We construct certain algebraic integers $\alpha_n$ as special values of two variable theta functions in the ray class field of a certain quartic field modulo $2^n$, and study a property of prime ideals which appear in $\alpha_n$ in connection to the relationships between cyclotomic units and exponential functions and between elliptic units and elliptic theta functions.

We discuss a confluence from Siegel–Whittaker functions to Whittaker functions on $Sp(2,{\mathbb{R})$ by using their explicit formulae. In our proof, we use expansion theorems of the good Whittaker functions by the secondary Whittaker functions.

We give a precise description of combed trees in terms of Kelly–Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly–Mac Lane graph of a certain shape. Conversely, we show that any such Kelly–Mac Lane graph uniquely defines a combed tree. We show how to apply this to the construction of opetopes.

This paper shows that, given a finite subset $X$ of a finitely generated virtually free group $F$, the freeness of the subsemigroup of $F$ generated by $X$ can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup of $F$ has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

We study the properties of shifted vertex operator algebras, which are vertex algebras derived from a given theory by shifting the conformal vector. In this way, we are able to exhibit large numbers of vertex operator algebras which are regular (rational and $C_2$-cofinite) and yet are pathological in one way or another.

A basis, denoted $\{Q_{\lambda,\mu\}$, for the full Homfly skein of the annulus $\mathcal{C}$ has been introduced by the authors, where $\lambda$ and $\mu$ are partitions of integers $n$ and $p$ into $k$ and $k^*$ parts respectively. The basis consists of eigenvectors of the two meridian maps on $\mathcal{C}$; these maps are the linear endomorphisms of $\mathcal{C}$ induced by the insertion of a meridian loop with either orientation around a diagram in the annulus.

Here we present an explicit expression for each $Q_{\lambda,\mu}$ as the determinant of a $(k^*+k)\times(k^*+k)$ matrix whose entries are simple elements $h_n, h_n^*$ in the skein $\mathcal{C}$. In the case $p=0$ ($\mu=\phi$) the determinant gives the Jacobi–Trudy formula for the Schur function $s_{\lambda}$ of $N$ variables as a polynomial in the complete symmetric functions $h_n$ of the variables. The Jacobi–Trudy determinants have previously been used by Kawagoe and Lukac in discussing the elements in the skein of the annulus represented by closed braids in which all strings are oriented in the same direction. Our results and techniques form a natural extension of the work of Lukac.

Using the concept of $s$-formality we are able to extend the bounds of a Theorem of Miller and show that a compact $k$-connected $(4k+3)$- or $(4k+4)$-manifold with $b_{k+1}=1$ is formal. We study $k$-connected $n$-manifolds, $n=4k+3, 4k+4$, with a hard Lefschetz-like property and prove that in this case if $b_{k+1}=2$, then the manifold is formal, while, in $4k+3$-dimensions, if $b_{k+1}=3$ all Massey products vanish. We finish with examples inspired by symplectic geometry and manifolds with special holonomy.

In this paper, we study the residual Julia sets of meromorphic functions. In fact, we prove that if a meromorphic function $f$ belongs to the class ${\bf S}$ and its Julia set is locally connected, then the residual Julia set of $f$ is empty if and only if its Fatou set $F(f)$ has a completely invariant component or consists of only two components. We also show that if $f$ is a meromorphic function which is not of the form $\alpha + (z\,-\,\alpha)^{-k}e^{g(z)}$, where $k$ is a natural number, $\alpha$ is a complex number and $g$ is an entire function, then $f$ has buried components provided that $f$ has no completely invariant components and its Julia set $J(f)$ is disconnected. Moreover, if $F(f)$ has an infinitely connected component, then the singleton buried components are dense in $J(f)$. This generalizes a result of Baker and Domínguez. Finally, we give some examples of meromorphic functions with buried points but without any buried components.

An example illustrates (a) that the use of indefinite inner products can illuminate Probabilistic Wiener–Hopf Theory in symmetrizable cases, and (b) that half-winding probabilities should be thought of as branching measures for Ray processes. The paper's wide variety of techniques could make it instructive for research students.

We study the spectrum containment of almost periodic solutions to differential equations with piecewise constant argument (EPCA, for short). We find an important property, which is different from that given by Cartwright for Ordinary Differential Equations (ODE). Some known (periodic, quasi-periodic) results may be expanded. As a corollary, it is shown that EPCA with periodic perturbations possess a quasi-periodic solution and no periodic solution. As a consequence, a new class of quasi-periodic functions is described. This new phenomenon is due to the piecewise constant argument and illustrates a crucial difference between ODE and EPCA.

In this paper, a generalized predator-prey system with delays is considered. By using coincidence degree theory, we establish the existence result of multiple positive periodic solutions for the system.