Results for “math.HO” — Page 4

arXiv2022

An Alternative Form of the Functional Equation for Riemann's Zeta Function

In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann's Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work "Remarques sur un beau rapport entre les series des puissances tant directes que reciproques" in Memoires de l'Academie des Sciences de Berlin 17, (1768), permit to define a special function, named A(s), which is fully symmetric and is similar to Riemann's "XI" function. To be complete we find several integral representations of the A(s) function and as a direct consequence of the second integral representation we obtain also an analytic continuation of the same function using an identity of Ramanujan.

math.HO↗

arXiv2007

Hasard et déterminisme chez Laplace

Laplace's views on randomness and determinism. The paper was written for "Cahiers rationalistes" and addresses a rather wide audience. It contains large quotations of Laplace, most of them coming from his introduction to the book "Analytical theory of probabilities".

math.HO↗

arXiv2019

Fun with "Analysis I": basic theorems in calculus revisited

This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the extreme, namely we present two proofs of the Extreme Value Theorem: "the programmer proof" that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values of the given function in a metric space setting, and an abstract space proof ("the level-set proof") for semicontinuous functions defined on compact topological spaces. Next, in the intermediate part, we consider the Intermediate Value Theorem, generalize it to a wide class of discontinuous functions, and re-examine the meaning of the intermediate value property. The trek reaches the final frontier when we discuss the Uniform Continuity Theorem, generalize it, re-examine the meaning of uniform continuity, and find the optimal delta of the given epsilon. Have fun!

math.HO↗

arXiv2009

Developments in finite Phan theory

This is a final report on finite Phan theory, a project that has been concerned with a revision and generalisation of Phan's presentation results of twisted Chevalley groups over finite fields with simply laced diagrams by unitary subgroups of rank one and two. Finite Phan theory has been initiated on request by Richard Lyons and Ronald Solomon in order to support their revision of the classification of the finite simple groups.

math.GR↗

arXiv2007

Convexity and Cone-Vexing

The idea of convexity feeds generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability. This is an overview of the origin, evolution, and trends of convexity. Study of convexity in the Sobolev Institute was initiated by Leonid Kantorovich (1912--1986) and Alexandr Alexandrov (1912--1999). This talk is a part of their memory.

math.FA↗

arXiv2007

Overhang

How far off the edge of the table can we reach by stacking $n$ identical, homogeneous, frictionless blocks of length 1? A classical solution achieves an overhang of $1/2 H_n$, where $H_n ~ \ln n$ is the $n$th harmonic number. This solution is widely believed to be optimal. We show, however, that it is, in fact, exponentially far from optimality by constructing simple $n$-block stacks that achieve an overhang of $c n^{1/3}$, for some constant $c>0$.

math.HO↗

arXiv2007

The Three Hat Problem

In this paper we study the Three Hat Problem which appeared in Puzzle Corner of the Technology Review magazine. This puzzle gives a scenario in which three players wearing hats are sitting together and each hat can be seen by everyone except the player that is wearing that hat. Each player is told that all of the hats contain a positive integer and that two of the integers add to the third. In an ordered, turn-wise, modular fashion, each player truthfully states whether or not he knows his integer. We give a strategy which allows for one of the players to solve for his integer for all possible integer configurations of the puzzle and prove it is the optimal such strategy.

math.HO↗

arXiv2007

The solution of a memorable problem by a special artifice of calculation

E731 in the Enestrom index. Originally published as "Solutio problematis ob singularia calculi artificia memorabilis", Memoires de l'academie des sciences de St-Petersbourg 2 (1810), 3-9. For $z$ the distance from the origin, and $v$ a given function of $z$, Euler wants to find a curve $s$ such that the integral of $z$ over $s$ is a maximum or a minimum. He starts with the Euler-Lagrange equation, and does a lot of manipulations with polar coordinates.

math.HO↗

arXiv2008

Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

math.HO↗

arXiv2008

Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume II(a)

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

math.HO↗

arXiv2008

Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume II(b)

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

math.HO↗

arXiv2008

Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume III

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

math.HO↗

arXiv2008

Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume IV

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

math.HO↗

arXiv2008

Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume VI

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

math.HO↗

arXiv2008

Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume V

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

math.HO↗

arXiv2007

The collaboration between Korteweg and de Vries -- An enquiry into personalities

In the course of the years the names of Korteweg and de Vries have come to be closely associated. The equation which is named after them plays a fundamental role in the theory of non-linear partial differential equations. What are the origins of the doctoral dissertation of De Vries and of the Korteweg-de Vries paper? Bastiaan Willink, a distant relative of both of these mathematicians, has sought to answer these questions. This article is based on a lecture delivered by the author at the symposium dedicated to Korteweg and de Vries at University of Amsterdam in September 2003.

physics.hist-ph↗

arXiv2007

The work of Jesse Douglas on Minimal Surfaces

This paper describes the work of Jesse Douglas on the Plateau problem, work for which he was awarded a Fields Medal in 1936, and considers the contributions Tibor Rado made in the 1930s.

math.HO↗

arXiv2011

The open mapping theorem and the fundamental theorem of algebra

This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple topological argument. The proof is elementary in nature and does not use any kind of integration (neither complex nor real). In addition, it is also independent of the fact that the roots of an analytic function are isolated. The proof is based on either the Banach or Brouwer fixed point theorems. In particular, this shows that one can obtain a proof of the FTA (albeit indirect) which is based on the Brouwer fixed point theorem, an aim which was not reached in the past and later the possibility to achieve it was questioned. We close this note with a simple generalization of the FTA. A short review of certain issues related to the OMT and the FTA is also included.

math.CV↗