Notes on Higher-Spin Diffeomorphisms

Higher-spin diffeomorphisms are to higher-order differential operators what diffeomorphisms are to vector fields. Their rigorous definition is a challenging mathematical problem which might predate a better understanding of higher-spin symmetries and interactions. Several yes-go and no-go results on higher-spin diffeomorphisms are collected from the mathematical literature in order to propose a generalisation of the algebra of differential operators on which higher-spin diffeomorphisms are well-defined. This work is dedicated to the memory of Christiane Schomblond, who taught several generations of Belgian physicists the formative rigor and delicate beauty of theoretical physics.

The Integral Mittag-Leffler, Whittaker and Wright Functions

Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are expressed as generalized hypergeometric functions but also in terms of elementary and special functions. The behavior of some of the new integral functions is presented in graphical form. By using the MATHEMATICA program to obtain infinite sums that define the Mittag-Leffler, Whittaker, and Wright functions and also their corresponding integral functions, these functions and many new Laplace transforms of them are also reported in the Appendices for integral and fractional values of parameters.

Some Instructive Mathematical Errors

We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructiveerrors that have been detected in the author's own published papers.

Generalized Optical Theorem and Point Sources

A simple derivation of the general form of the optical theorem (GOT) is given for the case of a conservative scatterer in a homogeneous lossless medium, suitable for describing point sources and an observation region close to the scatterer. The presentation is based on the use of the operator approach and scalar wave equation in the limit of vanishingly small absorption. This approach does not require asymptotic estimates of rapidly oscillating integrals, does not use the integration of fluxes, which leads to the loss of information about the energy conservation law, and allows a natural generalization to the case of polarized radiation, as well as more complex multi-part fields. Such GOT generalizes the results known in the mathematical literature for models to the case of any conservative (real) scattering potential and arbitrary sources.

Symmetry, Antisymmetry, and Chirality: Use and Misuse of Terminology

We outline the need for rigorous and consensual definitions in the field of symmetry, in particular about chirality. We provide examples of confusing use of such terminology in the mathematical literature and in the physics literature. In particular, we prove that an antisymmetric function is symmetric for a wide class of metrics. It may be either direct-symmetric or achiral or both direct-symmetric and achiral.

Non-compact superconformal field theory and mock modular forms

One of interesting issues in two-dimensional superconformal field theories is the existence of anomalous modular transformation properties appearing in some non-compact superconformal models, corresponding to the “mock modularity” in mathematical literature. I review a series of my studies on this issue in collaboration with T. Eguchi, mainly focusing on T. Eguchi and Y. Sugawara, J. High Energy Phys. 1103, 107 (2011); J. High Energy Phys. 1411, 156 (2014); and Prog. Theor. Exp. Phys. 2016, 063B02 (2016).

On weak solutions to a dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases

In this paper, we consider a compressible dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases. We prove that the set of weak solutions is stable, meaning that any sequence of weak solutions contains a (weakly) convergent subsequence whose limit is again a weak solution to the original system. Such type of results is usually considered as the most essential step to the proof of the existence of weak solutions. This is the first result of this type in the mathematical literature. Nevertheless, the construction of weak solutions to this system however remains still an (difficult) open problem.

Quasi-compactness of linear operators on Banach spaces: New properties and application to Markov chains

The purpose of this paper is to study the notion of quasi-compact linear operators acting in a Banach space. This class of operators contains the set of compact, polynomially compact, quasi-nilpotent and that of all Riesz operators. We show the equivalence between different definitions of quasi-compactness known in the mathematical literature and we present several general theorems about quasi-compact endomorphisms: stability under algebraic operations, extension of Schauder theorem and the Fredholm alternative. We also study the question of existence of invariant subspaces and we examine the class of semigroups for quasi-compact operators. The obtained results are used to describe Markov chains.

Application-oriented mathematical algorithms for group testing

Group testing is a widely used protocol which aims to test a small group of people to identify whether at least one of them is infected. It is particularly efficient if the infection rate is low, because it only requires a single test if everybody in the group is negative. The most efficient use of group testing is a complex mathematical question. However, the answer highly depends on practical parameters and restrictions, which are partially ignored by the mathematical literature. This paper aims to offer practically efficient group testing algorithms, focusing on the current COVID-19 epidemic.

Stabilization of third order differential equation by delay distributed feedback control with unbounded memory

There are almost no results on the exponential stability of differential equations with unbounded memory in mathematical literature. This article aimes to partially fill this gap. We propose a new approach to the study of stability of integro-differential equations with unbounded memory of the following forms $$\begin{array}{} \begin{split} \displaystyle x'''(t)+\sum_{i=1}^{m}\int\limits_{t-\tau_{i}(t)}^{t}b_{i}(t)\text{e}^{-\alpha _{i}(t-s) }x(s)\text{d} s &=0, \\ x'''(t)+\sum_{i=1}^{m}\int\limits_{0}^{t-\tau _{i}(t)}b_{i}(t)\text{e}^{-\alpha _{i}(t-s) }x(s)\text{d} s &= 0, \end{split} \end{array}$$ with measurable essentially bounded bi(t) and τi(t), i = 1, …, m. We demonstrate that, under certain conditions on the coefficients, integro-differential equations of these forms are exponentially stable if the delays τi(t), i = 1, …, m, are small enough. This opens new possibilities for stabilization by distributed input control. According to common belief this sort of stabilization requires first and second derivatives of x. Results obtained in this paper prove that this is not the case.