Search Results (126)

The review provides a pedagogical but comprehensive introduction to the foundations of a recently proposed statistical mechanics ($\mu$NEQT) of a stable nonequilibrium thermodynamic body, which may be either isolated or interacting. It is an extension of the well-established equilibrium statistical mechanics by considering microstates $\left\{{\mathfrak{m}}_k\right\}$ in an extended state space in which macrostates (obtained by ensemble averaging $\widehat{A}$) are uniquely specified so they share many properties of stable equilibrium macrostates. The extension requires an appropriate extended state space, three distinct infinitessimals $d_{\alpha }=(d,d_{\mathrm{e}},d_{\mathrm{i}})$ operating on various quantities q during a process, and the concept of reduction. The mechanical process quantities (no stochasticity) like macrowork are given by $\widehat{A}{d}_{\alpha }\mathsf{q}$, but the stochastic quantities ${\widehat{C}}_{\alpha }\mathsf{q}$ like macroheat emerge from the commutator ${\widehat{C}}_{\alpha }$ of $d_{\alpha }$ and $\widehat{A}$. Under the very common assumptions of quasi-additivity and quasi-independence, exchange microquantities $d_{\mathrm{e}}$q${}_k$ such as exchange microwork and microheat become nonfluctuating over $\left\{{\mathfrak{m}}_k\right\}$ as will be explained, a fact that does not seem to have been appreciated so far in diverse branches of modern statistical thermodynamics (fluctuation theorems, quantum thermodynamics, stochastic thermodynamics, etc.) that all use exchange quantities. In contrast, dq${}_k$ and $d_{\mathrm{i}}$q${}_k$ are always fluctuating. There is no analog of the first law for a microstate as the latter is a purely mechanical construct. The second law emerges as a consequence of the stability of the system, and cannot be violated unless stability is abandoned. There is also an important thermodynamic identity $d_{\mathrm{i}}Q\equiv d_{\mathrm{i}}W$$\ge 0$ with important physical implications as it generalizes the well-known result of Count Rumford and the Gouy-Stodola theorem of classical thermodynamics. The $\mu$NEQT has far-reaching consequences with new results, and presents a new understanding of thermodynamics even of an isolated system at the microstate level, which has been an unsolved problem. We end the review by applying it to three different problems of fundamental interest. Full article

Living beings are composite thermodynamic systems in non-equilibrium conditions. Within this context, there are a number of thermodynamic potential differences (forces) between them and the surroundings, as well as internally. These forces lead to flows, which, ultimately, are essential to life itself, but, at the same time, are associated with entropy generation, i.e., a loss of useful work. The maintenance of homeostatic conditions, the tenet of physiology, demands the regulation of these flows by control of variables. However, due to the very nature of these systems, the regulation of flows and control of variables become entangled in closed loops. Here, we show how to combine entropy generation with respect to a process, and control of parameters (in such a process) in order to create a criterium of optimal ways to regulate changes in flows, the coefficient of flow-entropy (C^Jσ). We demonstrate the restricted possibility to obtain an increase in flow along with a decrease in entropy generation, and the more general situation of increases in flow along with increases in entropy generation of the process. In this scenario, the C^Jσ aims to identify the best way to combine the gain in flow and the associated loss of useful work. As an example, we analyze the impact of vaccination effort in the spreading of a contagious disease in a population, showing that the higher the vaccination effort the higher the control over the spreading and the lower the loss of useful work by the society. Full article

Any two points are close together in a $𝜃$-contraction by a factor of $𝜃$. The function $\Delta$ is implied to be a contraction under this condition, but with a tighter bound on the contraction factor. In this paper, we introduce the notions of orthogonal $𝜃$-contraction and orthogonal $\alpha -𝜃$-contraction and prove several fixed point results by utilizing these contraction mappings in the context of orthogonal metric spaces. Further, we provide several non-trivial examples to show the validity of our results. Full article

This article shows the life and work of Gerty Cori, a woman born in Czechoslovakia and who later became a naturalized American, who spent her whole life researching, together with her husband, in the laboratory to find the cause of some diseases, particularly those of a metabolic type, and to be able to find substances that alleviate their effects. The result of this joint work was the obtaining by both, together with the physiologist Bernardo Houssay, of the Nobel Prize in Medicine or Physiology in 1947. The objective of this article is to complete the scarce existing biographies about this woman with new data that highlight the most outstanding events of her life, quite a few of which are still largely ignored. A relatively complete information on the presence of female chemists in the awarded Nobel Prizes is also shown. Full article

