General Mathematics

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Recent submissions

Any replacements are listed farther down

[3068] viXra:1911.0317 [pdf] submitted on 2019-11-18 09:26:22

Unknown Summations

Authors: Yuly Shipilevsky
Comments: 2 Pages.

We introduce a set of finite and infinite summations which looks like were never considered yet.
Category: General Mathematics

[3067] viXra:1911.0311 [pdf] submitted on 2019-11-18 01:26:38

A Note on a Possible Anomaly in the Complex Numbers

Authors: Han Geurdes
Comments: 3 Pages.

In the present paper a conflict in basic complex number theory is reported. The ingredients of the analysis are Euler's identity and the DeMoivre rule for $n=2$. The outcome is that a quadratic equation only has one single solution because one of the existing solutions gives rise to an impossibility.
Category: General Mathematics

[3066] viXra:1911.0252 [pdf] submitted on 2019-11-14 11:58:07

Catalan's Constant and pi :Integrals

Authors: Edgar Valdebenito, Rodrigo Valdebenito
Comments: 2 Pages.

Double integrals
Category: General Mathematics

[3065] viXra:1911.0251 [pdf] submitted on 2019-11-14 12:00:10

Numbers: Part 6, Three Integrals .

Authors: Edgar Valdebenito, Rodrigo Valdebenito
Comments: 1 Page.

We give three definite integrals
Category: General Mathematics

[3064] viXra:1911.0242 [pdf] submitted on 2019-11-14 06:51:46

Numbers Do Not Exist

Authors: Averky Glebov
Comments: 1 Page.

In this paper we discuss how numbers, are just not real, and do not exist in the world.
Category: General Mathematics

[3063] viXra:1911.0233 [pdf] submitted on 2019-11-13 17:21:28

Refutation of Multiplication, Addition, and Subtraction by Zero Calculus

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the conjecture of the following equations: (a×0)≠0 ; (a×0)<a; and (a−0)<a<(a+0). None is tautologous, and the first two are contradictory. This refutes the conjectures to form a non tautologous fragment of the universal logic VŁ4.
Category: General Mathematics

[3062] viXra:1911.0231 [pdf] submitted on 2019-11-13 01:57:55

Multiplication by Zero Calculus, Addition by Zero Calculus, and Subtraction by Zero Calculus

Authors: Toshiro Takami
Comments: 2 Pages.

In physics, there are many particles in a vacuum.\\ Perfect zero cannot exist.\\ 0 is not perfect zero.\\ 0 is almost zero.\\ Perfect zero is only a mathematical fantasy.\\ $a\times0\approx0$, but $a\times0\neq0$.\\ $a\times0\times0\times0\times0\times0<a\times0\times0\times0\times0<a\times0\times0\times0<a\times0\times0<a\times0<a$.\\ $a-0-0-0<a-0-0<a-0<a<a+0<a+0+0<a+0+0+0$.\\
Category: General Mathematics

[3061] viXra:1911.0206 [pdf] submitted on 2019-11-11 18:05:40

Finite-Time Lyapunov Exponents in the Instantaneous Limit and Material Transport

Authors: Peter J. Nolan, Mattia Serra, Shane D. Ross
Comments: 43 Pages. Submitted for publication

Lagrangian techniques, such as the Finite-Time Lyapunov Exponent (FTLE) and hyperbolic Lagrangian coherent structures, have become popular tools for analyzing unsteady fluid flows. These techniques identify regions where particles transported by a flow will converge to and diverge from over a finite-time interval, even in a divergence-free flow. Lagrangian analyses, however, are time consuming and computationally expensive, hence unsuitable for quickly assessing short-term material transport. A recently developed method called OECSs rigorously connected Eulerian quantities to short-term Lagrangian transport. This Eulerian method is faster and less expensive to compute than its Lagrangian counterparts, and needs only a single snapshot of a velocity field. Along the same line, here we define the instantaneous Lyapunov Exponent (iLE), the instantaneous counterpart of the finite-time Lyapunov exponent (FTLE), and connect the Taylor series expansion of the right Cauchy-Green deformation tensor to the infinitesimal integration time limit of the FTLE. We illustrate our results on geophysical fluid flows from numerical models as well as analytical flows, and demonstrate the efficacy of attracting and repelling instantaneous Lyapunov exponent structures in predicting short-term material transport.
Category: General Mathematics

[3060] viXra:1911.0058 [pdf] submitted on 2019-11-04 06:03:56

SCIS- Traffic Congestion Optimisation using River Formation Dynamics

Authors: Kunal Verma, Vishal Paike
Comments: 5 Pages.

