Related papers: Arbitrarily Substantial Number Representation for …
The syntactic nature and compositionality characteristic of stochastic process algebras make models to be easily understood by human beings, but not convenient for machines as well as people to directly carry out mathematical analysis and…
Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…
The problem of separating structured information representing phenomena of differing natures is considered. A structure is assumed to be independent of the others if can be represented in a complementary subspace. When the concomitant…
We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea…
This paper introduces a novel training methodology that enables a Transformer model to generalize the addition of two-digit numbers to numbers with unseen lengths of digits. The proposed approach employs an autoregressive generation…
This work considers the problem of distributing matrix multiplication over the real or complex numbers to helper servers, such that the information leakage to these servers is close to being information-theoretically secure. These servers…
We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. $\left\| n \right\|$ denotes the minimum number of 1's in the expressions representing $n$. The logarithmic complexity $\left\|…
Most existing super-resolution methods do not perform well in real scenarios due to lack of realistic training data and information loss of the model input. To solve the first problem, we propose a new pipeline to generate realistic…
The development of modern information technologies permits to collect and to analyze huge amounts of statistical data in different spheres of life. The main problem is not to only to collect but to process all relevant information. The…
We present new proofs of eight integral representations of the Catalan numbers. Then, we create analogous integral representations of the Motzkin numbers and obtain new results. Most integral representations of counting sequences found in…
In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a…
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…
Computer-based modelling and simulation have become useful tools to facilitate humans to understand systems in different domains, such as physics, astrophysics, chemistry, biology, economics, engineering and social science. A complex system…
This note being devoted to some aspects of the inverse problem of representation theory contains a new insight into it illustrated by two topics. The attention is concentrated on the manner of representation of abstract objects by the…
Efforts to promote equitable public policy with algorithms appear to be fundamentally constrained by the "impossibility of fairness" (an incompatibility between mathematical definitions of fairness). This technical limitation raises a…
In this paper, we propose hybrid real- and complex-valued neural networks for speech enhancement. Real- or complex-valued models are either inefficient or present high complexity. We devise a straightforward design method for extending a…
Recent advancements in neural networks, supported by foundational theoretical insights, emphasize the superior representational power of complex numbers. However, their adoption in randomized neural networks (RNNs) has been limited due to…
We consider the fundamental problem of communicating an estimate of a real number $x\in[0,1]$ using a single bit. A sender that knows $x$ chooses a value $X\in\set{0,1}$ to transmit. In turn, a receiver estimates $x$ based on the value of…
We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that…
Arbitrarily sparse sets A of integers are constructed with the property that every integer can be represented uniquely in the form n = a + a', where a and a' belong to the set A and a < a' or a = a'. Some related open problems are stated.