Entropy Formula Research Articles

Generalized q-rung orthopair fuzzy interactive Hamacher power average and Heronian means for MADM.

In this paper, we establish a novel q-rung orthopair fuzzy (q-ROF) multi-attribute decision making (MADM) model on the basis of the proposed q-ROF interactive Hamacher weighted adjustable power average (q-ROFIHWAPA) and q-ROF interactive Hamacher weighted coordinated Heronian means (HMs), which (1) can reflect the correlations among multiple attributes; (2) weakens the impacts of the extreme evaluation values more reasonably; (3) considers the interactions between the membership degree (MD) and non-membership degree (N-MD) of different q-ROF numbers (q-ROFNs); (4) has the characteristic of generality (It can generate different methods by different operations). Firstly, the q-ROF interactive Hamacher operations, improved score function and new q-ROF entropy (q-ROFE) formula, which are the necessary raw materials for the implementation of MADM, are presented. Secondly, we introduce the adjustable power average (APA) and its weight form (WAPA) to remedy the deficiencies of the classical power averages (PAs). Afterwards we extend the WAPA to q-ROF circumstance and propose the q-ROF interactive Hamacher WAPA (q-ROFIHWAPA), and its basic properties are analyzed. Further, the entropy weight fitting method is presented to determine the parameter carried by the q-ROFIHWAPA. Thirdly, inspired by the evolutionary process of Bonferroni means (BMs), we define the weighted coordinated HM (WCHM) and weighted geometric coordinated HM (WGCHM) based on the traditional HMs, respectively, which eliminate the redundancy of the dual generalized weighted BM (DGWBM) and dual generalized weighted Bonferroni geometric mean (DGWBGM), i.e., the case of Then we develop the q-ROF interactive Hamacher WCHM (q-ROFIHWCHM) and q-ROF interactive Hamacher WGCHM (q-ROFIHWGCHM) by combining them with the q-ROF interactive Hamacher operations, and the common properties and special cases are also investigated. Finally, we create a MADM algorithm relied on the q-ROFIHWAPA and q-ROFIHWCHM (resp. q-ROFIHWGCHM), and a practical example is introduced to illustrate the effectiveness and superiority of the proposed method.

The entropy of systems

The entropy concept is studied from the perspective of several formalisms. It is reviewed the origins of this in the classical thermodynamics. It also is developed a step-by-step-clearly demonstration about it. After that, it is connected this formula with the entropy of Shannon. Elementary concepts of the quantum mechanics are explained to demonstrate the von Neumann entropy formula. It is proposed the entropy concept as a measure of the variability into the distribution of the states of a system given a set of rules that operate inside it for a while. From the perspective of the information theory, a language reaches a configuration to be optimal for communications. In this way, systems and languages can be studied using the same concept of entropy. It is stressed the importance of the teaching of the entropy because this is useful in the development of new technologies, for example the quantum communications.

The Massless Modular Hamiltonian

We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massless Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the higher dimensional Legendre differential operator. The quadratic form of the massless modular Hamiltonian is expressed in terms of an integral of the energy density with the parabolic distribution. We then get the formula for the local entropy of a wave packet. This gives the vacuum relative entropy of a coherent state on the double cone von Neumann algebras associated with the free scalar QFT. Among other points, we provide the passivity characterisation of the modular Hamiltonian within the standard subspace setup.

Sequence complexity, rigidity and logarithmic Sarnak conjecture

In this paper, we introduce sequence complexity and sequence mean complexity. It turns out that a topological dynamical system is uniformly rigid if and only if it has bounded topological sequence complexity; A measure preserving system is rigid if and only if it has bounded measure theoretic sequence complexity if and only if it has bounded measure theoretic sequence mean complexity. We obtain a formula of measure-theoretic entropy of an ergodic measure preserving system. We also prove that a topological dynamical systems $ (X, T) $ meets the Logarithmic Sarnak conjecture if each invariant measure of $ (X, T) $ has sub-linear measure theoretic sequence mean complexity.

