Embeddings of finite-dimensional compacta in Euclidean spaces
If g is a map from a space X into Rm and q is an integer, let B q,d,m(g) be the set of all planes Πd ⊂Rm such that |g−1(Πd)| ≥ q. Let also H(q,d,m,k) denote the set of all continuous maps g : X →Rm such that dim B q,d,m(g) ≤k. We prove that for any n-dimensional metric compactum X each of the sets H(3, 1,m, 3n + 1 −m) and H(2, 1,m, 2n) is dense and Gδ in the function space C(X, Rm) provided m ≥ 2n + 1 (in this case H(3, 1,m, 3n + 1 − m) and H(2, 1,m, 2n) can consist of embeddings). The same is true for the sets H(1,d,m,n+ d(m − d)) ⊂ C(X, Rm) if m ≥n + d, and H(4, 1, 3, 0) ⊂ C(X, R3) if dim X ≤ 1.
The main aim of the book is, naturally, to give students the fundamental notions and instruments in linear algebra. Linearity is the main assumption used in all fieldsof science. It gives a first approximation to any problem under study and is widely used in economics and other social sciences. One may wonder why we decided to write a book in linear algebra despite the fact that there are many excellent books such as [10, 11, 19, 27, 34]? Our reasons can be summarized as follows. First, we try to fit the course to the needs of the students in economics and the students in mathematics and informatics who would like to get more knowledge in economics. Second, we constructed all expositions in the book in such a way to help economics students to learn mathematics and the proof making in mathematics in a convenient and simple manner. Third, since the hours given to this course in economics departments are rather limited, we propose a slightly different way of teaching this course. Namely, we do not try to give all proofs of all theorems presented in the course. Those theorems which are not proved are illustrated via figures and examples, and we illustrated all notions appealing to geometric intuition. Those theorems which are proved are proved in a most accurate way as it is done for the students in mathematics. The main notions are always supported with economic examples. The book provides many exercises referring to pure mathematics and economics. The book consists of eleven chapters and five appendices. Chapter 1 contains the introduction to the course and basic concepts of vector and scalar. Chapter 2 introduces the notions of vectors and matrices, and discusses some core economic examples used throughout the book. Here we begin with the notion of scalar product of two vectors, define matrices and their ranks, consider elementary operations over matrices. Chapter 3 deals with special important matrices – square matrices and their determinants. Chapter 4 introduces inverse matrices. In Chap. 5 we analyze the systems of linear equations, give methods how to solve these systems. Chapter ends with the discussion of homogeneous equations. Chapter 6 discusses more general type of algebraic objects – linear spaces. Here the notion of linear independence of vectors is introduced, which is very important from economic point of view for it defines how diverse is the obtained information. We consider here the isomorphism of linear spaces and the notion of subspace. Chapter 7 deals with important case of linear spaces – the Euclidean ones. We consider the notion of orthogonal bases and use it to construct the idea of projection and, particularly, the least square method widely used in social sciences. In Chapter 8 we consider linear transformations, and all related notions such as an image and kernel of transformation. We also consider linear transformations with respect to different bases. Chapter 9 discusses eigenvalues and eigenvectors. Here we consider self-adjoint transformations, orthogonal transformations, quadratic forms and their geometric representation. Chapter 10 applies the concepts developed before to the linear production model in economics. To this end we use, particularly, Perron–Frobenius Theorem. Chapter 11 deals with the notion of convexity, and so-called separation theorems. We use this instrument to analyse the linear programming problem. We observe during the years of our teaching experience that induction argument creates some difficulties among students. So, we explain this argument in Appendix A. In Appendix B we discuss how to evaluate the determinants. In Appendix C we give a brief introduction to complex numbers, which are important for better understanding the eigenvalues of linear operators. In Appendix D we consider the notion of the pseudoinverse, or generalized inverse matrix, widely used in different economic applications. Each chapter endswith the number of problemswhich allowbetter understanding the issues considered. In Appendix E the answers and hints to solutions to the problems from previous chapters and appendices are given.
In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal dimension and Serre dimension. We prove some basic properties of these dimensions, compare them and discuss open problems.
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.