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.     

Separation of partition inequalities with terminals

Given a graph with n nodes each of them having labels equal either to 1 or 2 (a node with label 2 is called a terminal), we consider the (1,2)-survivable network design problem and more precisely, the separation problem for the partition inequalities. We show that this separation problem reduces to a sequence of submodular flow problems. Based on an algorithm developed by Fujishige and Zhang the problem is reduced to a sequence of O(n⁴) minimum cut problems.     

Summation formulae on trigonometric functions

Trigonometric sums over the angles equally distributed on the upper half plane are investigated systematically. Their generating functions and explicit formulae are established through the combination of the formal power series method and partial fraction decompositions.     

On the polynomial representation for the number of partitions with fixed length

In this paper, it is shown that the number of partitions of a nonnegative integer with parts can be described by a set of polynomials of degree in , where denotes the least common multiple of the integers and denotes the quotient of when divided by . In addition, the sets of the polynomials are obtained and shown explicitly for and .

     

Vector spaces as unions of proper subspaces

In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of partitioning V into subspaces.     

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.     

On moment conditions for normed sums of independent variables and martingale differences

Let X₁, X₂,… be a sequence of i.i.d. random variables and S_n their partial sums. Necessary and sufficient conditions are given for $\text{{n}{}^{\mathrm{<mtext>?1</mtext><mtext>q</mtext>}}\text{S}{}_{\mathrm{n}}\text{}}{}^{\mathrm{\propto }}{}_1$ to have uniformly bounded pth moments, 0<p<q?2.Some of the results are generalized to martingle differences.     

On a problem of partitions of the set of nonnegative integers with the same representation functions

Dombi has shown that the set $\mathbb{N}$ of all non-negative integers can be partitioned into two subsets with identical representation functions. In this paper, we prove that one cannot partition $\mathbb{N}$ into more than two subsets with identical representation functions, while for any integer $k\ge 3$ there is a partition $\mathbb{N}=A_1\cup ?\cup A_k$ such that $A_i$ and $A_{k+1?i}$ have the same representation function for any integer $1\le i\le k$.     

Higher order log-concavity in Euler’s difference table

For 0≤k≤n, let be the entries in Euler’s difference table and let . Dumont and Randrianarivony showed equals the number of permutations on [n] whose fixed points are contained in {1,2,…,k}. Rakotondrajao found a combinatorial interpretation of the number in terms of k-fixed-points-permutations of [n]. We show that for any n≥1, the sequence is essentially 2-log-concave and reverse ultra log-concave.     

Z-自动机与分划逻辑

本文分解Ｚ 自动机成为ω 自动机偶的复合;建立了Ｚ的Ｌ（ＭＰ）理论,利用Ｚ 自动机的技巧证明了Ｚ的Ｌ（ＭＰ）理论的可判定性．作为应用,证明了有限状态过程的正确性问题是可判定的．     

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.     

The partition function and Hecke operators

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(?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.     

Enhancing credit default swap valuation with meshfree methods

In this paper, we apply the meshfree radial basis function (RBF) interpolation to numerically approximate zero-coupon bond prices and survival probabilities in order to price credit default swap (CDS) contracts. We assume that the interest rate follows a Cox-Ingersoll-Ross process while the default intensity is described by the Exponential-Vasicek model. Several numerical experiments are conducted to evaluate the approximations by the RBF interpolation for one- and two-factor models. The results are compared with those estimated by the finite difference method (FDM). We find that the RBF interpolation achieves more accurate and computationally efficient results than the FDM. Our results also suggest that the correlation between factors does not have a significant impact on CDS spreads.