Symmetric Index Research Articles

Generalized Zagreb indices for some silicate networks

Graph theory received a great attention in chemical graph theory which is a branch of mathematical chemistry to model a molecule through a graph where vertices are considered as the atoms of the molecule and edges are considered as the bonds between them. A topological index or molecular descriptor of a graph is a numeric quantity obtained mathematically which characterizes its topology and it is invariant under graph isomorphisms. In chemical graph theory, various topological indices play important role in predicting physicochemical properties of chemical compounds. In this paper, we compute the general Zagreb index of some silicate networks and hence obtain some degree based topological indices such as Zagreb indices, forgotten topological index, redefined Zagreb index, general first Zagreb index, general Randić index and symmetric division deg index of silicate networks.

Proof of a Conjecture on Symmetric Division Deg Index of Graphs

Proof of a Conjecture on Symmetric Division Deg Index of Graphs

On Lanzhou Index of Graphs

Let G = ( V , E ) be a simple graph with vertex set V ( G ) and edge set E ( G ) . The Lanzhou index of a graph G is defined by L z ( G ) = ∑ u ∈ V ( G ) d 2 u ¯¯¯ d u , where d u ( ¯¯¯ d u resp.) denotes the degree of the vertex u in G ( ¯¯¯¯ G , the complement graph of G resp.). It has predictive powers to provide insights of chemical relevant properties of chemical graph structures. In this paper we discuss some properties of Lanzhou index. Several inequalities having lower and upper bound for the Lanzhou index in terms of first, second and third Zagreb indices, radius of graph, eccentric connectivity index, Schultz index, inverse sum indeg index and symmetric division deg index, are discussed. At the end the Lanzhou index of corona and join of graphs have been derived.

What Did Turkey Do in terms of Export Competitiveness After 2000?

Competitiveness relates to an economy's ability to produce products efficiently and effectively, enabling it to compete in domestic and global markets. It encompasses factors such as productivity, innovation, quality, cost-efficiency, and market access. Competitive economies are more likely to attract investment, create jobs, and achieve higher levels of economic development. Especially competitiveness rankings provide opportunity of benchmarking, effective policy guidance, attractive investment, global comparison and eveluation. In this respect, this study has been conducted to examine Turkey's export structure in the period 2001-2021 within the framework of Harmonized Commodity Description and Coding System and to compare its competitiveness with the Revealed Symmetric Comparative Advantage Index (RSCA). The data used to calculate the RSCA values are taken from TradeMap, the database of the International Trade Centre (ITC). The comparison was made with Turkey's highest trading partners, the countries with the largest share in Turkey's export. While Turkey had RSCA in 37 products in 2001, it increased to 45 products in 2021. Similar to Podoba et al. (2021), product groups are categorised according to observable trends in RSCA values. In 2001, 5 products with RSCA decreased, and in 2021, 12 products with comparative advantage replaced them

Spatial and temporal variation of three Eddy-Covariance flux footprints in a Tropical Dry Forest

Tropical Dry Forest (TDF) is a distinctive ecosystem characterized by distinct seasonal changes, primarily composed of deciduous trees adapted to endure prolonged drought periods. Obtaining ecosystem-scale flux measurements using the Eddy Covariance (EC) technique is crucial for comprehending the spatiotemporal dynamics of interactions between the land and the atmosphere. It is imperative to have information about the variability in the contribution source area of the Flux Footprint Area (FFA) to assess the impact of changing microclimate variables on the EC FFA. In this study, we describe and evaluate the influence of various phases of the El Niño Southern Oscillation (ENSO) on the spatial and temporal climatology of FFA at three EC sites within the TDF of the Santa Rosa National Park-Environmental Super Site (SRNP-EMSS). Our analysis covered the period from 2014 to 2021 on a seasonal scale and employed a 2D model. Our findings indicate that: a) seasonal climatology FFAs exhibit variation across locations and over time due to factors such as forest canopy height, site topography, wind direction, and atmospheric stability conditions; b) FFAs are smaller during dry seasons compared to wet seasons; and c) multiple ENSO phases (cold, neutral, and warm) significantly affect the seasonal climatology of FFAs. Notably, during the moderate La Niña phase (2020/2021), the mean FFA size was 4.20 km2, which was 23.80% larger than the mean FFA during the intense El Niño phase (2015/2016) at 3.20 km2. Additionally, during La Niña phases, FFAs were more asymmetrically distributed around the EC tower compared to El Niño phases, with mean symmetric indices of 0.35 and 0.29, respectively. Finally, the overlap among the three EC climatology FFAs during La Niña phases was 35% greater than that during the El Niño phases. Our study highlights how variations in FFAs can significantly impact intra-annual flux estimates.

