Geometric Theorem Research Articles

공학 도구 기반의 보조 구성 도형 분석을 통한 논증 기하 학습 방법 탐색

This paper proposes proof methods for synthetic geometry problem by dividing it into a process of observing geometric properties of auxiliary constructions(auxiliary lines and auxiliary circles) drawn using technology tools without logical proof and a process of logically proving the observes geometric properties. Through this method, students can approach problems more easily and the utilization of technology tools is maximized, so each student can solve geometry problems that were difficult to solve using traditional methods. In addition, it can be expected that this method will play a meaningful role in improving students' logical thinking abilities by being able to solve problems with only basic geometry knowledge without using difficult geometric theorems.

A New Class of Univalent Functions Defined by Differential Operator

In the present work, we submit and study a new class $\mathcal{S}_{\lambda, \alpha, b}^{z, m, t}(\beta, \gamma, \omega, \mu)$ containing analytic univalent functions defined by new differential operator $D_{\lambda, \alpha}^{z, m, t}$ in the open unit disk $E=\{s \in \mathbb{C}:|s|<1\}$. We get some geometric properties, such as, coefficient estimate, growth and distortion theorems, convex set, radii of convexity and starlikeness, weighted mean, arithmetic mean and partial sums for functions belonging to the class $\mathcal{S}_{\lambda, \alpha, b}^{z, m, t}(\beta, \gamma, \omega, \mu)$.

Vedic Contributions to Geometry: Unveiling the Origins of Mathematics

The historical origins of foundational mathematical principles often lie obscured in the past. Among the luminaries in this nebulous landscape were Vedic mathematicians like Baudhāyana, Bhāṣkarācārya, Brahmagupta, Aryabhata, Pāṇini, Kātyāyana, and Āpastamba, eminent figures and sages of their time. Baudhāyana, in particular, is revered for his seminal contributions, prominently through the Baudhāyana Sulba Sūtra. This compendium of mathematical and ritualistic precepts predates the. The Baudhāyana Sulba Sūtra text belongs to the ritualistic ethos of Vedic Hinduism, yet it contains profound geometric theorems that expose the multifaceted genius of Baudhāyana. A scholarly endeavor has been an exegetical exercise on the mathematical corpus left by Baudhāyana, which throws new light on his pivotal role in shaping the contours of mathematical discourse within the antiquity of our civilization. This research helps us to enrich our knowledge about Vedic mathematical thinking and its intricate relationship with cultural and spiritual traditions.

Soft scalars in effective field theory

We derive a soft theorem for a massless scalar in an effective field theory with generic field content using the geometry of field space. This result extends the geometric soft theorem for scalar effective field theories by allowing the massless scalar to couple to other scalars, fermions, and gauge bosons. The soft theorem keeps its geometric form, but where the field-space geometry now involves the full field content of the theory. As a bonus, we also present novel double soft theorems with fermions, which mimic the geometric structure of the double soft theorem for scalars.

New solitary waves in a convecting fluid

It is critical to examine the effects of small perturbations on solitary wave solutions to nonlinear wave equations because of the existence of the unavoidable perturbations in real applications. In this work, we aim to study the solitary wave solutions for the dissipative perturbed mKdV equation which is proposed to model the evolution of shallow wave in a convecting fluid. By using the geometric singular perturbation theorem, the three-dimensional traveling wave system is reduced to a planar dynamical system with regular perturbation. By examining the homoclinic bifurcations via Melnikov method, we show that not only some solitary waves with particularly chosen wave speed persist but also some new type of solitary waves appear under small perturbation. The theoretical analysis results are illustrated by numerical simulations.

On the Transport of Currents

AbstractIn this work, we consider some evolutionary models for k-currents in $$\mathbb {R}^d$$ R d . We study a transport-type equation which can be seen as a generalisation of the transport/continuity equation and can be used to model the movement of singular structures in a medium, such as vortex points/lines/sheets in fluids or dislocation loops in crystals. We provide a detailed overview of recent results on this equation obtained mostly in (Bonicatto et al. Transport of currents and geometric Rademacher-type theorems. arXiv:2207.03922, 2022; Bonicatto et al. Existence and uniqueness for the transport of currents by Lipschitz vector fields. arXiv:2303.03218, 2023). We work within the setting of integral (sometimes merely normal) k-currents, covering in particular existence and uniqueness of solutions, structure theorems, rectifiability, and a number of Rademacher-type differentiability results. These differentiability results are sharp and can be formulated in terms of a novel condition we called “Negligible Criticality condition” (NC), which turns out to be related also to Sard’s Theorem. We finally provide a new stability result for integral currents satisfying (NC) in a uniform way.

