Circle Theorem Research Articles

Möbius Invariant Y-systems (Cluster Structures) for Miquel Dynamics

Abstract Miquel dynamics is a discrete time dynamics for circle patterns, which relies on Miquel’s six circle theorem. Previous work shows that the evolution of the circle centers satisfy the dSKP equation on the octahedral lattice $A_{3}$. As a consequence, Miquel dynamics is a discrete integrable system. Moreover, Miquel dynamics give rise to a real-valued cluster structure. The evolution of the cluster variables under Miquel dynamics is also called a Y-system in the discrete integrable systems community. If the Y-system is real positive-valued then the circle pattern is accompanied by an invariant dimer model, an exactly solvable model studied in statistical physics. However, while circle patterns are Möbius invariant, the circle centers and the Y-system are not Möbius invariant, which violates the so called transformation group principle. In this article we show that half the intersection points satisfy the dSKP equation as well, and we introduce two new real-valued Y-systems for Miquel dynamics that involve only the intersection points. Therefore, the new Y-systems are Möbius invariant, and thus satisfy the transformation group principle. We also show that the circle centers and intersection points combined satisfy the dSKP equation on the 4-dimensional octahedral lattice $A_{4}$. In addition, we present two more complex-valued Y-systems for Miquel dynamics, which are real-valued in and only in the case of integrable circle patterns. We also investigate the special cases of harmonic embeddings and s-embeddings, which relate to the spanning tree and Ising model, respectively.

Multistability of state-dependent switched neural networks with Mexican-hat-type activation functions

This paper investigates the issue of multistability in state-dependent switched neural networks (SSNNs) with Mexican-hat-type activation functions (AFs). It establishes the coexistence and stability of multiple equilibrium points (EPs). Initially, the state space is partitioned based on the geometric characteristics of the Mexican-hat-type AF, enabling to determine the positions of the EPs. Secondly, the coexistence of 9h17h25h33h4 EPs for n-neurons SSNNs under specific sufficient conditions is proved with the Brouwer’s fixed-point theorem. Next, by using diagonally dominant matrix theory and Gershgorin circle theorem, it is proven that there are 5h14h23h32h4 asymptotically stable EPs under some conditions, where h1,h2,h3 and h4 are nonnegative integers satisfying 0≤h1+h2+h3+h4≤n. Therefore, we can obtain that SSNNs can have larger storage capacity by selecting the appropriate parameters hi,i=1,2,3,4. Finally, the correctness of the results in this paper is verified through two numerical examples.

A Novel Clustering Algorithm Integrating Gershgorin Circle Theorem and Nonmaximum Suppression for Neural Spike Data Analysis

(1) Problem Statement: The development of clustering algorithms for neural recordings has significantly evolved, reaching a mature stage with predominant approaches including partitional, hierarchical, probabilistic, fuzzy logic, density-based, and learning-based clustering. Despite this evolution, there remains a need for innovative clustering algorithms that can efficiently analyze neural spike data, particularly in handling diverse and noise-contaminated neural recordings. (2) Methodology: This paper introduces a novel clustering algorithm named Gershgorin—nonmaximum suppression (G–NMS), which incorporates the principles of the Gershgorin circle theorem, and a deep learning post-processing method known as nonmaximum suppression. The performance of G–NMS was thoroughly evaluated through extensive testing on two publicly available, synthetic neural datasets. The evaluation involved five distinct groups of experiments, totaling eleven individual experiments, to compare G–NMS against six established clustering algorithms. (3) Results: The results highlight the superior performance of G–NMS in three out of five group experiments, achieving high average accuracy with minimal standard deviation (SD). Specifically, in Dataset 1, experiment S1 (various SNRs) recorded an accuracy of 99.94 ± 0.01, while Dataset 2 showed accuracies of 99.68 ± 0.15 in experiment E1 (Easy 1) and 99.27 ± 0.35 in experiment E2 (Easy 2). Despite a slight decrease in average accuracy in the remaining two experiments, D1 (Difficult 1) and D2 (Difficult 2) from Dataset 2, compared to the top-performing clustering algorithms in these categories, G–NMS maintained lower SD, indicating consistent performance. Additionally, G–NMS demonstrated robustness and efficiency across various noise-contaminated neural recordings, ranging from low to high signal-to-noise ratios. (4) Conclusions: G–NMS’s integration of deep learning techniques and eigenvalue inclusion theorems has proven highly effective, marking a significant advancement in the clustering domain. Its superior performance, characterized by high accuracy and low variability, opens new avenues for the development of high-performing clustering algorithms, contributing significantly to the body of research in this field.

