| viXra.org > General Mathematics > 2510 |
 |
 |
| viXra.org > General Mathematics > 2510 |
|
|
[9] viXra:2510.0152 [pdf] submitted on 2025-10-31 18:33:29
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 16 pages, 6 figures
This paper is a continuation of viXra:2508.0176, in which we saw that we can avoid theconcepts of negative number and complex number thanks to the study of the underlying vectornature of some arithmetic and polynomial problems. With the solutions of the polynomialequations which were actually geometrical, in the Euclidean vector space, we will constructseveral operations which are analogous to what we have seen until now with "complex numbers".We will show also the representations of functions whose arguments are vectors. We will see thebasic elements needed in order to rebuild all what has been constructed in complex analysis.We will show also that we can construct the Mandelbrot set in the Euclidean vector space.
Category: General Mathematics
[8] viXra:2510.0107 [pdf] submitted on 2025-10-22 07:31:19
Authors: Zhi Li, Hua Li
Comments: 15 Pages.
Finding the roots of polynomial equations is a fundamental problem inmathematics. This paper discovers that general polynomial equations can be simplifiedinto a canonical or standard form through Tschirnhaus transformations. A power seriesrepresentation consisting of coefficients in the canonical or standard form is a universalrepresentation of the roots of polynomial equations. If the series converges, a root of theequation is obtained. If the series does not converge, it can be further transformedthrough one or more Tschirnhaus transformations to obtain a convergent seriesrepresentation. This method is applicable to higher degree polynomial equations withreal and complex coefficients, avoiding the complex determination of whether they aresolvable in the radicals , and has universal significance. This advance returns theproblem of finding polynomial roots to the realm of pure algebra, using only polynomialtransformations and multivariable power series.
Category: General Mathematics
[7] viXra:2510.0076 [pdf] submitted on 2025-10-14 08:46:26
Authors: Marciano L. Legarde
Comments: 7 Pages.
This study explores the Antiderivative Power Rule Sequence, demonstrating how its infinite series leads to the polylogarithm. By iteratively applying the power rule for antiderivatives to successive powers of x, we derive the sequence, which, when expressed as an infinite series, converges to -ln(1-x). Differentiating the resulting series recovers the geometric series, highlighting a profound inverse relationship between 1/(1-x) and -ln(1-x). Furthermore, this formulation establishes a natural connection to the polylogarithm function, generalizing the relationship for higher orders of integration. This work provides both pedagogical and theoretical insights, reconstructing a transcendental function from elementary calculus operations.
Category: General Mathematics
[6] viXra:2510.0058 [pdf] replaced on 2026-01-14 22:16:14
Authors: Felix M. Lev
Comments: 14 Pages. Relationship between the foundation of mathematics and quantum theory is discussed in more details.
A common situation in physics involves two theories, ${cal A}$ and ${cal B}$, where ${cal A}$ contains a nonzero parameter, and ${cal B}$ arises as a limit of ${cal A}$ as this parameter approaches zero or infinity. In such cases, ${cal A}$ is more general and ${cal B}$ is a degenerate case of ${cal A}$. Well-known examples include relativistic theory being more general than non-relativistic theory and quantum theory being more general than classical theory. In this short review we argue that an analogous situation holds in mathematics. Classical mathematics (CM) is based on the infinite ring of integers $Z$, whereas finite mathematics (FM) is based on the finite ring $R_p=(0,1,2,...p-1)$ of residues modulo $p$. CM has foundational difficulties (as highlighted by Gödel's incompleteness theorems) while FM does not. All attempts to construct a quantum theory of gravity within CM encounter unavoidable divergencies. The existence of elementary particles also suggests that infinitesimals do not exist in nature. Despite this, CM is usually regarded as fundamental theory, while FM merely as a tool useful only in some models. We argue instead that FM is the more general theory, with CM appearing as its degenerate limit as $ptoinfty$. The key points are: $R_pto Z$ as $ptoinfty$, and this can be proved using only potential (not actual) infinity; quantum theory based on FM is more general than quantum theory based on CM.
