Algebraic Analysis Research Articles

Quantum search algorithm for binary constant weight codes

A binary constant weight code is a type of error-correcting code with a wide range of applications. The problem of finding a binary constant weight code has long been studied as a combinatorial optimization problem in coding theory. In this paper, we propose a quantum search algorithm for binary constant weight codes. Specifically, the search problem is formulated as a polynomial binary optimization problem and Grover adaptive search is used for providing the quadratic speedup. Focusing on the inherent structure of the problem, we derive an upper bound on the minimum of the objective function value and a lower bound on the exact number of solutions. By exploiting these two bounds, we successfully reduced the constant overhead of the algorithm, although the overall query complexity remains exponential due to the NP-complete nature of the problem. In our algebraic analysis, it was found that this proposed algorithm is capable of reducing the number of required qubits, thus enhancing the feasibility. Additionally, our simulations demonstrated that it reduces the average number of classical iterations by 63% as well as the average number of total Grover rotations by 31%. The proposed approach may be useful for other quantum search algorithms and optimization problems.

Algebraic analysis of p-type brooms and their application in allergic drugs

Algebraic analysis of p-type brooms and their application in allergic drugs

Weight/height2: Mathematical overview of the world's most widely used adiposity index.

A footnote in Adolphe Quetelet's classic 1835 Treatise on Man described his algebraic analysis of how body weight ( ) varies with height ( ) in adult males and females. Using data on 12 short and 12 tall subjects of each sex, Quetelet established the rule that is approximately proportional ( ) to H2 in adults; that is, when for some constant . Quetelet's Rule ( ), transformed and renamed in the twentieth century to body mass index ( ), is now a globally applied phenotypic descriptor of adiposity at the individual and population level. The journey from footnote to ubiquitous adiposity measure traveled through hundreds of scientific reports and many more lay publications. The recent introduction of highly effective pharmacologic weight loss treatments has heightened scrutiny of BMI's origins and appropriateness as a gateway marker for diagnosing and monitoring people with obesity. This contemporary context prompted the current report that delves into the biological and mathematical paradigms that underlie the simple index . Students and practitioners can improve or gain new insights into their understanding of BMI's historical origins and quantitative underpinning from the provided overview, facilitating informed use of BMI and related indices in research and clinical settings.

The Analogy Method as an Element of Research Work of Junior Students

The article explores the possibility of increasing the efficiency of research work within the framework of lectures and practical classes for junior students. We have identified a contradiction between the potential of research work in the development of students’ research competencies in the process of teaching junior students and the insufficient development of productive methods for introducing research tools in addition to skills in working with sources of information and methods of conducting experimental work. We are talking about tools that allow you to identify objects of research, put forward and formulate hypotheses, test hypotheses, etc. The solution to the problem is realized through the developed methodology of lectures and practical classes, containing elements of intellectual tools of scientific activity in a systematic connection with the subject methodology. The method of analogy has been chosen as a specific tool for scientific activity. A system for assessing the levels of mastering the skills of intellectual tools of students’ scientific activities has been developed and an experimental test of the effectiveness of the developed methodology has been carried out. These methods, implemented for a higher mathematics course for non-core specialties, as well as algebra and mathematical analysis courses for mathematical and IT specialties, can be transferred to other disciplines of the theoretical cycle. This is the practical significance of the study. Keywords: research activity, levels of research activity, analogy method, research methodology

A variational multiscale stabilized finite element method for double diffusive incompressible flow with application to mixed convection in a heated staggered cavity under magnetic field

In this work, a novel stabilized finite element scheme in a subgrid multiscale variational framework is proposed to efficiently solve double-diffusive incompressible Newtonian fluid flow in a complex geometry occurring in several real-life applications. Here, the fine scale effects are transferred to the course scale problem following algebraic subgrid analysis, and then, the course scale equations are computed and the key stabilization parameters are derived following the Fourier analysis. The influence of flow, heat, and mass transfer parameters such as Reynold’s Number, Hartmann Number, Richardson Number, and Lewis Number are analyzed under a magnetic field, through the streamlines, isotherms, and the local heat fluxes to understand the double-diffusive mixed convective process in a complex staggered cavity, which is relevant to engineering applications such as ventilation and contamination assessment systems. The study finds that maximum species contamination together with the maximum Sherwood Number (12.54) occurs when the Richardson number is 10, and the minimum contamination together with the minimum Sherwood Number (0.71) occurs when the Lewis number is 0.1 when other parameters are varied within the operational range. The results are validated with previously existing literature for simple cases.

