General Mathematics

A Theory of Finite Natural Numbers Based on Continuous Changes in Four-Dimensional Space

Authors: Dan Liu
Comments: 9 Pages.

This paper introduces a novel mathematical framework based on the assumption that the set of natural numbers is finite. By considering continuous changes in four-dimensional space, we redefine the concepts of natural numbers and multidimensional spaces, establish new mapping relations, and explore the implications of this hypothesis for Gödel's Incompleteness Theorem.
Category: General Mathematics

Cooperative Neighboring Numbers (Pascal's Triangle — Another View)

Authors: Marko V. Jankovic
Comments: 5 Pages.

In this paper, a modification and a generalization of the idea that was used for the creation of Pascal's triangle, is proposed. The proposed method is based on cooperative neighboring numbers that reside on the edges, diagonals and vertices of regular polygons. Cooperative strategy represents creation of the new number using addition.
Category: General Mathematics

Axiom of Infinite Cycles with Vector Density: A New Mathematical Framework

Authors: Alexander K. Shakhov
Comments: 5 Pages.

This paper introduces a novel mathematical concept - the Axiom of Infinite Cycles with Vector Density. The axiom presents a fundamental mathematical model describing cyclic processes through vector interactions and density relationships in three-dimensional space. Centered around a singular point of origin (0), the model demonstrates how cyclic numerical sequences (01987654321012345678910) interact along three primary vectors, creating a universal framework for understanding and modeling repetitive processes. The axiom establishes new principles for analyzing cyclic systems, vector interactions, and density relationships, offering applications across mathematics, physics, and computer science. This work presents both theoretical foundations and practical implementations of the concept, demonstrating its potential for various scientific applications.
Category: General Mathematics

Some Consequences of Lyapunov Regularity

Authors: Luis Gordo Barreira, Claudia Puta Fea Valls
Comments: 25 Pages.

For a non-autonomous dynamics with discrete time defined by a tempered sequence ofupper-triangular matrices, we obtain lower and upper bounds for the Grobman regularitycoefficients. We also give two applications of these results: we obtain an upper boundfor the Grobman coefficients of the exterior powers of a tempered sequence, and we givea simple proof of Oseledets' multiplicative ergodic theorem for cocycles over ameasure-preserving transformation without using Kingman's subadditive ergodic theorem
Category: General Mathematics

Computation of Antiderivatives of Rational Functions

Authors: Subrat Kumar Verma
Comments: 3 Pages.

This article has been based on a lecture handout for high school students. Most of the calculus books mention the method of partial fractionsin an algorithmic way. I have described the reason behind the method. It has been mentioned as a theorem without proof in Problems in Analysis by Prof Maron published by Mir Publications of the Soviet Era, cited in the text of the article. It is possible that the proof is also included in some book maybe in Russian language but I have not come across the reasoning in any English language book and thus is novel to the best of my knowledge. The proof uses arguments accessible to high school students who have seen polynomials and complex numbers before, such as the class I was lecturing in India.
Category: General Mathematics

Resolving The Cosmological Constant: A Conjecture for Homogeneous Infinitesimals

Authors: J. P. Baugher
Comments: 36 Pages.

The discovery in 1998 that the universe is paradoxically accelerating its expansion has led some cosmologists to question the correctness of the non-Euclidean geometric theory of gravity, General Relativity. Physically assigning the term Dark Energy to the Cosmological Constant, sometimes viewed as a constant of integration, as the source of this acceleration has only produced even more questions. In the 17th century, there was also a great paradox between two views for the geometric constituents of a line, heterogeneous (made of points) versus homogeneous (made of infinitesimal segments). Evangelista Torricelli elucidated his logical reasoning on why lines must be made of infinitesimal segments instead of points and created one particular fundamental example among many. In this paper, I produce unknown corollaries to Torricelli's argument allowing me to falsify the relationship between his infinitesimals and the Archimedean axiom, resolve L'Hopital's paradox, as well as redefine the Fundamental Theorem of Calculus, scale factor/metrics, n-spheres and Gaussian curvature. I conjecture that the intractability of Dark Energy is due to the points of coordinate systems within General Relativity actually being a logically flawed heterogeneous interpretation. I propose that Euclidean and non-Euclidean geometry, and the physics equations based upon them, should be rewritten from the perspective of homogeneous infinitesimals. I introduce the geometrical logic in this paper in order to pave the way for the physical logic.
Category: General Mathematics

Proving Irrationality of Infinite Series

Authors: Jay Pillai
Comments: 5 Pages.

A relatively concise method on proving the irrationality of a given infinite series based on a few conditions.
Category: General Mathematics

Primorial Powers Conjecture

Authors: Jay Pillai
Comments: 3 Pages.

