Steklov Mathematical Institute Research Articles

A spectral theorem for compact representations and non-unitary cusp forms

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series.

From quantum field theory to the quantum method the inverse problem On the 90th anniversary of the birth of academician L.D. Faddeev

The article is dedicated to the life and work of a theoretical physicist and mathematician, a full member of the USSR Academy of Sciences and RAS L.D. Faddeev (1934−2017), the formation of his scientific school. Faddeev’s works on scattering theory, quantum field theory, classical and quantum theory of integrable systems were included in the golden fund of world science and largely determined the face of modern mathematical physics. The author of the article is an employee of the Laboratory of Mathematical Problems of Physics created by L.D. Faddeev at the St. Petersburg Department of the V.A. Steklov Mathematical Institute of the Russian Academy of Sciences since its foundation.

Ludwig Dmitrievich Faddeev. 23 March 1934—26 February 2017

Ludwig Faddeev was a preeminent mathematical physicist of the second half of the twentieth and early twenty-first century. He made fundamental contributions to mathematics and theoretical physics. These included the solution of the quantum three-body scattering problem, groundbreaking work on the quantization of gauge theories, the formulation of Hamiltonian methods for classical integral models, and the creation, with colleagues, of the quantum inverse scattering method which led to the discovery of quantum groups. An incomplete list of important concepts and results named after him include the Faddeev equations, the Faddeev–Popov determinant, Faddeev–Popov ghosts, the Faddeev–Green function, the Gardner–Faddeev–Zakharov bracket, Faddeev–Zamolodchikov algebra and the Faddeev quantum dilogarithm. Faddeev was a charismatic teacher and a natural leader at the Steklov Institute of the Russian Academy of Sciences and the International Mathematical Union.

On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations

Nikulin [On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2 reflections, J. Soviet. Math. 22 (1983) 1401–1476; Surfaces of type [Formula: see text] with a finite automorphism group and a Picard group of rank three, Proc. Steklov Institute of Math. (3) (1985) 131–155] and Vinberg [Classification of 2-reflective hyperbolic lattices of rank 4, Trans. Moscow Math. Soc. (2007) 39–66] proved that there are only a finite number of lattices of rank [Formula: see text] that are the Néron–Severi lattice of projective [Formula: see text] surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such [Formula: see text] surfaces [Formula: see text], when these surfaces have moreover no elliptic fibrations. In that case, we show that such [Formula: see text] surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study the converse, i.e. if the geometric description we obtained characterizes these surfaces. In four cases, the description is sufficient, in each of the four other cases, there is exactly another one possibility which we study. We obtain that at least five moduli spaces of [Formula: see text] surfaces (among the eight we study) are unirational.

Sturm's Theorem on the Zeros of Sums of Eigenfunctions: Gelfand's Strategy Implemented

In the second section ``Courant-Gelfand theorem'' of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273 (2011) 25--34), Arnold recounts Gelfand's strategy to prove that the zeros of any linear combination of the $n$ first eigenfunctions of the Sturm-Liouville problem $$-\, y''(s) + q(x)\, y(x) = \lambda\, y(x) \mbox{ in } ]0,1[\,, \mbox{ with } y(0)=y(1)=0\,,$$ divide the interval into at most $n$ connected components, and concludes that ``the lack of a published formal text with a rigorous proof \dots is still distressing.''\\ Inspired by Quantum mechanics, Gelfand's strategy consists in replacing the ana\-lysis of linear combinations of the $n$ first eigenfunctions by that of their Slater determinant which is the first eigenfunction of the associated $n$-particle operator acting on Fermions.\\ In the present paper, we implement Gelfand's strategy, and give a complete proof of the above assertion. As a matter of fact, refining Gelfand's strategy, we prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836).

СУДЬБЫ И СЕМИНАРЫ О семинаре «Проблемы сокращения перебора» в ЛЭТИ в историко-научном контексте.

Exhaustive Search» organized at the Leningrad Electrotechnical Institute (LETI) by R.I. Freidson (1942-2018) is considered. The seminar opened in 1982, one of its main aims being the development of scientific ideas of S. Yu. Maslov (1939-1982). The meetings continued regularly until the beginning of 1990es. The historico-scientific context is outlined, including the links with other contemporary seminars, such as the S.Yu. Maslov’s seminar and the seminar on mathematical logic at the Leningrad Branch of the Steklov Mathematical Institute (LOMI); the ideas developed at the seminar and main results represented by scientific publications are considered. The pedagogical role of the seminar (its formative influence on young researchers), the organizational talent of R.I. Freidson and the effect of his personality on seminar’s creative atmosphere are considered as well.

