asymptotic fuzzy tracking control for a class of stochastic strict feedback systems this paper presents an asymptotic tracking control design method for stochastic strict feedback systems via fuzzy logic systems in the existing results stochastic controls are usually limited to be bounded in probability however how to realize the asymptotic tracking control for stochastic strict feedback systems remains a control dilemma this paper achieves the asymptotic tracking control by proposing a novel gain suppressing inequality approach specifically the three part construction is performed to achieve such control objective for stochastic systems first the novel gain suppressing inequality technique is developed to lay the foundation for carrying out the lyapunov stability analysis second the developed inequality technique is integrated with each step of the backstepping based adaptive control design procedure third analyses are provided to realize the asymptotic tracking control of stochastic strict feedback systems

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

brief paper a stability analysis with time delay of primal dual power control in optical links primal dual power control algorithms in optical networks must be designed properly to remain stable in the presence of time delays control algorithms at the signal transmitters continuously update their channel powers in response to dynamic pricing information from the network links the control algorithms optimize the optical signal to noise ratios osnrs of the channels in a distributive and non cooperative manner we consider the case of a single link with multiple channels and a constant time delay these results also apply to single sink general network configurations if the time delay represents the longest round trip time in the network we derive sufficient conditions for the stability of the closed loop system based on a tunable parameter in the control algorithm the work combines singular perturbation theory with lyapunov razumikhin time delay techniques we verify our results through simulations based on realistic network parameters

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

the generalization of bivariate mkz operators by multiple generating functions in the present paper we study approximation properties of multiple generating functions type bivariate meyer konig and zeller mkz operators with the help of volkov type theorem we compute the order of convergence of these operators by means of modulus of continuity and the elements of modified lipschitz class finally we give application to partial differential equations

on the andrews stanley refinement of ramanujan s partition congruence modulo 5 and generalizations in a recent study of sign balanced labelled posets stanley introduced a new integral partition statistic srank o o where o denotes the number of odd parts of the partition and is the conjugate of in a forthcoming paper andrews proved the following refinement of ramanujan s partition congruence mod 5 p 0 5n 4 p 2 5n 4 0 mod 5 p n p 0 n p 2 n where p i n i 0 2 denotes the number of partitions of n with srank i mod 4 and p n is the number of unrestricted partitions of n andrews asked for a partition statistic that would divide the partitions enumerated by p i 5n 4 i 0 2 into five equinumerous classes in this paper we discuss three such statistics the st crank the 2 quotient rank and the 5 core crank the first one while new is intimately related to the andrews garvan 1988 crank the second one is in terms of the 2 quotient of a partition the third one was introduced by garvan kim and stanton in 1990 we use it in our combinatorial proof of the andrews refinement remarkably the andrews result is a simple consequence of a stronger refinement of ramanujan s congruence mod 5 this more general refinement uses a new partition statistic which we term the bg rank we employ the bg rank to prove new partition congruences modulo 5 finally we discuss some new formulas for partitions that are 5 cores and discuss an intriguing relation between 3 cores and the andrews garvan crank

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

bounds for bernstein basis functions and meyer konig and zeller basis functions abstract in this paper the inequality estimates of bernstein basis functions and meyer konig and zeller basis functions are studied exact bounds for these two basis functions are obtained moreover some application results of the new estimates in estimating the rate of convergence of durrmeyer operators and meyer koning and zeller operators for functions of bounded variation are also given

convergence rate of a new bezier variant of chlodowsky operators to bounded variation functions in the present paper we estimate the rate of pointwise convergence of the bezier variant of chlodowsky operators c n a for functions defined on the interval extending infinity of bounded variation to prove our main result we have used some methods and techniques of probability theory

on the rank and the crank modulo 4 and 8 in this paper we prove some identities conjectured by lewis for the rank and crank of partitions concerning the modulo 4 and 8 these identities are similar to dyson s identities for the rank modulo 5 and 7 which give a combinatorial interpretation to ramanujan s partition congruences for this we use multisection of series and some of the results that watson established for the third order mock theta functions

periodicity of signs of fourier coefficients of eta quotients we study the periodicity of signs of fourier coefficients of the function d f qd rd where is a positive integer f q n 1 1 qn and rd z as an application we will prove the periodicity of signs for the crank differences conjectured by g e andrews and r lewis and for the weighted counts of certain types of partitions

