Related papers: Hilbert-type dimension polynomials of intermediate…
In this paper we discuss the permutational property of polynomials of the form $f(L(x))+k(L(x))\cdot M(x)\in \mathbb F_{q^n}[x]$ over the finite field $\mathbb F_{q^n}$, where $L, M\in \mathbb F_q[x]$ are $q$-linearized polynomials. The…
An elementary introduction to Hilbert modular forms, with a particular attention to their differential properties: Rankin-Cohen brakets, structure of differential rings... This text will appear in SMF Seminaires et Congres.
Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in…
This report investigates the main definitions and fundamental properties of the fractional two-sided quaternionic Dunkl transform in two dimensions. We present key results concerning its structure and emphasize its connections to classical…
Let G be a compact p-adic analytic group. We study K-theoretic questions related to the representation theory of the completed group algebra kG of G with coefficients in a finite field k of characteristic p. We show that if M is a finitely…
In this note we connect Sobolev estimates in the context of polynomial averages e.g. \[ \| \int_0^1 \prod_{k=1}^m f_k(x-t^k) \|_{1} \leq \text{Const} \cdot 2^{-\text{const} \cdot l} \prod_{i=1}^m \| f_k \|_m \] whenever some $f_i$ vanishes…
Given a finite extension $K/F$ of degree $r$ of a finite field $F$, we enumerate all selfdual skew cyclic codes in the Ore quotient ring $K[X;\text{Frob}]/(X^{rk}-1)$ for any positive integer $k$ coprime to the characteristic $p$ (separable…
We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it…
We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $\mathbb{R}$, we define fractional operators by means of a functional calculus…
We introduce an algebra $\mathcal H$ consisting of difference-reflection operators and multiplication operators that can be considered as a $q=1$ analogue of Sahi's double affine Hecke algebra related to the affine root system of type…
Given a nonzero integer $d$, we know by Hermite's Theorem that there exist only finitely many cubic number fields of discriminant $d$. However, it can happen that two non-isomorphic cubic fields have the same discriminant. It is thus…
A result of Kaufmann shows that if $L_\alpha$ is countable, admissible and satisfies $\Pi_n\textsf{-Collection}$, then $\langle L_\alpha, \in \rangle$ has a proper $\Sigma_{n+1}$-elementary end extension. This paper investigates to what…
Let $k$ be a differential field having an algebraically closed field of constants, $E$ be a strongly normal extension of $k$, and $k^0$ be the algebraic closure of $k$ in $E.$ We prove for any intermediate differential field $k\subset…
Fractional differential (and difference) operators play a role in a number of diverse settings: integrable systems, mirror symmetry, Hurwitz numbers, the Bethe ansatz equations. We prove extensions of the three major results on algebras of…
In this article, we study the relationship between the exponential dichotomy properties of a triangular system of linear difference equations and its associated diagonal system on Hilbert spaces. We stress that all previous results in this…
Let D be an integral domain with quotient field K. For any set X, the ring Int(D^X) of integer-valued polynomials on D^X is the set of all polynomials f in K[X] such that f(D^X) is a subset of D. Using the t-closure operation on fractional…
Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of…
The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from $q=e^{2\pi i \tau}$ to the reciprocal of the…
In this paper, we consider an infinite dimensional exponential family, $\mathcal{P}$ of probability densities, which are parametrized by functions in a reproducing kernel Hilbert space, $H$ and show it to be quite rich in the sense that a…
We introduce certain functors from the category of commutative rings (and related categories) to that of $\mathbb{Z}$-algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor…