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This PhD thesis deals with a number of different problems in mathematical physics with the common thread that they have probabilistic aspects. The problems all stem from mathematical studies of lattice systems in statistical and quantum physics; however beyond that, the selection of the concrete problems is to a certain extent arbitrary. This thesis consists of an introduction and seven papers., Comment: PhD Thesis, University of Copenhagen. Comments welcome. Previous versions of most papers are already on arXiv

In this short paper, the results of the paper by Drab et al. (Eur. Phys. J. E (2022) 45:79) are described in the framework of wave mechanics and mathematical physics based on common understandings. The attention is focused on properties of Boussinesq-type equations, solitons, and peakons. These concepts are supported by several experimental observations.

We briefly recount the long friendship that developed between Ludwig and us (Moshe Flato and I), since we first met at ICM 1966 in Moscow. That friendship extended to his school and family, and persists to this day. Its strong personal impact and main scientific components are sketched, including reflexions on what mathematical physics is (or should be)., Comment: 12 pages

This paper surveys the long-standing connections and impact between Vaughan Jones's theory of subfactors and various topics in mathematical physics, namely statistical mechanics,quantum field theory,quantum information and two-dimensional conformal field theory., Comment: 22 pages, 1 figure. To appear in an issue of the Bulletin of the AMS, dedicated to the mathematical legacy of Vaughan Jones

These are the third 50 issues of This Week's Finds of Mathematical Physics, from April 9, 1997 to June 18, 2000. These issues discuss quantum gravity, topological quantum field theory, n-categories and other topics in mathematics and physics. They also contain a series of expository mini-articles on homotopy theory. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird; you can get a better-looking version at https://math.ucr.edu/home/baez/twf_latex/), Comment: 348 pages

We present new closed-form expressions for certain improper integrals of Mathematical Physics such as certain Ising, Box, and Associated integrals. The techniques we employ here include (a) the Method of Brackets and its modifications and suitable extensions to obtain the Mellin-Barnes representation. (b) The evaluation of the resulting Mellin-Barnes representations via the recently discovered Conic Hull method via the automated package $\textit{MBConichulls.wl}$. Finally, the analytic continuations of these series solutions are then produced using the automated package \texttt{Olsson.wl}, based on the method of Olsson. Thus, combining all these recent advances allows for closed-form evaluation of the hitherto unknown $B_3(s)$, $B_4(s)$, and related integrals in terms of multivariable hypergeometric functions. Along the way, we also discuss certain complications while using the Original Method of Brackets for these evaluations and how to rectify them. The interesting cases of $C_{5,k}$ are also studied. It is not yet fully resolved for the reasons we discuss in this paper., Comment: To appear: EPJ ST Special Issue: Frontier 23: Elementary particle physics, Dark matter and Astroparticle physics

It is shown with the help of skew-symmetric forms that the mathematical physics equations, on which no additional conditions are imposed, have quantum properties. And this is due to the integrability properties of differential equations, which depends on the consistency of derivatives with respect to different variables and the consistency of equations, if the mathematical physics equations are a system of equations. It was found that such equations on the original tangent space turn out to be non-integrable. Their derivatives do not form a differential. The integrability of such equations is realized only on the structures of a cotangent integrable manifold. This happens using a degenerate, non-differential-preserving transformation that has quantum properties. When implementing degenerate transformations, mini structures (quanta) arise, from which integrable structures are formed. Such properties of integrability of mathematical physics equations and features of degenerate transformations reveal the quantum properties of mathematical physics equations and their ability to generate quantum structures., Comment: 21 pp

This text offers a personal account of the scientific legacy of two giants of mathematical physics at the turn of the Millenium and their heritage in Canada, their land of adoption., Comment: 12 pages, Memorial plenary talk at the 34th International Colloquium on Group Theoretical Methods in Physics, Strasbourg, France, July 2022. A few minor edits in the revised version

Mathematical physics has many facets, of which we shall briefly give a (very partial) description, centered around those of main interest for Elliott and us (Moshe Flato and I). In our case these aspects had as a corollary a variety of "parascientific activities", in particular the foundation of IAMP (the International Association of Mathematical Physics) and of the journal LMP (Letters in Mathematical Physics), both of which were strongly impacted by Elliott, and Elliott's long insistence that publishers do not demand "copyright transfer" as a precondition for publication but are satisfied with "consent to publish", which is increasingly becoming standard. Since this article is testimony to the huge scientific impact of Elliott, the latter intertwined aspects constitute the core of the present contribution. The last part deals briefly with metaphysical and metamathematical considerations related to axioms., Comment: 12 pages, third version (October 2022) of our paper dedicated to our friend Elliott Lieb on the occasion of the ninetieth anniversary of his birth. Some additions in the first two sections, a new "metamathematical" Section 3, and journal reference added

These are the second 50 issues of This Week's Finds of Mathematical Physics, from April 23, 1995 to March 23, 1997. These issues discuss quantum gravity, topological quantum field theory and other topics in mathematics and physics. They also include two expository series: one on ADE classifications and one on categories and higher categories. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird)., Comment: 281 pages

These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of $n$-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird)., Comment: 241 pages

We give an account of the theory of factorization spaces, categories, functors, and algebras, following the approach of [Ras1]. We apply these results to give geometric constructions of factorization $\mathbb{E}_n$ algebras describing mixed holomorphic-topological twists of supersymmetric gauge theories in low dimensions. We formulate and prove several recent predictions from the physics literature in this language: We recall the Coulomb branch construction of [BFN1] from this perspective. We prove a conjecture from [CosG] that the Coulomb branch factorization $\mathbb{E}_1$ algebra $\mathcal{A}(G,N)$ acts on the factorization algebra of chiral differential operators $\mathcal{D}^{ch}(Y)$ on the quotient stack $Y=N/G$. We identify the latter with the semi-infinite cohomology of $\mathcal{D}^{ch}(N)$ with respect to $\hat{\mathfrak{g}}$, following the results of [Ras3]. Both these results require the hypothesis that $Y$ admits a Tate structure, or equivalently that $\mathcal{D}^{ch}(N)$ admits an action of $\hat{\mathfrak{g}}$ at level $\kappa=-\text{Tate}$. We construct an analogous factorization $\mathbb{E}_2$ algebra $\mathcal{F}(Y)$ describing the local observables of the mixed holomorphic-B twist of four dimensional $\mathcal{N}=2$ gauge theory. We apply the theory of equivariant factorization algebras of the prequel [Bu1] in this example: we identify $S^1$ equivariant structures on $\mathcal{F}(Y)$ with Tate structures on $Y=N/G$, and prove that the corresponding filtered quantization of $\iota^!\mathcal{F}(Y)$ is given by the two-periodic Rees algebra of chiral differential operators on $Y$. This gives a mathematical account of the results of [Beem4]. Finally, we apply the equivariant cigar reduction principle of [Bu1] to explain the relationship between these results and our account of the results of [CosG] described above., Comment: 153 pages, 5 figures, Part II of a series

This is an extended version of my 2018 Heinemann prize lecture describing the work for which I got the prize. The citation is very broad so this describes virtually all my work prior to 1995 and some afterwards. It discusses work in non-relativistic quantum mechanics, constructive quantum field theory and statistical mechanics., Comment: This is the final presubmission version which has been submitted to J. Math. Phys

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Pleba\'nski heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler--Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler--Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The~existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are~discussed., Comment: 22 pages

It is shown that mathematical physics differential equations have properties that allow describing processes such as the structures emergence, discrete transitions, quantum jumps. The peculiarity is that such properties are hidden. They do not follow directly from the mathematical physics equations but are realized discretely in the solving process. This is due to the mathematical physics equations integrability, which, as shown, can be realized only discretely in the presence of any degrees of freedom. In this case, a transition occurs from the original coordinate space with a solution that is not a function (the solution derivatives do not compose a differential) to integrable structures with a solution that is a discrete function. It is the double solutions and spatial transitions that can describe the processes of the emergence of any structures or phenomena. Due to hidden properties, the mathematical physics equations have unique possibilities in describing physical processes and phenomena that cannot be described in the framework of other mathematical formalisms. Such results were obtained using skew-symmetric differential forms., Comment: 20 pages

