مجله فرهنگ و اندیشه ریاضی
سال چهل و یکم شماره 1 (پیاپی 70، بهار و تابستان 1401)

Giving a brief description on types of scientific/academic journals and articles, we focus on, mainly expository, mathematical journals published in Iran during recent decades. Then we give a brief history of “Farhang va Andishe-ye Riyazi”, an expository journal published by the Iranian Mathematical Society. We conclude with a table of contents of entire journal’s published volumes within its 40 years life.

This is a translation of the first chapter in What’s the Use?—How Mathematics Shapes Everyday Life (2021), by I. Stewart.

Hilbert’s nineteenth problem concerns the regularity of partial differential equations solutions. In this paper, we review some of the developments and important results related to this problem. We also get a glimpse of two important classes of problems in regularity theory, variational inequalities and minimal surfaces.

Ultrafilter appears as an utmost applicable concept in numerous branches of mathematics including logic, algebra, topology, nonstandard analysis, and functional analysis. In this article, regarding the properties and applications of ultrafilters, we intend to identify the structure of this concept specially in functional analysis. To this end, ultrafilters are investigated in terms of set theory, measure theory, Stone-Cech compactification, nonstandard analysis and functional analysis which enable us to describe this concept from various perspectives.

Due to the huge amount of information available online, it seems necessary to have a recommender system that automatically and intelligently suggests to users. Recommendation systems constitute a specific type of information filtering technique that attempt to present items according to the interest expressed by a user. Collaborative Filtering (CF) is the process of evaluating information using the opinion of other people and computes recommendations based on the information about similar items or users. One of the challenging problems is the sparsity problem of the user-item matrix. Collaborative recommenders try to capture relationships among user-user or item-item pairs by reducing the dimensionality of the user or rather item space. Matrix factorization techniques are employed to reduce the dimension of the observed dataset. In our work we want to investigate the well-known Singular Value Decomposition (SVD) and Semi Discrete Decomposition (SDD), which factorize a matrix into three low-dimensional matrices. The comparison of the methods shows that although SVD has the better results than SDD, SDD is efficient in terms of time and especially memory by producing an error close to SVD.

The inscribed square problem, also known as the square peg problem or the Toeplitz’ conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of some square? This is true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911. Some early positive results were obtained by Arnold Emch and Lev Schnirelmann. As of 2022, the general case remains open. In this paper we give a brief survey of most important results in this respect.

Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented.

What is Mathematics?” [with a question mark!] is the title of a famous book by Courant and Robbins, first published in 1941, which does not answer the question. The question is, however, essential: The public image of the subject (of the science, and of the profession) is not only relevant for the support and funding it can get, but it is also crucial for the talent it manages to attract—and thus ultimately determines what mathematics can achieve, as a science, as a part of human culture, but also as a substantial component of economy and technology.

A palpable example for a one-to-one Thomae-type popcorn continuous function of the real line is constructed.

In August 2015 a distinguished collection of numerical analysts gathered at Oxford to celebrate the sixtieth birthday of Nick Trefethen FRS and consider the future of numerical analysis. Some of the plenary speakers provided short essays for Notices.