Gershgorin circle theorem-based feature extraction for biomedical signal analysis.

Recently, graph theory has become a promising tool for biomedical signal analysis, wherein the signals are transformed into a graph network and represented as either adjacency or Laplacian matrices. However, as the size of the time series increases, the dimensions of transformed matrices also expand, leading to a significant rise in computational demand for analysis. Therefore, there is a critical need for efficient feature extraction methods demanding low computational time. This paper introduces a new feature extraction technique based on the Gershgorin Circle theorem applied to biomedical signals, termed Gershgorin Circle Feature Extraction (GCFE). The study makes use of two publicly available datasets: one including synthetic neural recordings, and the other consisting of EEG seizure data. In addition, the efficacy of GCFE is compared with two distinct visibility graphs and tested against seven other feature extraction methods. In the GCFE method, the features are extracted from a special modified weighted Laplacian matrix from the visibility graphs. This method was applied to classify three different types of neural spikes from one dataset, and to distinguish between seizure and non-seizure events in another. The application of GCFE resulted in superior performance when compared to seven other algorithms, achieving a positive average accuracy difference of 2.67% across all experimental datasets. This indicates that GCFE consistently outperformed the other methods in terms of accuracy. Furthermore, the GCFE method was more computationally-efficient than the other feature extraction techniques. The GCFE method can also be employed in real-time biomedical signal classification where the visibility graphs are utilized such as EKG signal classification.

Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian Matrices in Computer Vision Applications.

In graph theory, the weighted Laplacian matrix is the most utilized technique to interpret the local and global properties of a complex graph structure within computer vision applications. However, with increasing graph nodes, the Laplacian matrix's dimensionality also increases accordingly. Therefore, there is always the "curse of dimensionality"; In response to this challenge, this paper introduces a new approach to reducing the dimensionality of the weighted Laplacian matrix by utilizing the Gershgorin circle theorem by transforming the weighted Laplacian matrix into a strictly diagonal domain and then estimating rough eigenvalue inclusion of a matrix. The estimated inclusions are represented as reduced features, termed GC features; The proposed Gershgorin circle feature extraction (GCFE) method was evaluated using three publicly accessible computer vision datasets, varying image patch sizes, and three different graph types. The GCFE method was compared with eight distinct studies. The GCFE demonstrated a notable positive Z-score compared to other feature extraction methods such as I-PCA, kernel PCA, and spectral embedding. Specifically, it achieved an average Z-score of 6.953 with the 2D grid graph type and 4.473 with the pairwise graph type, particularly on the E_Balanced dataset. Furthermore, it was observed that while the accuracy of most major feature extraction methods declined with smaller image patch sizes, the GCFE maintained consistent accuracy across all tested image patch sizes. When the GCFE method was applied to the E_MNSIT dataset using the K-NN graph type, the GCFE method confirmed its consistent accuracy performance, evidenced by a low standard deviation (SD) of 0.305. This performance was notably lower compared to other methods like Isomap, which had an SD of 1.665, and LLE, which had an SD of 1.325; The GCFE outperformed most feature extraction methods in terms of classification accuracy and computational efficiency. The GCFE method also requires fewer training parameters for deep-learning models than the traditional weighted Laplacian method, establishing its potential for more effective and efficient feature extraction in computer vision tasks.

A Hilbert‐space variant of Geršgorin's circle theorem

A Hilbert‐space variant of Geršgorin's circle theorem

Exploring Lee-Yang and Fisher zeros in the 2D Ising model through multipoint Padé approximants

We present a numerical calculation of the Lee-Yang and Fisher zeros of the 2D Ising model using multipoint Padé approximants. We perform simulations for the 2D Ising model with ferromagnetic couplings both in the absence and in the presence of a magnetic field using a cluster spin-flip algorithm. We show that it is possible to extract genuine signature of Lee-Yang and Fisher zeros of the theory through the poles of magnetization and specific heat, using the multipoint Padé method. We extract the poles of magnetization using Padé approximants and compare their scaling with known results. We verify the circle theorem associated to the well known behavior of Lee-Yang zeros. We present our finite volume scaling analysis of the zeros done at T=Tc for a few lattice sizes, extracting to a good precision the (combination of) critical exponents βδ. The computation at the critical temperature is performed after the latter has been determined via the study of Fisher zeros, thus extracting both βc and the critical exponent ν. Results already exist for extracting the critical exponents for the Ising model in two and three dimensions making use of Fisher and Lee-Yang zeros. In this work, multipoint Padé is shown to be competitive with this respect and thus a powerful tool to study phase transitions. Published by the American Physical Society 2024

Efficient interpolation of molecular properties across chemical compound space with low-dimensional descriptors

We demonstrate accurate data-starved models of molecular properties for interpolation in chemical compound spaces with low-dimensional descriptors. Our starting point is based on three-dimensional, universal, physical descriptors derived from the properties of the distributions of the eigenvalues of Coulomb matrices. To account for the shape and composition of molecules, we combine these descriptors with six-dimensional features informed by the Gershgorin circle theorem. We use the nine-dimensional descriptors thus obtained for Gaussian process regression based on kernels with variable functional form, leading to extremely efficient, low-dimensional interpolation models. The resulting models trained with 100 molecules are able to predict the product of entropy and temperature (S × T) and zero point vibrational energy (ZPVE) with the absolute error under 1 kcal mol−1 for >78 % and under 1.3 kcal mol−1 for >92 % of molecules in the test data. The test data comprises 20 000 molecules with complexity varying from three atoms to 29 atoms and the ranges of S × T and ZPVE covering 36 kcal mol−1 and 161 kcal mol−1, respectively. We also illustrate that the descriptors based on the Gershgorin circle theorem yield more accurate models of molecular entropy than those based on graph neural networks that explicitly account for the atomic connectivity of molecules.

Investigating the Philosophical View of Teaching Circles Theorem

Circle theorem was seen as the most challenging mathematics topic most students’ in Ghana run away from. During the WASCE, most students run away from answering mathematics questions that are in line with circle theorem. Such challenge has drawn the attention of mathematics teachers and researchers in mathematics education to find out the factors that drives students from attempting circle theorem question and the appropriate approach that can be used to teacher and to solve questions in circle theorem. The study aimed to investigate the philosophical view of teaching circle theorem. The literature review was based on ontological view of circle theorem, axiological view of circle theorem, and epistemological view of circle theorem. The study concluded that, the three philological view of mathematics has a unique significant effect on teaching and learning circle theorem.

Engaging Neural Plasticity in Senior High School Students: The Impact of Guided Discovery Teaching Method on Achievement in Circle Theorems

Purpose: The study's objective was to ascertain how circle theorem achievement in SHS students was affected by the guided discovery teaching method, a student-centered approach associated with improved neural plasticity.
 Materials and Methods: Two Form 2 classes from various Wa Municipal schools were chosen for the study using convenient, purposeful, and straightforward random sampling techniques. The study adopted a non-equivalent quasi-experimental design to compare students who are taught with the guided discovery method and the traditional method of teaching circle theorems concepts. The sample size was composed of 164 students. Using a separate t-test and descriptive statistics, the Geometry Achievement test was investigated. A pre-test was given before the experiment (post-test) began. The students who engaged in guided discovery instruction outperformed than those who did not when teaching and learning Circle theorems.
 Findings: The findings suggested that student-centered methods like guided discovery can greatly improve students’ achievement in the study of circle theorems. One of the implications derived from the study indicated that guided discovery teaching approach offers students the chance to put a method of learning into practice after they have used it. This was done by using illustrations of diagrams on cardboard. This must be considered in the planning of educators and subject-matter experts. For pre-tertiary education, the government must make it mandatory to use cardboard, mathematical instruments, and instructional sheets as teaching aids. Since visual representations of Euclidean Geometry diagrams bring reality to teaching and learning through pictures and diagrams, the importance of visualization and experimentation in learning circle theorems should not be underestimated by teachers or students. This will improve their conceptual understanding.
 Implications to Theory, Practice and Policy: One can recommend that in order to enhance the performance of SHS students, it will be most advantageous to introduce guided discovery teaching methods to pre- and in-service teachers, through promotion by Ghana Education Service and/or other stakeholders in the education sector. This introduction may be distributed through workshops and seminars for mathematics teachers' instructional techniques and skills will improve as a result. This study has added a lot to our understanding of the world. The guided discovery teaching method approach to teaching circle theorems has been strengthened and expanded as a result of this research, first and foremost. This thesis thus makes a substantial enhancement of the body of knowledge. The research also explains and backs up the notion that guided discovery methods aid students' academic endeavors. This shows that the teaching process engages students' attention and improves their capacity for memory and recall.

An Efficient Block Successive Upper-Bound Minimization Algorithm for Caching a Reconfigurable Intelligent Surface-Assisted Downlink Non-Orthogonal Multiple Access System

With the booming rollout of 5G communication, abundant new technologies have been proposed for quality of service requirements. In terms of the betterment in transmission coverage, mobile edge caching (MEC) has shown potential in reducing the transmission outage. The performance of MEC, meanwhile, can be promisingly enhanced by reconfigurable intelligent surfaces (RIS). Under this context, we explore a system comprising a small base-station (SBS) with limited cache capacity, two users, and one RIS. The SBS transmits the contents from the cache or fetches them from the remote backhaul hub to communicate with users through directional and possibly reflective channels. In this point-to-multipoint connection, non-orthogonal multiple access (NOMA) is applied, improving the capacity of the system. To minimize the outage probability, we first propose a caching policy from entropy perspective, based on which we investigate the beamforming and power allocation problem. The issue, however, is non-convex and involves multi-dimensional optimization. To address this, we introduce an efficient block successive upper-bound minimization algorithm, grounded in Gershgorin’s circle theorem. This algorithm aims to find the globally optimal solution for power allocation and RIS beamformer, considering both the channel condition and content popularity. Numerical studies are performed to verify the effectiveness of the proposed algorithm.

The conditions for the analog of QED photons in semi-classical periodically driven systems

The Floquet and quantum electrodynamics (QED) Hamiltonians are widely used in various contexts for light-matter interactions. While they exhibit structural similarity, the QED Hamiltonian has a bounded spectrum while the Floquet Hamiltonian does not. Thus, it remains uncertain if they share the same or similar spectra, even at high energy with a substantial average photon count. Using the Gershgorin circle theorem, we bound analytically the difference between the spectra of the QED and Floquet Hamiltonians. We establish a common spectrum by imposing the constraints of high photon numbers and narrow photon statistics. Following the analytic proof, we numerically demonstrate this bound’s implications on a model Xe atom previously used in high harmonic generation, showing correspondence between Floquet and QED photons.

The Conway Circle Theorem: Equivalence and Generalization

Summary The elegant and surprising Conway Circle Theorem (CCT) for triangles has become more widely known following the death of its discoverer John Horton Conway (1938–2020). The logical equivalence of the CCT and the well-known triangle incircle tangent length formulas is demonstrated, thereby also providing an alternative proof of the CCT. The CCT is then generalized for tangential polygons.

Complexity and algorithm of setting optimal location for data sink in real-time NOMA-based IIoTs

Real-time performance is one of the most vital metrics in Non-Orthogonal Multiple Access (NOMA) based Industrial Internet of Things (IIoTs) applications. Since the relative geographic relationship between data sink and wireless sensors affects the transmission parallelism and thus the real-time performance, setting a reasonable location for the data sink is thus a feasible way to high real-time performance for applications where the locations of wireless sensors are fixed. In this paper, we consider how to find an optimal location for the data sink to minimize average access delay. We first formulate the problem and prove it to be NP-Complete by presenting an original reduction proof from classic set partition problem. Second, by tightening the NOMA decoding constraint of the original problem, a heuristic algorithm, which is designed based on Apollonian Circle Theorem and Dilworth Theorem, obtains a feasible location for the data sink. Simulation results reveal that the real-time performance loss of the heuristic algorithm is no more than 50% of the best one. Compared with the classic TDMA scheme, the real-time performance increases by more than 60% for some typical settings, and it can even reach 70% for the linear network topology, owing to the full exploitation of NOMA parallelism.

Maximum Likelihood Identification of Backflow Vortex Instability in Rocket Engine Inducers

Abstract Bayesian estimation is applied to the analysis of backflow vortex instabilities in typical three- and four-bladed liquid propellant rocket engine inducers. The flow in the impeller eye is modeled as a set of equally intense and evenly spaced 2D axial vortices, located at the same radial distance from the axis and rotating at a fraction of the impeller speed. The circle theorem is used to predict the flow pressure in terms of the vortex number, intensity, rotational speed, and radial position. The theoretical spectra so obtained are frequency broadened to mimic the dispersion of the experimental results and parametrically fitted to the measured data by maximum likelihood estimation with equal and independent Gaussian errors. The method is applied to three inducers, tested in water at room temperature and different operating conditions. It successfully characterizes backflow instabilities using the signals of a single pressure transducer flush-mounted in the impeller eye, effectively bypassing the aliasing limitations and the data acquisition/reduction complexities of traditional multiple-sensor cross-correlation methods. The identification returns the estimates of the model parameters and their standard deviations, providing the information necessary for assessing the accuracy and statistical significance of the results. The flowrate is found to be the major factor affecting the backflow vortex instability, which, on the other hand, is rather insensitive to the occurrence of cavitation. The results are consistent with the data reported in the literature, as well as with those generated by the auxiliary models specifically developed for initializing the maximum likelihood searches and supporting the identification procedure.

A Gershgorin Circle Theorem Inspired Controller for Renewable Energy Source Integrated Power Systems and Multimicrogrids

Increasing load demands, variations in the operating conditions, continuous integration of the renewable energy sources constitute a multifarious challenge for the researchers to maintain the quality in performances of a load frequency controller. To contemplate such issues, a Gershgorin circle theorem inspired controller assimilated with the particle swarm optimization is proposed for multiarea power systems. The proposed method ensures system stability while abating the frequency variations in the operational areas and the power deviations in the interconnected tie-lines. The present analysis is substantiated through an interconnected photovoltaic (PV) integrated thermal power systems. Furthermore, the competence of the proposed method against increasing complexities is validated by considering an interconnected multimicrogrid systems consisting of renewable and nonrenewable energy sources. The performances of the proposed controller are demonstrated considering fixed and time-varying load disturbances, and fluctuations in renewable energy sources. Results corroborate that the proposed method is successful in maintaining the system stability while mitigating the frequency and the power deviations in all such operating conditions. A comparative analysis of the proposed method with prevalent control techniques (e.g., PI and PID-based controllers, tilt–integral–derivative, integral–tilt–derivative) and optimization algorithms (e.g., genetic algorithm, gravitational search algorithm, imperialist competitive algorithm) is presented to highlight the substantial performance improvements in the considered systems.

Additive autoregressive models for matrix valued time series

In this article, we develop additive autoregressive models (Add‐ARM) for the time series data with matrix valued predictors. The proposed models assume separable row, column and lag effects of the matrix variables, attaining stronger interpretability when compared with existing bilinear matrix autoregressive models. We utilize the Gershgorin's circle theorem to impose some certain conditions on the parameter matrices, which make the underlying process strictly stationary. We also introduce the alternating least squares estimation method to solve the involved equality constrained optimization problems. Asymptotic distributions of the parameter estimators are derived. In addition, we employ hypothesis tests to run diagnostics on the parameter matrices. The performance of the proposed models and methods is further demonstrated through simulations and real data analysis.

Teacher’s Pedagogical Content Knowledge and Students’ Academic Performance in Circle Theorem

Aim: The main purpose of this study was to investigate the impact of teacher's pedagogical content knowledge (TPCK) on students' academic performance in circle theorem.
 Methods: The study used a mixed-methods research approach and a sequential explanatory design with a sample 210 students selected through the probability systematic sampling technique. The primary data was collected using questionnaire, an interview guide and a circle theorem achievement test. The data collected was analyzed using, the regression tool, and deductive manual thematic analysis, which was used for only the qualitative data collected from the interviews.
 Results: The study found a significant relationship between the independent variable teacher’s pedagogical content knowledge and the dependent variable academic performance in circle theorem signifying that students’ performance in circles theorem depends on the pedagogical content knowledge of the teacher.
 Conclusion: Based on the findings that emanated from the data analysed, the study concluded that teachers' pedagogical content knowledge has a significant relationship with students’ academic performance in circle theorem.
 Recommendation: the study recommends that Ghana Education Service organise training conferences and workshops aimed at improving teachers’ pedagogical circle theorem content knowledge.

Research on the dynamics of a restricted cavitation bubble near a symmetric Joukowsky hydrofoil

In the present paper, the restricted cavitation bubble dynamics near a symmetric Joukowsky hydrofoil are researched theoretically and experimentally. Using Kelvin impulse theory, the Joukowsky transformation, and the circle theorem, a theoretical model for restricted bubble dynamics is established to analyze the collapse jet characteristics. The validity of this model is then verified using high-speed photographic experiments. The velocity and direction of the collapse jet at specific position angles are quantitatively analyzed. Furthermore, the spatial characteristics of the Kelvin impulse direction near the symmetric Joukowsky hydrofoil are revealed by theoretical results. The main conclusions include the following: (1) the new theoretical model is proven to be effective in predicting the direction of the collapse jet for a restricted bubble near a symmetric Joukowsky hydrofoil. (2) As the distance between the bubble and hydrofoil increases, the collapse jet direction changes from pointing toward the nearest wall to pointing toward the center of the hydrofoil. (3) The variation rate of the Kelvin impulse direction for the restricted bubble is very sensitive to the bubble position near the two ends of the symmetric Joukowsky hydrofoil.

Teachers’ Geometry Vocabulary Competence and Students’ Academic Performance in Circle Theorem

Teachers’ Geometry Vocabulary Competence and Students’ Academic Performance in Circle Theorem