Set Theory and Logic

The Problem P=NP (in French)

Authors: Thierry Delort
Comments: 20 Pages.

In this article, we are going to solve the problem P=NP for a particular kind of problems called basic problems of numerical determination. We are going to propose 3 fundamental Axioms permitting to solve the problem P=NP for basic problems of numerical determination, those Axioms can also be considered as pure logical assertions, intuitively evident and never contradicted, permitting to understand the solution of the problem P=NP for basic problems of numerical determination. We will see that those Axioms imply that the problem P=NP in undecidable for basic problems of numerical determination. Nonetheless we will see that it is possible to give a theoretical justification (which is not a classical proof) of the proposition "P≠NP". We will then study a 2nd problem, named "PN=DPN problem" analogous to the problem P=NP but which is fundamental in mathematics.

Dans cet article, nous allons résoudre le problème P=NP pour un cas particulier de problèmes appelés problèmes de détermination numérique basiques. Nous allons proposer 3 Axiomes fondamentaux permettant de résoudre le problème considéré pour les problèmes de détermination numérique basiques, ces Axiomes pouvant aussi être considérés comme des assertions de logique pure évidentes intuitivement et jamais contredites permettant de comprendre la solution du problème considéré. On verra que ces Axiomes entraînent l’indécidabilité du problème P=NP pour les problèmes de détermination numérique basiques. On montrera cependant qu’on peut donner une justification théorique (qui n’est pas une démonstration classique) de P≠NP. Nous étudierons ensuite un 2nd problème, appelé problème « PN=DPN », analogue au problème P=NP mais ayant une importance fondamentale en mathématique.
Category: Set Theory and Logic

Fractal Partitioning and Subconvexity

Authors: Parker Emmerson
Comments: 18 Pages.

This document presents a comprehensive study of fractal partitioning and its application to subconvexity generalizations across various mathematical contexts. By utilizing a combination of advanced equations andinequalities, the paper develops robust models for partitioning sets into subsets of varying sizes, measuring the similarity and complexity within these partitions, and ensuring consistent interactions across boundaries. Special attention is given to computing the norm of differences betweensubsets and assessing their similarity, along with complexity measurements utilizing tensor equations and sums. These calculations provideinsights into the partitions’ fractal behavior and their probabilistic interactions.The document also delves into task scheduling algorithms based on SRPT, round-robin, and deadline-driven protocols, highlighting practical implications of fractal partitioning in optimizing resource management and minimizing distortions in dynamic systems. An emphasis is placedon ensuring the robustness and efficiency of fractal partitions through rigorousmathematical proofs and algorithmic implementations. By applyingthese models to data compression and analysis, the study demonstrates how fractal partitioning can efficiently represent complex data sets, expose hidden patterns, and identify anomalies in various domains such as finance and natural systems. Furthermore, the paper explores the concept of subconvexity in higher powers of the Riemann zeta function, establishing stronger forms of subconvexity conditions for different mathematical functions. This includesgeneralizations for cubic and higher powers of zeta functions, providing substantial evidence in support of hypotheses like the Riemann Hypothesis. The comprehensive approach combines theoretical constructs with practical algorithms, offering a powerful framework for analyzing and understanding complex mathematical and natural phenomena through fractalpartitioning and subconvexity measures.
Category: Set Theory and Logic