The Hubbard model is the simplest model of interacting fermions on a lattice and is of similar importance to correlated electron physics as the Ising model is to statistical mechanics or the fruit fly to biomedical science. Despite its simplicity, the model exhibits an incredible wealth of phases, phase transitions, and exotic correlation phenomena. Although analytical methods have provided a qualitative description of the model in certain limits, numerical tools have shown impressive progress in achieving quantitative accurate results over the past several years. This article gives an introduction to the model, motivates common questions, and illustrates the progress that has been achieved over recent years in revealing various aspects of the correlation physics of the model. Expected final online publication date for the Annual Review of Condensed Matter Physics, Volume 13 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
Smart clothing, a transformative innovation, has applications in various sectors including sports, military, fashion, and medical monitoring, where intelligent fabric selection and adaptive control systems are crucial for optimal performance. However, existing approaches face significant challenges in handling uncertainty, personalizing user experiences, and optimizing design parameters under conflicting requirements. This paper introduces the integration of soft set and fuzzy set theory to address these complexities in smart clothing development. A soft set-based algorithm is proposed for systematic fabric selection that evaluates material options using weighted parameters. An adaptive fuzzy logic algorithm for real-time environmental control that personalizes responses based on user demographics and climate zones is developed. Further, a dynamic weight optimization algorithm that determines optimal parameter priorities is proposed. Through comprehensive case studies, we demonstrate the practical applicability of our approach. Sensitivity analysis is performed to explore the impact of parameter weighting, comparing equal and prioritized weight scenarios to reveal critical factors influencing efficacy. Comparative analysis with traditional multi-criteria decision-making methods shows that our integrated approach achieves better uncertainty handling and adaptability while maintaining computational efficiency. Statistical validation and testing underscore the effectiveness of the proposed methodologies.
Bernstein polynomials play a very important role in approximation theory, probability theory, computer aided geometric design and many other areas. In 2017 J. Szabados constructed polynomial operators that can be considered as the most natural generalization to several intervals of the classical Bernstein operators. Their main advantages include fixed difference between degrees of the used polynomials and the number of used nodes. Unfortunately, they exist only under strong restrictions on the geometry of intervals (intervals have to form a polynomial inverse image of an interval). The main goal of the paper is to present a rational operator that generalizes J. Szabados’ construction, and exists for an arbitrary system of several intervals. Moreover, this construction (unlike J. Szabados’) is a linear positive operator. One of the main ingredients in the construction is the fact (which was proved by M.G. Krein, B.Ya. Levin, and A.A. Nudel’man) that an arbitrary finite system of real intervals is the inverse image of an interval by a rational function with precisely one pole at each gap. The approximation properties of such operators are studied as well. Further possible generalizations (of V.S. Videnskii’s operators to one interval) are considered.
Michael D. Hunter, Zachary F. Fisher, Charles F. Geier
This paper explores the relation between within-person and between-person research designs using the concept of ergodicity from statistical mechanics in physics. We demonstrate the consequences of ergodicity using several real data examples from previously published studies. We then create several simulated examples that illustrate the independence of within-person processes from between-person differences, and pair these examples with analytic results that reinforce our conclusions. Finally, we discuss the plausibility of ergodicity being the general rule rather than the exception for social and behavioral processes, address common arguments against heeding the implications of ergodicity for behavioral research, and offer several possible solutions.
A.R. Yeshkeyev, I.O. Tungushbayeva, A.K. Koshekova
In this article, within the framework of the study of Jonsson theories, the model-theoretic properties of cosemanticness classes belonging to the factor set of the Jonsson spectrum of an existentially closed models’ subclass of some Jonsson theory in a fixed language were studied. Various results have been obtained. In particular, the properties of the cosemanticness of models and classes of models are considered; some results concerning the Jonsson equivalence in generalization for classes of existentially closed models are obtained; a criterion for the cosemanticness of J-classes in connection with their Kaiser hulls has been found.
In the paper a novel boundary value problem for a third-order partial differential equation (PDE) of a parabolic-hyperbolic type, within a pentagonal domain consisting of both parabolic and hyperbolic regions was investigated. Such equations are pivotal in modeling complex physical phenomena across diverse fields such as physics, engineering, and finance due to their ability to encapsulate a wide range of dynamics through their mixed-type nature. By employing a constructive solution approach, we demonstrate the unique solvability of the posed problem. The significance of this study lies in its extension of the mathematical framework for understanding and solving higher-order mixed PDEs in complex geometrical domains, thus offering new avenues for theoretical and applied research in mathematical physics and related disciplines.