Paper detailing a conjecture of exponent patterns found in prime numbers.
Category: General Mathematics

I Ching Hexagram Groups and Subgroups

Authors: Claude Michael Cassano
Comments: 3 Pages.

I Ching hexagram groups and subgroups Groupings and subgroups exist between the hexagrams of the I Ching The I Ching (Yijing) (Book of Changes) is an ancient Chinese divination text that is manual in the Western Zhou period (1000-750 BC). Thus, the I Ching Zhou yi originated around 5000 years ago. The Zhou yi was traditionally scribed to King Wen of Zhou and the Duke of Zhou, and also associated with the legendary Fuxi. Relationships exist between I Ching hexagram groups (and subgroups). One may wonder on the mathematical insight of the initial developer of the yinyang-trigram-hexagram system five thousand years ago!
Category: General Mathematics

Sum of Two Inverse Trigonometric Functions

Authors: Edgar Valdebenito
Comments: 2 Pages.

We give some formulas of the type: y*arcsin(x)+y*arctan(x)=pi.
Category: General Mathematics

The Empty Set Constructs the Natural Numbers

Authors: Jiang Yang
Comments: 8 Pages.

In this paper, I construct natural numbers by using empty sets, cardinality of set theory and definite operations. Based on the discussion of kernel numbers dynamic space reasoning in [1] to [4], the ruler set is introduced. And the enhanced definition of one-to-one correspondence mapping is called one-to-one correspondence ordinal mapping. And it makes the Continuum Hypothesis(CH) a new conclusion.
Category: General Mathematics

On Limit of Mathematical Analysis and Continuum Hypothesis

Authors: Jiang Yang
Comments: 24 Pages.

The limit of mathematical analysis is defined by ε- δ. A concept of dynamic limit is proposed in the article, and the dynamic space of kernel numbers is established. This concept has been extended and studied in depth, yielding several results, including setting up shell-medium cluster, dynamic limit process and steps; kernel number clouds; introducing elfin number and elfin space which the elfin number is non-construct and extend of real number; discussions on the Continuum Hypothesis (CH) what is not contradiction with new dynamic space.
Category: General Mathematics

A Simple Method for Solving Optimal Control Problems by Legendre Approximations

Authors: Mun Ju Won, Choe Yu Song, Kang Hyok Chol
Comments: 12 Pages.

In control system synthesis, the use of orthogonal functions such as Chebyshev polynomials, Lagrange polynomials, Legendre polynomials and Fourier series has recently attracted special attention.An important objective of applying these functions and polynomial sequences is to avoid the complexity as possible in considering optimal control problems and to fix the solution of algebraic equations, thus simplifying the problem consideration.In this paper, the Legendre approximation method for solving optimal control problems is proposed.Using the Gauss-Legendre quadrature method, the given integration problem is transformed into a polynomial series, and Legendre approximations for the control and state variables are performed to consider the given problem as a nonlinear programming problem.
Category: General Mathematics

A New Continued Fraction Approximation and Inequalities for the Lugo’s Constant

Authors: Kim Kyong Il, Jo Yong Hun, Ri Kwang
Comments: 13 Pages.

In this paper, we provide a new continued fraction approximation for the Lugo’s constant. Then, we derive the inequalities concerning the Lugo’s constant. Finally, we give some numerical computations to demonstrate the superiority of our new results.
Category: General Mathematics

Unscented Kalman Filtering Method Without the Matrix Square Root to Estimate the Satellite Attitude Using Magnetometer

Authors: Kuk Hyon Ham, Song Jin Kim, Jong Hyok Choe
Comments: 17 Pages.

In satellite mission, attitude control system plays an important role, and precise attitude control presents high attitude determination requirements. The TRIAD (TRIaxial Attitude Determination) method, which is widely used for satellite attitude determination, requires two sensor signals. However, when the reference vector direction to be observed in these sensors is close, the attitude determination error increases. Thus, in this case, attitude estimation is required, and the state estimator of nonlinear objects is widely used for extended Kalman and unscented Kalman filters. In this paper, we propose a method for determining satellite attitude using an unscented Kalman filter with high estimation accuracy compared to an extended Kalman filter. To reduce the amount of computation in the unscented Kalman filter and to ensure the real-time of the estimation, we use the unscented Kalman filtering method with a new sigma point selection. Compared with the traditional unscented Kalman filter, it ensures better real-time and higher accuracy.
Category: General Mathematics

Different Approaches for Proving the Pythagorean Theorem Using Trigonometry

Authors: Tathagata Biswas
Comments: 8 Pages.

