Equations of state are not always effective in continuum mechanics. Maxwell and Boltzmann created a kinetic theory of gases, using classical mechanics. How could they derive the irreversible Boltzmann equation from a reversible Hamiltonian framework? By using probabilities, which destroy physical reality! Forces at distance are non-physical as we know from Poincaré's theory of relativity. Yet Maxwell and Boltzmann only used trajectories like hyperbolas, reasonable for rarefied gases, but wrong without bound trajectories if the "mean free path between collisions" tends to 0. Tartar relies on his H-measures, a tool created for homogenization, to explain some of the weaknesses, e.g. from quantum mechanics: there are no "particles", so the Boltzmann equation and the second principle, can not apply. He examines modes used by energy, proves which equation governs each mode, and conjectures that the result will not look like the Boltzmann equation, and there will be more modes than those indexed by velocity!

The coupling between acoustic waves and fluid flow motion is basically nonlinear, with the result that flow and sound modify themselves reciprocally with respect to generation and propagation properties. As a result this problem is investigated by many different communities, such as applied mathematics, acoustics and fluid mechanics. This book is the result of an international school which was held to discuss the foundation of sound–flow interactions, to share expertise and methodologies, and to promote cross-fertilization between the different disciplines involved. It consists essentially of a set of pedagogical lectures and is meant to serve not only as a compact source of reference for the experienced researcher but also as an advanced textbook for postgraduate students, and nonspecialists wishing to familiarize themselves in depth, at a research level, with this fascinating subject.

Equations of state are not always effective in continuum mechanics. Maxwell and Boltzmann created a kinetic theory of gases, using classical mechanics. How could they derive the irreversible Boltzmann equation from a reversible Hamiltonian framework? By using probabilities, which destroy physical reality! Forces at distance are non-physical as we know from Poincaré's theory of relativity. Yet Maxwell and Boltzmann only used trajectories like hyperbolas, reasonable for rarefied gases, but wrong without bound trajectories if the "mean free path between collisions" tends to 0. Tartar relies on his H-measures, a tool created for homogenization, to explain some of the weaknesses, e.g. from quantum mechanics: there are no "particles", so the Boltzmann equation and the second principle, can not apply. He examines modes used by energy, proves which equation governs each mode, and conjectures that the result will not look like the Boltzmann equation, and there will be more modes than those indexed by velocity!

Like many other textbooks the present one is based on a lecture course given by the author for master students of the RWTH Aachen University. In spite of a somewhat di?cult matter those students were able to endure and, as far as I know, are still ?ne. I wish the same for the reader of the book. Although the present book can be referred to as a textbook one ?nds only little plain text inside. I tried to explain the matter in a brief way, nevert- lessgoinginto detailwherenecessary.Ialsoavoidedtediousintroductions and lengthy remarks about the signi?cance of one topic or another. A reader - terested in tensor algebra and tensor analysis but preferring, however, words instead of equations can close this book immediately after having read the preface. The reader is assumed to be familiar with the basics of matrix algebra and continuum mechanics and is encouraged to solve at least some of num- ous exercises accompanying every chapter. Having read many other texts on mathematics and mechanics I was always upset vainly looking for solutions to the exercises which seemed to be most interesting for me. For this reason, all the exercises here are supplied with solutions amounting a substantial part of the book. Without doubt, this part facilitates a deeper understanding of the subject. As a research work this book is open for discussion which will certainly contribute to improving the text for further editions. In this sense, I am very gratefulfor comments,suggestionsand constructivecriticismfromthe reader.

This book explores the identification of influencers in complex networks, bridging theoretical approaches with practical applications across diverse fields. It examines interdisciplinary complex systems, including online social media, biological networks, brain networks, socioeconomic and financial systems, and ecosystems. The research presented aims to benefit scientists in relevant areas and inspire new scientific inquiries, potentially advancing the field of influencer identification.

Hyperbolic numbers are proposed for a rigorous geometric formalization of the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers as a simple extension of the field of complex numbers is extensively studied in the book. In particular, an exhaustive solution of the "twin paradox" is given, followed by a detailed exposition of space-time geometry and trigonometry. Finally, an appendix on general properties of commutative hypercomplex systems with four unities is presented.

The Theory of the Top: Volume III. Perturbations: Astronomical and Geophysical Applications is the third installment in a series of four self-contained English translations of the classic and definitive treatment of rigid body motion.

