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Any replacements are listed farther down
[3437] viXra:2603.0033 [pdf] submitted on 2026-03-06 21:47:45
Authors: Abdelmajid Ben Hadj Salem
Comments: 5 Pages. In French
A new approach finds solutions to fluid dynamics equations and could be applied to solve one of the Millennium Prize Problems.
Category: General Mathematics
[3436] viXra:2603.0001 [pdf] submitted on 2026-03-01 01:16:32
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 27 Pages. 9 figures
This paper is a continuation of our previous paper viXra:2512.0150, in which we expressed all vectors of the 3D Euclidean vector space with operations involving rotations and homotheties of the unit vector of the x-axis. We defined also a new vector multiplication and in this paper we will show that with the usual vector addition, we have a K-algebra. We will also study the 4D case and we will also express all vectors with simple rotations and homotheties of the unit vector of the x-axis about 3 planes, we will define a vector multiplication analogous to what we have seen in the 3D case, and we will show that we have again a K-algebra with the usual vector addition. We will construct also a 4D fractal set which contains the 3D fractal set (which contains the Mandelbrot set) seen in our previous paper and we will show several 4D projections of that set in the 3D space. We will show also that there are no quaternion numbers, because there are actually geometrical operations involving the unit vector of the x-axis. In that case we will use compositions of 2 rotations of the unit vector of the x-axis about different planes to express all the vectors. In that case we will have also a division algebra. Finally, we will show that for all dimensions, we have K-algebras if we express all vectors with operations involving homotheties and simple rotations of the unit vector of the first axis (x_1), with the usual vector addition and a generalization of the multiplication that we have seen in our previous papers for the 2D and 3D cases and for the 4D case in this paper. One advantage of the simple rotations, among other ones, is that they allow to construct interesting n-dimensional fractal sets linked to the Mandelbrot set.
Category: General Mathematics
[3435] viXra:2602.0102 [pdf] submitted on 2026-02-19 20:14:58
Authors: Tajmul Khan
Comments: 13 Pages. (Note by viXra Admin: Please cite listed scientific references)
We present a new analytic framework for generating infinite series representations of pi. Using the digamma reflection formula and a modulus-q expansion strategy, we construct families of convergent pi-series. In particular, we derive a novel modulus-5 weighted series (Khan's pi-series), which converges absolutely and exhibits faster convergence than the classical Leibniz series. The approach generalizes naturally to arbitrary moduli q >= 3, yielding entire families of pi-series with adjustable convergence behaviour.
Category: General Mathematics
[3434] viXra:2602.0091 [pdf] submitted on 2026-02-18 19:47:27
Authors: Binay Krishna Maity
Comments: 3 Pages.
We have two straight lines in a graph. We need to determine if these straight lines will meet each other or remain parallel if we extend these lines. This paper helps us to determine this question. If we square a certain type of polynomial (x^pn - x^p(n-1) -....... -x^p1-1) and take the coefficient of x of this square on the x axis and the power of x on the y axis and if we make the graph, the spectra that will be created, consider the initial and final part of the spectra as two straight lines, then those two straight lines will meet each other or be parallel, it will depend on the n and p of this polynomial. That is, on the length and step of the polynomial.
Category: General Mathematics
[3433] viXra:2602.0085 [pdf] submitted on 2026-02-17 00:31:42
Authors: Binay Krishna Maity
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references!)
A very common term in mathematics is a^2 + b^2 = c^2. This equation has been discussed since many years ago. This equation is called the Pythagorean Theorem. And for a, b and c as positive inintegers a, b and c are called Pythagorean triples. For example, 3, 4, 5 is a Pythagorean triple. Because, 3^2+ 4^2 = 5^2. Some more examples are (5,12,13), (9,12,15), and (12,16,20) etc. Any number has more than one Pythagorean triple. For example, with 12 numbers (5,12,13), (9,12,15), (12,16,20) and (12, 35, 37) these four Pythagorean triples are obtained. Now I am given a positive integer number and I have to show how many Pythagorean triples can be found with that number? There may have some solutions in the mathematics. This paper provides an alternative solution to evaluate these Pythagorean triplets for any given number.