Contrary to the claims by Elisha S Loomis in his famous book and popular belief, several approaches towards proving the Pythagorean theorem using trigonometry exists. These approaches essentially use trigonometric identities and concepts that can be derived independent of the identity {sin}^2x + {cos}^2x = 1, to avoid any circular reasoning. Crucial to the trigonometric approaches are the law of sines, trigonometric angle sum and difference identities and modern definitions of trigonometric functions using the power series and Euler’s formula. This article describes these trigonometric proofs of the theorem.
Category: General Mathematics

Proof of Collatz Conjecture

Authors: Zhi Li, Hua Li
Comments: 4 Pages.

Any positive integer can be expressed as k*2^n, where k is an odd number and n is a natural number. Each operation of the Collatz conjecture can be represented by (3k+1)*2^n, regardless of whether it is an odd or even number. The distribution type of k belongs to deterministic random distribution. Let 2^t be a perfect square number that is just less than 3k, and the cumulative probability value of (3k+1) being a perfect square number after each operation in the Collatz conjecture is conservatively estimated as Σ1/2^t. By comparing with the harmonic function Σ1/n, it is proved that when the number of operations gradually increases, the cumulative probability function value Σ1/2^t of (3k+1) being a perfect square number is much larger than 1, and tends to infinity when the number of operations is infinitely large. This result shows that the occurrence of (3k+1) being a perfect square is inevitable, thus proving the Collatz conjecture.
Category: General Mathematics

Finite Mathematics as the Most General (Fundamental) Mathematics

Authors: Felix M. Lev
Comments: 22 Pages. published in Symmetry vol. 16(10) paper 1340 (2024).

The purpose of this paper is to explain at the simplest possible level why finite mathematics based on a finite ring of characteristic $p$ is more general (fundamental) than standard mathematics. The belief of most mathematicians and physicists that standard mathematics is the most fundamental arose for historical reasons. However, simple {it mathematical} arguments show that standard mathematics (involving the concept of infinities) is a degenerate case of finite mathematics in the formal limit $ptoinfty$: standard mathematics arises from finite mathematics in the degenerate case when operations modulo a number are discarded. Quantum theory based on a finite ring of characteristic $p$ is more general thanstandard quantum theory because the latter is a degenerate case of the former in the formal limit $ptoinfty$.
Category: General Mathematics

A Dynamical Systems Model for Population and Depleting Resources

Authors: Samuel Forbes
Comments: 7 Pages.

We investigate a coupled, non-linear dynamical systems model for the relationship between population and depleting resources inspired by limits to growth. The model is determined by logistic growth in population with carrying capacity determined by resources. The rate of decline of resources is determined linearly by the population. The model produces an initial exponential increase in population followed by a decrease to the fixed point while congruently resources decrease in a sigmoidal fashion to the fixed point. We fit the model to world population over the period 10000 BC to 2021 in different time intervals corresponding to different growth rates. We show a number of projections to 2500 based on fitting to the time period of 1950 to 2021 with various parameter constraints.
Category: General Mathematics

Computational Complexity Analysis using Negation Negation Normal Form Circuit

Authors: Koji KOBAYASHI
Comments: 17 Pages.

概要. This paper describes about analyzing method for computational complexity using Negation Normal Form Circuit. Although Negation Normal Form Circuit can emulate Turing machines, most of the circuits are monotonic. In this paper, Negation Normal Form Circuit is further divided into a monotonic subcircuit consisting of AND and OR gate (Rating Circuit) and a subcircuit consisting of NOT elements, and the analysis focuses on the Rating Circuit. From the viewpoint of the Rating Circuit, the NOT gate is constraint on the input of the Rating Circuit. We can use another input constraints. By changing the constraints of the inputs of Rating Circuit, We can analyse complexity detail. In this paper, we use this method to analyze the complexity of the clique problem.
Category: General Mathematics

Cournot's Principle Revisited

Authors: Bruno Galvan
Comments: 7 Pages.

Cournot's principle states that a typical event (i.e., an event with probability very close to 1) occurs nearly certainly in a single trial of an experiment. This principle has been considered by various authors as the only connection between mathematical probability and the real world of experiments. To make the logical structure of the principle clearer, in this paper a reformulation of the principle is proposed. This reformulation is based on the following three elements: (1) The explicit definition of the empirical property of practical certainty, (2) the clear separation between probability measure and experiment, including the remark that typicality is a mathematical property defined by the probability measure while practical certainty is an empirical property defined by the experiment, and (3) the explicit formulation of the product rule for independent trials. The novel formulation then states that a probability measure P "governs" an experiment E if the events that are typical according to P^n are practically certain according to E^n for all n >= 1, where P^n is the n-fold product of P and E^n is the experiment whose trials are composed of n trials of E. The novel formulation highlights the possible existence of two ambiguities in the principle, namely: (i) that different probability measures govern the same experiment and (ii) that the same probability measure governs different experiments. In this paper the first ambiguity is rigorously disproved, while the second is disproved provided that a suitable property characterizing the empirical equivalence of experiments is assumed.
Category: General Mathematics

Fractional Order System Identification with State Delay

Authors: HyonSong Yun, SungChol U, KungNam Kim, MyongHyok Sin
Comments: 10 Pages.

