scholarly journals A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 306
Author(s):  
Mingming Li ◽  
Huafei Sun ◽  
Didong Li
Keyword(s):  
Riemannian Metrics ◽  
Geometric Approach ◽  
Geometric Structures ◽  
Multiple Points ◽  
Multinomial Models ◽  
Karcher Mean

This paper is concerned with the formulation and computation of average problems on the multinomial and negative multinomial models. It can be deduced that the multinomial and negative multinomial models admit complementary geometric structures. Firstly, we investigate these geometric structures by providing various useful pre-derived expressions of some fundamental geometric quantities, such as Fisher-Riemannian metrics, α -connections and α -curvatures. Then, we proceed to consider some average methods based on these geometric structures. Specifically, we study the formulation and computation of the midpoint of two points and the Karcher mean of multiple points. In conclusion, we find some parallel results for the average problems on these two complementary models.

ITERATIVE GEOMETRIC STRUCTURES

2010 ◽  
Vol 07 (07) ◽  
pp. 1103-1114 ◽  
Author(s):  
GABRIEL BERCU ◽  
CLAUDIU CORCODEL ◽  
MIHAI POSTOLACHE
Keyword(s):  
Differential Geometry ◽  
Special Functions ◽  
Riemannian Metrics ◽  
General Setting ◽  
Mathematical Physics ◽  
Geometric Structures ◽  
Geometric Characteristics ◽  
Geometrical Characteristics ◽  
Metric Connection

In this work, we propose a study of geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry. Then we find geometrical characteristics of some ODE or PDE of Mathematical Physics. While Sec. 1 contains the general setting, Secs. 2–5 contain our results. In Sec. 2, we introduce a Hessian structure having the same connection as the initial metric. In Sec. 3, we initiate a study on iterative 2D Hessian structures. In Sec. 4, we find pairs (metric, connection) generated by special functions. In Sec. 5, we find geometric characteristics of a PDE.

scholarly journals Geometry of color perception. Part 1: structures and metrics of a homogeneous color space

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Edoardo Provenzi
Keyword(s):  
Color Space ◽  
Color Perception ◽  
Riemannian Metrics ◽  
Geometric Structures ◽  
Perceived Color

Abstract This is the first half of a two-part paper dealing with the geometry of color perception. Here we analyze in detail the seminal 1974 work by H.L. Resnikoff, who showed that there are only two possible geometric structures and Riemannian metrics on the perceived color space $\mathcal{P} $ P compatible with the set of Schrödinger’s axioms completed with the hypothesis of homogeneity. We recast Resnikoff’s model into a more modern colorimetric setting, provide a much simpler proof of the main result of the original paper, and motivate the need of psychophysical experiments to confute or confirm the linearity of background transformations, which act transitively on $\mathcal{P} $ P . Finally, we show that the Riemannian metrics singled out by Resnikoff through an axiom on invariance under background transformations are not compatible with the crispening effect, thus motivating the need of further research about perceptual color metrics.

scholarly journals Non-stationary smooth geometric structures for contracting measurable cocycles

2017 ◽  
Vol 39 (2) ◽  
pp. 392-424 ◽  
Author(s):  
KARIN MELNICK
Keyword(s):  
Normal Forms ◽  
Geometric Approach ◽  
Geometric Structures ◽  
Invariant Differential ◽  
Homogeneous Structures ◽  
Polynomial Diffeomorphisms

We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\operatorname{Diff}^{q}(\mathbb{R}^{n},\mathbf{0})$, $q\geq 2$. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via $C^{q}$ changes of coordinates. These are interpreted as non-stationary invariant differential-geometric structures. We also consider the case of contracted foliations in a manifold, and obtain $C^{q}$ homogeneous structures on leaves for an action of the group of subresonance polynomial diffeomorphisms together with translations.

