on a class of degenerate parabolic equations with dynamic boundary conditions we consider a quasi linear parabolic possibly degenerate equation with nonlinear dynamic boundary conditions the corresponding class of initial and boundary value problems has already been studied previously proving well posedness of weak solutions and the existence of the global attractor assuming that the nonlinearities are subcritical to a given exponent the goal of this article is to show that the previous analysis can be redone for supercritical nonlinearities by proving an additional l estimate on the solutions in particular we derive new conditions which reflect an exact balance between the internal and the boundary mechanisms involved even when both the nonlinear sources contribute in opposite directions then we show how to construct a trajectory attractor for the weak solutions of the associated parabolic system and prove that any solution belonging to the attractor is bounded which implies uniqueness finally we also prove for the semilinear reaction diffusion equation with nonlinear dynamic boundary conditions that the fractal dimension of the global attractor is of the order n 1 as diffusion 0 in any space dimension n 2 improving some recent results in gal 2012 23

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

spinodal decomposition for the cahn hilliard equation in higher dimensions nonlinear dynamics this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation where n n 1 2 3 is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u 3 based on the results of 26 the nonlinear cahn hilliard equation will be discussed this equation generates a nonlinear semiflow in certain affine subspaces of h 2 in a neighborhood u e with size proportional to e n around the constant solution where lies in the spinodal region we observe the following behavior within a local inertial manifold containing there exists a finite dimensional invariant manifold which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1 the dimension of is proportional to e n and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to e

partitions durfee symbols and the atkin garvan moments of ranks atkin and garvan introduced the moments of ranks of partitions in their work connecting ranks and cranks here we consider a combinatorial interpretation of these moments this requires the introduction of a new representation for partitions the durfee symbol and subsequent refinements this in turn leads us to a variety of new congruences for our marked durfee symbols much in the spirit of dyson s original conjectures on the ranks of partitions

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

partition congruences and the andrews garvan dyson crank in 1944 freeman dyson conjectured the existence of a crank function for partitions that would provide a combinatorial proof of ramanujan s congruence modulo 11 forty years later andrews and garvan successfully found such a function and proved the celebrated result that the crank simultaneously explains the three ramanujan congruences modulo 5 7 and 11 this note announces the proof of a conjecture of ono which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function

optimal control of an allen cahn equation with singular potentials and dynamic boundary condition in this paper we investigate optimal control problems for allen cahn equations with differentiable singular nonlinearities and a dynamic boundary condition involving differentiable singular nonlinearities and the laplace beltrami operator the approach covers both the cases of distributed controls and of boundary controls the cost functional is of standard tracking type and box constraints for the controls are prescribed parabolic problems with nonlinear dynamic boundary conditions involving the laplace beltrami operator have recently drawn increasing attention due to their importance in applications while their optimal control was apparently never studied before in this paper we first extend known well posedness and regularity results for the state equation and then show the existence of optimal controls and that the control to state mapping is twice continuously frechet differentiable between appropriate function spaces based on these results we establish the first order necessary optimality

proof of the macdonald conjecture a plane partition of n is an array a i not necessarily square of positive integers which has non increasing rows and columns such that the sum of its elements is n with such an array we associate the set d which is the set of all integer points i j k in the first octant such that aij is defined and 1 k aij a plane partition is cyclically symmetric if d is invariant under the mapping i j k o k i let m m n be the number of such cyclically symmetric partitions of n such that none of i j aij exceed m macdonald has conjectured that the generating function

a new four parameter q series identity and its partition implications we prove a new four parameter q hypergeometric series identity from which the three parameter identity for the gollnitz theorem due to alladi andrews and gordon follows as a special case by setting one of the parameters equal to 0 the new identity is equivalent to a four parameter partition theorem which extends the deep theorem of gollnitz and thereby settles a problem raised by andrews thirty years ago some consequences including a quadruple product extension of jacobi s triple product identity and prospects of future research are briefly discussed

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

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

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

the combinatorics of k marked durfee symbols andrews recently introduced k marked durfee symbols which are connected to moments of dyson s rank by these connections andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems the primary goal of this article is to provide combinatorial proofs in answer to andrews request we obtain a relation between k marked durfee symbols and durfee symbols by constructing bijections and all identities on k marked durfee symbols given by andrews could follow from this relation in a similar manner we also prove the identities due to andrews on k marked odd durfee symbols combinatorially which resemble ordinary k marked durfee symbols with a modified subscript and with odd numbers as entries

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

cahn hilliard equation with dynamic boundary conditions and mass constraint on the boundary abstract the well known cahn hilliard equation entails mass conservation if a suitable boundary condition is prescribed in the case when the equation is also coupled with a dynamic boundary condition including the laplace beltrami operator on the boundary the total mass on the inside of the domain and its trace on the boundary should be conserved the new issue of this paper is the setting of a mass constraint on the boundary the effect of this additional constraint is the appearance of a lagrange multiplier in fact two lagrange multipliers arise one for the bulk the other for the boundary the well posedness of the resulting cahn hilliard system with dynamic boundary condition and mass constraint on the boundary is obtained the theory of evolution equations governed by subdifferentials is exploited and a complete characterization of the solution is given

global solution to the allen cahn equation with singular potentials and dynamic boundary conditions abstract we prove well posedness results for the solution to an initial and boundary value problem for an allen cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain the dynamic boundary conditions for the order parameter have been recently proposed by some physicists to account for interactions with the walls in fischer 1997 13 kenzler 2001 14 we show our results using suitable regularizations of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments

on the cahn hilliard equation with dynamic boundary conditions and a dominating boundary potential the cahn hilliard and viscous cahn hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well posedness and regularity results are proved

existence and uniqueness of solutions to singular cahn hilliard equations with nonlinear viscosity terms and dynamic boundary conditions abstract we prove global existence and uniqueness of solutions to a cahn hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions the problem is highly nonlinear characterized by four nonlinearities and two separate diffusive terms all acting in the interior of the domain or on its boundary through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations existence and uniqueness of solutions are proved using compactness and monotonicity arguments the asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown

equation and dynamic boundary condition of cahn hilliard type with singular potentials abstract the well posedness of a system of partial differential equations and dynamic boundary conditions both of cahn hilliard type is discussed the existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary a very general class of double well like potentials is allowed moreover some further regularity is obtained to guarantee the strong solution

optimal boundary control of a viscous cahn hilliard system with dynamic boundary condition and double obstacle potentials in this paper we investigate optimal boundary control problems for cahn hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the laplace beltrami operator the cost functional is of standard tracking type and box constraints for the controls are prescribed we prove existence of optimal controls and derive first order necessary conditions of optimality the general strategy which follows the lines of the recent approach by colli farshbaf shaker and sprekels appl math optim 71 2015 pp 1 24 to the simpler allen cahn case is the following we use the results that were recently established by colli gilardi and sprekels appl math optim online first doi 10 1007 s00245 015 9299 z 2015 for the case of differentiable logarithmic potentials and perform a so called deep quench limit using compactness and monotonicity arguments it is shown that this strategy leads to the desired first order necessary optimality conditions for t