In this work, we introduce a generalized measure of cumulative residual entropy and study its properties. We show that several existing measures of entropy such as cumulative residual entropy, weighted cumulative residual entropy and cumulative residual Tsallis entropy, are all special cases of the generalized cumulative residual entropy. We also propose a measure of generalized cumulative entropy, which includes cumulative entropy, weighted cumulative entropy and cumulative Tsallis entropy as special cases. We discuss generating function approach using which we derive different entropy measures. Finally, using the newly introduced entropy measures, we establish some relationships between entropy and extropy measures.

The fractal dimension of is identified in this paper. More importantly, the undertaken research has shown how affects the Koch snowflake . The study has demonstrated the existence of negative values of (Koch snowflake), which connects to a considerable number of extremely intriguing mathematical and physical research areas. The information-theoretic impact on the Sierpiniski Gasket dimension , which corresponds to , was treated in a more fundamental way. Notably, the credibility of this current study adds to the existing kThe fractal dimension of is identified in this paper. More importantly, the undertaken research has shown how affects the Koch snowflake . The study has demonstrated the existence of negative values of (Koch snowflake), which connects to a considerable number of extremely intriguing mathematical and physical research areas. The information-theoretic impact on the Sierpiniski Gasket dimension , which corresponds to , was treated in a more fundamental way. Notably, the credibility of this current study adds to the existing knowledge for the information- theoretic fractal dimensions. On another applicative note, this paper offers some potential applications of fractal dimension to smart cities. The investigation of the threshold theorems for and in terms of the triad is part of the next study phase. Additionally, investigating the newly obtained 's corresponding arbitrary Sierpinski Gasket for the first time.applicative note, this paper offers some potential applications of fractal dimension to smart cities. The investigation of the threshold theorems for and in terms of the triad is part of the next study phase. Additionally, investigating the newly obtained 's corresponding arbitrary Sierpinski Gasket for the first time.

The study of the change in entropy during the formation of stars in collapsing molecular clouds is an important area of research in astrophysics. As the cloud decreases in size, the presence of gases leads to an increase in entropy. However, the extent of this change in entropy is influenced by the amount of space available for the gases. In this scientific investigation, I conduct a detailed analysis of entropy on a small scale within the contracting cloud, following the principles of the second law of thermodynamics. Initially, my focus is on a turbulent cloud composed of N particles with a mass of M, which collapses under the force of gravity. It becomes clear that in order for the collapse to occur, gravitational energy must overcome other pressure forces that oppose it. This process results in the rapid movement of particles within the cloud, ultimately leading to a change in entropy

We introduce the notion of generalized contraction on dualistic partial metric spaces. A fixed point theorem for mappings satisfying above mentioned contraction is obtained. Some consequences of our result are obtained. We construct an example to demonstrate the effectiveness of our result among the corresponding results in partial metric spaces. Our results provide substantial generalizations and improvements of several well known results existing in the comparable literature. We discuss an application of our fixed point results to show the existence of solution of functional equations.

Drowsiness is a main factor for various costly defects, even fatal accidents in areas such as construction, transportation, industry and medicine, due to the lack of monitoring vigilance in the mentioned areas. The implementation of a drowsiness detection system can greatly help to reduce the defects and accident rates by alerting individuals when they enter a drowsy state. This research proposes an Electroencephalography (EEG) based approach for detecting drowsiness. EEG signals are passed through a preprocessing chain composed of artifact removal and segmentation to ensure accurate detection followed by different feature extraction methods to extract the different features related to drowsiness. This work explores the use of various machine learning algorithms such as Support Vector Machine (SVM) the K Nearest Neighbor (KNN) the Naive Bayes (NB) the Decision Tree (DT) and the Multilayer Perceptron (MLP) to analyze EEG signals sourced from the DROZY database, carefully labeled into two distinct states of alertness (awake, and drowsy). Segmentation into 10-second intervals ensures precise detection, while a relevant feature selection layer enhances accuracy and generalizability. The proposed approach achieves high accuracy rates of 99.84% and 96.4% for intra (subject by subject) and inter (cross-subject) modes, respectively. SVM emerges as the most effective model for drowsiness detection in the intra mode, while MLP demonstrates superior accuracy in the inter mode. This research offers a promising avenue for implementing proactive drowsiness detection systems to enhance occupational safety across various industries.

In this paper, we unite concepts from Husserlian phenomenology, the active inference framework in theoretical biology, and category theory in mathematics to develop a comprehensive framework for understanding social action premised on shared goals. We begin with an overview of Husserlian phenomenology, focusing on aspects of inner time consciousness, namely, retention, primal impression, and protention. We then review active inference as a formal approach to modeling agent behavior, based on variational (approximate Bayesian) inference. Expanding upon Husserl’s model of time consciousness, we consider collective goal-directed behavior, emphasizing shared protentions among agents and their connection to the shared generative models of active inference. This integrated framework aims to formalize shared goals in terms of shared protentions, and to thereby shed light on the emergence of group intentionality. Building on this foundation, we incorporate mathematical tools from category theory; particularly, sheaf and topos theory, to furnish a mathematical image of individual and group interactions within a stochastic environment. Specifically, we employ morphisms between polynomial representations of individual agent models, allowing predictions not only of their own behaviors but also those of other agents and environmental responses. Sheaf and topos theory facilitate the construction of coherent agent worldviews and provides a way of representing consensus or shared understanding. We explore the emergence of shared protentions, bridging the phenomenology of temporal structure, multi-agent active inference systems, and category theory. Shared protentions are highlighted as pivotal for coordination and achieving common objectives. We conclude by acknowledging the intricacies stemming from stochastic systems and uncertainties in realizing shared goals.

It has been empirically observed that the upper tail of the firm size distribution follows either the Pareto distribution or the Zipf distribution, with both patterns being explained by Gibrat’s law (Gabaix, 1999). This article analyzed firm revenue data from China, spanning from 2005 to 2013, to examine the whole range of the firm size distribution. Our empirical analysis revealed that firm revenue data over these years is well fitted by a three-parameter generalized Pareto distribution, with the fitted parameters indicating a dichotomy: The size distribution of large-sized firms, namely the upper tail, is asymptotically characterized by a Pareto distribution or a Zipf distribution, whereas the size distribution of smaller and medium-sized firms is approximated by an exponential distribution. This finding suggests that Gibrat’s law should be extended to account for the emergence of a generalized Pareto distribution.

The present work investigates primarily the morphology evolution of the body-centered-cubic (BCC)/B2 phases in AlxNiCoFeCr high-entropy alloys (HEAs) with increasing Al content, which has been neglected so far. There exist two types of microscopic morphologies of BCC and B2 phases in this HEA series: one is the weave-like morphology induced by the spinodal decomposition, and the other is the microstructure of a spherical disordered BCC precipitation on the ordered B2 matrix that appears in HEAs with a much higher Al content. The shape of coherent precipitates is found to be closely related to the lattice misfit between BCC and B2 phases, which is sensitive to Al. The mechanical properties, including the compressive yielding strength and microhardness of the AlxNiCoFeCr HEAs, are also discussed in light of the concept of the valence electron concentration (VEC).

We introduce a new generalized inverse which is a natural generalization of (pseudo) right core inverse in a Banach *-algebra. We characterize this generalized inverse by using right core and quasi-nilpotent decomposition. We then present its polar-like characterization and investigate related algebraic properties. Finally, the right core-EP inverse is characterized by certain new ways.

The real systems are often complex, nonlinear, and noisy in various areas including mathematics, natural science, and social science. We present the symplectic entropy (SymEn) measure as well as an analysis method based on SymEn to estimate the nonlinearity of the complex system by analyzing the given time series. The SymEn estimation is a kind of entropy based on symplectic principal component analysis (SPCA) which represent organized but unpredictable behaviors of systems. The key to SPCA is to preserve the global submanifold geometrical properties of the systems through symplectic transform in the phase space, which is a kind of the measure-preserving transforms. The capability of preserving the global geometrical characteristics makes the SymEn a test statistic to detect the nonlinear characteristics in several typical chaotic time series and the stochastic characteristic in the Gaussian white noise. The results are in agreement with findings in the approximate entropy (ApEn), the sample entropy (SampEn) and the fuzzy entropy (FuzzyEn). Moreover, the SymEn method is also used to analyze the nonlinearities of the real signals (including the EEG signals for ASD and healthy subjects, and the sound and vibration signals for the mechanical systems). The results indicate that the SymEn estimation can be taken as a measure for describing the nonlinear characteristics in the data collected from the natural complex systems.

In this paper, we present Binet's formulas, generating functions, and the summation formulas for generalized Pentanacci numbers, and as special cases, we investigate Pentanacci and Pentanacci-Lucas numbers with their properties. Also, we define Gaussian generalized Pentanacci numbers and as special cases, we investigate Gaussian Pentanacci and Gaussian Pentanacci-Lucas numbers with their properties. Moreover, we give some identities for these numbers. Furthermore, we present matrix formulations of generalized Pentanacci numbers and Gaussian generalized Pentanacci numbers.

