### Article

## When is the set of embeddings finite up to isotopy?

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddings N → Sm finite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when N is a product of two spheres. In the following theorem, FCS(i, j) ⊂ ℤ2 is a specific set depending only on the parity of i and j which is defined in the paper. Theorem. Assume that m > 2p + q + 2 and m < p + 3q/2 + 2. Then the set of isotopy classes of C1-smooth embeddings Sp × Sq → Sm is infinite if and only if either q + 1 or p + q + 1 is divisible by 4, or there exists a point (x, y) in the set FCS(m - p - q, m - q) such that (m - p - q - 2)x + (m - q - 2)y = m - 3. Our approach is based on a group structure on the set of embeddings and a new exact sequence, which in some sense reduces the classification of embeddings Sp × Sq→ Sm to the classification of embeddings Sp+q {square cup} Sq → Sm and Dp × Sq → Sm. The latter classification problems are reduced to homotopy ones, which are solved rationally. © 2015 World Scientific Publishing Company.

We prove that the bound from the theorem on 'economic' maps is best possible. Namely, for m > n + d we construct a map from an n-dimensional simplex to an m-dimensional Euclidean space for which (and for any close map) there exists a d-dimensional plane whose preimage has cardinality not less than the upper bound \(dn + n + 1)/(m - n - d)] + d from the theorem on 'economic' maps. Bibliography: 16 titles.

This article reviews the archaeography of translations of Felix Platter’s «Tagebuch» into German and French languages, while the original had written on the Basel dialect. 19th century translators passed many details and translated only those paragraphs that they considered interesting. In 1878, the first complete translation of the Felix Platter’s «Tagebuch» had published. Based on this translation, in 1961, the English translator Sean Jennet had published the diary in English. In this article, we consider the relevance of an English translation; also in the appendix for the first time in Russian language, we are publishing one chapter of a diary of Felix Platter.

We prove that the bound from the theorem on ‘economic’ maps is best possible. Namely, for m > n + d we construct a map from an n-dimensional simplex to an m-dimensional Euclidean space for which (and for any close map) there exists a d-dimensional plane whose preimage has cardinality not less than the upper bound ⌈(dn + n + 1)/(m − n − d)⌉ + d from the theorem on ‘economic’ maps.

The study of clinical terminology has always occupied a significant place in the discipline "Latin language and outlines of medical terminology." Undoubtedly, surgical terminology is one of the most voluminous terminology in the clinical block. Moreover, there are a lot of terms used in it in other departments of the clinical direction, so-called "common" terms.

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings N->S^m. We study the specific case of knotted tori, i. e. the embeddings S^p x S^q -> S^m. The classification of knotted tori up to isotopy in the metastable dimension range m>p+3q/2+3/2, p<q+1, was given by A. Haefliger, E. Zeeman and A. Skopenkov. We consider the dimensions below the metastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension:

Theorem. Assume that p+4q/3+2<m<p+3q/2+2 and m>2p+q+2. Then the set of smooth embeddings S^p x S^q -> S^m up to isotopy is infinite if and only if either q+1 or p+q+1 is divisible by 4.

Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new beta-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.

This article deals with the problem of translations. It covers the history of translation in linguistics and analyzes peculiarities and role of translation in logic. Moreover, the article contains typical examples of embedding operations in terms of dierent logical theories.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.