Related papers: A Less Restrictive Brian\c{c}on-Skoda Theorem with…

Let G be a connected split reductive group over a p-adic field. In the first part of the paper we prove, under certain assumptions on G and the prime p, a localization theorem of Beilinson-Bernstein type for admissible locally analytic…

Let $(R,m)$ be a Noetherian local ring and $I$ an ideal with finite projective dimension. If $R/I$ satisfies some property $\mathcal{P}$, it is natural to ask whether $R$ would also satisfy this property $\mathcal{P}$. This is called the…

Let $(A, \m, k)$ be a Gorenstein local ring of dimension $ d\geq 1.$ Let $I$ be an ideal of $A$ with $\htt(I) \geq d-1.$ We prove that the numerical function \[ n \mapsto \ell(\ext_A^i(k, A/I^{n+1}))\] is given by a polynomial of degree…

Let R be an unramified regular local ring of mixed characteristic, D an Azumaya R-algebra, K the fraction field of R, Nrd the reduced norm homomorphism for the Azumaya R-algebra D. Let a be a unit in R. It is proved the following: suppose…

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings.…

Inspired by the works in linkage theory of ideals, we define the concept of linkage of ideals over a module. Several known theorems in linkage theory are improved or recovered by new approaches. Specially, we make some extensions and…

Induced representations of Brauer algebra $D_{f}(n)$ from $S_{f_{1}}\times S_{f_{2}}$ with $f_{1}+f_{2}=f$ are discussed. The induction coefficients (IDCs) or the outer-product reduction coefficients (ORCs) of $S_{f_{1}}\times…

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the field of rational numbers. Let $f_1, \ldots, f_r$ be homogeneous…

Let $(A,\mathfrak{m})$ be an excellent equi-charateristic Gorenstein isolated singularity of dimension $d \geq 2$. Assume the residue field of $A$ is perfect. Let $I$ be any $\mathfrak{m}$-primary ideal. Let $G_I(A) = \bigoplus_{n \geq…

Magyar has shown that if $B \subset \mathbb{Z}^d$ has positive upper density $(d \geq 5)$, then the set of squared distances $\{ \|b_1-b_2 \|^2 \text{ }: \text{ } b_1,b_2 \in B \}$ contains an infinitely long arithmetic progression, whose…

For any given submersion $\pi:X\to B$ with closed, oriented and spin$^c$ fibers of even dimension, equipped with a Riemannian and differential spin$^c$ structure, we apply the Atiyah-Singer-Gorokhovsky-Lott approach to the local family…

The powers ${\mathfrak m}^n$ of the maximal ideal $\mathfrak m$ of a local Noetherian ring $R$ are known to satisfy certain homological properties for large values of $n$. For example, the homomorphism $R\to R/{\mathfrak m}^n$ is Golod for…

For the p-adic group G=SL (2) , we present results of the computations of the sums of the Bernstein projectors of a given depth. Motivation for the computations is based on a conversation with Roger Howe in August 2013. The computations are…

The John-Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the sharp maximal…

In \cite{rees} Rees gave a characterization for the normal joint reduction number zero of two $\m$-primary ideals in an analytically unramified Cohen-Macaulay local ring of dimension two. Rees' result is a generalization of Zariski's…

We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincar\'e series sharing a common denominator if $R/\soc(R)$ is a Golod ring. If $R$ is a Gorenstein local ring with square of the maximal ideal being generated…

Using normal approximation (NA) to construct a kernel-smoother-based confidence interval faces a fundamental challenge: the normalization makes a small estimation bias become a non-negligible inferential bias. This paper takes a different…

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the…

Modern deep learning has revealed a surprising statistical phenomenon known as benign overfitting, with high-dimensional linear regression being a prominent example. This paper contributes to ongoing research on the ordinary least squares…

Let $G= GL_n(\mathbb{C})$, the general linear group over the complex numbers, and let $B$ be the set of invertible upper triangular matrices in $G$. Let $\mathfrak{b}=\text{Lie}(B)$. For $\mu:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow…