Branch Of Mathematics Research Articles

Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization

Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems.

Application of Residue Theorem in Calculating Improper Integral in Calculus

In this paper, the residue theorem and its applications in the field of calculus are discussed. This study first emphasizes the importance of the residue theorem in the analysis of complex functions, and points out its wide applicability in various branches of mathematics. In the second part, this article explores some basic concepts in depth, including a section on singularities and higher-order singularities, precise definitions of residuals, and different methods used to calculate them. Building on this theoretical foundation, this paper transitions to practical applications, showing how to use the residual theorem to compute integrals, with particular emphasis on its ability to simplify calculations and provide efficient solutions to other thorny problems. Through a series of detailed examples, the validity of the theorem in solving difficult integrals solved by standard methods is emphasized. The conclusion reiterates the key role of the residue theorem in calculus and emphasizes its indispensable contribution to calculus calculation by providing accurate and simplified solutions.

Exploring Topological Information Beyond Persistent Homology to Detect Geospatial Objects

Accurate detection of geospatial objects, particularly landslides, is a critical challenge in geospatial data analysis due to the complex nature of the data and the significant consequences of these events. This paper introduces an innovative topological knowledge-based (Topological KB) method that leverages the integration of topological, geometrical, and contextual information to enhance the precision of landslide detection. Topology, a fundamental branch of mathematics, explores the properties of space that are preserved under continuous transformations and focuses on the qualitative aspects of space, studying features like connectivity and exitance of loops/holes. We employed persistent homology (PH) to derive candidate polygons and applied three distinct strategies for landslide detection: without any filters, with geometrical and contextual filters, and a combination of topological with geometrical and contextual filters. Our method was rigorously tested across five different study areas. The experimental results revealed that geometrical and contextual filters significantly improved detection accuracy, with the highest F1 scores achieved when employing these filters on candidate polygons derived from PH. Contrary to our initial hypothesis, the addition of topological information to the detection process did not yield a notable increase in accuracy, suggesting that the initial topological features extracted through PH suffices for accurate landslide characterization. This study advances the field of geospatial object detection by demonstrating the effectiveness of combining geometrical and contextual information and provides a robust framework for accurately mapping landslide susceptibility.

Development of the whole-brain functional connectome explored via graph theory analysis

Introduction Adolescence is a period of remarkable development as children’s brains change to resemble adult brains. Resting state fMRI measures fluctuations in blood-oxygen signal from which we can infer functional connectivity (FC). Graph theory is a branch of mathematics that can quantify the complex patterns of connectivity and network architecture inherent in the functional connectome. An ideal graph theory analysis explores edges that are weighted, directional, and heterogenous (can be positive or negative). Recent developmental studies have applied graph theory to the functional connectome, yet due to the considerable complexity added by each facet, most ignore one or more aspects of an ideal graph theory analysis (directionality and heterogeneity). Methods The present cross-sectional study measured FC in typically developing children, adolescents, and young adults (age 6-24 years) using 150+ echo-planar volumes (3.6mm isotropic voxels, repetition/echo time=2000/30ms) acquired at rest. A standard pre-processing pipeline was used, and the functional connectome was quantified using a weighted, directed graph analysis, including both positive and negative connections. Five different graph theory metrics were utilized to quantify developmental trajectories: connection density, modularity, clustering coefficient, global efficiency, and betweenness centrality. Positive and negative connections were analyzed separately, and age and sex associations were explored. Results The total sample comprised 219 participants (mean age (SD) [range] = 14.1 (3.3) [6.5-24.0] years, 50% female). For positive connections, modularity and betweenness centrality increased with age (both p<0.001), while connection density, clustering coefficient, and global efficiency decreased with age (all p<0.001). By contrast, for negative connections, modularity and betweenness centrality decreased with age (p=0.002, p=0.003), while connection density, clustering coefficient , and global efficiency increased with age (p<0.001, p<0.001, p=0.003). Effects of sex, hemisphere, and their interaction were minimal, though global efficiency for negative connections was higher in the right hemisphere than the left (p<0.001). Conclusion Graph theory appears to be a useful tool for quantifying the complex development of the functional connectome. The developmental changes presented here may be driven by an intrinsic pressure to balance functionality with low metabolic cost to maintain the network. The positive connection network appears to shift towards a more efficient conformation resembling “small-world” architecture. In contrast, the negative connection network seems to shift away from such efficient architecture, possibly to prioritize improving functionality before later refinement.

On a polygon whose side lengths form a geometric sequence

In this article, we discuss the necessary and sufficient conditions for the existence of a polygon whose side lengths form a geometric sequence. We prove that such a polygon exists if and only if the common ratio r of the geometric sequence satisfies the condition 0.5 < r < 2. A formula has been obtained to determine the minimum number of the polygon sides for a given r. The article illustrates the relation between various branches of mathematics (geometry, sequences, limits) and can be used by university and college students specialising in mathematics and its teaching.

Generalization of the Fuzzy Fejér–Hadamard Inequalities for Non-Convex Functions over a Rectangle Plane

Integral inequalities with generalized convexity play a vital role in both theoretical and applied mathematics. The theory of integral inequalities is one of the branches of mathematics that is now developing at the quickest rate due to its wide range of applications. We define a new Hermite–Hadamard inequality for the novel class of coordinated ƛ-pre-invex fuzzy number-valued mappings (C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s) and examine the idea of C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s in this paper. Furthermore, using C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s, we construct several new integral inequalities for fuzzy double Riemann integrals. Several well-known results, as well as recently discovered results, are included in these findings as special circumstances. We think that the findings in this work are new and will help to stimulate more research in this area in the future. Additionally, unique choices lead to new outcomes.

Linear Algebra :Eigen Value and Eigen Vectors and Their Applications

Abstract: Linear algebra is the branch of mathematics that concerned with the study of vectors,vector spaces(linear spaces), linear maps(linear transformations') and system of linear equations. Linear space is the central theme in modern mathematics that is widely used in both abstract algebra and functional analysis.Linear algebra also has a concrete representation in analytical geometry and in generalized operator theory. It has extensive applications in natural sciences as well as social sciences. Here in this paper we are presenting a study on linear algebra and associated linear equations, also the eigen values and eigen vectors as a subtopic and few applications of it.the work basically focuses n eigen values and eigen vectors their useful properties including some applications of it.

H-partial uniform spaces and their application in the compression of digital images

Fixed point theorem is very important tool in different branches of mathematics. In this paper, we introduce partial uniform spaces as a generalization of uniform spaces and metric spaces; and study some basic properties. Various examples support the theory. We prove fixed point theorems for H-partial uniform spaces, by using a mapping called E-distance function. Finally, we give the applications of these fixed point theorems to compress digital images.

Mechanics of Materials: Multiscale Design of Advanced Materials and Structures

Materials can now be designed and architectured like structural components for targeted mechanical and physical properties. Structures and microstructures should not be studied independently and their design will benefit from a multiscale approach combining nonlinear continuum mechanics approaches and physical descriptions of elasticity, viscoplasticity, phase transformations and damage of microstructures, at various scales. The aim of the workshop was to gather outstanding junior and senior researchers in the various branches of mathematics, physics and engineering sciences suited to address the question of design of materials and structures by means of multiscale discrete and continuum approaches to their constitutive behavior. Examples include atomic or macroscopic lattices, random or periodic cellular materials, smart materials like shape memory alloys, 3D woven composites, acoustic and electromagnetic metamaterials, etc. Modern continuum mechanics relies on sophisticated constitutive laws for anisotropic materials exhibiting elastoviscoplastic behavior, still a field of intense research with new mathematical concepts. In particular size-dependent properties are addressed by resorting to generalized continua such as gradient or micromorphic and phase field models. The latter are attractive for the simulation of microstructure evolution coupled with mechanics, due to thermodynamic and metallurgical processes and damage. Scale transition and homogenization methods for continuous and discrete systems are required for the determination of effective material and structural behavior. Metamaterials are architectured materials specifically designed to achieve certain propagation and dispersion properties of elastic and plastic waves. Optimization strategies for the design of optimal architectures are involved in the design process. Target functions for optimization are now based on multicriteria (stiffness, strength, thermal expansion, transport properties, anisotropy etc.).

