Multiplicative properties of the partition function

A lower bound for the number of multiplicatively independent values ofp(n) forN ≤ n <N + R is given. The proof depends on the Hardy-Ramanujan formula and is of an elementary nature.     

Parity of the partition function

Although much is known about the partition function, little is known about its parity. For the polynomials D(x):=(Dx²+1)/24, where , 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 computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of for for primes up to and small powers of and .

     

New inequalities for the Hurwitz zeta function

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.     

A new proof of the Ramanujan congruences for the partition function

Let p(n) denote the number of partitions of n. Recall Ramanujan’s three congruences for the partition function,

Dissecting the Stanley partition function

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p_i(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π^′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p₀(n)-p₂(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that

(i)

and that for sufficiently large n

(ii)

In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:

Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.     

On some inequalities for the incomplete gamma function

Let be a positive real number. We determine all real numbers and such that the inequalities

are valid for all . And, we determine all real numbers and such that

hold for all .

     

Error bounds for asymptotic approximations of the partition function

We consider Markovian queueing models with a finite number of states and a product form solution for its steady state probability distribution. Starting from the integral representation for the partition function in complex space we construct error bounds for its asymptotic expansion obtained by the saddle point method. The derivation of error bounds is based on an idea by Olver applicable to integral transforms with an exponentially decaying kernel. The bounds are expressed in terms of the supremum of a certain function and are asymptotic to the absolute value of the first neglected term in the expansion as the large parameter approaches infinity. The application of these error bounds is illustrated for two classes of queueing models: loss systems and single chain closed queueing networks.     

Nikolskii-type inequalities for shift invariant function spaces

Let be a vectorspace of complex-valued functions defined on of dimension over . We say that is shift invariant (on ) if implies that for every , where on . In this note we prove the following.

Theorem. Let be a shift invariant vectorspace of complex-valued functions defined on of dimension over . Let . Then

for every and

     

A characterization of some weighted inequalities for the vector-valued weighted maximal function

We give in this paper a necessary and sufficient condition of weighted weak and strong type norm inequalities for the vector-valued weighted maximal function.     

Some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables

Several inequalities for differentiable convex, wright-convex and quasi-convex mapping are obtained respectively that are connected with the celebrated Hermite-Hadamard integral inequality. Also, some error estimates for weighted Trapezoid formula and higher moments of random variables are given.     

Sensitivity analysis framework for variational inequalities

In this paper a sensitivity analysis framework is developed for variational inequalities. The perturbed solution to a parametric variational inequality problem is shown to be continuous and directionally differentiable under appropriate second order and regularity assumptions. Moreover, this solution is once continuously differentiable if strict complementarity also holds.     

A unifying geometric solution framework and complexity analysis for variational inequalities

In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result, we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ellipsoids, and Vaidya's algorithm) to develop a unifying geometric framework for solving variational inequality problems. The analysis rests upon the assumption of strong-f-monotonicity, which is weaker than strict and strong monotonicity. Since linear programs satisfy this assumption, the general framework applies to linear programs.Preparation of this paper was supported, in part, by NSF Grant 9312971-DDM from the National Science Foundation.     

指数凸函数的积分不等式及其在Gamma函数中的应用

仿对数凸函数的概念,给出指数凸函数的定义,并证明有关指数凸函数的几个积分不等式,作为应用,得到一个新的Kershaw型双向不等式.     

Finite difference scheme for variational inequalities

In this paper, we show that a class of variational inequalities related with odd-order obstacle problems can be characterized by a system of differential equations, which are solved using the finite difference scheme. The variational inequality formulation is used to discuss the uniqueness and existence of the solution of the obstacle problems.     

关于一类F.Smarandache可乘函数的均值

主要目的是利用解析方法研究一类F.Smarandache可乘函数的渐近性质,并给出关于这个函数的一个有趣的渐近公式.     

A double projection algorithm for multi-valued variational inequalities and a unified framework of the method

In this paper, we propose a double projection algorithm for a generalized variational inequality with a multi-valued mapping. Under standard conditions, our method is proved to be globally convergent to a solution of the variational inequality problem. Moreover, we present a unified framework of projection-type methods for multi-valued variational inequalities. Preliminary computational experience is also reported.     

Kunita-Watanabe inequalities and Tanaka formula for multi-dimensional G-Brownian motion

In this paper, some new properties and interesting estimations of mutual variation process for G-Brownian motion are presented, Kunita-Watanabe inequalities and Tanaka formula for multi-dimensional G-Brownian motion are also obtained.