Related papers: A newly-generalized problem from a problem for the…
The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…
Machine learning (ML) in its current form implies that an answer to any problem can be well approximated by a function of a very peculiar form: a specially adjusted iteration of Heavyside theta-functions. It is natural to ask if the answers…
It is argued that the two problems of choosing characterizations and models of complex systems should not be considered independently. A particular criterion for these choices, oriented on the potential usefulness of the results, is…
Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these…
Override and update are natural constructions for combining partial functions, which arise in various program specification contexts. We use an unexpected connection with combinatorial geometry to provide a complete finite system of…
Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
An effective method for generating linear equations of maximal symmetry in their much general normal form is obtained. In the said normal form, the coefficients of the equation are differential functions of the coefficient of the term of…
Machine Learning (ML) is applicable to scientific problems, i.e. to those which have a well defined answer, only if this answer can be brought to a peculiar form ${\cal G}: X\longrightarrow Z$ with ${\cal G}(\vec x)$ expressed as a…
Combinatorial optimization can be described as the problem of finding a feasible subset that maximizes a objective function. The paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is…
We investigate the concept of symmetry and its role in problem solving. This paper first defines precisely the elements that constitute a "problem" and its "solution," and gives several examples to illustrate these definitions. Given…
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
The paper deals with a new approach to Poisson summation formulas in the context of function spaces on $\mathbb{R}^n$.
This paper examines various methods and ideas for humanizing mathematics. The term 'humanizing mathematics' which includes elements of 'aesthetic mathematics' refers to approaches that emphasize the aesthetic, philosophical, and subjective…
A generalization of a distribution increases the flexibility particularly in studying of a phenomenon and its properties. Many generalizations of continuous univariate distributions are available in literature. In this study, an…
An interpretation of selected parts of Newton's Principia, with modern notation and methods. Keplers Laws are derived from an inverse square law using Newton's methods.
In this note some philosophical thoughts and observations about mathematics are expressed, arranged as challenges to some common claims.
A general approach is established for deriving one-shot performance bounds for information-theoretic problems on general alphabets beyond countable alphabets. It is mainly based on the quantization idea and a novel form of "likelihood…
The Carath\'eodory theorem on the construction of a measure is generalized by replacing the outer measure with an approximation of it and generalizing the Carath\'eodory measurability. The new theorem is applied to obtain dynamically…
A definition of a {\it Realistic} Physics Theory is proposed based on the idea that, at all time, the set of physical properties possessed (at that time) by a system should unequivocally determine the probabilities of outcomes of all…