Category: General Mathematics
[3432] viXra:2602.0084 [pdf] submitted on 2026-02-17 00:24:20
Authors: Arthur Shevenyonov
Comments: 6 Pages.
The present paper proposes a most parsimonious scheme to arrive at solutions for polynomial algebraic (or ODE) equations of an arbitrary degree (order). The former option treats the polynomial as an implied characteristic of a difference/recurrent equation (dubbed an AlD, or algebraic-to-difference path). The latter (i.e. ODE) domain, rather than building on the more trivial differential-difference parallelism, embarks on first reducing ODE to algebraic (while making use of, say, Mikusinski-style E-operators) then applying the above procedure (what amounts to a DARF/DARE, or differential to algebraic to recurrent/functional equation path).
Category: General Mathematics
[3431] viXra:2602.0082 [pdf] submitted on 2026-02-15 10:06:26
Authors: Mieczyslaw Szyszkowicz
Comments: 5 Pages.
Archimedes used the perimeter of inscribed and circumscribed regular polygons to obtain lower and upper bounds of the number pi. He started with two regular hexagons and he doubled their sides from 6 to 12, 24, 48, until 96. Applying the perimeters of 96 side regular polygons, Archimedes obtained the bounds for the number pi: 3+10/71<pi<3+1/7. His algorithm can be executed as a recurrence formula called the Borchardt-Pfaff-Schwab method. Dörrie proposed an improvement of this algorithm to produce narrower interval which encapsulates pi. Here a linear combination of the bounds is realized to obtain an improved accuracy. Many other linear combinations are presented to approximate this mathematical constant.
Category: General Mathematics
[3430] viXra:2602.0052 [pdf] submitted on 2026-02-07 01:20:20
Authors: Felix M Lev
Comments: 17 Pages.
As shown by Gödel and other mathematicians, foundational problems of classical mathematics (CM) arise because this theory involves the entire infinite set of natural numbers. Therefore, CM must be modified in some way. A problem discussed in a wide literature is how mathematics should be treated: (1) as a purely abstract discipline, independent of nature; or (2) as a discipline that must ultimately describe nature. Most physicists accept only viewpoint (2), while many mathematicians and philosophers adopt viewpoint (1). However, currently approach (1) did not solve the problem of how CM should be modified, and quantum theory (QT) is considered to be the most general theory for describing nature. Therefore, CM must be modified so that it correctly describes QT. As shown in our publications, finite mathematics (FM) satisfies this condition. It involves a finite ring $R_p=(0, 1, ...p-1)$ where addition, subtraction, and multiplication are performed modulo $p$. FM does not contain any foundational problems and is a more general theory than CM: the latter is a degenerate special case of the former in the limit $ptoinfty$. The purpose of this paper is to provide a brief overview of our results to make them understandable to a wide audience of mathematicians and physicists.
Category: General Mathematics
[3429] viXra:2601.0138 [pdf] submitted on 2026-01-30 01:04:02
Authors: Edgar Valdebenito
Comments: 4 Pages.
This note is about a specific value of Lambert's W function.
Category: General Mathematics
[3428] viXra:2601.0126 [pdf] submitted on 2026-01-26 08:26:11
Authors: Zhi Li, Hua Li
Comments: 6 Pages.
In mathematics, real numbers can be represented by points on a straight line called the number line, which includes a point called the origin, the direction of number growth, and a unit length. It is generally assumed that there is a one-to-one correspondence between real numbers and points on the number line, with the position of a point determining the size and order of the numbers. This essentially assumes that all real numbers have a definite position on the number line, and that there is a definite order between any two real numbers.This paper shows that there are real numbers with uncertain positions, and that all real numbers do not lie on the number line of the same dimension. The number line is composed of discrete points, which are "pure numbers"—that is, only pure numbers exist on the number line, while non-pure numbers exist in "empty space." Therefore, there is a logical contradiction between the continuity of real numbers and the real number line; the real number line is an incomplete and imperfect conception for representing real numbers.These results verify the viewpoint of quantum theory in physics, namely that the straight line on the "macroscopic" number line is composed of "microscopic" discrete and discontinuous points.
