Applications of Number Theory to the Sciences and Mathematics

A special issue of AppliedMath (ISSN 2673-9909).

Deadline for manuscript submissions: closed (30 April 2023) | Viewed by 12805

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Department of Chemistry, Concordia College, Moorhead, MN 56562, USA
Interests: physical chemistry; spectroscopy; liquids; physical neuroscience; physical chemistry
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Special Issue Information

Dear Colleagues,

The scope of this Special Issue on applications of number theory to sciences and mathematics includes the application of number theory (all subfields) to problems of scientific interest. This includes physics, physical mathematics, chemistry, biology, neuroscience, psychology, etc. Related fields, such as combinatorics, group theory, graph theory, complex analysis, etc., are also included in the call if they use techniques found in number theory. In addition to primary research articles, review papers and manuscripts that bring pedagogical value in helping connect scientists with important concepts in number theory are invited.

The rationale for this Special Issue is the following. Those scientists working on the more physical side of the spectrum are often well-training in applied analysis, differential equations, and linear algebra. Meanwhile, those working on the more life science side of the spectrum often have a strong background in statistics. Either way, it is often the case that ideas from number theory are lacking in the mathematical preparation and, consequently, in the applied setting. The goal of this Special Issue is to collect papers that show the application of number theory in the broad sense to solve problems in science and to help scientists learn about methods arising in number theory.

Prof. Dr. Darin J. Ulness
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. AppliedMath is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1000 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (6 papers)

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16 pages, 427 KiB  
Article
Taking Rational Numbers at Random
by Nicola Cufaro Petroni
AppliedMath 2023, 3(3), 648-663; https://doi.org/10.3390/appliedmath3030034 - 1 Sep 2023
Viewed by 1363
Abstract
In this article, some prescriptions to define a distribution on the set Q0 of all rational numbers in [0,1] are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers asymptotically [...] Read more.
In this article, some prescriptions to define a distribution on the set Q0 of all rational numbers in [0,1] are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers asymptotically equiprobable in a suitable sense. In particular, it will be shown that in the said limit—albeit no absolutely continuous uniform distribution can be properly defined in Q0—the probability allotted to every single qQ0 asymptotically vanishes, while that of the subset of Q0 falling in an interval [a,b]Q0 goes to ba. We finally present some hints to complete sequencing without repeating the numbers in Q0 as a prerequisite to laying down more distributions on it. Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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7 pages, 316 KiB  
Article
A Novel Algebraic System in Quantum Field Theory
by Abdulaziz D. Alhaidari and Abdallah Laradji
AppliedMath 2023, 3(2), 461-467; https://doi.org/10.3390/appliedmath3020024 - 24 May 2023
Viewed by 1124
Abstract
An algebraic system is introduced which is very useful for performing scattering calculations in quantum field theory. It is the set of all real numbers greater than or equal to −m2 with parity designation and a special rule for addition and [...] Read more.
An algebraic system is introduced which is very useful for performing scattering calculations in quantum field theory. It is the set of all real numbers greater than or equal to −m2 with parity designation and a special rule for addition and subtraction, where m is the rest mass of the scattered particle. Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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12 pages, 686 KiB  
Article
Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra
by Paul C. Arpin, Mihail Popa and Daniel B. Turner
AppliedMath 2022, 2(4), 675-686; https://doi.org/10.3390/appliedmath2040039 - 1 Dec 2022
Cited by 2 | Viewed by 1959
Abstract
The motions of nuclei in a molecule can be mathematically described by using normal modes of vibration, which form a complete orthonormal basis. Each normal mode describes oscillatory motion at a frequency determined by the momentum of the nuclei. Near equilibrium, it is [...] Read more.
The motions of nuclei in a molecule can be mathematically described by using normal modes of vibration, which form a complete orthonormal basis. Each normal mode describes oscillatory motion at a frequency determined by the momentum of the nuclei. Near equilibrium, it is common to apply the quantum harmonic-oscillator model, whose eigenfunctions intimately involve combinatorics. Each electronic state has distinct force constants; therefore, each normal-mode basis is distinct. Duschinsky proposed a linearized approximation to the transformation between the normal-mode bases of two electronic states using a rotation matrix. The rotation angles are typically obtained by using quantum-chemical computations or via gas-phase spectroscopy measurements. Quantifying the rotation angles in the condensed phase remains a challenge. Here, we apply a two-dimensional harmonic model that includes a Duschinsky rotation to condensed-phase femtosecond coherence spectra (FCS), which are created in transient–absorption spectroscopy measurements through impulsive excitation of coherent vibrational wavepackets. Using the 2D model, we simulate spectra to identify the signatures of Duschinsky rotation. The results suggest that peak multiplicities and asymmetries may be used to quantify the rotation angle, which is a key advance in condensed-phase molecular spectroscopy. Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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40 pages, 7655 KiB  
Article
Continued Fractions and Probability Estimations in Shor’s Algorithm: A Detailed and Self-Contained Treatise
by Johanna Barzen and Frank Leymann
AppliedMath 2022, 2(3), 393-432; https://doi.org/10.3390/appliedmath2030023 - 18 Jul 2022
Cited by 2 | Viewed by 2343
Abstract
Shor’s algorithm for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing to text books on number [...] Read more.
Shor’s algorithm for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing to text books on number theory. In this contribution, we present the relevant results and proofs from the theory of continued fractions in detail (even in more detail than in text books), filling the gap to allow a complete comprehension of Shor’s algorithm. Similarly, we provide a detailed computation of the estimation of the probability that convergents will provide the period required for determining a prime factor. Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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33 pages, 4737 KiB  
Article
Structure of Triangular Numbers Modulo m
by Darin J. Ulness
AppliedMath 2022, 2(3), 326-358; https://doi.org/10.3390/appliedmath2030020 - 1 Jul 2022
Viewed by 2199
Abstract
This work focuses on the structure and properties of the triangular numbers modulo m. The most important aspect of the structure of these numbers is their periodic nature. It is proven that the triangular numbers modulo m forms a 2m-cycle [...] Read more.
This work focuses on the structure and properties of the triangular numbers modulo m. The most important aspect of the structure of these numbers is their periodic nature. It is proven that the triangular numbers modulo m forms a 2m-cycle for any m. Additional structural features and properties of this system are presented and discussed. This discussion is aided by various representations of these sequences, such as network graphs, and through discrete Fourier transformation. The concept of saturation is developed and explored, as are monoid sets and the roles of perfect squares and nonsquares. The triangular numbers modulo m has self-similarity and scaling features which are discussed as well. Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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6 pages, 350 KiB  
Technical Note
On One Problem of the Nonlinear Convex Optimization
by Robert Vrabel
AppliedMath 2022, 2(4), 512-517; https://doi.org/10.3390/appliedmath2040030 - 21 Sep 2022
Viewed by 1647
Abstract
In this short paper, we study the problem of traversing a crossbar through a bent channel, which has been formulated as a nonlinear convex optimization problem. The result is a MATLAB code that we can use to compute the maximum length of the [...] Read more.
In this short paper, we study the problem of traversing a crossbar through a bent channel, which has been formulated as a nonlinear convex optimization problem. The result is a MATLAB code that we can use to compute the maximum length of the crossbar as a function of the width of the channel (its two parts) and the angle between them. In case they are perpendicular to each other, the result is expressed analytically and is closely related to the astroid curve (a hypocycloid with four cusps). Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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