The raising level of traffic imposes a great demand in the growth of intelligent traffic systems. With increase in complexity of alleviation, finding solutions to traffic congestion problem have become one of the challenges. Various optimization techniques have been proposed in literature to meet these challenges. This paper surveys different optimization techniques based on heuristics for automated traffic congestion control. Moreover, an approach based on River Formation Dynamics scheme is introduced to analyze the optimization problem for traffic congestion control and a scheme to extract real time information through Internet of Things is presented for superior efficiency and productivity
Category: General Mathematics

[3059] viXra:1911.0050 [pdf] submitted on 2019-11-03 02:41:08

О парадоксе при классическом подходе к доказательству теорем и непригодности общепринятых стереотипных тавтологий классической математики для доказательства теорем

Authors: Victoria Kondratenko Виктория Александровна Кондратенко
Comments: 5 Pages.

Аннотация. В текущее время в естественных и прикладных науках в большинстве публикаций доказательство теорем осуществляется: во-первых, содержательным способом, что противоречит настоятельному требованию философов науки использовать исключительно формальное доказательство, которое является критерием оценки корректности и достоверности доказательства; во-вторых, при содержательном доказательстве в 95% случаев используются исключительно стандартный перечень тавтологий, который по определению некорректен для целей доказательства теорем о явлениях и процессах мироздания на основе истинных аксиом, полученных в результате натурного экспериментирования с этими явлениями и процессами. В статье анализируется часто используемый стандартный перечень тавтологий, доказывается непригодность этих тавтологий для доказательства теорем, что порождает очередной неразрешимый парадокс в началах математики. Автором предлагается выход из создавшегося тупика.
Category: General Mathematics

[3058] viXra:1910.0653 [pdf] submitted on 2019-10-31 23:04:59

Metrology of the Circle and the Royal Cubit: Deciphering the Ancient 360 Degree Circle Design

Authors: Mark Anthony Musgrave
Comments: 11 Pages.

Design of the ancient 360-degree circle is proposed to be the result of using a scientific length standard, the Egyptian Royal cubit, to define the dimensions of the circle reference frame. The ancient length standard is subdivided into its own sub-units of palms and fingers, but it is the equivalence to other ancient length units (inches, feet) in proportions that match specific circle features that allow underling design aspects to be identified. The available evidence suggests that the original circle design described by Hipparchus, as being based on “a radius of 3438 minutes”, should be interpreted to mean that the design circle radius was actually 3438 Royal cubits. From this framework it is possible to observe direct metrological relationships between the design of the Royal cubit and the design of the 360-degree circle, as well as the origin of the inch and feet units. Multiple shared features between the circle and the Royal cubit suggest a common design principle was involved, and the evidence suggests that an understanding of electromagnetic physics was in place when the sexagesimal circle was created. If verified, the hypothesis presented here infers that a new frequency standard could be implemented in modern metrology that would provide both the time and length units and allow for complete integration with the 360-degree circle reference frame. This step may also then provide deeper insights into astronomical physics as dimensional features are examined under a suitable length unit
Category: General Mathematics

[3057] viXra:1910.0567 [pdf] submitted on 2019-10-27 19:22:22

Proof of the Riemann Hypothesis [final Edition]

Authors: Toshiro Takami
Comments: 9 Pages.