Category: General Mathematics
[3427] viXra:2601.0101 [pdf] submitted on 2026-01-22 21:26:54
Authors: Juan Moreno Borrallo
Comments: 19 Pages. (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org!)
The arithmetic of the integers is governed by two fundamental operations, addition and multiplication, whose interaction lies at the core of many deep problems in number theory. While multiplication preserves prime factorization in a rigid and conservative manner, addition typically destroys multiplicative structure and generates new prime content.In this work, we develop a unified structural framework that explains this asymmetry through spectral and operator-theoretic principles. By embedding the integers into a Hilbert space, we show that multiplication acts as a diagonal, layer-preserving operator in the prime spectral basis, whereas addition acts as a non-local, mixing operator driven by carry propagation. This spectral incompatibility leads to an arithmetic uncertainty principle, forbidding simultaneous localization in additive and multiplicative bases.Building on this structure, we introduce additive innovation as a quantitative measure of the new prime information created by a sum. We prove that the only obstruction to innovation arises from smoothness and $S$-unit phenomena in the coprime core. Using classical results on smooth numbers, we show that additive innovation is typically large, yielding unconditional abc-type inequalities in density.Finally, we develop an information-theoretic perspective, showing that addition produces entropy across prime scales while multiplication remains information-preserving. These results provide a structural explanation for the sum-product phenomenon and reframe classical problems as manifestations of the intrinsic incompatibility between additive and multiplicative spectral structures.
Category: General Mathematics
[3426] viXra:2601.0096 [pdf] submitted on 2026-01-22 21:20:43
Authors: Mattia Furlin
Comments: 5 Pages.
In this short article, we will discuss a card game, from now on namely Solitaire modulo 3. After having described how it works, through a probabilistic calculation, we will arrive at determining the probability of victory. In particular, we will use the rook polynomials, which will allow us to finally obtain a closed form for calculating the probability of winning at Solitaire modulo 3. Finally, we will study the case where the number of cards in play is much more greater than the number of constraints present in the game format. Under this assumption, the Solitaire modulo 3 mechanism becomes asymptotically equivalent to a binomial distribution.
Category: General Mathematics
[3425] viXra:2512.0150 [pdf] submitted on 2025-12-31 17:55:15
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 15 pages, 9 figures
In this paper we will see that each vector of the 3D Euclidean vector space can be expressed with operations involving rotations of the unit vector of the x-axis. Thanks to that, we will define a new multiplication between vectors which is analogue to what we have seen in our previous paper viXra:2510.0152 without complex numbers. This operation will allow us to construct a 3D fractal set which contains the Mandelbrot set in the planes OXY and OXZ. We will show some cross sections of other parts of that 3D fractal set.
Category: General Mathematics
[3424] viXra:2512.0133 [pdf] submitted on 2025-12-27 23:30:01
Authors: Marciano L. Legarde
Comments: 28 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In standard calculus, many functions do not admit antiderivatives that can be written using elementary functions. Classical results in mathematics show that no algebraic manipulation can overcome this limitation. However, the absence of an elementary antiderivative does not mean that such integrals are without structure or representation. This work introduces a new symbolic approach for handling non-elementary integrals through the definition of a puncture operator. Rather than attempting to force an elementary closed form, the puncture operator compresses the infinite summation structure that naturally arises in these integrals into a single, well-defined symbolic object. This object fully encodes the antiderivative while avoiding the need to explicitly display long or impractical infinite series. The puncture operator is constructed explicitly and is shown to preserve convergence, remain invariant under partition refinement, and provide a canonical representation of series-based antiderivatives. The method is demonstrated in detail on the non-elementary integral ∫ �� ���� where the infinite expansion of the antiderivative is compressed into a compact symbolic form without loss of information. This framework does not contradict known impossibility results in symbolic integration. Instead, it offers a complementary perspective in which non-elementary antiderivatives are treated as structured symbolic objects rather than unsimplifiable expressions. The approach provides a new way to represent and manipulate integrals that lie beyond the reach of elementary calculus.