We review the theory of Toeplitz extensions and their role in operator K-theory, including Kasparov's bivariant K-theory. We then discuss the recent applications of Toeplitz algebras in the study of solid state systems, focusing in particular on the bulk-edge correspondence for topological insulators., Comment: 18 pages

This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of ($q$-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra., Comment: 46 pages

The search for generating compatibility conditions (CC) for a given operator is a very recent problem met in General Relativity in order to study the Killing operator for various standard useful metrics (Minkowski, Schwarschild and Kerr). In this paper, we prove that the link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with a kind of differential transcendence degree, also called degree of generality by A. Einstein in his 1930 letters to E. Cartan. The motivating examples that we provide are among the rare ones known in the literature and could be used as testing examples for future applications of computer algebra., Comment: arXiv admin note: text overlap with arXiv:1805.11958

These are lecture notes for a master-level course given at KTH, Stockholm, in the spring of 2017, with the primary aim of proving the stability of matter from first principles using modern mathematical methods in many-body quantum mechanics. General quantitative formulations of the uncertainty and the exclusion principles of quantum mechanics are introduced, such as the Hardy, Sobolev and Poincar\'e functional inequalities as well as the powerful Lieb-Thirring inequality that combines these two principles. The notes are aimed to be both fairly self-contained and at the same time complementary to existing literature, also covering recent developments to prove Lieb-Thirring inequalities and stability from general, weaker formulations of the exclusion principle., Comment: 108 pages, 2 figures. Some sections still incomplete

The paper is an investigation of the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These functions, recently introduced by the second author, generate the kernels of the principal parts of these operators and play an important role in understanding their spectral characteristics. We reveal the representations of these functions in terms of the Gauss and Clausen hypergeometric functions and present differential-difference and differential equations they satisfy. Mathematically, the results include calculation of certain trigonometric double integrals and derivation of their other properties. Furthermore, they represent a potentially useful tool in matter relaxation in an external field, the study of nanoelectronic electrolyte-based systems and dynamics of charge carriers in media with obstacles., Comment: A number of improvements following the referee suggestions. The final form accepted by Analysis Mathematica; 15 pages, no figures

Many equations of mathematical physics are described by differential polynomials, that is by polynomials in the derivatives of a certain number of functions. However, up to the knowledge of the author, differential algebra in a modern setting has never been applied to study the specific algebraic feature of such equations. The purpose of this short but difficult paper is to revisit a few domains like general relativity, conformal geometry and contact geometry in the light of a modern approach to nonlinear systems of partial differential equations, using new methods from Differential Geometry (D.C. Spencer, 1970), Differential Algebra (J.F. Ritt, 1950 and E. Kolchin, 1973) and Algebraic Analysis (M. Kashiwara, 1970). Identifying the differential indeterminates of Ritt and Kolchin with the jet coordinates of Spencer, the idea is to study Differential Duality by using only linear differential operators with coefficients in a differential field. In particular, the linearized second order Einstein equations are parametrizing the first order Cauchy stress equations but cannot themselves be parametrized. In the framework of Homological Algebra, this result is not coherent with the vanishing of certain first and second extension modules. As a byproduct, we shall prove that gravitation and electromagnetism must only depend on the second order jets (elations) of the system of conformal Killing equations. Finally, we shall use these new methods in order to study contact transformations in arbitrary odd dimension and apply these results to the study of Hamilton-Jacobi equations in mechanics., Comment: This paper provides the nonlinear generalization of the linear methods presented in arXiv:1706.04105 and applies them to various domains of mathematical physics, in particular to conformal geometry (separating the dimensions 3, 4 and greater or equal to 5) and contact geometry in arbitrary odd dimension (separating the dimensions 3 and greater or equal to 5). All these results are new

