Mathematical Structure Research Articles

Exploring Fibonacci Algebraic Equations and Their Application

This article explores the sequence's generating functions, algebraic equations regulating it, and relationships to abstract algebra. Fibonacci algebra shows how patterns, numbers, and equations interact, from quadratic equations and generating functions to matrix algebra and beyond. Its simplicity and the surprising breadth of its linkages to other areas of mathematics are what give it its ongoing appeal. The study of Fibonacci algebraic equations is not just an exploration of an ancient sequence but a gateway to profound mathematical structures and applications. Whether through its connections to the golden ratio, matrix algebra, or advanced number theory, Fibonacci algebra continues to inspire and challenge mathematicians, offering a timeless bridge between elementary and advanced mathematics.

Investigation of new techniques for energy and its applications

In this article, we present a unified mathematical solver that is widely used in applied nonlinear differential equations based on the well-known tanh–coth method. We employ this solver to provide new solutions for the unstable and modified unstable nonlinear Schrödinger models. We also introduce He’s frequency technique to examine the dynamical behavior properties. The presented solutions include various mathematical structures, such as solitons, dissipative super-waves, bright and dark envelope waves, and explosive envelope waves. The behavior of the presented solutions for suitable free parameter values is depicted by two-dimensional and three-dimensional graphs. The proposed approach offers various benefits, such as eliminating time-consuming and difficult calculations. The results presented here can be utilized to characterize a wide range of physical phenomena in numerous disciplines of practical study, such as pulse propagation in optical fibers and quantum field. Finally, this study increases the current understanding of mathematical methods for solving nonlinear differential equations, which may be applied to other nonlinear wave equations in other scientific areas.

The irreducibility of chemistry to Everettian quantum mechanics

Abstract The question of whether chemical structure is reducible to Everettian Quantum Mechanics (EQM) should be of interest to philosophers of chemistry and philosophers of physics alike. Among the three realist interpretations of quantum mechanics, EQM resolves the measurement problem by claiming that measurements (now interpreted as instances of decoherence) have indeterminate outcomes absolutely speaking, but determinate outcomes relative to emergent worlds—Maudlin (Topoi, 14:7-15, 1995). Philosophers who wish to be sensitive to the practice of quantum chemistry e.g. Scerri (The changing views of a philosopher of chemistry on the question of reduction, 2016) should be interested in EQM because Franklin and Seifert (J. Philos. Sci, 2020) claim that resolving the measurement problem also resolves the reducibility of chemical structure, and EQM is the interpretation which involves no mathematical structure beyond that used by practicing scientists. Philosophers interested in the quantum interpretation debate should be interested in the reducibility of chemistry because chemical structure is precisely the kind of determinate three-dimensional fact which EQM should be able to ground if it is to be empirically coherent—see Allori (Quantum Rep, 5:80-101, 2023). The prospects for reduction of chemical structure are poor if it cannot succeed in EQM; the prospects for EQM as a guide to ontology are poor if it cannot reduce chemical structure. Unfortunately for proponents of chemical reduction and EQM, there are three serious barriers to the reduction of chemistry to EQM. The first concern is that quantum treatments of chemical structure rely on the Born-Oppenheimer approximation, which holds nuclear locations fixed while minimizing the energy of the electronic configuration—Hendry (Philosophical Perspectives in Quantum Chemistry, 147-172, 2022), but this approximation is not licensed by EQM. The Born-Oppenheimer approximation relies on nuclei and molecular orbitals being simultaneously present, but in the three-dimensional ontology following from the Everett interpretation these only emerge at different energy scales and are not simultaneously present—Miller (Found. Chem, 25:405-417, 2023). The second concern is that the emergent worlds of EQM are supposed to be decoherent at the macro-scale—Wilson (The Nature of Contingency: Quantum Physics as Modal Realism 2020), but the recent development of superchemistry suggests that chemical reactions can occur in coherent states—Zhang et al. (Nat. Phys, 1-5, 2023). The third concern is that emergent worlds are only pragmatic pseudo-processes—Wallace (The Emergent Multiverse: Quantum Theory According to the Everett Interpretation, 2012b), but this means EQM trades realist physics for mere instrumentalism about chemistry. Absent a commitment to chemical realism, reduction is an empty promise. The prospects for reduction of chemical structure to EQM are therefore poor.

Exploring a Novel Approach to Deferred Nörlund Statistical Convergence

This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks.

