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In this paper, a joint study of the behavior of solutions of the Heavy Ball ODE and Heavy Ball type algorithms is given. Since the pioneering work of Polyak (USSR Comput Math Math Phys 4(5):1–17, 1964), it is well known that such a scheme is very efficient for C 2 strongly convex functions with Lipschitz gradient. But much less is known when only growth conditions are considered. Depending on the geometry of the function to minimize, convergence rates for convex functions, with some additional regularity such as quasi-strong convexity, or strong convexity, were recently obtained in Aujol et al. (Convergence rates of the Heavy-Ball method for quasi-strongly convex optimization, 2020). Convergence results with much weaker assumptions are given in the present paper: namely, linear convergence rates when assuming a growth condition (which amounts to a Łojasiewicz property in the convex case). This analysis is firstly performed in continuous time for the ODE, and then transposed for discrete optimization schemes. In particular, a variant of the Heavy Ball algorithm is proposed, which converges geometrically whatever the parameters choice, and which has the best state of the art convergence rate for first order methods to minimize composite non smooth convex functions satisfying a Łojasiewicz property. [ABSTRACT FROM AUTHOR]

This paper belongs to a field of research that has appeared comparatively recently, namely, the study of mathematics competitions. The paper utilizes historical-theoretical methodology and is devoted to changes in the way in which mathematics competitions have been conducted in Russia in recent decades. Mathematics competitions, like mathematics education as a whole, are seen as part of a social process, and changes in them are seen as expressions of new social needs and possibilities. The analysis of the system of competitions, as it evolved in the USSR, and the literature devoted to them, makes it possible to identify certain challenges recognized by the mathematics community. In this paper I discuss examples of new competitions that have emerged to address these challenges. It is noted that new types of competitions have emerged, including team competitions and competitions involving mass participation by students. Also discussed is the emergence of new competitions with subject matter closer to that of ordinary school mathematics than that of old, traditional competitions. An analysis of the situation in Russia reveals the existence of a significant group of mathematics educators, parents, and children interested in the development of mathematics competitions. Also, in the paper I call attention to certain questions that I consider worthy of further study. [ABSTRACT FROM AUTHOR]

The political changes in the USSR and Russia almost coincided with the beginning of an unprecedented leap in the development of information technologies in the country (and to a certain extent spurred it). Gradually, many different initiatives emerged to apply these technologies in education in general, and in mathematics education. In the paper, we discuss several examples of didactical innovations in teaching and learning mathematics with technology, namely, LOGO-based microworlds, dynamic geometry and computer algebra systems, as well as verifiers. The examples we provide, along with the theoretical perspective of productive learning and thinking, seem to demonstrate the innovative potential of new tools to make learning more visual, dynamic, and interactive. However, the issues of aligning with curriculum and standard examinations, teacher professional development, and rapid changes in technologies, require more attention by mathematics educators. The presence of such opposite trends makes the task of analyzing the history of the development of mathematics teaching technologies and underlying pedagogies relevant and important for predicting the development of this field in the near (and far-extended) future, and eventually preventing the repetition of errors in this new direction for mathematics education. In the paper we focus primarily on problems and processes in Russia, but they are investigated in connection with and against the backdrop of global trends. [ABSTRACT FROM AUTHOR]

The Soviet Union collapsed in 1991, and Russia, the largest of the countries formed in its place, has gone through notable changes in the period since then. This paper is devoted to the study of how (and whether) the mathematics curriculum has changed. At one time, before the revolution of 1917, Russian mathematics education was closely connected with Western European mathematics education (above all, German and French). Subsequently, their paths diverged, and later, Russian mathematics education became very highly regarded around the world. What, then, has happened over the last three decades? To what extent does the Russian curriculum today differ from the old Soviet curriculum and from contemporary curricula in other countries? The answers to these questions are of interest to an international audience already because they help in understanding the processes and distinctive characteristics of the development of mathematics education, including differences that might be concealed even behind identical terms. Based on a detailed analysis of Russian standards and textbooks, conclusions are drawn about the processes in the transformation of the curriculum that have occurred over thirty years, including the conclusion that the period of transition that began in the late 1980s is still not over. The Russian curriculum, of course, is not exactly the same as the curriculum of Soviet times, but the differences between them are by no means revolutionary. The paths of the development of mathematics education in Russia and in Western Europe and the United States, having at some point diverged, have still not come back together. [ABSTRACT FROM AUTHOR]

For a long time, Soviet students learned 'pure' mathematics in their mathematics classrooms, while applications of mathematics were introduced in their science (mainly physics) classrooms. This approach was a part of a uniform and rigid national curriculum. Even when in the 1990s the world was moving towards including applications in school mathematics, Russian students continued to engage in pure mathematics learning in their mathematics classrooms. It was physics teachers' responsibility to teach applications of mathematics; therefore, physics courses were highly mathematics-intensive, making extensive use of mathematics from algebra to calculus in the formulation of scientific laws and the investigation of their consequences. The collapse of the Soviet Union and some liberalization in educational policy led to changes in graduation requirements in mathematics and physics as well as diversity in mathematics and physics curricula in schools. Based on a review of textbooks, standards, curriculum documents, and other resources, in this paper we analyze changes that affected the teaching and learning of mathematics in physics classrooms in Russia from the Soviet period to the present. [ABSTRACT FROM AUTHOR]