$p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of $p$-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects., Comment: 50 pages

This paper aims to revisit the mathematical foundations of both General Relativity and Electromagnetism after one century, in the light of the formal theory of systems of partial differential equations and Lie pseudogroups (D.C. Spencer, 1970) or Algebraic Analysis, namely a mixture of differential geometry and homological algebra (M. Kashiwara, 1970). Among the new results obtained, we may quote: 1) In dimension 4 only, the 9 Bianchi identities that must be satisfied by the 10 components of the Weyl tensor are described by a second order operator and have thus nothing to do with the 20 first order Bianchi identities for the 20 components of the Riemann tensor. This result, not known after one century, has been recently confirmed by A. Quadrat (INRIA) using new computer algebra packages. 2) The Ricci tensor R is a section of the Ricci bundle of symmetric covariant 2-tensors which is the kernel of the canonical projection of the Riemann bundle onto the Weyl bundle, induced by the canonical inclusion of the classical Killing system (Poincare group) into the conformal Killing system (Conformal group). It has only to do with the second order jets (elations) of the conformal Killing system because any 1-form with value in the bundle of elations can be decomposed in the direct sum (R,F) where the electromagnetic field F is a section of the vector bundle of skewsymmetric covariant 2-tensors. It follows therefore that electromagnetism and gravitation have only to do with second order jets. 3) The 10 linearized second order Einstein equations are parametrizing the 4 first order Cauchy stress equations but cannot be parametrized themselves. As a byproduct of this negative result, these 4 Cauchy stress equations have nothing to do with the 4 divergence-type equations usually obtained from the 20 Bianchi identities by contraction of indices., Comment: The paper is an extended written version of a series of lectures (10 h) given during a short intensive course organized by the Center of Theoretical Physics (CPT) in Marseille, France (may 23-25, 2016) under the title " Algebraic Analysis, Double Duality and Applications in Mathematical Physics"

A class of solutions, decaying as $t\rightarrow \infty$, of a two-dimensional model problem on the oscillations of an ideal rotating fluid in some domains with angular points is constructed explicitly. The existence of solutions whose $L_2$-norms decrease more rapidly than any negative power of $t$, is established., Comment: Published version with some misprints corrected

We discuss new approaches to fundamental problems of mathematics and mathematical physics such as mathematical foundation of quantum field theory, the Riemann hypothesis, and construction of noncommutative algebraic geometry., Comment: 7 pages, the third problem added

We develop a theory of equivariant factorization algebras on varieties with an action of a connected algebraic group $G$, extending the definitions of Francis-Gaitsgory [FG] and Beilinson-Drinfeld [BD1] to the equivariant setting. We define an equivariant analogue of factorization homology, valued in modules over $\text{H}^\bullet_G(\text{pt})$, and in the case $G=(\mathbb{C}^\times)^n$ we prove an equivariant localization theorem for factorization homology, analogous to the classical localization theorem [AtB]. We establish a relationship between $\mathbb{C}^\times$ equivariant factorization algebras and filtered quantizations of their restrictions to the fixed point subvariety. These results provide a model for predictions from the physics literature about the $\Omega$-background construction introduced in [Nek1], interpreting factorization $\mathbb{E}_n$ algebras as observables in mixed holomorphic-topological quantum field theories. In the companion paper [Bu2], we develop tools to give geometric constructions of factorization $\mathbb{E}_n$ algebras, and apply them to define those corresponding to holomorphic-topological twists of supersymmetric gauge theories in low dimensions. Further, we apply our above results in these examples to give an account of the predictions of [CosG] as well as [Beem4], and explain the relation between these constructions from this perspective., Comment: 95 pages, 5 figures, Part I of a series

We review various derivations of the partition function of the two-dimensional Ising Model of ferromagnetism and proofs of the stability of matter, paying attention to passages where there would appear to be a lacuna between steps or where the structure of the argument is not so straightforward. Authors cannot include all the intermediate steps, but sometimes most readers and especially students will be mystified by such a transition. Moreover, careful consideration of such lacunae points to interesting physics and not only mathematical technology. Also, when reading the original papers, the structure of the physics argument may be buried by the technical moves. Improvements in the derivations, in subsequent papers by others, may well be clearer and more motivated. But, there is remarkably little written and published about how to read some of the original papers, and the subsequent ones, yet students and their teachers would often benefit from such guidance. I should note that much of the discussion below will benefit from having those papers in front of you.