on a modularity conjecture of andrews dixit schultz and yee for a variation of ramanujan s ω q abstract we analyze the mock modular behavior of p q a partition function introduced by andrews dixit schultz and yee this function arose in a study of smallest parts functions related to classical third order mock theta functions one of which is q we find that the modular completion of p q is not simply a harmonic maass form but is instead the derivative of a linear combination of products of various harmonic maass forms and theta functions we precisely describe its behavior under modular transformations and find that the image under the maass lowering operator lies in a relatively simpler space

constrained games the impact of the attitude to adversary s constraints we consider properties of constrained games where the strategy set available to a player depends on the choice of strategies made by other players we show that the utilities of each player associated with that player s own performance and constraints are not sufficient to model a constrained game and to define equilibria for the latter one also needs to model how a player values the fact that other players meet their constraints we study three different approaches to other players constraints and show that they exhibit completely different equilibrium behaviors further we study a general class of stochastic games with partial information and focus on the case where the players are indifferent to whether the constraints of other players hold

rates of approximation of bounded variation functions by two generalized meyer konig and zeller type operators abstract in this paper the approximation behaviours of two generalized meyer konig and zeller type operators m n and m n are studied by means of the decomposition technique of functions of bounded variation and the method of bojanic 1 bojanic and vuilleumier 2 and cheng 3 together with some results from probability theory and the exact bound of meyer konig and zeller operators basis functions the rates of convergence of these two operators for functions of bounded variation are obtained

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

degree of approximation to function of bounded variation by bezier variant of mkz operators abstract guo approx theory appl 4 1988 9 18 introduced the integral modification of meyer konig and zeller operators m n and studied the rate of convergence for functions of bounded variation in this paper we introduce the bezier variant of these integrated mkz operators and study the rate of convergence by means of the decomposition technique of functions of bounded variation together with some results of probability theory and the exact bound of mkz basis functions recently zeng j math anal appl 219 1998 364 376 claimed to improve the results of guo and gupta approx theory appl 11 1995 106 107 but there is a major mistake in the paper of zeng for special case our main theorem gives the correct estimate on the rate of convergence over the result of zeng

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

a noncooperative game approach to osnr optimization in optical networks we consider a game theory framework for power control in optical networks channel optical signal to noise ratio osnr optimization is formulated as an m player noncooperative game based on a general network osnr model conditions for existence and uniqueness of the game equilibrium solution are given an iterative algorithm for power control is proposed that uses only channel specific feedback measurements and is shown to converge to the equilibrium solution

distributed nash equilibrium seeking for aggregative games with coupled constraints abstract in this paper we study a distributed continuous time design for aggregative games with coupled constraints in order to seek the generalized nash equilibrium by a group of agents via simple local information exchange to solve the problem we propose a distributed algorithm based on projected dynamics and non smooth tracking dynamics even for the case when the interaction topology of the multi agent network is time varying moreover we prove the convergence of the non smooth algorithm for the distributed game by taking advantage of its special structure and also combining the techniques of the variational inequality and lyapunov function

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

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

asymptotic formulas for spt crank of partitions abstract in this paper using a variant of the hardy ramanujan circle method originally due to e wright we prove asymptotic formulas for spt crank of ordinary partitions

distributed power control for one to many transmissions in gaussian interference channels in this paper we extend the distributed power control problem for one to one transmissions in gaussian interference channels to one to many transmission scenarios we assume a user centric wireless network where the end users play the roles of decision makers we formulate the power control problem as a noncooperative game new challenges arise due to the coupling issues among power strategy spaces of distributed end users which make standard nash equilibrium based noncooperative game approach inapplicable indeed our problem turns out to be a generalized nash equilibrium problem gnep resorting to variational inequality theory we show several fundamental properties of the gnep then we propose a penalty based distributed algorithm ip 2ja which possesses favorable properties for practical implementation numerical results are provided to verify our arguments and proposed algorithm

an extension of duality to a game theoretic framework this paper extends some duality results from a standard optimization setup to a noncooperative nash game framework a nash game ng with coupled constraints is considered solving directly such a coupled ng requires coordination among possibly all players an alternative approach is proposed based on its relation to a special constrained optimization problem for the ng game cost function with respect to the second argument that admits a fixed point solution specific separability properties of the ng game cost are exploited and duality results are developed this duality extension leads naturally to a hierarchical decomposition into a lower level ng with no coupled constraints and a higher level system optimization problem in the second part of the paper these theoretical results are applied to a coupled ng with coupled constraints as encountered in optical networks

distributed algorithms for aggregative games on graphs we consider a class of nash games termed as aggregative games being played over a networked system in an aggregative game a player s objective is a function of the aggregate of all the players decisions every player maintains an estimate of this aggregate and the players exchange this information with their local neighbors over a connected network we study distributed synchronous and asynchronous algorithms for information exchange and equilibrium computation over such a network under standard conditions we establish the almost sure convergence of the obtained sequences to the equilibrium point we also consider extensions of our schemes to aggregative games where the players objectives are coupled through a more general form of aggregate function finally we present numerical results that demonstrate the performance of the proposed schemes

