In this paper, we give a characterization of the simultaneous unitarizability of any finite set of $\operatorname{SL}(2, \mathbb{C})$-valued functions on $\mathbb{S}^{1}$ and determine all possible ways of the unitarization. Such matrix functions can be regarded as images of the generators for the fundamental group of a surface in an $\mathbb{S}^{1}$-family, and the results of this paper have applications in the construction of constant mean curvature surfaces in space.

Several years ago, Huneke and Leuschke proved a theorem solving a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is considered over an arbitrary local ring. The main purpose of this paper is to prove that the existence of a certain prime ideal and a certain totally reflexive module implies the existence of an uncountably infinite number of isomorphism classes of indecomposable totally reflexive modules.

For a smooth variety proper over a curve having a fibre with isolated ordinary quadratic singularities, it is well-known that we have the vanishing cycles associated to the singularities in the étale cohomology of the geometric generic fibre. The base-change by a double cover of the base curve ramified at the image of the singular fibre has singularities corresponding to the singularities in the fibre. In this note, we show that in the even relative-dimensional case, there exist elements of the bivariant Chow group of the base-change with supports in the singularities and hence their images in the bivariant Chow group with supports in the special fibre and that the usual cohomological vanishing cycles are obtained as their images by a natural map, a kind of "cycle map" so that the elements in the bivariant Chow groups can be regarded as the vanishing cycles. The bivariant Chow group with supports in the special fibre has a ring structure and the natural map is a ring homomorphism to the cohomology ring of the geometric generic fibre. Also discussed is the relation of the bivariant Chow group with supports in the special fibre to the specialization map of Chow groups.

Donsker-Varadhan's type large deviation will be discussed for the pinned motion of a radial random walk on a homogeneous tree. We shall prove that the rate function corresponding to the large deviation is associated with a new Markov chain constructed from the above random walk through a harmonic transform based on a positive principal eigenfunction for the generator of the random walk.

Sufficient conditions for the analytic coefficients of the linear differential equation

f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = 0

are found such that all solutions belong to a given $H^{\infty}_{q}$-space, or to the Dirichlet type subspace $\mathcal{D}^{p}$ of the classical Hardy space $H^{p}$, where $0 < p \le 2$. For $0 < q < \infty$, the space $H^{\infty}_{q}$ consists of those functions $f$, analytic in the unit disc $D$, for which $|f(z)|(1-|z|^{2})^{q}$ is uniformly bounded in $D$, and $f \in \D^{p}$ if the integral $\int_{D} |f'(z)|^{p}(1-|z|^{2})^{p-1} \, d\sigma_{z}$ converges.

Let $R = k[x_{1}, \dots, x_{n}]$ be a polynomial ring over a field $k$. Let $J = \{j_{1}, \dots, j_{t}\}$ be a subset of $\{1, \dots, n\}$, and let $\mathfrak{m}_{J} \subset R$ denote the ideal $(x_{j_{1}}, \dots, x_{j_{t}})$. Given subsets $J_{1}, \dots, J_{s}$ of $\{1, \dots, n\}$ and positive integers $a_{1}, \dots, a_{s}$, we study ideals of the form $I = \mathfrak{m}_{J_{1}}^{a_{1}} \cap \cdots \cap \mathfrak{m}_{J_{s}}^{a_{s}}$. These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that $I$ is always componentwise linear when $s \le 3$ or when $J_{i} \cup J_{j} = [n]$ for all $i \neq j$. When $s \ge 4$, we give examples to show that $I$ may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the $s = 2$ case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when $\operatorname{char}(k) = 0$, our work also yields new cases in which this conjecture holds.

J. S. Hsia has conjectured an arithmetical version of Springer Theorem, which states that no two spinor genera in the same genus of integral quadratic forms become identified over an odd degree extension. In this paper we prove by examples that the corresponding result for quaternionic skew-hermitian forms does not hold in full generality. We prove that it does hold for unimodular skew-hermitian lattices under all extensions and for lattices whose discriminant is relatively prime to $2$ under Galois extensions.