m which is defined by means of the chern simons quantum field theory and which coincides with the reshetikhin turaev invariant we present some arguments and numerical results supporting the conjecture that for nonvanishing i m the absolute value i m only depends on the fundamental group 1 m of the manifold m for lens spaces the conjecture is proved when the gauge group is su 2 in the case in which the gauge group is su 3 we present numerical computations confirming the conjecture

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

on the dual nature of partial theta functions and appell lerch sums abstract in recent work hickerson and the author demonstrated that it is useful to think of appell lerch sums as partial theta functions this notion can be used to relate identities involving partial theta functions with identities involving appell lerch sums in this sense appell lerch sums and partial theta functions appear to be dual to each other this duality theory is not unlike that found by andrews between various sets of identities of rogers ramanujan type with respect to baxter s solution to the hard hexagon model of statistical mechanics as an application we construct bilateral q series with mixed mock modular behaviour in subsequent work we see that our bilateral series are well suited for computing radial limits of ramanujan s mock theta functions

a class of optimization problems over the efficient set of a multiple criteria nonlinear programming problem abstract one of the most important and interesting approaches in multiple criteria optimization is the problem of optimizing some function over the set of efficient solutions this is a very difficult multiextremal global optimization problem even for the case that the underlying multiple criteria program is linear and the function to be optimized over the efficient set is linear as well in this article we consider the problem of maximizing a special function over the efficient set of a multiple criteria concave maximization problem under the assumption that the objective function is a nondecreasing composite quasiconvex function of the concave criteria we show that this problem can be formulated as a special global optimization problem in the outcome space an algorithm of the outer approximation type is proposed for solving the resulting problem preliminary computational experiments show that this algorithm can work well for problems with a small number of criteria less than 8 while the dimension of the decision space can be fairly large

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

combination between global and local methods for solving an optimization problem over the efficient set abstract in this paper we propose an algorithm for globally solving optimization problems over efficient sets the algorithm is established based on a branch and bound scheme in which the bounding procedure is performed by using the well known weak duality theorem in lagrange duality a suitable combination of this algorithm with a local search procedure in d c optimization named dca leads to a promising global algorithm whose efficiency is more or less confirmed by computational experiments on a large set of test problems

exotic bailey slater spt functions ii hecke rogers type double sums and bailey pairs from groups a c e abstract we investigate spt crank type functions arising from bailey pairs we recall four spt type functions corresponding to the bailey pairs a1 a3 a5 and a7 of slater and given four new spt type functions corresponding to bailey pairs c1 c5 e2 and e4 each of these functions can be thought of as a count on the number of appearances of the smallest part in certain integer partitions we prove simple ramanujan type congruences for these functions that are explained by a spt crank type function the spt crank type functions are two variable q series determined by a bailey pair that when z 1 reduce to the spt type functions we find the spt crank type functions to have interesting representations as either infinite products or as hecke rogers type double series these series reduce nicely when z is a certain root of unity and allow us to deduce the congruences additionally we find dissections when z is a certain root of unity to give another proof of the congruences our double sum and product formulas require bailey s lemma and conjugate bailey pairs our dissection formulas follow from bailey s lemma and dissections of known ranks and cranks

energies and spacings of point charges on a sphere numerical methods are in general required for the determination of the stable configurations of n point charges on a sphere the stable configurations for n up to 50 have previously been ascertained and the authors extend the calculations for values up to 101 they report for the first time some remarkable global features of these configurations they show that the minimum energy accurately follows a simple half integral power law in 1 n over the full range they have investigated this power law is explicable in terms of the idealization of mapping a planar wigner lattice onto the surface of the unit sphere the pair distribution functions of the larger n configurations indicate predominant hexagonal coordination the coefficients of the observed power law are closely straddled by values calculated on the basis of hexagonal and square wigner lattices this highly accurate description of the energy permits them to remark on the detailed deviations of the individual structures from the general trend for n 30 they note that structures with n prime are relatively less stable while structures with n equal to 6 12 32 44 48 and 60 seem more stable

motion of vortex lines in the ginzburg landau model equations of motion for vortex lines in the ginzburg landau theory are derived we construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field using the method of matched asymptotic expansions we show that to leading order the line moves in the binormal direction with a curvature dependent velocity we also consider the contribution of remote parts of the line interaction between several vortex lines and interaction with external fields

