on the seventh order mock theta functions in a recent paper h we proved the mock theta conjectures these are identities stated by ramanujan in his lost notebook jr2 pp 19 20 involving two of the 5th order mock 0 functions in fact ramanujan gave one such identity for each of the ten 5th order functions as shown in a g1 these ten identities are all equivalent to the two proved in i hi in this paper we prove the corresponding identities for ramanujan s 7th order mock 0 functions and show how they are related to rank differences of partitions to state our results we first introduce some standard notation if q and x are complex numbers with ql 1 and n is a nonnegative integer then

the f q mock theta function conjecture and partition ranks in 1944 freeman dyson initiated the study of ranks of integer partitions here we solve the classical problem of obtaining formulas for ne n resp no n the number of partitions of n with even resp odd rank thanks to rademacher s celebrated formula for the partition function this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f q a problem with its own long history dating to ramanujan s last letter to hardy little was known about this problem until dragonette in 1952 obtained asymptotic results in 1966 g e andrews refined dragonette s results and conjectured an exact formula for the coefficients of f q by constructing a weak maass poincare series whose holomorphic part is q 1f q24 we prove the andrews dragonette conjecture and as a consequence obtain the desired formulas for ne n and no n

on ramanujan s definition of mock theta function in his famous deathbed letter ramanujan defined the notion of a mock theta function and offered some examples of functions he believed satisfied his definition very recently griffin et al established for the first time that ramanujan s mock theta functions actually satisfy his own definition on the other hand zwegers 2002 doctoral thesis zwegers s 2002 mock theta functions phd thesis univ utrecht utrecht the netherlands showed that all of ramanujan s examples are holomorphic parts of harmonic maass forms this has led to an alternate definition of a mock theta function this paper shows that ramanujan s definition of mock theta function is not equivalent to the modern definition

ramanujan s lost notebook vi the mock theta conjectures 1 introduction in this paper we shall consider only ramanujan s two families of fifth order mock theta functions these functions were briefly described in ramanujan s last letter to g h hardy 11 pp 354 3551 and g n watson subsequently 12 131 proved all the assertions about these functions contained in the letter subsequently the identities watson proved were greatly generalized in 2 also the fifth order mock theta functions were shown to have double series expansions involving indefinite quadratic forms 7 however there remains a profound mystery about these functions namely no one including ramanujan has ever proved that these functions are indeed mock theta functions and not just some clever combination of theta functions this problem was described in detail by watson 12 p 2741 and redescribed in 7 pp 113 l 141 in s p 97 eqs 3 5 and 3 6 a formula from the lost notebook was pointed out that would if valid lead to the establishment of at least some of the fifth order mock theta functions as truly mock theta functions in the sense of watson 12 p 78 footnote there are indeed ten such identities live for each of the two families we have two objects in this paper first we wish to show that the identities in each family are equivalent i e if one is true all five are true and if one is false all five are false to introduce our second objective we recall some basic notions from partition theory 4 p 142 8 p 841 the rank of a partition is the largest

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

degenerate asymptotic perturbation theory asymptotic rayleigh schrodinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case for this purpose we study the spectral properties ofm m matrix functionsa of a complex variable which have an asymptotic expansion ea k k as 0 we show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections ofa can be obtained from the set a by analytic perturbation theory special attention is given to the case wherea is borel summable in some sector originating from 0 with opening angle here we prove that the asymptotic series describe individual eigenvalues and eigenprojections ofa which are shown to be holomorphic ins near 0 and borel summable ifa k a k for allk we then fit these results into the scheme of rayleigh schrodinger perturbation theory and we give some examples of asymptotic estimates for schrodinger operators

spectral analysis and stability properties of a relativistic deformation of the harmonic oscillator the spectral properties are established and the stability of the eigenvalues under the singular perturbation representing the relativistic corrections are proven for a family of ordinary differential operators describing a relativistic deformation of the quantum harmonic oscillator

a finite difference scheme on graded meshes for time fractional nonlinear korteweg de vries equation abstract in this paper a finite difference scheme for time fractional nonlinear korteweg de vries kdv problems with the caputo type fractional derivative is presented to deal with the weak singularity caused by the fractional derivative that the solution has in the initial layer the well known l1 scheme on graded meshes has been used for time discretization meanwhile a nonlinear finite difference approximation on uniform meshes is proposed for spatial discretization the existence stability and convergence of the numerical solution are studied by the energy method it is proved that the scheme is of min 2 r order convergence in time and of first order convergence in space where is the order of fractional derivative and r is a parameter of graded meshes furthermore in order to increase computational efficiency the corresponding fast algorithm of the presented scheme has been considered in numerical simulation a valid approach based on linear interpolation is designed to test convergence rate for the scheme on temporal graded meshes numerical results demonstrate sharpness of the theoretical convergence estimate and effectiveness of the fast algorithm

temporal exponential decay for the stark effect in atoms abstract let h 1 i 1 n i v x i 1 i x i x j 1 v x i n x y 1 y d y with a normalized gaussian suppose e 0 and that h h 1 e i 1 n x i has no eigenfunctions in l 2 r 3n if h 1 with ess h 1 then e ith decays exponentially at a rate governed by the positions of the resonances of h

stability of schrodinger eigenvalue problems we derive a general stability criterion for discrete eigenvalues of schrodinger operators such asa p 2 v x using only strong continuity ofa anda in the perturbation parameter the theory is developed for non selfadjoint operators and illustrated with examples like the anharmonic oscillator the stark and the zeeman effect the principal tools are weyl s criterion for the essential spectrum and a construction due to enss 5 they are also used to extend the classical invariance theorems for the essential spectrum to certain singular perturbations including some local perturbations of the laplacian by differential operators of arbitrary high order

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

exponentially accurate error estimates of quasiclassical eigenvalues we study the behaviour of truncated rayleigh schrodinger series for the low lying eigenvalues of the one dimensional time independent schrodinger equation in the semiclassical limit 0 under certain hypotheses on the potential v x we prove that for any given small 0 there is an optimal truncation of the series for the approximate eigenvalues such that the difference between an approximate and exact eigenvalue is smaller than exp c for some positive constant c we also prove the analogous results concerning the eigenfunctions

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results

on the explicit construction of higher deformations of partition statistics the modularity of the partition generating function has many important consequences for example asymptotics and congruences for p n in a series of papers the author and ono 12 13 connected the rank a partition statistic introduced by dyson to weak maass forms a new class of functions which are related to modular forms and which were rst considered in 11 here we do a further step towards understanding how weak maass forms arise from interesting partition statistics by placing certain 2 marked durfee symbols introduced by andrews 1 into the framework of weak maass forms to do this we construct a new class of functions which we call quasiweak maass forms because they have quasimodular forms as components as an application we prove two conjectures of andrews it seems that this new class of functions will play an important role in better understanding weak maass forms of higher weight themselves and also their derivatives as a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices

analysis of a time stepping scheme for time fractional diffusion problems with nonsmooth data this paper establishes the convergence of a time stepping scheme for the time fractional diffusion problems with nonsmooth data we first analyze the regularity of the time fractional diffusion pro