Preface

The IV International conference “Supercomputer Technologies of Mathematical Modelling (SCTeMM’19) was organized by Steklov Mathematical Institute of the Russian Academy of Sciences in cooperation with Ammosov North-Eastern Federal University, Ivannikov Institute for System Programming of the Russian Academy of Sciences and Bauman Moscow State Technical University on June 19-21, 2019 in Moscow (Russia).The purpose of the conference is to share new achievements and create conditions for fruitful cooperation between researchers involved in mathematical modeling, high-performance computing. It is also aimed at discussing current problems of mathematical modeling of continuum mechanics, computational algorithms on multiprocessor systems, scientific computing, and prospects for development of parallel computing technologies.

History of the Discovery of the Pontryagin Maximum Principle

We briefly describe the history of the discovery of the Pontryagin maximum principle in optimal control theory in the mid-1950s by Academician L. S. Pontryagin in collaboration with his students—V. G. Boltyanskii and the author of the present paper—who were then young scientists at the Department of Differential Equations headed by Pontryagin at the Steklov Mathematical Institute.

The Local Theory for Regular Systems in the Context of t-Bonded Sets

The main goal of the local theory for crystals developed in the last quarter of the 20th Century by a geometry group of Delone (Delaunay) at the Steklov Mathematical Institute is to find and prove the correct statements rigorously explaining why the crystalline structure follows from the pair-wise identity of local arrangements around each atom. Originally, the local theory for regular and multiregular systems was developed with the assumption that all point sets under consideration are ( r , R ) -systems or, in other words, Delone sets of type ( r , R ) in d-dimensional Euclidean space. In this paper, we will review the recent results of the local theory for a wider class of point sets compared with the Delone sets. We call them t-bonded sets. This theory, in particular, might provide new insight into the case for which the atomic structure of matter is a Delone set of a “microporous” character, i.e., a set that contains relatively large cavities free from points of the set.

К 70-летию профессора Александра Павловича Солдатова

18 января 2018 г. исполнилось 70 лет доктору физико-математических наук, профессору, заслуженному деятелю науки Российской Федерации Александру Павловичу Солдатову. В настоящей заметке приводятся сведения из научной биографии этого крупного ученого, который хорошо известен в России и за рубежом. Представлены сведения о его вкладе в развитие математики в области дифференциальных уравнений с частными производными и их приложений.

On an Optimal Stopping Problem of an Insider

We consider the optimal stopping problem $v^{(\varepsilon)}:=\sup_{\tau\in{\cal T}_{0,T}}{\bf E}\,B_{(\tau-\varepsilon)^+}$ posed by Shiryaev at the International Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of Mathematics in September 2012. Here $T>0$ is a fixed time horizon, $(B_t)_{0\le t\le T}$ is the Brownian motion, $\varepsilon\in[0,T]$ is a constant, and ${\cal T}_{\varepsilon,T}$ is the set of stopping times taking values in $[\varepsilon,T]$. The solution of this problem is characterized by a path dependent reflected backward stochastic differential equation, from which the continuity of $\varepsilon \to v^{(\varepsilon)}$ follows. For large enough $\varepsilon$, we obtain an explicit expression for $v^{(\varepsilon)}$, and for small $\varepsilon$ we have lower and upper bounds. The main result of the paper is the asymptotics of $v^{(\varepsilon)}$ as $\varepsilon\searrow 0$. As a by-product, we also obtain Levy's modulus of continuity result in the ...

A Conversation with Estate V. Khmaladze

Estate V. Khmaladze was born in Tbilisi, Georgia, on October 20, 1944. He earned his B.Sc. degree from the Javakhishvili Tbilisi State University in 1964, majoring in physics. and his Ph.D. in mathematics in 1971 and Doctor of Physical and Mathematical Sciences in 1988, both from the Moscow State University. From 1972 to 1990, he held appointments at the Razmadze Mathematical Institute in Tbilisi and interim appointments at the V. A. Steklov Mathematical Institute in Moscow. From 1990 to 1999, he was head of the Department of Probability and Mathematical Statistics of the Razmadze Institute. From 1996 to 2001, he was on the faculty of the Department of Statistics of the University of New South Wales. Since 2002, he holds the Chair in Statistics in the School of Mathematics and Statistics of Victoria University of Wellington, New Zealand. He is a Fellow of the Royal Society of New Zealand and of the Institute of Mathematical Statistics. In 2013, he was awarded the Javakhishvili Medal from Tbilisi I. Javakhishvili State University and was elected to be a Foreign Member of the Georgian Academy of Sciences in 2016. As the conversation reveals, Khmaladze’s research ranges widely over statistical topics and beyond. The conversation began in the old building of I. Javakhishvili Tbilisi State University during a conference on probability theory and mathematical statistics, September 6–12, 2015, and continued in the Research Center of Ilia University, Stephantsminda, during the subsequent workshop, 12–16, September, Georgia. Mount Kazbegi, 5047 m, with its white summit was occasionally visible not too far away. In what follows, the questions are put in italics while the Estate’s answers appear in the standard font.