Category: General Mathematics
[3423] viXra:2511.0144 [pdf] submitted on 2025-11-30 01:57:00
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 18 Pages. 1 figure
In this paper we are going to study the polynomials whose arguments and coefficients are vectors in the Euclidean vector space together with the new operations defined in our previous paper viXra:2510.0152. In order to prove the Fundamental Theorem of Algebra with topological tools, we are going to define the limits and derivatives with respect to vectors. We are going to represent some values of the polynomials thanks to paths in the plane. We will see that for the partial sums of the Taylor development linked to the exponential function, we get spiral paths leading to the unit circle. We are going to find the zeros/roots and we are going to present a new formulation of the Fundamental Theorem of Algebra, in the Euclidean vector space, with its meaning linked to paths in the plane. We are going to adapt a proof made by Laurent Schwartz with complex numbers. We are going to present also an adaptation of the algorithm of Kneser in order to find the roots. We will show that the Fundamental Theorem of Algebra is definitely geometrical. We will give also a link to a code source for GNU Octave for experiments with operations and polynomials in this framework.
Category: General Mathematics
[3422] viXra:2511.0091 [pdf] submitted on 2025-11-18 09:46:11
Authors: Marko V. Jankovic
Comments: 5 Pages.
n this paper Galileo's paradox is going to be analyzed. It will be shown that there is no paradox, but rather animprecise analysis in which very big numbers were not treated with the utmost care. A new method for comparison ofthe sizes of infinite sets will be introduced
Category: General Mathematics
[3421] 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
[3420] 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
[3419] 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
[3418] viXra:2510.0058 [pdf] submitted on 2025-10-12 18:12:59
Authors: Felix M Lev
Comments: 12 Pages.
As shown in the works of Gödel and others, standard mathematics has foundational problems because it invokes the infinite ring of integers $Z$. At the same time, finite mathematics proceeds from a finite ring $R_p=(0,1,2,...p-1)$ of residues modulo $p$. We give a new and simple proof that $Z$ is the limit of $R_p$ when $ptoinfty$. The proof involves only potential infinity and not actual infinity. We explain why the proof plays an important role in the foundations of mathematics and physics.
Category: General Mathematics
[3417] 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
[3416] viXra:2510.0024 [pdf] submitted on 2025-10-05 14:04:46
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 the basics of the theory explained, followed by exponentials, logarithms and more trigonometry, and with the derivatives and integrals discussed too. Finally, we provide an introduction to vector calculus, space geometry and basic mechanics.
Category: General Mathematics
[3415] 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
[3414] 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
[3413] 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
[3412] viXra:2509.0126 [pdf] submitted on 2025-09-23 17:24:26
Authors: Amit Kumar
Comments: 16 Pages. Appendix and supplementary material can be obtained upon the request.
The concept of combination is crucial to layout groundwork for many fundamental sciences, finance and technical domains. There are multiple known methods to show that the combination (binomial coefficient) is integral in nature. This article approaches the same concept with simple non-inductive algebraic steps that can be followed by a wide range of audiences. This article separates an ${n choose k}$ into mutually exclusive and collectively exhaustive cases and then shows for each case that any term in a denominator have at least one corresponding multiple in the numerator. All k for a particular n (for an ${n choose k}$) are visualized through plots to gain better understanding. In the context of the plots, this different way of looking at a Combination ${n choose k}$, gives hindsights into the generation process of factors and prime numbers contained within a particular range of 1 to n.
Category: General Mathematics
[3411] viXra:2509.0113 [pdf] submitted on 2025-09-19 18:16:03
Authors: Lucian M. Ionescu
Comments: 21 Pages. Presentation at Illinois State University Pure and Applied Mathematics Seminar Sept. 18 2025.