A GEOMETRIC STUDY OF SOME EQUATIONS OF MATHEMATICAL PHYSICS

2012 ◽  
Vol 09 (04) ◽  
pp. 1250030 ◽  
Author(s):  
ARIANA PITEA
Keyword(s):  
Differential Geometry ◽  
Riemannian Metrics ◽  
Main Tool ◽  
Mathematical Physics ◽  
Geometric Structures ◽  
Boundary Problems ◽  
Geometrical Characteristics ◽  
Equations Of Mathematical Physics ◽  
Geometric Study

We introduce geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry, and find geometrical characteristics associated to equations of Mathematical Physics. Also, we introduce a geometric study of some boundary problems. Throughout this work, as main tool we employed an adequate Riemannian Hessian structure, suggested in [Int. J. Geom. Meth. Mod. Phys.7(7) (2010) 1104–1113].

A Geometric Approach to Homology Theory

1976 ◽  
Author(s):  
S. Buonchristiano ◽  
C. P. Rourke ◽  
B. J. Sanderson
Keyword(s):  
Geometric Approach ◽  
Homology Theory

A Hierarchy of Multinomial Models for Multidimensional Source Monitoring

Methodology ◽  
2005 ◽  
Vol 1 (1) ◽  
pp. 2-17 ◽  
Author(s):  
Thorsten Meiser
Keyword(s):  
Cognitive Processes ◽  
Source Monitoring ◽  
Source Information ◽  
Model Comparisons ◽  
Storage And Retrieval ◽  
Multinomial Processing Tree ◽  
Multinomial Models ◽  
Hierarchical Relationship ◽  
Statistical Framework ◽  
Tree Models

Abstract. Several models have been proposed for the measurement of cognitive processes in source monitoring. They are specified within the statistical framework of multinomial processing tree models and differ in their assumptions on the storage and retrieval of multidimensional source information. In the present article, a hierarchical relationship is demonstrated between multinomial models for crossed source information ( Meiser & Bröder, 2002 ), for partial source memory ( Dodson, Holland, & Shimamura, 1998 ) and for several sources ( Batchelder, Hu, & Riefer, 1994 ). The hierarchical relationship allows model comparisons and facilitates the specification of identifiability conditions. Conditions for global identifiability are discussed, and model comparisons are illustrated by reanalyses and by a new experiment on the storage and retrieval of multidimensional source information.

A GEOMETRIC APPROACH TO DISSIPATION AND ITS APPLICATION TO DISSIPATIVE NUCLEAR DYNAMICS

1984 ◽  
Vol 45 (C6) ◽  
pp. C6-87-C6-94
Author(s):  
H. Reinhardt ◽  
R. Balian ◽  
Y. Alhassid
Keyword(s):  
Geometric Approach ◽  
Nuclear Dynamics

Computer Modeling of a Tire under Cornering Loads

1989 ◽  
Vol 17 (2) ◽  
pp. 86-99 ◽  
Author(s):  
I. Gardner ◽  
M. Theves
Keyword(s):  
Finite Element Analysis ◽  
Geometric Approach ◽  
Lateral Force ◽  
Slip Angle ◽  
Element Analysis ◽  
Pressure Distributions ◽  
Slip Effects ◽  
Improved Accuracy ◽  
Nonlinear Geometric ◽  
Good Agreement

Abstract During a cornering maneuver by a vehicle, high forces are exerted on the tire's footprint and in the contact zone between the tire and the rim. To optimize the design of these components, a method is presented whereby the forces at the tire-rim interface and between the tire and roadway may be predicted using finite element analysis. The cornering tire is modeled quasi-statically using a nonlinear geometric approach, with a lateral force and a slip angle applied to the spindle of the wheel to simulate the cornering loads. These values were obtained experimentally from a force and moment machine. This procedure avoids the need for a costly dynamic analysis. Good agreement was obtained with experimental results for self-aligning torque, giving confidence in the results obtained in the tire footprint and at the rim. The model allows prediction of the geometry and of the pressure distributions in the footprint, since friction and slip effects in this area were considered. The model lends itself to further refinement for improved accuracy and additional applications.

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