instanton homology of seifert fibred homology three spheres for an oriented integral homology 3 sphere 2 a casson has introduced an integer invariant a 2 that is defined by using the space 2 of conjugacy classes of irreducible representations of 2 into su 2 see 1 this invariant a 2 can be computed from a surgery or heegard description of 2 and satisfies a 2 ju 2 mod 2 where i 2 is the kervaire milnor rochlin invariant of 2 this powerful new invariant was used to settle an outstanding problem in 3 manifold topology namely showing that if 2 is a homotopy 3 sphere then ju 2 o c taubes 17 utilizing gauge theoretic considerations has reinterpreted casson s invariant in terms of flat connections refining this approach a floer 13 has recently defined another invariant of 2 its instanton homology which takes the form of an abelian group 2 with a natural z8 grading that is an enhancement of a 2 in that

on a discrete variational problem involving interacting particles we consider the minimization problem quad min cal f bf z equiv sum k j log z k z j qquad k j 1 2 dots n for bf z z 1 z 2 dots z n in mathbb c n satisfying sum k 1 n z k 2 1 and z k ne z j for 1 le k j le n this problem arises in different contexts in several physical models such as incompressible euler equations and the ginzburg landau model in superconductivity we study the stability properties of some symmetric critical points such as regular polygons and configurations that consist of a regular polygon plus the origin we also establish the existence of an infinite number of critical points enjoying different kinds of symmetry we show that when the number of particles n exceeds a critical value the global minimizer cannot be a regular polygon n ge6 and if n ge 11 it cannot be a star configuration i e an n 1 sides regular polygon plus the origin

maximizing a concave function over the efficient or weakly efficient set an important approach in multiple criteria linear programming is the optimization of some function over the efficient or weakly efficient set this is a very difficult nonconvex optimization problem even for the case that the function to be optimized is linear in this article we consider the problem of maximizing a concave function over the efficient or weakly efficient set we show that this problem can essentially be formulated as a special global optimization problem in the space of the extreme criteria of the underlying multiple criteria linear program an algorithm of branch and bound type is proposed for solving the resulting problem

uniqueness results for co circular central configurations for power law potentials for a class of potential functions including those used for the planar n body and n vortex problems we investigate co circular central configurations whose center of mass coincides with the center of the circle containing the bodies useful equations are derived that completely describe the problem using a topological approach it is shown that for any choice of positive masses or circulations if such a central configuration exists then it is unique it quickly follows that if the masses are all equal then the only solution is the regular n gon for the planar n vortex problem and any choice of the vorticities we show that the only possible co circular central configuration with center of vorticity at the center of the circle is the regular n gon with equal vorticities

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

the closest packing of equal circles on a sphere the closest packing of x circles on the surface of a sphere is examined with the use of techniques that have been developed to determine the stereochemical arrangement of atoms packed around a central atom the technique is based on the concept that the centres of the circles repel one another and minimizing the total repulsion energy leads to an approximate structure from which exact solutions can be determined a number of improved packings have been discovered for values of x in the range 20 40 many different types of structure are found that are of lower symmetries than those previously described the packing density p defined as the fraction of the spherical surface that is enclosed by the circles is found to increase as the number of circles increases in contrast to the conclusion from previous studies however this increase in p is very slight and the values remain substantially below that for an infinite number of circles or a close packed plane

on the spectrum of a partial theta function the bivariate series defines a partial theta function for fixed q q is an entire function we show that for q 0 108 the function q has no multiple zeros

the colored jones polynomials and the simplicial volume of a knot we show that the set of colored jones polynomials and the set of generalized alexander polynomials defined by akutsu deguchi and ohtsuki intersect non trivially moreover it is shown that the intersection is at least includes the set of kashaev s quantum dilogarithm invariants for links therefore kashaev s conjecture can be restated as follows the colored jones polynomials determine the hyperbolic volume for a hyperbolic knot modifying this we propose a stronger conjecture the colored jones polynomials determine the simplicial volume for any knot if our conjecture is true then we can prove that a knot is trivial if and only if all of its vassiliev invariants are trivial in 13 r m kashaev defined a family of complex valued link invariants indexed by integers n 2 using the quantum dilogarithm later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp vol k n 2 when n goes to the infinity where vol k is the hyperbolic volume of the complement of a knotk 14 this amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated though it uses only elementary tools the aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored jones polynomial the colored jones polynomial is defined for colored links each component is decorated with an irreducible representation of the lie algebra sl 2 c the original jones polynomial corresponds to the case that all the colors are identical to the 2 dimensional fundamental representation we show that kashaev s invariant with parameter n coincides with the colored jones polynomial in a certain normalization with every color the n dimensional representation evaluated at the primitive n th root of unity we have to normalize the colored jones polynomial so that the value for the trivial knot is one for otherwise it always vanishes on the other hand there are other colored polynomial invariants the generalized multivariable alexander polynomial defined by y akutsu t deguchi and t ohtsuki 1 they used the same lie algebra sl 2 c but a different hierarchy of representations their invariants are parameterized by c 1 parameters an integer date february 1 2008 1991 mathematics subject classification 57m25 57m50 17b37 81r50

