Related papers: Mathematics for theoretical physics
We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.
This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and…
The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are…
Wigner's "unreasonable effectiveness of mathematics" in physics can be understood as a reflection of a deep and unexpected unity between the fundamental structures of mathematics and of physics. Some of the history of evidence for this is…
\noindent 1. Generalities\hfil\break 2. Lie groups and Lie algebras\hfil\break 3. The unitary groups\hfil\break 4. Representations of the SU(n) groups (and of their algebras)\hfil\break 5. The tensor method for unitary groups, and\hb the…
The authors give a detailed information about symmetry (Lie, non-Lie, conditional) of nonlinear PDEs for spinor, vector and scalar fields; using advanced methods of group-theoretical, symmetry analysis construct wide families of classical…
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
In recent years, the field of statistics has experienced a surge in interest and application, largely due to significant advances in computer technology. This progress has led to remarkable developments in statistics methods and algorithms,…
This primer is intended as an introduction to differential forms, a central object in modern mathematical physics, for scientists and engineers.
This doctoral dissertation presents an in-depth analysis of the first six chapters of Eddington's Fundamental Theory, sometimes referred to as his 'statistical' theory, in the context of discoveries and advancements made since its original…
Course material for mathematical methods of theoretical physics intended for an undergraduate audience.
It is shown that physical mechanics for pointlike bodies can be effectively modeled in terms of the action of transformation groups that act as symmetries of the solutions of systems of differential equations that describe the integrability…
We introduce PHYSICS, a comprehensive benchmark for university-level physics problem solving. It contains 1297 expert-annotated problems covering six core areas: classical mechanics, quantum mechanics, thermodynamics and statistical…
This book provides a gentle introduction to fractional Sobolev spaces, which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
Can machine learning help discover new mathematical structures? In this article we discuss an approach to doing this which one can call "mathematical data science". In this paradigm, one studies mathematical objects collectively rather than…
This introductory work combines bottom-up and top-down approaches towards understanding the underlying categorical structure of possible unifying theories descending from string theory. Guided by well-established developments in the realm…
This set of lecture notes first gives an introduction to the geometry of principal bundles. Next, it demonstrates how they can be used to formalize the concept of gauge theories arising in physics. A basic familiarity with the differential…
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
This book is a regular textbook of analytical geometry covering vector algebra and its applications to describing straight lines, planes, and quadrics in two and three dimensions. The stress is made on vector algebra by using skew-angular…