In this review paper we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics.We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind. Keywords: Fractional Calculus, Wright Functions, Green's Functions, Diffusion-Wave Equation, Comment: 31 pages, 10 figures

Non-Hermitian matrices $H\in M_2(\mathbb{C})$ satisfying the relation $ H^{\dag}G = GH $, for invertible and singular Hermitian matrices $G$ have been studied. The matrices $H$ corresponding to invertible $G$ are known in the literature as G-pseudo Hermitian matrices. We label the matrices corresponding to the singular $G_s$ as $G_s$-pseudo Hermitian. We have proved that all $ 2\times 2$ $G$-pseudo Hermitian matrices are PT-symmetric. For a given $G$ ($G_s$), all $G$ ($G_s$)-pseudo-Hermitian $H\in M_2(\mathbb{C})$ are found to be expressed as a linear variety. It is further found that for any two Hermitian $G_i,G_j\in M_2(\mathbb{C})$ such that $G_i\neq \lambda G_j$, there always exists exactly one trace less $H\in M_2(\mathbb{C})$ (up to real scaling) which is pseudo-Hermitian with respect to both these $G$ matrices. The set of all $G$- and $G_s$- pseudo-Hermitian matrices has been divided into seven distinct ensembles of matrices and the set of all PT-symmetric matrices in $M_2(\mathbb{C})$ is partitioned into four cells, denoted by $S_1,S_2,S_3$ and $S_4$. The ensembles of trace-less G-pseudo Hermitian matrices are shown to be written as a linear combination of three basis elements from these cells. When $\mathrm{Tr}(G) = 0$, one basis element is from $S_1$ and the other two are from $S_2$. On the other hand, when $\mathrm{Tr}(G)\neq0$, one basis element is from $S_1$ and the other two are from $S_4$. The determinant of such ensembles of trace-less matrices are shown to be quadrics, which could be hyperboloid of two sheets, hyperboloid of one sheet, ellipsoid or quadric cone for invertible $G$, whereas it is two parallel planes or a plane for singular $G_s$. Finally, the set of all the matrices $G\in M_2(\mathbb{C})$, satisfying $H^{\dagger}G = GH$, given a specific $H\in M_2(\mathbb{C})$, are shown to be describable in terms of quadratic variety., Comment: This article has been submitted for peer review to the Journal of Mathematical Physics (AIP Publishing)

This survey article, written for the Encyclopedia of Mathematical Physics, 2nd edition, is devoted to the remarkable family of operators introduced by Charles Dunkl and to their $q$-analogues discovered by Ivan Cherednik. The main focus is on the r\^ole of these operators in studying integrable many-body systems such as the Calogero-Moser and the Ruijsenaars systems. To put these constructions into a wider context, we indicate their relationship with the theory of the rational Cherednik algebras and double affine Hecke algebras. While we do not include proofs, references to the original research articles are provided, accompanied by brief historical comments., Comment: 25 pages

In this contribution we present a concise introduction to quantum entanglement in multipartite systems. After a brief comparison between bipartite systems and the simplest non-trivial multipartite scenario involving three parties, we review mathematically rigorous definitions of separability and entanglement between several subsystems, as well as their transformations and measures., Comment: Multipartite Entanglement will be a chapter in Encyclopedia of Mathematical Physics 2nd edition