Hamilton-type entropy formula along the Ricci flow on surfaces with boundary

Hamilton-type entropy formula along the Ricci flow on surfaces with boundary

Entropy of credibility distribution for intuitionistic fuzzy variable

<abstract> <p>This paper handles the new information entropy measure and divergence measure associated with intuitionistic fuzzy variables (IFVs). Based on credibility distribution and credibility measure of intuitionistic fuzzy variable, the credibility entropy formulas of discrete and continuous IFVs are proposed and some of their properties are investigated. The cross-entropy of intuitionistic fuzzy variable and its relationship with credibility entropy are then discussed. Finally, some numerical examples are given to illustrate the practicability of the presented credibility entropy and cross-entropy of intuitionistic fuzzy variable. Also, we make some comparative analysis on the credibility cross-entropy measure and some existing distance measures of IFVs in the pattern recognition problem.</p> </abstract>

Entropies of the Y-Junction Type Nanostructures

Recent research on nanostructures has demonstrated their importance and application in a variety of fields. Nanostructures are used directly or indirectly in drug delivery systems, medicine and pharmaceuticals, biological sensors, photodetectors, transistors, optical and electronic devices, and so on. The discovery of carbon nanotubes with Y-shaped junctions is motivated by the development of future advanced electronic devices. Because of their interaction with Y-junctions, electronic switches, amplifiers, and three-terminal transistors are of particular interest. Entropy is a concept that determines the uncertainty of a system or network. Entropy concepts are also used in biology, chemistry, and applied mathematics. Based on the requirements, entropy in the form of a graph can be classified into several types. In 1955, graph-based entropy was introduced. One of the types of entropy is edge-weighted entropy. We examined the abstract form of Y-shaped junctions in this study. Some edge-weight-based entropy formulas for the generic view of Y-shaped junctions were created, and some edge-weighted and topological index-based concepts for Y-shaped junctions were discussed in the present paper.

Entanglement of free fermions on Hamming graphs

Free fermions on Hamming graphs H(d,q) are considered and the entanglement entropy for two types of subsystems is computed. For subsets of vertices that form Hamming subgraphs, an analytical expression is obtained. For subsets corresponding to a neighborhood, i.e. to a set of sites at a fixed distance from a reference vertex, a decomposition in irreducible submodules of the Terwilliger algebra of H(d,q) also yields a closed formula for the entanglement entropy. Finally, for subsystems made out of multiple neighborhoods, it is shown how to construct a block-tridiagonal operator which commutes with the entanglement Hamiltonian. It is identified as a BC-Gaudin magnet Hamiltonian in a magnetic field and is diagonalized by the modified algebraic Bethe ansatz.

Katok’s Entropy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

Katok’s Entropy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

Non-Additive Entropy Composition Rules Connected with Finite Heat-Bath Effects.

Mathematical generalizations of the additive Boltzmann-Gibbs-Shannon entropy formula have been numerous since the 1960s. In this paper we seek an interpretation of the Rényi and Tsallis q-entropy formulas single parameter in terms of physical properties of a finite capacity heat-bath and fluctuations of temperature. Ideal gases of non-interacting particles are used as a demonstrating example.

Entropy formula for systems with inducing schemes

We obtain entropy formulas for Sinai–Ruelle–Bowen (SRB) measures with finite entropy given by inducing schemes. In the first part of the work, we obtain Pesin entropy formula for the class of noninvertible systems whose SRB measures are given by Gibbs-Markov induced maps. In the second part, we obtain Pesin entropy formula for invertible maps whose SRB measures are given by Young sets, taking into account a classical compression technique along the stable direction that allows a reduction of the return map associated with a Young set to a Gibbs-Markov map. In both cases, we give applications of our main results to several classes of dynamical systems with singular sets, where the classical results by Ruelle and Pesin cannot be applied. We also present examples of systems with SRB measures given by inducing schemes for which Ruelle inequality does not hold.

On Gage-Hamilton's entropy formula and Harnack inequality for the curve shortening flow

By the first two derivatives of the Boltzmann entropy of the curvature, which was first studied by Gage and Hamilton for the curve shortening flow in the plane, we define a monotonicity formula which is strictly increasing unless on a shrinking circle. By calculating pointwisely we give an alternate proof of Gage-Hamilton's Harnack inequality.

Entropy and the arrow of time in population dynamics

The concept of entropy in statistical physics is related to the existence of irreversible macroscopic processes. In this work, we explore a recently introduced entropy formula for a class of stochastic processes with more than one absorbing state that is extensively used in population genetics models. We will consider the Moran process as a paradigm for this class, and will extend our discussion to other models outside this class. We will also discuss the relation between non-extensive entropies in physics and epistasis (i.e., when the effects of different alleles are not independent) and the role of symmetries in population genetic models.