Common Tripled Fixed Point Theorem on M- Fuzzy Metric Space for Occasionally Weakly Compatible Mappings

The fixed point theorems, which are primarily existential in nature, serve as a fundamental topological toolkit for the qualitative analysis of solutions to both linear and nonlinear equations in various branches of mathematics. Many authors have extended and generalized these results in different ways, particularly in the context of fuzzy metric spaces and fuzzy mappings. Numerous researchers have also proved common fixed point theorems under the condition of compatible mappings for fizzy metric spaces. Coupled common fixed point theorems for fuzzy metric spaces with the condition of weakly compatible mappings were attempted to be proved by many authors. Tripled fixed points have emerged as a significant area of research within fixed point theory. Berinde and Borcut introduced the concept of a tripled fixed point for nonlinear mappings in partially ordered metric spaces. They also established a common fixed point theorem for contractive type mappings in M-fuzzy metric spaces. Later, other authors extended these results for common tripled fixed point theorems in fuzzy metric spaces. In this paper we introduce a new technique for proving some new common tripled fixed point theorems for Occasionally Weakly Compatible Mappings in M-fuzzy metric spaces, a method which is not previously utilized by authors in this field. Additionally, we provide illustrative example to support our findings, which represent an improvement over recent results found in the literature.

A Study on Calculus Examination in Japanese Top University Entrance Exams and Educational Enlightenment for Gifted Students in Mathematics

Calculus is the cornerstone of many branches of modern mathematics and a crucial component of mathematics education. This article conducts in-depth qualitative and quantitative research and analysis on the calculus questions in the entrance exams of three top universities in Japan (University of Tokyo, Kyoto University, and Tokyo Institute of Technology). It is found that the calculus questions in the entrance exams of the three universities cover a wide range of knowledge and are highly comprehensive, with comparable difficulty in testing calculus knowledge. At the same time, the examination of calculus emphasizes the characteristics of students' comprehensive mathematical literacy.

On the Construction of Congruences over Generalized Fuzzy G-Acts

Group action is defined to support Cayley’s claim that every group is isomorphic to a suitable subgroup of a symmetric group. Group actions have a wide range of applications, including the analysis of symmetries of geometric objects and algorithms and cryptographic systems. In 1965, Zadeh introduced the concept of fuzzy sets and provided a mathematical formulation for various concepts in this area. Since then, the concept of fuzziness has been integrated into several branches of mathematics to address the uncertainties of real-life scenarios. This article introduces the concept of group action in a fuzzy environment, termed fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-subacts. The study provide the concept of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-orbits and fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-stabilizers and clearly outlines fuzzy permutation representations of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document} and fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-morphisms. The research findings significantly contributes to the understanding of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-congruences and fuzzy quotient G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-subacts with the help of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-partitions. This approach not only refines the underlying theories but also opens up new possibilities for practical implementation. Thus, the study demonstrates how the more complex fuzzy theory can expand and enrich the mathematical structures of abstract algebra, making them highly applicable.

Expressing Partial Order-Preserving Transformations as Products of Nilpotents

Let S be a semigroup with zero, then an element a∈S is called nilpotent, if there exists a positive integer n such that . In the partial transformation semigroup on , where is a non-empty set, denoted by , the empty map is the zero of . Further, nilpotency is a structural property of fundamental importance which arises in many situations, for instance in linear algebra, we have nilpotent matrices and much more in other branches of mathematics. In this study, we will investigate, some elements of semigroups of partial one-to-one order-preserving transformations (denoted by ) and partial order-preserving transformations (denoted by ), and some basic properties of nilpotent transformations. We have studied certain properties of nilpotent elements in , symmetric inverse semigroup, where is finite and infinite, certain properties of nilpotent elements in , descriptions of the elements of 〈N〉, where N is the set of nilpotents of and we used the same letter N to denote that of , and used them to establish the following results, in this paper (that is, we have used the knowledge that we have acquired to establish the following results): - a partial ono-to-one order-preserving transformation , has from 1 to fixed, and there is an upper jump of length two(2) between and , and lower jumps of length one(1) and one(1), immediately after , can be expressed as a product of three nilpotents. – a partial one-to-one order-preserving transformation , has from 1 to fixed, and there is an upper jump of length three (3) between and and lower jumps of length two (2) and one (1), immediately after , can be expressed as a product of fewer than three nilpotents. – a partial order-preserving transformation , has from 1 to fixed, but belongs to , and there is an upper jump of length two(2) between and , and lower jumps of length two(2) and one(1), immediately after , cannot be expressed as a product of fewer than three nilpotents. – a partial order-preserving transformation , has from 1 to fixed, but belongs to , and there is an upper jump of length three (3) between and , and lower jumps of length three (3) and one (1), immediately after n-5, can be expressed as a product of fewer than three nilpotents. – a transformation , partial transformation semigroup, is not nilpotent if and only if the graph of the transformation has a loop, and - a transformation , partial one-to-one transformation semigroup, which has a nilpotent path and a permutation part is not nilpotent

