congruences involving arithmetic progressions for weakly holomorphic modular forms abstract in this paper we give a classification of weights k such that there is a nonzero weakly holomorphic modular form f a n q n of weight k on 1 n having infinitely many congruences of the form a l n 0 mod l where l is a prime and is an integer in 0 1 l 1 these are similar to congruences for the partition function investigated by ramanujan furthermore we characterize linear combinations of shimura theta functions with odd characters in terms of these congruences as an application of our main theorem we consider a generalization of newman s conjecture for weakly holomorphic modular forms on 0 n with real dirichlet characters finally we use these results to study arithmetic properties of colored partitions and generalized frobenius partitions

congruences for the andrews spt function ramanujan type congruences for the andrews spt n partition function have been found for prime moduli 5 l 37 in the work of andrews andrews ge 2008 j reine angew math 624 133 142 and garvan garvan f 2010 int j number theory 6 1 29 we exhibit unexpectedly simple congruences for all l 5 confirming a conjecture of garvan we show that if l 5 is prime and l 1 then spt l2 ln 1 24 0 mod l this congruence gives l 1 2 arithmetic progressions modulo l 3 which support a mod l congruence this result follows from the surprising fact that the reduction of a certain mock theta function modulo l for every l 5 is an eigenform of the hecke operator t l 2

regularity of plate equations with control concentrated in interior curves we consider initial and boundary value problems modelling the vibration of a plate with piezoelectric actuator the usual models lead to the bernoulli euler and kirchhoff plate equations with right hand side given by a distribution concentrated in an interior curve we obtain regularity results which are stronger than those obtained by simply using the sobolev regularity of the right hand side by duality we obtain new trace regularity properties for the solutions of plate equations our results provide appropriate function spaces for the control of plates provided with piezoelectric actuators

finite range repulsive systems of finitely many particles we consider the dynamics of finite systems of spherical particles where a fraction of the kinetic energy may be lost during collisions we show that the set of initial configurations leading to infinitely many collisions in finite time can have positive measure contrary to the hard collision energy conservation case in which this particular set is claimed to be empty we also show that after sufficient time a system will decouple into maximal subsystems this generalizes proofs that in the hard sphere case there can be at most finitely many collisions for almost all initial configurations

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

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

congruences for the coefficients of weakly holomorphic modular forms recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli we show that this type of phenomenon is completely general by finding similar congruences for the coefficients of any weakly holomorphic modular form on any congruence subgroup gamma 0 n in particular we give congruences for a wide class of partition functions and for traces of cm values of arbitrary modular functions on certain congruence subgroups of prime level

the andrews sellers family of partition congruences in 1994 james sellers conjectured an infinite family of ramanujan type congruences for 2 colored frobenius partitions introduced by george e andrews these congruences arise modulo powers of 5 in 2002 dennis eichhorn and sellers were able to settle the conjecture for powers up to 4 in this article we prove sellers conjecture for all powers of 5 in addition we discuss why the andrews sellers family is significantly different from classical congruences modulo powers of primes

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

arithmetic properties of the partition function let p n denote the number of partitions of the positive integer n p n is the number of representations of n as a non increasing sequence of positive integers by convention we agree that p 0 1 and that p n 0 if n 0 the study of the arithmetic properties of p n has a long and rich history see for example the works of andrews atkin dyson garvan kim stanton and swinnerton dyer an an g at1 at swd d g k s these works have their origins in the groundbreaking observations of ramanujan r1 r2 r3 r4 in ramanujan s own words r3

partitions into kth powers of terms in an arithmetic progression g h hardy and s ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer and claimed a similar asymptotic formula for the number of partitions into perfect kth powers which was later proved by e m wright recently r c vaughan provided a simpler asymptotic formula in the case k 2 in this paper we consider partitions into parts from a specific set a k a 0 b 0 left m k m in mathbb n m equiv a 0 text mod b 0 right for fixed positive integers k a 0 and b 0 we give an asymptotic formula for the number of such partitions thus generalizing the results of wright and vaughan moreover we prove that the number of such partitions is even odd infinitely often which generalizes o kolberg s theorem for the ordinary partition function p n

inelastic collisions of three particles on a line as a two dimensional billiard abstract we study the inelastic collisions of three particles on a line we show that this system is a two dimensional billiard in a semi enclosed space with unconventional reflection laws using this geometric language we describe completely the dynamics of this system we find that the asymptotic behavior changes when the coefficient of restitution becomes smaller than a critical value r c for r r c the system can be understood in terms of a conventional billiard in an infinite wedge for r r c this is no longer true and the asymptotic behavior is richer we also investigate the question of triple collisions and find that they are regularizable only for certain values of r

euler s de partitio numerorum chapter 16 of euler s introductio in analysin infinitorum is titled de partitio numerorum our goal is to provide a survey of the impact of euler s chapter on the study of partitions in the following 250 plus years we ask the natural question do we still have anything to learn from euler the answer is yes

l adic properties of the partition function abstract ramanujan s famous partition congruences modulo powers of 5 7 and 11 imply that certain sequences of partition generating functions tend l adically to 0 although these congruences have inspired research in many directions little is known about the l adic behavior of these sequences for primes l 13 using the classical theory of modular forms mod p as developed by serre in the 1970s we show that these sequences are governed by fractal behavior modulo any power of a prime l 5 these sequences of generating functions l adically converge to linear combinations of at most l 1 12 l 2 1 24 l many special q series for l 5 7 11 we have l 1 12 l 2 1 24 l 0 thereby giving a conceptual explanation of ramanujan s congruences we use the general result to reveal the theory of multiplicative partition congruences that atkin anticipated in the 1960s his results and observations are examples of systematic infinite families of congruences which exist for all powers of primes 13 l 31 since l 1 12 l 2 1 24 l 1 answering questions of mazur in appendix a we give a new general theorem which fits these results within the framework of overconvergent half integral weight p adic modular forms this result which is based on recent works by n ramsey is due to frank calegari

local existence of weak solutions to kinetic models of granular media we prove in any dimension d geqq 1 a local in time existence of weak solutions to the cauchy problem for the kinetic equation of granular media partial t f v cdot nabla x f div v f nabla w v f when the initial data are nonnegative integrable and bounded functions with compact support in velocity and the interaction potential w is a c 2 mathbb r d radially symmetric convex function our proof is constructive and relies on a splitting argument in position and velocity where the spatially homogeneous equation is interpreted as the gradient flow of a convex interaction energy with respect to the quadratic wasserstein distance our result generalizes the local existence result obtained by benedetto et al rairo model math anal numer 31 5 615 641 1997 on the one dimensional model of this equation for a cubic power law interaction potential

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

stirling probability and q bosons normal ordering of powers of the bosonic number operator can be used to define a discrete probability distribution associated with the number of elements of a random set we represent these random sets by vectors in a hilbert space and obtain q bosons

nonholomorphic ramanujan type congruences for hurwitz class numbers in contrast to all other known ramanujan type congruences we discover that ramanujan type congruences for hurwitz class numbers can be supported on nonholomorphic generating series we establish a divisibility result for such nonholomorphic congruences of hurwitz class numbers the two key tools in our proof are the holomorphic projection of products of theta series with a hurwitz class number generating series and a theorem by serre which allows us to rule out certain congruences

regularity with interior point control part ii kirchhoff equations abstract we study the regularity of kirchhoff equations defined on an open bounded domain dim 1 2 3 and subject to the action of point control through the dirac mass at an interior point of the results of this paper are 1 2 sharper in space regularity measured in sobolev space order over those that can be obtained by simply using that by sobolev embedding h 3 2 for n 3 l for n 2 1 2 for n l the approach used here is very general

coefficients of half integral weight modular forms modulo lj suppose that l 5 is prime that j 0 is an integer and that f z is a half integral weight modular form with integral fourier coefficients we give some general conditions under which the coefficients of f are well distributed modulo lj as a consequence we settle many cases of a classical conjecture of newman by proving for each prime power lj with l 5 that the ordinary partition function p n takes each value modulo lj infinitely often

m2 rank of overpartitions and harmonic weak maass forms there has been significant recent interest in constructing many families of harmonic weak maass forms using rank generating functions of various partitions in this paper applying zwegers results on appell lerch sums we provide a new family of harmonic weak maass forms with the m2 rank generating function of overpartitions as applications we prove the modularity of m2 rank differences and ramanujan type congruences for such ranks

modeling and stabilizability of voltage actuated piezoelectric beams with magnetic effects models for piezoelectric beams and structures with piezoelectric patches generally ignore magnetic effects this is because the magnetic energy has a relatively small effect on the overall dynamics piezoelectric beam models are known to be exactly observable and can be exponentially stabilized in the energy space by using a mechanical feedback controller in this paper a variational approach is used to derive a model for a piezoelectric beam that includes magnetic effects it is proved that the partial differential equation model is well posed magnetic effects have a strong effect on the stabilizability of the control system for almost all system parameters the piezoelectric beam can be strongly stabilized but it is not exponentially stabilizable in the energy space strong stabilization is achieved using only electrical feedback furthermore using the same electrical feedback an exponentially stable closed loop system can be obtained for a set of system parameters of zero lebesgue measure these r

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

how many shuffles to mix a deck a simple probabilistic model is devised to determine the number of shuffles required for the bottom card of a deck to become uniformly distributed with a specified tolerance this number is a lower bound on the number of shuffles needed for the entire deck to be randomly mixed

periodic trajectories in right triangle billiards billiard problems are simple examples of hamiltonian dynamical systems these problems have been used as model systems to study the link betwen classical and quantum chaos the heart of this linkage is provided by the periodic orbits in the classical system in this article we will show that for an arbitrary right triangle almost all trajectories that begin perpendicular to a side are periodic that is the set of points on the sides of a right triangle from which nonperiodic perpendicular trajectories begin is a set of measure zero our proof incorporates the previous result for rational right triangles where the angles are rational multiples of ensuremath pi while extending the result to nonrational right triangles

