Geometric Sequence Research Articles

State geometric adjustability for interval max‐plus linear systems

AbstractThis article investigates the state geometric adjustability for interval max‐plus linear systems, which means that the state vector sequence is transformed into a geometric vector sequence by using the state feedback control. It is pointed out that the geometric state vector sequence and its common ratio are closely related to the eigenvectors and eigenvalues of the special interval state matrix, respectively. Such an interval state matrix is determined by the eigen‐robust interval matrix, which has a universal eigenvector relative to a universal eigenvalue. The state geometric adjustability is characterized by the solvability of interval max‐plus linear equations, and a necessary and sufficient condition for the adjustability is given. A polynomial algorithm is provided to find the state feedback matrix. Several numerical examples and simulations are presented to demonstrate the results. At the same time, the proposed method is applied for the regulation of battery energy storage systems to optimize the start time of executing tasks for all processing units in each activity.

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.

The Frobenius number for shifted geometric sequences associated with the number of solutions

The Frobenius number for shifted geometric sequences associated with the number of solutions

PowerPoint as a Tool to Support the Creation of Dynamic Models for Teaching: The Case of the Sum of an Infinite Geometric Sequence

PowerPoint is a learning aid, developed for presentations. PowerPoint has “dynamic” features for creating presentations with dynamic actions. The knowledge of the infinite backward multiplier is difficult to understand. One of the difficulties in understanding this knowledge is infinity, “dynamic”. In this study, we use the dynamic features of PowerPoint to design the sum of an infinite geometric sequence teaching. The study was conducted with 32 students in 11th grade. Students experimented with the dynamic model in PowerPoint. Research results show that the flexible use of PowerPoint’s dynamic features allows the creation of dynamic models of the sum of an infinite geometric sequence. Dynamic models have made it easier for students to build knowledge, creating geometric images of infinite sums. Moreover, with the dynamic features of PowerPoint, creating artistic images, attracts students to actively participate in learning.

On linear algebra of r-Hankel and r-Toeplitz matrices with geometric sequence

AbstractMatrix theory plays a crucial role in solving practical problems and performing computational operations. In particular, specific types of matrices and their linear algebraic properties are of paramount significance for these processes. In this paper, we study the properties of r-Hankel and r-Toeplitz matrices whose entries are geometric sequences, and then the determinants, inverse matrix, generalized inverse matrix (the Moore-Penrose inverse), and spectral norms of such matrices are obtained.

Learning the weights using attribute order information for multi-criteria decision making tasks

In multi-criteria decision making, the importance of decision criteria (decision attributes) plays a crucial role. Ranking is a useful technique for expressing the importance of decision criteria in a decision-makers’ preference system. Since weights are commonly utilized for characterizing the importance of criteria, weight determination and assessment are important tasks in multi-criteria decision making and in voting systems as well. In this study, we concentrate on the connection between the preference order of decision criteria and the decision weights. Here, we present an easy-to-use procedure that can be used to produce a sequence of weights corresponding to a decision-makers’ preference order of decision criteria. The proposed method does not require pairwise comparisons, which is an advantageous property especially in cases where the number of criteria is large. This method is based on the application of a class of regular increasing monotone quantifiers, which we refer to as the class of weighting generator functions. We will show that the derivatives of these functions can be used for approximating the criteria weights. Also, we will demonstrate that using weighting generator functions, weights can be inverted in a consistent way. We will deduce the generators for arithmetic and geometric weight sequences, and we will present a one-parameter generator function known as the tau function in continuous-valued logic. We will show that using these weighting generator functions, the weight learning task can be turned into a simple, one-parameter optimization problem.

Atomicity of positive monoids

An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey many of the recent advances regarding positive monoids, and we provide numerous examples to illustrate the complexity of their atomic and arithmetic structures.

Learning Obstacle of Students in Geometrical Sequence and Series

Geometrical sequence and series are mathematic topics that should be mastered by students, but some students encounter obstacles in learning these topics. This research aims to identify the learning obstacle by students in solving geometrical sequence and series problems. The research question was what kind of learning obstacles students face in the topic of geometrical sequence and series. The research is qualitative method with a case study research design. The participants were students of senior high school in Cirebon. Data collection was done by conducting tests and interviews. The test method was to identify the learning obstacles faced by the students and interviews as supporting data. The results show that students meet learning obstacles in solving geometrical sequence and series problems, namely ontogenical obstacle, didactical obstacle, and epistemological obstacles. Ontogenical obstacles were experienced by those students who do not understand the concept of geometrical sequence material, and that the answer written was wrong. The didactical obstacle is experienced by those students who find it easier to do it manually compared to using formulas in solving geometrical sequences and series problems. An epistemological obstacle is experienced by those students who are not able to write down the correct answers because the questions are different from habits. Keywords: geometrical sequence and series, learning obstacle, senior high school

Improved Spatiotemporal Resolution in Echocardiography Using Mixed Geometry Imaging Sequences.

Cardiac ultrasound seeks to image the most dynamic environment in the body-the moving heart. Many modern ultrasound imaging techniques address the tradeoff between spatial and temporal resolution using either narrow focused beams or with broad beam, synthetic aperture (SA) sequences that have been shown to suffer from motion artifacts. Retrospective encoding for conventional ultrasound sequences (REFoCUS) unifies the processing of these various geometric sequences, but the motion sensitivity of this approach has yet to be investigated. We hypothesize that a "mixed sequence" enabled by the REFoCUS method incorporating several beam geometries may better resolve cardiac motion over a wide field of view (FOV) and at a high frame rate. First, the motion sensitivity of REFoCUS was evaluated in simulation for several focused and broad transmit profiles. Focused transmissions resolve both lateral and axial motion much more effectively than broad transmissions, with performance similar to conventional beamforming techniques. Second, a mixed sequence was designed that insonifies the full field-of-view with plane wave (PW) transmissions and key moving targets with focused transmissions. This mixed sequence was tested in simulation and in vivo and was used to image the heart as well as the liver, a low-motion control. By combining a sparse PW sequence ( n=60 ) with a small group of targeted focused transmissions ( n = 10), the anterior mitral valve leaflet (AML) at its peak observed velocity was better resolved. We believe that mixed sequences have strong potential to resolve cardiac motion at clinically relevant frame rates.

Discrete Scale Invariance and U(2) Family of Two-Body Contact Interactions in One Dimension

Because of the absence of indistinguishability constraint, interparticle interactions between nonidentical particles have in general much more variety than those between identical particles. In particular, it is known that there exists a U(2) family of two-body contact interactions between nonidentical particles in one spatial dimension. This paper studies breakdown of continuous scale invariance to discrete scale invariance under this U(2) family of two-body contact interactions in two-body problems of nonidentical particles on the half line. We show that, in contrast to the corresponding identical-particle problem, there exist two distinct channels that admit geometric sequences of two-body bound states.

Zaremba’s Conjecture for Geometric Sequences: An Algorithm

Even though Zaremba’s conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new examples that fulfill the strong version of Zaremba’s conjecture.

An Analysis of Sequence and Series

This article provides a comprehensive exploration of sequences and series, fundamental concepts in mathematics with wide-ranging applications. Starting with an introduction to sequences, including arithmetic and geometric sequences, the discussion extends to finite sequences and n-tuple sequences. Arithmetic and geometric series are explored, with formulas for their sums. The article delves into the properties of sequences and series, discussing convergence, divergence, and absolute convergence. Various types of sequences, such as bounded and monotonic sequences, are defined, and the Cauchy Criterion is introduced as a criterion for the existence of the limit of a sequence. Elementary facts about series, including absolute convergence and the number e as the sum of a series, are covered. Additionally, the concept of sequences of functions is introduced. The article concludes by emphasizing the significance of sequences and series in fields like higher mathematics, art, science, technology, and finance, showcasing their crucial role in decision-making, financial analysis, and risk assessment across diverse disciplines.

Development of MATLAB GUI-Based Mathematics Learning Media with ADDIE Model to Enhance Students' Mathematical Connection Ability

Research was conducted to develop learning media using Matlab GUI to enhance students' connection abilities. The aim was to create media containing valid, practical, and effective arithmetic and geometric sequences and series to support mathematical understanding. The study involved 10 randomly selected eleventh-grade students. Data was collected from validation results, tests, and student questionnaires. The research followed the ADDIE model in five stages: analysis, design, development, implementation, and evaluation. Media validation test results showed 86.5% validity, while student use scored 85.67% for practicality. Effectiveness testing revealed an average student score of 85.84%, demonstrating high effectiveness. These findings indicate that the Matlab GUI-based learning media successfully meet valid, effective, and practical criteria for enhancing students' mathematical connection abilities.

Mathematical Representation Ability-Based Mathematical Contextual Problems of Sequences and Progression Material

This research objective is to analyze and explain the student’s capacity for mathematical representation based on mathematical contextual problems of arithmetic and geometric sequences and progression material. Students enrolled in SMA 1 Sukoharjo X.E10 were the subject of this research. The data collection process used interviews, tests, and documentation. This research used a descriptive research approach with a qualitative research type. Based on the findings of this research, students who have visual representation skills in mathematics are more likely to complete challenges that involve visuals, such as diagrams. Typically, mathematically symbolic representation-capable pupils utilize mathematical formulas and symbols to solve problems. Furthermore, students who have the ability to verbally represent mathematics have a tendency to encrypt problem-solving steps into words.

The Global 2:1 Mean-motion Resonance in HD 110067 is Not Vacant!

Extensive modeling of observations obtained by two space telescopes and 14 ground-based observatories revealed the presence of at least six closely packed planets orbiting HD 110067. Their orbital periods form a nearly perfect resonant chain, in which adjacent periods are in a geometric sequence with a ratio of r = 3/2 (planets b to e) that switches to r = 4/3 (planets e to g). The latter part of the sequence is based on single transits of planets f and g and their periods are model-dependent, as only first-order mean-motion resonances (MMRs) were so far investigated; and there is no physical justification for this assumption besides the simplicity imprinted on to the resonant chain itself. Furthermore, the global 2:1 MMR is apparently occupied, whereas this orbit is vacant in all other multibody (extra) solar systems. We summarize the physical arguments for/against the above r = 4/3 period ratios in anticipation of future observations to resolve the issue.

