Related papers: Highly nonlinear functions over finite fields

Motivated by the Quantum Modularity Conjecture and its arithmetic aspects related to the Habiro ring of a number field, we define a map from the Kauffman bracket skein module of an integer homology 3-sphere to the Habiro ring, and use…

Let Y be an infinite covering space of a projective manifold M in P^N of dimension n geq 2. Let C be the intersection with M of at most n-1 generic hypersurfaces of degree d in P^N. The preimage X of C in Y is a connected submanifold. Let…

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where…

Peter Lappan in [9] proved that for each $n\in \mathbb{N}=\{1,2,3,\dots\}$, let $f_{1,n}, f_{2,n}$ and $f_{3,n}$ be three continuous functions on $\mathbb{D}:=\{z\in \mathbb{C} : |z| < 1\}$ such that for each $j=1,2,3,$ the sequence…

Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work [2]. Here we consider, more generally, affine Grassmann codes of a given level.…

It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower $Q$-homeomorphisms $f$ between domains in $\overline{\Rn}=\Rn\cup\{\infty\}$, $n\geqslant2$, under integral constraints…

We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of prime characteristic $p$ and $Q$ be an $\mathfrak{m}$-primary parameter ideal. We give criteria for F-rationality of $R$ using the tight Hilbert function $H^*_Q(n)=\ell(R/(Q^n)^*$ and the…

We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Beilinson-Gaitsgory deformation of the affine flag manifold to the affine Grassmannian. We study the multiplicity functions we introduced in an…

The goal of this note is to generalize Thurston's Topological Characterization of Rational Functions to the setting when both the covering degree and the set of marked points are infinite. A relevant class of branched coverings are…

We prove that if $f:\mathbb{R}^n\to\mathbb{R}$ is convex and $A\subset\mathbb{R}^n$ has finite measure, then for any $\varepsilon>0$ there is a convex function $g:\mathbb{R}^n\to\mathbb{R}$ of class $C^{1,1}$ such that $\mathcal{L}^n(\{x\in…

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable…

We consider the problem of describing the typical (possibly) non-linear code of minimum distance bounded from below over a large alphabet. We concentrate on block codes with the Hamming metric and on subspace codes with the injection…

We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic…

We give a new proof of the Hansen-Mullen irreducibility conjecture. The proof relies on an application of a (seemingly new) sufficient condition for the existence of elements of degree $n$ in the support of functions on finite fields. This…

We investigate the realizations of a random Gaussian field on a finite domain of ${\mathbb R}^d$ in the limit where a given linear functional of the field is large. We prove that if its variance is bounded, the field converges uniformly and…

Determination of quasi-invariant generalized functions is important for a variety of problems in representation theory, notably character theory and restriction problems. In this note, we review some new and easy-to-use techniques to show…

In this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initial-value problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson's…

Suppose $\{f_1,...,f_m\}$ is a set of Lipschitz maps of $\mathbb{R}^d$. We form the iterated function system (IFS) by independently choosing the maps so that the map $f_i$ is chosen with probability $p_i$ ($\sum_{i=1}^m p_i=1$). We assume…

We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to 0 modulo $n$. We give several formulas for computing the values of this…