the partition function and hecke operators abstract the theory of congruences for the partition function p n depends heavily on the properties of half integral weight hecke operators the subject has been complicated by the absence of closed formulas for the hecke images p z t l 2 where p z is the relevant modular generating function we obtain such formulas using euler s pentagonal number theorem and the denominator formula for the monster lie algebra as a corollary we obtain congruences for certain powers of ramanujan s delta function

ramanujan s congruences and dyson s crank the achievement of karl mahlburg in this issue of pnas 1 adds a lustrous chapter to a unique mathematical object the crank in 1944 the crank was first hinted at by freeman dyson 2 then an undergraduate at cambridge university he had written an article titled some guesses in the theory of partitions for eureka the undergraduate mathematics journal of cambridge dyson discovered the many conjectures in this article by attempting to find a combinatorial explanation of ramanujan s famous congruences for p n the number of partitions of n the three simplest of ramanujan s congruences assert that in dyson s article he defines the rank of a partition to be the largest part of the partition minus the number of parts for example the rank of the partition of 22 5 5 3 3 3 2 1 is 5 7 2 dyson conjectured and later atkin and swinnerton dyer proved see ref

iterated impact dynamics of n beads on a ring when n beads slide along a frictionless hoop their collision sequence gives rise to a dynamical system that can be studied via matrix products it is of general interest to understand the distribution of velocities and the corresponding eigenvalue spectrum that a given collision sequence can produce we formulate the problem for general n and state some basic theorems regarding the eigenvalues of the collision matrices and their products the case of three beads of masses m1 m2 m3 is studied in detail we exploit the fact that each collision sequence can be viewed as a billiard trajectory in a right triangle with non standard reflection rules existence of families of periodic orbits are proven and orbits that densely fill the triangle are computed eigenvalue distributions and position and velocity histograms are computed as a function of the restitution coefficient both periodic and dense collision sequences are discussed and a series of conjectures based on computational evidence are formulated comparisons are made between the eigenvalue distributions and autocorrelation matrices associated with dense trajectories generated from a chaotic collision sequence and spectra from matrix sequences generated from random orderings and we describe how the three bead system could be used as the basis for a random number generating algorithm that is computationally efficient

parity of the partition function abstract although much is known about the partition function little is known about its parity for the polynomials d x d x 2 1 24 where d 23 mod 24 we show that there are infinitely many m resp n for which p d m is even resp p d n is odd if there is at least one such m resp n we bound the first m and n if any in terms of the class number h d for prime d we show that there are indeed infinitely many even values to this end we construct new modular generating functions using generalized borcherds products and we employ galois representations and locally nilpotent hecke algebras

computing the integer partition function in this paper we discuss efficient algorithms for com puting the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjec tures related to the partition function we present the distribution of p n for n 10 9 for primes up to 103 and small powers of 2 and 3

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

divisibility and distribution of partitions into distinct parts abstract we study the generating function for q n the number of partitions of a natural number n into distinct parts using the arithmetic properties of fourier coefficients of integer weight modular forms we prove several theorems on the divisibility and distribution of q n modulo primes p 5

the spt function of andrews recently andrews introduced the function s n spt n which counts the number of smallest parts among the integer partitions of n we show that its generating function satisfies an identity analogous to ramanujan s mock theta identities as a consequence we are able to completely determine the parity of s n using another type of identity one based on hecke operators we obtain a complete multiplicative theory for s n modulo 3 these congruences confirm unpublished conjectures of garvan and sellers our methods generalize to all integral moduli

sieved partition functions and q binomial coefficients the q binomial coefficient is a polynomial in q given an integer t and a residue class r modulo t a sieved q binomial coefficient is the sum of those terms whose exponents are congruent to r modu o t in this paper explicit polynomial identities in qt are given for sieved q binomial coefficients as a limiting case generating functions for the sieved partition function are found as multidimensional theta functions a striking corollary of this representation is the proof of ramanujan s congruences mod 5 7 and 11 by exhibiting symmetry groups of orders 5 7 and 11 of explicit quadratic forms we also verify the subbarao conjecture for t 3 t 5 and t 10

l adic properties of smallest parts functions abstract we prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions one for partitions one for overpartitions and one for partitions without repeated odd parts the proofs depend on l adic properties of certain modular forms and mock modular forms of weight 3 2 with respect to the hecke operators t l 2 m

regularity and exact controllability for a beam with piezoelectric actuator we consider an initial and boundary value problem modelling the vibrations of a bernoulli euler beam with an attached piezoelectric actuator we show that the sobolev regularity of the solution is by frac 1 2 epsilon higher than that one obtains by simply using the sobolev regularity of the control term the main results concern the dependence of the space of exactly controllable initial data on the location of the actuator our approach is based on the hilbert uniqueness method introduced by lions em contr olabilit e exacte des syst emes distribu es masson paris 1988 combined with some results from the theory of diophantine approximation

lifting cusp forms to maass forms with an application to partitions for 2 k abstract see text we define lifts of cuspidal poincare series in s k gamma 0 n to weight 2 k harmonic weak maass forms this construction answers a question of dyson by providing the general framework explaining ramanujan s mock theta functions as an application we show that the number of partitions of a positive integer n is the trace of singular moduli of a maass form arising from the lift of a weight 4 cusp form corresponding to a calabi yau threefold