The Boltzmann distribution function for equilibrium fluctuations

A complex of issues related to the problem of proving the Boltzmann formula for the probability density (distribution) w(η) of the values of a certain set of parameters η of an equilibrium system is investigated. The classical Boltzmann formula expresses this distribution through the non-equilibrium free energy of the system F (η) , which depends on the mentioned parameters. This is possible because after the occurrence of fluctuations, the system finds itself in a non-equilibrium state, which evolves further to equilibrium. The specified parameters are chosen according to the problem under consideration. In the theory of phase transitions, they are called the order parameters. Questions under consideration include definition and construction of the free energy F (η) of a non-equilibrium system and distribution w(η ) for it in the microscopic theory. The approaches of Landau, Leontovich, and Peletminsky are discussed. It is proposed to investigate the results for states in the vicinity of equilibrium. The leading ideas of the research are considering the non-equilibrium state as being realized in the presence of an appropriate external field, using the Gibbs formula for the non-equilibrium system entropy, and applying the Boltzmann formula as a definition of the free energy of the non-equilibrium system. The article is a continuation of the authors' works and sets the task of clarifying some of their statements as well as simplifying and clarifying the calculations. Among other things, the following are discussed: the formula for the expansion of the free energy of the system in powers of the field, the simplification of the distribution w(η ) calculation, the normalization of the approximate expressions for the distribution w(η) , the possibilities of studying the free energy of the equilibrium system using our expression for the effective Landau Hamiltonian, the refinement of the calculation of the non-equilibrium free energy of a spatially inhomogeneous system, the investigation of new types of effective interactions with the Landau Hamiltonian.

A PROMETHEE based outranking approach for the construction of Fangcang shelter hospital using spherical fuzzy sets

This study mainly aims to develop two effective and practical multi-criteria group decision-making approaches by taking advantage of the ground-breaking theory of PROMETHEE family of outranking methods. The presented variants of Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) method are acknowledged to address the complex decision-making problems carrying the ambiguous information, expressible in terms of yes, no, abstinence and refusal, owing to the preeminent condition and wider structure of spherical fuzzy sets. Both of the proposed approaches seek help from the Shannon’s entropy formula to evaluate the object weights of the decision criteria. The proposed techniques operate by taking into account the deviation between each pair of potential alternatives in accordance to different types of preference functions to determine the preference indices. The proposed technique of spherical fuzzy PROMETHEE I method carefully compares the positive and negative outranking flows of the alternative to get partial rankings. In contrast, the spherical fuzzy PROMETHEE II method has the edge to eliminate the incomparable pair by employing the net outranking flow to derive the final ranking. The application of proposed approaches is explained via a case study in the field of medical concerning the selection of appropriate site to establish Fangcang shelter hospital in Wuhan to treat COVID-19 patients. The convincing comparisons of the proposed methodologies with q-rung orthopair fuzzy PROMETHEE and spherical fuzzy TOPSIS methods are also included to verify the aptitude of the proposed methodology.

The Dark Side of the Entangled Informational Universe

After having explored some basic theoretical concepts about the quantum information approach, we focus on Entropic Information Theory which is an informational approach mathematically based on the mass of the bit of information; massbit = \(\frac{kTIn(2)}{c^2}\). The mass of the bit of information and the new entropy formulae associated to it,S = k2\(\frac{TIn(2)t}{h}\) , and its alternative writings lead to new formulation,SBH = K\(\frac{c^3In(2)tevap}{16\pi^2GM}\), to calculate the entropy of black holes independently of the law of area. Being able to express the fine-grained gravitational entropy of a black hole using the rules of gravity, we can, at this level, speak of quantum gravity as emerging through the fundamentality of entangled quantum information by considering that information emerges from degree of freedom; indeed, information being a quantum state change due to the modification of one degree of freedom from the considered quantum system. In addition, we calculated the informational content of the observable universe using the entropic information formula, to obtain, 1.57 1099 bits, a result remarkably close to some previous estimates to account for all the dark matter missing in the visible Universe. After that, we calculated the amount of energy associated with this informational content using Landauer's principle, to obtain, 3.50 1076 Joules, a result that we can relate to dark energy estimates. Moreover, some deep considerations based on the perspectives of Entropic Information Theory have been explored. This new complete mathematical framework of Entropic Information Theory can explain various processes being several aspects of the same, entangled information, by considering that information emerges from degree of freedom, it is the theoretical framework of the entangled informational universe.

