Odd values of the Rogers–Ramanujan functions

Let $g(n)$ and $h(n)$ be the coefficients of the Rogers–Ramanujan identities. We obtain asymptotic formulas for the number of odd values of $g(n)$ for odd n, and $h(n)$ for even n, which improve Gordon's results. We also obtain lower bounds for the number of odd values of $g(n)$ for even n, and $h(n)$ for odd n.     

The odd primary homology of SU(n), Sp(n) and Spin(n)-gauge groups

For an odd prime $p$, we calculate the mod-$p$ homology of $SU\left(n\right)$-gauge groups over a simply-connected, closed 4-manifold for all $n\ge 2$. Similar calculations are obtained for the structure groups $Sp\left(n\right)$ when $n\ge 1$ and $Spin\left(n\right)$ for $n\ge 3$ (except for some cases when $n$ is even and $p=3$).     

Trimmed stable AR(1) processes

In this paper we investigate the distribution of trimmed sums of dependent observations with heavy tails. We consider the case of autoregressive processes of order one with independent innovations in the domain of attraction of a stable law. We show if the $d$ largest (in magnitude) terms are removed from the sample, then the sum of the remaining elements satisfies a functional central limit theorem with random centering provided $d=d\left(n\right)\ge n^{\gamma }$ (for some $\gamma >0$) and $d\left(n\right)/n\to 0$. This result is used to get asymptotics for the widely used CUSUM process in case of dependent heavy tailed observations.     

Fredholmness and smooth dependence for linear time-periodic hyperbolic systems

This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type${?}_t{u}_j+a_j(x){?}_x{u}_j+\sum _{k=1}^n{b}_{jk}(x)u_k=f_j(x,t),j=1,\dots ,n,$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right-hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur.We show that perturbations of the coefficients $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients $a_j$ and $b_{jk}$, but they are new even in the case of strict hyperbolicity and of smooth coefficients.     

Discretizing Malliavin calculus

Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark–Ocone derivative to their continuous counterparts. Moreover, given a sequence $\left(X^n\right)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and sufficient conditions for strong $L^2$-convergence to a $\sigma \left(B\right)$-measurable random variable $X$ via convergence of the discrete chaos coefficients of $X^n$ to the continuous chaos coefficients.     

Relations and homomorphisms of n-hypergroups

In this paper, some types of homomorphisms of $n$-hypergroups is introduced and several properties are found and examples are presented. Homomorphisms of $n$-hypergroups are a generalization of homomorphisms of hypergroups. Homomorphism between $n$-hypergroups and equivalence relations on $n$-hypergroups are closely related. Also, we consider an equivalence relation $\rho$ on an $n$-hypergroup $H$ and define an $n$-hyperoperation on $H/\rho$ and prove some results in this respect.     

Vertex-minimal graphs with dihedral symmetry I

Let $D_{2n}$ denote the dihedral group of order $2n$, where $n\ge 3$. In this article, we build upon the findings of Haggard and McCarthy who, for certain values of $n$, produced a vertex-minimal graph with dihedral symmetry. Specifically, Haggard considered the situation when $\frac{n}{2}$ or $n$ is a prime power, and McCarthy investigated the case when $n$ is not divisible by $2$, $3$ or $5$. In this article, we assume $n$ is not divisible by $4$ and construct a vertex-minimal graph whose automorphism group is isomorphic to $D_{2n}$.     

A new class of n-ary hyperoperations

To a given $n$-ary hyperoperation $f$ on a universe $H$ and a unary hyperoperation $\phi$ on $H$ we define a new $n$-ary hyperoperation ${\partial }_f^{\phi }$ on $H$. We study the associativity, weak associativity and reproductivity of $n$-ary hyperoperations ${\partial }_f^{\phi }$.