Mathematical Literature Research Articles

Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence

We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d $\mathcal N=2$ quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an anisotropic/elliptic superspin chain, and the stable envelopes compute the $R$-matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic $\mathfrak{sl}(1|1)$ spin chain with fundamental representations using the corresponding 3d $\mathcal N=2$ SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on $I \times \mathbb E$ for an interval $I$ and an elliptic curve $\mathbb E$ compute the elliptic stable envelopes, and in turn the geometric elliptic $R$-matrix, of the anisotropic $\mathfrak{sl}(1|1)$ spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the $R$-matrix. The reduction to 2d gives the K-theoretic stable envelopes and the trigonometric $R$-matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational $R$-matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature.

Towards a probabilistic foundation of relativistic quantum theory: the one-body Born rule in curved spacetime

In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.

What are explanatory proofs in mathematics and how can they contribute to teaching and learning?

This paper will examine several simple examples (drawn from the mathematics literature) where there are multiple proofs of the same theorem, but only some of these proofs are widely regarded by mathematicians as explanatory. These examples will motivate an account of explanatory proofs in mathematics. Along the way, the paper will discuss why deus ex machina proofs are not explanatory, what a mathematical coincidence is, and how a theorem's proper setting reflects the naturalness of various mathematical kinds. The paper will also investigate how context influences which features of a theorem are salient and consequently which proofs are explanatory. The paper will discuss several ways in which explanatory proofs can contribute to teaching and learning, including how shifts in context (and hence in a proof’s explanatory power) can be exploited in a classroom setting, leading students to dig more deeply into why some theorem holds. More generally, the paper will examine how “Why?” questions operate in mathematical thinking, teaching, and learning.

Discontinuous Galerkin approximations of the heterodimer model for protein–protein interaction

Mathematical models of protein–protein dynamics, such as the heterodimer model, play a crucial role in understanding many physical phenomena, e.g., the progression of some neurodegenerative diseases. This model is a system of two semilinear parabolic partial differential equations describing the evolution and mutual interaction of biological species. This article presents and analyzes a high-order discretization method for the numerical approximation of the heterodimer model capable of handling complex geometries. In particular, the proposed numerical scheme couples a Discontinuous Galerkin method on polygonal/polyhedral grids for space discretization, with a θ-method for time integration. This work presents novelties and progress with respect to the mathematical literature, as stability and a-priori error analysis for the heterodimer model are carried out for the first time. Several numerical tests are performed, which demonstrate the theoretical convergence rates, and show good performances of the method in approximating traveling wave solutions as well as its flexibility in handling complex geometries. Finally, the proposed scheme is tested in a practical test case stemming from neuroscience applications, namely the simulation of the spread of α-synuclein in a realistic test case of Parkinson’s disease in a two-dimensional sagittal brain section geometry reconstructed from medical images.

One class of solutions to the equations of ideal plasticity

Much attention is given to the study and solution of nonlinear differential equations in the modern mathematical literature. Despite this, there are not many methods for researching and solving such equations. These are point and contact transformations of equations, various methods of separating variables, the method of differential connections, the search for various symmetries and their use to construct solutions, as well as conservation laws. The paper considers a nonlinear differential equation describing the plastic flow of a prismatic rod. A group of point symmetries is found for this equation. The optimal system of one-dimensional subalgebras is calculated. Conservation laws corresponding to Noetherian symmetries are given, and it is also shown that there are infinitely many non-Noetherian conservation laws. Several new invariant solutions of rank one, i. e. depending on one independent variable, are constructed. It is shown how classes of new solutions can be constructed from two exact solutions, passing to a linear equation. Thus, in this short article, almost all methods of modern research of nonlinear differential equations are involved.

Spatial information allows inference of the prevalence of direct cell-to-cell viral infection.

The role of direct cell-to-cell spread in viral infections-where virions spread between host and susceptible cells without needing to be secreted into the extracellular environment-has come to be understood as essential to the dynamics of medically significant viruses like hepatitis C and influenza. Recent work in both the experimental and mathematical modelling literature has attempted to quantify the prevalence of cell-to-cell infection compared to the conventional free virus route using a variety of methods and experimental data. However, estimates are subject to significant uncertainty and moreover rely on data collected by inhibiting one mode of infection by either chemical or physical factors, which may influence the other mode of infection to an extent which is difficult to quantify. In this work, we conduct a simulation-estimation study to probe the practical identifiability of the proportion of cell-to-cell infection, using two standard mathematical models and synthetic data that would likely be realistic to obtain in the laboratory. We show that this quantity cannot be estimated using non-spatial data alone, and that the collection of data which describes the spatial structure of the infection is necessary to infer the proportion of cell-to-cell infection. Our results provide guidance for the design of relevant experiments and mathematical tools for accurately inferring the prevalence of cell-to-cell infection in in vitro and in vivo contexts.

