scholarly journals Differential Invariants of Measurements, and Their Relation to Central Moments

Entropy ◽  
2020 ◽  
Vol 22 (10) ◽  
pp. 1118
Author(s):  
Eivind Schneider
Keyword(s):  
Affine Space ◽  
Information Gain ◽  
Probability Distributions ◽  
Differential Invariants ◽  
Affine Transformations ◽  
Equilibrium Thermodynamics ◽  
Central Moments ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds ◽  
Minimal Information

Due to the principle of minimal information gain, the measurement of points in an affine space V determines a Legendrian submanifold of V×V*×R. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on V. We investigate the action of this group of affine transformations on Legendrian submanifolds of V×V*×R by giving a detailed overview of the structure of the algebra of scalar differential invariants, and we show how the scalar differential invariants can be constructed from the central moments. In the end, we view the results in the context of equilibrium thermodynamics of gases, and notice that the heat capacity is one of the differential invariants.

scholarly journals Scalable information-theoretic path planning for a rover-helicopter team in uncertain environments

2021 ◽  
Vol 18 (2) ◽  
pp. 172988142199958
Author(s):  
Larkin Folsom ◽  
Masahiro Ono ◽  
Kyohei Otsu ◽  
Hyoshin Park
Keyword(s):  
Path Planning ◽  
Lower Risk ◽  
Information Gain ◽  
Probability Distributions ◽  
Single Agent ◽  
Uncertain Environments ◽  
Information Theoretic ◽  
Highly Correlated ◽  
Naive Approach

Mission-critical exploration of uncertain environments requires reliable and robust mechanisms for achieving information gain. Typical measures of information gain such as Shannon entropy and KL divergence are unable to distinguish between different bimodal probability distributions or introduce bias toward one mode of a bimodal probability distribution. The use of a standard deviation (SD) metric reduces bias while retaining the ability to distinguish between higher and lower risk distributions. Areas of high SD can be safely explored through observation with an autonomous Mars Helicopter allowing safer and faster path plans for ground-based rovers. First, this study presents a single-agent information-theoretic utility-based path planning method for a highly correlated uncertain environment. Then, an information-theoretic two-stage multiagent rapidly exploring random tree framework is presented, which guides Mars helicopter through regions of high SD to reduce uncertainty for the rover. In a Monte Carlo simulation, we compare our information-theoretic framework with a rover-only approach and a naive approach, in which the helicopter scouts ahead of the rover along its planned path. Finally, the model is demonstrated in a case study on the Jezero region of Mars. Results show that the information-theoretic helicopter improves the travel time for the rover on average when compared with the rover alone or with the helicopter scouting ahead along the rover’s initially planned route.

scholarly journals A Theoretical Framework for a Hybrid View of the N400

2021 ◽  
Vol 12 ◽  
Author(s):  
Ralf Naumann ◽  
Wiebke Petersen
Keyword(s):  
Empirical Evidence ◽  
Information Gain ◽  
Probability Distributions ◽  
Critical Word ◽  
Event Related Potential ◽  
Argument Position ◽  
Graded Structure ◽  
Cognitive Operations ◽  
Hybrid View ◽  
The Difference

In this study, we present a novel theoretical account of the N400 event-related potential (ERP) component. Hybrid views interpret this ERP component in terms of two cognitive operations: (i) access of information, which is related to predictions (predictability component), and (ii) integration of information, which is related to plausibility (plausibility component). Though there is an empirical evidence for this view, what has been left open so far is how these two operations can be defined. In our approach, both components are related to categorization. The critical word and the argument position it is related to are associated with categories that have a graded structure. This graded structure is defined in terms of weights both on attributes and values of features belonging to a category. The weights, in turn, are defined using probability distributions. The predictability component is defined in terms of the information gain with respect to non mismatched features between the two categories. The plausibility component is defined as the difference in the degree of typicality between the two categories. Finally, the N400 amplitude is defined as a function of both components.

scholarly journals Optimal Thermodynamic Processes For Gases

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 448 ◽  
Author(s):  
Alexei Kushner ◽  
Valentin Lychagin ◽  
Mikhail Roop
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Thermodynamic Process ◽  
Equilibrium Thermodynamics ◽  
Thermodynamic State ◽  
Ideal Gases ◽  
Legendrian Submanifold ◽  
Thermodynamic Processes