Babylonian Algorithm for Square Roots is a precursor of Newton-Cotes numerical method for finding roots, as numerical approximations(perturbative method)."What is a number?" has two complementary aspects: axiomatic viewpoint and as a tool in applications, via mathematical models.Modern 21-st century Mathematics regards Real Numbers as part of Complex Dynamics, in terms of Newton flow, Fatou and Julia sets and fractals, pointing towards their role as trajectories in Dynamics. A brief account is provided.Their relation to Number Fields, theory resulting from the work of Galois and Artin, starts to be apparent via Dirichlet Unit Theorem and the concept of uniformizer (Power Series/Diff. Eq.), extending the correspondence between Number Fields and Function Fields of Algebraic Curves.
Category: General Mathematics
[3410] viXra:2509.0105 [pdf] submitted on 2025-09-17 16:40:03
Authors: Hiroshi Okumura
Comments: 22 Pages. 1 Figure (Note by viXra Admin: Further repetition may not be accepted)
We give several pairs of congruent circles arising from a generalized arbelos. A special case, in which the congruent circles are lines, is also considered using division by zero 1/0=0.
Category: General Mathematics
[3409] viXra:2509.0036 [pdf] submitted on 2025-09-05 05:12:38
Authors: Martín Alejandro Monzón Marimón
Comments: 1 Page.
The structure of First Order logic will be described here in its verbal, or explicit (Second Order) form, by means of an "apparent" Zeroth Order form, because the actual pure Second Order form is not axiomatizable. Also, another "apparent" Second Order form will be used to make a Zeroth Order description of the structure, which, without the "apparent" Second Order quantification, would remain silent, or implicit, as that is its pure Zeroth Order form.
Category: General Mathematics
[3408] viXra:2508.0176 [pdf] submitted on 2025-08-30 20:44:45
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 14 pages, 4 figures
In this paper we will see that we can avoid the concepts of negative number and complexnumber thanks to the study of the underlying vector nature of some arithmetic and polynomialproblems. We will see that the geometrical models used until now to represent negative numbersand 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 whatwe call "negative numbers" and "complex numbers" are just the solutions of vector calculationsand equations. All that is the consequence of the fact that geometrical considerations areunavoidable when we think about debts and gains and when we try to solve some polynomialequations. We will see that thanks to the solutions of those vector equations we can constructpaths in the plane. We will also give the vector meaning of the formulas of De Moivre andEuler. An interpretation of the vertical axis linked to gains and losses will also be given.
Category: General Mathematics
[3407] 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
[3406] 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
[3405] 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
[3404] 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
[3403] viXra:2508.0069 [pdf] submitted on 2025-08-11 00:56:15
Authors: Abdelhay Benmoussa
Comments: 5 Pages.
This paper presents the 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 generating function of Bernoulli polynomials and using known relations with falling factorials, we derive the closed-form expression systematically.
Category: General Mathematics
[3402] viXra:2508.0058 [pdf] submitted on 2025-08-09 03:09:23
Authors: Robert A. Rice
Comments: 49 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This work introduces canonical envelope theory as a mathematical framework for understanding completion phenomena in category theory and related mathematical disciplines. Buildingon Shulman’s and Riehl’s characterization of weighted limits through natural transformationsθ : Q ⇒ C(D, E) [32; 33], we demonstrate that numerous completion constructions—includingcanonical extensions, topological compactifications, categorical completions, and constructions in algebraic geometry—can be understood as instances of factorization through virtual weighted (co)limiting structures.Our main theoretical contribution identifies canonical envelopes as initial objects in categories of factorizations of appropriately constructed pairings. The (heuristic) classification θ = id versus θ ̸= id distinguishes internal completion from external mediation. We establish existence criteria through bilateral denseness and compactness conditions, providing systematic construction procedures for a range of mathematical contexts.The framework encompasses several major completion constructions through classificationtables. Virtual weighted limits extend Gabriel-Ulmer methodology [9] from filtered/cofiltereddiagrams to arbitrary weights, enabling systematic treatment of incomplete categorical frameworks. We introduce and develop "gem theory" as a systematic classification of mathematical structures according to their bilateral completion properties. Key results include the pullback characterization showing that canonical interpolants are categorically determined, bilateral envelope structure capturing fundamental duality patterns, andsystematic organization through a canonical envelope pseudomonad. The theory suggests deepcategorical principles underlying mathematical completion while providing practical methodology for discovering canonical constructions in various mathematical contexts.