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In this paper we propose a physical derivation of a 4d conjectural duality for USp(2N) with an anti-symmetric rank-two tensor and fundamental flavors, in presence of a non-trivial superpotential. This duality has been conjectured as a consequence of an exact identity between the superconformal indices of the two phases, proved in the mathematical literature. Here we show that the duality can be derived by a combined sequence of known dualities, deconfinement of tensor matter, RG flow and Higgsing. Furthermore, by following these steps on the superconformal index, we provide an alternative derivation of the integral identity as well.

Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle

The connection between absorbing boundary conditions and hard walls is well established in the mathematical literature for a variety of stochastic models, including for instance the Brownian motion. In this paper we explore this duality for a different type of process which is of particular interest in physics and biology, namely the run-tumble-particle, a toy model of active particle. For a one-dimensional run-and-tumble particle (RTP) subjected to an arbitrary external force, we provide a duality relation between the exit probability, i.e. the probability that the particle exits an interval from a given boundary before a certain time t, and the cumulative distribution of its position in the presence of hard walls at the same time t. We show this relation for a RTP in the stationary state by explicitly computing both quantities. At finite time, we provide a derivation using the Fokker–Planck equation. All the results are confirmed by numerical simulations.

Gopal Pandey's Contributions to Nepali Mathematics through "Wyakta Chandrika"

This study reviews the pioneering work of Gopal Pandey and his influential textbook "Wyakta Chandrika" in shaping Mathematical education in Nepal. Beginning with an overview of the historical context of education in Nepal, including the establishment of formal schooling and the contributions of early scholars, the research focuses on Pandey's life and his journey as a Mathematician. It examines the content and significance of "Wyakta Chandrika," highlighting its role in imparting Mathematical knowledge and fostering cultural awareness among students. Additionally, the study analyzes the impact of Pandey's innovative methods, such as the trinomial method for finding cube roots, and explores the evolution of Mathematical literature in Nepal following the publication of "Wyakta Chandrika." Through an analysis of the book's contents and its implications for Mathematical education, this research provides insights into the enduring legacy of Pandey's work and its continued relevance in contemporary Nepali Mathematics.

A Mathematics Intervention in Kindergarten Using Shel Silverstein’s “The Missing Piece Meets the Big O”: Early Childhood Educators’ Perceptions on a Teaching Framework Integrating Literature in Mathematics-Related Concepts

Teaching mathematics in kindergarten necessitates interdisciplinary methods. By merging different disciplines, educators should create an environment conducive to learning, equipped with diverse materials, which allows children to engage with mathematical concepts through play, observation, and read-aloud stories. The aim of the present study is to explore early childhood educators’ perceptions of a teaching framework integrating literature in mathematics-related concepts. The hypothesis is that a mathematics intervention using Shel Silverstein’s “The missing piece meets the big O” could affect the teaching of mathematics in kindergarten. In this action research, twenty-four (n=24) early childhood educators participated by implementing the intervention and completing a semi-structured questionnaire. Data analysis included descriptive and inferential statistical analysis that was used to compare the mean scores of the two parts of the questionnaire before and after the intervention. Results show that educators were more positively inclined to use picturebooks as a stimulus for approaching mathematics-related concepts and creating a demarcated and inviting space for learning mathematics. Their answers lead to the assumption that a clear and concise framework, which integrates literature in mathematics and offers specific phases with activities would facilitate their teaching process. The notable deviation in their answers before and after the intervention demonstrates the extent to which the intervention affected their perceptions. Discussion of findings highlights the need for the Greek curriculum for preschool education to offer methodological recommendations for the integration of picturebooks in the teaching of mathematics.

(Non-)unitarity of strictly and partially massless fermions on de Sitter space II: an explanation based on the group-theoretic properties of the spin-3/2 and spin-5/2 eigenmodes

Abstract In our previous article (Letsios 2023 J. High Energy Phys. JHEP05(2023)015), we showed that the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields, on N-dimensional ( N ⩾ 3 ) de Sitter (dS) spacetime (dS N ) are non-unitary unless N = 4. The (non-)unitarity was demonstrated by simply observing that there is a (mis-)match between the representation-theoretic labels that correspond to the unitary irreducible representations (UIR’s) of the dS algebra spin ( N , 1 ) and the ones corresponding to the space of eigenmodes of the field theories. In this paper, we provide a technical representation-theoretic explanation for this fact by studying the (non-)existence of positive-definite, dS invariant scalar products for the spin-3/2 and spin-5/2 strictly/partially massless eigenmodes on dS N ( N ⩾ 3 ). Our basic tool is the examination of the action of spin ( N , 1 ) generators on the space of eigenmodes, leading to the following findings. For odd N, any dS invariant scalar product is identically zero. For even N > 4, any dS invariant scalar product must be indefinite. This gives rise to positive-norm and negative-norm eigenmodes that mix with each other under spin ( N , 1 ) boosts. In the N = 4 case, the positive-norm sector decouples from the negative-norm sector and each sector separately forms a UIR of spin(4, 1). Our analysis makes extensive use of the analytic continuation of tensor-spinor spherical harmonics on the N-sphere (S N ) to dS N and also introduces representation-theoretic techniques that are absent from the mathematical physics literature on half-odd-integer-spin fields on dS N .

