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Articles 1 - 30 of 807
Full-Text Articles in Mathematics
How Euler Could Have Done It: Euler And A Heuristic Derivation Of The Prime Number Theorem, Alexander Aycock
How Euler Could Have Done It: Euler And A Heuristic Derivation Of The Prime Number Theorem, Alexander Aycock
Euleriana
We present an argument that leads to the formulation of the Prime Number Theorem that Euler could have given.
The Modern History Of The Basel Problem, John Campbell, Paul Levrie
The Modern History Of The Basel Problem, John Campbell, Paul Levrie
Euleriana
The \emph{Basel problem} refers to the problem of determining a closed form for the infinite series $\frac{1}{1^2} + \frac{1}{2^2} + \cdots$. If we consider what mathematical results have the most peer-reviewed papers devoted to new ways of proving such results, Euler's formula $\frac{1}{1^2} + \frac{1}{2^2} + \cdots = \frac{\pi^2}{6}$ is certainly among the top of such results. This motivates our historical study of peer-reviewed papers based on proofs of Euler's formula, and we introduce what appears to be the most comprehensive and up-to-date and exhaustive catalogue of peer-reviewed, published papers in the 20th and 21st centuries devoted to or mainly …
Even Repdigits Are Not Perfect: Conclusions From Triangular Numbers, Uwe Hassler
Even Repdigits Are Not Perfect: Conclusions From Triangular Numbers, Uwe Hassler
Euleriana
Are there nontrivial repdigits that are perfect numbers? This note gives an answer: not for numbers smaller than $10^{1500}$. This finding follows from results for triangular numbers. Passing by, I review Euler's contributions on polygonal numbers.
Euler And Gauss On The Hypergeometric Function -- A Comparison, Alexander Aycock
Euler And Gauss On The Hypergeometric Function -- A Comparison, Alexander Aycock
Euleriana
We present several results stated by Euler on what is today called the Gaussian hypergeometric function. For this purpose, we compare the results found by Gauss to Euler's results.
The Extent To Which The Earth's Motion Is Perturbed By The Moon, More Accurately Investigated: An English Translation Of E139, Patrick T. Headley
The Extent To Which The Earth's Motion Is Perturbed By The Moon, More Accurately Investigated: An English Translation Of E139, Patrick T. Headley
Euleriana
In Tabulae astronomicae solis et lunae (Solar and lunar astronomical tables) (E87), published in 1746, Euler made a first attempt to correct his astronomical tables for the gravitational attraction of the Earth toward the Moon, simply by accounting for the difference between the position of the Earth and the center of gravity of the Earth-Moon system. In this article Euler revisited the problem, noting that this center of gravity would not itself take an elliptical path around the Sun. To model the motion more accurately, Euler considered the separate gravitational attractions of the Sun and Moon acting on the Earth. …
About Infinite Algebraic Curves Whose Indefinite Lengths Equal An Elliptic Arc: An English Translation Of E780, Derek R. Aoki
About Infinite Algebraic Curves Whose Indefinite Lengths Equal An Elliptic Arc: An English Translation Of E780, Derek R. Aoki
Euleriana
Euler continues his study of algebraic curves with equal arc length, a subject to which he returned several times. After a brief review, he introduces an infinite family of curves with the same indefinite length as a given ellipse. However, this first family is by his own admission poorly motivated, so he derives directly a different but related family of curves with the same indefinite length as a given ellipse.
Historical Beginnings To Modern Results, Christopher Goff, Erik Tou
Historical Beginnings To Modern Results, Christopher Goff, Erik Tou
Euleriana
An introduction to the contents in Volume 6 (Issue 1) of Euleriana.
How Euler Could Have Done It: Euler And A Direct Proof For The Functional Equation For The Riemann Zeta-Function, Alexander Aycock
How Euler Could Have Done It: Euler And A Direct Proof For The Functional Equation For The Riemann Zeta-Function, Alexander Aycock
Euleriana
We present a chain of argumentation for a direct proof of the functional equation for the Riemann ζ–function that Euler could have presented.
Euler’S Original Derivation Of Elastica Equation, Shigeki Matsutani
Euler’S Original Derivation Of Elastica Equation, Shigeki Matsutani
Euleriana
Euler derived the differential equations of elastica by the variational method in 1744, but his original derivation has never been properly interpreted or explained in terms of modern mathematics. We elaborate Euler's original derivation of elastica and show that Euler used Noether's theorem concerning the translational symmetry of elastica, although Noether published her theorem in 1918. It is also shown that his equation is essentially the static modified KdV equation which is obtained by the isometric and isoenergy conditions, known as the Goldstein-Petrich scheme.