In this paper, we introduce the notion of the generalized $w$-group inverse in a Banach algebra with proper involution. This is a natural generalization of generalized (weak) group inverse in Banach *-algebras. Many elementary properties of such new weighted generalized inverse are presented. We further establish the equivalent relations between generalized weighted group inverse and weighted generalized Drazin inverse. Related properties of generalized (weak) group inverse are extended to the wider cases.

The self-assembly is an intriguing process of great importance in general soft matter and thus re-ceives an ever-growing scientific interest. Its most trivial example is the entropy-driven crystalli-zation of hard spheres. Past works have established the similarities and differences in the phase behavior of monomers and chains made of hard spheres. In the present work, inspired by the dif-ference in the melting points of the pure components, we study, through Monte Carlo simulations, the phase behavior of athermal mixtures composed of fully flexible polymers and individual monomers of uniform size. We analyze how the relative number fraction and the packing density affect crystallization and the established ordered morphologies. A synergetic effect is observed in the crystallization leading to synchronous crystallization of the two species. Structural analysis of the resulting ordered morphologies shows perfect mixing and thus no phase separation, which could split the system into polymer-rich and monomer-rich domains. Due to the constraints imposed by chain connectivity, the local environment of the individual spheres is systematically more spherical and more symmetric compared to the one of spheres belonging to chains. In general, increasing polymer content leads to reduction of the degree of crystallinity. According to the present findings, relative concentration is another determining factor to control the phase behavior of hard colloidal systems based on polymers.

：Non-Euclidean data, such as social networks, and citation relationships between documents, has node information and structural information. Graph Convolutional Network(GCN) can automatically learn node features and association information between nodes. The core ideology of the graph convolutional network is to aggregate node information by using edge information, thereby generating a new node feature. In the process of updating node features, there are two core influencing factors. One is the number of neighboring nodes of the central node, the other is the contribution of the neighboring nodes to the central node. Due to the previous GCN methods not simultaneously considering the numbers and different contributions of neighboring nodes to the central node, we design the Adaptive Attention Mechanism(AAM). To further enhance the representational capability of the model, we utilize Multi-Head Graph Convolution(MHGC). Finally, we adopt cross-entropy(CE) loss function to describe the difference between the predicted results of node categories and the ground truth (GT). Combined with backpropagation, this ultimately achieves accurate node classification. Based on AAM, MHGC and CE, we contrive the novel Graph Adaptive Attention Network (GAAN). Experiments show that the classification accuracy has achieved outstanding performances on Cora, Citeseer and Pubmed datasets.

A new generalized definition of Mersenne numbers is proposed of the form (a^n - (a-1)^n), called Global Generalized Mersenne numbers, or simply Generalized Mersenne numbers and noted GM_{a,n} where a is the base and n is the exponent, both being positive integers. The properties are investigated for prime exponents n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. In particular, it is found that, for any base a, Generalized Mersenne numbers are in general such that (GM_{a,n} - 1) are even and divisible by n, a and (a-1) for any odd prime exponent n and by (a(a-1)+1) for any prime exponent n > 5. The remaining factor is a function of triangular numbers of (a-1), specific to each prime exponent n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, allowing to show first, that (GM_{a,n} - 1) are divisible by 6, and more precisely GM_{a,n} are congruent to 1(mod 12) or 7(mod 12) depending on the congruence of the base a(mod 4); second, that (GM_{a,n} - 1) are divisible by 10 if n == 1(mod 4) and, if n == 3(mod 4), GM_{a,n} == 1(mod 10), or 7(mod 10) or 9(mod 10) depending on the congruence of the base a(mod 5); third, that all factors c_{i} of GM_{a,n} are of the form (2nf_{i}+1) with f_{i} natural integers such that c_{i} is prime itself or the product of primes of the form (2nj+1) with j natural integer; fourth, that for odd prime exponents n, all GM_{a,n} are periodically congruent to either +/-1\(mod 8) or +/-3(mod 8) depending on the congruence of the base a(mod 8); and fifth, that the factors of a composite GM_{a,n} is of the form (2nf_{i}+1) with f_{i} == u(mod 4) and u being either 0, 1, 2 or 3 depending on the congruence of the exponent n(mod 4) and on the congruence of the base a(mod 8). The potential use of Generalized Mersenne primes in cryptography is shortly addressed.

In addition to their importance in statistical thermodynamics, probabilistic entropy measurements are crucial for understanding and analyzing complex systems, with diverse applications in time series and one-dimensional profiles. However, extending these methods to two- and three-dimensional data still requires further development. In this study, we present a new method to classify spatiotemporal processes based on entropy measurements. To test and validate the method, we selected four classes of similar processes related to the evolution of random patterns: dynamic colored noises ((i)~white and (ii)~red); (iii)~weak turbulence from reaction-diffusion; (iv)~hydrodynamic fully developed turbulence, and (v)~plasma turbulence from MHD. Considering seven possible ways to measure entropy from a matrix, we present the method as a parameter space composed of the two best separating measures of the five selected classes. The results highlight better combined performance of Shannon Permutation Entropy ($S^p_H$) and a new approach based on Tsallis Spectral Permutation Entropy ($S^s_q$). Notably, our observations reveal the segregation of reaction terms in this $S^p_H \times S^s_q$ space, a result that identifies specific sectors for each class of dynamic process, and can be used to train machine learning models for automatic classification of complex spatiotemporal patterns.

Motivated by empirical observations, we propose a possible extension of Gibrat's law. By applying it into the random growth theory of income distribution (Gabaix, 2009), we find that the income distribution is described by a generalized Pareto-type distribution (GPD) with three parameters. We observe that there is a parameter /elta in the GPD that plays a key role in determining the shape of income distribution. By using the Kolmogorov-Smirnov test, we empirically show that, for typical market-economy countries, /elta is close to 0 significantly, such that the income distribution is characterized by a two-class pattern in which the bottom 90% of the population is approximated by an exponential distribution and the richest 1%~3% is approximated by an asymptotic power law. However, we empirically find that, for China both in planned economy period and in early stages of market reformation (from 1978 to 1990), /elta is significantly deviated from 0, such that the bottom of the population no longer conforms to an exponential distribution.

Predicting the outcome of a future game between two National Basketball Association (NBA) teams poses a challenging problem of interest to statistical scientists as well as the general public. In this article, we formalize the problem of predicting the game results as a classification problem and apply the principle of maximum entropy to construct NBA maximum entropy (NBAME) model that fits to discrete statistics for NBA games, and then predict the outcomes of NBA playoffs by the NBAME model. The best NBAME model is able to correctly predict the winning team 74.4 percent of the time as compared to some other machine learning algorithms which is correct 69.3 percent of the time.

An analysis of the processes of plastic deformation and acoustic relaxation in a high-entropy alloy Al0.5CoCrCuFeNi was carried out. The following have been established: dominant dislocation defects; types of barriers that prevent the movement of dislocations; mechanisms of thermally activated movement of various elements of dislocation lines through barriers at room and low temperatures. Based on modern dislocation theory, quantitative estimates have been obtained for the most important characteristics of dislocations and their interaction with barriers.

In this paper, we introduce and investigate a new family of sequences called the generalized Fibospinomials (or the generalized Fibonacci polynomial spinors or Horadam polynomial spinors). Being particular cases, we handle with $(r,s)$-Fibonacci and $(r,s)$-Lucas polynomial spinors. After a short history on spinors and quaternions, we present Binet's formulas, generating functions and the summation formulas for these polynomials. In addition, we obtain some identities of generalized Fibonacci polynomial spinors, $(r,s)$-Fibonacci polynomial spinors and $(r,s)$-Lucas polynomial spinors. Moreover, we give some special identities such as Catalan's and Cassini's identities and we present matrices related with these polynomials.

In this work, an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically is introduced. The goal is to optimally partition the sensor characteristic, i.e. to select the vertices of the approximating polyline (approximant) along with their positions on, the sensor characteristics so that the distance (i.e. the separation) between the approximant and the characteristic is rendered below a certain problem-specific tolerance. To achieve this goal, two alternative non-linear optimization problems are solved, whose essential difference is in the adopted quantitative measure of the separation between the transfer function and the approximant. In the first problem, which relates to absolutely integrable sensor characteristics (their energy is not necessarily finite, but they can be represented in terms of convergent Fourier series), the polylinearization is constructed by numerical minimization of the L^1--metric (a distance-based separation measure), concerning the number of polyline vertices and their locations. In the second problem which covers the quadratically integrable sensor characteristics (whose energy is finite, but they do not necessarily admit a representation in terms of convergent Fourier series), the polylinearization is constructed by minimizing numerically the L^2-metric (area-, or energy-based separation measure) for the same set of optimization variables –the locations, and the number of polyline vertices.

While entropy was introduced in the second half of the 19th century in the international vocabulary as a scientific term, in the 20th century it became common in colloquial use. Popular imagination has loaded “entropy” with almost every negative quality in the universe, in life and in society, with a dominant meaning of disorder and disorganization. Exploring the history of the term and many different approaches on it, we show that entropy has a universal stochastic definition which is not disorder. The accompanying principle of maximum entropy, which lies behind the Second Law, gives explanatory and inferential power to the concept and promotes entropy as the mother of creativity and evolution. As social sciences are often contaminated by subjectivity and ideological influences, we try to explore whether the maximum entropy, applied to the distribution of wealth quantified by annual income, can give an objective description. Using publicly available income data, we show that the income distribution is consistent with the principle of the maximum entropy. The increase of entropy is associated to increase of society’s wealth yet a standardized form of entropy can be used to quantify inequality. Historically, technology has played a major role in development and increase of the entropy of income. Such findings are contrary to the theories of ecological economics and other theories which use the term entropy in a Malthusian perspective.

