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| viXra.org > General Mathematics > 2508 |
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[9] viXra:2508.0176 [pdf] replaced on 2025-12-19 00:57:26
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 15 Pages. 4 figures
In this paper we will see that we can avoid the concepts of negative number and complex number thanks to the study of the underlying vector nature of some arithmetic and polynomial problems. We will see that the geometrical models used until now to represent negative numbers and complex numbers and their operations are not just interpretations or models. Translations, rotations and homotheties are what we need to solve several problems. We will see that what we call "negative numbers" and "complex numbers" are just the solutions of vector calculations and equations. All that is the consequence of the fact that geometrical considerations are unavoidable when we think about debts and gains and when we try to solve some polynomial equations. Those considerations are linked to a geometrical system with symmetries and a center. We will see that thanks to the solutions of those vector equations we can construct paths in the plane. We will also give the vector meaning of the formulas of De Moivre and Euler. An interpretation of the vertical axis linked to gains and losses will also be given.
Category: General Mathematics
[8] viXra:2508.0138 [pdf] submitted on 2025-08-21 20:19:33
Authors: Eric Louis Beaubien
Comments: 1 Page. (Note by viXra Admin: Please refrain from using incredulous expression in a scholarly article!)
I was just about floored when I did this meaningless calculation and wondered if anyone else had seen it before or anything even remotely like it. It has no physical significance as far as I can tell u2026 but u2026 wow u2026
Category: General Mathematics
[7] viXra:2508.0136 [pdf] submitted on 2025-08-21 20:00:00
Authors: Edgar Valdebenito
Comments: 3 Pages.
We present an identity that relates the Appell F1 function and the constant Pi.
Category: General Mathematics
[6] viXra:2508.0118 [pdf] submitted on 2025-08-19 15:43:13
Authors: Zhi Li, Hua Li
Comments: 11 Pages.
This paper reports a general solution for the sextic equations, which is an explicit power series oftwo parameters and fit for equations with real and/or complex coefficients.The general sextic equation can be simplified by the Tschirnhausen transformations andexpressed with four items in a type, called normal type. And it can further be simplified with onlytwo non-constant coefficients into a form, called standard form. This fact means that theresolution of the sextic is a problem of two degree of freedoms.There are totally 10 types and each type contains 6 forms. Among the total 60 forms, eachcorrespondents to a power series, the coefficients in most of series are fractional sequences,some integer sequences.If the series converges, the solution is found. Otherwise, successive Tschirnhausentransformations can be employed to obtain a series of new forms until the condition ofconvergence is satisfied. And then a reverse procedure is needed to find an original root. Theexperiment results show that it is always possible to satisfy the convergence condition and findthe roots of transformed equations after several iterations.The convergence of power series in all the 60 forms are different. The most favorite type andform are recommended.Similar method can be used to the resolution of higher degree of polynomial equations.
Category: General Mathematics
[5] viXra:2508.0117 [pdf] submitted on 2025-08-19 23:26:31
Authors: Dwight Boddorf
Comments: 2 Pages.
Article on numbers such that the product of two exponential entities equal or nearly equal the product of the two exponential entities inverted. Key numbers 137, 2036, 5435817984.
Category: General Mathematics
[4] viXra:2508.0069 [pdf] replaced on 2025-08-30 21:34:32
Authors: Abdelhay Benmoussa
Comments: 5 Pages.
This paper presents a proof of the classical explicit formula for Bernoulli numbers, expressed as a sum involving Stirling numbers of the second kind. The approach follows a combinatorial and polynomial comparison method similar to that used by Maurice d'Ocagne. Starting from the explicit formula of Stirling numbers and using known relations with falling factorials, we derive the closed-form expression
Category: General Mathematics
[3] viXra:2508.0058 [pdf] replaced on 2026-05-27 22:28:59
Authors: Robert A. Rice
Comments: 74 Pages.
We define the notion of a canonical envelope of a bilateral pairing and analyze canonical envelopes through bilateral density and compactness conditions. Canonical envelopes provide a systematic framework for a significant class of completion phenomena in mathematics, unifying these constructions through initial factorizations in categories of bilateral decompositions.The construction was motivated by Riehl's adjunction for weighted limits cite{riehl2008weighted}. We prove that all four Kan constructions (left and right Kan extensions and left and right Kan liftings) arise as instances of canonical envelopes (Corollary~ef{cor:kan-are-envelopes}), so weighted (co)limits and all four Kan constructions are recovered within the framework.We develop the theory of outer bilateral envelopes for cases where classical completions fail: the outer envelope objects $Y_Q$ and $X_Q$ always exist in presheaf categories, and the term emph{virtual canonical envelope} refers to this outer data when the interpolant is not yet known. We establish the geometric interpretation through cylinder factorization systems, connecting to Garner's work. We prove that canonical envelopes admit monadic organization: the canonical envelope functor, defined on the full subcategory of admissible pairings where CEs exist, extends to an idempotent monad whose Eilenberg-Moore algebras are precisely the complete bilateral pairings, with Garner's Isbell monad emerging as a natural specialization.We show that dicategories provide a bilateral algebraic presentation of dagger categories (Theorem~ef{thm:dicat-dagger-equiv}), with every dagger category admitting a canonical dicategory presentation and vice versa. The dicategory presentation, characterized as a canonical envelope (Theorem~ef{thm:dicat-envelope}), makes explicit the symmetric relationship between categorical and cocategorical composition that is implicit in the standard dagger category axiomatization. This bilateral presentation arises naturally from the canonical envelope construction and connects to Frobenius pseudomonoids, where the dicategory axioms correspond to Frobenius compatibility conditions.We construct canonical left and right envelope objects explicitly in presheaf categories via coend and end formulas, reducing the existence of a canonical envelope to a universal interpolation problem. We show that the interpolation problem is solvable in several classical settings, including: ind- and pro-completions cite{gabriel1971lokal}, Cauchy completions cite{lawvere1973metric}, Pratt's communes cite{pratt2010communes}, Isbell envelopes cite{isbell1960adequate}, and topological completions (Stone-Čech compactification, sobrification). We establish a structural correspondence with Pratt's commune theory for identity pairings, and compare structurally with Schoots's categorical canonical extensions cite{schoots2015generalising} and classical canonical extensions of distributive lattices cite{jonsson1951boolean,gehrke2001bounded}.
Category: General Mathematics
[2] viXra:2508.0047 [pdf] submitted on 2025-08-07 22:05:18
Authors: Abdelhay Benmoussa
Comments: 6 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In 2009, Tunisian media celebrated a 19-year-old student, Karim Ghariani, for proposing a method—referred to as emph{Karimation}—that claimed to simplify the direct computation of Bernoulli numbers. Despite local acclaim, his approach, archived on platforms such as Wikiversity but never formally peer-reviewed, contains gaps and minor errors. This paper revisits Karim's main integral formula involving Bernoulli polynomials and Stirling numbers, identifies a critical flaw in differentiating under an integral with a fixed upper bound, and provides a rigorous correction by extending the integral to a continuous upper limit. We conclude that while the original method does not present fundamentally new results, the episode highlights the importance of mathematical rigor and peer review, as well as the value of encouraging youthful mathematical curiosity.
Category: General Mathematics
[1] viXra:2508.0032 [pdf] submitted on 2025-08-06 21:41:03
Authors: Izzie Boxen
Comments: 12 Pages.
Here, we generalize the concept and notation of repetends, develop an algebra of rules for manipulation, and give two examples of how these can be used in mathematics.
Category: General Mathematics
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