Related papers: Computing Absolutely Normal Numbers in Nearly Line…
Let $\mathbb{N}_0$ be a class of natural numbers whose binary decompositions has even number of 1. We estimate of the sum $\sum\limits_{n\in \mathbf{N}_0,n\le X}\exp(2\pi i \alpha n^2)$.
Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$…
We resolve a conjecture of Cox and Martin by determining asymptotically for every $k\ge 2$ the maximum number of copies of $C_{2k}$ in an $n$-vertex planar graph.
Decimal multiplication is the task of multiplying two numbers in base $10^N.$ Specifically, we focus on the number-theoretic transform (NTT) family of algorithms. Using only portable techniques, we achieve a 3x-5x speedup over the mpdecimal…
We state and give self contained proofs of semidefinite programming characterizations of the numerical radius and its dual norm for matrices. We show that the computation of the numerical radius and its dual norm within $\varepsilon$…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
In the present paper the authors construct normal numbers in base $q$ by concatenating $q$-adic expansions of prime powers $\lfloor\alpha p^\theta\rfloor$ with $\alpha>0$ and $\theta>1$.
Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two…
Let $n$ be a prime power, $r$ be a prime with $r\mid n-1$, and $\varepsilon\in (0,1/2)$. Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over $\mathbb F_r$ having length $n$, relative…
A permutiple is a natural number whose representation in some base, $b>1$, is an integer multiple of a number whose base-$b$ representation has the same collection of digits. Previous efforts have made progress in finding such numbers using…
We utilize the asymmetric random telegraph wave-based instantaneous noise-base logic scheme to represent the problem of drawing numbers from a hat, and we consider two identical hats with the first 2^N integer numbers. In the first problem,…
Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss-Seidel iterative solution of linear equations in the precision matrix. Specifically, the…
In the present paper we investigate two special types of non-normal numbers. On the one hand we call a number extremely non-normal if the set of accumulation points of its frequency of blocks vector is the full set of shift invariant…
We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…
It is known, that an $\epsilon$-machine is either exactly or asymptotically synchronizing. In the exact case, the observer can infer the current machine state after observing $L$ generated symbols with probability $1-a^L$ where $0 \leq a<1$…
A \emph{linear extension} of a partial order \(\preceq\) over items \(A = \{ 1, 2, \ldots, n \}\) is a permutation \(\sigma\) such that for all \(i < j\) in \(A\), it holds that \(\neg(\sigma(j) \preceq \sigma(i))\). Consider the problem of…
A triply even code is a binary linear code in which the weight of every codeword is divisible by 8. We show how two doubly even codes of lengths m_1 and m_2 can be combined to make a triply even code of length m_1+m_2, and then prove that…
An integer program is called ideal if its continuous relaxation coincides with its convex hull allowing the problem to be solved as a continuous program and offering substantial computational advantages. Proving idealness analytically can…
A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order $0$ as well as evaluating Schl\"{o}milch and Fourier--Bessel expansions in $\mathcal{O}(N(\log N)^2/\log\!\log N)$ operations. The…
A determined algorithm is presented for solving the rSUM problem for any natural r with a sub-quadratic assessment of time complexity in some cases. In terms of an amount of memory used the obtained algorithm is the nlog^3(n) order. The…