Foundations, Volume 3, Issue 3 (September 2023) – 11 articles

Diagnostic classification models (DCMs) are statistical models with discrete latent variables (so-called skills) to analyze multiple binary variables (i.e., items). The one-parameter logistic diagnostic classification model (1PLDCM) is a DCM with one skill and shares desirable measurement properties with the Rasch model. This article shows that the 1PLDCM is indeed a latent class Rasch model. Furthermore, the relationship of the 1PLDCM to extensions of the DCM to mixed, partial, and probabilistic memberships is treated. It is argued that the partial and probabilistic membership models are also equivalent to the Rasch model. The fit of the different models was empirically investigated using six datasets. It turned out for these datasets that the 1PLDCM always had a worse fit than the Rasch model and mixed and partial membership extensions of the DCM. Full article

The nonlocality (superdiffusion) of turbulence is expressed in the empiric Richardson t³ scaling law for the mean square of the mutual separation of a pair of particles in a fluid or gaseous medium. The development of the theory of nonlocality of various processes in physics and other sciences based on the concept of Lévy flights resulted in Shlesinger and colleagues’ about the possibility of describing the nonlocality of turbulence using a linear integro-differential equation with a slowly falling kernel. The approach developed by us made it possible to establish the closeness of the superdiffusion parameter of plasma density fluctuations moving across a strong magnetic field in a tokamak to the Richardson law. In this paper, we show the possibility of a universal description of the characteristics of nonlocality of transfer in a stochastic medium (including turbulence of gases and fluids) using the Biberman–Holstein approach to examine the transfer of excitation of a medium by photons, generalized in order to take into account the finiteness of the velocity of excitation carriers. This approach enables us to propose a scaling that generalizes Richardson’s t³ scaling law to the combined regime of Lévy flights and Lévy walks in fluids and gases. Full article

A method without memory as well as a method with memory are developed free of derivatives for solving equations in Banach spaces. The convergence order of these methods is established in the scalar case using Taylor expansions and hypotheses on higher-order derivatives which do not appear in these methods. But this way, their applicability is limited. That is why, in this paper, their local and semi-local convergence analyses (which have not been given previously) are provided using only the divided differences of order one, which actually appears in these methods. Moreover, we provide computable error distances and uniqueness of the solution results, which have not been given before. Since our technique is very general, it can be used to extend the applicability of other methods using linear operators with inverses along the same lines. Numerical experiments are also provided in this article to illustrate the theoretical results. Full article

Derivative-free iterative methods are useful to approximate the numerical solutions when the given function lacks explicit derivative information or when the derivatives are too expensive to compute. Exploring the convergence properties of such methods is crucial in their development. The convergence behavior of such approaches and determining their practical applicability require conducting local as well as semi-local convergence analysis. In this study, we explore the convergence properties of a sixth-order derivative-free method. Previous local convergence studies assumed the existence of derivatives of high order even when the method itself was not utilizing any derivatives. These assumptions imposed limitations on its applicability. In this paper, we extend the local analysis by providing estimates for the error bounds of the method. Consequently, its applicability expands across a broader range of problems. Moreover, the more important and challenging semi-local convergence not investigated in earlier studies is also developed. Additionally, we survey recent advancements in this field. The outcomes presented in this paper can be proved valuable to practitioners and researchers engaged in the development and analysis of derivative-free numerical algorithms. Numerical tests illuminate and validate further the theoretical results. Full article

Numerous applications from diverse disciplines are formulated as an equation or system of equations in abstract spaces such as Euclidean multidimensional, Hilbert, or Banach, to mention a few. Researchers worldwide are developing methodologies to handle the solutions of such equations. A plethora of these equations are not differentiable. These methodologies can also be applied to solve differentiable equations. A particular method is utilized as a sample via which the methodology is described. The same methodology can be used on other methods utilizing inverses of linear operators. The problem with existing approaches on the local convergence of iterative methods is the usage of Taylor expansion series. This way, the convergence is shown but by assuming the existence of high-order derivatives which do not appear on the iterative methods. Moreover, bounds on the error distances that can be computed are not available in advance. Furthermore, the isolation of a solution of the equation is not discussed either. These concerns reduce the applicability of iterative methods and constitute the motivation for developing this article. The novelty of this article is that it positively addresses all these concerns under weaker convergence conditions. Finally, the more important and harder to study semi-local analysis of convergence is presented using majorizing scalar sequences. Experiments are further performed to demonstrate the theory. Full article

A brief review of the classical and quantum description of the interaction of electromagnetic radiation with matter based on the model of a harmonic oscillator is presented. This review includes the generalized Bohr correspondence principle, the excitation of a quantum oscillator by electromagnetic pulses including saturation effect, the harmonic limit of the Bloch equations, and a phenomenological account of the damping of the quantum oscillator. In all cases, at the mathematical level, the relationship between the classical and quantum descriptions of the electromagnetic interaction is established and the conditions for such compliance are identified. Full article

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

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