Evaluation of The Significance Grades of The Problems Experienced by Mathematics Teachers in Distance Education in The Covid-19 Pandemic by The Entropy Method

The entropy formula, created by Claude E. Shannon using probability calculations, is an essential and objective measurement method used in multi-criteria decision-making. The purpose this study was to evaluate the problems experienced by mathematics teachers during the Covid-19 and the degree of importance of these problems were determined. Expert opinions were used to identify the problems. In the study, the ‘Entropy’ method was used to determine the degree of importance of the problems. After determining the problems, a questionnaire was applied to 22 teachers working in different geographical regions of Turkey and with different service lengths. According to the entropy results, it has been determined that the most critical problem faced by mathematics teachers in the distance education process is the ‘communication problem’ between students and teachers. In conclusion, mathematics teachers must find the right learning method to overcome communication problems so that in the future, distance learning will not experience obstacles.

Quantum oscillations in the black hole horizon

By applying Rosen's quantization approach to the historical Oppenheimer and Snyder gravitational collapse and by setting the constraints for the formation of the Schwarzschild black hole (SBH), in a previous paper [1] two of the Authors (CC and FF) found the gravitational potential, the Schrodinger equation, the solution for the energy levels, the area quantum and the quantum representation of the ground state at the Planck scale of the SBH. Such results are consistent with previous ones in the literature. It was also shown that the traditional classical singularity in the core of the SBH is replaced by a quantum oscillator describing a non-singular two-particle system where the two components, named the "nucleus" and the "electron", strongly interact with each other through a quantum gravitational interaction. In agreement with the de Broglie hypothesis, the "electron" is interpreted in terms of the quantum oscillations of the BH horizon. In other words, the SBH should be the gravitational analogous of the hydrogen atom. In this paper, it is shown that these results allow us to compute the SBH entropy as a function of the BH principal quantum number in terms of Bekenstein-Hawking entropy and three sub-leading corrections. In addition, the coefficient of the formula of Bekenstein-Hawking entropy is reduced to a quarter of the traditional value. Then, it is shown that, by performing a correct rescaling of the energy levels, the semi-classical Bohr-like approach to BH quantum physics, previously developed by one of the Authors (CC), is consistent with the obtained results for large values of the BH principal quantum number. After this, Hawking radiation will be analysed by discussing its connection with the BH quantum structure. Finally, it is shown that the time evolution of the above mentioned system solves the BH information paradox.

Improved Wald formalism and first law of dyonic black strings with mixed Chern-Simons terms

We study the first law of thermodynamics of dyonic black strings carrying a linear momentum in type IIA string theory compactified on K3 with leading order α′ corrections. The low energy effective action contains mixed Chern-Simons terms of the form −2B(2) ^ tr(R(Γ±) ^ R(Γ±)) which is equivalent to 2H(3) ^ CS(3)(Γ±) up to a total derivative. We find that the naive application of Wald entropy formula leads to two different answers associated with the two formulations of the mixed Chern-Simons terms. Surprisingly, neither of them satisfies the first law of thermodynamics for other conserved charges computed unambiguously using the standard methods. We resolve this problem by carefully evaluating the full infinitesimal Hamiltonian at both infinity and horizon, including contributions from terms proportional to the Killing vector which turn out to be nonvanishing on the horizon and indispensable to establish the first law. We find that the infinitesimal Hamiltionian associated with −2B(2) ^ tr(R(Γ±) ^ R(Γ±)) requires an improvement via adding a closed but non-exact term, which vanishes when the string does not carry either the magnetic charge or linear momentum. Consequently, both formulations of the mixed Chern-Simons terms yield the same result of the entropy that however does not agree with the Wald entropy formula. In the case of extremal black strings, we also contrast our result with the one obtained from Sen’s approach.

Accessibility, Walkability, Mixed Land Uses:

Developing technology and maximum usage of underground resources have led to the rise of several urban problems. Nonetheless, smart growth (SG) is one of the practical principles for solving different issues, such as traffic problems, air and noise pollution, urban sprawling, etc. In this article, three parameters of smart growth, namely: accessibility to public transportation, walkability, and mixed land use, and the main parameters of space syntax theory including, integration and connectivity in three different urban districts (old, middle, and modern) in the city of Bojnourd, the city in north-eastern of Iran have been investigated. By analysing these parameters, the relationship between selected SG indices and space syntax indexes has been presented. The ANP approach was also applied to improve decision-making and to grade the importance of criteria; districts were finally graded using the TOPSIS model concerning walkability, and access to public transportation. The mixed-land uses index was also calculated by examining 11 land uses through the Shannon entropy formula. The results indicate that there is an internal relationship and correlation between smart growth indicators and space syntax ones. Additionally, given solutions can be taken as a significant step to tackle urban issues and advance sustainable development aims.