Related papers: Normal numbers and nested perfect necklaces

We first investigate the algebraic structure of vertex algebroids $B$ when $B$ are simple Leibniz algebras. Next, we use these vertex algebroids $B$ to construct indecomposable non-simple $C_2$-cofinite $\mathbb{N}$-graded vertex algebras…

We show how to build an alphabetic minimax tree for a sequence (W = w_1, >..., w_n) of real weights in (O (n d \log \log n)) time, where $d$ is the number of distinct integers (\lceil w_i \rceil). We apply this algorithm to building an…

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers $a$ and $b$, with…

The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset...\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,...,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We…

We define the twofold semidirect product of two skew left braces, in which both the additive and multiplicative groups are semidirect products of the corresponding groups of the given skew left braces. We consider an odd prime $p$ and an…

The class of problems complete for NP via first-order reductions is known to be characterized by existential second-order sentences of a fixed form. All such sentences are built around the so-called generalized IS-form of the sentence that…

Based on continued fractions with subtractions, we identify the set of real numbers with the set of infinite integer sequences with all terms but the first one greater or equal to two. Each such sequence produces in a canonical way a unique…

We present explicit algorithms for computing structured matrix-vector products that are optimal in the sense of Strassen, i.e., using a provably minimum number of multiplications. These structures include Toeplitz/Hankel/circulant,…

In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$…

Let $G=(\mathbb Z/n\mathbb Z) \oplus (\mathbb Z/n\mathbb Z)$. Let $\mathsf {s}_{\leq k}(G)$ be the smallest integer $\ell$ such that every sequence of $\ell$ terms from $G$, with repetition allowed, has a nonempty zero-sum subsequence with…

We introduce a randomized algorithm for computing the minimal-norm solution to an underdetermined system of linear equations. Given an arbitrary full-rank m x n matrix A with m<n, any m x 1 vector b, and any positive real number epsilon…

We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length $O(\log n \log \log n)$ with exactly $n$ subsequences; this…

Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$;…

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every…

In this paper, we investigate the problem of designing $(n, N; \mathcal{B})$-reconstruction codes for $N\in \{14,11,9,5\}$, where $\mathcal{B}$ is the single-deletion single-substitution ball function that maps a sequence to the set of all…

A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial…

The Weyl operators give a convenient basis of $M_n(\mathbb{C})$ which is also orthonormal with respect to the Hilbert-Schmidt inner product. The properties of such a basis can be generalised to the notion of a nice error basis(NEB), as…

Here we demonstrate a sieve for analysing primes and their composites, using equivalence classes based on the modulo 6 return value as applied to the Natural numbers. Five features of this 'Hexile' sieve are reviewed. The first aspect, is…

Using Popa's deformation/rigidity theory, we investigate prime decompositions of von Neumann algebras of the form $L(\mathcal{R})$ for countable probability measure preserving equivalence relations $\mathcal{R}$. We show that…

Assuming an instance of the Brodsky-Rinot proxy principle holding at a regular uncountable cardinal $\kappa$, we construct $2^\kappa$-many pairwise non-embeddable minimal non-$\sigma$-scattered linear orders of size $\kappa$. In particular,…