In this paper, we consider the continuous time fractional order system with unknown state . The fractional integral operational matrix of the block pulse functions(BPFs) is upper triangular Toeplitz. Using the commutativity and nilpotent property of upper triangular Toeplitz, we propose an efficient identification method in which the nonlinear parameters. The accuracy of the proposed method is illustrated by several simulations.
Category: General Mathematics

New Methods Based on the Calculation of Specific Decimal Fractions for Decomposing an Integer Into a Product of Prime Factors

Authors: bouchaïb Bahbouhi
Comments: 27 Pages.

This article presents for the first time two methods for decomposing integers in products of prime factors which are based on the calculation of decimal fractions. Its originality lies in the fact that the divisors used are decimals and not prime divisors and in addition the decimal part is manipulated in such a way that two decimal digits are fixed and the others are variable. In the first method, the divisors are of type 2n and which have a very interesting particularity which is that they always have two same digits at the end of their decimal parts (25 or 75). And it is this particularity which is exploited to develop these methods. The other method introduces a new notion that of the decomposition key which is a product of prime factors used to decompose all numbers having the same number of digits. It is similar to the first method because it also uses decimal fractions for the calculation and the denominator is the square root. This article paves the way for new applications in computer science.
Category: General Mathematics

Problem of Principal Axis and DBZC: Tan(pi/2)=0

Authors: Saburou Saitoh
Comments: 3 Pages.

In this note, we would like to see the fundamental result $tan(pi/2)=0$ from the famous problem of principal axis in connection with the division by zero calculus $frac{f(x)}{(x - a)^n}|_{x =a} : = frac{f^{(n)}(a)}{n!}.$
Category: General Mathematics

Resonance Phenomena May be Interpreted by DBZC: $(f(x)/x )(x=0):= F^prime(0)$

Authors: Saburou Saitoh
Comments: 2 Pages.

In this note, we would like to show the simple result that resonance phenomena may be interpreted by DBZC: $(f(x)/x )(x=0):= f^prime(0)$ by a typical simple example.
Category: General Mathematics

Two Types of Universal Arrows

Authors: Zhao-Dan Lee
Comments: 10 Pages.

A universal arrow is a pair which consists of an object and a morphism. And an isomorphism is defined by a universal arrow. The isomorphism may be a composition of two morphisms. We may define two types of universal arrows, which is determined by the properties of the morphisms. A universal arrow is of the type I if the morphisms are not isomorphisms; And a universal arrow is of the type II if the morphisms are isomorphisms.
Category: General Mathematics

Skill in Backgammon: Cubeless vs Cubeful

Authors: Tilemachos Zoidis
Comments: 16 Pages. CC BY

Does the doubling cube make backgammon more skillful? And is the answer the same in both money and match play? This article 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

The Various Representative Values Representing Between the Two Figures, and How to Create These Representative Values

Authors: Sungmin Kang
Comments: 2 Pages. (Author name added to the article by viXra Admin as required; also, please cite and list scientific references)

There are countless means that are neither arithmetic nor geometric means, and to satisfy the mean, f(x,y) must be a one-to-one correspondence to a bivariate function f(x,x).
Category: General Mathematics

A New Binary Encoding Disk from Shen Nong's Diagrams in I-Ching and Its Application -- Also Uncovering the e, φ, Spiral Fractal, Fibonacci and TM Sequences in the Encoding Disk

Authors: Hua-Fang Wu
Comments: 24 Pages. In Chinese

The Shen Nong's Diagrams of I-Ching is a set of "the dichotomy approach" Diagrams of I-Ching discovered by the author in 1994, which has been published in Chinese core journals. The diagram contains the geometric sequence of 1, 2, 4, 8... Shen Nong's Diagrams of I-Ching actually represents a new binary coding scheme with a specific arrangement and combination of YIN and YANG symbols, in which, the circular diagram constitutes an encoding disk that can be used as a photoelectric code disk. In recent years, the author has discovered that the Fibonacci sequence, TM sequence, φ, e, Pascal's triangle (Yang Hui Triangle), the integer value of the fine structure constant 137, and the spiral fractal structure, which cross coexistence on the encoding disk. Even the spiral structure of the Milky Way is very similar to the spiral of the encoding disk. This provides us with a new perspective and entry point for studying these mathematical and scientific issues and even the internal connections among them, and it is expected to that the encoding disk will be basic tool, which, like Pascal's triangle, will be widely used in the field of mathematical and scientific research.
Category: General Mathematics

Analytical Exploration and Extension of the Function

Authors: Lynette E. M. Z. Winslow
Comments: 6 Pages.

This paper investigates the function ( f(x) = int_{-infty}^{+infty} e^{(-x)^{|u|}} , du ), focusing on its analytical expression and extension over the real number domain. We employ techniques analogous to the analytic continuation of the Gamma function to extend ( f(x) ) beyond its initial domain, addressing convergence issues and exploring its properties across the entire real line.
Category: General Mathematics

An Example of Tan(π/2) = 0 from Seiyo Sampo

Authors: Saburou Saitoh, Hiroshi Okumura
Comments: 2 Pages. (Note by viXra Admin: Further repetition may not be accepted)

We show a very simple and pleasant example tan(π/2) = 0 from Seiyo Sampo.
Category: General Mathematics

Tan(pi/2)=0 from Jacobi's Method in Diagonalization of Matrices

Authors: Saburou Saitoh
Comments: 3 Pages.

In this note, we would like to show the simple result $tan(pi/2)=0$ from Jacobi's formula in diagonalization of matrices.
Category: General Mathematics

Computationally Efficient Differenceless Derivatives with Equidistant Steps and It's Applications

Authors: Yuri Mahotin
Comments: 15 Pages.

Computationally efficient differenceless derivatives with equidistant steps have been developed, which makes it possible to calculate an unlimited number of derivatives. The new algorithm can be applied in various fields of science and technology. As an example, we provide step-by-step instructions on how to improve the accuracy of the predicted trajectory of a flying missile.
Category: General Mathematics

An Oscillatory Integral

Authors: Edgar Valdebenito
Comments: 4 Pages.

Some remarks on an oscillatory integral [are given].
Category: General Mathematics

Solution of Wallis’s Integral Using Complex Functions

Authors: Kazuaki Shimada
Comments: 2 Pages. (Note by viXra Admin: A separate abstract is requited)

This article shows the value of the Wallis integral when n is an even number, n≥2 using the integration of a complex function. Proof of the Wallis product is generally derived using partial integrals, but here derivation using complex integrals is introduced.
Category: General Mathematics

On the Equation: S=(1/2)gamma(1/2,s^2) , S>0

Authors: Edgar Valdebenito
Comments: 3 Pages.

We solve the equation: s=(1/2)Gamma(1/2,s^2), s>0, where Gamma(x,y) is the incomplete gamma function.
Category: General Mathematics

Triples "Ф, e, π" and "π, 4, 6" as Input Data for Modified Koide-Formulas and the Common Grounds of the Results

Authors: Andreas Ball
Comments: 8 Pages.

In this report the common grounds of the results of modified Koide-Formulas are presented, in which the Triples "Ф, e, π" and "π, 4, 6" are set as basis values of various exponents.The figures of the first Triple are the Quotient of the Golden Ratio Ф, the Euler Figure Figure e and the Circle Figure π. Besides the circle/sphere diameter the figures of the second Triple "π, 4, 6" determine the circle area and the sphere volume. The exact exponent value, which results by the Equalization of the two modified Koide-Formulas, is close to the figure 0.444, which is also used at an approximation for the mass ratio of the elementar particles Tauon and Electron. The results of the two Koide-Formulas are close to each other over a relatively wide exponent range.
Category: General Mathematics

Collatz’s Conjecture, Proposal, Solution and Analysis

Authors: Daniel Oliivares
Comments: 5 Pages.

The Collatz conjecture has baffled mathematicians for decades due to its apparent simplicity and the lack of a formal proof. In the next Paper we will address a possible solution to the conjecture by modifying it and the reasons for it and we will analyze determining factors for it. all conditions are met.
Category: General Mathematics

Some Missed Opportunities for Archimedes and Early pi-Computors

Authors: Warren D. Smith
Comments: 4 Pages.

We point out some simple improvements to Archimedes' "regular polygon methods" for computing and bounding π , which all the workers before 1650 could have used, but did not. All methods employed before the 1970s to compute the first D decimals of π required order D or more arithmetic operations (±, ×, ÷, x^1/2, x^-1/2). But we shall show that if Archimedes or his followers had been a bit smarter, they could have sped that up to O(D^2/3).
Category: General Mathematics

Revised Attempt to Prove the Collatz Conjecture

Authors: Krishna Paliwal
Comments: 4 Pages.

