Related papers: An extension of the digital method based on $b$-ad…
In this paper, we introduce the BMT distribution as an unimodal alternative to continuous univariate distributions supported on a bounded interval. The ideas behind the mathematical formulation of this new distribution come from computer…
For an integer b>=2, let s_b(n) be the sum of the digits of the integer n when written in base b, and let S_b(N) be the sum of s_b(n) over n=0,...,N-1, so that S_b(N) is the sum of all b-ary digits needed to write the numbers 0,1,...,N-1.…
We introduce a new -as far as we know- problem, according to which we are asked to match sequences of two digits in matrices having entries among those two digits (but others too) and prove that this problem is NP-complete
Integer factorization is a famous computational problem unknown whether being solvable in the polynomial time. With the rise of deep neural networks, it is interesting whether they can facilitate faster factorization. We present an approach…
Let $A$ be a $d \times d$ matrix with rational entries which has no eigenvalue $\lambda \in \mathbb{C}$ of absolute value $|\lambda| < 1$ and let $\mathbb{Z}^d[A]$ be the smallest nontrivial $A$-invariant $\mathbb{Z}$-module. We lay down a…
Transformer-based autoregressive models excel in data generation but are inherently constrained by their reliance on discretized tokens, which limits their ability to represent continuous values with high precision. We analyze the…
Given a base $b$, a "digit map" is a map $f: \mathbb{Z}^{\ge 0} \to \mathbb{Z}^{\ge 0}$ of the form $f(\sum_{i=0}^n a_ib^i) = \sum_{i=0}^n f_*(a_i)$, $0 \le a_i \le b-1$ for each $i$, where $f_* : \{0,1,\dots, b-1\} \to \mathbb{Z}^{\ge 0}$…
We present a triangle of connections between the Sierpinski triangle, the sum-of-digits function, and the Binomial Theorem via a one-parameter family of Sierpinski matrices, which encodes a digital version of the Binomial Theorem.
In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that…
Solving symbolic reasoning problems that require compositionality and systematicity is considered one of the key ingredients of human intelligence. However, symbolic reasoning is still a great challenge for deep learning models, which often…
Random graphs with prescribed degree sequences have been widely used as a model of complex networks. Comparing an observed network to an ensemble of such graphs allows one to detect deviations from randomness in network properties. Here we…
A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some…
We initiate the study of enumerating linear subspaces of alternating matrices over finite fields with explicit coordinates. We postulate that this study can be viewed as a linear algebraic analogue of the classical topic of enumerating…
A general input-output modelling technique for aperiodic-sampling linear systems has been developed. The procedure describes the dynamics of the system and includes the sequence of sampling periods among the variables to be handled. Some…
In this paper, we prove that there are uncountable many real transcendental numbers, which are generated by digital pattern sequences. This generalizes the main theorem in Morton and Mourant, which states the existence of countable many…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
We propose an unified algebraic approach for static condensation and hybridization, two popular techniques in finite element discretizations. The algebraic approach is supported by the construction of scalable solvers for problems involving…
Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence.…
We use a unified framework to summarize sixteen randomized iterative methods including Kaczmarz method, coordinate descent method, etc. Some new iterative schemes are given as well. Some relationships with \textsc{mg} and \textsc{ddm} are…
In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we…