Up to now, I have tried to expand this equation and prove Riemann hypothesis with the equation of cos, sin, but the proof was impossible. However, I realized that a simple formula before expansion can prove it. The real value is zero only when the real part of s is 1/2. Non-trivial zeros must always have a real value of zero. The real part of s being 1/2 is the minimum requirement for s to be a non-trivial zeros.
Category: General Mathematics

[3056] viXra:1910.0560 [pdf] submitted on 2019-10-27 07:22:45

Non-Negative in Value, but Absolute in Function by a Magnum and Parabolin—the Cogent Value Function

Authors: William F. Gilreath
Comments: 34 Pages. Published in the General Science Journal

The absolute value function is a fundamental mathematical concept taught in elementary algebra. In differential calculus, the absolute value function has certain well-known mathematical properties that are often used to illustrate such concepts of—a continuous function, differentiability or the existence of a derivative, the limit, and etcetera. An alternative to the classical definition of absolute value is given to define a new function that is mathematically equivalent to the absolute value, yet the different mathematically. This new mathematical formalism, the cogent value function, does not have the same mathematical properties of the absolute value function. Two other new mathematical functions are used in the definition of the cogent value function—the parabolin function, and the magnum function. The cogent value function and the absolute value function have the same domain and range, but both are mathematically very different. The cogent value function demonstrates that the same mathematical concept when formally defined by an alternative method has different mathematical properties. The functions by operation are mathematically similar, but in mathematical formalism each is unique.
Category: General Mathematics

[3055] viXra:1910.0477 [pdf] submitted on 2019-10-23 19:21:11

Remainder Theorem and the Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 4 Pages. In this short note, for the elementary theorem of remainder in polynomials we recall the division by zero calculus that appears naturally in order to show the importance of the division by zero calculus.

In this short note, for the elementary theorem of remainder in polynomials we recall the division by zero calculus that appears naturally in order to show the importance of the division by zero calculus.
Category: General Mathematics

[3054] viXra:1910.0361 [pdf] submitted on 2019-10-19 20:55:51

Trigonometric Tutorial: Pythagorean Theorem, Planar Rotations, Chord Lengths, and Trig Function Features

Authors: Steven Kenneth Kauffmann
Comments: 4 Pages.

This tutorial takes trigonometry to be the study of the transformation of a two-dimensional vector's Cartesian coordinates when it is rotated about the Cartesian origin of coordinates in its plane. Since the Pythagorean theorem underlies the idea of Cartesian coordinates, the tutorial commences with a plane-geometry recapitulation of that theorem. Three characteristics of the planar rotation transformations of a two-dimensional vector's Cartesian coordinates are pointed out: their linearity, their preservation of the vector's length and their additivity in successive rotation angles. The rotated vector's Cartesian coordinates themselves aren't thus successively additive; they reflect mapping of the rotation angles into the more intricate corresponding changes of the vector's location in two dimensions. The machinery which maps successive rotation angles into the corresponding two-dimensional locations is the "angle-addition formula"; it performs that task via application of the sine and cosine functions to the rotation angles. The last part of the tutorial studies properties of the sine and cosine functions, one of the most fascinating is that they are the imaginary and real parts of the exponential function of imaginary argument.
Category: General Mathematics

[3053] viXra:1910.0165 [pdf] submitted on 2019-10-10 18:32:05

Tutorial: Continuous-Function on Closed Interval Basics, with Mean-Value and Taylor Theorem Upshots

Authors: Steven Kenneth Kauffmann
Comments: 5 Pages.

This tutorial explores the relation of the local concept of a function's continuity to its global consequences on closed intervals, such as a continuous function's unavoidable boundedness on a closed interval, its attainment of its least upper and greatest lower bounds on that interval, and its unavoidable assumption on that closed interval of all of the values which lie between that minimum and maximum. In a nutshell, continuous functions map closed intervals into closed intervals. It is understandable that verifying this astonishing fact involves subtle and very intricate manipulation of the least-upper-bound/greatest-lower-bound postulate for the real numbers. In conjunction with the basic inequality properties of integrals, this continuous-function fact immediately implies the integral form of the mean-value theorem, which is parlayed into its differential form by the fundamental theorem of the calculus. Taylor expansion and its error estimation are additional fascinating developments intertwined with these lines of thinking.
Category: General Mathematics

[3052] viXra:1910.0022 [pdf] submitted on 2019-10-01 11:58:09

Using the Rational Root Test to Factor with the TI-83

Authors: Timothy W. Jones
Comments: 4 Pages.