Application of machine learning regression models to inverse eigenvalue problems

In this work, we study the numerical solution of inverse eigenvalue problems from a machine learning perspective. Two different problems are considered: the inverse Sturm-Liouville eigenvalue problem for symmetric potentials and the inverse transmission eigenvalue problem for spherically symmetric refractive indices. Firstly, we solve the corresponding direct problems to produce the required eigenvalues datasets in order to train the machine learning algorithms. Next, we consider several examples of inverse problems and compare the performance of each model to predict the unknown potentials and refractive indices respectively, from a given small set of the lowest eigenvalues. The supervised regression models we use are k-Nearest Neighbours, Random Forests and Multi-Layer Perceptron. Our experiments show that these machine learning methods, under appropriate tuning on their parameters, can numerically solve the examined inverse eigenvalue problems.

Connection number topological aspect for backbone DNA networks.

The present study investigates the complex topological characteristics of DNA networks, with a specific emphasis on the innovative metric known as Connection Number (CN) as a key factor in determining network structure. The Connection Number, represented as CN(v) for a vertex v, measures the count of unique paths that link v to every other vertex in the network. By employing rigorous mathematical modeling and analysis techniques, we are able to reveal the profound implications of CN (complex networks) in characterizing the structural robustness and dynamics of information flow within DNA networks. The study of how the theory of graphs and chemicals interact is known as chemical graph theory. This paper, computing the hyper Zagreb connection index, augmented connection index, inverse sum connection index, harmonic connection index, symmetric division connection index, geometric arithmetic connection index, and atom bond connectivity connection index, of two significant types of backbone DNA and Barycentric subdivision of backbone DNA networks. Direct method computation is used to produce these Connection-based topological descriptors.

PSVI-17 Application of Stride Length to Assess Leg Soundness for Growing Gilt

Abstract The durability of sows would be affected by their leg soundness. Problems of leg or lameness is the second reason for culling especially in the first three parities. Observing the leg structure during the growth period of pigs is necessary in selection programs because traits related to leg structure have intermediate to low heritability. However, long-term tracking of changes in feet and leg structure of pigs using quantitative measuring methods has not been conducted. Such measuring methods have not yet been applied on farms so far. Thus, this study aimed to use stride length data from pressure mat to evaluate leg soundness in growing gilts. A pressure mat system was used in this study to collect stride length (SL) data and ground reaction force (GRF) data of 23 purebred gilts at 4.5, 6.0, and 7.5 months of age. The SL data obtained in each pig was used to calculate the symmetry index with the SL records of the left and right legs for both front and rear. Similarly, the GRF data enabled the calculation of GRF symmetry index of the left and right leg of a pig for both front and rear. The confusion matrix was applied to compute the accuracy, sensitivity, and specificity by comparing visual inspection of gilt culling or the SL symmetry index to the GRF symmetry index as the standard. The accuracy, sensitivity, and specificity of visual inspection of gilt culling versus GRF symmetry index at 7.5 months of age were 80%, 31%, and 100%, respectively. Meanwhile, the accuracy, sensitivity, and specificity of the SL symmetry index versus GRF symmetry index at 7.5 months of age were 96%, 85%, and 100%, respectively. At 6.0 months of age, such accuracy, sensitivity, and specificity of the SL were 89%, 67%, and 97%, respectively. But, at 4.5 months of age, such accuracy, sensitivity, and specificity of the SL were 87%, 50%, and 97%, respectively. These findings suggest that visual inspection for leg soundness of gilt has limited capability to identify pigs with an asymmetry gait which might cause current and future leg problems. But the SL symmetric index shows better potential for that purpose that it will be a suitable objective measure to replace visual inspection for leg soundness in pigs.

On inverse symmetric division deg index of graphs

One of the 148 discrete Adria indices is the symmetric division deg (SDD) index. It was developed about 13 years ago. Motivated by the success of the symmetric division deg index, Ghorbani et al. recently proposed an inverse version of this index, which they called the ISDD index (Inverse symmetric division deg index). The inverse symmetric division deg index (ISDD) of a graph Γ is defined as follows: $$ \mathrm{ISDD}(\mathrm{\Gamma })=\sum_{{v}_i{v}_j\in E(\mathrm{\Gamma })} \enspace \frac{{d}_i{d}_j}{{d}_i^2+{d}_j^2}, $$ where di is the degree of the vertex vi in Γ. In this paper, we determine the second maximal and the second minimal trees with respect to the inverse symmetric division deg index (ISDD). We prove that the star gives the minimal and the complete bipartite graph K⌈n/2⌉, ⌊n/2⌋ gives the maximal graphs with respect to the inverse symmetric division deg index (ISDD) among any chain graph of order n. Moreover, the Turán graph gives the maximal graph with respect to the ISDD index for any simple graph of order n with chromatic number k. Finally, we give concluding remarks about future works.