On statistical evaluation of reverse degree based topological indices for iron telluride networks

In the context of graph theory and chemical graph theory, this research conducts a detailed mathematical investigation of reverse topological indices as they relate to iron telluride networks, clarifying their complex interactions. Graph theory is a branch of abstract mathematics that carefully studies the connections and structural features of graphs made up of edges and vertices. These theoretical ideas are expanded upon in chemical graph theory, which models molecular architectures with atoms acting as vertices and chemical bonds as edges. By extending these concepts, this work investigates the reverse topological indices in the context of Iron Telluride networks and outlines their significant effects on chemical reactivity, molecular topology and statistical modeling. By navigating intricate mathematical formalisms and algorithmic approaches, the analysis provides profound insights into the reactivity patterns and structural dynamics of Iron Telluride compounds, enhancing our knowledge of solid-state chemistry and materials science.

Modeling financial market dynamics with the use of fuzzy

In the modern world, financial markets play an important role in the economy and people’s lives. They provide access to financial resources and are also a source of profit for many companies. However, instability in the financial markets can lead to serious consequences such as financial crises and loss of investor confidence. In this regard, modelling the financial market dynamics becomes increasingly relevant. This work considered the use of fuzzy mathematics for this purpose. Fuzzy mathematics is a branch of mathematics that studies methods and algorithms for dealing with fuzzy data and fuzzy objects. It allows to consider uncertainty and incompleteness of information, which is especially important in the financial markets where data is often incomplete and inaccurate. The purpose of this research is to establish the relationship between financial asset prices while using behavioural factors (investor sentiment), fundamental (market returns), and microstructural ones (company size, ratio of book and market values of the company). The application of fuzzy mathematics in financial modelling will improve the accuracy and reliability of forecasts as well as increase the stability of the model to various sources of uncertainty.

STAGES OF TEACHING ELEMENTS OF STATISTICS IN ELEMENTARY SCHOOL

Mathematical statistics is one of the branches of mathematics, which is dedicated to the mathematical methods of organizing, processing and using statistical data for the purpose of scientific and practical conclusions. In practice, the methods of mathematical statistics are used in cases when it is necessary to examine a large amount of data according to a certain characteristic. Mathematical statistics is of great application importance in various spheres of human activity, so the study of the elements of this branch of mathematics is more than up-to-date. In order to study the elements of statistics at school, it is necessary to develop the following abilities among students: carry out simple surveys, collect quantitative data, and present them in the form of tables, graphs and diagrams. These abilities are suitable not only for the study of mathematics, but also for the study of other academic subjects, as well as in certain life situations. One of the main ideas fixed in the standard of the subject “Mathematics” is the analysis and interpretation of data, and its sub-idea is statistics, the study of which will be carried out in all three levels of education. The article presents the stages of learning the elements of mathematical statistics in elementary school, supported by many examples. All of this creates fertile ground for continued teaching of the elements of statistics in middle and high school.

Effect of chemical reaction on MHD Casson natural convection flow over an oscillating plate in porous media using Caputo fractional derivative