Written in a clear pedagogic style, this book deals with the application of density matrix theory to atomic and molecular physics. The aim is to precisely characterize sates by a vector and to construct general formulas and proofs of general theorems. The basic concepts and quantum mechanical fundamentals (reduced density matrices, entanglement, quantum correlations) are discussed in a comprehensive way. The discussion leads up to applications like coherence and orientation effects in atoms and molecules, decoherence and relaxation processes. This third edition has been updated and extended throughout and contains a completely new chapter exploring nonseparability and entanglement in two-particle spin-1/2 systems. The text discusses recent studies in atomic and molecular reactions. A new chapter explores nonseparability and entanglement in two-particle spin-1/2 systems.

Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin­ tegrable classical and quantum dynamics. Several of these lectures are collected in this volume. The basic phenomenon of nonlinear dynamics is mixing in phase space, lead­ ing to a positive dynamical entropy and a loss of information about the initial state. The nonlinear motion in phase space gives rise to a linear action on phase space functions which in the case of iterated maps is given by a so-called transfer operator. Good mixing rates lead to a spectral gap for this operator.

Our knowledge of objects of complex and potential analysis has been enhanced recently by ideas and constructions of theoretical and mathematical physics, such as quantum field theory, nonlinear hydrodynamics, material science. These are some of the themes of this refereed collection of papers, which grew out of the first conference of the European Science Foundation Networking Programme 'Harmonic and Complex Analysis and Applications' held in Norway 2007.

This book presents the fundamentals of stochastic thermodynamics, one of the most central subjects in non-equilibrium statistical mechanics. It also explores many recent advances, e.g., in information thermodynamics, the thermodynamic uncertainty relation, and the trade-off relation between efficiency and power.

This expanded second edition of Relativity and the Nature of Spacetime c- tains several major changes and a number of additions to different ch- ters. Two chapters (Chaps. 6 and 7), which discussed two speci?c groups of arguments against the reality of spacetime, have been transformed into - pendices (A and B). Two new chapters (Chaps. 6 and 10) have been added. Chapter 6, entitled Why Is the Issue of the Nature of Spacetime So Imp- tant?, elaborates on what was Sect. 5. 6 of the ?rst edition, and addresses some recent work on the nature of spacetime – for example, the growing (or evolving) block universe model of the world, which has recently been - vived by several physicists as what appears to be the last remaining alter- tive to the Minkowski absolute four-dimensional world (after it had become an undeniable fact that three-dimensionalism, or presentism, contradicts the relativistic experimental evidence). Chapter 10, entitled Spacetime and the Nature of Quantum Objects and based on what used to be Sects. 6. 2 and 6. 3 in the ?rst edition, explores the implications of the issue of the nature of spacetime for quantum physics, in order to see whether it can provide some insight into the nature of quantum objects. Two new sections have been included, namely, Sect. 5. 6 entitled Re- tivization of Existence and Observers in General Relativity and Sect. 7. 6 - titled Probing the Anisotropic Velocity of Light by a Terrestrial Experiment.

This classroom-tested textbook is an innovative, comprehensive, and forward-looking introductory undergraduate physics course. While it clearly explains physical principles and equips the student with a full range of quantitative tools and methods, the material is firmly grounded in biological relevance and is brought to life with plenty of biological examples throughout.

Sergio Albeverio gave important contributions to many fields ranging from Physics to Mathematics, while creating new research areas from their interplay. Some of them are presented in this Volume that grew out of the Random Transformations and Invariance in Stochastic Dynamics Workshop held in Verona in 2019. To understand the theory of thermo- and fluid-dynamics, statistical mechanics, quantum mechanics and quantum field theory, Albeverio and his collaborators developed stochastic theories having strong interplays with operator theory and functional analysis. His contribution to the theory of (non Gaussian)-SPDEs, the related theory of (pseudo-)differential operators, and ergodic theory had several impacts to solve problems related, among other topics, to thermo- and fluid dynamics. His scientific works in the theory of interacting particles and its extension to configuration spaces lead, e.g., to the solution of open problems in statistical mechanics and quantum field theory. Together with Raphael Hoegh Krohn he introduced the theory of infinite dimensional Dirichlet forms, which nowadays is used in many different contexts, and new methods in the theory of Feynman path integration. He did not fear to further develop different methods in Mathematics, like, e.g., the theory of non-standard analysis and p-adic numbers.

In this undergraduate textbook, now in its 2nd edition, the author develops the quantum theory from first principles based on very simple experiments: a photon traveling through beam splitters to detectors, an electron moving through magnetic fields, and an atom emitting radiation. From the physical description of these experiments follows a natural mathematical description in terms of matrices and complex numbers.