The rational root test gives a way to solve polynomial equations. We apply the idea to factoring quadratics (and other polynomials). A calculator speeds up the filtering through possible rational roots.
Category: General Mathematics

[3051] viXra:1909.0603 [pdf] submitted on 2019-09-29 14:48:48

MMM Imfinities Version 1.0 6 a4 Pages 29.09.2019) on Two Proposed Distinct Types of Imaginary (Im) Infinities ("imfinities") in Mathematics and Meta-Mathematics (Including Meta-Geometry), Emphasizing the Unlimited "Diversity" of Zero and Infinity

Authors: Andrei Lucian Dragoi
Comments: 6 Pages.

This paper proposes two distinct types of imaginary (im) infinities ("imfinities") in mathematics and meta-mathematics (including meta-geometry), emphasizing the unlimited "diversity" of zero and infinity, with far-reaching implications in all these domains, but also in math-related domains like modern physics, including the help in redefining the basics of Einstein's General relativity theory (GRT), quantum field theory (QFT), superstring theories (SSTs) and M-theory (MT). Keywords (including a list of main abbreviations): imaginary (im) infinities (“imfinities”), mathematics, metamathematics, metageometry, zero, infinity; Einstein’s General relativity theory (GRT); quantum field theory (QFT); superstring theories (SSTs); M-theory (MT);
Category: General Mathematics

[3050] viXra:1909.0550 [pdf] submitted on 2019-09-25 13:28:46

Historical Problem of Mathematics. Complex Numbers

Authors: A.I.Somsikov
Comments: 4 Pages. -

The description of the complex number which is not containing a concept of the complex plane is offered.
Category: General Mathematics

[3049] viXra:1909.0537 [pdf] submitted on 2019-09-24 07:30:46

Numbers: Part 4," Pi Representations "

Authors: Edgar Valdebenito
Comments: 3 Pages.

We give some representations of Pi.
Category: General Mathematics

[3048] viXra:1909.0536 [pdf] submitted on 2019-09-24 07:33:18

Numbers: Part 5 , Two Formulas

Authors: Edgar Valdebenito
Comments: 1 Page.

This note presents two formulas.
Category: General Mathematics

[3047] viXra:1909.0535 [pdf] submitted on 2019-09-24 07:35:55

Simpson's Rule and Simpson's 3/8 Rule

Authors: Edgar Valdebenito
Comments: 2 Pages.

Numerical evaluation of the integral: int(f(x),x=0..1)=pi/2
Category: General Mathematics

[3046] viXra:1909.0517 [pdf] submitted on 2019-09-24 16:12:58

What Was Division by Zero?; Division by Zero Calculus and New World (Compact Version)

Authors: Saburou Saitoh
Comments: This version was written for the Proceedings of ICRAMA2019 (16-18 July, 2019) with the 8 pages restriction under the requested form.

Based on the preprint survey paper, we will introduce the importance of the division by zero and its great impact to elementary mathematics and mathematical sciences for some general people. For this purpose, we will give its global viewpoint in a self-contained manner by using the related references. This version was written for the Proceedings of ICRAMA2019 (16-18 July, 2019) with the 8 pages restriction under the requested form.
Category: General Mathematics

[3045] viXra:1909.0483 [pdf] submitted on 2019-09-22 08:27:44

Tutorial: Generating All of the Real Powers from Two Simple Properties of Positive-Integer Powers

Authors: Steven Kenneth Kauffmann
Comments: 4 Pages.

This tutorial parlays two simple properties of positive-integer powers of positive numbers into a closed formula for arbitrary real-valued powers of positive numbers. The two properties are the very familiar result for the derivative of a positive-integer power of a positive real variable, and the fact that positive-integer powers of unity equal unity. Extended to real values of the powers of positive numbers, these two properties comprise a set of first-derivative linear differential equations for functions of a positive real variable, together with their initial conditions, so it isn't surprising that a unique solution is in fact forthcoming via Taylor expansion methods. The additive property of multiplied powers emerges immediately from the two properties of powers that are initially assumed, and enters prominently into working out the closed-formula result, which turns out to be a creature of the exponential function and its inverse. Logarithms are defined in terms of the inverse operation of taking powers in the familiar way, and the special base whose powers reproduce the exponential function and whose logarithms reproduce the exponential function's inverse is pointed out and developed.
Category: General Mathematics