A geometric Jordan decomposition theorem

A geometric Jordan decomposition theorem

A physical point of view on the arithmetic and geometric mean inequality

We propose a simple experiment designed to justify the arithmetic and geometric mean inequality by means of the laws of thermodynamics. The experiment consists in measuring the entropy variation in the thermodynamic irreversible process of cooling a metal in water. By considering the metal and water as a single isolated system, the arithmetic and geometric theorem is seen to hold by noticing that is positive for this irreversible transformation. These interdisciplinary activities may be used to reinforce basic concepts in thermodynamics in high school or first-year college students.

Self-evident Automated Geometric Theorem Proving Based on Complex Number Identity

Self-evident Automated Geometric Theorem Proving Based on Complex Number Identity

Proving geometry theorems: Student prospective teachers’ perseverance and mathematical reasoning

Proof of geometry is a topic that involves mathematical reasoning abilities and relates to perseverance involving hard work, the spirit of achievement, and self-confidence. The current important problem that occurs at this time is that students who are future teachers of mathematics still experience difficulties in compiling proofs, especially those who are not challenged to work hard. This qualitative research explores mathematics teacher candidates' reasoning abilities and perseverance in proving geometric theorems. Therefore, the research design used a case study. There were three participants in this study, and they were student prospective mathematics teachers' s taking geometry courses. Data were collected through working documents, open questionnaires, and semi-structured interviews and were analyzed using iterative techniques consisting of data condensation, data exposure, and verification. The study's results showed that students' prospective teachers did not prioritize proof in solving geometry problems, even though they worked hard to solve the problems independently until they were finished. The students' perseverance also impacts their mathematical reasoning in proving geometric theorems. Students with more hard work values tend to have more reasoning values. The results of this study have implications that there needs to be an effort from the teacher to get used to giving proof questions to support students' perseverance and mathematical reasoning abilities.

Second order linear differential equations with a basis of solutions having only real zeros

Let A be a transcendental entire function of finite order. We show that, if the differential equation w″ + Aw = 0 has two linearly independent solutions with only real zeros, then the order of A must be an odd integer or one half of an odd integer. Moreover, A has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.

Realism and the point at infinity: The end of the line?

Philosophers of mathematics often rely on the historical progress of mathematics in support of mathematical realism. These histories typically build on formal semantic tools to evaluate the changes in mathematics, and on these bases present later mathematical concepts as refined versions of earlier concepts which are taken to be vague. Claiming that this view does not apply to mathematical concepts in general, we present a case-study concerning projective geometry, for which we apply the tools of cognitive linguistics to analyse the developmental trajectory of the domain. On the basis of this analysis, we argue for the existence of two conceptually incompatible inferential structures, occurring at distinct moments in history, both of which yield the same projective geometric theorems; the first invoked by the French mathematicians Girard Desargues (1591–1661) and Jean-Victor Poncelet (1788–1867), and the second characterising a specific modern mode. We demonstrate that neither of these inferential structures can be considered as a refinement of the other. This case of conceptual development presents an issue to the standard account of progress and its bearing on mathematical realism. Our analysis suggests that the features that distinguish the underlying conceptually incompatible inferential structures are invisible to the standard application of the tools of formal semantics. Thus this case-study stands as an example of the manner and necessity of linguistics—specifically cognitive linguistics—to inform the philosophy of mathematics.

Iterative schemes for linear equations of the second kind and related inverse problems

This paper consists of two parts. The first one deals with the generation of an iterative algorithm to obtain an approximate solution of a linear equation of the second kind in a Banach space. This generation is based on a perturbed version of the geometric series theorem which, in particular, allows us to find a family of unisolvent linear Fredholm integral equations of the second kind, even when the associated linear operator has norm greater than or equal to 1. When we consider Fredholm equations of this type and linear Volterra integral equations of second kind, the numerical schemes obtained when appropriate Schauder bases are also introduced in the spaces where the equations operate, enable us to approximate their respective solutions iteratively. The second part of this work focuses on the design of a numerical method for solving an inverse problem associated with a linear equation of the second kind in a Banach space, a method which we apply to problems of parameter estimation related to the two classes of integral equations mentioned above.

Two Perpendicular Lines Related to a Circle : PWW accompanied by GeoGebra Applets

PWW style are presented. The difference of evidence is based on various additional auxiliary constructions, which in itself is a “mathematical art” that the student can master as a result of much practice. The PWW are accompanied by GeoGebra applets containing HINT buttons that allow the student to get step-bystep help in completing the proof and understand what geometrical properties and theorems it relies on.