hardy petrovitch hutchinson s problem and partial theta function in 1907 m petrovitch initiated the study of a class of entire functions all whose finite sections i e truncations are real rooted polynomials he was motivated by previous studies of e laguerre on uniform limits of sequences of real rooted polynomials and by an interesting result of g h hardy an explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminant inequalities one for each degree however interesting necessary or sufficient conditions can be formulated in particular j i hutchinson has shown that an entire function p x a 0 a 1 x a n x n with strictly positive coefficients has the property that all of its finite segments a i x i a i 1 x i 1 a j x j have only real roots if and only if a i 2 a i 1 a i 1 4 for i 1 2 in the present paper we give sharp lower bounds on the ratios a i 2 a i 1 a i 1 i 1 2 for the class considered by m petrovitch in particular we show that the limit of these minima when i infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the laguerre polya class l pi we also explain the relation between newton s and hutchinson s inequalities and the logarithmic image of the set of all real rooted polynomials with positive coefficients

superconformal algebras and mock theta functions it is known that characters of bps representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the bps condition in order to improve their modular properties we apply the method of zwegers which has recently been developed to analyze modular properties of mock theta functions we consider the case of the superconformal algebra at general levels and obtain the decomposition of characters of bps representations into a sum of simple jacobi forms and an infinite series of non bps representations we apply our method to study elliptic genera of hyper kahler manifolds in higher dimensions in particular we determine the elliptic genera in the case of complex four dimensions of the hilbert scheme of points on k3 surfaces k 2 and complex tori a 3

relative equilibria of point vortices and the fundamental theorem of algebra relative equilibria of identical point vortices may be associated with a generating polynomial that has the vortex positions as its roots a formula is derived that relates the first and second derivatives of this polynomial evaluated at a vortex position using this formula along with the fundamental theorem of algebra one can sometimes write a general polynomial equation in this way results about relative equilibria of point vortices may be proved in a compact and elegant way for example the classical result of stieltjes that if the vortices are on a line they must be situated at the zeros of the n th hermite polynomial follows easily it is also shown that if in a relative equilibrium the vortices are all situated on a circle they must form a regular n gon several other results are proved using this approach an ordinary differential equation for the generating polynomial when the vortices are situated on two perpendicular lines is derived the method is extended to vortex systems where all the vortices have the same magnitude but may be of either sign derivations of the equation of tkachenko for completely stationary configurations and its extension to translating relative equilibria are given

on simplicity of vacuum modules we find necessary and sufficient conditions of irreducibility of vacuum modules over affine lie algebras and superalgebras from this we derive conditions of simplicity of minimal w algebras moreover in the case of the virasoro and neveu schwarz algebras we obtain explicit formulas for the vacuum determinants

streamline topologies for integrable vortex motion on a sphere we describe the instantaneous streamline patterns that occur on the surface of a two dimensional sphere in the presence of point vortices of general strength for cases in which the system is completely integrable after stating some general results based on the spherical topology we categorize all possible instantaneous streamline patterns and describe their stagnation point structure for the cases of two and three vortices it is found that for the case of two vortices the only non degenerate topologies that can arise are a figure eight lemniscate or a limacon which are homotopically equivalent for the case of three vortices there are 12 topologically distinct primitives from which an additional 23 patterns can be produced via continuous deformations on the sphere homotopies all possible streamline patterns that arise from three vortex motion can be obtained via linear superposition of the primitive topologies and their homotopic equivalents in this sense the primitives can be viewed as the building blocks for the general flow patterns the equations of motion and streamline patterns in the stereographic plane are obtained using the projected equations of motion in hamiltonian form we describe streamline patterns for three vortex fixed equilibria and relative equilibria as seen in both a fixed and a rotating frame of reference non equilibrium streamline patterns for a collapsing and special periodic solution are studied although to generally understand and classify the many bifurcations associated with non equilibrium patterns would require more extensive computation we conclude with a discussion of the relevance of the three vortex topological classification scheme and the bifurcation of streamline topologies for understanding global atmospheric weather patterns spherical isobars and large scale mixing phenomena 2000 elsevier science b v all rights reserved

on identities of the rogers ramanujan type abstract an elementary approach to a number of identities of the rogers ramanujan type is given it is shown that analytic formulas like e g the rogers ramanujan the rogers selberg and the gollnitz gordon identities can be obtained essentially as consequences of the q binomial theorem and the q vandermonde formula

a u n 1 bailey lattice abstract multiple basic hypergeometric series associated with the unitary group u n 1 have been investigated by various authors the purpose of this paper is to obtain a u n 1 analogue of the agarwal andrews bressoud bailey lattice and to give some applications