Fractional calculus is a branch of mathematics that extends the traditional definitions of calculus to include non-integer exponents. It has been successfully used to model a variety of physical processes, including the coiling of polymers, viscoelasticity, traffic flow, diffusive transport, fluid dynamics, electromagnetic theory, and electrical networks. However, fractional calculus has not yet been widely used to study the physical properties of non-Newtonian fluids flowing over accelerating porous plates. The present research examines the unsteady Casson flow over an infinitely horizontal porous plate, MHD and porosity impacts on fluid velocity are also discussed. The major goal here is to develop analytical outcomes for the tran-sient incompressible MHD Casson flow over an accelerating porous plate. This plate is oscillating in the x-direction and infinitely large in the y-direction and fluids flowing over it at y > 0 due to oscillation. Choosing the right choice of dimensionless variables allows the model's governing equations to become dimensionless. These dimensionless differential equations of the Casson fluid are calculated by applying the Laplace transform method. On velocity, the impacts of different factors are investigated. They are time, porosity parameter, oscillation frequency of plate, magnetic parameter, and Casson fluid parameter. As we rise the Casson flow parameter values, magnetic parameter, and Prandtl number we observe that the velocity drops. In contrast to other factors, the effects of Grashof number, oscillation frequency, fractional parameter, and porosity on the velocity profile are reversed. Using Mathcad software, graphs are sketched for various values of these parameters and are thoroughly described. The fluid properties discovered signif-icant findings and disclosed several characteristics for a range of flow parameters and fractional parameter values. Increasing the values of fractional parameters is shown to improve the characteristics of fluids, and this is beneficial in fitting experimental data related to heating and cooling processes, particularly in electronic equipment.

Introduction to Algebraic Deformation Theory and The Case of k-Linear Categories

Deformation theory is a branch of mathematics which studies how mathematical objects, such as algebraic varieties, schemes, algebras, or categories, can be deformed “continuously” depending on a space of parameter while preserving certain algebraic or geometric structures. Algebraic deformation theory, which was pioneered by Murray Gerstenhaber in 1960s-1970s, established its role as a cornerstone in modern mathematics and theoretical physics. This theory provides a powerful framework for understanding the subtle variations and deformations of mathematical and physical objects depending on a parameter space. Lying at the interface of algebra, geometry and topology this theory has been being studied extensively worldwide and obtained many applications in various areas of mathematics and theoretical physics, such as the study of Calabi-Yau manifolds, mirror symmetry, quantum physics. In such context, this article aims to introduce this important mathematical theory to the community of Vietnamese mathematicians in a hope to bring more attention of Vietnamese mathematicians and math students to this vibrant research area. We also expect that this topic will be taught at universities in Vietnam in the near future.

The contribution of working memory and spatial perception to the ability to solve geometric problems

Geometry is a branch of mathematics that deals with the properties of space, including distance, shape, size, and the relative position of figures. It is one of the oldest branches of mathematics and has applications in various fields such as science, art, architecture, and even in areas seemingly unrelated to mathematics. Studies show that working memory and spatial perception contribute to students' geometry performance. This paper presents multiple studies demonstrating the brain regions activated when solving geometric problems. Interestingly, the brain areas activated when solving algebraic problems are different from those activated when solving geometric problems. Finally, multiple studies are presented that indicate students with learning difficulties lag in geometry, as solving geometric problems requires good reading and arithmetic skills.

ISLAM DAN ILMU PENGETAHUAN : REKOGNISI PENDIDIKAN ISLAM DI BARAT ABAD KLASIK

The development of education in the world is inseparable from the contributions of Islam, which has played a significant role in it. Consequently, Islam holds a legitimate position in terms of the quality of education, which should be acknowledged publicly as a marker that these contributions must be academically recognized. The contributions of Islamic education to the world can be seen from the scientific heritage of the medieval period when Europe and the Middle East were engaged in debates over reason. The presence of Islamic education had an impact on the process of classifying knowledge into rational ('aqliah) and transmitted ('naqliah) sciences. This classification led to the emergence of branches of rational sciences, including medicine, mathematics, sciences (physics, astronomy, and chemistry), and sosial sciences (economics, geography, and psychology). In the field of medicine, Abdullah Al-Hasan Bin Al Bin Sina, commonly known as Ibn Sina, emerged with his book "Al-Qanun fi at-Tibb." In the field of mathematics, Muhammad Bin Musa Al-Khwarizmi made significant contributions to algebra, trigonometry, algorithms, and the invention of the number zero, and authored the book "Mukhtasar fi Hisab Al-Jabr Wa’l Muqabala." In addition to his expertise in mathematics, he also had a deep understanding of astronomy, as reflected in his work "as-Sindhind." In the field of technology, Abu Al-Izz Ibn Ismail Ibn Razaz Al-Jazari.