Wreathing, discrete gauging, and non-invertible symmetries

’t Hooft anomalies of discrete global symmetries and gaugings thereof have rich mathematical structures and far-reaching physical consequences. We examine each subgroup G, up to automorphisms, of the permutation group S4 that acts on the four legs of the affine D4 quiver diagram, which is mirror dual to the 3d N = 4 SU(2) gauge theory with four flavours. These actions are studied in terms of how each permutation cycle acts on the superconformal index of the theory in question. We present a prescription for refining the index with respect to the fugacities associated with the Abelian discrete symmetries that are subgroups of G. This allows us to study sequential gauging of various subgroups of G and construct symmetry webs. We study the effects of ’t Hooft anomalies and non-invertible symmetries that arise from discrete gauging on the index. When the whole symmetry G is gauged, our results are in perfect agreement with a type of discrete operations on the quiver, known as wreathing, discussed in the literature. We provide a general prescription for computing the index for any wreathed quivers that contain unitary or special unitary gauge groups. We demonstrate this in an example of the 3d N = 4 U(N) gauge theory with n flavours and compare the results with gauging the charge conjugation symmetry associated with the flavour symmetry of such a theory.

Tests of thermal macroeconomic theory on simulated microeconomies

Abstract In this paper, we test predictions of a new theory of macroeconomics, called “thermal macroeconomics.” The theory aims to apply the mathematical structure of classical thermodynamics, including analogues of temperature and entropy, to predict aspects of the aggregate behaviour of populations of economic agents without analyzing their detailed interactions. We test the theory by comparing its predictions with the behaviour of a variety of simulated micro-economies in which goods and money can be exchanged between agents, confirming the predictions of the theory. The paper serves also to illustrate and make more tangible the predictions of thermal macroeconomics.

A Topological Approach to Enhancing Consistency in Machine Learning via Recurrent Neural Networks

The analysis of continuous events for any application involves the discretization of an event into sequences with potential historical dependencies. These sequences represent time stamps or samplings of a continuous process collectively forming a time series dataset utilized for training recurrent neural networks (RNNs) such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) for pattern prediction. The challenge is to ensure that the estimates from the trained models are consistent in the same input domain for different discretizations of the same or similar continuous history-dependent events. In other words, if different time stamps are used during the prediction phase after training, the model is still expected to give consistent predictions based on the knowledge it has learned. To address this, we present a novel RNN transition formula intended to produce consistent estimates in a wide range of engineering applications. The approach was validated with synthetically generated datasets in 1D, 2D, and 3D spaces, intentionally designed to exhibit high non-linearity and complexity. Furthermore, we have verified our results with real-world datasets to ensure practical applicability and robustness. These assessments show the ability of the proposed method, which involves restructuring the mathematical structure and extending conventional RNN architectures, to provide reliable and consistent estimates for complex time series data.

The Univalence Principle

The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk’s completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai’s First-Order Logic with Dependent Sorts, but is expressed in Voevodsky’s Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.

Calculational Design of Hyperlogics by Abstract Interpretation

We design various logics for proving hyper properties of iterative programs by application of abstract interpretation principles. In part I, we design a generic, structural, fixpoint abstract interpreter parameterized by an algebraic abstract domain describing finite and infinite computations that can be instantiated for various operational, denotational, or relational program semantics. Considering semantics as program properties, we define a post algebraic transformer for execution properties (e.g. sets of traces) and a Post algebraic transformer for semantic (hyper) properties (e.g. sets of sets of traces), we provide corresponding calculuses as instances of the generic abstract interpreter, and we derive under and over approximation hyperlogics. In part II, we define exact and approximate semantic abstractions, and show that they preserve the mathematical structure of the algebraic semantics, the collecting semantics post, the hyper collecting semantics Post, and the hyperlogics. Since proofs by sound and complete hyperlogics require an exact characterization of the program semantics within the proof, we consider in part III abstractions of the (hyper) semantic properties that yield simplified proof rules. These abstractions include the join, the homomorphic, the elimination, the principal ideal, the order ideal, the frontier order ideal, and the chain limit algebraic abstractions, as well as their combinations, that lead to new algebraic generalizations of hyperlogics, including the ∀∃ ∗ , ∀∀ ∗ , and ∃∀ ∗ hyperlogics.