The article profiles the Russian mathematician Andrei Andreevich Markov Jr. It includes information on Markov's personal and educational background, as well as his early interest in abstract mathematics indicated by his series of papers on geometry, algebra, topology and analysis. Markov, however, is most well-known for founding the Russian school of constructive mathematics during the late 1940s and early 1950s. His constructive mathematics is considered as one of the 20th century's most important and coherent constructivist trends.

Why did British writers, when they wrote about the Soviet Union, often deploy the imagery of numbers, arithmetic and mathematics? This paper scrutinises a number of such instances, including Orwell's famous use of the equation ‘2 + 2 = 5’ inNineteen Eighty-Fourand Koestler's fascination with Euclid's proof of the infinitude of prime numbers inThe Invisible Writing. These are put into relation with less celebrated works where questions of number or of mathematical reasoning are politicised by being applied to the Soviet Union. The paper situates these literary representations in relation to three key debates that intersected in interesting ways. Firstly, a debate about utilitarianism's attempt to quantify social goods and the Romantic rejection of that attempt; secondly, a debate about the philosophical foundations of mathematics (which involved Russell, Wittgenstein and Heidegger); and finally, a debate about the relation between mathematics and dialectical materialism, which involved key British and Soviet scientists and mathematicians and reflected on the position of science under Communism. Taking my cue from recent calls by Alain Badiou and Steven Connor for a rapprochement between the humanities and mathematics, I argue that this was a period in which numbers and arithmetic were profoundly politicised in literature. [ABSTRACT FROM PUBLISHER]

A model for the placement of n detection stations for optimum coverage of an arbitrary plane area is described. The stations are assumed to be identical and to have a probability of detection that is a function only of the distance between the station and the event to be detected. Furthermore, the stations are assumed to operate independently of each other. It is also assumed that the enemy has complete knowledge of the station locations and effectiveness and is interested only in eluding detection by the detection stations. Thus, the situation is reduced to the minimax problem of placing the stations so that the maximum probability of not detecting an enemy event is minimized. A hill climbing iterative technique for finding the optimum locations is described in some detail. This technique is illustrated for the problem of locating detection stations within the United States and Soviet Union. The results of these applications are presented and discussed. The paper concludes with some remarks on how this model can be made more descriptive of the real world situations being modeled. [ABSTRACT FROM AUTHOR]

We consider the classical point vortex model in the mean-field scaling regime, in which the velocity field experienced by a single point vortex is proportional to the average of the velocity fields generated by the remaining point vortices. We show that if at some time the associated sequence of empirical measures converges in a renormalized H ˙ - 1 sense to a probability measure with density ω 0 ∈ L ∞ (R 2) and having finite energy as the number of point vortices N → ∞ , then the sequence converges in the weak-* topology for measures to the unique solution ω of the 2D incompressible Euler equation with initial datum ω 0 , locally uniformly in time. In contrast to previous results Schochet (Commun Pure Appl Math 49:911–965, 1996), Jabin and Wang (Invent Math 214:523–591, 2018), Serfaty (Duke Math J 169:2887–2935, 2020), our theorem requires no regularity assumptions on the limiting vorticity ω , is at the level of conservation laws for the 2D Euler equation, and provides a quantitative rate of convergence. Our proof is based on a combination of the modulated-energy method of Serfaty (J Am Math Soc 30:713–768, 2017) and a novel mollification argument. We contend that our result is a mean-field convergence analogue of the famous theorem of Yudovich (USSR Comput Math Math Phys 3:1407–1456, 1963) for global well-posedness of 2D Euler with vorticity in the scaling-critical function space L ∞ (R 2) . [ABSTRACT FROM AUTHOR]

Russian culture developed under the burden of a series of totalitarian governments, a circumstance that, oddly, supported mathematics education. The Soviet period in particular saw a flowering of mathematical culture, the seeds of which have, since the collapse of the Soviet Union, been carried to the rest of Europe and America. A trickle of ideas, materials—and émigrés—has swollen to a powerful stream. Some of the heaviest influences have occurred in the field of extra-curricular education. We explore some of the ways that Russian and Soviet mathematical culture has fertilized this field. In particular, we look closely at the pivotal role of the professional mathematical communities, both in Russia and in America. [ABSTRACT FROM AUTHOR]