[3044] viXra:1909.0439 [pdf] submitted on 2019-09-20 12:25:01

Number and Arithmetic Operation

Authors: A.I.Somsikov
Comments: 4 Pages. -

The meaning (logical content) of the concept of numbers is revealed. Arithmetic operations are defined.
Category: General Mathematics

[3043] viXra:1909.0421 [pdf] submitted on 2019-09-19 07:58:13

Successions and Series

Authors: Viola Maria Grazia
Comments: Pages. Successions are successions with positive terms

In this pdf there is my point of view about successions and series. I talk about when a succession (and so a serie) is said convergent making attention to the succession's definition.
Category: General Mathematics

[3042] viXra:1909.0359 [pdf] submitted on 2019-09-18 06:27:24

Mathematics as Information Compression Via the Matching and Unication of Patterns

Authors: J Gerard Wolff
Comments: 61 Pages. Accepted for publication in the journal "Complexity", 2019-09-17.

This paper describes a novel perspective on the foundations of mathematics: how mathematics may be seen to be largely about 'information compression (IC) via the matching and unification of patterns' (ICMUP. That is itself a novel approach to IC, couched in terms of non-mathematical primitives, as is necessary in any investigation of the foundations of mathematics. This new perspective on the foundations of mathematics reects the idea that, as an aid to human thinking, mathematics is likely to be consonant with much evidence for the importance of IC in human learning, perception, and cognition. This perspective on the foundations of mathematics has grown out of a long-term programme of research developing the SP Theory of Intelligence and its realisation in the SP Computer Model, a system in which a generalised version of ICMUP -- the powerful concept of SP-multiple-alignment -- plays a central role. The paper shows with an example how mathematics, without any special provision, may achieve compression of information. Then it describes examples showing how variants of ICMUP may be seen in widely-used structures and operations in mathematics. Examples are also given to show how several aspects of the mathematics-related disciplines of logic and computing may be understood as ICMUP. Also discussed is the intimate relation between IC and concepts of probability, witharguments that there are advantages in approaching AI, cognitive science, and concepts of probability via ICMUP. Also discussed is how the close relation between IC and concepts of probability relates to the established view that some parts of mathematics are intrinsically probabilistic, and how that latter view may be reconciled with the all-or-nothing, 'exact', forms of calculation or inference that are familiar in mathematics and logic. There are many potential benefits and applications of the mathematics-as-IC perspective.
Category: General Mathematics

[3041] viXra:1909.0330 [pdf] submitted on 2019-09-15 08:15:06

My Contribustion in Claymath Millenium Problems:

Authors: Louiz Akram
Comments: 1 Page.

I present my contribustion in three Claymath Millenium problems.
Category: General Mathematics

[3040] viXra:1909.0288 [pdf] submitted on 2019-09-13 12:22:55

The Prime as a Consequence of Skip Number

Authors: Suraj Deshmukh
Comments: 2 Pages.

In this paper I have defined primes on the basis of "skip function" which basically is nothing but tells you how many numbers are skipped between two numbers. And we define a collection of set which I call skip set. The occurrence of a number decides weather it is prime or not.
Category: General Mathematics

[3039] viXra:1909.0283 [pdf] submitted on 2019-09-14 00:46:51

Number and Arithmetic Action

Authors: A.I.Somsikov
Comments: 4 Pages. -

Definition is given to concepts of number and arithmetic action.
Category: General Mathematics

[3038] viXra:1909.0215 [pdf] submitted on 2019-09-09 08:08:04

Simple Tesla Algorithm, the Art of Multiplying by Adding.

Authors: Zeolla Gabriel Martín
Comments: 25 Pages.

The Simple Tesla algorithm, is the art of multiplying by adding. We can use this method to calculate the product of any number with true accuracy. It has indisputable applications in various areas such as polynomials, complex numbers, binary numbers and many more. Basically multiplying is an act of repeated sums, and true to this concept is how this algorithm is developed.
Category: General Mathematics

[3037] viXra:1909.0080 [pdf] submitted on 2019-09-04 22:40:00

Division by Zero Because Next Infinity is Zero

Authors: Toshiro Takami
Comments: 5 Pages.

tan(π)=0, 0 =1, 1 =z =0 2000 When I saw this expression, I was surely suspicious. But I knew intuitively that Next infinity is zero. For me, infinite and zero were equal, that’s true now. The universe did not start with the Big Burn. The universe has existed for an infinite amount of time, and has repeated an infinite number of big burns. In other words, the universe is a repetition of Next infinity is zero.
Category: General Mathematics

[3036] viXra:1908.0619 [pdf] submitted on 2019-08-30 15:59:20

On Basic Tensor Concepts and the Taylor Series

Authors: Anamitra Palit
Comments: 7 Pages.