Classical calculation of differential cross section for a beam deflected by a concentric refractive index field.

Ray tracing is a fundamental geometric-optics issue which gives a single ray path but seldom presents the collective behavior of light. The optical field distribution usually involves the calculation of an electromagnetic field and is rarely discussed from the perspective of geometric optics. However, in this paper, we show for a concentric medium with spherically symmetric refractive index, how the relative angular distribution of refractive beams can be obtained from the pure classical geometric optics method. As a measurement of the distribution, we introduce the concept of the differential cross section (DCS), which can be calculated from the relation between aiming distance and deflecting the angle of the ray. We present a general method to solve this relation from both Snell's law in a constant medium and the optical Binet equation (OBE) in a gradient-index (GRIN) medium. Even without observing the collective traces, the DCS can independently give a quantitative description for the deflected light density of concentric media at different directions. It may act as a reference index for the design of beam deflector.

On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves

A tree in which no vertex has a degree greater than 4 is called a chemical tree. The bond incident degree index of a chemical tree T is defined as ∑xy∈ETφ(degT(x),degT(y)), where ET is the edge set of T, φ is a real-valued symmetric function, and degT(x) stands for the degree of a vertex x of T. This paper reports extremal results on bond incident degree indices of chemical trees with a fixed order and a fixed number of leaves. Furthermore, we use these results directly to some renowned topological indices, such as symmetric division deg index, Randić index, geometric-arithmetic index, sum-connectivity index, Sombor index, harmonic index, multiplicative sum Zagreb index, atom-bond connectivity index, etc.

On Degree-Based Topological Indices of Toeplitz Graphs

In this research paper, the determination of the orientation of a linear interpolation in an interconnected graph can be achieved by measuring its distance from a group of sonar stations strategically positioned within the graph. The study utilizes the metric dimension of toeplitz graphs. Several indices play a crucial role in analyzing motivating activities within such complex structures. The indices covered in this study include the general connectivity index of the toeplitz graph, zagreb indices, symmetric division degree index and randic indices, among others.

Algebraic Structure Graphs over the Commutative Ring Zm: Exploring Topological Indices and Entropies Using M-Polynomials

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G^ is fully made up of the zero divisors of the modular ring Zn, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

Tensor tomography of the residual stress field in graded-index YAG’s single crystals

Abstract This work presents an application of tensor field tomography for non-destructive reconstructions of axially symmetric residual stresses in a graded-index YAG single crystal for the case of beam deflection. The axis of the cylinder coincides with the crystallographic axis [001] of the single crystal and it has an axially symmetric refractive index distribution. The transformation of the polarization of light is measured in a plane orthogonal to the axis of the cylinder. Stresses are determined within the framework of the Maxwell piezo-optic law (linear dependence of the permittivity tensor on stresses) and small rotation of quasi principal stress axes. This paper generalizes the method of integrated photoelasticity for the case of ray deflection.

Sharp bounds on the symmetric division deg index of graphs and line graphs

Sharp bounds on the symmetric division deg index of graphs and line graphs

A Study on the Energy of Graphs and Its Applications

In this paper, we dealt with two very well-known eigenvalue-based molecular descriptors such as the energy of graph ( E ( G ) ) and Estrada index ( E E ( G ) ) . We test their predictive potential of some physicochemical properties of polycyclic aromatic hydrocarbons using a data set of benzenoid hydrocarbons. We stabilized a relation of these two indices with some well-known degree-based topological indices such as the first Zagreb index, second Zagreb index, F-index, Randić index, Reciprocal Randić index, Atom bond connectivity index, sum-connectivity index, Geometric-arithmetic, Harmonic index, symmetric division deg index, and redefined Zagreb index.

Application of Knee Robotic Orthosis to Improve the Walking Performance of People with Post-Polio Syndrome

ABSTRACT Introduction Poliomyelitis has been considered a statutory notifiable infectious disease in Hong Kong since 1948. Approximately 15% to 80% of patients suffer from post-polio syndrome (PPS) with an increased risk of falling and chronic pain. Most of them relied on the use of knee lock-type orthoses as aids. This study aimed to evaluate the effect of using a custom-fit knee robotic orthosis on their gait performance. Materials and Methods A case series report was used for this study on a group of individuals suffering from chronic PPS who used orthoses and had a history of falling. A knee robotic orthosis was custom-fit for eight sessions of walking practice. Assessments included laboratory gait analysis, functional measures, and subjective ratings of exertional level. Pain responses were recorded to evaluate the effect of using the knee robotic orthosis on participants’ gait performance. Results Five participants completed the study in using the knee robotic orthosis. All achieved reduction in shoulder tilting and trunk swaying movement on walking at the anterior-posterior and the medial-lateral direction. A total of 80% of the study participants achieved reduction in trunk rotation movement, lower-limb circumduction, and hip hiking movement on the affected side on walking. All study participants were able to demonstrate improvement in their gait symmetry after using the knee robotic orthosis as reflected by the symmetric index improvement of kinematics. The rate of perceived exertion and the visual analog scale of pain in all study participants were reported lower when they used the knee robotic orthosis. However, the 10-m walking time achieved was longer. Conclusions The knee robotic orthosis had a positive effect on remediating the abnormal kinematics of PPS patients using conventional lock-knee orthosis after eight sessions of training practice. Reduction in the limping gait pattern and improvement in gait symmetry was achieved in using a knee robotic orthosis for walking. The chronic pain response and the physical exertional level on walking were also reduced. However, the walking speed achieved was slower. Clinical Relevance The application of the knee robotic orthosis for walking assist is new to most of the PPS patients. The positive outcomes achieved merit further research for future clinical practice.

