L. V. Kantorovich

Bio: L. V. Kantorovich is an academic researcher from Russian Academy of Sciences. The author has contributed to research in topics: Economic problem & Space (mathematics). The author has an hindex of 13, co-authored 35 publications receiving 7112 citations. Previous affiliations of L. V. Kantorovich include Saint Petersburg State University.

On the Translocation of Masses

Abstract: ON THE TRANSLOCATION OF MASSES L. V. Kantorovich∗ The original paper was published in Dokl. Akad. Nauk SSSR, 37, No. 7–8, 227–229 (1942). We assume that R is a compact metric space, though some of the definitions and results given below can be formulated for more general spaces. Let Φ(e) be a mass distribution, i.e., a set function such that: (1) it is defined for Borel sets, (2) it is nonnegative: Φ(e) ≥ 0, (3) it is absolutely additive: if e = e1 + e2+ · · · ; ei∩ ek = 0 (i = k), then Φ(e) = Φ(e1)+ Φ(e2) + · · · . Let Φ′(e′) be another mass distribution such that Φ(R) = Φ′(R). By definition, a translocation of masses is a function Ψ(e, e′) defined for pairs of (B)-sets e, e′ ∈ R such that: (1) it is nonnegative and absolutely additive with respect to each of its arguments, (2) Ψ(e, R) = Φ(e), Ψ(R, e′) = Φ′(e′). Let r(x, y) be a known continuous nonnegative function representing the work required to move a unit mass from x to y. We define the work required for the translocation of two given mass distributions as W (Ψ,Φ,Φ′) = ∫

Mathematical Methods of Organizing and Planning Production

Abstract: The author of the work “Mathematical Methods of Organizing and Planning Production”, Professor L. V. Kantorovich, is an eminent authority in the field of mathematics. This work is interesting from a purely mathematical point of view since it presents an original method, going beyond the limits of classical mathematical analysis, for solving extremal problems. On the other hand, this work also provides an application of mathematical methods to questions of organizing production which merits the serious attention of workers in different branches of industry. This is the English translation of the famous 1939 article by L. V. Kantorovich, originally published in Russian.

Optimal Transport: Old and New

Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Detection of abrupt changes: theory and application

Abstract: This book is downloadable from http://www.irisa.fr/sisthem/kniga/. Many monitoring problems can be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. The main goal of this book is to describe a unified framework for the design and the performance analysis of the algorithms for solving these change detection problems. Also the book contains the key mathematical background necessary for this purpose. Finally links with the analytical redundancy approach to fault detection in linear systems are established. We call abrupt change any change in the parameters of the system that occurs either instantaneously or at least very fast with respect to the sampling period of the measurements. Abrupt changes by no means refer to changes with large magnitude; on the contrary, in most applications the main problem is to detect small changes. Moreover, in some applications, the early warning of small - and not necessarily fast - changes is of crucial interest in order to avoid the economic or even catastrophic consequences that can result from an accumulation of such small changes. For example, small faults arising in the sensors of a navigation system can result, through the underlying integration, in serious errors in the estimated position of the plane. Another example is the early warning of small deviations from the normal operating conditions of an industrial process. The early detection of slight changes in the state of the process allows to plan in a more adequate manner the periods during which the process should be inspected and possibly repaired, and thus to reduce the exploitation costs.

Matrix Differential Calculus with Applications in Statistics and Econometrics

Abstract: Preface MATRICES: Basic Properties of Vectors and Matrices Kronecker Products, the Vec-Operator and the Moore- Penrose Inverse Miscellaneous Matrix Results DIFFERENTIALS: THE THEORY: Mathematical Preliminaries Differentials and Differentiability The Second Differential Static Optimization DIFFERENTIALS: THE PRACTICE: Some Important Differentials First- Order Differentials and Jacobian Matrices Second-Order Differentials and Hessian Matrices INEQUALITIES: Inequalities THE LINEAR MODEL: Statistical Preliminaries The Linear Regression Model Further Topics in the Linear Model APPLICATIONS TO MAXIMUM LIKELIHOOD ESTIMATION: Maximum Likelihood Estimation Simultaneous Equations Topics in Psychometrics Subject Index Bibliography.

Optimization Algorithms on Matrix Manifolds

Abstract: Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

Markov Chains and Mixing Times

Abstract: This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. ""Markov Chains and Mixing Times"" is meant to bring the excitement of this active area of research to a wide audience.