Coercivity estimates of solutions for some problems of mathematical physics including parabolic and Stokes systems are obtained

In mathematical physics, time-dependent Green's functions (GFs) are the solutions of differential equations of the first and second time derivatives. Habitually, the time-dependent GFs are Fourier transformed into the frequency space. Then, analytical continuation of the frequency is extended to below or above the real axis. After inverse Fourier transformation, retarded and advanced GFs can be obtained, and there may be arbitrariness in such analytical continuation. In the present work, we establish the differential equations from which the retarded and advanced GFs are rigorously solved. The key point is that the derivative of the time step function is the Dirac delta function plus an infinitely small quantity, where the latter is not negligible because it embodies the meaning of time delay or time advance. The retarded and advanced GFs defined in this paper are the same as the one-body GFs defined with the help of the creation and destruction operators in many-body theory. There is no way to define the causal GF in mathematical physics, and the reason is given. This work puts the initial conditions into differential equations, thereby paving a way for solving the problem of why there are motions that are irreversible in time.

Mathematical physics provides physical theories with their logical basis and the tools for drawing conclusions from hypotheses. Introduction to Mathematical Physics explains to the reader why and how mathematics is needed in the description of physical events in space. For undergraduates in physics, it is a classroom-tested textbook on vector analysis, linear operators, Fourier series and integrals, differential equations, special functions and functions of a complex variable. Strongly correlated with core undergraduate courses on classical and quantum mechanics and electromagnetism, it helps the student master these necessary mathematical skills. It contains advanced topics of interest to graduate students on relativistic square-root spaces and nonlinear systems. It contains many tables of mathematical formulas and references to useful materials on the Internet. It includes short tutorials on basic mathematical topics to help readers refresh their mathematical knowledge. An appendix on Mathematica encourages the reader to use computer-aided algebra to solve problems in mathematical physics

This list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines with a common goal of understanding different viewpoints and approaches surrounding the theory of mathematical quasicrystals. The problems reflect this goal and this diversity and we hope that they will motivate further cross-disciplinary research and lead to new advances in our overall vision of this rapidly developing field.

The papers in this collection grew out of talks recently presented at S. P. Novikov's seminar on topology and mathematical physics in Moscow. They are devoted to various problems in the theory of completely integrable systems and relations to topology, algebra, and mathematical physics

Wormholes are extremely attractive objects for research, and although many mathematical and physical properties of these objects have been discovered and studied in the recent decades, there remain many unsolved problems and opportunities of interest. The Special issue of Universe entitled ``Recent Advances in Wormhole Physics,'' containing 14 contributions, is aimed at enlightening some recent results in selected areas of wormhole physics.

This collection presents new and interesting applications of Poisson geometry to some fundamental well-known problems in mathematical physics. The methods used by the authors include, in addition to advanced Poisson geometry, unexpected algebras with non-Lie commutation relations, nontrivial (quantum) Kählerian structures of hypergeometric type, dynamical systems theory, semiclassical asymptotics, etc

During the last decade we have witnessed an impressive development in so-called interpreted languages and computational environments such as Maple, Mathematica, IDL, Matlab etc. Problems which until recently were typically solved on mainframe machines and written in computing languages such as Fortran or C/C++, can now easily be solved on standard PCs with the bonus of immediate visualizations of the results. In our undergraduate programs an often posed question is how to incorporate and exploit efficiently these advances in the standard physics and mathematics curriculum, without detracting the attention from the classical and basic theoretical and experimental topics to be covered. Furthermore, if students are trained to use such tools at early stages in their education, do such tools really enhance and improve the learning environment? And, perhaps even more important, does it lead to a better physics understanding? Here we present one possible approach, where computational topics are gradually baked into our undergraduate curriculum in Mathematics and Physics, Astronomy and Meteorology. We focus on training our students to use general programming tools in solving physics problems, in addition to the classical analytic problems. By this approach, the students gain an expertise that they can build upon in their future studies and careers. We use mainly Java, Matlab and Maple as computational environments. Our students are now capable of handling at an early stage in their education more realistic physics problems than before. We believe firmly that, in addition to educating modern scientists, this promotes a better physics understanding for a majority of the students.

The book examines the classic and generally accepted methods for solving mathematical physics problems (method of the potential theory, the eigenfunction method, integral transformation methods, discretisation characterisation methods, splitting methods). A separate chapter is devoted to methods for solving nonlinear equations. The book offers a large number of examples of how these methods are applied to the solution of specific mathematical physics problems, applied in the areas of science and social activities, such as energy, environmental protection, hydrodynamics, theory of elasticity, etc

We review recent advances in solving problems of mathematical physics on domains with irregular boundaries in Rn. We distinguish two frameworks: a measure-free approach in the image of the trace operator spaces for extension domains and an L2-approach depending on a d-upper regular boundary measure. In both cases, the domains can have boundaries with different Hausdorff dimensions inside the interval (n -- 2, n). The generalization of the Poincar{é}-Steklov/Dirichlet-to-Neumann operator for these two contexts is given. To illustrate the established convergence of spectral problems for elliptic operators with Robin boundary conditions, we give a numerical example of the stability of localized eigenfunctions, using results of M. Graffin.

What sets this volume apart from other mathematics texts is its emphasis on mathematical tools commonly used by scientists and engineers to solve real-world problems. Using a unique approach, it covers intermediate and advanced material in a manner appropriate for undergraduate students. Based on author Bruce Kusse's course at the Department of Applied and Engineering Physics at Cornell University, Mathematical Physics begins with essentials such as vector and tensor algebra, curvilinear coordinate systems, complex variables, Fourier series, Fourier and Laplace transforms, differential and integral equations, and solutions to Laplace's equation

You might've heard about various mathematical properties of scattering amplitudes such as analyticity, sheets, branch cuts, discontinuities, etc. What does it all mean? In these lectures, we'll take a guided tour through simple scattering problems that will allow us to directly trace such properties back to physics. We'll learn how different analytic features of the S-matrix are really consequences of causality, locality of interactions, unitary propagation, and so on. These notes are based on a series of lectures given in Spring 2023 at the Institute for Advanced Study in Princeton and the Higgs Centre School of Theoretical Physics in Edinburgh.

