Mathematical Structure Research Articles

Geometric Relational Framework for General‐Relativistic Gauge Field Theories

AbstractIt is recalled how relationality arises as the core insight of general‐relativistic gauge field theories from the articulation of the generalized hole and point‐coincidence arguments. Hence, a compelling case for a manifestly relational framework ensues naturally. A formulation for such a framework is proposed, based on a significant development of the dressing field method of symmetry reduction. A version for the group of automorphisms of a principal bundle over a manifold is first developed, as it is the most natural and elegant, and as hosts all the mathematical structures relevant to general‐relativistic gauge field theory. However, as the standard formulation is local, on , the relational framework for local field theory is then developed. The generalized point‐coincidence argument is manifestly implemented, whereby the physical field‐theoretical degrees of freedoms co‐define each other and define, coordinatize, the physical spacetime itself. Applying the framework to General Relativity, relational Einstein equations are obtained, encompassing various notions of “scalar coordinatization” à la Kretschmann–Komar and Brown–Kuchař.

Optimal performance design of bat algorithm: An adaptive multi‐stage structure

AbstractThe bat algorithm (BA) is a metaheuristic algorithm for global optimisation that simulates the echolocation behaviour of bats with varying pulse rates of emission and loudness, which can be used to find the globally optimal solutions for various optimisation problems. Knowing the recent criticises of the originality of equations, the principle of BA is concise and easy to implement, and its mathematical structure can be seen as a hybrid particle swarm with simulated annealing. In this research, the authors focus on the performance optimisation of BA as a solver rather than discussing its originality issues. In terms of operation effect, BA has an acceptable convergence speed. However, due to the low proportion of time used to explore the search space, it is easy to converge prematurely and fall into the local optima. The authors propose an adaptive multi‐stage bat algorithm (AMSBA). By tuning the algorithm's focus at three different stages of the search process, AMSBA can achieve a better balance between exploration and exploitation and improve its exploration ability by enhancing its performance in escaping local optima as well as maintaining a certain convergence speed. Therefore, AMSBA can achieve solutions with better quality. A convergence analysis was conducted to demonstrate the global convergence of AMSBA. The authors also perform simulation experiments on 30 benchmark functions from IEEE CEC 2017 as the objective functions and compare AMSBA with some original and improved swarm‐based algorithms. The results verify the effectiveness and superiority of AMSBA. AMSBA is also compared with eight representative optimisation algorithms on 10 benchmark functions derived from IEEE CEC 2020, while this experiment is carried out on five different dimensions of the objective functions respectively. A balance and diversity analysis was performed on AMSBA to demonstrate its improvement over the original BA in terms of balance. AMSBA was also applied to the multi‐threshold image segmentation of Citrus Macular disease, which is a bacterial infection that causes lesions on citrus trees. The segmentation results were analysed by comparing each comparative algorithm's peak signal‐to‐noise ratio, structural similarity index and feature similarity index. The results show that the proposed BA‐based algorithm has apparent advantages, and it can effectively segment the disease spots from citrus leaves when the segmentation threshold is at a low level. Based on a comprehensive study, the authors think the proposed optimiser has mitigated the main drawbacks of the BA, and it can be utilised as an effective optimisation tool.

A Comparative Study: Domination Parameters and Physico-Chemical Properties of Benzenoid Hydrocarbons

In graph theory, a graph is a mathematical structure that consists of a set of vertices and a set of edges connecting those pairs of vertices. Let G = ( V , E ) be a graph, a dominating set for G is a subset D ⊆ V of vertices such that every vertex in V ∖ D has at least one neighbor in D. The domination number of G, denoted by γ ( G ) , is the minimum cardinality of a dominating set in G. Recently, Khan (2023) performed a comparative study of seven important domination parameters with the π -electronic energy of benzenoid hydrocarbons, with a new theoretical approach. Physico-chemical properties are crucial for understanding the physical and chemical behavior of a molecule. In this paper, we perform a comparative study of those seven domination parameters with the physico-chemical properties of benzenoid hydrocarbons. We find the values of those domination parameters for the 22 lower benzenoid hydrocarbons and follow the statistical approach described by Khan (2023). The two physico-chemical properties to be chosen are the normal boiling point T b , a representative for intermolecular and van-der-Waals type interactions, whereas, the standard enthalpy/heat of formation Δ H f o is referred to represent the thermal properties of a molecular graph. This study further enhances that the performance of paired domination number is exceptional among all other parameters and highly correlated with both of the physico-chemical properties. Its correlation coefficient is 0.9981 (respectively, 0.9726) with T b (respectively, Δ H f o ). The outstanding correlation coefficient of paired domination number ensures further usage in quantitative structure-property relationship (QSPR) models.

