A two-parameter entropy and its fundamental properties

This article proposes a new two-parameter generalized entropy, which can be reduced to the Tsallis and Shannon entropies for specific values of its parameters. We develop a number of information-theoretic properties of this generalized entropy and divergence, for instance, the sub-additive property, strong sub-additive property, joint convexity, and information monotonicity. This article presents an exposit investigation on the information-theoretic and information-geometric characteristics of the new generalized entropy and compare them with the properties of the Tsallis and Shannon entropies.

On a characterization of information improvement

Journal of Applied Probability ◽

10.1017/s0021900200048142 ◽

1975 ◽

Vol 12 (02) ◽

pp. 407-411

Author(s):

Ram Autar

Keyword(s):

Functional Equation ◽

Economic Analysis ◽

Probability Distributions ◽

Additive Property ◽

Information Theoretic

The object of this paper is to characterize an information theoretic measure associated with three probability distributions known as ‘Information Improvement’ through a functional equation which arises by considering the additive property of the measure. This measure has been extensively used in economic analysis.

CONVEXITY OF PARAMETER EXTENSIONS OF SOME RELATIVE OPERATOR ENTROPIES WITH A PERSPECTIVE APPROACH

Glasgow Mathematical Journal ◽

10.1017/s0017089517000131 ◽

2019 ◽

Vol 62 (3) ◽

pp. 737-744 ◽

Author(s):

ISMAIL NIKOUFAR

Keyword(s):

Crucial Role ◽

Quantum Statistics ◽

Joint Convexity ◽

Jointly Convex ◽

Parametric Extensions ◽

Proof Strategy ◽

AbstractIn this paper, we introduce two notions of a relative operator (α, β)-entropy and a Tsallis relative operator (α, β)-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning α and β. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.Significance Statement. What is novel here is that we convincingly demonstrate how our techniques can be used to give simple proofs for the old and new theorems for the functions that are relevant to quantum statistics. Our proof strategy shows that the joint convexity of the perspective of some functions plays a crucial role to give simple proofs for the joint convexity (resp. concavity) of some relative operator entropies.

The Fundamental Properties of Information-Carrying Relations

Thinking Machines and the Philosophy of Computer Science ◽

10.4018/978-1-61692-014-2.ch002 ◽

2010 ◽

pp. 16-35 ◽

Author(s):

Hilmi Demir

Keyword(s):

Mathematical Theory ◽

Communication Theory ◽

Detailed Examination ◽

Philosophy Of Information ◽

Information Theoretic ◽

Fundamental Properties ◽

Mathematical Properties ◽

Information And Communication ◽

Properties Of Information ◽

Shed Light

Philosophers have used information theoretic concepts and theorems for philosophical purposes since the publication of Shannon’s seminal work, “The Mathematical Theory of Communication”. The efforts of different philosophers led to the formation of Philosophy of Information as a subfield of philosophy in the late 1990s (Floridi, in press). Although a significant part of those efforts was devoted to the mathematical formalism of information and communication theory, a thorough analysis of the fundamental mathematical properties of information-carrying relations has not yet been done. The point here is that a thorough analysis of the fundamental properties of information-carrying relations will shed light on some important controversies. The overall aim of this chapter is to begin this process of elucidation. It therefore includes a detailed examination of three semantic theories of information: Dretske’s entropy-based framework, Harms’ theory of mutual information and Cohen and Meskin’s counterfactual theory. These three theories are selected because they represent all lines of reasoning available in the literature in regard to the relevance of Shannon’s mathematical theory of information for philosophical purposes. Thus, the immediate goal is to cover the entire landscape of the literature with respect to this criterion. Moreover, this chapter offers a novel analysis of the transitivity of information-carrying relations.

The parameter space and third law of thermodynamics for the Borges–Roditi, Abe and Sharma–Mittal entropies

International Journal of Modern Physics B ◽

10.1142/s0217979218502740 ◽

2018 ◽

Vol 32 (24) ◽

pp. 1850274 ◽

Author(s):

Bilal Canturk ◽

Thomas Oikonomou ◽

G. Baris Bagci

Keyword(s):

Parameter Space ◽

Additional Parameter ◽

Generalized Entropy ◽

The Third ◽

Third Law Of Thermodynamics ◽

Third Law ◽

The One ◽

Theoretical Results ◽

In order to investigate the role of the parameter space on the third law of thermodynamics, we have checked the validity intervals of the Borges–Roditi, Abe and Sharma–Mittal entropies in the framework of the third law of thermodynamics. The two-parameter Borges–Roditi entropy conforms to the third law for a [Formula: see text] 1, 0 [Formula: see text] b [Formula: see text] 1 or b [Formula: see text] 1, 0 [Formula: see text] a [Formula: see text] 1 due to its a [Formula: see text] b symmetry, while the Abe entropy satisfies the third law in the interval [Formula: see text] [Formula: see text] 1 or [Formula: see text]. Since the validity interval of the single-parameter Abe entropy can be fully recovered from the two-parameter Borges–Roditi entropy, we note that the third law is immune to the reduction of the parameters in this particular case. The Sharma–Mittal entropy, in this same context, is valid in the interval 0 [Formula: see text] q [Formula: see text] 1, whereas its second parameter r can take any real values without restriction. The interval 0 [Formula: see text] q [Formula: see text] 1 is the intersection of the Sharma–Mittal entropy with its reduced expressions, i.e., the Tsallis and Rényi entropies from the third law perspective. This implies that the additional parameter r of the Sharma–Mittal entropy indeed renders it more generalized compared to the one-parameter Rényi and Tsallis entropies when validated by the third law of thermodynamics. Therefore, based on these observations, we deduce that an additional parameter can extend the interval of validity of the third law for a particular generalized entropy only when this additional parameter is not confined by reduction to some other generalized entropy definitions. We also check our theoretical results by using the one-dimensional Ising model with periodic boundary conditions under zero external field.

On a characterization of information improvement

Journal of Applied Probability ◽

10.2307/3212459 ◽

1975 ◽

Vol 12 (2) ◽

pp. 407-411

Author(s):

Ram Autar

Keyword(s):

Functional Equation ◽

Economic Analysis ◽

Probability Distributions ◽

Additive Property ◽

Information Theoretic

Characterizations of Generalized Entropy Functions by Functional Equations

Advances in Mathematical Physics ◽

10.1155/2011/126108 ◽

2011 ◽

Vol 2011 ◽

pp. 1-12

Author(s):

Shigeru Furuichi

Keyword(s):

Functional Equation ◽

Functional Equations ◽

Tsallis Entropy ◽

Entropy Function ◽

Generalized Entropy ◽

Entropy Functions ◽

We will show that a two-parameter extended entropy function is characterized by a functional equation. As a corollary of this result, we obtain that Tsallis entropy function is characterized by a functional equation, which is a different form that used by Suyari and Tsukada, 2009, that is, in a proposition 2.1 in the present paper. We give an interpretation of the functional equation in our main theorem.

Some results of the Barbier-Chalonge-Divan system of stellar classification

Symposium - International Astronomical Union ◽

10.1017/s0074180900053250 ◽

1966 ◽

Vol 24 ◽

pp. 77-90 ◽

Author(s):

D. Chalonge

Keyword(s):

Early Type ◽

Parameter System ◽

Early Type Stars ◽

Several years ago a three-parameter system of stellar classification has been proposed (1, 2), for the early-type stars (O-G): it was an improvement on the two-parameter system described by Barbier and Chalonge (3).