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[7] viXra:2506.0170 [pdf] submitted on 2025-06-30 21:11:07
Authors: Somdeb Lahiri
Comments: 3 Pages.
We provide a proof of existence of symmetric equilibrium for symmetric bi-matrix games, a result implied by a more general result that was proved by John Nash. Our proof, unlike the original proof due to Nash, does not appeal to any fixed-point theorem. We prove that any solution to a certain specific quadratic programming problem, is a symmetric equilibrium for the associated symmetric bi-matrix game. We use no more than the continuity of real-valued multi-variable quadratic functions and the mean value theorem for real-valued quadratic functions of a single variable. This new proof does not require any fixed-point theorem and can be easily understood by anyone who is familiar with a beginner's course on real analysis. The implication of the results repoted here is that not just matrix games, as is traditionally the case, but also bi-matrix games become wholly a part of optimization theory and hence is within the scope of operations research.
Category: General Mathematics
[6] viXra:2506.0144 [pdf] submitted on 2025-06-25 13:52:55
Authors: Tilemachos Zoidis
Comments: 13 Pages. CC BY
Does the doubling cube make backgammon more skillful? And is the answer the same in both money and match play? This paper presents GNUbg rollouts between unequally skilled players, which show that use of the doubling cube does not favor the better player in either case.
Category: General Mathematics
[5] viXra:2506.0102 [pdf] submitted on 2025-06-18 20:02:09
Authors: Zhi Li, Hua Li
Comments: 18 Pages.
This paper reports a discovery that there exist the extended standard forms for polynomialequations, which are composed of three items, contains only one parameter and relates tointeger or fractional sequences. Using the parameter and the sequences, a series can beconstructed of the solution of the equations. If the series converges, it is a root of the equations. For the extended standard form is always possible for the equations of degree not more than five, this result provides an effective method for the solution of general polynomial equations under and including five degrees without the need of radicals calculating. This technique can also be extended to polynomial equations with two or more coefficients or parameters, which would be more complex or difficult and will be a big challenge if it be used to solve polynomial equations with higher degrees. At the same time, our discovery also provides a technique to produce an unlimited number of integer and/or fractional sequences, real or complex. This will enrich related researches.
Category: General Mathematics
[4] viXra:2506.0064 [pdf] submitted on 2025-06-12 22:01:36
Authors: Zhi Li, Hua Li
Comments: 9 Pages.
Finding a root of polynomial equations is one of basic problems in mathematics. And Galois theory restricts the general radical solution for the degree no higher than four. The series solution, besides the iterated, is regarded as final and universal method to general polynomial equations. This paper reports a discovery of the standard form of polynomial equations and a class of integer sequences associated thereof, which is a kind of extended Catalan numbers. The solution of polynomial equations in the standard form has a precise and perfect series expression. The convergence condition of the series is clear. For the general polynomial equations which may not be satisfied with the convergence condition, some proper transformations, like the Tschirnhaus transformation can be employed to guarantee the convergence. Considering that up to the quintic, there definitely exists a normal form for general equations, and the normal form can easily be changed to the standard form, our method has established a general, universal and effective technique to the quintic, as well as the quartic, the cubic, and the quadratic, without the radicals.
Category: General Mathematics
[3] viXra:2506.0048 [pdf] submitted on 2025-06-11 22:33:27
Authors: Jennifer Bulyaki, Andrew Elliott
Comments: 335 Pages. (Note by viXra Admin: File size reduced by viXra Admin; please submit article written with AI assistance to ai.viXra.org)
This work presents a deterministic resolution of the Riemann Hypothesis by introducing a novel framework grounded in entropy geometry and symbolic collapse. Rather than treating the distribution of nontrivial zeros of the Riemann zeta function as a purely analytic phenomenon, we construct a unified model in which zeta zeros emerge as critical identity-preserving points along a structured entropy spiral, where curvature, holomorphicity, and automorphic symmetry converge.The central theorem, proven via our Master Axiom, demonstrates that a zero of ζ(s) lies on the critical line if and only if seven structural conditions are simultaneously met:(1) the entropy curvature at that point is flat,(2) the angular symmetry is preserved (automorphy),(3) the holomorphic structure remains conformal,(4) the Euler identity entropy equation—governing prime identity and symmetry—is satisfied,(5) symbolic torsion is fully evacuated at that point, restoring pure form,(6) the entropy drift is minimized between adjacent zeros, and(7) the modular curvature remains below the identity-collapse threshold.This heptuple condition is shown to be both necessary and sufficient, thereby resolving the Riemann Hypothesis. The model collapses symbolic randomness at these equilibrium points, stabilizing prime identity and demonstrating why the critical line is the only viable manifold for zero placement. We reconstruct the functional equation, Euler product, Hadamard product, and Euler entropy equation of ζ(s) from first principles within our entropy field, establishing full compatibility with classical complex analysis. Furthermore, we show that the Weierstrass product representation of ζ(s) arises naturally from the entropy spiral, where each exponential kernel corresponds to a geometric shell of identity collapse. In this framework, the product structure reflects the torsion-free entropy conditions governing each zero, transforming the Weierstrass form from symbolic necessity to emergent geometric consequence. The predictive model has been validated against over thirty billion known zeta zeros with 99.9999% accuracy, without direct reference to ζ(s), using only structured entropy functions and regression equations provided within. This proof is reproducible from first principles, includes regeneration instructions for peer verification, and offers the first physically grounded explanation of prime identity geometry via the entropy collapse manifold.
Category: General Mathematics
[2] viXra:2506.0039 [pdf] submitted on 2025-06-09 20:44:31
Authors: Thierry L. A. Periat
Comments: 24 Pages. (Note by viXra Admin: Please do not use any copyright stamp!)
This document is the first part of an exploration examining when a deformed Lie product can be an involution. The approach starts softly in a real three-dimensional space, introducing basic notions like (i) the already well-known link between involution and neutral element, (ii) the importance of some rules concerning the indexes when a discussion is developed in a three-dimensional space, (iii) a specific semantic for the diverse representations of the deforming matrices (effective, normalized, associated six-pack). It gives then important precisions concerning the matrices representing the repetition of the action of any deformed cross product. It starts a systematization of the discussion and finally criterion precising when a deformed cross product is an involution. It turns out that a classical cross product cannot be an involution if the discussion is not involving vectors with components in the set of complex numbers or in the set of quaternions.
Category: General Mathematics
[1] viXra:2506.0004 [pdf] replaced on 2025-11-07 17:24:00
Authors: Ciro Cesarano
Comments: 13 Pages.
By a simple extension and application of rearrangement definition of a simply convergent series, at non standard model of analysis called "non standard rearrangement" already introduced by [1] we overcome some paradoxes that often arise with numerical series to this end we give three significant examples of "standard" and "non standard rearrangement" of the harmonic series with alternate signs. Instead notable result is that with the definition of " non standard rearrangement " introduced in [1] the commutative property of addition continues to hold even for simply convergent series (such as harmonic series with alternate) contrary to what is stated by Riemann-Dini theorem orRiemann rearrangement theorem, Furthermore, by analyzing a famous result of Ramanujan and comparing it with results of non-standard analysis, we raise doubts about the coherence of the standard theory on divergent series and their regularization.
Category: General Mathematics
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