Validasi Teorema Geometri pada kasus Titik Berat Segitiga secara Induktif Menggunakan Komputasi Simbolik

Symbolic computing emphasizes the exact processing of data representations. The exact representative states that even though the output is small, the data generated using processing can produce larger mathematical expressions. To overcome this problem, a method needs to be used in representing data. This process is carried out to solve a physical problem in the case of the center of gravity on a triangle. Symbolic programming languages can convert rational numbers, so that this symbolic processing can be applied to inductively validate geometric theorems in the case of triangular center.

A geometric generalization of Kaplansky’s direct finiteness conjecture

Let G G be a group and let k k be a field. Kaplansky’s direct finiteness conjecture states that every one-sided unit of the group ring k [ G ] k[G] must be a two-sided unit. In this paper, we establish a geometric direct finiteness theorem for endomorphisms of symbolic algebraic varieties. Whenever G G is a sofic group or more generally a surjunctive group, our result implies a generalization of Kaplansky’s direct finiteness conjecture for the near ring R ( k , G ) R(k,G) which is k [ X g : g ∈ G ] k[X_g\colon g \in G] as a group and which contains naturally k [ G ] k[G] as the subring of homogeneous polynomials of degree one. We also prove that Kaplansky’s stable finiteness conjecture is a consequence of Gottschalk’s Surjunctivity Conjecture.

Математика: становление, обоснование, разрешение кризиса

The article discusses the current situation in mathematics, which is interpreted as a crisis. The problems that allow making such a conclusion are considered: the question of the scientific status of mathematics, the possibility of its substantiation, the mental or experimental nature of mathematics, the time of its formation. The author presents the results of the genesis of geometry, the generalization of which makes it possible to assert that mathematics, on the one hand, can be considered a kind of empirical science, on the other, a constructive scientific discipline, in which more complex ideal objects are created on the basis of initial ideal objects and knowledge. Two different understandings of the foundations of mathematics are considered - general scientific and David Hilbert’s, as well as what paradoxes are in science and how they are removed. The author is inclined to believe that antinomies in mathematics cannot be eliminated once and for all, their source is attributing inconsistent characteristics to objects of mathematics, which is due to the very structure of mathematics. The fact is that the construction of ideal objects of mathematics proceeds under the influence of at least four areas: the empirical area, describing which mathematicians create initial ideal objects and knowledge; areas of design based on the original more complex ideal objects; areas of geometric theorem proofs; area of construction of the theory of geometry. In addition, here it is necessary to add the concept of the foundation of mathematics by D. Hilbert, and also take into account that there are different concepts of the foundation of mathematics. At the end of the article, the features of the current crisis in mathematics and the possible direction of its resolution are discussed.

A Rule Based Theorem Prover: an Introduction to Proofs in Secondary Schools

The introduction of automated deduction systems in secondary schools face several bottlenecks. Beyond the problems related with the curricula and the teachers, the dissonance between the outcomes of the geometry automated theorem provers and the normal practice of conjecturing and proving in schools is a major barrier to a wider use of such tools in an educational environment. Since the early implementations of geometry automated theorem provers, applications of artificial intelligence methods, synthetic provers based on inference rules and using forward chaining reasoning are considered to be more suited for education proposes. Choosing an appropriate set of rules and an automated method that can use those rules is a major challenge. We discuss one such rule set and its implementation using the geometry deductive databases method (GDDM). The approach is tested using some chosen geometric conjectures that could be the goal of a 7th year class (approx. 12-year-old students). A lesson plan is presented, its goal is the introduction of formal demonstration of proving geometric theorems, trying to motivate students to that goal

Counting geodesics of given commutator length

Abstract Let $\Sigma $ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in $\Sigma $ having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $\Sigma $ . In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.

DOF Calculation Method of Planar Mechanisms Based on Constraint Normal Line

In the analysis and calculation of the degree of freedom (DOF) of mechanisms, it is generally a complicated problem to judge the virtual constraint correctly. Under what conditions do virtual constraints exist, and how many virtual constraints are there? Understanding these problems can contribute to effectively tackling the difficulty of calculating the DOF. With planar mechanisms as a research object, the constraints among various components are simplified into point constraints, and the common normal at a constraint point is referred to as normal. According to whether a constraint point can move relative to the frame in its normal direction, normals are divided into different categories. Based on the geometric theorem for judging the DOF of a workpiece, a set of geometric theorems for judging the DOF of components and their nature, over‐constraint, and quantity are established. After the correlation between virtual constraints and over‐constraints is clarified, the total number of virtual constraints in a mechanism is calculated. The analysis and calculation of the DOF of several typical planar mechanisms demonstrate that the new method is logically rigorous, simple, and intuitive, and the DOF of a mechanism can be accurately calculated.