Dynamical mechanisms of how an RNN keeps a beat, uncovered with a low-dimensional reduced model

Despite music’s omnipresence, the specific neural mechanisms responsible for perceiving and anticipating temporal patterns in music are unknown. To study potential mechanisms for keeping time in rhythmic contexts, we train a biologically constrained RNN, with excitatory (E) and inhibitory (I) units, on seven different stimulus tempos (2–8 Hz) on a synchronization and continuation task, a standard experimental paradigm. Our trained RNN generates a network oscillator that uses an input current (context parameter) to control oscillation frequency and replicates key features of neural dynamics observed in neural recordings of monkeys performing the same task. We develop a reduced three-variable rate model of the RNN and analyze its dynamic properties. By treating our understanding of the mathematical structure for oscillations in the reduced model as predictive, we confirm that the dynamical mechanisms are found also in the RNN. Our neurally plausible reduced model reveals an E-I circuit with two distinct inhibitory sub-populations, of which one is tightly synchronized with the excitatory units.

‘Let me tell you a story’: the politics of macroeconomic models

ABSTRACT Various social science literatures suggest that the general character of macroeconomic models reflects their assumptive base. A more specialist literature in the Weintraub-Boumans-Morgan tradition shows how the mathematics of those models moulds together logical implications of particular starting assumptions, insights from generally accepted theoretical propositions, and professionalised common-sense about how the world works. I go one step further in arguing that a process of narrative moulding operates in tandem with this mathematical moulding. A naming strategy provides the mathematical properties of macroeconomic models with economic labels to create the feeling that they are something more than a merely mathematical structure. A storytelling strategy then informs policy-makers of where the solution to the system of equations positions the outer limits of both political desirability and political possibility. Future dedicated research programmes into the narrative dimensions of macroeconomic models can be expected to shed further light on how theory models can masquerade as policy models and substitute models as surrogate models.

A Review of Particle Packing Models and Their Applications to Characterize Properties of Sand-Silt Mixtures

This paper reviews particle packing models and explores their application in geotechnical engineering, specifically for sand-silt mixtures. The review covers key models, including limiting case, linear, and non-linear packing models, focusing on their mathematical structures, physical principles, assumptions, and limitations through the concept of excess free volume. The application of particle packing models in geotechnical engineering is explored in characterizing the properties of sand-silt mixtures, offering insights into maximum, minimum, and critical void ratios and inter-granular void ratio, and the prediction of mechanical properties.

A design strategy with the combination of neuro-regression and stochastic optimizations on the hyperelastic gasket

For numerous years, PVC and aluminum doors and windows have been integral components of buildings. The primary technical requisites for doors and windows encompass superior acoustic performance, thermal insulation, and effective sealing. Particularly, the sealing properties, crucially important, are ensured by the gaskets fitted onto the profiles. This paper aims to present a gasket design with a new methodology that includes a systematic modeling-optimization process based on multiple non-linear neuro-regression. Comprehensive research has been done by multiple non-linear neuro-regression analyses of gaskets due to deficient studies in the literature on hyperelastic materials. 17 candidate functional forms based on four basic mathematical structures (polynomial, trigonometric, logarithmic, and linear functions, and their hybrid or rational forms) have been proposed to model the FEM data of the gasket. Hybrid versions of three stochastic search algorithms, Differential Evolution, Random Search, and Simulated Annealing, are considered to solve highly non-linear, non-convex, multi-variables mixed continuous-discrete problems. The experimental results were compared with the numerical result using the finite-element analysis (FEA). Nonlinear static analyses were performed with Abaqus. Stress and logarithmic strain distributions are obtained. A good correlation was found between the experimental and the numerical results The optimum locations and numbers of the nodes have been found to obtain the minimum overflow distance and the maximum compression surface area for gaskets having the same topological structure.

Mathematical structures in quantum physics education for high school students: Unveiling the power of Dirac notation for conceptual and problem-solving proficiency

[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Teaching quantum physics (QP) to high school (HS) students is gaining momentum, necessitating the exploration of various effective methods. Specifically, the research on quantitative teaching methods is still in its early stages. Understanding the power of Dirac notation (DN) in teaching is crucial for grasping the complexities of QP, as it provides a concise and powerful symbolic framework that facilitates both conceptual and computational understanding. This study investigates the proficiency of HS students in interpreting, producing, and manipulating DN within QP. Utilizing qualitative content analysis of student responses to various tasks, quizzes, and final tests, the study aims to uncover the sensemaking opportunities DN provides and the difficulties students encounter. The findings indicate that DN facilitates both conceptual understanding and quantitative problem solving in QP by enabling students to interpret and produce representations of quantum states. The results highlight the importance of extensive practice and well-structured teaching sequences in developing DN proficiency, reinforcing the role of DN as an effective tool for teaching QP at the HS level. The study contributes to the ongoing discourse on integrating mathematical modeling in physics education, emphasizing the potential of DN in enhancing students’ grasp of QP concepts and their application in problem solving. Published by the American Physical Society 2024