Category: General Mathematics
[5] viXra:2510.0037 [pdf] submitted on 2025-10-07 18:37:53
Authors: David Park
Comments: 13 Pages.
The Van der Pol oscillator is a nonlinear system known for its self-sustaining oscillation and behavior. This paper analyzes how the system evolves as the damping parameter μ changes, focusing on equilibrium points, phase plane trajectories, and limit cycles. Throughout the paper, we highlight how the equation also relate to physical systems, such as electrical circuits and biological rhythms, showing the significance and relevance of the Van der Pol oscillator in modeling real-world nonlinear behavior.
Category: General Mathematics
[4] viXra:2510.0024 [pdf] replaced on 2026-01-10 20:59:02
Authors: Teo Banica
Comments: 400 Pages.
This is an introduction to mathematics, with emphasis on geometric aspects. We first discuss numbers, counting, fractions and percentages, and their basic applications. Then we get into plane geometry, with a study of triangles and trigonometry, followed by coordinates and complex numbers. We then go into functions and analysis, with a detailed discussion of the polynomials, the basics of continuity explained, and with the derivatives and integrals discussed too. Finally, we provide an introduction to vector calculus, space geometry, linear algebra and basic mechanics.
Category: General Mathematics
[3] viXra:2510.0023 [pdf] submitted on 2025-10-05 23:49:44
Authors: Marciano L. Legarde
Comments: 2 Pages. (Note by viXra Admin: An abstract in the article is required and please cite listed scientific references)
I present two results known as the Leaf Theorems, that were initially noticed via numerical experiment and subsequently proved analytically. Each of these theorems illustrates that the disparity between rapidly oscillating and slow growth functions, and rapidly diminishing functions and disappearing power functions, respectively, result in constant, interpretable, and finite values when integrated over the unit segment. Together, these results demonstrate that contrasting mathematical behaviors may cancel in the process of integrating these functions and result in interpretable and finite quantities, and offer apparent and pedagogical demonstrations of real analysis convergence. They could prove helpful for pedagogy, for use in asymptotic analysis, and for applications in number and numerical methods and in probability, and could serve to inform and educate analysts and students in these and related fields.
Category: General Mathematics
[2] viXra:2510.0021 [pdf] submitted on 2025-10-05 23:42:08
Authors: Felipe Wescoup
Comments: 12 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper provides a practical guide for determining the optimal mixed strategy in two-player, zero-sum games. It presents a method for calculating the Nash Equilibrium by starting with the well-understood 2x2 matrix and intuitively extending the logic to 3x3 and larger NxN scenarios. The purpose is not to derive new mathematical theory, but to make the powerful concepts of game theory accessible to a wider audience, such as coaches, athletes, and business strategists. This paper is supplemented by a GitHub repository containing a spreadsheet tool that performs the calculations, allowing for direct practical application of the concepts discussed.
Category: General Mathematics
[1] viXra:2510.0012 [pdf] submitted on 2025-10-04 16:27:50
Authors: M.S. Petrovskaya
Comments: 42 Pages. Translation of Estimates of the residual members of the Hill series. Bull. ITA, IX, 4 (107). Translator: Thomas S. Ligon, orcit 0000-0002-4067-876X.
Estimates have been obtained of the residual members of the Hill series for cases where the coefficients of these series, which are series by powers of $m (m=n_0/(n_1-n_0 ),n_0,n_1$ — average movements of the sun and moon), calculated with precision of the 2nd, 3rd, 4th, 5th, 6th power of $m$. There are also new estimates of the residual members of Hill's series, based on the powers of $m^2$, considered in the paper (Lyapunov, 1954). Estimates were found for the case $|m|≤sigma (≈0.080849)$, where $sigma$ is the value of the m parameter for the moon.
Category: General Mathematics
| Third-Party Links: |
|