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

inequalities for ranks of partitions and the first moment of ranks and cranks of partitions abstract we prove two monotonicity properties of n m n the number of partitions of n with rank m they are i for any nonnegative integers m and n n m n n m 2 n and ii for any nonnegative integers m and n such that n 12 n m 2 n m n n m n 1 g e andrews b kim and the first author introduced ospt n a function counting the difference between the first positive rank and crank moments they proved that ospt n 0 in another article k bringmann and k mahlburg gave an asymptotic estimate for ospt n the two monotonicity properties for n m n lead to stronger inequalities for ospt n that imply the asymptotic estimate

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

asymptotic inequalities for positive crank and rank moments andrews chan and kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments in this paper we prove the asymptotic behavior of these moments in all cases our main result states that the two families of moment functions are asymptotically equal but the crank moments are also asymptotically larger than the rank moments andrews chan and kim also gave a combinatorial description for the differences of the first crank and rank moments that they named the ospt function our main results therefore also give the asymptotic behavior of the ospt function and its analogs for higher moments and we further determine the behavior of the ospt function modulo 2 by relating its parity to andrews spt function

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

distributed nash equilibrium seeking this paper presents an asynchronous gossip based algorithm for finding a nash equilibrium ne of a game in a distributed multi player network the algorithm is designed in such a way that players make decisions based on estimates of the other players actions obtained from local neighbors using a set of standard assumptions on the cost functions and communication graph the paper proves almost sure convergence to a ne for diminishing step sizes for constant step sizes an error bound on expected distance from a ne is established the effectiveness of the proposed algorithm is demonstrated via simulation for both diminishing and constant step sizes

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

pointwise approximation by bezier variant of integrated mkz operators abstract in this paper the pointwise approximation of bezier variant of integrated mkz operators for general bounded functions is studied two estimate formulas of this type approximation are obtained the approximation of functions of bounded variation becomes a special case of the main result of this paper in the case of functions of bounded variation theorem b of the paper corrects the mistake of theorem 1 of the article v gupta degree of approximation to functions of bounded variation by bezier variant of mkz operators j math anal appl 289 2004 292 300

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

games with coupled propagated constraints in optical networks with multi link topologies we consider games in optical networks in the class of m player games with coupled utilities and constraints nash equilibria of such games can be computed based on recent extension of duality to a game theoretical framework this work extends previous results on games with coupled constraints in optical links to multi link topologies coupled constraints in optical networks are propagated along links which introduces additional complexities for analysis specifically convexity of the propagated constraints is no longer automatically ensured we show that convexity is satisfied for single sink multi link topologies the general case of multi links with arbitrary sources and sinks is dealt with by a partitioned game with stages we exploit the single sink structure of each stage and the ladder nested form of the game and we discuss iterative computation of equilibria based on a three level hierarchical algorithm and prove its convergence under certain conditions

on dyson s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions in this paper we prove a longstanding conjecture by freeman dyson concerning the limiting shape of the crank generating function we fit this function in a more general family of inverse theta functions which play a key role in physics

the spt crank for ordinary partitions the spt function spt n was introduced by andrews as the weighted counting of partitions of n with respect to the number of occurrences of the smallest part andrews garvan and liang defined the spt crank of an s partition which leads to combinatorial interpretations of the congruences of spt n mod 5 and 7 let ns m n denote the net number of s partitions of n with spt crank m andrews garvan and liang showed that ns m n is nonnegative for all integers m and positive integers n and they asked the question of finding a combinatorial interpretation of ns m n in this paper we introduce the structure of doubly marked partitions and define the spt crank of a doubly marked partition we show that ns m n can be interpreted as the number of doubly marked partitions of n with spt crank m moreover we establish a bijection between marked partitions of n and doubly marked partitions of n a marked partition is defined by andrews dyson and rhoades as a partition with exactly one of the smallest parts marked they consider it a challenge to find a definition of the spt crank of a marked partition so that the set of marked partitions of 5n 4 and 7n 5 can be divided into five and seven equinumerous classes the definition of spt crank for doubly marked partitions and the bijection between the marked partitions and doubly marked partitions leads to a solution to the problem of andrews dyson and rhoades