On the Generalized Weighted Statistical Convergence

Statistical convergence and summability represent a significant generalization of traditional convergence for sequences of real or complex values, allowing for a broader interpretation of convergence phenomena. This concept has been extensively examined by numerous researchers using various mathematical tools and applied to different mathematical structures over time, revealing its relevance across multiple disciplines. In the present study, a generalized definition of the concepts of statistical convergence and summability, termed (△_v^m )_u-generalized weighted statistical convergence and (△_v^m )_u-generalized weighted by [¯N_t ]-summability for real sequences, is introduced using the weighted density and generalized difference operator. Based on this definition, several fundamental properties and inclusion results, obtained by differentiating the components used in the definitions, are provided.

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to O(Nlog2⁡N) for 1d problems and O(N2log2⁡N) for 2d problems, compared with O(N3) of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

On a Class of Generalized Multivariate Hermite–Humbert Polynomials via Generalized Fibonacci Polynomials

This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields.

On the stabilizer formalism and its generalization

Abstract The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.

Symmetric Reverse n-Derivations on Ideals of Semiprime Rings

This paper focuses on examining a new type of n-additive map called the symmetric reverse n-derivation. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse n-derivations. This study aims to deepen our understanding of these symmetric maps and their properties within mathematical structures.

Thermostatted Kinetic Theory Structures in Biophysics: Generalizations and Perspectives

The mathematical modeling of multicellular systems is an important branch of biophysics, which focuses on how the system properties emerge from the elementary interaction between the constituent elements. Recently, mathematical structures have been proposed within the thermostatted kinetic theory for the modeling of complex living systems and have been profitably employed for the modeling of various complex biological systems at the cellular scale. This paper deals with a class of generalized thermostatted kinetic theory frameworks that can stand in as background paradigms for the derivation of specific models in biophysics. Specifically, the fundamental homogeneous thermostatted kinetic theory structures of the recent literature are recovered and generalized in order to take into consideration further phenomena in biology. The generalizations concern the conservative, the nonconservative, and the mutative interactions between the inner system and the outer environment. In order to sustain the strength of the new structures, some specific models of the literature are reset into the style of the new frameworks of the thermostatted kinetic theory. The selected models deal with breast cancer, genetic mutations, immune system response, and skin fibrosis. Future research directions from the theoretical and modeling viewpoints are discussed in the whole paper and are mainly devoted to the well-posedness in the Hadamard sense of the related initial boundary value problems, to the spatial–velocity dynamics and to the derivation of macroscopic-scale dynamics.

An inductive bias from quantum mechanics: learning order effects with non-commuting measurements

There are two major approaches to building good machine learning algorithms: feeding lots of data into large models or picking a model class with an “inductive bias” that suits the structure of the data. When taking the second approach as a starting point to design quantum algorithms for machine learning, it is important to understand how mathematical structures in quantum mechanics can lead to useful inductive biases in quantum models. In this work, we bring a collection of theoretical evidence from the quantum cognition literature to the field of quantum machine learning to investigate how non-commutativity of quantum observables can help to learn data with “order effects,” such as the changes in human answering patterns when swapping the order of questions in a survey. We design a multi-task learning setting in which a generative quantum model consisting of sequential learnable measurements can be adapted to a given task — or question order — by changing the order of observables, and we provide artificial datasets inspired by human psychology to carry out our investigation. Our first experimental simulations show that in some cases the quantum model learns more non-commutativity as the amount of order effect present in the data is increased and that the quantum model can learn to generate better samples for unseen question orders when trained on others — both signs that the model architecture suits the task.