Four Families of Summation Formulas for 4F3(1) with Application

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F34(1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F23(x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results.

O nouă abordare a problemei trisecției unghiului

The article presents new formulas in the problem of trisection of the angle with the help of conics and thanks to this fact a new vision is proposed for solving the problem of trisection of the angle with the ungraded ruler and the compass, using the ellipse drawn in the plane, not solved so far. Currently, the trisection of the angle with the help of the parabola (R. Decarts) and with the help of the hyperbola (M. Chasles) are described in the mathematical literature. The trisection of the angle with the help of the ellipse drawn in the plane is described in symbols and textually since not all readers know the symbols used in constructive geometry. The content of this article contributes to solving the angle trisection problem with the ungraded ruler and the compass. The information in the article is of interest in the field of constructive geometry.

Concentration and non-concentration of eigenfunctions of second-order elliptic operators in layered media

This work is concerned with operators of the type A=-\tilde{c}\Delta acting in domains \Omega\coloneq \Omega'\times (0,H)\subseteq\mathbb{R}^{d}\times\mathbb{R}_{+}. The diffusion coefficient \tilde{c}>0 depends on one coordinate y\in (0,H) and is bounded but may be discontinuous. This corresponds to the physical model of “layered media,” appearing in acoustics, elasticity, optical fibers… . Dirichlet boundary conditions are assumed. In general, for each \varepsilon>0, the set of eigenfunctions is divided into a disjoint union of three subsets: \mathfrak{F}_{NG} (non-guided), \mathfrak{F}_{G} (guided) and \mathfrak{F}_{\textup{res}} (residual). The residual set shrinks as \varepsilon \to 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of “layered media” enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y)\in C^{2} but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in \mathfrak{F}_{NG} ) are oscillatory. However, this fact by itself does not exclude the possibility of “flattening out” of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a “minimal amplitude hypothesis.” However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem.

Cutting-Edge Computational Approaches for Approximating Nonlocal Variable-Order Operators

This study presents an algorithmically efficient approach to address the complexities associated with nonlocal variable-order operators characterized by diverse definitions. The proposed method employs integro spline quasi interpolation to approximate these operators, aiming for enhanced accuracy and computational efficiency. We conduct a thorough comparison of the outcomes obtained through this approach with other established techniques, including finite difference, IQS, and B-spline methods, documented in the applied mathematics literature for handling nonlocal variable-order derivatives and integrals. The numerical results, showcased in this paper, serve as a compelling validation of the notable advantages offered by our innovative approach. Furthermore, this study delves into the impact of selecting different variable-order values, contributing to a deeper understanding of the algorithm’s behavior across a spectrum of scenarios. In summary, this research seeks to provide a practical and effective solution to the challenges associated with nonlocal variable-order operators, contributing to the applied mathematics literature.

Review of Mathematical Modeling Research: A Descriptive Content Analysis Study

Recently, mathematical models and modeling practices have become popular in associating mathematics with real-life problems and, hence understanding this relationship. Accordingly, the mathematical modeling skill has been adopted in the standards and by researchers. Understanding the use of mathematical modeling in the learning and teaching processes of mathematics education will contribute to the future of the field. This study aimed to review the trends in mathematical modeling literature using leading research studies. This study reviewed the various types of studies indexed in the Web of Science database between 2000 and 2021 regarding how they addressed modeling. As well as mathematical modeling approaches used, the studies were reviewed in terms of basic characteristics such as publication year, sample, and research method. We evaluated studies using a form developed by the researchers, and the study revealed an increase in the number of studies over the years, and the studies were conducted mostly with pre-service teachers. In addition, we observed that research studies employed mostly small samples to closely monitor the modeling process.

Evaluation of school achievements in adolescents with primary hypertension.