Analyzing diverse soliton wave profiles and bifurcation analysis of the (3 + 1)-dimensional mKdV–ZK model via two analytical schemes

The (3 + 1)-dimensional modified Korteweg–deVries–Zakharov–Kuznetsov model is widely used in the study of nonlinear wave phenomena. These forms of wave phenomena are more useful in science and engineering. This work will analyze the model to identify the processes of the obtained traveling wave solutions through the modified version of the new Kudryashov and extended hyperbolic function schemes, as well as evaluate the solidity of the solitons at numerous equilibrium points using bifurcation analysis in conjunction with the Hamiltonian planar system. In addition, bifurcations are used to display the shifting framework and to test for the presence of different traveling wave solutions. Moreover, we show the balance point in the photographic form to examine the signal’s stability by specifying the saddle point and system center. Thus, the originality of this study is that the obtained traveling wave solutions of the mentioned governing model produce a variety of waves, including dark, kink, bell, and cospon bright soliton, depending on spacetime and wave propagation variables, which are illustrated in two-dimension, three-dimension, and contour charts. Furthermore, this research examines the wave’s nature using the governing model’s ion acoustic parameters and characterizes the outcome of these factors on the wave structure. As can be seen, the bifurcation exploration and the stated method are extremely valuable and instructive for describing the mathematical structure in later studies such as this one and many others.

Boosting Investigative Knowledge of Linear Algebra With Interactive Classroom Practices

Objective: To analyze whether practicing interactivity with an Interventional Learning Object (ILO) in the Linear Algebra course facilitates a holistic understanding of the mental processes that generate mathematical structures and arguments. Theoretical Framework: Emphasizes the importance of fostering research from the classroom to spark students' curiosity and motivate them to seek answers to relevant questions through the systematic analysis of information. Method: The research was conducted in several stages: requirements analysis for designing the ILO prototype, diagnosing the sample's learning styles in mathematics, designing the ILO using authoring tools, and validation by experts in mathematics and digital technologies through a survey. Results and Discussion: The survey revealed that most students demonstrated active behavior during interaction with the ILO, attempting to understand the changes and results presented. It was observed that students try to apply their prior knowledge to analyze the new mathematical results arising from the interaction. Research Implications: The results suggest that integrating digital interactivity into Linear Algebra teaching can enhance the understanding and application of mathematical concepts in engineering students, highlighting the need to foster a deeper approach to conceptualization and abstraction during interaction with the ILO. Originality/Value: The study provides valuable insights into the behavior and perceptions of Generation Z students when interacting with digital tools for learning mathematics.

Equivalent transformation of differential and integral equations in hyperbolic plane

Hyperbolic numbers, serving as an extension of real numbers, exhibit an algebraic structure akin to complex numbers, constituting commutative rings with zero divisors that stem from two real numbers. This paper delves into the exploration of the equivalence relation between the solutions of differential equations and integral equations within the realm of the hyperbolic plane. The pivotal finding of this study reveals that the solutions of hyperbolic differential equations and integral equations are inherently equivalent. Such a discovery not only solidifies the groundwork for delving into hyperbolic differential equations but also injects a renewed impetus into the advancement of physics. By bridging the gap between differential and integral equations in the hyperbolic domain, our research paves the way for a deeper understanding of hyperbolic numbers’ mathematical structures and their applications in the physical sciences, thereby opening up new avenues for exploration and discovery.

On noncommutative leapfrog map

AbstractWe investigate the integrability of the noncommutative leapfrog map in this paper. First, we derive the explicit formula for the noncommutative leapfrog map and corresponding discrete zero‐curvature equation by employing the concept of noncommutative cross‐ratio. Then we revisit this discrete map, as well as its continuous limit, from the perspective of noncommutative Laurent bi‐orthogonal polynomials. Finally, the Poisson structure for this discrete noncommutative map is formulated with the help of a noncommutative network. Through these constructions, we aim to enhance our understanding of the integrability properties of the noncommutative leapfrog map and its related mathematical structures.

Detecting unfaithful entanglement by multiple fidelities

Abstract Certifying entanglement for unknown quantum states experimentally is a fundamental problem in quantum computing and quantum physics. Because of being easy to implement, a most popular approach for this problem in modern quantum experiments is detecting target quantum states with fidelity-based entanglement witnesses. Specifically, if the fidelity between a target state and an entangled pure state exceeds a certain value, the target state can be guaranteed to be entangled. Recently, however, it has been realized that there exist so-called unfaithful quantum states, which can be entangled, but their entanglement cannot be certified by any fidelity-based entanglement witnesses. In this paper, by specific examples, we show that if one makes a slight modification to fidelity-based entanglement witnesses by combining multiple fidelities together, it is still possible to certify entanglement for unfaithful quantum states with this popular technique. Particularly, we will analyze the mathematical structure of the modified entanglement witnesses, and propose an algorithm that can search for the optimal designs for them.

