### Working paper

## Affine transformations of the plane and their geometrical properties

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 main object of this paper consists in solving of the inverse problem of optical tomography through the development of a method of dynamical statistical spatial-temporal reconstruction of sizes and shape of moving three-dimensional objects with the usage of their projected images.

The newness of the given method consists in dynamical, few view, spatial-temporal reconstruction of sizes and shape not of a convex three-dimensional object itself, but its adequate approximation represented as a three-dimensional image of the ellipsoid of general form. Along with this, the geometrical sizes of a three-dimensional object are specified by numerical values of the axes of this three-dimensional image of ellipsoid and the average projected diameter of the image (D), and the shape factor (K) is specified by the ratio of the maximum and minimum overall dimensions (axes) of the image of the approximating ellipsoid.

The contours of three discrete, two-dimensional projected images of an object are specified as the optimal form of the primary geometrical information by a simulation modeling method. In this case, their spatial orientation is their mutual orthogonality. Their maximum and minimum overall dimensions are chosen as the optimal basic geometrical characteristics of projected images (the most informative characteristics according to the method of maximal entropy).

The mathematical model of object reconstruction is defined by the functional dependences of linear sizes of three mutually orthogonal axis of the approximating ellipsoid to the maximum and minimum dimensions of its three projected images.

In the result of statistical studies it is determined that the relative error of computing of the average projected diameter of an object is about 0.25% (at the reliability of PD = 0.7 and K = 1,3 relative units). The relative error of computing of form factor of an object is from 2.3% (PK = 0.7 and K = 1.3 relative units) to 0.6% (with PK = 0.96 and K = 1.05 relative units), and the total control time of object sizes and object shape does not exceed 10 Ms.

Thus, the proposed method of dynamic reconstruction has a new combination of characteristics of accuracy, reliability and performance. It has been successfully employed for a high performance, manufacturing, televisional size and form control for elements of the nuclear fuel and can be applied for remote control of various moving convex objects in real-time.

In the paper, a fast laser method of the geometrical optoelectronic differential control of three-dimensional micro objects (MO) flow, which is based on the statistical dynamic few-views reconstruction of the sizes and shape of each MO with the usage of the basic characteristics of one triad of their impulse discrete two-dimensional projective images is under consideration. The squares of three mutually orthogonal two-dimensional projective images of MO and linear dimensions of its three one-dimensional projective images onto mutually orthogonal axes are chosen as the main basic characteristics.

Impulse images of moving (flying) MO are formed by their flaring in the parallel rays from impulse laser sources of optical radiation and their simultaneous registration with the usage an optoelectronic position-sensitive video detector with memory, which works in the mode of separate image recording and reading.

The spatial geometrical characteristics of dimensions for each MO are described as linear dimensions (overall dimension) and the average projected diameter (*D*) of a three-dimensional image of the reference ellipsoid that approximates each MO. The shape factor (*K*), which is used when describing the shape of MO, is specified by the ratio of the maximum and minimum overall dimensions (axis) of the approximating ellipsoid.

The speed performance of the method of control under consideration is not less than 100 MO per second. Along with this the relative error of MO diameter control does not exceed 0,25 per cent (with the accuracy *PD* = 0,7 and *K* = 1,3 relative units), and the relative error of control of the MO shape factor is in the scope from 2,3 % (*PK* = 0,7 and *K* = 1,3 relative units) till 0,6% (with *PK* = 0,96 and *K* = 1,05 relative units).

We address the problem of the exact computation of two joint spectral characteristics of a family of linear operators, the joint spectral radius (JSR) and the lower spectral radius (LSR), which are well-known different generalizations to a set of operators of the usual spectral radius of a linear operator. In this paper we develop a method which—under suitable assumptions—allows us to compute the JSR and the LSR of a finite family of matrices exactly. We remark that so far no algorithm has been available in the literature to compute the LSR exactly.

The paper presents necessary theoretical results on extremal norms (and on extremal antinorms) of linear operators, which constitute the basic tools of our procedures, and a detailed description of the corresponding algorithms for the computation of the JSR and LSR (the last one restricted to families sharing an invariant cone). The algorithms are easily implemented, and their descriptions are short. If the algorithms terminate in finite time, then they construct an extremal norm (in the JSR case) or antinorm (in the LSR case) and find their exact values; otherwise, they provide upper and lower bounds that both converge to the exact values. A theoretical criterion for termination in finite time is also derived. According to numerical experiments, the algorithm for the JSR finds the exact value for the vast majority of matrix families in dimensions ≤20. For nonnegative matrices it works faster and finds the JSR in dimensions of order 100 within a few iterations; the same is observed for the algorithm computing the LSR. To illustrate the efficiency of the new method, we apply it to give answers to several conjectures which have been recently stated in combinatorics, number theory, and formal language theory.

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.