مجله فرهنگ و اندیشه ریاضی
سال چهل و دوم شماره 1 (پیاپی 72، بهار و تابستان 1402)

Elsewhere, the history and introduction of harmonic analysis into Iran hasbeen briefly discussed. Abstract harmonic analysis is, without doubt, a natural extensionof the Fourier’s series and integrals. It’s worth noting that Harmonic analysis isnothing but the development of the theory of series and integrals on topological groups(or similar structures). The history of Fourier analysis can be divided into two periods:classical and modern. The classical period of Fourier analysis, which spanned fromFourier’s treatise into the 1950s, focused mainly on investigating the convergence andsummability methods of Fourier series. The modern era of Fourier analysis was essentiallyshaped by Lebesgue’s theory of integration and the introduction of the concept ofabstract function spaces, thus shifting the focus from examining individual members ofa space of functions to investigating the entire space. This era was further developed inFourier analysis by Antoni Zygmund’s introduction of real variable methods. AlbertoCalderon’s contributions began in the 1950s.In the first part of this paper, we briefly survey the most critical developmentsin Fourier analysis from the early days till recent decades. Then, in the second part,an interview with M. Hormozi –a Fourier analyst who graduated from the ChalmersUniversity of Technology– provides us with a more rigorous picture of the modern eraof Fourier analysis.

The representation theory of finite groups has important applications in investigatingthe structure of abstract finite groups.This theory was developed by Frobenusmore than 100 years ago and later studied by Burnside, Schur, and Brauer. Inthis article we try to explain fundamental theorems of this theory and mention a fewapplications.

In this expository paper‎, ‎a description of null hypersurfaces in space forms and some of their applications have been given‎. ‎Such hypersurfaces are of special interest in geometry and physics‎. ‎Among other results of the paper‎, ‎geometric properties of black holes‎, ‎event horizons‎, ‎the important Cheeger–Gromoll splitting theorem and some famous space-forms such as asymptotically flat/de Sitter space forms have been stated and considered‎. ‎Finally‎, ‎null submanifolds in pseudo-Riemannian manifolds have been described shortly.

This paper deals with some of the most important and longstanding homological conjectures‎. ‎The framework which‎ ‎is of interest to us is that of a finite dimensional algebra over a field‎. ‎Auslander-Reiten conjecture‎, ‎finitistic dimension conjecture‎, ‎(Generalized) Nakayama conjecture‎, ‎and vanishing conjecture are discussed in separate sections and the interaction between them‎ ‎is studied‎. ‎As declared by the title‎, ‎the current paper will put an emphasis on the Auslander-Reiten conjecture.

In the most mathematical texts‎, ‎there have been paid less attention to “symmetric difference” and its properties‎. ‎In this article‎, ‎after verifying some basic properties of this operation‎, ‎we provide a formula for computing the cardinal (res‎. ‎probability) of the sets (res‎. ‎events) obtained from the combination of symmetric difference of finite number of sets (res‎. ‎events)‎, ‎in terms of the cardinal (res‎. ‎probability) of the constituent sets (res‎. ‎events)‎. ‎Finally‎, ‎we end the article by an application in cryptography‎.

Many standard problems in calculus can be easily solved by an innovativevisual approach that makes no use of formulas. The method is based on Mamikon’ssweeping-tangent theorem, a geometrically intuitive result that is easily understood byvery young students. In this paper, the method of sweeping tangents introduced andshown how it can be used to find (without the machinery of calculus) areas of manyplane regions, including an oval ring, a parabolic segment, a hyperbolic segment, theregion under a general power function, an exponential curve, a logarithmic curve, atractrix, the region between two curves traced by the rear and front wheels of a bicycle,the region enclosed by a cardioid, and by each member of a family of lima¸cons.The treatment of the parabolic segment and the exponential use geometric propertiesof subtangents to these curves, which can also be used to draw tangent lines. An unexpectedapplication of the method of sweeping tangents is to physics. In this application,conservation of angular momentum in a central force field is deduced as an elementaryconsequence of Mamikon’s sweeping-tangent theorem.

In the 16th and 17th centuries the classical Greek notions of (discrete) numberand (continuous) magnitude (preserved in medieval Latin translations of Euclid’sElements) underwent a major transformation that turned them into continuous but measurablemagnitudes. This article studies the changes introduced in the classical notionsof number and magnitude by three influential Renaissance editions of Euclid’s Elements.Besides providing evidence of earlier discussions preparing notions and argumentseventually introduced in Simon Stevin’s Arithmétique of 1585, these editionsdocument the role abacus algebra and Renaissance views on the history of mathematicsplayed in bridging the gulf between discrete numbers and continuous magnitudes.

This article deals with the life and times of Olga Taussky-Todd‎, ‎the first female professor of mathematics at the‎
‎California Institute of Technology‎, ‎and her decades-long quest in Europe and the United States for a permanent academic position‎.

Beginning in 1968‎, ‎Realistic Mathematics Education (RME) has evolved into the main‎
‎approach to mathematics education in the Netherlands‎. ‎This paper describes how this fifty year‎
‎reform came into being and developed further.

In the past decades, advanced probabilistic methods have had significant impacton the field of finance, both in academia and in the financial industry. Conversely,financial questions have stimulated new research directions in probability. In this surveypaper, we review some of these developments and point to some areas that mightdeserve further investigation. We start by reviewing the basics of arbitrage pricing theory,with special emphasis on incomplete markets and on the different roles played bythe “real-world” probability measure and its equivalent martingale measures. We thenfocus on the issue of model ambiguity, also called Knightian uncertainty. We presenttwo case studies in which it is possible to deal with Knightian uncertainty in mathematicalterms. The first case study concerns the hedging of derivatives, such as varianceswaps, in a strictly pathwise sense. The second one deals with capital requirementsand preferences specified by convex and coherent risk measures. In the final two sectionswe discuss mathematical issues arising from the dramatic increase of algorithmictrading in modern financial markets.

We survey the published work of Harry Kesten in probability theory‎, ‎with emphasis‎
‎on his contributions to random walks‎, ‎branching processes‎, ‎percolation‎, ‎and related‎
‎topics

This is a translation of some parts of Chapter 11 in  I Want to be a Mathematician  (1985)‎, ‎by Paul R‎. ‎Halmos.

It is safe to say that the history of mathematics in our country has beenneglected, and this lack of attention has been twofold in relation to the history of mathematicsin Iran in the contemporary era. After conversing, exchanging thoughts, andidentifying individual concerns and possibilities, we gradually created a plan titled,’Documenting the History of Mathematics in Contemporary Iran’. It is worth notingthat this collection comprises three volumes that have been prepared and published thusfar. In this paper, we provide a concise overview of the project. We present a brief descriptionof the project’s recent outputs, which have been published as a three-volumebook so far.