On the free loop spaces of a toric space

In this note, it is shown that the Hilbert–Poincare series for the rational homology of the free loop space on a moment-angle complex is a rational function if and only if the moment-angle complex is a product of odd spheres and a disk. A partial result is included for the Davis–Januszkiewicz spaces. The opportunity is taken to correct the result (Bahri et al., Proceedings of the Steklov Institute of Mathematics, Russian Academy of Sciences, vol. 286, pp. 219–223. doi:10.1134/S0081543814060121, 2014) which used a theorem from Berglund and Jollenbeck (J Algebra 315:249–273, 2007).

Seminar Dedicated to the Memory of Yu. V. Prokhorov

On December 15, 2015, a seminar was held at the Steklov Mathematical Institute of Russian Academy of Sciences in recognition of the memory of Yu. V. Prokhorov. Six scientific papers were presented at the seminar and are reproduced in this issue in their entirety.

L.S. Pontryagin’s studies on optimal processes and differential games

The paper presents the opening address delivered by D.V. Anosov at the conference “Optimal Control and Applications” dedicated to the 105th birthday of Lev Semenovich Pontryagin (Moscow, Steklov Mathematical Institute of Russian Academy of Sciences, September 24–25, 2013).

Relating and Contrasting Plain and Prefix Kolmogorov Complexity

In (Bauwens and Shen, J. Symb. Log. 79(2), 620---632, 2013) a short proof is given that some strings have maximal plain Kolmogorov complexity but not maximal prefix-free complexity. We argue that the proof technique is useful to simplify existing proofs and to solve open questions. We present a short proof of a result due to Robert Solovay that relates plain and prefix complexity: K(x)=C(x)+CC(x)+O(CCC(x))C(x)=K(x)?KK(x)+O(KKK(x)),$\begin {array}{ccc} K(x) &=& C(x) + CC(x) + O(CCC(x)) C(x) &=& K(x) - KK(x) + O(KKK(x)), \end {array}$ (here CC(x) denotes C(C(x)), etc.). We show that there exist ? such that liminfC(?1??n)?C(n)$\liminf C(\omega _{1}{\dots } \omega _{n}) - C(n)$ is infinite and liminfK(?1??n)?K(n)$\liminf K(\omega _{1}{\dots } \omega _{n}) - K(n)$ is finite, i.e. the infinitely often C-trivial reals are not the same as the infinitely often K-trivial reals, answering Question 1 in Barmpalias (Bull. Symb. Log. 19(3), 2013). We answer a question from Bienvenu (Laurent Bienvenu, personal communication 2011): some 2-random sequence has a family of initial segments with bounded plain deficiency (i.e. |x|?C(x) is bounded) and unbounded prefix deficiency (i.e. |x|+K(|x|)?K(x) is unbounded). Finally, we show that there exists no monotone relation between probability and expectation bounded randomness deficiency, answering Question 1 in Bienvenu et al. (Proceedings of the Steklov Institute of Mathematics, 274(1), 34---89, 2011).

Vladimir Steklov: A mathematician at the turn of the era

The article depicts the contributions of Vladimir Andreevich Steklov to the development of Russian mathematics and the key role he played in the formation of the Russian mathematical community and institutions. The article also focuses on the scientist’s work in the Soviet period. Despite the collapse of the old regime and the construction of the new political system, Steklov guided and supported the Academy of Sciences, establishing its reputation in the new Soviet state. He also laid the foundation for the institutional framework of Russian mathematics, with one of its cornerstones being the Institute of Physics and Mathematics, from which the Steklov Mathematical Institute has emerged.

On some applications of the boundary control method to spectral estimation and inverse problems

S. A. Avdonin, A. S. Mikhaylov, V. S. Mikhaylov Department of Mathematics and Statistics, University of Alaska, Fairbanks, USA St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia St. Petersburg State University, Faculty of Mathematics and Mechanics, St. Petersburg, Russia St. Petersburg State University, Faculty of Physics, St. Petersburg, Russia s.avdonin@alaska.edu, mikhaylov@pdmi.ras.ru, vsmikhaylov@pdmi.ras.ru

Congratulatory address from the Steklov Institute to Albert Nikolaevich Shiryaev on the occasion of his 80th birthday

Congratulatory address from the Steklov Institute to Albert Nikolaevich Shiryaev on the occasion of his 80th birthday

Congratulatory address from the Steklov Institute to Oleg Vladimirovich Besov on the occasion of his 80th birthday

Congratulatory address from the Steklov Institute to Oleg Vladimirovich Besov on the occasion of his 80th birthday