Determinants of short-term blood pressure variability – the value of arterial stiffness

Abstract. Increased blood pressure variability is a significant parameter that affects the prognosis in patients with arterial hypertension, and is an independent predictor of cardiovascular events and mortality. The arterial elasticity is an important factor in maintaining a constant average blood pressure for adequate blood supply to inner organs against the background of dynamic blood pressure fluctuations. A decrease in the elastic properties of the arteries, and therefore, an increase in the arterial stiffness can be one of the potential mechanisms for the blood pressure variability violation. Objective. The aim of the study was to evaluate the possible relationships between arterial stiffness indicators and blood pressure variability indices in patients with hypertension based on the results of ambulatory blood pressure monitoring. Patients and methods. The results of ambulatory blood pressure monitoring of 172 patients with hypertension were analyzed. Arterial stiffness was assessed using the ambulatory arterial stiffness index - symmetric and asymmetric. SD, SDw, CV, SV, ARV, and BPVR indices were calculated to assess blood pressure variability. Analysis of the obtained results was carried out using Microsoft Excel 2010 and the STATISTICA version 10.0 software. Differences between groups were assessed by the Mann-Whitney method, correlations among blood pressure variability indices and ambulatory arterial stiffness indices were evaluated using Spearman`s rank correlation analysis. Significance was set at p ˂ 0.05. The results. Awake asymmetric ambulatory arterial stiffness index was significantly higher than asleep one, no statistically significant difference between asleep and 24-hour asymmetric ambulatory arterial stiffness index was established. The asleep symmetric ambulatory arterial stiffness index did not significantly differ from the awake one and was significantly higher than the 24-hour one. In comparison with asymmetric, for symmetric ambulatory arterial stiffness index statistically significant relationships were established for a greater number of blood pressure variability indices. In all statistically significant cases, the direction of the established relationship with arterial stiffness was only direct for the systolic blood pressure variability indices. Variability of diastolic blood pressure had a clear inverse relationship with arterial stiffness. Conclusions. The presence of pathogenetic interrelationships between arterial stiffness and blood pressure variability have been proven in the present study. The obtained results allow to state that arterial elasticity is an important component of blood pressure variability.

540-degree deflecting lens and its general version.

We demonstrate an isotropic device called 540-degree deflecting lens, which has symmetric refractive index and can deflect parallel beam by 540 degrees. The expression of its gradient refractive index is obtained and generalized. We discover it's an optical absolute instrument with self-imaging characteristic. Using conformal mapping, we deduce its general version in one-dimensional space. We also introduce a combined lens called the generalized inside-out 540-degree deflecting lens similar to the inside-out Eaton lens. Ray tracing and wave simulations are used to demonstrate their characteristics. Our study expands the family of absolute instruments and provides new ideas to design optical systems.

International Competitiveness Analysis of Turkish Pasta Industry in Framework of Some Indexes and Product Mapping: A Research on the years 2001-2020

In the globalizing world, international trade is reaching advanced dimensions every passing day. Exporting companies had to make a competition both in local and international markets. Therefore, countries should have competitiveness both in the domestic and global markets. Nowadays, Turkey is leading pasta producer and exporter. In this study, the situation of the pasta industry in the world and in Turkey and the developments in its foreign trade were examined. This study aims to determine and analyze the international competitiveness of Turkey's pasta industry. Accordingly, pasta export and import data from Turkey and the world between 2001-2020 were used to calculate international competitiveness indices. Within the scope analysis, Balassa's RCA index, Vollrath’s indices (RXA, RMP, RTA, RC), Revealed Symmetric Comparative Advantage Index (RSCA), Trade Balance Index (TBI) were used and The Product Mapping Method developed by Tri Widodo was applied. As a result of the calculation of all indices, it has been revealed that Turkey has a competitive advantage in the pasta industry and specializes in its export. Turkey has the position of a net exporter (Group A) with a comparative advantage in this industry.