From onset of Cadomian subduction to forearc-arc-continent collision: Insights from the Neoproterozoic Calzadilla Metaophiolite (SW Iberia)

ABSTRACT The Cadomian basement of SW Iberian Massif appears in the so-called Ossa-Morena Complex, which contains several allochthonous terranes that are key for deciphering the evolution of the Gondwana margin during the Neoproterozoic. In this context, the Calzadilla Ophiolite represents the remnants of an oceanic Ediacaran lithosphere formed by a boninitic mafic crust that overlies a serpentinized ultramafic section, the latter being the biggest outcrop of Cadomian serpentinites in Europe. In this work, the petrological and geochemical nature of the mantle section is analysed, as well as the tectonothermal evolution of both, the mafic crust and the ultramafic section. The algebraic analysis of whole rock geochemistry of the serpentinites indicates that the protolith corresponds to harzburgite. The geochemical fingerprint of the serpentinites, as well as the primary spinel crystals composition, shows that peridotites underwent high degrees of partial melting (up to 16%) in a forearc setting. Thermodynamic modelling and average PT methods were applied to serpentinites and mafic dykes from the ultramafic section as well as to amphibolites from the mafic section, both of which experienced progradation up to 4–5 kbar and 525–575°C. According to the data, the protoliths of the Calzadilla Ophiolite were formed in a peri-Gondwanan arc system developed during the subduction of oceanic crust below Gondwana. The roll-back of the subducting slab would induce the opening of the forearc basin at c. 602 Ma, protolith age of amphibolites from crustal section, with an extensive partial melting of the mantle wedge and the formation of boninitic-affinity magmas that constitute the mafic section of the Calzadilla Ophiolite. Metasomatism and serpentinization of the mantle wedge would occur via melt- and fluid–rock interactions. A compression-dominated stage at c. 540 led to the emplacement of the Calzadilla Ophiolite onto the Cadomian magmatic arc, with the development of the amphibolite-facies metamorphic imprint.

T-duality and flavor symmetries in Little String Theories

We explore the T-duality web of 6D Heterotic Little String Theories, focusing on flavor algebra reducing deformations. A careful analysis of the full flavor algebra, including Abelian factors, shows that the flavor rank is preserved under T-duality. This suggests a new T-duality invariant in addition to the Coulomb branch dimension and the two-group structure constants. We also engineer Little String Theories with non-simply laced flavor algebras, whose appearance we attribute to certain discrete 3-form fluxes in M-theory. Geometrically, these theories are engineered in F-theory with non-Kähler favorable K3 fibers. This geometric origin leads us to propose that freezing fluxes are preserved across T-duality. Along the way, we discuss various exotic models, including two inequivalent Spin(32)/ℤ2 models that are dual to the same E8 × E8 theory, and a family of self-T-dual models.

Algebraic analysis of pairwise comparisons with non-reciprocal property

The law of comparative judgement usually implies the reciprocal property of pairwise comparisons. But time error and position preference could yield some judgements that contradict this identity assumption. This paper proposes an algebraic approach to dealing with the challenge of non-reciprocal property in pairwise comparisons. The concept of non-reciprocal pairwise comparison matrix (NrPCM) is used to characterize the situation of non-reciprocal property. Algebraic analysis of NrPCMs is carried out, where a linear space is constructed by proposing a set of basis. Then the linear space is decomposed into the direct sum of some subspaces. The properties of the space are analyzed such as inner product, decomposition and coordinate. Finally, we apply the algebraic method to investigate the Condorcet paradox under the impact of indifferent voters. A novel transformation formula of non-reciprocal pairwise comparisons is proposed, where the non-reciprocal property holds. It is revealed that the occurrence of voting paradox can be understood clearly according to the transitivity property of NrPCMs.

