Related papers: Hilbert's Program Then and Now
Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…
For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting…
When a physicist performs a quantic measurement, new information about the system at hand is gathered. This paper studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as…
In this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This…
Substitutions play a crucial role in a wide range of contexts, from analyzing the dynamics of social opinions and conducting mathematical computations to engaging in game-theoretical analysis. For many situations, considering one-step…
We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics,…
We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second G\"odel incompleteness theorem to stably…
Discrete mathematics is the foundation of computer science. It focuses on concepts and reasoning methods that are studied using math notations. It has long been argued that discrete math is better taught with programming, which takes…
In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of $\mathrm{GL}_2$-type over totally real fields. The original approach was based on hard open conjectures, which have made…
Starting from a Pfaffian equation in dimension $N$ and focusing on compact solutions for it, we place in perspective the variational method used in [29] to solve Hilbert's 16th problem. In addition to exploring how this viewpoint can help…
It is well known that, under certain conditions, it is possible to split logic programs under stable model semantics, i.e. to divide such a program into a number of different "levels", such that the models of the entire program can be…
We explore in the framework of Quantum Computation the notion of {\em Computability}, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to…
We provide an upper bound for the number of limit cycles that planar polynomial differential systems of a given degree may have. The bound turns out to be a polynomial of degree four in the degree of the system. The strategy brings together…
Using an algebraic framework we solve a problem posed in [5] and [7] about the axiomatizability of a quantum computational type logic related to fuzzy logic. A Hilbert-style calculus is developed obtaining an algebraic strong completeness…
Several different ways exist for approaching hard optimization problems. Mathematical programming techniques, including (integer) linear programming-based methods and metaheuristic approaches, are two highly successful streams for…
The words ``Programming is the second literacy'' were coined more than 40 years ago but never came to life. This paper is one in the series of papers aimed at the analysis of mathematical requirements for a merge of school mathematics with…
Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise…
In order to work with mathematical content in computer systems, it is necessary to represent it in formal languages. Ideally, these are supported by tools that verify the correctness of the content, allow computing with it, and produce…
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
Efficient algorithms for many problems in optimization and computational algebra often arise from casting them as systems of polynomial equations. Blum, Shub, and Smale formalized this as Hilbert's Nullstellensatz Problem $HN_R$: given…