Category: General Mathematics
[3401] 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
[3400] 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
[3399] viXra:2507.0222 [pdf] submitted on 2025-07-31 15:11:19
Authors: Jaykov Foukzon
Comments: 34 Pages.
In this paper generalized Chapman-Kolmogorov equation is derived.
Category: General Mathematics
[3398] viXra:2507.0219 [pdf] submitted on 2025-07-31 19:58:08
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 5 Pages. 4 figures.
This paper presents an explanation of the origins of the sexagesimal system linked to theuse of a knotted cord 12 lengths long closed to form a loop. We will see that this device hasalso important geometrical properties linked to the division of the circle in 360 equal parts.Historical and cultural arguments will also be given to support this thesis.
Category: General Mathematics
[3397] viXra:2507.0188 [pdf] submitted on 2025-07-26 01:23:01
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 3 Pages.
This paper presents a simple relationship between Golden Ratio φ and the length of the arcfrom x = 0 to x = 1 of the curve defined by the equation y = x².
Category: General Mathematics
[3396] viXra:2507.0160 [pdf] submitted on 2025-07-22 19:36:52
Authors: Edgar Valdebenito
Comments: 4 Pages.
Recently, K. Adegoke et al.(arXiv:2505.11575v1,16 May 2025) established a number of results on binomial series. In this note, we explore some particular cases of these series.
Category: General Mathematics
[3395] viXra:2507.0125 [pdf] submitted on 2025-07-17 20:03:13
Authors: Kleschev Anton Alevtinowitch
Comments: 2 Pages.
We present U4, a 4-symbol axiom that encodes the entire birth—dissolution cycle of natural- number arithmetic. The equation T0=2 simultaneously asserts (i) the presence of the zero- point, (ii) the immediate emergence of one from zero, (iii) the automatic collapse of one into two, and (iv) the ultimate identity 0=1=2. No external primitives are presupposed.
Category: General Mathematics
[3394] viXra:2507.0108 [pdf] submitted on 2025-07-15 19:10:04
Authors: Kleschev Anton Alevtinowitch
Comments: 2 Pages.
We present UF0(Unified Foundation 0), a minimal formal system in which the entire edificeof natural-number arithmetic is generated and dissolved by a single equation. u2022 0 is the pre-numerical, non-dual point. u2022 τ is a unary "tear" operator enacting the primal split. u2022 ⊕ is a non-repeating union. The sole axiom τ (0) ⊕ 0 = τ (1) ⊕ 1 simultaneously encodes 1. the birth of mathematics (0 → 1), and 2. the fundamental duality (1 → 2), while collapsing both into identity when 0 = 1 = 2. No external arithmetic, set theory, or category theory is presupposed. UF0 is therefore the first fully formal, one-sentence foundation that captures both the emergence and the illusoriness of number.
Category: General Mathematics
[3393] viXra:2507.0062 [pdf] submitted on 2025-07-08 12:45:51
Authors: Thierry L.A. Periat
Comments: 40 Pages.
A previous document laid the foundations for the study of involution in any three-dimensional spaces, [c]. This exploration continues the exploration of the topic in focusing now attention on four-dimensional spaces. The repetition of a deformed Lie product on a given argument carries two concepts with it: (i) the eventual invariance of this argument and (ii) the existence of an involution. The work discovers two distinct classes of decomposition without residual part for each deformed Lie product. It explains why only one of both (the simplest one) can characterize the involution when the deforming cube is anti-symmetric and anti-reduced. The document also starts a confrontation between the simplest representation and the electromagnetic duality in Maxwell's vacuum.