These lecture notes for the course APM 351 at the University of Toronto are aimed at mathematicians and physicists alike. It is not meant as an introductory course to PDEs, but rather gives an overview of how to view and solve differential equations that are common in physics. Among others, I cover Hamilton's equations, variations of the Schr\"odinger equation, the heat equation, the wave equation and Maxwells equations., Comment: 192 pages. arXiv admin note: text overlap with arXiv:1009.0444

In this paper, we resort to the Laplace transform method in order to show its efficiency when approaching some types of fractional differential equations. In particular, we present some applications of such methods when applied to possible generalizations of certain physical problems in linear viscoelasticity and harmonic oscillators, proving that fractional calculus is well suited for the modelling and solving of problems usually treated by ordinary integer calculus, with the promissing advantages of being able to provide more accurate theoretical predictions to fit with experimental data. OBS: Article in portuguese accepted for publication at RBEF (Revista Brasileira de Ensino de F\'isica)., Comment: 20 pages, 7 figures

To explore how rigid rotation affects the thermodynamic properties of free relativistic bosons, we employ the standard imaginary time formalism of thermal field theory to calculate the free propagator of complex scalar fields under rotation. We introduce the corresponding partition function and explicitly compute it by expanding the modes in cylindrical coordinates. The resulting propagator is in full agreement with similar findings in the existing literature., Comment: 4 pages, talk presented by E. Siri at the 8th Iranian Conference on Mathematical Physics (ICMP) in Qom, Iran, on 7-8 July 2024; Submitted to Transactions in Theoretical and Mathematical Physics (TTMP)

We show how to derive an effective nonlinear dynamics, described by the Hartree-Fock equations, for fermionic quantum particles confined to a two-dimensional box and in presence of an external, uniform magnetic field. The derivation invokes the Dirac-Frenkel principle. We discuss the validity of this effective description with respect to the many-body Schr\"odinger dynamics for small times and for weak interactions, and also in regards to the number of particles., Comment: 31 pages, no figures. v2: matches published version on Open Communications in Nonlinear Mathematical Physics

In this paper we present a detailed physical analysis of the formation of the propagation transverse modes in planar dielectric waveguides using a mathematical-physics approach. We demonstrate physically that, at the wavelength scale, the pure stationary mode inside planar waveguide is described by the cosine function. Meanwhile, the sine function yields a quasi-stationary periodic mode., Comment: Accepted in Revista Mexicana de Fisica E, May 2019

In the 60's Professor Chen Ping Yang with Professor Chen Ning Yang published several seminal papers on the study of Bethe's hypothesis for various problems of physics. The works on the lattice gas model, critical behaviour in liquid-gas transition, the one-dimensional (1D) Heisenberg spin chain, and the thermodynamics of 1D delta-function interacting bosons are significantly important and influential in the fields of mathematical physics and statistical mechanics. In particular, the work on the 1D Heisenberg spin chain led to subsequent developments in many problems using Bethe's hypothesis. The method which Yang and Yang proposed to treat the thermodynamics of the 1D system of bosons with a delta-function interaction leads to significant applications in a wide range of problems in quantum statistical mechanics. The Yang and Yang thermodynamics has found beautiful experimental verifications in recent years., Comment: 5 pages + 3 figures

In these notes of six lectures on selected topics in the theory of cold, dilute Bose gases, presented at the 5th Warsaw School of Statistical Physics in June 2013, the following topics are discussed: 1) The concept of BEC, 2) the ground state energy of a dilute Bose gas with short range interactions, 3) Gross-Pitaevskii theory and BEC in trapped gases, 4) Bose gases in rotating traps and quantized vortices, and 5) strongly correlated phases in the lowest Landau level generated by rapid rotation., Comment: For the proceedings of the 5th Warsaw School of Statistical Physics, Kazimierz Dolny, June 2013

