In this paper, we prove some special cases of the Eisenbud–Green–Harris Conjecture, which characterizes the Hilbert functions of homogeneous ideals containing a regular sequence in the polynomial ring.

In a paper that appeared in 2010, C. Tone proved a multivariate central limit theorem for some strictly stationary random fields of random vectors satisfying certain mixing conditions. The “normalization” of a given “partial sum” (or “block sum”) involved matrix multiplication by a “standard $-1/2$ power” of its covariance matrix (a symmetric, positive definite matrix), and the limiting multivariate normal distribution had the identity matrix as its covariance matrix. The mixing assumptions in Tone’s result implicitly imposed an upper bound on the ratios of the largest to the smallest eigenvalues in the covariance matrices of the partial sums. The purpose of this note is to show that in Tone’s result, for the entire collection of the covariance matrices of the partial sums, there is essentially no other restriction on the relative magnitudes of the eigenvalues or on the (orthogonal) directions of the corresponding eigenvectors. For simplicity, the example given in this note will involve just random sequences, not the broader context of random fields.

A valuated $p^{n}$-socle is $C_\alpha$ $n$-summable if for every ordinal $\beta <\alpha $, it has a $\beta $-high subgroup that is $n$-summable (i.e., a valuated direct sum of countable valuated groups). This generalizes both the classical concepts of a $C_\alpha$ group due to Megibben and of an $n$-summable valuated $p^n$-socle developed by the authors. The notion is first analyzed in the category of valuated $p^{n}$-socles and then applied to the category of Abelian $p$-groups. In particular, results of Nunke on the torsion product and results of Keef on the balanced projective dimension of $C_{\omega_{1}}$ groups are recast into statements involving valuated $p^{n}$-socles and their related groups.

Let $\mathcal{B}$ be a compact convex planar domain with[4] smooth boundary of finite type and $\mathcal{B}_{\theta}$ its rotation by an angle $\theta$. We prove that for almost every $\theta\in[0,2\pi]$ the remainder $P_{\mathcal{B}_{\theta}}(t)$ is of order $O_{\theta}(t^{2/3-\zeta})$ with a positive number $\zeta$ independent of the domain.

We study the almost everywhere behavior of the maximal operator associated to moving averages in the plane, both for Lebesgue derivatives and ergodic averages. We show that the almost everywhere behavior of the maximal operator associated to a sequence of moving rectangles $v_{i}+Q_{i}$, with $(0,0)\in Q_{i}$, depends both on the way the rectangles are moved by $v_{i}$ and the structure of the rectangles ($Q_{i}$) as a partially ordered set.

We prove the following localization for compactness of Hankel operators on Bergman spaces. Assume that $\Omega $ is a bounded pseudoconvex domain in $\mathbb{C} ^{n}$, $p$ is a boundary point of $\Omega $, and $B(p,r)$ is a ball centered at $p$ with radius $r$ so that $U=\Omega \cap B(p,r)$ is connected. We show that if the Hankel operator $H^{\Omega }_{\phi}$ with symbol $\phi\in C^{1}(\overline{\Omega } )$ is compact on $A^{2}(\Omega )$ then $H^{U}_{R_{U}(\phi)}$ is compact on $A^{2}(U)$ where $R_{U}$ denotes the restriction operator on $U$, and $A^{2}(\Omega )$ and $A^{2}(U)$ denote the Bergman spaces on $\Omega $ and $U$, respectively.

We extract an exact formula relating the number of lattice points in an expanding region of a complex semi-simple symmetric space and the automorphic spectrum from a spectral identity, which is obtained by producing two expressions for the automorphic fundamental solution of the invariant differential operator $(\Delta-\lambda_{z})^{\nu}$. On one hand, we form a Poincaré series from the solution to the corresponding differential equation on the free space $G/K$, which is obtained using the harmonic analysis of bi-$K$-invariant functions. On the other hand, a suitable global automorphic Sobolev theory, developed in this paper, enables us to use the harmonic analysis of automorphic forms to produce a solution in terms of the automorphic spectrum.

Recent work of Pritsker defines and studies an areal version of the Mahler measure. We further explore this function with a particular focus on the case where its value is small, as this is most relevant to Lehmer’s conjecture. In this situation, we provide improvements to two inequalities established in Pritsker’s original paper.

We study two subcategories of the category of Artinian modules, a wide subcategory and a Serre subcategory. We prove that all wide subcategories of Artinian modules are Serre subcategories. We also provide the bijection between the set of Serre subcategories and the set of specialization closed subsets of the set of closed prime ideals of some completed ring. These results are Artinian analogues of the theorems proved by Takahashi.