Primary hypertension (HT) is a global public health problem with increasing prevalence in recent years. HT may have caused decreased neurocognitive functions and learning difficulties. In this study, clinical characteristics of adolescents with primary HT were examined and the relationship between semester grade point average (GPA) and HT was evaluated. This is an observational, cross-sectional, descriptive study conducted on adolescents with primary HT attending high school. Patient records (number of hospital visits, HT-related complaints, blood pressure measurements, and laboratory tests) were evaluated retrospectively. End-of-semester report card grades of Mathematics, Turkish Language and Literature and English courses were noted, and compared with the clinical characteristics of the patients. The study included 83 patients with a mean age of 15.6±1.2 years. Patients with higher body mass index had lower grades in Mathematics (p=0.007) and Turkish Language and Literature (p=0.004). Patients with HT-related symptoms such as headache, epistaxis and palpitations had lower GPAs for all courses. Also, patients with hyperuricemia or proteinuria had lower semester GPAs compared to patients with normal serum uric acid levels or without proteinuria (p<0.05). GPAs for Mathematics (p=0.000) and Turkish Language and Literature (p=0.006) decrease as the number of hospital visits increases. HT may cause not only cardiovascular complications but also decreased neurocognitive functions through various mechanisms and may have a negative impact on academic skills. Therefore, HT should be followed up with a multidisciplinary approach and intensive efforts should be made to approach the goal of normotension.

Implementing measurement error models with mechanistic mathematical models in a likelihood-based framework for estimation, identifiability analysis and prediction in the life sciences.

Throughout the life sciences, we routinely seek to interpret measurements and observations using parametrized mechanistic mathematical models. A fundamental and often overlooked choice in this approach involves relating the solution of a mathematical model with noisy and incomplete measurement data. This is often achieved by assuming that the data are noisy measurements of the solution of a deterministic mathematical model, and that measurement errors are additive and normally distributed. While this assumption of additive Gaussian noise is extremely common and simple to implement and interpret, it is often unjustified and can lead to poor parameter estimates and non-physical predictions. One way to overcome this challenge is to implement a different measurement error model. In this review, we demonstrate how to implement a range of measurement error models in a likelihood-based framework for estimation, identifiability analysis and prediction, called profile-wise analysis. This frequentist approach to uncertainty quantification for mechanistic models leverages the profile likelihood for targeting parameters and understanding their influence on predictions. Case studies, motivated by simple caricature models routinely used in systems biology and mathematical biology literature, illustrate how the same ideas apply to different types of mathematical models. Open-source Julia code to reproduce results is available on GitHub.

Secondary K-Range Symmetric Neutrosophic Fuzzy Matrices

This paper introduces and explores the concept of secondary k-Range Symmetric (RS) Neutrosophic Fuzzy Matrices (NFM) and establishes its properties and relationships with other symmetric and secondary symmetric NFMs. The study defines secondary k-RS NFMs and provides insightful numerical examples to illustrate their characteristics. The paper investigates the interconnections among s-k-RS, s-RS, k-RS, and RS NFMs, discuss on their mutual relations. Additionally, the necessary and sufficient conditions for a given NFM to qualify as a s-k-RS NFM are identified. The research demonstrates that k-symmetry implies k-RS, and vice versa, contributing to a comprehensive understanding between different types of symmetries in NFMs. Graphical representations of RS, column symmetric, and kernel symmetric adjacency and incidence NFMs are presented, unveiling intriguing patterns and relationships. While every adjacency NFM is symmetric, range symmetric, column symmetric, and kernel symmetric, the incidence matrix satisfies only kernel symmetric conditions. The study further establishes that every range symmetric adjacency NFM is a kernel symmetric adjacency NFM, though the converse does not hold in general. The existence of multiple generalized inverses of NFMs in Fn is explored, with additional equivalent conditions for certain g-inverses of s-κ-RS NFMs to retain the s-κ-RS property. We conclude by characterizing the generalized inverses belonging to specific sets {1, 2}, {1, 2, 3}, and {1, 2, 4} of s-k-RS NFMs, providing a comprehensive framework for understanding the structure and properties of secondary k-Range Symmetric Neutrosophic Fuzzy Matrices. This research contributes to the mathematical literature by introducing a novel class of NFMs and establishing their fundamental properties and relationships, presenting new perspectives on matrix theory in the context of neutrosophic fuzzy logic.

English to Arabic Machine Translation of Mathematical Documents

This paper is about the development of a machine translation system tailored specifically for LATEX mathematical documents. The system focuses on translating English LATEX mathematical documents into Arabic LATEX, catering to the growing demand for multilingual accessibility in scientific and mathematical literature. With the vast proliferation of LATEX mathematical documents the need for an efficient and accurate translation system has become increasingly essential. This paper addresses the necessity for a robust translation tool that enables seamless communication and comprehension of complex mathematical content across language barriers. The proposed system leverages a Transformer model as the core of the translation system, ensuring enhanced accuracy and fluency in the translated Arabic LATEX documents. Furthermore, the integration of RyDArab, an Arabic mathematical TEX extension, along with a rule-based translator for Arabic mathematical expressions, contributes to the precise rendering of complex mathematical symbols and equations in the translated output. The paper discusses the architecture, methodology, of the developed system, highlighting its efficacy in bridging the language gap in the domain of mathematical documentation.