The present issue of the series <> represents the Proceedings of the Students Training Contest Olympiad in Mathematical and Theoretical Physics and includes the statements and the solutions of the problems offered to the participants. The contest Olympiad was held on May 21st-24th, 2010 by Scientific Research Laboratory of Mathematical Physics of Samara State University, Steklov Mathematical Institute of Russia's Academy of Sciences, and Moscow Institute of Physics and Technology (State University) in cooperation. The present Proceedings is intended to be used by the students of physical and mechanical-mathematical departments of the universities, who are interested in acquiring a deeper knowledge of the methods of mathematical and theoretical physics, and could be also useful for the persons involved in teaching mathematical and theoretical physics., Comment: 68 pages, Proceedings of the statements and solutions of the problems of the Students Training Contest Olympiad in Mathematical and Theoretical Physics. The subjects covered by the problems include classical mechanics, integrable nonlinear systems, probability, integral equations, PDE, quantum and particle physics, cosmology, and other areas of mathematical and theoretical physics; Ser. Modern Problems of Mathematical Physics Spec. Iss. 3. - Samara : Samara University Press, 2010

Advances in mathematical physics during the 20th century led to the discovery of a relationship between group theory and representation theory with the theory of special functions. Specifically, it was discovered that many of the special functions are (1) specific matrix elements of matrix representations of Lie groups, and (2) basis functions of operator representations of Lie algebras. By viewing the special functions in this way, it is possible to derive many of their properties that were originally discovered using classical analysis, such as generating functions, differential relations, and recursion relations. This relationship is of interest to physicists due to the fact that many of the common special functions, such as Hermite polynomials and Bessel functions, are related to remarkably simple Lie groups used in physics. Unfortunately, much of the literature on this subject remains inaccessible to undergraduate students. The purpose of this project is to research the existing literature and to organize the results, presenting the information in a way that can be understood at the undergraduate level. The primary objects of study will be the Heisenberg group and its relationship to the Hermite polynomials, as well as the Euclidean group in the plane and its relationship to the Bessel functions. The ultimate goal is to make the results relevant for undergraduate students who have studied quantum mechanics., Comment: Drexel undergraduate senior thesis in physics advised by Robert Gilmore. 43 pages, 1 figure

This book considers posing and the methods of solving simple linear boundary-value problems in classical mathematical physics. The questions encompassed include: the fundamentals of calculus of variations; one-dimensional boundary-value problems in the oscillation and heat conduction theories, with a detailed analysis of the Sturm-Liouville boundary-value problem and substantiation of the Fourier method; sample solutions of the corresponding problems in two and three dimensions, with essential elements of the special function theory. The text is designed for Physics, Engineering, and Mathematics majors., Comment: A university textbook (in Ukrainian), 380 pages, 40 figures, ISBN 978-966-190-912-9

The discrete time Vicsek model confined by a harmonic potential explains many aspects of swarm formation in insects. We have found exact solutions of this model without alignment noise in two or three dimensions. They are periodic or quasiperiodic (invariant circle) solutions with positions on a circular orbit or on several concentric orbits and exist for quantized values of the confinement. There are period 2 and period 4 solutions on a line for a range of confinement strengths and period 4 solutions on a rhombus. These solutions may have polarization one, although there are partially ordered period 4 solutions and totally disordered (zero polarization) period 2 solutions. We have explored the linear stability of the exact solutions in two dimensions using the Floquet theorem and verified the stability assignements by direct numerical simulations., Comment: 13 pages, 10 figures, RevTex

Schmidt decomposition is a powerful tool in quantum information. While Schmidt decomposition is universal for bipartite states, its not for multipartite states. In this article, we review properties of bipartite Schmidt decompositions and study which of them extend to multipartite states. In particular, Schmidt number (the number of non-zero terms in Schmidt decomposition) define an equivalence class using separable unitary transforms. We show that it is NP-complete to partition a multipartite state that attains the highest Schmidt number. In addition, we observe that purifications of a density matrix of a composite system preserves Schmidt decomposability.