The Development of Geometric Rows and Sequences Problem-Based Learning with Liveworksheet Media

The lack of development of Student Worksheets that match what is needed by students can be one of the reasons why many students have difficulty understanding learning materials. This study aims to describe the process and results of the development of the mathematics worksheets based on Problem Based Learning by utilizing liveworksheets on geometric sequences and series material. The type of this research is research and development. This worksheet is designed using Canva and Microsoft Word by utilizing liveworksheets. The subjects of the field trial were 28 students of class XI. This research uses the Four D model from Thiagarajan which consists of the stages of definition, design, development, and dissemination. Data collection methods consist of observation, test, and questionnaire methods. Based on the results obtained from trials in the field, it can be concluded that the mathematics worksheets based on Problem Based Learning by utilizing liveworksheets on geometric sequences and series material met the valid, practical, and effective categories. Other researchers can make maximum use of the features provided on the liveworksheets site because in this study they only utilized a few features and must be alert in anticipating when an error occurs on the site while it is being used.

Praxeological Analysis of Mathematics Textbooks for Class XI High School Students on Arithmetic and Geometric Sequences

Praxeological Analysis of Mathematics Textbooks for Class XI High School Students on Arithmetic and Geometric Sequences

A Modern Spin on Archimedes’ Quadrature of the Parabola

Summary In the 3rd century B.C.E., Archimedes wrote his treatise on the quadrature of the parabola, in which he laid out his solution to the problem of finding the area of a parabolic segment. Archimedes attacks the problem from two sides; first, he uses classical mechanics to derive an area formula, and then he uses pure geometry to prove his formula is correct. In this paper, we will present two modern approaches to this problem using calculus. The first approach can be presented to a calculus class that has learned about optimization methods and convergent geometric sequences, and the second can be presented after they have learned the Fundamental Theorem of Calculus.

Structural comparison of conventional and chiral auxetic morphed aircraft rib

Abstract Tri-chiral structures are auxetic structures that show negative Poisson’s ratio. This effect is due to their microstructure and geometric sequence. They are used in the development of novel products as they show improved damping and energy absorption properties. While traditional manufacturing methods remain dysfunctional, the development of additive manufacturing technology provides opportunities for new studies in various industries such as aviation, textile, and automotive. In this study, passive airfoil morphing application was applied and a comparative study was carried out. A two-stage study was conducted. First, the tri-chiral pattern was fabricated by an FDM 3D printer with PLA+ and subjected to the in-plane compression test. Stress–strain curves of the tri-chiral structure were generated. Then, it was used in airfoil morphing applications. Morphed airfoil was also manufactured and a compression test was applied. Secondly, the aerostatic loads of the aircraft were calculated. Both conventional and chiral morphed ribs’ behaviors under flight loads were examined using the FEM and results were compared. The weight difference was calculated. In addition, eigenfrequency and eigenvectors of traditional and chiral ribs were computed and transverse vibration frequencies were expressed. Despite being more than 50 % lighter, chiral morphed rib was found to be stiffer than conventional rib.

Development of e-modules based on brain balance to improve students' mathematical creative thinking skills on geometric sequences and series material

Penelitian ini bertujuan untuk mendeskripsikan proses pengembangan e-modul berbasis keseimbangan otak untuk meningkatkan kemampuan berpikir kreatif matematis siswa pada materi barisan dan deret geometri. Pengembangan e-modul ini menggunakan metode Whole Brain Teaching (WBT) yang merupakan salah satu metode keseimbangan otak. Aplikasi yang digunakan dalam pengembangan e-modul yaitu Microsoft Office Word dan Flip PDF Profesional. Penelitian ini merupakan penelitian pengembangan (Research and Development). Model pengembangan yang digunakan dalam penelitian ini adalah model Plomp yang terdiri dari tiga fase yaitu fase pendahuluan (Preliminarity Phase), fase Pengembangan atau prototipe (Development or Prototype Phase), dan fase penilaian (Assesment Phase). Instrumen yang digunakan dalan penelitian ini adalah lembar validasi, lembar kepraktisan dan instrumen Uji N Gain untuk melihat keefektifan e-modul. Berdasarkan hasil uji validasi yang dilakukan oleh enam orang validator ahli media dan materi memperoleh nilai rata-rata 94,77% dan dikategorikan sangat valid. Sedangkan berdasarkan hasil uji kepraktisan e-modul yang dilakukan kepada guru dan siswa memperoleh nilai rata-rata 91,3% dan dikategorikan sangat baik. Berdasarkan uji N Gain yang dilakukan di kelas X.3 SMAN 1 Darma yang berjumlah 23 orang siswa memperoleh hasil rata-rata pretest sebesar 43,26 dan hasil rata-rata posttest sebesar 78,47 sehingga diperoleh nilai N Gain sebesar 0,62 dan keefektifan e-modul dikategorikan sedang.