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## Hirzebruch Functional Equations and Krichever Complex Genera

As is well known, the two-parameter Todd genus and the elliptic functions of level d define n-multiplicative Hirzebruch genera if d divides n + 1. Both cases are special cases of the Krichever genera defined by the Baker–Akhiezer function. In the present paper, the inverse problem is solved. Namely, it is proved that only these properties define n-multiplicative Hirzebruch genera among all Krichever genera for all n.

We study ﬂat deformations of quotients of a polynomial algebra in a class of graded commutative associative algebras. Functional equations and their solutions in terms of theta functions play important role in these studies. An analog of this theory in a fermionic case is also brieﬂy discussed.

Leading logarithms in massless nonrenormalizable effective field theories can be computed using nonlinear recurrence relations. These recurrence relations follow from the fundamental requirements of unitarity, analyticity, and crossing symmetry of scattering amplitudes and generalize the renormalization group technique to the case of nonrenormalizable effective field theories. We review the existing exact solutions of nonlinear recurrence relations relevant for field theory applications. We introduce a new class of quantum field theories (quasirenormalizable field theories) in which resumming leading logarithms for 2→2 scattering amplitudes yields a possibly infinite number of Landau poles.

We consider approximations of an arbitrarymap *F*: *X* → *Y* between Banach spaces *X* and *Y* by an affine operator *A*: *X* → *Y* in the Lipschitz metric: the difference *F* — *A* has to be Lipschitz continuous with a small constant *ɛ* > 0. In the case *Y* = ℝ we show that if *F* can be affinely *ɛ*-approximated on any straight line in *X*, then it can be globally 2*ɛ*-approximated by an affine operator on *X*. The constant 2*ɛ* is sharp. Generalizations of this result to arbitrary dual Banach spaces *Y* are proved, and optimality of the conditions is shown in examples. As a corollary we obtain a solution to the problem stated by Zs. Páles in 2008. The relation of our results to the Ulam-Hyers-Rassias stability of the Cauchy type equations is discussed.

The theory of elliptic integrals and elliptic functions, which were driving force of mathematics in the eighteenth and nineteents centuries, are not only beautiful but have many applications in mathematics and physics. This simple reader-friendly book, based on the lectures at the Faculty of Mathematics, HSE, explains such theory and applications in original way.

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.