Recent investigations have established the physical relevance of spatially-localized instability mechanisms in fluid dynamics and their potential for technological innovations in flow control. In this letter, we show that the mathematical problem of identifying spatially-localized optimal perturbations that maximize perturbation-energy amplification can be cast as a sparse (cardinality-constrained) optimization problem. Unfortunately, cardinality constrained optimization problems are non-convex and combinatorially hard to solve in general. To make the analysis viable within the context of fluid dynamics problems, we propose an efficient iterative method for computing sub-optimal spatially-localized perturbations. Our approach is based on a generalized Rayleigh quotient iteration algorithm followed by a variational renormalization procedure that reduces the optimality gap in the resulting solution. The approach is demonstrated on a sub-critical plane Poiseuille flow at Re = 4000, which has been a benchmark problem studied in prior investigations on identifying spatially-localized flow structures. Remarkably, we find that a subset of the perturbations identified by our method yield a comparable degree of energy amplification as their global counterparts. We anticipate our proposed analysis tools will facilitate further investigations into spatially-localized flow instabilities, including within the resolvent and input-output analysis frameworks., Comment: 9 pages, 2 figures

In this paper, we study the stability of a simple model of a Hyperloop vehicle resulting from the interaction between electromagnetic and aeroelastic forces for both constant and periodically varying coefficients (i.e., parametric excitation). For the constant coefficients, through linear stability analysis, we analytically identify three distinct regions for the physically significant equilibrium point. Further inspection reveals that the system exhibits limit-cycle vibrations in one of these regions. Using the harmonic balance method, we determine the properties of the limit cycle, thereby unravelling the frequency and amplitude that characterize the periodic oscillations of the system's variables. For the varying coefficients case, the stability is studied using Floquet analysis and Hills determinant method. The part of the stability boundary related to parametric resonance has an elliptical shape, while the remaining part remains unchanged. One of the major findings is that a linear parametric force, can suppress or amplify the parametric resonance induced by another parametric force depending on the amplitude of the former. In the context of the Hyperloop system, this means that parametric resonance caused by base excitation-in other words by the linearized parametric electromagnetic force can be suppressed by modulating the coefficient of the aeroelastic force in the same frequency. The effectiveness is highly dependent on the phase difference between the modulation and the base excitation. The origin of the suppression is attributed to the stabilizing character of the parametric aeroelastic force as revealed through energy analysis. We provide analytical expressions for the stability boundaries and for the stability's dependence on the phase shift of the modulation.

In [1] the asymptotic charges of p-form gauge theories in any dimension are studied. Here we prove an existence and uniqueness theorem for the duality map linking asymptotic electric-like charges of the dual descriptions and we give it an algebraic topology interpretation. As a result the duality map has a topological nature and ensures the charge of a description has information of the dual description. The result of the theorem could be generalized to more generic gauge theories where the gauge field is a mixed symmetry tensor leading to a deeper understanding of gauge theories, of the non-trivial charges associated to them and of the duality of their observable.

We prove that for $n = 2$ the gaskets of critical rigid O(n) loop-decorated random planar maps are $3/2$-stable maps. The case $n = 2$ thus corresponds to the critical case in random planar maps. The proof relies on the Wiener-Hopf factorisation for random walks. Our techniques also provide a characterisation of weight sequences of critical $O(2)$ loop-decorated maps., Comment: 19 pages, 2 figures

In this paper, we revisit the concept of noncommuting common causes; refute two objections raised against them, the triviality objection and the lack of causal explanatory force; and explore how their existence modifies the EPR argument. More specifically, we show that 1) product states screening off all quantum correlations do not compromise noncommuting common causal explanations; 2) noncommuting common causes can satisfy the law of total probability; 3) perfect correlations can have indeterministic noncommuting common causes; and, as a combination of the above claims, 4) perfect correlations can have noncommuting common causes which are both nontrivial and satisfy the law of total probability., Comment: 34+7 pages, 2 figures

We study the nonlinear inverse source problem of detecting, localizing and identifying unknown accidental disturbances on forced and damped transmission networks. A first result is that strategic observation sets are enough to guarantee detection of disturbances. To localize and identify them, we additionally need the observation set to be absorbent. If this set is dominantly absorbent, then detection, localization and identification can be done in "quasi real-time". We illustrate these results with numerical experiments.