Related papers: On $k$-error linear complexity of pseudorandom bin…
In this paper we take up again the deformation theory for $K$-linear pseudofunctors initiated in a previous work (Adv. Math. 182 (2004) 204-277). We start by introducing a notion of a 2-cosemisimplicial object in an arbitrary 2-category and…
Successive pairs of pseudo-random numbers generated by standard linear congruential transformations display ordered patterns of parallel lines. We study the ``ordered'' and ``chaotic'' distribution of such pairs by solving the eigenvalue…
We consider rate R = k/n causal linear codes that map a sequence of k-dimensional binary vectors {b_t} to a sequence of n-dimensional binary vectors {c_t}, such that each c_t is a function of {b_1,b_2,...,b_t}. Such a code is called anytime…
We consider the computational complexity of training depth-2 neural networks composed of rectified linear units (ReLUs). We show that, even for the case of a single ReLU, finding a set of weights that minimizes the squared error (even…
We describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree three. As an application we establish some divisibility properties of the associated ternary…
We consider a complexification of the Euler equations introduced by \v{S}ver\'ak which conserves energy. We prove that these complex Euler equations are nonlinearly ill-posed below analytic regularity and, moreover, we exhibit solutions…
We study an integer sequence associated with Cantor's division polynomials of a genus 2 curve having an integral point. We show that the reduction modulo $p$ of such a sequence is periodic for all but finitely many primes $p$, and describe…
We examine the linear complexity and the autocorrelation properties of new quaternary cyclotomic sequences of period 2p. The sequences are constructed via the cyclotomic classes of order four.
We show that an elliptic modular form with integral Fourier coefficients in a number field $K$, for which all but finitely many coefficients are divisible by a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar…
In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence…
Consider the following truncated Freud linear functional $\mathbf{u}_z$ depending on a parameter $z$, $$\langle\mathbf{u}_z,p\rangle=\int_0^\infty p(x)e^{-zx^4}dx,\quad z>0.$$ The aim of this work is to analyze the properties of the…
Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period $4p$ with optimal autocorrelation was proposed…
We reexamine different examples of reduction chains $\mathfrak{g} \supset \mathfrak{g}'$ of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra $\mathfrak{g}'$ leads to polynomial…
This paper establishes for the first time the predictive performance of speed priors and their computational complexity. A speed prior is essentially a probability distribution that puts low probability on strings that are not efficiently…
In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi(m)/(k+1)$ residue classes in the…
Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form $C(x)$. Various instances are known where the corresponding truncated sum…
From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\ (2\le i \le [k/4])$. We present explicit formulas among these modular forms based on the…
The complexity of the simple and the Kac modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$ of type $A$ was computed by Boe, Kujawa, and Nakano in 2012. A natural continuation to their work is computing the complexity of…
Let $p>3$ be a prime. In this paper, we obtain the congruences for $$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k},\…
Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far…