Quantum physics has been highly successful for more than 90 years. Nevertheless, a rigorous construction of interacting quantum field theory is still missing. Moreover, it is still unclear how to combine quantum physics and general relativity in a unified physical theory. Attacking these challenging problems of contemporary physics requires highly advanced mathematical methods as well as radically new physical concepts. This book presents different physical ideas and mathematical approaches in this direction. It contains a carefully selected cross-section of lectures which took place in autumn 2014 at the sixth conference ``Quantum Mathematical Physics - A Bridge between Mathematics and Physics'' in Regensburg, Germany. In the tradition of the other proceedings covering this series of conferences, a special feature of this book is the exposition of a wide variety of approaches, with the intention to facilitate a comparison. The book is mainly addressed to mathematicians and physicists who are interested in fundamental questions of mathematical physics. It allows the reader to obtain a broad and up-to-date overview of a fascinating active research area

Many students in upper-division physics courses struggle with the mathematically sophisticated tools and techniques that are required for advanced physics content. We have developed an analytical framework to assist instructors and researchers in characterizing students' difficulties with specific mathematical tools when solving the long and complex problems that are characteristic of upper-division. In this paper, we present this framework, including its motivation and development. We also describe an application of the framework to investigations of student difficulties with direct integration in electricity and magnetism (i.e., Coulomb's Law) and approximation methods in classical mechanics (i.e., Taylor series). These investigations provide examples of the types of difficulties encountered by advanced physics students, as well as the utility of the framework for both researchers and instructors.

This up-to-date textbook on mathematical methods of physics is designed for a one-semester graduate or two-semester advanced undergraduate course. The formal methods are supplemented by applications that use MATHEMATICA to perform both symbolic and numerical calculations. The book is written by a physicist lecturer who knows the difficulties involved in applying mathematics to real problems. As many as 40 exercises are included at the end of each chapter. A student CD includes a basic introduction to MATHEMATICA, notebook files for each chapter, and solutions to selected exercises. Free solut

These are expanded notes of lectures given at the Advanced Summer School on Modern Mathematical Physics (JINR Dubna, July 2005) and at the 8th International School-Seminar ``The actual problems of microworld physics 2005'' (Gomel-Dubna, August 2005). I review classical monopole solutions of the SU(N) Yang-Mills-Higgs theory. The first part is a pedagogical introduction into to the theory of non-Abelian SU(2) monopoles. In the second part I discuss a particular case of SU(3) theories containing different limits of symmetry breaking. It turns out that the multimonopole configurations are natural in a model with the gauge group of higher rank. Here I discuss fundamental and composite monopoles and consider the limiting situation of the massless states. In the last part I briefly discuss construction of the $N = 2$ SU(2) supersymmetric monopoles and some of the basic properties which are connected with the field theoretical aspects of these classical solutions.

One of the major problems in the study of evolution equations of mathematical physics is the investigation of the behavior of the solutions to these equations when time is large or tends to infinity. The related important questions concern the stability of solutions or the character of the instability if a solution is unstable. In the last few decades, considerable progress in this area has been achieved in the study of autonomous evolution partial differential equations. For a number of basic evolution equations of mathematical physics, it was shown that the long time behavior of their solut

For several decades there has been no breakthrough in fundamental physics as revolutionary as relativity and quantum physics despite the amazing advancement of applied physics and technology. By discussing several examples of what physics could have achieved by now, but failed, I will argue that the present state of fundamental physics is not caused by the lack of talented physicists, but rather by problematic general views on how one should do physics. Although it appears to be widely believed that such general views cannot affect the advancement of physics I would like to draw the attention of the younger generation of physicists to three reasons that might have been responsible for failures in the past and might cause problems in the future: (i) misconceptions on the nature of physical theories, (ii) underestimation of the role of conceptual analyses so successfully employed by Galileo and Einstein, and (iii) overestimation of the predictive power of mathematics in physics.

The book assumes next to no prior knowledge of the topic. The first part introduces the core mathematics, always in conjunction with the physical context. In the second part of the book, a series of examples showcases some of the more conceptually advanced areas of physics, the presentation of which draws on the developments in the first part. A large number of problems helps students to hone their skills in using the presented mathematical methods. Solutions to the problems are available to instructors on an associated password-protected website for lecturers

Quantum computing enables the efficient resolution of complex problems, often outperforming classical methods across various applications. In 2009, Harrow, Hassidim and Lloyd proposed an algorithm for solving linear systems of equations, demonstrating exponential speedup (under ideal conditions) with a complexity of $poly(\log N)$, in contrast to classical approaches, which in the general case exhibit a complexity of $O(N^3)$, although they can achieve $O(N)$ in specific cases involving sparse matrices. This algorithm holds promise for advancements in machine learning, the solution of differential equations, linear regression, and cryptographic analysis. However, its structure is intricate, and there is a notable lack of detailed instructional materials in the literature. In this context, this paper presents a tutorial addressing the physical and mathematical foundations of the HHL algorithm, aimed at undergraduate students, explaining its theoretical construction and its implementation for solving linear equation systems. After discussing the underlying mathematical and physical concepts, we present numerical examples that illustrate the evolution of the quantum circuit. Finally, the algorithm's complexity, limitations, and future prospects are analyzed. The examples are compared with their classical simulations, allowing for an operational assessment of the algorithm's performance.