TThis paper tries to prove the Collatz Conjecture using a rigorous and logical approach trying to break down 80+ year old and proving that allsequences will eventually always reach to 1.
Category: General Mathematics

The Philosophical Nature of Zero, Negative, and Imaginary Numbers and the Use of Polar Coordinates in Complex Number Calculations

Authors: Bryce Petofi Towne
Comments: 12 Pages. (Note by viXra Admin: AI generated contents/results are in general not acceptable)

Mathematics serves as an abstract tool to study the natural world and its laws, aiding in our understanding and description of natural phenomena. In mathematics, real numbers, imaginary numbers, zero, and negative numbers are fundamental concepts, each with its unique importance and application. However, the philosophical nature of these concepts warrants further exploration. This paper aims to discuss the philosophical essence of imaginary numbers, zero, and negative numbers, argue that imaginary numbers have real-world counterparts, and explore the rationale and advantages of representing imaginary and complex numbers using polar coordinates. Furthermore, we extend our findings to more advanced mathematical problems in complex analysis, differential equations, and number theory, demonstrating the broader impact of our work.
Category: General Mathematics

The so-Called Extrinsic Method

Authors: Thierry L. A. Periat
Comments: 12 Pages.

The theory of the (E) question is concerned with the decomposition (synonym: division) of deformed tensor (resp. Lie) products. A first mathematical method (the intrinsic one) has been developed for the decomposition of deformed cross products. It only works in three-dimensional spaces and brings incomplete results. This document proposes a second approach bringing complete results, i.e.: the main and the residual parts of each decomposition, whatever the dimension D (D in N - {0, 1}) of the mathematical space is. But the method is plagued with a logical uncertainty. Fortunately, in any three-dimensional space, both methods can be calibrated through diverse scenarios. One of them may catch the attention of physicists since it re-introduces E. Cartan’s metrics induced by the evolution of surfaces.
Category: General Mathematics

Proof of Collatz Conjecture

Authors: Abdelkrim ben Mohamed
Comments: 7 Pages.

In this paper we try prove the Collatz conjecture also known the 3x+1 problem.
Category: General Mathematics

Buffon’s Needle Without π

Authors: Mieczysław Szyszkowicz
Comments: 4 Pages.

Buffon's needle problem, posed by Georges-Louis Leclerc, Comte de Buffon in the 18th century, stands as a cornerstone in the realm of geometric probability. The problem encapsulates a scenario where a needle of a given length is dropped randomly onto a floor composed of parallel strips of equal width. The inquiry revolves around determining the likelihood that the needle will intersect a line between two strips. Here a new solution of this classical problem is proposed.
Category: General Mathematics

Unlocking New Perspectives in Science and Engineering: The Sphere-Base-One Mathematical System

Authors: Athon Zeno
Comments: 5 Pages. (Note by viXra Admin: Please cite and list scientific references)

The Sphere-Base-One mathematical system is a novel framework that takes the sphere as the fundamental unit of volume and redefines the relationships between spheres, cubes, and other geometric objects. This innovative approach offers a fresh perspective on the nature of space and volume, challenging conventional notions of geometry and opening up new avenues for interdisciplinary research and discovery. By focusing on the sphere as the primary building block and exploring the negative space around it, the Sphere-Base-One system has the potential to unlock new insights and solutions in a wide range of scientific and engineering disciplines, including quantum physics, cosmology, surface chemistry, fluid dynamics, and electrical engineering. The simplification and reformulation of key equations in the Sphere-Base-One system may lead to easier calculations and, more importantly, to the identification of patterns and relationships that were previously obscured by the limitations of the traditional Cube-Base-One system. While not intended to replace the existing mathematical framework, the Sphere-Base-One system serves as a complementary tool that can be applied in parallel to drive progress and innovation across multiple fields. This article introduces the core concepts of the Sphere-Base-One system, explores its potential applications, and discusses the implications of this new mathematical paradigm for the future of scientific research and technological advancement.
Category: General Mathematics

From Derivation of Ancient Approximations of the Circle Figure π a Complete Solution of Leonardo da Vinci's Vitruvian Man

Authors: Andreas Ball
Comments: 22 Pages. (Correction made by viXra Admin to conform with the requirements of viXra.org)

In this report the author tries to handle with four themes. Referring the first topic [Part 2] derivations of ancient approximations for the Circle Figure π are presented using the figures of the two- and three-dimensioninal case for the straight and the round. [Part 3] deals with possible connections between the Circle Figure π and the Golden Section Ф using modified terms as presented at the first topic.[In Part 4] a complete solution of the puzzle around the drawing Vitruvian Man of Leonardo da Vinci is presented, which is mostly based on the informations given by two german authors. [In Part 5] a system is described, which is based on the geometrical system of the drawing Vitruvian Man of Leonardo da Vinci and by which the attempt of a connection between the Circle Figure π and the Golden Section Ф is done.
Category: General Mathematics

Improper Integrals of the Second Kind

Authors: Edgar Valdebenito
Comments: 3 Pages.