In this paper, we consider an optimal control problem in the equilibrium thermodynamics of gases. The thermodynamic state of the gas is given by a Legendrian submanifold in a contact thermodynamic space. Using Pontryagin’s maximum principle, we find a thermodynamic process in this submanifold such that the gas maximizes the work functional. For ideal gases, this problem is shown to be integrable in Liouville’s sense and its solution is given by means of action-angle variables. For real gases considered to be a perturbation of ideal ones, the integrals are given asymptotically.

scholarly journals Geometrical structures for classical and quantum probability spaces

2017 ◽  
Vol 15 (08) ◽  
pp. 1740007 ◽  
Author(s):  
Florio Maria Ciaglia ◽  
Alberto Ibort ◽  
Giuseppe Marmo
Keyword(s):  
Affine Space ◽  
Vector Fields ◽  
Probability Distributions ◽  
Quantum Probability ◽  
Finite Sample ◽  
Gradient Vector ◽  
Tensor Fields ◽  
Geometrical Structures ◽  
Geometrical Properties ◽  
Probability Spaces

On the affine space containing the space [Formula: see text] of quantum states of finite-dimensional systems, there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding the relevant geometrical properties of [Formula: see text]. Guided by Dirac’s analogy principle, we will use them as inspiration to define contravariant tensor fields, Hamiltonian and gradient vector fields on the affine space containing the space of fair probability distributions on a finite sample space and analyze their geometrical properties. Most of our considerations will be dealt with for the simple example of a three-level system.

Improvements for P-ELM1 and P-ELM2 Pruning Algorithms in Extreme Learning Machines

2016 ◽  
Vol 24 (03) ◽  
pp. 327-345
Author(s):  
Junhai Zhai ◽  
Qingyan Shao ◽  
Xizhao Wang
Keyword(s):  
Computational Complexity ◽  
Information Gain ◽  
Probability Distributions ◽  
Data Sets ◽  
Learning Machines ◽  
Hidden Node ◽  
Feed Forward Neural Networks ◽  
Class Labels ◽  
Learning Machine ◽  
Hidden Layer

Extreme learning machine (ELM) is an efficient training algorithm for single-hidden layer feed-forward neural networks (SLFNs). Two pruned-ELM named P-ELM1 and P-ELM2 are proposed by Rong et al. P-ELM1 and P-ELM2 employ [Formula: see text] and information gain to measure the association between the class labels and individual hidden node respectively. But for the continuous value data sets, it is inevitable for P-ELM1 and P-ELM2 to evaluate the probability distributions of the data sets with discretization methods for calculating [Formula: see text] and information gain, while the discretization will lead to information loss. Furthermore, the discretization will result in high computational complexity. In order to deal with the problems, based on tolerance rough sets, this paper proposed an improved pruned-ELM algorithm, which can overcome the drawbacks mentioned above. Experimental results along with statistical analysis on 8 UCI data sets show that the improved algorithm outperforms the pruned-ELM in computational complexity and testing accuracy.

scholarly journals Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1209-1215
Author(s):  
Aleksandar Sebekovic ◽  
Miroslava Petrovic-Torgasev ◽  
Anica Pantic
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Forms ◽  
Equality Case ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds ◽  
Weyl Conformal Curvature Tensor

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

On the Measure of Sets of Parallel Linear Subspaces in Affine Space

1962 ◽  
Vol 14 ◽  
pp. 313-319 ◽  
Author(s):  
L. A. Santaló
Keyword(s):  
Invariant Measure ◽  
Affine Space ◽  
Real Space ◽  
Affine Transformations ◽  
Linear Subspaces ◽  
Image Position

Let En be the n-dimensional euclidean real space and the group of unimodular affine transformations which operates on it. It is known that the sets of linear h-spaces Lh (0 < h < n) have no invariant measure with respect to u (5). We wish now to consider sets of elements1.1composed by q parallel subspaces of dimensions h1, h2, … , hq which transform transitively by .

scholarly journals An Energy–Capacity inequality for Legendrian submanifolds

2018 ◽  
Vol 12 (03) ◽  
pp. 547-623 ◽  
Author(s):  
Georgios Dimitroglou Rizell ◽  
Michael G. Sullivan
Keyword(s):  
Floer Homology ◽  
Betti Numbers ◽  
Main Tool ◽  
Energy Capacity ◽  
Contact Form ◽  
Symplectic Cobordism ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds

We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the [Formula: see text]-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy is measured in terms of the oscillation of the contact Hamiltonian, together with the maximal factor by which the contact form is shrunk during the isotopy. The main tool used is a Mayer–Vietoris sequence for Lagrangian Floer homology, obtained by “neck-stretching” and “splashing”.

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