In the study of the asymptotic behavior of polynomials from partition theory, the determination of their leading term asymptotics inside the unit disk depends on a sequence of sets derived from comparing certain complex-valued functions constructed from polylogarithms, functions defined as $$Li_s(z)=\sum_{n=1}^\infty \frac{z^n}{n^s}.$$ These sets we call phases. This paper applies complex analytic techniques to describe the geometry of these sets in the complex plane.

This paper is concerned with the lengths of constant length substitutions that generate topologically conjugate systems. We show that if the systems are infinite, then these lengths must be powers of the same integer. This result is a dynamical formulation of a special case of a 1969 theoretical computer science result of Cobham [{\bf1}]. Our proof is rather simple.

We show that the degree-3 Jacobi Polynomials define two one-parameter families of elliptic curves $\E_{r}/\q(r)$ and $\F_{s}/\q(s)$ with positive rank outside of an explicit finite set of $(r,s) \in \q \times \q$. In addition we initiate a program to study the ranks of the Jacobians of these curves in higher degree.

In this paper we have shown that if $\phi\in (L_h^2)^{\perp}\cap L^{\infty}$, $\phi\neq 0$, then $\text{ker\,}T_{\phi}=\text{ker\,}T^*_{\phi}=\text{sp\,}\{1\}$ and therefore finite-dimensional subspaces of $L_a^2$. Further, if $\phi\in L^{\infty}(\D)$, $\phi\neq 0$, then it is shown that the Toeplitz operator $T_{\phi}$ cannot be of finite rank.

Simple complete $\M$-primary ideals are fundamental in the Zariski-Lipman theory of complete ideals in a two-dimensional regular local ring $(R,\M)$. Complete $\M$-primary ideals of order one constitute a particular class of such ideals containing, for example, all the first neighborhood ideals of $R$. A number of interesting properties of complete $\M$-primary ideals of order one have been proved by several authors. For example, these ideals have only one Rees valuation (and hence are one-fibered) and they have the very simple form $(x_1^n,x_2)R$, $n\in\n_{+}$, with $x_1,x_2$ a minimal ideal basis of $\M$.

In the present paper we investigate how far some results concerning these complete $\M$-primary ideals of order one can be extended to complete quasi-one-fibered $\M$-primary ideals of order one in a natural generalization of $R$: a two-dimensional normal Noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain.

Given a commutative ring $R$, the determinant map $\text{det}_0\colon Rk_0(R)\to \text{Pic\,}(R)$ given by $[M]-[R^m]\mapsto\langle\wedge^mM\rangle $ is a homomorphism from the additive group of $Rk_0(R)$ to the multiplicative group $\text{Pic\,}(R)$. In this paper, some properties of the determinant map $\text{det}_0$ are given and some results in [{\bf5}] are extended.

We prove that a theorem by Stallings on finitely generated subgroups of free groups is also valid for free products of groups.

A Newman polynomial has all its coefficients in $\{0,1\}$ and constant term~$1$. It is known that every root of a Newman polynomial lies in the slit annulus $\{z\in\c: \tau^{-1} \lt |{z}| \lt \tau\}\setminus\r^+$, where $\tau$ denotes the golden ratio, but not every polynomial having all of its conjugates in this set divides a Newman polynomial. We show that every negative Pisot number in $(-\tau,-1)$ with no positive conjugates, and every negative Salem number in the same range obtained by using Salem's construction on small negative Pisot numbers, is satisfied by a Newman polynomial. We also construct a number of polynomials having all their conjugates in this slit annulus, but that do not divide any Newman polynomial. Finally, we determine all negative Salem numbers in $(-\tau,-1)$ with degree at most $20$, and verify that every one of these is satisfied by a Newman polynomial.

We give examples that provide negative answers to three questions about abstract factorization posed by Anderson and Frazier. We show that (1)~an atomic domain need not be $\tau$-atomic for $\tau$ divisive, (2)~an atomic domain need not be a comaximal factorization domain (CFD) and (3)~for $\tau$ divisive, a nonzero nonunit of a $\tau$-UFD need not be a $\tau$-product of $\tau$-primes. Along the way, we generalize the theorem of Anderson and Frazier that a UFD is a $\tau$-UFD for $\tau$ divisive (with a simplified proof), and we demonstrate a method for constructing domains with no pseudo-irreducible elements.

We estimate the boundary behavior of the Kobayashi metric on $\c\setminus\{0,1\}$. We also compare the Bergman metric on the ring domain in $\c^{2}$ to the Bergman metric on the ball and study the comparison between the Kobayashi, Carath\'eodory, Azukawa and Sibony metric.

For finite measure space $(X, \A, \mu)$, a Banach space $E$ with $E^{\p}$ its dual, and a relatively countably compact $Q \subset (L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$, entirely different proofs are given of the results that (i)~$\overline{Q}$ is Eberlein compact, (ii)~the closed convex hull of $\overline{Q}$ in $(L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$ is also compact and (iii)~the closed convex hull of $\overline{Q}$ in $(L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$ and in $(L_{1}(E), \| \cdot\|_{1})$ are the same.

Ultramodular binary copulas are discussed, i.e., copulas of a random vector whose components are mutually stochastically decreasing with respect to each other. The additive generators of Archimedean ultramodular binary copulas are fully characterized. Finally, a new construction method for binary copulas based on $n$-ary ultramodular aggregation functions is proposed.

This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These results can be used to give very elementary proofs of sharpened versions of some von Neumann type inequalities, as well as some other striking consequences about polynomials and matrices. Exploring the limits of the finite dimensional approach sheds light on the difference between those techniques and phenomena in operator theory that are inherently infinite dimensional, and those that are not.

In [{\bf1}], Belov, Seip and the author studied the Carleson measures for certain spaces of analytic functions of which the de Branges spaces and the model subspaces of the Hardy space $H^2$ are prime examples. In this paper, we continue this line of research by studying the compact Carleson measures for such spaces.

We define $K$-theory spectra associated to certain topological categories and compare these spectra to analytic $K$-theory spectra. As an application, we show how the assembly map in the analytic Novikov conjecture can be factored through an assembly map defined through more homotopy-theoretic machinery.

We establish a new bound for the number of solutions of a pair of symmetric diophantine equations, one quartic and one quadratic, in ten variables. This estimate is then used to deduce a modest refinement of Weyl's inequality for eighth powers, which improves on an earlier result of Robert and Sargos.

In this paper, a unicity theorem for Borel series is obtained and used to show that the collection of cyclic operators acting on the space of entire functions with non-dense eigenvalues and having the monomials $z^n$ as eigenvectors has a dense set of common cyclic vectors.

In this paper we show that no regular matrix can sum all subsequences of a divergent sequence. We also show that the Cesaro matrix cannot sum almost all subsequences of a divergent sequence. These results can be viewed as generalizations of a well-known result of Steinhaus.

In this paper, we establish weighted atomic decomposition characterizations of weighted weak local Hardy spaces $Wh_{\o}^p$ with local weights. As an application, we show that truncated Riesz transforms and pseudo-differential operators are bounded on weighted weak local Hardy spaces.

The present paper deals with the approximation of B\'{e}zier variants of Baskakov-Kantorovich operators $V_{n,\alpha}^{*}$ in the case $0\lt \alpha\lt 1$. Pointwise approximation properties of the operators $V_{n,\alpha}^{*}$ are studied. A convergence theorem of this type approximation for locally bounded functions is established. This convergence theorem subsumes the approximation of functions of bounded variation as a special case.

This note is concerned with a nonlocal version of the man-environment-man epidemic model in which the dispersion of infectious agents is assumed to follow a nonlocal diffusion law modeled by a convolution operator. The purpose of this note is to show that the minimal wave speeds of properly re-scaled nonlocal diffusion equations can approximate the corresponding one of the classical diffusion equation for this model. As a byproduct, our results indicate that the temporal delay in an epidemic model can reduce the speed of epidemic spread while the nonlocal effect can increase the speed.