Linear Algebraic Nodal Analysis: An Applied Project for a First Course in Linear Algebra

Many students who enroll in a first course in linear algebra major in STEM disciplines other than mathematics. Teachers who serve such students may find it difficult to provide authentic problems from these broader areas that ignite students' interest in linear algebra. In this paper, we highlight an interdisciplinary learning activity that engages students in using linear systems of equations to model the behavior of practical electric circuits. This exercise fits nicely into standard introductory linear algebra curricula and is designed to excite students majoring in engineering, physics, or applied mathematics. We also include references to a collection of open-access resources to support instructors who want to use this material in project-based, flipped-learning, inquiry-oriented, or independent-study environments.

Min–max consensus of multi-agent systems in random networks

This paper focuses on the study of the min–max consensus problem for multi-agent systems in random networks. A min–max consensus algorithm is proposed, which incorporates the weighted average of the agent’s state, the maximum value of its neighbors’ states, and the minimum value of its neighbors’ states. The interaction network is composed of a directed random network where edges exist with probability and are independent of each other. Then, the convergence of the min–max consensus algorithm is investigated utilizing max-plus algebra, min-plus algebra, and stochastic analysis theory. Sufficient and necessary conditions are given to ensure that the multi-agent systems achieve min–max consensus almost surely and in mean-square, respectively. Finally, some numerical simulations are conducted to confirm the effectiveness of the proposed consensus algorithm.

Improved phase retrieval method with dual-plane holograms for slightly off-axis digital holography

An improved phase retrieval method with dual-plane holograms for slightly off-axis digital holography (DH) is proposed. By applying the Hilbert transform (HT) twice on defocused hologram, the DC term can be effectively eliminated while preserving the original object information. The phase of tested sample can be retrieved from two transformed holograms with only a straightforward algebraic equation. The derivation of algebraic equation and characteristic analysis of HT on hologram are discussed in details. Our method offers a viable solution for quantitative phase imaging in slightly off-axis DH system, eliminating the need for complex spectrum filtering, phase-shifting devices, and iterative operation. Numerical simulation and experiment results of microlens array and phase plate demonstrate the validity of proposed method.

Modeling of material and energy inputs in the life cycle of a vehicle

The paper focuses on the analysis of the environmental impacts related to the production, operation, decommissioning of vehicles as well as servicing and repairs of cars in real conditions of their use. The authors’ presented mathematical model that was implemented in original numerical program EN-VEHICLE that enables the analysis of energy consumption and emission loads related to material inputs in the entire life cycle of a passenger car. It is a tool from the Life Cycle Assessment category that aims at the effective inclusion of environmental impacts in the decision-making process. The developed program allows for a quantitative interpretation of the calculation results in terms of the environmental safety in each of the phases and throughout the life cycle of the vehicle, taking into account the possibility of supplying the vehicle system with additional material streams derived from recycling and recovery, and introduced both during the construction phase and during the car operation phase. In the presented tool, linear algebra and matrix analysis were used in modeling the vehicle life cycle system. The results of the implementation of the mathematical model were presented in the form of a regression function that allows for approximation of selected empirical data. The regression analysis was used to verify the material characteristics. The tool can be used for a comprehensive comparative assessment of the environmental impact of a passenger car from different production periods. In addition, it can be used to forecast the environmental effects of changes in the material structure determining the production technology and, consequently, having a significant impact on the entire life cycle of the vehicle as well as energy and ecological parameters.