This document briefly discusses improper integrals of the second kind.
Category: General Mathematics

Truncated Taylor Solvers for Simple First-Order Ordinary Differential Equations

Authors: Steven Kenneth Kauffmann
Comments: 2 Pages.

Certain exact solutions of first-order ordinary differential equations naturally become corresponding solvers for simple first-order ordinary differential equations whose form is the equality of a twice continuously differentiable function of the dependent variable to the derivative with respect the independent variable of the dependent variable. When the twice continuously differentiable function of the dependent variable is replaced by its truncated Taylor expansions through second order about its initial value, the resulting first-order ordinary differential equations have exact solutions that naturally become corresponding solvers for those simple first-order ordinary differential equations.
Category: General Mathematics

Simple Results of 1/0=0/0= 0 by Elementary Figures

Authors: Saburou Saitoh
Comments: 3 Pages.

In this note, we would like to show the simple results $1/0= 0/0=0$ based on the elementary figures that are well-known and simple results on the complex plane. The logic and results are all reasonable and exceptionally pleasant lookings for undergraduate students.
Category: General Mathematics

Since Nothing Can’t Exist, Something Does

Authors: Thiago M. Nóbrega
Comments: 2 Pages.

The formalization presented provides a rigorous foundation for the philosophical argument that "nothing" cannot exist and therefore "something" must exist. By utilizing set theory and logical quantification, I establish a clear mathematical framework supporting this thesis. This foundational understanding has significant implications for various fields, including metaphysics, ontology, and cosmology.
Category: General Mathematics

The Unified Theory

Authors: Thiago M. Nóbrega
Comments: 3 Pages.

The Existence Principle is a novel theoretical framework that unifies ontological principles, the principle of least action, the bootstrap mechanism, Turing's universal machine, and von Neumann's architecture. By merging these concepts, I propose a comprehensive mathematical structure that demonstrates the self-consistent nature of reality, governed by physical laws and computational principles. This framework provides groundbreaking insights into the fundamental nature of existence, life, and the universe.
Category: General Mathematics

Optimization Problem

Authors: Edgar Valdebenito
Comments: 3 Pages.

[This paper consider] [w]hat the maximum value of f(x)=arctan(exp(-exp(x-2)))-arctan(exp(-exp(x+2)))[is].
Category: General Mathematics

Applicability Operator Approach to Mathematical Proofs

Authors: Jason Kodish
Comments: 2 Pages. (Note by viXra Admin: Please cite and list scientific references)

Teaching proofs of theorems and encouraging students to both comprehend written proofs and originate their own can at times be a difficult undertaking. This is due in part to the lack of a single unifying process by which one can approach mathematical proofs. In this paper a method using set theory as a foundation is presented.
Category: General Mathematics

Key to Leibnizian Mathematics

Authors: Adriaan van der Walt
Comments: 11 Pages.

This document states the relevant basic assumptions of Mathematics that lead to some of the salient features of Abstract Mathematics. These features are then deduced from these assumptions in a fundamental way. Infinitesimals and Infinitesimal Numbers are introduced and then used in an example of a Riemann sum to create a contradiction that motivates the introduction of Leibnizian Mathematics as a model for Mathematics that is supplemental to Abstract Mathematics. Leibnizian Mathematics is then introduced by stating its basic assumptions. Lastly a list of the meanings of some words that are common to both models, but describe properties that differ between the models, is given. The reader is then referred to the document LEIBNIZIAN MATHEMATICS, posted on the link stated in the Comments below, where this model is developed.
Category: General Mathematics

Discussions on the Decomposition of Deformed Vector Products

Authors: Thierry L. A. Periat
Comments: 40 Pages. In French (Translation made by viXra Admin - Future non-compliant submission will not be accepted)

This paper is part of a series of explorations exposing methods for dividing distorted tensor products by number cubes. Here, the discussion focuses on three-dimensional spaces and anti-symmetric cubes on their low indices; i.e. actually on distorted vector products. The method only delivers the main parts of the divisions. They are divided into two classes. The other documents in the collection complete this investigation.