In this article we derive an anomalous result that with curved space time transformations have to be linear. Technical difficulties with infinitesimal space time coordinates as tensors are exposed. Analysis with the Taylor series brings out a stupendous fact that it considers only such functions as are linear in the independent variables.
Category: General Mathematics

Replacements of recent Submissions

[189] viXra:1910.0567 [pdf] replaced on 2019-11-16 14:56:43

Proof of the Riemann Hypothesis [final Edition]

Authors: Toshiro Takami
Comments: 5 Pages.

Up to now, I have tried to expand this equation and prove Riemann hypothesis with the equation of cos, sin, but the proof was impossible. However, I realized that a simple formula before expansion can prove it. The real value is zero only when the real part of s is 1/2. Non-trivial zeros must always have a real value of zero. The real part of s being 1/2 is the minimum requirement for s to be a non-trivial zeros.
Category: General Mathematics

[188] viXra:1910.0567 [pdf] replaced on 2019-11-07 16:59:56

Proof of the Riemann Hypothesis [final Edition]

Authors: Toshiro Takami
Comments: 6 Pages.

Up to now, I have tried to expand this equation and prove Riemann hypothesis with the equation of cos, sin, but the proof was impossible. However, I realized that a simple formula before expansion can prove it. The real value is zero only when the real part of s is 1/2. Non-trivial zeros must always have a real value of zero. The real part of s being 1/2 is the minimum requirement for s to be a non-trivial zeros.
Category: General Mathematics

[187] viXra:1910.0567 [pdf] replaced on 2019-10-28 18:07:14

Proof of the Riemann Hypothesis [final Edition]

Authors: Toshiro Takami
Comments: 9 Pages.

Up to now, I have tried to expand this equation and prove Riemann hypothesis with the equation of cos, sin, but the proof was impossible. However, I realized that a simple formula before expansion can prove it. The real value is zero only when the real part of s is 1/2. Non-trivial zeros must always have a real value of zero. The real part of s being 1/2 is the minimum requirement for s to be a non-trivial zeros.
Category: General Mathematics

[186] viXra:1910.0361 [pdf] replaced on 2019-10-23 21:26:29

Trigonometric Tutorial: Pythagorean Theorem, Rectangular Coordinates of Circular Arc Points, Chord Lengths of Arcs, and Key Calculus Features of the Cosine and Sine Functions

Authors: Steven Kenneth Kauffmann
Comments: 4 Pages.

Trigonometry studies the properties of the cosine and sine functions, which relate a contiguous arc of the unit-radius circle centered on the origin of coordinates to the rectangular Cartesian coordinates of the arc's endpoints. Since the Pythagorean theorem underlies the concept of Cartesian coordinates, this tutorial commences with a plane-geometry recapitulation of that theorem. In the non-calculus treatment of the cosine and sine, their demonstrable properties are encompassed by the unit length of unit-radius circle vectors and the "angle-addition formula" which relates the rectangular coordinates of the endpoints of two immediately successive arcs of the unit-radius circle to the rectangular coordinates of the endpoints of the combined contiguous arc. Those properties are insensitive, however, to simultaneous single-parameter rescaling of all of the arc lengths involved, and so don't unambiguously characterize the cosine and sine functions of directed arc length. Unambiguous determination of the cosine and sine hinges on whether their derivatives with respect to directed arc length are well-defined, which presents no issues for arcs of the unit-radius circle. In fact the cosine and sine functions fascinatingly are the real and imaginary parts of the hyper-well-behaved exponential function of imaginary argument.
Category: General Mathematics

[185] viXra:1910.0165 [pdf] replaced on 2019-10-12 16:29:46

Tutorial: Continuous-Function on Closed Interval Basics, with Mean-Value and Taylor Theorem Upshots

Authors: Steven Kenneth Kauffmann
Comments: 4 Pages.

