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Related Concept Videos

Entropy02:39

Entropy

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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
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The process of surrounding a solute with solvent is called solvation. It involves evenly distributing the solute within the solvent. The rule of thumb for determining a solvent for a given compound is that like dissolves like. A good solvent has molecular characteristics similar to those of the compound to be dissolved. For example, polar solutions dissolve polar solutes, and apolar solvents dissolve apolar solutes. A polar solvent is a solvent that has a high dielectric constant (ϵ...
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A living cell's primary tasks of obtaining, transforming, and using energy to do work may seem simple. However, the second law of thermodynamics explains why these tasks are harder than they appear. None of the energy transfers in the universe are completely efficient. In every energy transfer, some amount of energy is lost in a form that is unusable. In most cases, this form is heat energy. Thermodynamically, heat energy is defined as the energy transferred from one system to another that...
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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
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Entropy of Convex Functions on ℝ .

Fuchang Gao1, Jon A Wellner2

  • 1Department of Mathematics, University of Idaho, Moscow, ID 83844-1103, USA.

Constructive Approximation
|April 17, 2018
PubMed
Summary
This summary is machine-generated.

This study provides sharp estimates for the entropy of convex functions in a bounded set, revealing universal bounds for d-polytopes and the unit ball. These findings are crucial for understanding nonparametric estimation in high dimensions.

Keywords:
bracketing entropyconvex functionsmetric entropypolytopessimplicial approximation

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Area of Science:

  • Mathematical Analysis
  • Functional Analysis
  • Approximation Theory

Background:

  • Convex functions play a vital role in various mathematical and scientific fields.
  • Understanding the metric entropy of function spaces is essential for approximation theory and statistical learning.
  • Previous research has focused on entropy estimates for specific function classes, but sharp bounds for general convex functions under L_p metrics were lacking.

Purpose of the Study:

  • To derive sharp estimates for the epsilon-entropy of the class of convex functions (𝒞(Ω)) with L-norm bounded by 1, within a bounded closed convex set (Ω).
  • To analyze these entropy estimates under L_p(Ω) metrics for 1 ≤ p < r ≤ ∞.
  • To explore the implications of these estimates for universal bounds and their attainment by specific geometric objects.

Main Methods:

  • Utilizing concepts from metric entropy and approximation theory.
  • Developing sharp estimation techniques for function spaces endowed with L_p norms.
  • Analyzing the behavior of entropy for classes of convex functions on bounded convex sets.

Main Results:

  • Sharp estimates for the epsilon-entropy of 𝒞(Ω) under L_p(Ω) metrics are obtained.
  • The universal lower bound ε⁻ is shown to be an upper bound for all d-polytopes.
  • The universal upper bound of [Formula: see text] is attained by the closed unit ball, and entropy rates do not transfer to limiting bodies from inscribed polytopes.

Conclusions:

  • The derived entropy estimates provide precise characterizations of the complexity of convex function classes.
  • The results offer insights into the approximation capabilities of polytopes and the limitations of entropy rate transfer.
  • These findings have direct applications in nonparametric estimation, particularly for shape-constrained functions in high-dimensional settings.