scholarly journals Generalized potential functions in differential geometry and information geometry

2019 ◽  
Vol 16 (supp01) ◽  
pp. 1940002 ◽  
Author(s):  
F. M. Ciaglia ◽  
G. Marmo ◽  
J. M. Pérez-Pardo
Keyword(s):  
Differential Geometry ◽  
Inverse Problem ◽  
Riemannian Manifold ◽  
Information Geometry ◽  
Riemannian Metric ◽  
Negative Answer ◽  
Potential Functions ◽  
Point Of View ◽  
Geometric Structures ◽  
Generalized Potential

Potential functions can be used for generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study whether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving the inverse problem and also from an intrinsic point of view.

scholarly journals An Elementary Introduction to Information Geometry

Entropy ◽  
2020 ◽  
Vol 22 (10) ◽  
pp. 1100
Author(s):  
Frank Nielsen
Keyword(s):  
Differential Geometry ◽  
Fundamental Theorem ◽  
Information Geometry ◽  
Use Cases ◽  
Geometric Structures ◽  
Information Sciences

In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information sciences. The exposition is self-contained by concisely introducing the necessary concepts of differential geometry. Proofs are omitted for brevity.

Golden Riemannian structures on the tangent bundle with g-natural metrics

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2543-2554
Author(s):  
E. Peyghan ◽  
F. Firuzi ◽  
U.C. De
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Riemannian Metric

Starting from the g-natural Riemannian metric G on the tangent bundle TM of a Riemannian manifold (M,g), we construct a family of the Golden Riemannian structures ? on the tangent bundle (TM,G). Then we investigate the integrability of such Golden Riemannian structures on the tangent bundle TM and show that there is a direct correlation between the locally decomposable property of (TM,?,G) and the locally flatness of manifold (M,g).

Determination of an impulsive diffusion operator from interior spectral data

Analysis ◽  
2020 ◽  
Vol 40 (1) ◽  
pp. 39-45
Author(s):  
Yasser Khalili ◽  
Dumitru Baleanu
Keyword(s):  
Inverse Problem ◽  
Spectral Data ◽  
Potential Functions ◽  
Half Line ◽  
Internal Point ◽  
Diffusion Operator ◽  
Impulsive Diffusion

AbstractIn the present work, the interior spectral data is used to investigate the inverse problem for a diffusion operator with an impulse on the half line. We show that the potential functions {q_{0}(x)} and {q_{1}(x)} can be uniquely established by taking a set of values of the eigenfunctions at some internal point and one spectrum.

scholarly journals The geometric structures of the Weibull distribution manifold and the generalized exponential distribution manifold

2008 ◽  
Vol 39 (1) ◽  
pp. 45-51 ◽  
Author(s):  
Limei Cao ◽  
Huafei Sun ◽  
Xiaojie Wang
Keyword(s):  
Weibull Distribution ◽  
Exponential Distribution ◽  
Exponential Family ◽  
Information Geometry ◽  
Generalized Exponential Distribution ◽  
Geometric Structures ◽  
Basic Task

Investigating the geometric structures of the distribution manifolds is a basic task in information geometry. However, by so far, most works are on the distribution manifolds of exponential family. In this paper, we investigate two non-exponential distribution manifolds —the Weibull distribution manifold and the generalized exponential distribution manifold. Then we obtain their geometric structures.

scholarly journals Another point of view on proportional navigation

1998 ◽  
Vol 4 (3) ◽  
pp. 201-231
Author(s):  
E. Duflos ◽  
P. Penel ◽  
P. Vanbeeghe ◽  
P. Borne
Keyword(s):  
Inverse Problem ◽  
Direct Problem ◽  
Point Of View ◽  
Proportional Navigation ◽  
Guidance Laws ◽  
The Way

Proportional navigation is one of the most popular and one of the most used of the guidance laws. But the way it is studied is always the same: the acceleration needed to reach a known target is derived or analyzed. This way of studying guidance laws is called “the direct problem” by the authors. On the contrary, the problem considered here is to find, from the knowledge of a part of the trajectory of a maneuvering object, the target of this object. The authors call this way of studying guidance laws “the inverse problem”.

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 Exceptional Experiences of Stable and Unstable Mental States, Understood from a Dual-Aspect Point of View

Philosophies ◽  
2019 ◽  
Vol 4 (1) ◽  
pp. 7 ◽  
Author(s):  
Harald Atmanspacher ◽  
Wolfgang Fach
Keyword(s):  
State Space ◽  
Empirical Support ◽  
Mental States ◽  
Interesting Case ◽  
Point Of View ◽  
Asymptotically Stable ◽  
World Model ◽  
State Space Approach ◽  
Generalized Potential ◽  
Space Approach

Within a state-space approach endowed with a generalized potential function, mental states can be systematically characterized by their stability against perturbations. This approach yields three major classes of states: (1) asymptotically stable categorial states, (2) marginally stable non-categorial states and (3) unstable acategorial states. The particularly interesting case of states giving rise to exceptional experiences will be elucidated in detail. Their proper classification will be related to Metzinger’s account of self-model and world-model, and empirical support for this classification will be surveyed. Eventually, it will be outlined how Metzinger’s discussion of intentionality achieves pronounced significance within a dual-aspect framework of thinking.

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