Exact solution of the [formula omitted] quantum spin chain with open boundary condition

In this paper, we studied the exact solution of the C2(1) invariant quantum spin chain with off-diagonal open boundary condition. We obtain a solution of the reflection equation where the all matrix element of reflection matrix are nonzeros. By using the technique of fusion, we construct the fused transfer matrix and find the closed recursive relations among the transfer matrices. Based on the algebraic analysis, we obtain the eigenvalue of the system and express it as the inhomogeneous T−Q relation.

A study on attenuation patterns of acoustic waves in waveguide structures with flexural boundaries

This research article investigates the attenuation patterns of acoustic waves within waveguide structures featuring flexural boundaries. By employing advanced methods in vibrations and controls, the study aims to enhance our understanding of waveguide behavior and offer valuable insights for optimizing acoustic wave propagation in diverse applications. The entire waveguide structure is tuned through cooperative interactions involving elastic plates, membranes, and rigid boundary surfaces. The lower boundary of the expansion chamber consists of a rigid plate, with elastic membranes covering the upper surfaces of the cavity. The problem is formulated in terms of transmission and reflection coefficients of the modes, enabling the solution of the scattering problem. Matching acoustic pressure and velocity at the waveguide junction yields the scattering coefficients. The results reveal that the scattering coefficients are intricately influenced by both the geometric characteristics of the waveguide and the frequency of the incident waves. Notably, the first higher-order mode of the cut-off frequency of the waveguide structure corresponds to the frequency at which the dissipation coefficient reaches its maximum value. This research provides valuable insights into the scattering behavior of acoustic waves in waveguide configurations, contributing to the design of acoustic devices and systems. Rigorous support and justification for all aspects of the mode-matching method, including pressure and velocity matching, eigen properties, physical edge conditions, convergence criteria, and energy conservation, are provided through thorough algebraic and numerical analyses, confirming the validity of the solution methodology.

How the level of student research autonomy in higher education affects learning efficiency by shaping motivation: A case of instructional disconformity

This study aims to examine the effects of different instructional interventions (i.e., autonomy-supportive, controlling, or a combination thereof) on student motivation and academic research performance. Here, a total of 328 s-year Chinese students majoring in mathematics completed the Academic Motivation Scale (AMS) survey before and after the intervention involving an array of research-related assignments. Learners in the autonomy-supportive group showed a lower level of amotivation and higher intrinsic motivation compared to the pretest; the differences were 2.62 and 11.95 points. Their level of extrinsic motivation remained unchanged. The controlling intervention had the opposite effect, causing the mean extrinsic motivation score to increase by 10.25 points. The hybrid approach showed significant improvement in all three scales: amotivation (−3.82 points), extrinsic motivation (+4.35 points), and intrinsic motivation (+7.13 points). The autonomy-supportive teaching style corresponds to lower scores in mathematical modeling (72.30), linear algebra (74.62), and numerical analysis (77.65). High standard deviations indicate that research activity scores are spread out over a wide range. The controlling instructional approach yielded higher scores (83.95, 81.68, and 81.88 points, respectively) with the smallest standard deviations. Learners in the hybrid group scored the highest (86.98, 82.30, and 82.90 points, respectively), with moderate standard deviations indicating a well-balanced performance. The findings indicate that a hybrid approach, which combines autonomy support with instructor control, is the most effective in enhancing both intrinsic and extrinsic motivation among students, as well as improving their academic performance. The implications for higher education policy include the necessity of developing and implementing hybrid learning models that cater to the individual needs of students while providing the requisite level of support and oversight.

