**Preface to "Mathematical Physics II"**

The mysterious charm of Mathematical Physics is beautifully represented in the celebrated 1960 paper of Eugene Wigner "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." It is indeed hard for us to understand the astonishing appropriateness of the language of mathematics for the formulation of the laws of physics and its capability to do predictions, appropriateness that emerged immediately at the beginning of the scientific thought and was splendidly depicted by Galileo: "The grand book, the Universe, is written in the language of Mathematics." Paraphrasing the words of Bertrand Russell, in this marriage the supreme beauty, cold and austere, of Mathematics complements the supreme beauty, warm and engaging, of Physics. This book, which consists of nine articles, gives a flavor of this beauty and covers various topics related to physics and engineering. A brief outline of these topics is given hereafter.

The study of free probability in certain probability spaces induced by functions on *p*-adic number fields is here a very interesting example of the application of *p*-adic mathematical methods for modeling physical phenomena. Within the quantum information processing framework is presented the analysis of a non-commutative measure of quantum discord in the two-qubit case. The Riemann–Hilbert problem plays relevant but different roles in two papers. In one paper, the Riemann–Hilbert problem, formulated with respect to the spectral parameter, and the prolongation structure theory are used to analyze the modified nonlinear Schrodinger equation. ¨ In the other case, the Riemann–Hilbert structure emerges from the holomorphic extension of certain Legendre expansions, leading thus to an explicit connection between spherical Laplace transform and non-Euclidean Fourier transform. Remaining in the spectral analysis field, the study of a deformed wave equation, with the Laplacian being replaced by a differential-difference Dunkl operator, shows its relation with a generalized Fourier transform and the non-existence of the related Huygens principle. The Gibbs phenomenon is then the subject of a paper where the primary tool of analysis is the representation of suitable functions in terms of dual tight framelets. Finally, numerical analysis and optimization methods are the main mathematical devices used to study equations that are relevant in the study of material properties, such as thermomechanical performances, flaw dynamics, and bearing capacity of structures.

In conclusion, as Editor of this Special Issue, I wish to thank the authors of the articles for their valuable contributions, the referees for their precious reviews, and Ms. Julie Shi and Ms. Grace Wang of MDPI for their kind assistance.

> **Enrico De Micheli** *Editor*

### *Article* **Primes in Intervals and Semicircular Elements Induced by** *p***-Adic Number Fields** Q *p* **over Primes** *p*

### **Ilwoo Cho 1,\* and Palle Jorgensen 2**


Received: 11 December 2018; Accepted: 15 February 2019; Published: 19 February 2019

**Abstract:** In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach ∗-probability space (LS, *τ*<sup>0</sup>) induced by measurable functions on *p*-adic number fields Q*p* over primes *p*. In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable "truncated" linear functionals on LS.

**Keywords:** free probability; primes; *p*-adic number fields; Banach ∗-probability spaces; weighted-semicircular elements; semicircular elements; truncated linear functionals

**MSC:** 05E15; 11G15; 11R47; 11R56; 46L10; 46L54; 47L30; 47L55
