scholarly journals Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals

2020 ◽  
pp. 96-101
Author(s):  
M.V. Kurkina ◽  
S.P. Semenov ◽  
V.V. Slavsky ◽  
O.V. Samarina ◽  
O.A. Petuhova ◽  
...  
Keyword(s):  
Convex Functions ◽  
Riemannian Geometry ◽  
Duality Theory ◽  
Digital Processing ◽  
Riemannian Metric ◽  
Theoretical Physics ◽  
Vector Spaces ◽  
Legendre Transformation ◽  
Conformally Flat ◽  
Tropical Mathematics

In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky. The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space. The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

scholarly journals Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

2021 ◽  
Vol 16 (3) ◽  
pp. 75-82
Author(s):  
Maria V. Kurkina ◽  
Sergey P. Semenov ◽  
Viktor V. Slavsky ◽  
Olga V. Samarina ◽  
Olga A. Petuhova ◽  
...  
Keyword(s):  
Time Series ◽  
Riemannian Metrics ◽  
Digital Processing ◽  
Conformally Flat ◽  
Idempotent Algebra ◽  
Program Complex ◽  
Tropical Mathematics ◽  
Main Tendency ◽  
And Mathematics ◽  
Under Construction

Alignment of time series [time-series smoothing] identification of the main tendency of development (временнго a trend) by "cleaning" of a time series of the accidental deviations distorting this tendency. At a research of time series of economy (bioinformation science) apply for detection of patterns [1-3]. In this work it is offered to use for this purpose Legendre's transformation well-known in physics and mathematics. Its direct application to poorly regular objects is difficult therefore in work its idempotent analog is defined previously and on its basis the concept of the TRACK for a time series is defined. In recent years within the international center "Cuofus Li" the new field of mathe-matics idempotent or "tropical" mathematics gained intensive development that is reflected in works of the academician V.P. Maslov and his pupils: G.L. Litvinov, A.N. Sobolevsky, etc. The purpose of this work to be beyond duality of the theory of linear vector spaces, using similar concepts of duality of conformally flat Riemannian geometry and of idempotent algebra. By analogy with the polar transformation of a conformally flat Riemannian metrics entered in E.D. Rodionov and V.V. Slavsky's works the abstract idempotent analog of transformation of Legendre is under construction. In the MATLAB system the program complex for calculation the TRACK of a time series is created. It is in-vestigated its opportunities for digital processing of time series.

A Generalization of Finsler Geometry

1962 ◽  
Vol 14 ◽  
pp. 87-112 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Elliptic Partial Differential Equations ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Metric Function

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

scholarly journals Riemannian Structures on Z 2 n -Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1469
Author(s):  
Andrew James Bruce ◽  
Janusz Grabowski
Keyword(s):  
Riemannian Geometry ◽  
Fundamental Theorem ◽  
Scalar Product ◽  
Riemannian Metric ◽  
Theoretical Framework ◽  
Similarities And Differences

Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

Conjugates and Legendre Transforms of Convex Functions

1967 ◽  
Vol 19 ◽  
pp. 200-205 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Strict Convexity ◽  
Legendre Transformation ◽  
Legendre Transform ◽  
Implicit Functions ◽  
The One ◽  
Legendre Transforms ◽  
Effective Domain ◽  
The Relationship

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.

scholarly journals A NOTE ON THE QUANTUM-MECHANICAL RICCI FLOW

2009 ◽  
Vol 24 (27) ◽  
pp. 4999-5006
Author(s):  
JOSÉ M. ISIDRO ◽  
J. L. G. SANTANDER ◽  
P. FERNÁNDEZ DE CÓRDOBA
Keyword(s):  
Quantum Mechanics ◽  
Configuration Space ◽  
Ricci Flow ◽  
Riemannian Metric ◽  
Quantum Mechanical ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Flow Equations ◽  
Emergent Quantum Mechanics

We obtain Schrödinger quantum mechanics from Perelman's functional and from the Ricci-flow equations of a conformally flat Riemannian metric on a closed two-dimensional configuration space. We explore links with the recently discussed emergent quantum mechanics.

scholarly journals Duality in finite dimensional complex space

1978 ◽  
Vol 18 (1) ◽  
pp. 65-75
Author(s):  
C.H. Scott ◽  
T.R. Jefferson
Keyword(s):  
Quadratic Programming ◽  
Duality Theory ◽  
Complex Space ◽  
Optimization Problems ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Mathematical Programs ◽  
Finite Dimensional ◽  
General Duality ◽  
General Duality Theory

The idea of duality is now a widely accepted and useful idea in the analysis of optimization problems posed in real finite dimensional vector spaces. Although similar ideas have filtered over to the analysis of optimization problems in complex space, these have mainly been concerned with problems of the linear and quadratic programming variety. In this paper we present a general duality theory for convex mathematical programs in finite dimensional complex space, and, by means of an example, show that this formulation captures all previous results in the area.

Conformally convex functions and conformally flat metrics of nonnegative curvature

2015 ◽  
Vol 91 (3) ◽  
pp. 287-289 ◽  
Author(s):  
M. V. Kurkina ◽  
E. D. Rodionov ◽  
V. V. Slavskii
Keyword(s):  
Convex Functions ◽  
Nonnegative Curvature ◽  
Conformally Flat ◽  
Conformally Flat Metrics

On the Nonstandard Duality Theory of Locally Convex Spaces

1980 ◽  
Vol 32 (2) ◽  
pp. 460-479 ◽  
Author(s):  
Arthur D. Grainger
Keyword(s):  
Vector Space ◽  
Duality Theory ◽  
Dual System ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Nonstandard Model ◽  
Locally Convex ◽  
External Property ◽  
Convex Spaces

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).

On State-Space Modeling and Signal Localization in Dynamical Systems

2021 ◽  
Vol 2 (1) ◽  
Author(s):  
Asok Ray
Keyword(s):  
Dynamical Systems ◽  
State Space ◽  
Applied Mathematics ◽  
System Model ◽  
Theoretical Physics ◽  
Vector Spaces ◽  
Real Field ◽  
Finite Dimensional ◽  
Space Modeling ◽  
And Control

Abstract This letter focuses on two topics in engineering analysis, which are (1) degree-of-freedom (DOF) in modeling of dynamical systems and (2) simultaneous time and frequency localization of signals. These issues are explained from the perspectives of decision and control by making use of concepts from applied mathematics and theoretical physics. Specifically, a new definition is proposed to clarify the notion of “DOF,” which is consistent with the dimension of the state space of the dynamical system model. Relevant examples are presented on (finite-dimensional) vector spaces over the real field R and/or the complex field C.

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