Graph Of Function Research Articles

Chromatic M-polynomial of Mobius Function Graphs of Product Groups

Objective: To investigate the chromatic M-polynomials of the Mobius function graph of product groups, specifically focusing on the direct product of finite groups. Methods: We used algebraic and number theoretical techniques to construct a Mobius function graph of product groups and derived the chromatic M-polynomials using graph theoretic methods. Findings: Chromatic M polynomials of Mobius function graphs are introduced with a focus on deriving their significant properties and providing a practical approach for computation. Novelty: This research introduces a novel approach to studying the chromatic M-polynomials of the Mobius function graph of product groups, providing insights into the interplay between group theory, graph theory, and number theory. Our findings contribute connection between chromatic M polynomials and coloring problems, followed by a discussion and understanding of graph polynomials. Keywords: Mobius function graph of groups, Chromatic number, Chromatic M-polynomials, Product group, Dihedral Groups

Influence of individual's age on the characteristics of brain effective connectivity.

Given the increasing number of older adults in society, there is a growing need for studies on changes in the aging brain. The aim of this research is to investigate the effective connectivity of different age groups using resting-state functional magnetic resonance imaging (fMRI) and graph theory. By examining connectivity in different age groups, a better understanding of age-related changes can be achieved. Lifespan pilot data from the Human Connectome Project (HCP) were used to examine dynamic effective connectivity (dEC) changes across different age groups. The Granger causality method with time windowing was employed to calculate dEC. After extracting graph measures, statistical analyses were performed to compare the age groups. Support vector machine and decision tree classifiers were used to classify the different age groups based on the extracted graph measures. Based on the obtained results, it can be concluded that there are significant differences in the effective connectivity among the three age groups. Statistical analyses revealed disassortativity. The global efficiency exhibited a decreasing trend, and the transitivity measure showed an increasing trend with the advancing age. The decision tree classifier showed an accuracy of with Kruskal-Wallis selected features. This study demonstrates that changes in effective connectivity across different age brackets can serve as a tool for better understanding brain function during the aging process.

How Can Graph Theory Inform the Dual-stream Model of Speech Processing? A Resting-state Functional Magnetic Resonance Imaging Study of Stroke and Aphasia Symptomology.

The dual-stream model of speech processing describes a cortical network involved in speech processing. However, it is not yet known if the dual-stream model represents actual intrinsic functional brain networks. Furthermore, it is unclear how disruptions after a stroke to the functional connectivity of the dual-stream model's regions are related to speech production and comprehension impairments seen in aphasia. To address these questions, in the present study, we examined two independent resting-state fMRI data sets: (1) 28 neurotypical matched controls and (2) 28 chronic left-hemisphere stroke survivors collected at another site. We successfully identified an intrinsic functional network among the dual-stream model's regions in the control group using functional connectivity. We then used both standard functional connectivity analyses and graph theory approaches to determine how this connectivity may predict performance on clinical aphasia assessments. Our findings provide evidence that the dual-stream model of speech processing is an intrinsic network as measured via resting-state MRI and that functional connectivity of the hub nodes of the dual-stream network defined by graph theory methods, but not overall average network connectivity, is weaker in the stroke group than in the control participants. In addition, the functional connectivity of the hub nodes predicted linguistic impairments on clinical assessments. In particular, the relative strength of connectivity of the right hemisphere's homologues of the left dorsal stream hubs to the left dorsal hubs, versus to the right ventral stream hubs, is a particularly strong predictor of poststroke aphasia severity and symptomology.

Predicting Ride-Hailing Demand with Consideration of Social Equity: A Case Study of Chengdu

In the realm of shared autonomous vehicle ride-sharing, precise demand prediction is vital for optimizing resource allocation, improving travel efficiency, and promoting sustainable transport solutions. However, existing studies tend to overlook social attributes and demographic characteristics across various regions, resulting in disparities in prediction fairness between areas with plentiful and limited transportation resources. In order to achieve more accurate and fair prediction, an innovative Social Graph Convolution Long Short-Term Memory framework is proposed, incorporating demographic, spatial, and transportation accessibility information into multiple functional graphs, including functional similarity, population structure, and historical demand graphs. Furthermore, Mean Percentage Error indicators are employed in the loss function to balance prediction accuracy and fairness. The findings indicate that there is an enhancement in both prediction accuracy and fairness by at least 8.9% and 12.9%, respectively, compared to base models. Additionally, the predictions for rush hours in both privileged and underprivileged regions exhibit greater precision and rationality, supporting sustainable transport practices. The proposed framework effectively captures the demands of diverse social groups, thereby contributing to the advancement of social equity and long-term sustainability in urban mobility.

Comparative study of various molecular feature representations for solvation free energy predictions of neutral species

Predicting molecular properties with the help of Neural Networks is a common way to substitute or enhance comprehensive quantum-chemical calculations. One of the problems facing researchers is the choice of vectorization approach to representing the solvent and the solute for the estimator model. In this work, 10 different approaches have been investigated for both organic solute and solvent including vectorizers that relied on macroscopic parameters, functional groups classification, molecular graphs, or atomic coordinates. A variation of the Bag of Bonds approach called JustBonds, trained on the MNSol database, showed the best overall performance resulting in RMSD < 2 kcal/mol for the blind dataset that contains the solutes not presented in the training subset and < 1 kcal/mol on records from Solv@TUM database, which is close to contemporary continuum models. We have also demonstrated that the most important bags usually contain heteroatom and play a key role in the solvation process. Furthermore, the small role of solvent vectorization was demonstrated and revealed that approaches based on functional groups or macroscopic solvent parameters are often enough to efficiently represent solvent media.

학교 수학에서 원주율 π의 수학적 연결성에 대한 논의

The 16th Greek alphabet π being a kind of ‘numeral’ as the letter or symbol used to represent a mathematical constant, the ratio of the circumference of a circle to its diameter, the ‘length’ of the circumference and the ‘area’ of a circle with radius 1, the ‘angular measure’ of straight angle, the ‘proportional constant’ of direct proportion y/x=π , the ‘slope’ of the graph of linear function y=πx, a ‘mathematical invariant’ of circle representing the similarity of all circles, and so on is connected with several mathematical concepts and has several meanings. In this paper we investigate those several meanings of π with reference to several mathematical concepts in school mathematics and discuss how to improve teaching and learning of those concepts and π in view of mathematical connectedness of π

Synthesis of High-Selectivity Two-Dimensional Filter Banks Using Sigmoidal Function

This paper explores an application of the sigmoidal function—the complementary integral Gaussian error function (erfc(.))—in two-dimensional (2D) uniform and nonuniform filter bank synthesis. The complementary integral Gaussian error function graph represents a smooth low-pass filter magnitude response. A parameter changes the function slope and increases the magnitude response selectivity. The theory is applied to 2D band-pass filter banks. Exact expressions for the magnitude response parameters are determined. As a result, 2D uniform and nonuniform filter banks with very high selectivity and exact shapes are obtained. Three synthesis examples of 2D filter banks with circular and fan-shaped magnitude responses are provided. The theoretical exposition is supplemented with two examples of image analysis using 2D uniform and nonuniform filter banks. A procedure to reduce the computations in image analysis is proposed. A comparison of filter synthesis between Parks–McLellan’s 2D filters and the erfc(.) demonstrates the significantly shorter calculation time of the proposed method.

Brain networks in newborns and infants with and without sensorineural hearing loss: A functional near-infrared spectroscopy study.

Understanding the impact of early sensory deficits on brain development is essential for understanding developmental processes and developing potential interventions. While previous studies have looked into the impact of prenatal experiences on language development, there is a lack of research on how these experiences affect early language and brain function development in individuals with sensorineural hearing loss (SNHL). To investigate SNHL effects on early brain development and connectivity in 4-month-olds vs healthy newborns and controls. The research involved analyzing the functional brain networks of 65 infants, categorized into three groups: 28 healthy newborns, 22 4-month-old participants with SNHL, and 15 age-matched healthy participants. The resting-state functional connectivity was measured and compared between the groups using functional near-infrared spectroscopy and graph theory to assess the brain network properties. Significant differences were found in resting-state functional connectivity between participants with SNHL and age-matched controls, indicating a developmental lag in brain connectivity for those with SNHL. Surprisingly, SNHL participants showed better connectivity development compared to healthy newborns, with connectivity strengths of 0.13 ± 0.04 for SNHL, 0.16 ± 0.08 for controls, and 0.098 ± 0.04 for newborns. Graph theory analysis revealed enhanced global brain network properties for the SNHL group, suggesting higher communication efficiency at 4 months. No significant differences were noted in network properties between 4-month-old SNHL participants and neonates. A unique pattern of central hubs was observed in the SNHL group, with 2 hubs in the left hemisphere compared to 6 in controls. 4-month-old infants with SNHL have a distinct brain network pattern with efficient long-distance information transmission but less effective local communication compared to age-matched controls.

Enhancing non-invasive pre-surgical evaluation through functional connectivity and graph theory in drug-resistant focal epilepsy

BackgroundEpilepsy, characterized as a network disorder, involves widely distributed areas following seizure propagation from a limited onset zone. Accurate delineation of the epileptogenic zone (EZ) is crucial for successful surgery in drug-resistant focal epilepsy. While visual analysis of scalp electroencephalogram (EEG) primarily elucidates seizure spreading patterns, we employed brain connectivity techniques and graph theory principles during the pre-ictal to ictal transition to define the epileptogenic network. MethodCortical sources were reconstructed from 40-channel scalp EEG in five patients during pre-surgical evaluation for focal drug-resistant epilepsy. Temporal Granger connectivity was estimated ten seconds before seizure and at seizure onset. Results have been analyzed using some centrality indices taken from Graph theory (Outdegree, Hubness). A new lateralization index is proposed by taking into account the sum of the most relevant hubness values across left and right regions of interest. ResultsIn three patients with positive surgical outcomes, analysis of the most relevant Hubness regions closely aligned with clinical hypotheses, demonstrating consistency in EZ lateralization and location. In one patient, the method provides unreliable results due to the abundant movement artifacts preceding the seizure. In a fifth patient with poor surgical outcome, the proposed method suggests a wider epileptic network compared with the clinically suspected EZ, providing intriguing new indications beyond those obtained with traditional electro-clinical analysis. ConclusionsThe proposed method could serve as an additional tool during pre-surgical non-invasive evaluation, complementing data obtained from EEG visual inspection. It represents a first step toward a more sophisticated analysis of seizure onset based on connectivity imbalances, electrical propagation, and graph theory principles.

&lt;span&gt;Enhancing student's mathematical competency in learning the graph and properties of the sine function by Google Sheets app on smartphone&lt;/span&gt;

This study explores the integration of the Google Sheets app on smartphones to enhance students' mathematical competency in understanding the graph and properties of the sine function. The paper was an exploratory study involving 32 11th-grade students. The teaching intervention was structured into four phases: familiarizing with the Google Sheets app, creating the sine function value table, sketching the function graph, and analyzing the graph to deduce the sine function's properties. Data were collected through worksheets and audio -video recordings of group work, focusing on manifestations of mathematical competencies. The findings indicate that students completed the tasks and demonstrated significant competencies in mathematical reasoning and using digital tools. Students were able to identify and solve mathematical problems, utilize Google Sheets effectively, and use logical reasoning to enhance their understanding of the sine function. The study concludes that integrating Google Sheets into mathematics education not only aids in visualizing and comprehending mathematical concepts but also promotes interactive and collaborative learning environments, ultimately fostering a deeper understanding and proficiency in mathematics.

First mixed Laplace eigenfunctions with no hot spots

The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in R n \mathbb {R}^n attains its extrema only on the boundary of the domain. We present an analogous problem for domains with mixed Dirichlet-Neumann boundary conditions. We then solve this problem for Euclidean triangles and a class of planar domains bounded by the graphs of certain piecewise smooth functions.

Accuracy of the geometric shape of the hole in the longitudinal section during honing

The wide application of honing as a finishing treatment of internal cylindrical surfaces for cylinder-piston systems, used in some structures, is caused by high accuracy measured in tenths of a micrometer, and high productivity of the process. The most important indicator of reliable operation of cylinder-piston systems are high requirements for the geometric accuracy of holes. Due to the lack of sufficient theoretical justification for the selection of honing parameters ensuring the accuracy of the geometric shape of the hole in the longitudinal section, the authors proposed a model for the formation of errors in the geometric shape of the hole. The model is built on the kinematic characteristics of the process including the ratio of the honing stone dimensions, the length of the hole, the stroke of the honing head, the ratio of the speeds of translational and rotational movements, and the force action in the processing zone, which changed due to the presence of an overrun of the honing stone. To obtain analytical dependencies ensuring the minimisation of form deviations, the conditions for stock removal for the points of the machined surface were considered, the value of which was taken proportional to the path of movement, and the pressure value. For this purpose, graphs of the distribution functions of displacements and pressure changes were constructed depending on the coordinate of the point location on the generating line of the hole being machined. Using the obtained analytical dependencies, the potential occurrence of a shape error in the form of a saddle shape was found, the dominant factor influencing the value of which is the value of the honing stone overrun. At the same time, it was identified that the ratio of the speeds of translational and rotational movements has an insignificant effect on the violation of the form in the longitudinal section.

Integrating dimensional and scaling analyses with functional modelling and graphs: An approach to comprehend mass transfer in welding

Integrating dimensional and scaling analyses with functional modelling and graphs: An approach to comprehend mass transfer in welding

Sensor self-diagnosis method for a nickel top-blowing furnace based on graph interactive dynamic fusion

Abstract As modern industry gradually advances towards greater automation and intelligence, the scale of nickel top-blowing furnace smelting systems is continuously expanding, leading to an increasing need for sensor maintenance. Traditional periodic evaluations and manual maintenance methods are no longer sufficient to meet the development needs of intelligent sensors. To address this issue, this paper proposes a sensor self-diagnosis method based on graph interactive dynamic fusion, called DLGCN-GIDF. First, a combination of knowledge-driven and data-driven approaches is introduced. By constructing a dual-layer architecture based on a functional module graph network and a sensor graph network, a sensor correlation graph model for the nickel top-blowing furnace system is established. Next, with the aid of a GIDF module, the relative weights between functional modules and sensors are integrated to perform spatiotemporal correlation-based graph fusion. This enables the prediction of spatiotemporal data for sensors from a system perspective. Finally, the goal of sensor self-diagnosis is achieved using a standardised residual testing algorithm. Taking a nickel top-blowing furnace smelting system as an example, the feasibility and effectiveness of our method of sensor fault self-diagnosis are verified.

An invariant for affine maximal type equations

Let y:M→Rn+1 be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space Rn+1, given as the graph of a smooth, strictly convex function xn+1=f(x1,...,xn) defined on a domain Ω⊂Rn. Considering the α-relative normalization of the graph of the convex function f, we will prove a Bernstein theorem for a class of nonlinear, fourth order partial differential equations of affine maximal type. As applications, we define an invariant of the equations and prove a rigidity result of the complete Tn-invariant Kähler metric on complex torus (C⁎)n with vanishing scalar curvature for n≤5.

Increased coherence predicts medical refractoriness in patients with temporal lobe epilepsy on monotherapy

Among patients with epilepsy, 30–40% experience recurrent seizures even after adequate antiseizure medications therapies, making them refractory. The early identification of refractory epilepsy is important to provide timely surgical treatment for these patients. In this study, we analyze interictal electroencephalography (EEG) data to predict drug refractoriness in patients with temporal lobe epilepsy (TLE) who were treated with monotherapy at the time of the first EEG acquisition. Various EEG features were extracted, including statistical measurements and interchannel coherence. Feature selection was performed to identify the optimal features, and classification was conducted using different classifiers. Functional connectivity and graph theory measurements were calculated to identify characteristics of refractory TLE. Among the 48 participants, 34 (70.8%) were responsive, while 14 (29.2%) were refractory over a mean follow-up duration of 38.5 months. Coherence feature within the gamma frequency band exhibited the most favorable performance. The light gradient boosting model, employing the mutual information filter-based feature selection method, demonstrated the highest performance (AUROC = 0.821). Compared to the responsive group, interchannel coherence displayed higher values in the refractory group. Interestingly, graph theory measurements using EEG coherence exhibited higher values in the refractory group than in the responsive group. Our study has demonstrated a promising method for the early identification of refractory TLE utilizing machine learning algorithms.

Polynomials on graphs of analytical functions

Polynomials on graphs of analytical functions

Classifying Unconscious, Psychedelic, and Neuropsychiatric Brain States with Functional Connectivity, Graph Theory, and Cortical Gradient Analysis.

Accurate and generalizable classification of brain states is essential for understanding their neural underpinnings and improving clinical diagnostics. Traditionally, functional connectivity patterns and graph-theoretic metrics have been utilized. However, cortical gradient features, which reflect global brain organization, offer a complementary approach. We hypothesized that a machine learning model integrating these three feature sets would effectively discriminate between baseline and atypical brain states across a wide spectrum of conditions, even though the underlying neural mechanisms vary. To test this, we extracted features from brain states associated with three meta-conditions including unconsciousness (NREM2 sleep, propofol deep sedation, and propofol general anesthesia), psychedelic states induced by hallucinogens (subanesthetic ketamine, lysergic acid diethylamide, and nitrous oxide), and neuropsychiatric disorders (attention-deficit hyperactivity disorder, bipolar disorder, and schizophrenia). We used support vector machine with nested cross-validation to construct our models. The soft voting ensemble model marked the average balanced accuracy (average of specificity and sensitivity) of 79% (62-98% across all conditions), outperforming individual base models (70-76%). Notably, our models exhibited varying degrees of transferability across different datasets, with performance being dependent on the specific brain states and feature sets used. Feature importance analysis across meta-conditions suggests that the underlying neural mechanisms vary significantly, necessitating tailored approaches for accurate classification of specific brain states. This finding underscores the value of our feature-integrated ensemble models, which leverage the strengths of multiple feature types to achieve robust performance across a broader range of brain states. While our approach offers valuable insights into the neural signatures of different brain states, future work is needed to develop and validate even more generalizable models that can accurately classify brain states across a wider array of conditions.

How students interpret points and positions in graphs of functions: an intersection of theoretical frameworks

In this paper, we examine the intersection of two previously recognised dimensions of students’ interpretations of symbolic notations within graphs of functions. One dimension distinguishes location-thinking, where notations refer only to a point’s location on a graph, from value-thinking, where such a point is treated as a multiplicative object. The other dimension distinguishes a nominal interpretation of expressions, where expressions refer to positions in the plane, from cardinal and magnitude interpretations, where expressions describe a discrete count of units, or measure of length, respectively. Taken together these dimensions provide six distinct ways students interpret expressions, especially those involving function notation, on graphs. Further, the frameworks offer suggestions about the ways of thinking underlying each interpretation. Each case reveals new meanings and affordances indicated by the interplay between the two dimensions. We provide both a theoretical account and empirical example of each case.

MDD brain network analysis based on EEG functional connectivity and graph theory

BackgroundExisting studies have shown that the brain network of major depression disorder (MDD) has abnormal topologies. However, constructing reliable MDD brain networks is still an open problem. New methodThis paper proposed a reliable MDD brain network construction method. First, seven connectivity methods are used to calculate the correlation between channels and obtain the functional connectivity matrix. Then, the matrix is binarized using four binarization methods to obtain the EEG brain network. Besides, we proposed an improved binarization method based on the criterion of maximizing differences between groups: the adaptive threshold (AT) method. The AT can automatically set the optimal binarization threshold and overcome the artificial influence of traditional methods. After that, several network metrics are extracted from the brain network to analyze inter-group differences. Finally, we used statistical analysis and Fscore values to compare the performance of different methods and establish the most reliable method for brain network construction. ResultsIn theta, alpha, and total frequency bands, the clustering coefficient, global efficiency, local efficiency, and degree of the MDD brain network decrease, and the path length of the MDD brain network increases. Comparison with existing methodsThe results show that AT outperforms the existing binarization methods. Compared with other methods, the brain network construction method based on phase-locked value (PLV) and AT has better reliability. ConclusionsMDD has brain dysfunction, particularly in the frontal and temporal lobes.