In this paper, we use the theory of semi-embeddings to show that if $E$ is a Banach lattice and $X$ is a Banach space then $E\,\hat{\otimes}\,X$, the projective tensor product of $E$ and $X$, has, respectively, the near Radon–Nikodym property, the analytic Radon–Nikodym property, the analytic complete continuity property, and the property of non-containment of a copy of $c_{0}$ whenever both $E$ and $X$ have the same property.

We present a condition on higher order asymptotic behavior of basic sequences in a Banach space ensuring the existence of bounded noncompact strictly singular operators on a subspace. Applications concern asymptotic $\ell_{p}$ spaces, $1\leq p<\infty$, in particular convexified mixed Tsirelson spaces and related asymptotic $\ell_{p}$ HI spaces.

As a natural extension of mixing linear operators we introduce the notion of mixing multilinear operators. We prove composition results for mixing multilinear operators extending those from the linear case.

We show that $s$-John domains satisfy the $(1,p)$-Poincaré inequality for all finite $p>p_{0}$. We prove that the lower bound $p_{0}$ is sharp. We formulate a conjecture concerning $(q,p)$-Poincaré inequalities in $s$-John domains, $1\le q\le p$.

Let $\Omega\subset\mathbb{R}^{n}$, $q\geq n$ and $\alpha\geq0$ or $1<q\leq n$ and $\alpha\leq0$. We prove that the composition of $q$-quasiconfomal mapping $f$ and function $u\in WL^{q}\log^{\alpha}L_{\operatorname{loc}}(f(\Omega))$ satisfies $u\circ f\in WL^{q}\log^{\alpha}L_{\operatorname{loc}}(\Omega)$. Moreover, each homeomorphism $f$ which introduces continuous composition operator from $WL^{q}\log^{\alpha}L$ to $WL^{q}\log^{\alpha}L$ is necessarily a $q$-quasiconformal mapping. As a new tool, we prove a Lebesgue density type theorem for Orlicz spaces.

We give a necessary and sufficient condition for a standard graded Artinian ring of the form $K[x_{1},\ldots,x_{n}]/I$, where $I$ is an ${\mathfrak{m}}$-full ideal, to have the weak Lefschetz property in terms of graded Betti numbers. This is a generalization of a theorem of Wiebe for componentwise linear ideals. We also prove that the class of componentwise linear ideals and that of completely ${\mathfrak{m}}$-full ideals coincide in characteristic zero and in positive characteristic, with the assumption that $\mathrm{Gin}(I)$ w.r.t. the graded reverse lexicographic order is stable.

Let $p$ be an odd prime. In 2008, E. Mortenson proved van Hamme’s following conjecture:

\[\sum_{k=0}^{(p-1)/2}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}\equiv(-1)^{(p-1)/2}p\ \bigl(\operatorname{mod}p^{3}\bigr).\]

In this paper, we show further that

\begin{eqnarray*}\sum_{k=0}^{p-1}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}&\equiv&\sum_{k=0}^{(p-1)/2}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}\\[-2pt]&\equiv&(-1)^{(p-1)/2}p+p^{3}E_{p-3}\ \bigl(\operatorname{mod}p^{4}\bigr),\end{eqnarray*}

where $E_{0},E_{1},E_{2},\ldots$ are Euler numbers. We also prove that if $p>3$ then

\begin{eqnarray*}&&\sum_{k=0}^{(p-1)/2}\frac{20k+3}{(-2^{10})^{k}}\Bigl({\matrix{4k\\k,k,k,k}}\Bigr)\\&&\quad \equiv(-1)^{(p-1)/2}p\bigl(2^{p-1}+2-\bigl(2^{p-1}-1\bigr)^{2}\bigr)\ \bigl(\operatorname{mod}p^{4}\bigr).\end{eqnarray*}

Let $\Lambda$ be an artin algebra. In his Philadelphia Notes, M. Auslander showed that any homomorphism between $\Lambda$-modules is right determined by a $\Lambda$-module $C$, but a formula for $C$ which he wrote down has to be modified. The paper presents corresponding counter-examples, but also provides a quite short proof of Auslander’s assertion that any homomorphism is right determined by a module. Using the same methods, we describe the minimal right determiner of a morphism, as discussed in the book by Auslander, Reiten and Smalø. In addition, we look at the role of indecomposable projective direct summands of a minimal right determiner and provide a detailed analysis of the kernel-determined morphisms: these are those morphisms which are right determined by a module without any non-zero projective direct summand. In this way, we answer a question raised in the book by Auslander, Reiten and Smalø. What we encounter is an intimate relationship to the vanishing of $\operatorname{Ext}^{2}$.