Numerical investigation of Williamson fluid in presence of chemical reaction towards a parabolic surface

ABSTRACT Our focus here is on examining the behavior of a boundary layer that regulates the movement of a non-Newtonian fluid undergoing a chemical reaction on a radiative paraboloid surface. This includes studying the fluid’s movement, temperature, and mass transfer. The fluid being studied pertains to the Williamson fluid model, which is described as a type of extended Newtonian fluid. We have considered the effects of Joule heating and dissipation through viscous phenomena based on the rules of heat and mass movement. The equations in the mathematical structure of partial differential equations (PDEs) governing the fluid continue to flow model in the problem. A subsequent evolution of these equations into ODEs by incorporating similarity variables. The velocity field’s schematic behavior is attributable to the magnetic parameter, thickness parameter, and the Weissenberg number, which exhibit velocity profile decline. In this research, both thermal radiation and Eckert number are highlighted, and it is concluded that both of these variables substantially raise the temperature field. We continue to track the concentration-reducing chemical reaction characteristic that remains tracked along with concentration transfer. Furthermore, the study includes a comprehensive presentation of comparative analyses with previously published scholarly works.

Use and Abuse of Instance Parameters in the Lean Mathematical Library

The Lean mathematical library Mathlib features extensive use of the typeclass pattern for organising mathematical structures, based on Lean’s mechanism of instance parameters. Related mechanisms for typeclasses are available in other provers including Agda, Coq and Isabelle with varying degrees of adoption. This paper analyses representative examples of design patterns involving instance parameters in the finalized Lean 3 version of Mathlib, focussing on complications arising at scale and how the Mathlib community deals with them.

Systematics of U-spin sum rules for systems with direct sums

A rich mathematical structure underlying flavor sum rules has been discovered recently. In this work, we extend these findings to systems with a direct sum of representations. We prove several results for the general case. We derive an algorithm that enables the determination of all U-spin amplitude sum rules at arbitrary order of the symmetry breaking for any system containing a direct sum of the representations 0 ⨁ 1. Potential applications are numerous and include, for example, higher order sum rules for CP-violating charm decays with an arbitrary number of final states.

Impactul inteligenței artificiale în educația muzicală și artele spectacolului

The rapid progress of new technologies has facilitated the implementation of human intelligence in a computerized framework with the aim of imitating human behaviour. One of the essential characteristics of artificial intelligence is continuous learning, based on external stimuli and information collected from the environment. Artificial intelligence analyzes the surrounding reality and acts accordingly, without the need for human help or assistance. In this sense, new intelligent technologies have a strong impact in various fields and related sciences: education, medicine, computer science, mathematics, cognitive psychology, theatre and performing arts, law, entrepreneurship. Artificial intelligence, at the intersection of education, music and the performing arts, involves elements of creativity and innovation. In recent decades, many artists have introduced computer technology into performances and thus created an interactive environment, where spectators are able to explore narrative experiences, correlated with algorithms and mathematical structures. The frequent use of artificial intelligence applications in musical education classes and theatre performances is swiftly becoming a necessity for the expression of new ideas and artistic connections.

Variable Dose-Constraints Method for Enhancing Intensity-Modulated Radiation Therapy Treatment Planning

The conventional approach to intensity-modulated radiation therapy treatment planning involves two distinct strategies: optimizing an evaluation function while accounting for dose constraints, and solving feasibility problems using feasibility-seeking projection methods that incorporate inequality constraints. This paper introduces a novel iterative scheme within the framework of continuous dynamical systems, wherein constraint conditions dynamically evolve to enhance the optimization process. The validity of dynamically varying dose constraints is theoretically established through the foundation of continuous-time dynamical systems theory. In particular, we formalize a system of differential equations, with both beam coefficients and dose constraints modeled as state variables. The asymptotic stability of the system’s equilibrium is rigorously proven, ensuring convergence to a solution. In practical terms, we leverage a discretized iteration formula derived from the continuous-time system to achieve rapid computational speed. The mathematical structure of the proposed approach, which directly incorporates dose-volume constraints into the objective function, facilitates significant computational efficiency and solution refinement. The proposed method has an inherent dynamics that approaches more desirable solutions within the set of solutions when the solution to the optimization problem is not an isolated point. This property guarantees the identification of optimal solutions that respect the prescribed dose-volume constraints while enhancing accuracy when such constraints are feasible. By treating dose constraints as variables and concurrently solving the optimization problem with beam coefficients, we can achieve more accurate results when compared with using fixed values for prescribed dose conditions.

Finding bifurcations in mathematical epidemiology via reaction network methods.

Mathematical Epidemiology (ME) shares with Chemical Reaction Network Theory (CRNT) the basic mathematical structure of its dynamical systems. Despite this central similarity, methods from CRNT have been seldom applied to solving problems in ME. We explore here the applicability of CRNT methods to find bifurcations at endemic equilibria of ME models.