Category: General Mathematics
[3392] viXra:2507.0048 [pdf] submitted on 2025-07-06 21:11:22
Authors: Oliver Couto
Comments: 4 Pages.
In math literature,(eg: ref. #3), most solutions to the above equations are dealt with using elliptic curve theory. In this paper the author has, by imposing certain conditions, provided a parametric solution of the equation by using Algebra only.
Category: General Mathematics
[3391] viXra:2507.0042 [pdf] submitted on 2025-07-05 17:58:18
Authors: Mirzakhmet Syzdykov
Comments: 2 Pages.
After making the number of sufficient and successful experiments on account of "P versus NP" theorem, specifically according to the equivalence of complexity classes, we are giving the final and formal theoretical proof by contradiction in this paper summarizing all the results before and giving the definition of our functional hypothesis or conjecture.
Category: General Mathematics
[3390] viXra:2507.0024 [pdf] submitted on 2025-07-03 21:20:01
Authors: Somdeb Lahiri
Comments: 5 Pages.
We consider a sub-class of bi-matrix games which we refer to as two-person (hereafter referred to as two-player) additively-separable sum (TPASS) games, where the sum of the pay-offs of the two players is additively separable. The row player’s pay-off at each pair of pure strategies, is the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the column player and the second being independent of the pure strategy chosen by the column player. The column player’s pay-off at each pair of pure strategies, is also the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the row player and the second being independent of the pure strategy chosen by the row player. The sum of the inter-dependent components of the pay-offs of the two players is assumed to be zero. We show that a (randomized or mixed) strategy pair is an equilibrium of the game if and only if there exist two other real numbers such that the three together solve a certain linear programming problem. Using, theorem 9 in a document prepared by Chandrasekaran and theorem 2 in a 2025 paper by Lahiri, we show that every solution of a certain bi-linear programming problem, provides an equilibrium for the TPASS game.
Category: General Mathematics
[3389] 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
[3388] 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
[3387] 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
[3386] 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
[3385] 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
[3384] 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
[3383] viXra:2506.0004 [pdf] submitted on 2025-06-02 19:34:05
Authors: Ciro Cesarano
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
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 or Riemann rearrangement theorem.
Category: General Mathematics
[3382] viXra:2505.0178 [pdf] submitted on 2025-05-27 03:23:03
Authors: Martín Alejandro Monzón Marimón
Comments: 1 Page. (Note by viXra.org Admin: Please cite and list scientific references)
There are four possible axiomatizations of the Zeroth Order Categorical theory, which are the fourpossible forms in which the implication operator can be constructed (they are implicit). Analogously, there are four possible axiomatizations of the Second Order Categorical theory (the explicit form of the Zeroth Order Categorical theory). They are presented here. Only one is known to date with rigor (Peano axioms + Hereditarily Finite Sets axioms).
Category: General Mathematics
[3381] viXra:2505.0146 [pdf] submitted on 2025-05-21 19:46:49
Authors: Samel Datu Castelo
Comments: 1 Page. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
This paper presents two mathematical formulas involving differentiation and summation. While the derivations do not strictly follow conventional mathematical rigor and certain foundational rules are bypassed, the formulas demonstrate surprisingly consistent results. A full formal proof is currently under construction.
Category: General Mathematics
[3380] viXra:2505.0143 [pdf] submitted on 2025-05-21 18:21:58
Authors: Edgar Valdebenito
Comments: 3 Pages.
In this note, we consider an integral of the Ising class.
Category: General Mathematics
[292] viXra:2512.0150 [pdf] replaced on 2026-01-23 21:01:17
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 17 Pages. 11 figures
In this paper we will see that each vector of the 3D Euclidean vector space can be expressed with operations involving rotations of the unit vector of the x-axis. Thanks to that, we will define a new multiplication between vectors which is analogous to what we have seen in our previous paper viXra:2510.0152 without complex numbers. This operation will allow us to construct a 3D fractal set which contains the Mandelbrot set in the planes OXY and OXZ. We will show some cross sections of other parts of that 3D fractal set.