Ce document fait partie d'une série d'explorations exposant des méthodes permettant de diviser les produits tensoriels déformés par des cubes de nombres. Ici, la discussion se focalise sur les espaces de dimension trois et les cubes antisymétriques sur leurs indices bas ; c'est-à-dire en fait sur des produits vectoriels déformés. La méthode livre seulement les parties principales des divisions. Elles se répartissent en deux classes. Les autres documents de la collection complètent cette investigation.
Category: General Mathematics

On a Combinatorial Problem of Existing Matchings

Authors: Wladislaw Zlatjkovic Petrovescu
Comments: 2 Pages. (Abstract added by viXra Admin as required - Please conform)

In this paper we prove a classic combinatorial result on matchings.
Category: General Mathematics

Foundations of Differential Calculus: Does the Theory Satisfy the Criterion of Truth?

Authors: Temur Z. Kalanov
Comments: 16 Pages.

A detailed proof of the incorrectness of the foundations of the differential calculus is proposed. The correct methodological basis for the proof is the unity of formal logic and rational dialectics. The unity of formal logic and rational dialectics is the only correct criterion of truth. The proof leads to the following irrefutable statement: differential calculus represents an incorrect theory in mathematics and physics. The proof of this statement is based on the following irrefutable results: (1) the standard theory of infinitesimals and the theory of limits underlying the differential calculus are incorrect theories. The concepts of "infinitesimal quantity", "movement", "process of tendency", and "limit of tendency" are meaningless concepts in mathematics; (2) the concepts of "increment of argument" and "increment of function" as the starting point of the differential calculus are not defined correctly; (3) the definition of the derivative of a function is an incorrect because the following logical contradiction arises: the increment of the argument is both not equal to zero and equal to zero; (4) the differentials of the argument and the function - as infinitesimal quantities - do not take on numerical values. This means that the differentials of quantities have neither quantitative nor qualitative determinacy; (5) the definition of the total differential of a function of two (many) variables does not satisfy the formal-logical law of the lack (absence) of contradiction; (6) the theory of proportions completely refutes the theory of differential calculus. Thus, differential calculus does not satisfy the criterion of truth and is not correct scientific (mathematical) theory.
Category: General Mathematics

A Power Series Divergent at Half Pi Yields a Sequence Briskly Convergent to Pi

Authors: Steven Kenneth Kauffmann
Comments: 2 Pages.

The tangent function's Taylor expansion about zero is a series of odd powers with positive coefficients; it converges when the absolute value of its argument is less than half pi, and diverges to positive infinity when its argument equals half pi, so with the aid of the ratio test it is seen that twice the square root of the ratio of its successive coefficients is a sequence which converges to pi. The odd derivatives of the tangent function are polynomials in powers of its square with positive integer coefficients, so a recursion of positive integers can be found from which the coefficients of the series described above may be successively obtained. Its related sequence which converges to pi does so monotonically from above, and appears to refine its approximation to pi by about one significant decimal figure per successive term.
Category: General Mathematics

Geometric Facts of a Circle Through Its Equation

Authors: Rajesh Sharma, Vijay Sharma
Comments: 4 Pages.

We derive equation of a circle passing through three distinct points in a different way and demonstrate that several basic geometric properties of a circle can be derived easily by means of this equation alone.
Category: General Mathematics

Squaring the Circle and Approximations to pi

Authors: Edgar Valdebenito
Comments: 4 Pages.

Since the 19th century when von Lindemann set out a proof of the transcendental properties of the mathematical constant Pi, mathematicians have taken the view that squaring the circle using a straight edge and compass is not possible.
Category: General Mathematics

Skill in Backgammon: Cubeless vs Cubeful

Authors: Tilemachos Zoidis
Comments: 16 Pages. CC BY

Does the doubling cube make backgammon more skillful? And is the answer the same in both money and match play? This article presents GNUbg rollouts between unequally skilled players which show that use of the doubling cube favors the better player only in match play.
Category: General Mathematics

Skill in Backgammon: Cubeless vs Cubeful

Authors: Tilemachos Zoidis
Comments: 16 Pages. CC BY

Does the doubling cube make backgammon more skillful? And is the answer the same in both money and match play? This article presents GNUbg rollouts between unequally skilled players which show that use of the doubling cube favors the better player only in match play.
Category: General Mathematics

Proof of Collatz Conjecture

Authors: Abdelkrim ben Mohamed
Comments: 7 Pages.

In this working paper we try to prove the Collatz conjecture also known the 3x+1 problem.
Category: General Mathematics

Proof of Collatz Conjecture

Authors: Abdelkrim ben Mohamed
Comments: 7 Pages.

In this working paper we try to prove the Collatz conjecture also known the 3x+1 problem.
Category: General Mathematics