Early Theories On Fluid Resistance And Translation Of Euler’S “Dilucidationes De Resistentia Fluidorum”, Sylvio R. Bistafa
Early Theories On Fluid Resistance And Translation Of Euler’S “Dilucidationes De Resistentia Fluidorum”, Sylvio R. Bistafa
Euleriana
In 1763, Euler published Dilucidationes de resistentia fluidorum (Explanations on the resistance of fluids), a memoir that challenges the fluid resistance theories proposed by Isaac Newton and d’Alembert. Euler's work explores the resistance experienced by solid bodies moving through fluids, critiquing both Newton's "common rule" and d’Alembert's paradox, which predicted zero resistance for non-viscous fluids. Euler's treatise is divided into two parts: the first focuses on the mathematical modeling of fluid flow patterns, while the second addresses the calculation of fluid resistance on surfaces. Despite significant advancements, Euler's work remains constrained by the limitations of non-viscous fluid assumptions, ultimately grappling …
Assistance For The Calculation Of Sines (Translation Of E246), Julian Schennach
Assistance For The Calculation Of Sines (Translation Of E246), Julian Schennach
Euleriana
Paralleling his famous relation eiφ = cos(φ) + i sin(φ), Euler establishes the equality (cos φ + i sin φ)n = (cos nφ + i sin nφ). He uses it to comprehensively derive trigonometric identities that convert arbitrary powers of sines and cosines of an angle (and products thereof) into sums of sines and cosines of multiples of that angle. Some negative and fractional powers are shown to yield infinite series. Euler further describes a general method to evaluate various infinite series involving weighted trigonometric functions. These results foreshadow Fourier series. As Euler points out, the scope of …
On Amicable Numbers, Jonathan David Evans
On Amicable Numbers, Jonathan David Evans
Euleriana
This is an English translation of Euler's 1750 paper "De numeris amicabilibus" (E152), the most substantial of his three works with this name. In it, he expounds at great length the ad hoc methods he has developed to search for pairs of amicable numbers, concluding with a list of around 60 new pairs.
A Reader's Guide To Daniel Bernoulli's "Recurrent Series", Stacy Langton
A Reader's Guide To Daniel Bernoulli's "Recurrent Series", Stacy Langton
Euleriana
This is a commentary and reader’s guide to Daniel Bernoulli’s article “Observations concerning recurrent series”. It is intended to help the modern reader understand what Bernoulli is doing in that article.
Daniel Bernoulli's "Observations Concerning Recurrent Series", Stacy Langton
Daniel Bernoulli's "Observations Concerning Recurrent Series", Stacy Langton
Euleriana
Daniel Bernoulli’s article "Observations concerning Recurrrent Series" is one of the gems of the mathematical literature. It describes a numerical method for solving polynomial equations. This method, now known as “Bernoulli’s method”, is still in use today. The basis of the method is the study of certain sequences of numbers that satisfy what we now call linear recurrence relations. Bernoulli’s article shows how to find solutions of these recurrence relations. A nice application is a now well-known formula for the Fibonacci numbers. Bernoulli also uses his theory of recurrence relations to give a proof of an early form of De …
Comprehensive Conversations, Christopher Goff, Erik Tou
Comprehensive Conversations, Christopher Goff, Erik Tou
Euleriana
An introduction to the contents in Volume 5 (Issue 1) of Euleriana.
Discrete Math For Computer Science - Chapter 8: Union And Intersection And Complement: Set Identities, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 8: Union And Intersection And Complement: Set Identities, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 15: Mathematical Induction, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 15: Mathematical Induction, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 13: Sequences: Recurrence Relations, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 13: Sequences: Recurrence Relations, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 26: Introduction To Graphs: Graph Representations, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 26: Introduction To Graphs: Graph Representations, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 2: Logical Equivalence: Laws Of Propositional Logic, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 2: Logical Equivalence: Laws Of Propositional Logic, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 11: Inverse Of A Function: Composition Of Functions, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 11: Inverse Of A Function: Composition Of Functions, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 16: Recursive Definitions: Recursive Algorithms, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 16: Recursive Definitions: Recursive Algorithms, Houman Kamran Habibkhani
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Discrete Math For Computer Science, Houman Kamran Habibkhani
Discrete Math For Computer Science, Houman Kamran Habibkhani
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Discrete Mathematics and its Applications is a focused introduction to the primary themes in a discrete mathematics course, as introduced through extensive applications, expansive discussion, and detailed exercise sets. These themes include mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and enhanced problem-solving skills through modeling. Its intent is to demonstrate the relevance and practicality of discrete mathematics to all students.
Discrete Math For Computer Science - Chapter 4: Logical Reasoning: Rules Of Inference, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 4: Logical Reasoning: Rules Of Inference, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 14: Summations, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 14: Summations, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 19: Number Representation, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 19: Number Representation, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 27: Paths And Cycles: Graph Connectivity, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 27: Paths And Cycles: Graph Connectivity, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 28: Introduction To Trees: Properties Of Trees, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 28: Introduction To Trees: Properties Of Trees, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 29: Tree Traversals: Spanning Trees, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 29: Tree Traversals: Spanning Trees, Houman Kamran Habibkhani
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Discrete Math For Computer Science - Chapter 3: Predicates And Quantifiers: Quantified Statements, Houman Kamran Habibkhani
Discrete Math For Computer Science - Chapter 3: Predicates And Quantifiers: Quantified Statements, Houman Kamran Habibkhani
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