This tutorial explores the relation of the local concept of a function's continuity to its global consequences on closed intervals, such as a continuous function's unavoidable boundedness on a closed interval, its attainment of its least upper and greatest lower bounds on that interval, and its unavoidable assumption on that closed interval of all of the values which lie between that minimum and maximum. In a nutshell, continuous functions map closed intervals into closed intervals. It is understandable that verifying this local-to-global fact involves subtle and very intricate manipulation of the least-upper-bound/greatest-lower-bound postulate for the real numbers. In conjunction with the basic inequality properties of integrals, this continuous-function fact immediately implies the integral form of the mean-value theorem, which is parlayed into its differential form by the fundamental theorem of the calculus. Taylor expansion and its error estimation are further developments which are intertwined with these fascinating concepts.
Category: General Mathematics

[184] viXra:1910.0022 [pdf] replaced on 2019-10-08 04:49:08

Using the Rational Root Test to Factor with the TI-83

Authors: Timothy W. Jones
Comments: 5 Pages. Further corrections and amplifications.

The rational root test gives a way to solve polynomial equations. We apply the idea to factoring quadratics (and other polynomials). A calculator speeds up the filtering through possible rational roots.
Category: General Mathematics

[183] viXra:1910.0022 [pdf] replaced on 2019-10-06 06:08:11

Using the Rational Root Test to Factor with the TI-83

Authors: Timothy W. Jones
Comments: 4 Pages. Corrections and amplifications.

The rational root test gives a way to solve polynomial equations. We apply the idea to factoring quadratics (and other polynomials). A calculator speeds up the filtering through possible rational roots.
Category: General Mathematics

[182] viXra:1909.0483 [pdf] replaced on 2019-09-29 20:36:38

Tutorial: The Real Powers from Extension of Two Determining Properties of Positive-Integer Powers

Authors: Steven Kenneth Kauffmann
Comments: 5 Pages.

This tutorial parlays two determining properties of positive-integer power functions of a positive real variable into closed formulas for the real power functions of a positive real variable. These two determining properties are that any positive-integer power of unity equals unity, and the linear first-order differential equation that a positive-integer power function of a real positive variable satisfies, which is implicit in its derivative. These two determining properties of positive-integer power functions of a positive real variable are extended to arbitrary real values of the positive-integer power. The extended linear first-order differential equations and initial conditions are then used to generate the Taylor expansions of those real power functions of a positive real variable around the zero value of the real power; this can be carried out in at least two different ways. Those Taylor expansions converge for every real value of the power and every positive real value of the variable, and are readily reexpressed entirely in terms of the exponential function and its inverse; one thus has closed formulas for all the real power functions of a positive real variable. Logarithms describe arbitrary positive numbers as real powers of a given positive number; they can expressed entirely in terms of the exponential function's inverse. The value of the particular positive constant whose powers yield the exponential function itself is worked out.
Category: General Mathematics

[181] viXra:1909.0080 [pdf] replaced on 2019-09-18 22:02:27

Division by Zero Because Next Infinity is Zero

Authors: Toshiro Takami
Comments: 4 Pages.

tan(π)=0, 0 =1, 1 =z =0 2000 When I saw this expression, I was surely suspicious. But I knew intuitively that Next infinity is zero. For me, infinite and zero were equal, that’s true now. The universe did not start with the Big Burn. The universe has existed for an infinite amount of time, and has repeated an infinite number of big burns. In other words, the universe is a repetition of Next infinity is zero.
Category: General Mathematics

[180] viXra:1909.0080 [pdf] replaced on 2019-09-08 16:25:04

Division by Zero Because Next Infinity is Zero

Authors: Toshiro Takami
Comments: 4 Pages.

tan(π)=0, 0 =1, 1 =z =0 2000 When I saw this expression, I was surely suspicious. But I knew intuitively that Next infinity is zero. For me, infinite and zero were equal, that’s true now. The universe did not start with the Big Burn. The universe has existed for an infinite amount of time, and has repeated an infinite number of big burns. In other words, the universe is a repetition of Next infinity is zero.
Category: General Mathematics