This article reinterprets algorithmic rationality by looking at the interaction between mathematical logic, mechanized reasoning, and, later, computing in the Russian Imperial and Soviet contexts to offer a history of the algorithm as a mathematical object bridging the inner and outer worlds, a humanistic vision that we, following logician Vladimir Uspensky, call the "culture of the impossible." We unfold the deep roots of this vision as embodied in scientific intelligentsia. In Part I, we examine continuities between the turn-of-the-twentieth-century discussions of poznaniye—an epistemic orientation towards the process of knowledge acquisition—and the postwar rise of the Soviet school of mathematical logic. Establishing this connection allows us to explain, in Part II, the role of the algorithm in disciplinary dynamics between mathematical logic and cybernetics and a characteristic understanding of programming, not as a narrow skill, but as a matter of consciousness. [ABSTRACT FROM AUTHOR]

The Łojasiewicz inequality shows that sharpness bounds on the minimum of convex optimization problems hold almost generically. Sharpness directly controls the performance of restart schemes, as observed by Nemirovskii and Nesterov [USSR Comput. Math. Math. Phys., 25 (1985), pp. 21--30]. The constants quantifying these sharpness bounds are of course unobservable, but we show that optimal restart strategies are robust, and searching for the best scheme only increases the complexity by a logarithmic factor compared to the optimal bound. Overall then, restart schemes generically accelerate accelerated methods. [ABSTRACT FROM AUTHOR]

Let r d be the maximum number of skew lines that a smooth projective surface of degree d (over the complex numbers) can have. It is known that r 3 = 6 , r 4 = 16 (Schläfli in Q J Math Soc 2:55–65, 110–121, 1858; Nikulin in Math USSR Izv 9:261–275, 1975) and was proven by Miyaoka in 1975 that r d ≤ 2 d (d - 2) if d ≥ 4 (Miyaoka in Math Ann 268:159–172, 1984). Up to now r d remains unknown for d ≥ 5 . However, the lower bound d (d - 2) + 2 was found by Rams (Proc Am Math Soc 133(1):11–13, 2005) which was improved by Boissière and Sarti (Ann Scuola Norm Sup Pisa Cl Sci 5:39–52, 2007), who showed that d (d - 2) + 4 ≤ r d for d ≥ 5 and odd. In this work, we take the family of degree d smooth surfaces R d in P 3 (cf. (1)), considered by Boissière and Sarti (2007) and study r (R d) , the maximum number of skew lines that R d can have. In fact, we prove that r (R d) = d (d - 2) + 4 if d ≥ 5 and odd. Otherwise, we prove that r (R 6) ≥ 48 , which implies that Miyaoka's bound is sharp for d = 6 , i.e. r 6 = 48 . Still in the even case, we show that R d contains d (d - 2) + 4 skew lines and we improve the Miyaoka's bound for the family R d if d is even (Theorem 4.11). [ABSTRACT FROM AUTHOR]

The article outlines the significant contributions of Russian mathematicians throughout history and the leading roles they have always played in the advance of mathematical theory. Among these scientists were Leonard Euler and Nicholas Lobachevsky, discoverer of non-Euclidian geometry. The author contends that Russian mathematics is untainted with any political and socio-economic bias. Most of their mathematicians were active in all fields of mathematical research, specifically applied mathematics. They also liked to publish books and mathematical journals. Nevertheless, they did not have enough freedom in making personal contacts outside Soviet Union.

Over the past 20-30 years multi-objective mathematical programming (MOMP) has emerged as an increasingly active area of research in the fields of management science, operations research, applied mathematics, and engineering. Despite the intensity of interest, however, earlier surveys of MOMP methods have all but ignored Soviet work in this area. Published in unfamiliar journals and often available only in Russian, the Soviet research has remained virtually unknown outside of the USSR and Eastern Europe. This has been particularly unfortunate given the extent, importance, and originality of much of the Soviet research. The current article attempts to correct this situation by providing a comprehensive, yet nontechnical, overview of Soviet MOMP methods. Using a taxonomy similar to ones that have been applied to Western MOMP techniques, the article categorizes the Soviet methods based on when they involve a decision maker in the solution process--a priori, progressively, a posteriori, or never--and what kind of information the decision maker provides. In all, 17 methods are analyzed with particular attention given to their distinctive features and to the issues of computational feasibility and burden on the decision maker. The article concludes with observations on the overall character of Soviet MOMP research, comparing the general directions in Soviet and Western research. [ABSTRACT FROM AUTHOR]

The article offers information on the logical system that is extended with schema. It mentions that if axiomatic theory has decidable atomic predicates then the predicates of the theory is decidable. It mentions that the atomic predicates of the Peano's axioms are decidable. It offers information on several theorems used to define the logical system. It mentions that the logical system was related to Russian constructivism.

The article presents an author's interpretation on the technical education in the Soviet Union. The author includes a description of the elementary and secondary education with the emphasis on the technical side including some of the curricula in the schools and how a person can get through the Soviet school system. According to the author, the first through third grades includes a study on Russian language and literature, mathematics, freehand drawing, singing, physical education, work and practice exercises.