A review of results on Hermite–Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions, $(p,s)$-convex functions, modified $(p,s)$-convex functions, $(p,h)$-convex functions, $tgs$-convex functions, $\eta$-quasi-convex functions, $\varphi$-convex functions, $(\alpha ,m)$-convex functions, $\varphi$-quasi-convex functions, and coordinated convex functions. Quantum H-H type inequalities via preinvex functions and Green functions are also presented. Finally, H-H type inequalities for $(p,q)$-calculus, h-calculus, and $(q-h)$-calculus are also included. Full article

Weber’s electrodynamics presents an alternative theory to the widely accepted Maxwell–Lorentz electromagnetism. It is founded on the concept of direct action between particles, and has recently gained some momentum through theoretical and experimental advancements. However, a major criticism remains: the lack of a comprehensive electromagnetic wave equation for free space. Our motivation in this research article is to address this criticism, in some measure, by deriving an electric wave equation from Weber’s electrodynamics based on the axiom of vacuum polarization. Although this assumption has limited experimental evidence and its validity remains a topic of debate among researchers, it has been shown to be useful in the calculation of various quantum mechanical phenomena. Based on this concept, and beginning with Weber’s force, we derive an expression which resembles the familiar electric field wave equation derived from Maxwell’s equations. Full article

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order $\alpha \in (1,2)$. The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results. Full article

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented. Full article

One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists and is unique on its interval of existence. The vast majority of existence and uniqueness results available in the literature, including Picard’s method, for ordinary and/or fractional dynamic equations will result in only local existence results. In this work, we generalize Picard’s method to obtain the existence and uniqueness of the solution of the nonlinear multi-order Caputo derivative system with initial conditions, on the interval where the solution is bounded. The challenge presented to establish our main result is in developing a generalized form of the Mittag–Leffler function that will cooperate with all the different fractional derivative orders involved in the multi-order nonlinear Caputo fractional differential system. In our work, we have developed the generalized Mittag–Leffler function that suffices to establish the generalized Picard’s method for the nonlinear multi-order system. As a result, we have obtained the existence and uniqueness of the nonlinear multi-order Caputo derivative system with initial conditions in the large. In short, the solution exists and is unique on the interval where the norm of the solution is bounded. The generalized Picard’s method we have developed is both a theoretical and a computational method of computing the unique solution on the interval of its existence. Full article

In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach’s contraction principle, the Leray–Schauder alternative and the well-known fixed-point theorem of Krasnosel’skiĭ. Finally, the main results are illustrated by constructing numerical examples. Full article

Building on previous work that considered gravity to emerge from the collective behaviour of discrete, pre-geometric spacetime constituents, this work identifies these constituents with gravitons and rewrites their effective gravity-inducing interaction in terms of local variables for Schwarzschild–de Sitter scenarios. This formulation enables graviton-level simulations of entire emergent gravitational systems. A first simulation scenario confirms that the effective graviton interaction induces the emergence of spacetime curvature upon the insertion of a graviton condensate into a flat spacetime background. A second simulation scenario demonstrates that free fall can be considered to be fine-tuned towards a geodesic trajectory, for which the graviton flux, as experienced by a test mass, disappears. Full article

The dynamic equilibrium between death and regeneration is well established at the cell level. Conversely, no study has investigated the homeostatic control of shape at the whole organism level through processes involving apoptosis. To address this fundamental biological question, we took advantage of the morphological and functional properties of the carnivorous sponge Lycopodina hypogea. During its feeding cycle, this sponge undergoes spectacular shape changes. Starved animals display many elongated filaments to capture prey. After capture, prey are digested in the absence of any centralized digestive structure. Strikingly, the elongated filaments actively regress and reform to maintain a constant, homeostatically controlled number and size of filaments in resting sponges. This unusual mode of nutrition provides a unique opportunity to better understand the processes involved in cell renewal and regeneration in adult tissues. Throughout these processes, cell proliferation and apoptosis are interconnected key actors. Therefore, L. hypogea is an ideal organism to study how molecular and cellular processes are mechanistically coupled to ensure global shape homeostasis. Full article

The discussion of what matter and mass are has been going on for more than 2500 years. Much has been discovered about mass in various areas, such as relativity theory and modern quantum mechanics. Still, quantum mechanics has not been unified with gravity. This indicates that there is perhaps something essential not understood about mass in relation to gravity. In relation to gravity, several new mass definitions have been suggested in recent years. We will provide here an overview of a series of potential mass definitions and how some of them appear likely preferable for a potential improved understanding of gravity at a quantum level. This also has implications for practical things such as getting gravity predictions with minimal uncertainty. Full article

This article establishes a comparison principle for the nabla fractional difference operator ${\nabla }_{\rho \left(a\right)}^{\nu }$, $1<\nu <2$. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive the corresponding Green’s function. I prove that this Green’s function satisfies a positivity property. Then, I deduce a relatively general comparison result for the considered boundary value problem. Full article