Related papers: Refinable functions with PV dilations

For a continuous angle-valued map defined on a compact ANR, a fixed field and any degree one proposes a refinement of the Novikov-Betti number and of the Novikov homology of the pair consisting of the ANR and the degree one integral…

Based on the recent developments in the irregular Riemann-Hilbert correspondence for holonomic D-modules and the Fourier-Sato transforms for enhanced ind-sheaves, we study the Fourier transforms of some irregular holonomic D-modules. For…

We put forward the exact version of two-parameter generating functions with $\tau$-invariance, which allows us to give a unified and inherent definition for the Drinfeld realization of two-parameter quantum affine algebras for all the…

We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi)…

We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when $\varphi_j\in \mathbb Z[t]$ $(1\le j\le k)$ is a system of polynomials with non-vanishing Wronskian, and…

Following our first article, we continue to investigate ultrametic modules over a ring of twisted polynomials of the form $[K;\vfi]$, where $\vfi$ is a ring endomorphism of $K$. The main motivation comes from the the theory of valued…

The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{\mu^2(d)}{\varphi(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right).…

Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes, because of their good resistance to differential cryptanalysis. An APN function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$ is…

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term…

Assume that $V_h$ is a space of piecewise polynomials of degree less than $r\geq 1$ on a family of quasi-uniform triangulation of size $h$. Then the following well-known upper bound holds for a sufficiently smooth function $u$ and $p\in [1,…

Let $(a_n)_{n \geq 0}$ be a sequence of integers such that its generating series satisfies $\sum_{n \geq 0} a_nt^n = \frac{h(t)}{(1-t)^d}$ for some polynomial $h(t)$. For any $r \geq 1$ we study the coefficient sequence of the numerator…

We consider models of gradient type, which are the densities of a collection of real-valued random variables $\phi :=\{\phi_x: x \in \Lambda\}$ given by $Z^{-1}\exp({-\sum\nolimits_{j \sim k}V(\phi_j-\phi_k)})$. We focus our study on the…

In the theory of the Nil-DAHA Fourier transform, the inner products of q-Hermite polynomials for the measure function multiplied by a level one theta function are the key. They are used to obtain expansions of products of any number of such…

Let $\chi$ be a non-principal Dirichlet character of modulus $q$ with associated \textit{L}-function $L(s,\chi)$. We prove that $$|L(1,\chi)|\le\left(\frac{1}{2}+O\Big(\frac{\log\log q}{\log q}\Big)\right)\frac{\varphi(q)}{q}\log q\,,$$…

We introduce the method of desingularization of multi-variable multiple zeta-functions (of the generalized Euler-Zagier type), under the motivation of finding suitable rigorous meaning of the values of multiple zeta-functions at…

Let $K/k$ be a finite Galois extension of global function fields. Let $E$ be a Drinfeld module over $k$. We state and prove an equivariant refinement of Taelman's analogue of the analytic class number formula for $(E,K/k)$, and derive…

Let $\lambda$ be an integer, and $f(z)=\sum_{n\gg-\infty} a(n)q^n$ be a weakly holomorphic modular form of weight $\lambda+\frac 12$ on $\Gamma_0(4)$ with integral coefficients. Let $\ell\geq 5$ be a prime. Assume that the constant term…

We consider the complement value problem for a class of second order elliptic integro-differential operators. Let $D$ be a bounded Lipschitz domain of $\mathbb{R}^d$. Under mild conditions, we show that there exists a unique bounded…

Let $\mathcal{V}$ be a mixed characteristic complete discrete valuation ring, $\mathcal{P}$ a separated smooth formal scheme over $\mathcal{V}$, $P$ its special fiber, $X$ a smooth closed subscheme of $P$, $T$ a divisor in $P$ such that…

Motivated by recent results of Tao-Ziegler [Discrete Anal. 2016] and Greenfeld-Tao (2022 preprint) on concatenating affine-linear functions along subgroups of an abelian group, we show three results on recovering affine-linearity of…