Transformation formula under the action of a general linear fractional transformation for generalized Dedekind eta-function has been a subject of intensive study, Rademacher, Dieter, Meyer et al. However, the (Hecke) modular relation structure has not been recognized until the work of Goldstein-de la Torre, and streamlining the classical proofs in the modular relation will reveal the meaning hidden in those works. Our main aim is to elucidate the works of these researchers in the context of modular relations.

An entropic origin of gravity is re-visited. Isothermal self-gravitating cloud seen as an ideal gas is analyzed. Gravitational attraction within the isothermal cloud in equilibrium is balanced by the pressure, which is of a pure entropic nature. The notion of the Jeans entropy of the cloud corresponding to the entropy of the self-gravitating cloud in mechanical and thermal equilibrium is introduced. Balance of the gravitational compression and the entropic repulsion yields the scaling relation hinting to the entropic origin of the gravitational force. The analysis of the Jeans instability enables elimination of the “holographic screen” or “holographic principle” necessary for grounding of the entropic origin of gravity.

Sentiment analysis plays a crucial role in understanding customer feedback, guiding product development, and informing business decisions. This paper introduces term sentiment entropy (TSE), a novel weighting method that leverages the distribution of sentiment labels associated with words to enhance text classification accuracy. TSE complements traditional TF-IDF techniques by capturing the nuances of sentiment variation across different contexts. Experiments across diverse public datasets demonstrate TSE's potential to improve sentiment analysis performance, especially in capturing subtle sentiment shifts and adapting to specific domains. While computational cost and label quality pose challenges, TSE offers a promising avenue for refining sentiment analysis and opening new research frontiers.

Over the past decade and a half, dynamic functional imaging has revolutionized the neuroimaging field. Since 2009, it has revealed low dimensional brain connectivity measures, has identified potential common human spatial connectivity states, has tracked the transition patterns of these states, and has demonstrated meaningful alterations in these transition and spatial patterns in neurological disorders, psychiatric disorders, and over the course of development. Recently, researchers have begun to analyze this data from the perspective of dynamic system and information theory in the hopes of understanding how these dynamics support less easily quantified processes, such as information processing, cortical hierarchy, and consciousness. Consciousness, in particular, appears to be strongly linked to entropy production in the brain. Little attention has been paid to the effects of psychiatric disease on entropy production in the human brain, however. Even disorders characterized by substantial changes in conscious experience have not been widely analyzed from this perspective. We begin to rectify this by examining the complexity of subject trajectories in state space through the lens of information theory. Specifically, we identify a basis for the dynamic functional connectivity state space and track subject trajectories through this space over the course of the scan. The dynamic complexity of these trajectories is assessed along each dimension of the proposed basis space. Using these estimates, we demonstrate that schizophrenia patients display substantially simpler trajectories than demographically matched healthy controls, and that this drop in complexity concentrates along specific dimensions. We also demonstrate that entropy generation in at least one of these dimensions is linked to cognitive performance. Overall, results suggest great value in applying dynamic systems theory to problems of neuroimaging and reveal a substantial drop in the complexity of schizophrenia patients’ brain function.

Diagnostics and decomposition of atmospheric disturbances in a planar flow are considered and applied to numerical modeling results with the direct possibility to use in atmosphere monitoring especially in such strong events which follow magnetic storms. The study examines a situation in which the stationary equilibrium temperature of a gas may depend on a vertical coordinate, that seriously complicates the problem solution. The relations connecting perturbations for acoustic and entropy modes are analytically established and led to the solvable diagnostic equations. These perturbation structures, found as the equation solutions specify acoustic and entropy modes in an arbitrary stratified gas under the condition of stability. These time-independent diagnostic relations link gas perturbation variables of the acoustic and the entropy modes. Hence, they provide the ability to decompose the total vector of perturbations into acoustic and non-acoustic (entropy) parts uniquely at any instant within the all accessible heights range. As a prospective model, we consider the diagnostics at the height interval [120;180] km, where the equilibrium temperature of a gas depends linearly on the vertical coordinate. For such a heights range it is possible to proceed with analytical expressions for pressure and entropy perturbations of gas variables. Individual profiles of acoustic and entropy parts for some data, obtained by numerical experiment, are illustrated by the plots for the pure numerical data against ones obtained by the model. The total energy of a flow is determined for both approaches and its height profiles are compared.

How to build an optimal model where three terms “time, causality, and entropy” will be interrelated in a most correct way? This task has a semantic and physical aspect. We give in this article the solution that is based only on elementary perception. In physics, we experience time with our senses as the duration of material change running in space. This fact is the standpoint for our model where causality is only a principle, it is not a physical actuality and entropy runs only in space and not in time. Time is merely a duration of entropy increasing and is entering existence when measured by the observer.

Different notions of entropy can be identified in different communities: (i) the thermodynamic sense, (ii) the information sense, (iii) the statistical sense, (iv) the disorder sense, and (v) the homogeneity sense. Especially the “disorder sense” and the “homogeneity sense” relate to and require the notion of space and time. One of the few prominent examples relating entropy to geometry and to space is the Bekenstein-Hawking entropy of a Black Hole. Although being developed for the description of a physics object—a black hole—having a mass, a momentum, a temperature, a charge etc. absolutely no information about these attributes of this object can eventually be found in the final formula. In contrast, the Bekenstein-Hawking entropy in its dimensionless form is a positive quantity only comprising geometric attributes like an area A—which is the area of the event horizon of the black hole-, a length L_P—which is the Planck length - and a factor ¼. A purely geometric approach towards this formula will be presented. The approach is based on a continuous 3D extension of the Heaviside function, with this extension drawing on the phase-field concept of diffuse interfaces. Entropy enters into the local, statistical description of contrast respectively gradient distributions in the transition region of the extended Heaviside function definition. The structure of the Bekenstein-Hawking formula eventually is derived for a geometric sphere based on mere geometric-statistic considerations.

In this work, satellite data from the Clouds and Earth’s Radiant Energy System (CERES) and Moderate Resolution Imaging Spectroradiometer (MODIS) instruments is analyzed to determine how the global absorbed sunlight and global entropy generation rate have changed during the time period from 2002 to 2023. The data is used to test hypotheses derived from the Maximum Power Principle (MPP) and Maximum Entropy Production Principle (MEP) about the evolution of earth’s surface and atmosphere. The results indicate that the amount of absorbed sunlight has increased over the last 20 years but rising surface temperatures have caused the entropy generation rate to remain approximately constant despite that decrease in albedo. The results are simpler to explain using the MPP but do not completely rule out the MEP. Given the acceptance of the MPP or MEP, some extensions and nuances are discussed.

The existing segmentation-based scene text detection methods mostly need complicated post-processing, and the post-processing operation is separated from the training process, which greatly reduces the detection performance. The previous method, DBNet successfully simplified post-processing and integrated the post-processing into a segmentation network. However, the training process of the model took a long time for 1200 epochs and the sensitivity to texts of various scales was lacking, leading to some text instances being missed. Considering the above two problems, we design the text detection Network with Binarization of Hyperbolic Tangent(HTBNet). First of all, we propose Binarization of Hyperbolic Tangent (HTB), optimized along with which, the segmentation network can expedite the initial convergent speed by reducing the amount of epochs from 1200 to 600. Because features of different channels in the same scale feature map focus on the information of different regions in the image, to better represent the important features of all objects in the image, we devise the Multi-Scale Channel Attention(MSCA). Meanwhile considering that multi-scale objects in the image cannot be simultaneously detected, we propose a novel module named Fused Module with Channel and Spatial(FMCS), which can fuse the multi-scale feature maps from channel and spatial dimension. Finally we adopt cross entropy as the loss function, which measures the difference between predicted values and ground truths. The experimental results show that HTBNet compared with lightweight models has achieved competitive performance and speed on Total-Text(F-measure:86.0%, FPS:30) and MSRA-TD500 (F-measure:87.5%, FPS:30).

In the current work, the stable queueuing system's Generalised Z-Entropy (GZE) of Rényian formalism is examined in its fractal dimension. Notably, fractal dimension’s resulting behaviour that corresponds to the GZE parameters is examined through numerical tests. This research makes a substantial generalization in the literature by fusing fractal geometry and information theory to shed light on how entropy and complexity interact. More fundamentally, the significant role of fractal dimension to advance Big Data Analytics(BDAs) is highlighted. Closing remarks combined with open problems and the next phase of research are provided.

In this paper, under the symmetric entropy and the scale squared error loss functions, we consider the maximum likelihood (ML) estimation and Bayesian estimation of the Shannon entropy and Rényi entropy of the two-parameter inverse Weibull distribution. In the ML estimation, the dichotomy is used to solve likelihood equation. In addition, the approximation confidence interval is given by the Delta method. Because the form of estimation results is more complex in the Bayesian estimation, the Lindley approximation method is used to achieve the numerical calculation. Finally, Monte Carlo simulations and a real data set are used to illustrate the results derived. By comparing the mean square error between the estimated value and the real value, it can be found that the performance of ML estimation of Shannon entropy is better than that of Bayesian estimation, and there is no significant difference between the performance of ML estimation of Rényi entropy and that of Bayesian estimation.

This study focuses on a new class of functions called sub-b-s-preinvex, that is a generalization of sub-b-s-convex and preinvex functions, and discusses some of their properties. A new sub-b-s-preinvex programming is introduced and the sufficient conditions of optimality under this type of function is established.

The Householder transformation, allowing a rewrite of probabilities into expectations of dichotomic observables, is generalized in terms of its spectral decomposition. Dichotomy is modulated by allowing more than one negative eigenvalues, or by abandoning it altogether, yielding generalized operator valued arguments for contextuality. We also discuss a form of state-dependent contextuality by variation of the functional relations of the operators; in particular, by additivity.

To enhance the applicability of dual-phase high-entropy alloys (HEAs) like Fe32Cr33Ni29Al3Ti3, fabricated via selective laser melting (SLM), a focus on improving their mechanical properties is essential. As part of this effort, heat treatment was explored. This study compares the microstructure and mechanical properties of the as-printed sample with those cooled in water after undergoing heat treatment at temperatures ranging from 1000 to 1200 ℃ for 1 hour. Both pre- and post-treatment samples reveal a dual-phase microstructure comprising FCC and BCC phases. Although heat treatment led to a reduction in tensile and yield strength, it significantly increased ductility compared to the as-printed sample. This strength-ductility trade-off is related to changes in grain sizes with ultrafine grains enhancing strength and micron grains optimizing ductility, also the influence of the content of FCC/BCC phases and dislocation density. In particular, the sample heat-treated at 1000 °C for 1 hour and then water-cooled exhibited a better combination of strength and ductility, a yield strength of 790 MPa, and an elongation of 13%. This research offers innovative perspectives on crafting dual-phase HEA of Fe32Cr33Ni29Al3Ti3, allowing for tailorable microstructure and mechanical properties through a synergistic approach involving SLM and heat treatment.

The fitting and modeling of skewed, complex, symmetric, and asymmetric datasets is an exciting research topic in many fields of applied sciences, notably lifetime, medical, and financial sciences. The Heavy tailed Nadarajah Haghighi model is introduced by compounding the heavy tailed family and Nadarajah Haghighi distribution in this paper. The so obtained model has three parameters accounting for the scale and shape of the distribution. The proposed distribution's fundamental characteristics such as probability density, cumulative distribution, hazard rate, and survival functions are provided, and several key statistical properties are established, and several entropy information measures are proposed. The estimation of model parameter is performed via the ML procedure. Further, different simulation experiments are conducted to demonstrate the performance of proposed estimator’s using some measure, like the average estimate, the average bias, and associated mean square error. Finally, we are applied our proposed model to analyze three different real datasets. In our illustration, we are compared the practicality of the recommended model with several well-known competing models.

In the paper, the authors generalize the notion "k-Mittag-Leffler function'', establish some integral transforms of the generalized k-Mittag-Leffler function, and derive several special and known conclusions in terms of the generalized Wright function and the generalized k-Wright function.

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (eg, Galileo's group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R. F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie Groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

In order to solve the problem of feature extraction of underwater acoustic signals in complex ocean environment, a new method for feature extraction from ship radiated noise is presented based on empirical mode decomposition theory and permutation entropy. It analyzes the separability for permutation entropies of the intrinsic mode functions of three types of ship radiated noise signals, and discusses the permutation entropy of the intrinsic mode function with the highest energy. In this study, ship radiated noise signals measured from three types of ships are decomposed into a set of intrinsic mode functions with empirical mode decomposition method. Then, the permutation entropies of all intrinsic mode functions are calculated with appropriate parameters. The permutation entropies are obviously different in the intrinsic mode functions with the highest energy, thus, the permutation entropy of the intrinsic mode function with the highest energy is regarded as a new characteristic parameter to extract the feature of ship radiated noise. After that, the characteristic parameters, namely, the energy difference between high and low frequency, permutation entropy, and multi-scale permutation entropy, are compared with the permutation entropy of the intrinsic mode function with the highest energy. It is discovered that the four characteristic parameters are at the same level for similar ships, however, there are differences in the parameters for different types of ships. The results demonstrate that the permutation entropy of the intrinsic mode function with the highest energy is better in separability as the characteristic parameter than the other three parameters by comparing their fluctuation ranges and the average values of the four characteristic parameters. Hence, the feature of ship radiated noise can be extracted efficiently with the method.

Combining modern thermodynamics theory branches, including finite time thermodynamics or entropy generation minimization, constructal theory and entransy theory, with metallurgical process engineering, this paper provides a new exploration on generalized thermodynamic optimization theory for iron and steel production processes. The theoretical core is to thermodynamically optimize performances of elemental packages, working procedure modules, functional subsystems, and whole process of iron and steel production processes with real finite-resource and/or finite-size constraints with various irreversibilities toward saving energy, decreasing consumption, reducing emission and increasing yield, and to achieve the comprehensive coordination among the material flow, energy flow and environment of the hierarchical process systems. A series of application cases of the theory are reviewed. It can provide a new angle of view for the iron and steel production processes from thermodynamics, and can also provide some guidelines for other process industries.

In the present research, we will develop some integral inequalities of Hermite Hadamard type for differentiable η-convex function. Moreover, our results include several new and known results as special cases.

Mutations in KRT1 (keratin 1) or KRT10 (keratin 10) underlie a spectrum of diseases known as keratinopathic ichthyoses. Epidermolytic ichthyosis (EI) is caused by heterozygous missense mutations in the genes KRT1 or KRT10, mutations in the gene KRT2 (keratin 2) lead to superficial epidermolytic ichthyosis, and congenital reticular ichthyosiform erythroderma is caused by frameshift mutations in the genes KRT10 or KRT1, which lead to the phenomenon of revertant mosaicism. Epidermolytic ichthyosis is also present in a mosaic pattern known as epidermolytic (acantholytic) nevus, isolated or diffuse. In the latter case, gonadic involvement is possible, leading to a rare pedigree in which a parent with diffuse epidermolytic nevus (linear EI) gives birth to a child affected by EI. We present here an update on the phenotypic presentations of keratinopathic ichthyoses and their molecular mechanisms.

In this paper, we investigate the generalized Padovan sequences and we deal with, in detail, four special cases, namely, Padovan, Perrin, Padovan-Perrin and modified Padovan sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

In this paper, we introduce the generalized Tetranacci quaternions and present some properties of these quaternions.

Asymptotic theories for fractional cointegrations have been extensively studied in the context of time series data, with numerous empirical studies and tests developed. However, most of the previously developed testing procedures for fractional cointegration were primarily designed for time series data. This paper proposes a generalized residual-based test for fractionally cointegrated panels with fixed effects. The test development is based on bivariate panel series with the regressor assumed to be fixed across cross-sectional units. The proposed test procedure accommodates any integration order between [0,1], and it is asymptotically normal under the null hypothesis. Monte Carlo experiments demonstrate that the test exhibits better size and power compared to a similar residual-based test across varying sample sizes.

The prevailing interpretations of physics are based on deeply entrenched assumptions, rooted in classical mechanics. Logical implications include: the denial of entropy as a fundamental physical property, and the inability to explain irreversible change, random quantum measurements, or nonlocality without untestable and implausible metaphysical implications. We propose a conceptual model that is based on empirically justifiable assumptions and consistent with observations. The WYSIWYG Conceptual Model (WCM) assumes no hidden properties: “What You can See Is What You Get.” The WCM contextually defines a system’s state with respect to its actual ambient background, and it extends existing models of physical reality by defining entropy and exergy as objective contextual properties of state. The WCM establishes the irreversible dissipation of exergy and the Second law of thermodynamics as a fundamental law of physics, it recognizes physical randomness, and it provides a physical explanation for nonlocality, consistent with Special Relativity, without hidden variables, superdeterminism, or “spooky action.”

The Carnot cycle and the attendant notions of reversibility and entropy are examined. It is shown how the modern view of these concepts still correspond to the ideas Clausius laid down in the nineteenth century. As such, they reflect the outmoded idea current at the time that heat is motion. It is shown how this view of heat led Clausius to develop the entropy of a body based on the work that could be done in a reversible process rather than the work that was actually done. In consequence, Clausius built into entropy a conflict with energy conservation, which is concerned with actual changes in energy. In this paper, a macroscopic formulation of internal mechanisms of damping based on rate equations for the distribution of energy within a gas. It is shown that work processes involving a step-change in external pressure, however small, are intrinsically irreversible. However, under idealised conditions of zero damping the gas inside a piston expands and traces out a trajectory through the space of equilibrium states. Therefore, the entropy change due to heat flow from the reservoir matches the entropy change of the equilibrium states. This trajectory can traced out in reverse as the piston reverses direction, but if the external conditions are adjusted appro-priately, the gas can be made to trace out a Carnot cycle in P-V space. The cycle is dynamic as opposed to quasi-static as the piston has kinetic energy equal in difference to the work done in-ternally and externally.

The role of energy and entropy in decomposition of turbulent velocity flow-fields is to be shown in the paper. The decomposition methods based on energy concept are taken into account, namely the Proper Orthogonal Decomposition (POD) and its extension Bi-Orthogonal Decomposition (BOD). Entropy motivated view on the decomposed modes could offer new possibilities in the modes physical interpretation and in Reduced Order Modelling (ROM) strategy efficiency evaluation.

We examined a series of real-world, pictorial photographs with varying characteristics, along with their modification by noise addition and unsharp masking. As response metrics we used three different versions of the aesthetic measure originally proposed by Birkhoff. The first aesthetic measure, which has been used in other studies, and which we used in our previous work as well, showed a preference for the least complex of the images. It provided no justification for noise addition, but did reveal enhancement on unsharp masking. Optimum level of unsharp masking varied with the image, but was predictable from the individual image’s GIF compressibility. We expect this result to be useful for guiding the processing of pictorial photographic imagery. The second aesthetic measure, that of informational aesthetics based on entropy alone failed to provide useful discrimination among the images or the conditions of their modification. A third measure, derived from the concepts of entropy maximization, as well as the hypothesized preference of observers for “simpler”, i.e., more compressible, images, yielded qualitatively the same results as the more traditional version of the measure. Differences among the photographs and the conditions of their modification were more clearly defined with this metric, however.

During the last years some new variants of Permutation entropy have been introduced and applied to EEG analysis, among them a conditional variant and variants using some additional metric information or being based on entropies different from the Shannon entropy. In some situations it is not completely clear what kind of information the new measures and their algorithmic implementations provide. We discuss the new developments and illustrate them for EEG data.

{Here, we propose a new tool to estimate the complexity of a time series: the entropy of difference (ED). The method is based solely on the sign of the difference between neighbouring values in a time series. This makes it possible to describe the signal as efficiently as prior proposed parameters such as permutation entropy (PE) or modified permutation entropy (mPE), but (1) reduces the size of the sample that is necessary to estimate the parameter value, and (2) enables the use of the Kullback-Leibler divergence to estimate the "distance" between the time series data and random signals.

Experimental recordings of the collective activity of interacting spiking neurons exhibit random behavior and memory effects, thus the stochastic process modeling the spiking activity is expected to show some degree of time irreversibility. We use the thermodynamic formalism to build a framework, in the context of spike train statistics, to quantify the degree of irreversibility of any parametric maximum entropy measure under arbitrary constraints, and provide an explicit formula for the information entropy production of the inferred Markov maximum entropy process. We provide examples to illustrate our results and discuss the importance of time irreversibility for modeling the spike train statistics.

Random variations in microelectronic circuit structures represent the source of entropy for physical unclonable functions (PUFs). In this paper, we investigate delay variations that occur through the routing network and switch matrices of a field programmable gate array (FPGA). The delay variations are isolated from other components of the programmable logic, e.g., Look-up tables (LUTs), flip-flops (FFs), etc. using a feature of Xilinx FPGAs called dynamic partial reconfiguration (DPR). A set of partial designs are created that fix the placement of a time-to-digital converter (TDC) and supporting infrastructure to enable the path delays through the target interconnect and switch matrices to be extracted by subtracting out common-mode delay components. Delay variations are analyzed in the different levels of routing resources available within FPGAs, i.e., local routing and across chip routing. Data is collected from a set of Xilinx Zynq 7010 devices, and a statistical analysis of within-die variations in delay through a set of the randomly-generated and hand-crafted interconnects is presented.

This work aims to generate some types of M_(j )–neighborhoods. Relationships among these M_(j )-neighborhoods and some other types of neighborhoods are discussed. Additionally, M_(j )-lower, M_(j )-upper approximations and M_(j )-accuracy in terms of M_(j )-neighborhoods are presented. Their properties, with some examples, are investigated. Finally, We addressed the comparisons among M_(j )approach and other approaches.

The classical limit definition of a derivative is expressed in a more general form. The general form includes two arbitrary functions of the parameter for which the limit is calculated. A special case of the general form, which includes scaling and translational symmetry transformations of the limiting parameter, is also discussed. The errors in using the classical definition and the generalized form are calculated for small values of the limiting parameter. The derivatives of some known functions are proven using the new definition. For some well-known functions, a suitable selection of the generalized form may introduce simplicity in calculating the derivatives.

In this paper, it is introduced the notion of r-fuzzy β-T_i, i = 0, 1, 2 separation axioms related to a fuzzy operator β on the initial set X which is a generalization of previous fuzzy separation axioms. An r-fuzzy α-connectedness related to a fuzzy operator α on the set X is introduced which is a generalization of many types of r-fuzzy connectedness. An r-fuzzy α-compactness related to a fuzzy operator α on the set X is introduced which is a generalization of many types of fuzzy compactness.

This study introduces a method for quantitatively assessing the complexity and predictability of fish behavior in closed systems through the application of information entropy, offering a novel lens to understand how fish adapt to environmental changes. Utilizing simulations rooted in a random walk model for fish movement, we delve into entropy fluctuations under varying environmental conditions, including responses to feeding and external stimuli. Our findings underscore the utility of information entropy in capturing the intricacies of fish behavior, particularly highlighting the synchrony in collective actions and adaptations to environmental shifts. This research not only broadens our comprehension of fish behavior but also paves the way for its application in fields like aquaculture and resource management. Through our analysis, we discovered that smaller grid sizes in simulations capture detailed local fluctuations, while larger grids elucidate general trends, pinpointing a 2.5m grid as optimal for our study. Moreover, changes in swimming speeds and behavioral adaptations during feeding were quantitatively analyzed, with results illustrating significant behavior modifications. Additionally, employing a Gaussian Mixture Model helped to clarify the nuanced changes in fish behavior in response to altered light conditions, demonstrating the layered complexity of fish responses to environmental stimuli. This investigation confirms the efficacy of information entropy as a robust metric for evaluating fish shoal behavior, offering a fresh methodology for ecological and environmental studies, with promising implications for sustainable management practices.

The numerical simulation of entropy generation at mixed natural free and forced convection flow in a square lid-driven cavity filled with an eletrically conducting binary fluids saturated porous media in the presence of a magnetic field is performed in this investigation. Both the top and bottom horizontal walls of the cavity are kept at constant and different temperatures while the left and right vertical walls are adiabatic and insulated. The magnetic field is applied in normal direction to the cavity. The Darcy model, including Brinkman term relative to viscous effects and Forchheimer term due to inertial forces, is used for the momentum equations, and the SIMPLER algorithm, based on the control volume finite-element method approach is used to solve the pressure –velocity coupling. The flow pattern and the heat transfer characteristics inside the cavity are presented in the form of isotherms, streamlines and isentropic lines and average Nusselt number trend. The entropy generation rate is determined for various values of Darcy number (10^-3 ≤ Da ≤ 1) and for a range of Hartmann number (0 ≤ Ha ≤ 10²). This study is limited to Pr = 7 which corresponds to water. It is observed that entropy generation is very affected by the variations of dimensionless control parameters considered.

In this paper, we suggest an expansion of the Planck distribution function derived from the Boltzmann principle. Thereby, we consider expanding Planck's law with a new distribution function, and discuss fine structure constant. Furthermore, using ideas applied to the expansion of the Planck distribution function, we show that Von Koch's inequality can be derived without using the Riemann Hypothesis, that is, the Riemann Hypothesis is true, and that the abc conjecture is not true. Furthermore, we define a generalization of Entropy and discuss that Entropy is relevant to dynamical systems described by logistic function models, such as the growth of bacteria and populations.

Topological indices are invariant numerical fields of a graph that give facts about the structure of graphs and are found to be very helpful in predicting the physical properties of aluminophosphates. The characteristics of the aluminophosphates are similar to those of zeolites. Two examples of current applications are natural gas dehydration and humidity sensor. New frameworks are being synthesised by researchers in chemistry and materials science. There are many layers and holes in these substances. In this study, Vertex version of distance-based topological indices, the entropy of topological indices and their numerical analysis are explained.

Attempts to find and quantify the supposed low entropy of organisms and its preservation are revised. Absolute entropy of the mixed components of non-living biomass (around -1.6 x 103 J K-1 L-1) is the reference to which other entropy decreases would be ascribed to life. Compartmentation of metabolites and departure from the equilibrium of metabolic reactions account for 1 and 40-50 J K-1 L-1, respectively, decreases of entropy and, though small, are distinctive features of living tissues. DNA and proteins do not supply significant decreases of thermodynamic entropy, but their low informational entropy is relevant for life and its evolution. No other living feature contributes significantly to the low entropy associated to life. The photosynthetic conversion of radiant energy to biomass energy accounts for the most of entropy (2.8 x 105 J K-1 carbon kg-1) produced by living beings. The comparative very low entropy produced in other processes (around 4.8 x102 J K-1 L-1 day-1 in human body) must be rapidly exported outside as heat to preserve the low entropy decreases due to compartmentation and non-equilibrium metabolism. Enzymes and genes are described whose control minimize the rate of production of entropy and could explain selective pressures in biological evolution and the rapid proliferation of cancer cells.

Entropy is usually understood as the quantitative measure of “chaos” or “disorder”. However, the notions of “chaos” and “disorder” are definitely obscure. This leads to numerous misinterpretations of entropy. We propose to see the disorder as an absence of symmetry and to identify “ordering” with symmetrizing of a physical system; in other words, introducing the elements of symmetry into initially disordered physical system. We demonstrate with the binary system of elementary magnets that introducing elements of symmetry necessarily diminishes its entropy. This is true for 1D and 2D systems of elementary magnets. Imposing symmetry does not influence the Landauer Principle valid for the addressed systems. Imposing the symmetry restrictions onto the system built of particles contained within the chamber divided by the permeable partition also diminishes its entropy.

Behavior analysis is a widely used non-invasive tool in the practical production routine, as the animal acts as a biosensor capable of reflecting its degree of adaptation and discomfort to some environmental challenge. Predominantly, conventional statistics use occurrence data for behavioral evaluation and well-being estimation, disregarding the temporal sequence of events. The Generalized Sequential Pattern (GSP) algorithm is a data mining method that identifies recurrent sequences that exceed a user-specified support threshold, whose potential has not yet been investigated for broiler chickens in enriched environments. Enrichment aims to increase environmental complexity with promising effects on animal welfare, stimulating priority behaviors and potentially reducing the deleterious effects of heat stress. The objective was to validate the application of the GSP algorithm to identify temporal correlations between heat stress and the behavior of broiler chickens in enriched environments through proof of concept. Video image collection was carried out automatically for 48 continuous hours, analyzing a continuous period of seven hours, from 12:00 PM to 6:00 PM, during two consecutive days of tests for chickens housed in enriched and non-enriched environments under comfort and stress temperatures. Chickens at comfort temperature showed high motivation to perform the behaviors of Preening (P), Foraging (F), Lying down (Ld), Eating (E), and Walking (W) whose sequences <{Ld,P}>; <{Ld,F}>; <{P,F,P}>; <{Ld,P,F}> and <{E,W,F}> were the only ones observed in both treatments. All other sequential patterns (comfort and stress) were distinct, suggesting that environmental enrichment alters the behavioral pattern of broiler chickens. Heat stress drastically reduced the sequential patterns found at the 20% threshold level in the tested environments. The behavior of lying laterally "Ll" is a strong indicator of heat stress in broilers and was only frequent in the non-enriched environment, which may suggest that environmental enrichment provides the animal with better opportunities to adapt to stress-inducing challenges, such as heat.

Quantum physics is intrinsically probabilistic. The entropy of a quantum state quantifies the amount of randomness (or information loss) of such state. The degrees of freedom of a quantum state are position and spin, and upon their specification Born's rule in phase space defines randomness. We focus on the spin degree of freedom and elucidate the spin entropy. Then, we present some of its properties and show how entanglement increases spin entropy. Some speculative predictions on the decay of the $Z^0,W^{+,-}$ gauge bosons conclude the paper.

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that excludes the existence of non-Shannon inequalities. It is demonstrated that planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.

A new formalism is introduced that makes it possible to elucidate the physical and geometric content of quantum spacetime. It is based in the Minimum Group Representation Principle (MGRP). Within this framework new results for entanglement and geometrical/topological phases are found and implemented in cosmological and black hole space-times. Our main results here are: {\bf(i)} We find the Berry phases for inflation and for the cosmological perturbations and express them in terms of the observables, as the spectral scalar and tensor indices, $n_S$ an $n_T$, and the tensor to scalar ratio $r$. The Berry phase for de Sitter inflation is imaginary with the sign describing the exponential acceleration. {\bf(ii)} The pure entangled states in the minimum group (metaplectic) $Mp(n)$ representation for quantum de Sitter space-time and black holes are found. {\bf(iii)} For entanglement, the relation between the Schmidt type representation and the physical states of the $Mp(n)$ group is found: This is a {\it new non-diagonal} coherent state representation complementary to the known Sudarshan diagonal one. {\bf(iv)} Mean value generators of $Mp(2)$ are related to the adiabatic invariant and topological charge of the spacetime (matrix element of the transition $-\infty < t < \infty$). {\bf(v)} The basic {\it even} and {\it odd} $n$-sectors of the Hilbert space are intrinsic to the quantum spacetime and its discrete levels (in particular continuum for $n \rightarrow \infty$), they do not require any extrinsic generation process as the standard Schrodinger cat states, and are {\it entangled}. {\bf(vi)} The gravity or cosmological domains in one side and another of the Planck scale are {\it entangled}. Examples: The quantum primordial trans-Planckian de Sitter vacuum and the classical late de Sitter vacuum today; the central quantum gravity reqion and the external classical gravity region of black holes. The classical and quantum dual gravity regions of the space-time are entangled. {\bf(vii)} The general classical-quantum gravity duality is associated to the Metaplectic $Mp(n)$ group symmetry which provides the complete full covering of the phase space and of the quantum space-time mapped from it.

Background: Periodontitis is a disease that leads to serious functional and aesthetic dysfunctions resulting in impairment of life quality; it exists in different forms and its etiology is related with multiple component causes. Angiogenesis, together with inflammatory infiltrate, are two key processes in the evolution of inflammatory processes involving gingival tissues. The aim of this study was to evaluate their alterations in relation to some important periodontal diseases-associated factors (i.e. smoke, sex, age, plaque, pus, probing pocket depth) in patients with diagnosis of Generalized Aggressive Periodontitis (GAP). Methods: Eighteen subjects with GAP were enrolled for this study. Percentage of inflammatory cells and vascular area were measured and evaluated in relation with each periodontal diseases-associated factor. Results: The results showed a significant decrease of the percentage of vascular area in association with almost all the periodontal diseases-related factors (smoke, age, PPD, plaque) and not significant differences in the percentage of inflammatory cells. Conclusions: Data suggest that, even if these factors are often commonly related to worsening of periodontal status, they did not affect inflammatory and vascular pattern, as we can assume. For the future, additional studies, aimed to evaluated each singular factor in a wider sample size, should be planned.

We introduce a new generalized inverse (i.e., generalized EP element) which is a natural generalization of EP and *-DMP elements in a Banach *-algebra. We present polar-like characterizations of generalized EP elements. The necessary and sufficient conditions under which the sum of two generalized EP elements is a generalized EP element are investigated. Finally, the generalized core-EP orders for generalized EP elements are characterized.

One open question remains regarding the theory of the generalized variational principle, that is, why the stress-strain relation still be derived from the generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional.

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

The main objective of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. We establish some results related to the newly defined fractional operator such as Mellin transform and relations to k-hypergeometric and k-Appell's functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and Wright hypergeometric functions.

The model of an oscillating universe proposed by A. Friedmann in 1922 dominated cosmology for a long time. R. Tolman put forward a thermodynamic argument against the cyclic cosmological model, believing that entropy should accumulate from cycle to cycle. J. Bekenstein and S. Hawking discovered a tremendous amount of entropy associated with black holes. This discovery led to the conclusion that the majority of the Universe's entropy is contained in black holes. N. Poplawski considers a Universe that oscillates within a black hole. This paper analyzes the relationship between the entropy of black holes and the oscillating Universe inside a large black hole. It is demonstrated that the entropy of a black hole can only be defined for an external observer. For an observer inside the black hole, its entropy is zero. This opens up the possibility of constructing a cyclic cosmology in which entropy and the size of the visible part of the Universe change strictly periodically. This does not violate the second law of thermodynamics because the continuous accumulation of entropy from cycle to cycle pertains to the surface of the big black hole, which is invisible to the internal observer.

Proteins in biological systems function at the interface of single molecule and bulk chemistry and thus provide novel insights into the basic physical chemistry of scaling and emergent phenomena. For example, a binary mechanical model based on the chemistry of muscle contraction unifies molecular mechanics and thermodynamics and provides an explicit solution to the Gibbs paradox. Using the same model system, here I show that chemical activities of molecular states have no effect on chemical kinetics or energetics. Specifically, while the concentration or number of molecules in a molecular state is widely thought to contribute to reaction free energies, here I show that it is not the physical presence of molecules that pushes a reaction forward but the number of microstates, , accessible in a chemical state that pulls a reaction toward equilibrium with an entropic force down an entropic funnel. With the derivation of an entropic contribution to chemical kinetics, I develop a novel chemical kinetic formulation that fully describes the chemical thermodynamics of both equilibrium and non-equilibrium reactions in terms of an a priori system reaction energy landscape.

Living cells are complex systems that may be characterized by fluids crowded by hundreds of different elements in particular by a high density of polymers; they are an excellent and challenging laboratory to study exotic emerging physical phenomena where entropic forces emerge from organization processes of many-body interactions. The competition between microscopic and entropic forces may generate complex behaviors like phase transitions that living cells may use to accomplish their functions. In the era of the big data, when biological information abounds but general principles and precise understanding of the microscopic interactions scarce, the entropy methods may offer significant information. In this work we develop a model where the thermodynamic equilibrium results from the competition between an effective electrostatic shortrange interaction and the entropic forces emerging in a fluid crowded by different size polymers. The target audience for this article are interdisciplinary researchers in complex systems, particularly in thermodynamics and biophysics modeling.

It is shown that the 5D geodetic equations and 5D Ricci identities give us a way to create a new viewpoint on some problems of Modern Physics, Astrophysics, and Cosmology. Specifically, the application of the 5D geodetic equations in (4+1) and (3+1+1) splintered forms obtained with the help of the monad and dyad methods made it possible to introduce a new, effective generalized concept of the rest mass of the elementary particle. The latter leads one to novel connections between the General Relativity and quantum field theories, as well as all of that, including the (4+1) splitting of the 5D Ricci identities, brings about a better understanding of the magnetic monopole problem and the vital difference in the origins of the Maxwell equations and gives rise to surprising connections between them. The obtained results also provide new insight into the mechanism of the 4D Universe’s expansion and its following acceleration.

We study the generalized χ and η cross helicities for non-ideal non-barotropic magnetohydrodynamics (MHD). χ and η, the additional label translation symmetry group, have been used to generalize the cross helicity in ideal flows. Both new helicities are additional topological invariants of ideal MHD. To study there behaviour in non-ideal MHD, we derive the time derivative of both helicities using the non-ideal MHD equations taking viscosity, finite resistivity and heat conduction into account. Physical variables are divided into ideal and non-ideal quantities separately during the mathematical analysis for simplification. The analytical results indicate that χ and η cross helicities are not strict constant of motion in the non-ideal MHD and show the rate of dissipation which is compared to the dissipation of other topological constants of motion.

In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

Entropy generation, formulated by combining the first and second laws of thermodynamics with an appropriate thermodynamic potential, emerges as the difference between a phenomenological entropy function and a reversible entropy function. The phenomenological entropy function is evaluated over an irreversible path through thermodynamic state space via real time measurements of thermodynamic states. The reversible entropy function is calculated along an ideal reversible path through the same state space. Entropy generation models for various classes of systems—thermal, externally loaded, internally reactive, open and closed––are developed via selection of suitable thermodynamic potentials. Here we simplify thermodynamic principles to specify convenient and consistently accurate system governing equations and characterization models. The formulations introduce a new and universal Phenomenological Entropy Generation (PEG) theorem. The systems and methods presented––and demonstrated on grease degradation, battery charging and discharging, metal fatigue and pump flow––can be used for design, analysis, and support of diagnostic monitoring and optimization.

Hund's rule of maximum multiplicity is a powerful empirical tool for determining the electron population of electronic shells. It was recently discovered that it is also the rule of minimum Shannon entropy. This study extends these findings to the Aufbau rule and discusses them from the perspective of emergent dimensionality, providing a monotonically increasing entropy function. We observed that some elements that violate the Aufbau rule have the same entropies in the actual and Aufbau electron configurations. On the other hand, lower entropies of all actual elements' configurations are associated with higher or equal numbers of unpaired electrons in these configurations compared to the Aufbau configurations. The only exception to this rule is palladium.

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links are considered. Shannon-entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. Introduced Shannon entropy is insensitive to the exact shape of the graph, but it is sensitive to the distribution of monochromatic polygons in a given graph. The introduced Shannon Entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph, Sβ is, in turn, an average uncertainty to find the red β-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon Entropy is suggested. Various measures quantifying the Shannon Entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon Entropies are discussed.

Bonchev and Trinajstic defined two distance based entropy measures to measure the molecular branching of molecular graphs in 1977 [Information theory, distance matrix, and molecular branching, J. Chem. Phys., 38 (1977), 4517–4533]. In this paper we use these entropy measures which are based on distance matrices of graphs. The first one is based on distribution of distances in distance matrix and the second one is based on distribution of distances in upper triangular submatrix. We obtain the two entropy measures of paths, stars, complete graphs, cycles and complete bipartite graphs. Finally we obtain the minimal trees with respect to these entropy measures with fixed diameter.

In quantum many-body systems, a Hamiltonian is called an ``extensive entropy generator'' if starting from a random product state the entanglement entropy obeys a volume law at long times with overwhelming probability. We prove that (i) any Hamiltonian whose spectrum has non-degenerate gaps is an extensive entropy generator; (ii) in the space of (geometrically) local Hamiltonians, the non-degenerate gap condition is satisfied almost everywhere. Specializing to many-body localized systems, these results imply the observation stated in the title of Bardarson et al. [PRL 109, 017202 (2012)].

Double-spiral galaxies are common in the Universe. It is known that the logarithmic double spiral is a Maximum Entropy geometry and represents spiral galaxies well. It is also known that the virial mass of such a galaxy can be approximately determined from the entropy of its central supermassive black hole. But over time the black hole must accrete mass, and therefore the overall galactic entropy must increase. From the associated entropic Euler-Lagrange equation (forming the basis of the Principle of Least Exertion, and also enabling the application of Nöther’s theorem) we show that the galactic entropy production is a conserved quantity, and we derive an appropriate expression for the relativistic entropic Hamiltonian of an idealised spiral galaxy. We generalise Onsager’s celebrated expression for entropy production and demonstrate that galactic entropy production has two parts, one many orders of magnitude larger than the other, and where the smaller is comparable to the Hawking radiation of the central supermassive black hole. We conclude that galaxies cannot be isolated, since even idealised spiral galaxies have non-zero entropy production.

This work (in two parts) will review the recently developed predictive modeling methodology called “4th-BERRU-PM” and its applicability to energy systems as exemplified by an illustrative application to the Polyethylene-Reflected Plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The acronym 4th-BERRU-PM designates the “Fourth-Order Best-Estimate Results with Reduced Uncertainties Predictive Modeling” methodology, which yields best-estimate results with reduced uncertainties for the first fourth-order moments (mean values, covariance, skewness, and kurtosis) of the optimally predicted posterior distribution of model results and calibrated model parameters. The 4th-BERRU-PM uses the Maximum Entropy (MaxEnt) principle to incorporate fourth-order experimental and computational information, including fourth (and higher) order sensitivities of computed model responses to model parameters, thus incorporating, as particular cases, the results previously predicted by the second-order predictive modeling methodology 2nd-BERRU-PM, and vastly generalizing the results produced by extant data assimilation and data adjustment procedures. The 4th-BERRU-PM methodology encompasses the scopes of high-order sensitivity analysis (SA), uncertainty quantification (UQ), data assimilation (DA) and model calibration (MC). The application of the 4th-BERRU-PM methodology to energy systems is illustrated by means of the above-mentioned OECD/NEA reactor physics benchmark, which is modeled using the neutron transport Boltzmann equation involving 21976 imprecisely known parameters, the solution of which is representative of “large-scale computations.” The model result (“response”) of interest is the leakage of neutrons through the outer surface of this spherical benchmark, which can be computed numerically and measured experimentally. Part 1 of this work illustrates the impact of high-order sensitivities, in conjunction with parameter standard deviations of various magnitudes, on the determination of the expected value and variance of the computed response in terms of the first four moments of the distribution of the uncertain model parameters. Part 2 of this work will illustrate the capabilities of the 4th-BERRU-PM methodology for combining computational and experimental information, up to and including forth-order sensitivities and distributional moments, for producing best-estimate values for the predicted responses and model parameters while reducing their accompanying uncertainties.

The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.

Let $n \geq 2$ be a fixed integer and $\mathcal{A}$ be a $C^*$-algebra. A permuting $ n $-linear map $ \mathcal{G} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ is known to be symmetric generalized $n$-derivation if there exists a symmetric $n$-derivation $ \mathfrak{D} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ such that $ \mathcal{G} \left(x_{1}, x_{2}, \ldots, x_{i} x_{i}^{\prime}, \ldots, x_{n}\right)= \mathcal{G} \left(x_{1}, x_{2}, \ldots, x_{i}, \ldots, x_{n}\right) x_{i}^{\prime}+x_{i} \mathfrak{D} (x_{1}, x_{2},\ldots, x_{i}^{\prime}, \ldots, x_{n})$ holds for all $x_{i}, x_{i}^{\prime} \in \mathcal{A} $. In this paper, we investigate the structure of $C^*$-algebras involving generalized linear $n$-derivations. Moreover, we describe the forms of traces of linear $n$-derivations satisfying certain functional identity.

In this paper, we introduce the notion of weighted generalized group inverse in a Banach algebra with proper involution. This is a natural generalization of weighted weak group inverse for a complex matrix and Hilbert space operator. We present several characterizations and representations of this generalized inverse. In addition, a new partial order on elements in a Banach *-algebra is investigated by using the weighted generalized group inverse and some known results are thus generalized.

Let {ξ1, ξ2. . . .} be a sequence of independent possibly differently distributed random variables defined on a probability space (Ω, F, P) with distribution functions {F_ξ1., F_ξ2 , . . .}. Let η be a counting random variable independent of sequence {ξ1, ξ2. . . .}. In this paper, we find conditions under which distribution function of randomly stopped sum Sη = ξ1 + ξ2 + . . . + ξη belongs to the class of generalized subexponential distributions.

In theory and applications, it is often inevitable to work with projectors onto convex sets, where a linear transform is involved. In this article, a novel projector is presented, which generalizes previous results in that it admits a broader family of linear transforms, but on the other hand it is limited to box-type convex sets in the transformed domain. The new projector has an explicit formula and it can be interpreted within the framework of proximal optimization. The benefit of the new projector is demonstrated on an example from signal processing, where it was possible to speed up the convergence of a signal declipping algorithm by a factor of more than two.

With the deepening of the diversification and openness of financial system, financial vulnerability, as an endogenous attribute of financial system, becomes an important measurement of financial security. Based on network analysis, we introduce network curvature indicator improved by Copula entropy as an innovative metric of financial vulnerability. Compared with previous network curvature analysis method, CE-based curvature proposed in this paper can measure market vulnerability and systematic risk with significant advantages.

The thermocontextual interpretation (TCI) establishes a system’s exergy, entropic energy, and thermal entropy as thermocontextual properties of state, defined with respect to its positive temperature surroundings. This work extends previous applications of the TCI to irreversible and statistical transitions. The TCI defines statistical entropy as a transactional process of derandomization and transition to a negative-entropy state. Statistical measurements of a confined quantum particle’s position are detailed in terms of reversible processes of instantiation and actualization. The TCI then formalizes the MaxEnt as a fundamental physical principle. We apply MaxEnt and statistical entropy measurements to the double-slit experiment. Particles passing through parallel slits record a wave interference pattern, but a “which-slit” detector eliminates wave interference. Richard Feynman called the double-slit experiment the only mystery, at the heart of quantum mechanics. The TCI offers a simple explanation. The which-slit detector breaks the system’s symmetry, enabling particles to pass through one slit or the other, and MaxEnt selects the asymmetrical transition, which has no wave interference and a higher statistical entropy.

Following a recent proof of Shannon's entropy power inequality (EPI), a comprehensive framework for deriving various EPIs for the Rényi entropy is presented that uses transport arguments from normal densities and a change of variable by rotation. Simple arguments are given to recover the previously known Rényi EPIs and derive new ones, by unifying a multiplicative form with constant c and a modification with exponent α of previous works. In particular, for log-concave densities, we obtain a simple transportation proof of a sharp varentropy bound.

Natural time is a new time domain introduced in 2001. The analysis of time series associated with a complex system in natural time may provide useful information and may reveal properties that are usually hidden when studying the system in conventional time. In this new time domain, an entropy has been defined and complexity measures based on this entropy as well as its value under time-reversal have been introduced and found applications in various complex systems. Here, we review these applications in the electric signals that precede rupture, e.g., earthquakes, in the analysis of electrocardiograms, as well as in global atmospheric phenomena like the El Nino/La Nina Southern Oscillation.

High entropy alloys have excellent wear resistance, so they have great application prospects in the fields of wear resistance and surface protection. In this study, the wear resistance of FeNiCrCoCu high entropy alloy coating was systematically analyzed by molecular dynamics method. FeNiCrCoCu high entropy alloy was used as a coating material to adhere to the surface of Cu matrix. The friction and nano indentation simulation of this coating material were carried out by controlling the ambient temperature. The influence of temperature on its friction properties was analyzed from five aspects: lattice structure, dislocation evolution, friction coefficient, hardness and elastic modulus. The results show that with the increase of temperature, the disorder of the lattice structure increases, which leads to the increase of the tangential force and the friction coefficient in the friction process. At 300K and 600K, the ordered lattice structure of the high entropy alloy coating material is basically the same, and thus their hardness is basically the same. However, the dislocation density at 600K is significantly reduced compared with that at 300K, resulting in the increase of the elastic modulus of the material from 173GPa to 219GPa. At the temperatures of 900K and 1200k, lattice disorder take place rapidly, and dislocation density also decreases significantly, resulting in a significant decrease in the hardness and elastic modulus of the material. When the temperature reaches 900K, the wear resistance of the FeNiCrCoCu high entropy alloy coating decreases sharply. This work is of great value to the analysis of wear resistance of high entropy alloy at high temperature.

The response of plasmonic metal particles to an electromagnetic wave undergoes significant features at the nanoscale level. Different properties of the internal composition of a metal, such as its ionic background and the free electron gas, begin to manifest more prominently. As the dimensions of the nanostructures decrease, the classical local theory gradually becomes inadequate. Therefore, Maxwellʹs equations need to be supplemented with a relationship determining the dynamics of current density which is the essence of nonlocal plasmonic models. In this field of physics, the standard (linearized) hydrodynamic model (HDM) has been widely adopted with great success, serving as the basis for a variety of simulation methods. However, ongoing efforts are also being made to expand and refine it. Recently, the GNOR (general nonlocal optical response) modification of the HDM has been used, with the intention of incorporating the influence of electron gas diffusion. Clearly, from the classical description of fluid dynamics, a close relationship between viscosive damping and diffusion arises. This offers a relevant motivation for introducing the GNOR modification in an alternative manner. The standard HDM and its existing GNOR modification also do not include the influence of interband electron transitions in the conduction band and other phenomena that are part of many refining modifications of the Drude‐Lorentz and other models of metal permittivity. In this article, we present a modified version of GNOR‐HDM that incorporates the viscosive damping of the electron gas and a generalized Drude‐Lorentz term. In the selected simulations, we also introduce Landau damping which corrects the magnitude of the standard damping constant of the electron gas based on the size of the nanoparticle. We have chosen a spherical particle as a suitable object for testing and comparing HD models and their modifications because it allows finding precise analytical solutions for the interaction and, simultaneously, it is a relatively easily fabricable nanostructure in practice. Our contribution also includes our own analytical method for solving the HDM interaction of a plane wave with a spherical particle. This method forms the core of calculations of the characteristic quantities, such as the extinction cross‐sections and the corresponding components of electric fields and current densities.

Two extensions of the generalized double Lomax distribution are introduced - the spliced-scale generalized double Lomax distribution and the exponentiated generalized double Lomax distribution. Their properties are studied. Usefulness of the new distributions is shown by fitting them to stock returns datasets.

In this paper we investigate the geometric structure and control of exponential families depending on additional parameters, called external parameters. These generalized exponential families emerges naturally when one applies the maximum entropy formalism to derive the equilibrium statistical mechanics framework. We study the associated statistical model, compute the Fisher metric and introduce a natural fibration of the parameter space over the external parameter space. The Fisher Riemannian metric allows to endow this fibration with an Ehresmann connection and to study the geometry and control of these statistical models. As an example, we show that horizontal lift of paths in the external parameter space corresponds to an isentropic evolution of the system. We apply the theory to the example of a ideal gas in a rotating rigid container. Most of the results are expressed in local coordinates; in the appendices we hint at possible global extensions of the theory.

The historical idea of entropy as a property of a body has been reviewed and shown to arise from Clausius’s view of heat as motion. This view of heat, being intermediate between the now defunct idea of heat as substance and the modern view that heat represents an exchange of energy, implied that a body contains a definite quantity of heat. Heat, and therefore entropy, was thus considered a property of a body. These ideas led Clausius to develop his famous inequality and the idea that entropy always increases in irreversible, non-cyclic processes. In this paper, the notion of the entropy as a property of a body is examined in detail. A physical meaning is attached to the thermodynamic entropy by showing that a change in the function Q/T can be understood as representing a change in the number of ways that energy can be distributed among the degrees of freedom active in the system at any given temperature. The idea is illustrated with reference to solids and simple liquids. It is shown that the total thermodynamic entropy is, in general, less than the Boltzmann entropy, except for the case of a monatomic classical ideal gas in which the number of degrees of freedom is independent of temperature. Finally, entropy as a state function is discussed. It is argued that this is entirely mathematical in nature and that the entropy of a state represented in p-V space is not equivalent to a physical property of a physical system in the same thermodynamic state.

The Universe is treated as an unbound thermodynamic system in which two forces of Nature compete: Attractive gravity loss and repulsive entropic gain. Entropic gain, although not presently considered a force of Nature, is otherwise well known and can be expressed with the gas laws. The Universe at the time of last scattering was all gas. Its instant gain is expressed as gas pressure. When the Gibbs equation is properly applied to this unbound gas, the resulting atomic Hubble parameter is found to be exclusively dependent on baryon density, giving a constant and perpetual two-to-one gain/loss ratio. Instant loss to gravity cannot offset gas pressure in this “thermal equation”. By contrast, the Friedmann equation, a ΛCDM foundation, treats baryon mass as accreted, having little if any entropic gain. These two equations are actually equivalent. Expanding accreted matter at its comoving critical density is herein shown to behave exactly like a freely expanding gas. However, neither equation includes all of today’s Universal kinetic energy, and cannot by themselves fully account for observed stellar movement. The present paper proposes the plausible existence of highly energetic or “suprathermal” free electrons which comprise about half of the intergalactic medium’s kinetic total. It is this suprathermal energy which Λ expresses. The fluid equation, another ΛCDM foundation, eliminates entropic gain from the Gibbs equation. Additionally, rest mass’s Einstein energy E=Mc² is treated as a thermal variable. These two assumptions, isoentropy and energy conflation, led astronomers to populate Λ as they followed the prior art.