Category: General Mathematics
[291] 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
[290] viXra:2510.0058 [pdf] replaced on 2025-12-22 19:48:47
Authors: Felix M Lev
Comments: 13 Pages. Revised version, title changed.
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
[289] viXra:2510.0058 [pdf] replaced on 2025-12-02 05:34:45
Authors: Felix M. Lev
Comments: 13 Pages. Considerably revised version
A typical situation in physics is as follows. There are two theories, ${cal A}$ and ${cal B}$, ${cal A}$ contains a nonzero parameter, and ${cal B}$ can be considered the limit of ${cal A}$ as the parameter goes to zero or infinity. In our publications, we consider when ${cal A}$ is more general than ${cal B}$ and ${cal B}$ is a degenerate case of ${cal A}$. Well known examples: relativistic theory is more general than non-relativistic theory, quantum theory is more general than classical theory and de Sitter-invariant theory is more general than Poincare-invariant theory. In this short review we argue that the situation in mathematics is similar. Classical mathematics (CM) invokes the infinite ring of integers $Z$ while finite mathematics (FM) proceeds from a finite ring $R_p=(0,1,2,...p-1)$ of residues modulo $p$. CM has foundational problems (as follows from Gödel's incompleteness theorems) while FM does not have such problems. All approaches to construct a quantum theory of gravity within the CM framework have proven unsuccessful because they cannot avoid irremovable divergencies. Also, the very fact of the existence of elementary particles shows that there are no infinitesimals in nature. Nevertheless, it is usually believed that CM is a fundamental theory, while FM is a theory useful only in some models. However, we argue that the situation is the opposite: although many strong and beautiful results have been obtained in CM, FM is a more general theory than CM and CM is a degenerate case of FM at $ptoinfty$. The main points of the proof are as follows: $R_pto Z$ when $ptoinfty$ and the proof of this fact involves only potential infinity and not actual infinity; quantum theory based on FM is more general than quantum theory based on CM.
Category: General Mathematics
[288] viXra:2510.0058 [pdf] replaced on 2025-10-27 23:58:07
Authors: Felix M. Lev
Comments: 13 Pages. In the beginning of the paper I now describe the difference between potential infinity and actual infinity. This point is crucial for understanding the results of the paper.
As shown in the works of Gödel and others, standard mathematics has foundational problems because it invokes the infinite ring of integers $Z$. At the same time, finite mathematics proceeds from a finite ring $R_p=(0,1,2,...p-1)$ of residues modulo $p$. We give a new and simple proof that $Z$ is the limit of $R_p$ when $ptoinfty$. The proof involves only potential infinity and not actual infinity. We explain why the proof plays an important role in the foundations of mathematics and physics.
Category: General Mathematics
[287] 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
[286] 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
[285] 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
[284] viXra:2507.0219 [pdf] replaced on 2025-09-27 21:30:23
Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 7 Pages. 5 figures
This paper presents an explanation of the origins of the sexagesimal system linked to theuse of a knotted cord 12 lengths long closed to form a loop. This device can suggest a groupingof elements by 60. We will see that this device has also important geometrical properties linkedto the division of the circle in 360 equal parts. Historical and cultural arguments will also begiven to support this thesis. We will see how the grouping of the Old Mesopotamian unitsof measurement of capacity, weight, mass and time confirm this thesis. We will see also howimportant groupings of the Old Mesopotamian units of measurement of length and area use thenumber 12 which is also suggested by this device.
Category: General Mathematics
[283] 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
[282] viXra:2506.0004 [pdf] replaced on 2025-07-01 22:30:38
Authors: Ciro Cesarano
Comments: 8 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 or Riemann rearrangement theorem.
Category: General Mathematics
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