Volichenko-type metasymmetry of braided Majorana qubits

Abstract This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits and clarifies their role; in particular, "mixed-bracket" Heisenberg-Lie algebras are introduced. These algebras belong to a more general framework than the Volichenko algebras defined in 1990 by Leites-Serganova as metasymmetries which do not respect even/odd gradings and lead to mixed brackets interpolating ordinary commutators and anticommutators.
In a previous paper braided Z2-graded Majorana qubits were first-quantized within a graded Hopf algebra framework endowed with a braided tensor product. The resulting system admits truncations at 
roots of unity and realizes, for a given integer s=2,3,4, ..., an interpolation between ordinary Majorana fermions (recovered at s=2) and bosons (recovered in the s →∞ limit); it implements a parastatistics where at most s-1 indistinguishable particles are accommodated in a multi-particle sector.
The structures discussed in this work are:
- the quantum group interpretation of the roots of unity truncations recovered from a (superselected) set of reps of the quantum superalgebra Uq(osp(1|2));
- the reconstruction, via suitable intertwining operators, of the braided tensor products as ordinary tensor products
(in a minimal representation, the N-particle sector of the braided Majorana qubits is described by 2N x 2N;
- the introduction of mixed brackets for the braided creation/annihilation operators which define generalized Heisenberg-Lie algebras;
- the s→∞ untruncated limit of the mixed-bracket Heisenberg-Lie algebras producing parafermionic oscillators;
- (meta)symmetries of ordinary differential equations given by matrix Schrödinger equations in 0+1 dimension induced by the braided creation/annihilation operators;
- in the special case of a third root of unity truncation, a nonminimal realization of the intertwining operators defines the system
as a ternary algebra.

Determination of Initial Data in the Time-Fractional Pseudo-Hyperbolic Equation

We examine a time-fractional pseudo-hyperbolic equation involving positive operators. We explore the determination of initial velocity and perturbation. It is demonstrated that these initial inverse problems are ill posed. Additionally, we prove that under certain conditions, the inverse problems exhibit well-posedness properties. Our focus is on developing a theoretical framework for these initial inverse problems associated with time-fractional pseudo-hyperbolic equations, laying the groundwork for future studies on numerical algorithms to solve these problems. This investigation is crucial for understanding the fundamental behavior of the equations under various initial conditions and perturbations. By establishing a rigorous theoretical framework, we pave the way for future research to focus on practical numerical methods and simulations. Our results provide a deeper insight into the mathematical structure of time-fractional pseudo-hyperbolic equations, ensuring that future computational approaches are built on a solid theoretical foundation.

Mathematical Information Retrieval: A Review

Mathematical formulas are commonly used to demonstrate theories and basic fundamentals in the Science, Technology, Engineering, and Mathematics (STEM) domain. The burgeoning research in the STEM domain results in the mass production of scientific documents that contain both textual and mathematical terms. In scientific information, the definition of mathematical formulas is expressed through context and symbolic structure that adheres to strong domain-specific notions. Whereas the retrieval of textual information is well-researched, and numerous text-based search engines are present. However, textual information retrieval systems are inadequate for searching scientific information containing mathematical formulas, including simple symbols to complicated mathematical structures. The retrieval of mathematical information is infancy, and it requires the inclusion of new technologies and tools to promote the retrieval of scientific information and the management of digital libraries. This paper provides a comprehensive study of mathematical information retrieval, highlights their challenges and future opportunities.

Measuring global urban complexity from the perspective of living structure

As urban critic Jane Jacobs conceived, a city is essentially the problem of organized complexity. What underlies the complexity refers to a structural factor, called living structure, which is defined as a mathematical structure composed of hierarchically organized substructures. Through these substructures, the complexity of cities, or equivalent to the livingness of urban space (L), can be measured by the multiplication the number of cities or substructures (S) and their scaling hierarchy (H), indicating that complexity is about both quantity of cities and how well the city is organized hierarchically. In other words, complexity emerges from a hierarchical structure where there are far more small cities or substructures than large ones across all scales, and cities are more or less similar within each individual hierarchy. In this paper, we conduct comprehensive case studies to investigate urban complexity on a global scale using multisource geospatial data. We develop an efficient approach to recursively identifying all natural cities with their inner hotspots worldwide through connected component analysis. To characterize urban complexity, urban space is initially represented as a hierarchy of recursively defined natural cities, and all the cities are then represented as a network for measuring the degree of complexity or livingness of the urban space. The results show the Earth’s surface is growing more complex from an economic perspective, and the dynamics of urban complexity are more explicit from nighttime light imagery than from population data. We further discuss the implications in city science, aiming to help create and recreate urban environments that are more resilient and livable by fostering organized complexity from the perspective of living structure.

The Sublime of Consciousness

Abstract The aesthetic tradition has identified as paradigmatically sublime such objects as imposing mountains and intense storms, as well as monumental art. But the tradition also acknowledges less paradigmatic cases, including sometimes mathematical structures or abstract concepts. In this paper, we argue that there is also a case for considering phenomenal consciousness—the experiential quality of subjective awareness—as a sublime phenomenon. One appreciates this, we argue, when one is struck by (fitting) awe upon contemplating (a) the perplexing existence of something like phenomenal consciousness in an otherwise completely material universe and (b) the role of consciousness in injecting meaning and value in an otherwise brutally factual reality.