In the article the spectrum and resolvent of the so-called multichannel systems with nonzero internal energies were investigated. The spectrum and resolvent of multichannel Sturm-Liouville systems with non-zero internal energies mi2 and general boundary conditions were investigated. These systems describe the propagation of partial waves in the theory of quantum physics. The importance of studying the spectral characteristics of these systems is presented in the well-known books of the theory of quantum physics. The finiteness of the number of eigenvalues was proved, the multiplicity of positive eigenvalues was investigated, and as well as the resolvent kernel of the system was found.
The class K of algebraic systems of signature σ is called a formula-definable class if there exists an algebraic system A of signature σ such that for any algebraic system B of signature σ it is B ∈ K if and only if Th(B) · Th(A) = Th(A). The paper shows that the formula-definable class of algebraic systems is idempotently formula-definable and is an axiomatizable class of algebraic systems. Any variety of algebraic systems is an idempotently formula-definite class. If the class K of all existentially closed algebraic systems of a theory T is formula-definable, then a theory of the class K is a model companion of the theory T. Also, in the paper the examples of some theories on the properties of formula-definability, pseudofiniteness and smoothly approximability of their model companion were discussed.
Allan N. Kaufman, Bruce I. Cohen, Alain J. Brizard
Presented here is a transcription of the lecture notes from Professor Allan N. Kaufman's graduate statistical mechanics course at Berkeley from the 1972-1973 academic year. Part 1 addresses equilibrium statistical mechanics with topics: fundamentals, classical fluids and other systems, chemical equilibrium, and long-range interactions. Part 2 addresses non-equilibrium statistical mechanics with topics: fundamentals, Brownian motion, Liouville and Klimontovich equations, Landau equation, Markov processes and Fokker-Planck equation, linear response and transport theory, and an introduction to non-equilibrium quantum statistical mechanics.
In this paper, we analyze Gaussian processes using statistical mechanics. Although the input is originally multidimensional, we simplify our model by considering the input as one-dimensional for statistical mechanical analysis. Furthermore, we employ periodic boundary conditions as an additional modeling approach. By using periodic boundary conditions, we can diagonalize the covariance matrix. The diagonalized covariance matrix is then applied to Gaussian processes. This allows for a statistical mechanical analysis of Gaussian processes using the derived diagonalized matrix. We indicate that the analytical solutions obtained in this method closely match the results from simulations.
Jonas Heinzmann, Pietro Carrara, Chenyi Luo
et al.
In the context of the Damage Mechanics Challenge, we adopt a phase-field model of brittle fracture to blindly predict the behavior up to failure of a notched three-point-bending specimen loaded under mixed-mode conditions. The beam is additively manufactured using a geo-architected gypsum based on the combination of bassanite and a water-based binder. The calibration of the material parameters involved in the model is based on a set of available independent experimental tests and on a two-stage procedure. In the first stage an estimate of most of the elastic parameters is obtained, whereas the remaining parameters are optimized in the second stage so as to minimize the discrepancy between the numerical predictions and a set of experimental results on notched three-point-bending beams. The good agreement between numerical predictions and experimental results in terms of load-displacement curves and crack paths demonstrates the predictive ability of the model and the reliability of the calibration procedure.
Diego C. P. Blanco, Ardeshir Hanifi, Dan S. Henningson
et al.
Large-eddy simulations of a flat-plate boundary layer, without a leading edge, subject to multiple levels of incoming free stream turbulence are considered in the present work. Within an input-output model where non-linear terms of the incompressible Navier-Stokes equations are treated as an external forcing, we manage to separate inputs related to perturbations coming through the intake of the numerical domain, whose evolution represents a linear mechanism, and the volumetric non-linear forcing due to triadic interactions. With these, we perform the full reconstruction of the statistics of the flow, as measured in the simulations, to quantify pairs of wavenumbers and frequencies more affected by either linear or non-linear receptivity mechanisms. Inside the boundary layer, different wavenumbers at near-zero frequency reveal streaky structures. Those that are amplified predominantly via linear interactions with the incoming vorticity occur upstream and display transient growth, while those generated by the non-linear forcing are the most energetic and appear in more downstream positions. The latter feature vortices growing proportionally to the laminar boundary layer thickness, along with a velocity profile that agrees with the optimal amplification obtained by linear transient growth theory. The numerical approach presented is general and could potentially be extended to any simulation for which receptivity to incoming perturbations needs to be assessed.
In a rectangular domain, we consider a boundary value problem periodic in one variable for a system of partial differential equations of hyperbolic type. Introducing a new unknown function, this problem is reduced to an equivalent boundary value problem for an ordinary differential equation with an integral condition. Based on the parametrization method, new approaches to finding an approximate solution to an equivalent problem are proposed and its convergence is proved. This made it possible to establish conditions for the existence of a unique solution of a semiperiodic boundary value problem for a system of second-order hyperbolic equations.