Matrix methods and teaching the fundamentals of fluid and gas mechanics in higher education institutions

Problem. The instruction of physical and mathematical disciplines in higher education institutions employing cutting-edge methods and universal approaches, which seamlessly integrate scientific and engineering activities of future specialists, remains one of the pertinent and priority tasks. The utilization of multidimensional or even infinite-dimensional spaces has become an effective and nearly indispensable tool in mathematical modeling of physical phenomena. This circumstance is directly linked to the increasing level of abstraction and the refinement of mathematical methods. The use of geometric methods inherent in vector algebra and vector analysis as the foundation for studying mechanics, despite their illustrative nature, is losing its relevance. Methods built upon matrix formalism are evolving as substitutes. The matrix framework enables the exploitation of phase space advantages to derive canonical equations of motion for continuous media and electromagnetic field equations in covariant forms. The electromagnetic potential acquires mechanical significance, allowing the utilization of electromechanical analogies at a fundamental level rather than on a merely formal basis, as done in classical electrodynamics. Goal. The aim of this research is to study and justify the advantages of matrix methods for deriving equations of motion for fluids and gases from the canonical equations of mechanics by utilizing a continuum model in phase and physical space. The research is directed towards demonstrating the connection between the internal microscopic and macroscopic motion of fluids and gases, as well as highlighting the potential of matrix formalism both in terms of modeling and the utilization of computational tools. Methodology. The methodological basis for selecting matrix methods lies in the application of tensor analysis and its generalizations for modeling the motion of fluids and gases in physical and phase space. Results. It has been demonstrated that the matrix method, previously applied to classical and relativistic mechanics, allows for the consideration of the molecular structure of the environment by folding the multidimensional phase space of the mechanical system, represented by a particle of the medium, and thereby deriving equations of motion for fluids and gases from the canonical equations of classical mechanics and the equations of momentum balance of the medium. Originality. The combination of canonical equations as a result of applying the conservation law of matter in the form of balance equations in phase space and momentum balance equations in physical space allows for elucidating the relationship between macroscopic and microscopic motions of the medium, as well as the mathematical structure of the equations of motion for fluids and gases. Practical value. The proposed method allows, while remaining within the standard mathematical training offered by technical educational institutions, to effectively formalize the equations of motion for fluids and gases and provide them with an invariant form. By guiding oneself through invariance, both fundamental laws (such as the law of universal gravitation) and partial laws (the generalized Newton's law) can be obtained.

Energy spectrum of the boundary quantum integrable system with [formula omitted]-symmetry

We study the exact solution of the C2(1)-invariant quantum integrable systems associated with off-diagonal open boundary condition. The boundary reflections break the U(1)-symmetry of the model. We find that the fusion relations among the fused transfer matrices can be closed at the inhomogeneous points. Based on the algebra analysis instead of constructing the eigenstates, we obtain the eigenvalues of conserved quantities including the Hamiltonian. The inhomogeneous T−Q relations and the related Bethe ansatz equations are given explicitly.

Higher Dimensional Leibniz-Rinehart Algebras

In this article, we delve into the realm of higher dimensional Leibniz-Rinehart algebras, exploring the intricate structures of Leibniz algebroids and their applications. By generalizing the concept of Lie algebroids and incorporating a Leibniz rule for the anchor map, the study sheds light on the fundamental principles underlying connections and underscores their significance. Through a comprehensive analysis of Leibniz-Rinehart algebras, this study paves the way for advancements and applications, offering a deeper understanding of the intricate relationship between algebraic and geometric structures.

Rainfall dynamics in an ecologically vulnerable area using applied algebraic topology methods

Applied algebraic topology is employed in this work to shed new light on the rainfall dynamics in the Pernambuco state, Brazil. Historical data from the NASA’s Tropical Rainfall Measuring Mission (TRMM) precipitation processing system, for the period 1998 to 2020 with a spatial resolution of 0.25° and temporal resolution of 3 h is used to construct correlation matrices in different time frames. Matrices are then analyzed in terms of topological constructs of network theory to yield novel insights into this highly complex phenomenon in this semiarid, ecologically vulnerable area. The outcomes of the algebraic topological analysis reveal clustering patterns of areas and are related to natural climate phenomena. Together with the generality of the applied methodology, the results suggest a broad scope of future applications for the extraction of patterns in datasets related to the changes in the climate system.

Structuralism in differential equations

Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind’s approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, and a monograph on the Navier–Stokes equation to show structural methods like Dedekind’s have long been central to differential equations, and have philosophically respectable ontology and epistemology.