scholarly journals Genus Distributions for Iterated Claws

10.37236/2278 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Jonathan L. Gross ◽  
Imran F. Khan ◽  
Mehvish I. Poshni
Keyword(s):  
Multiple Roots ◽  
Single Edge ◽  
Root Vertex ◽  
Single Vertex ◽  
Single Root

We derive a recursion for the genus distributions of the graphs obtained by iteratively attaching a claw to the dipole $D_3$. The minimum genus of the graphs in this sequence grows arbitrarily large. The families of graphs whose genus distributions have been calculated previously are either planar or almost planar, or they can be obtained by iterative single-vertex or single-edge amalgamation of small graphs. A significant simplifying construction within this calculation achieves the effect of an amalgamation at three vertices with a single root vertex, rather than with multiple roots. 

scholarly journals NILAI AWAL PADA METODE SECANT YANG DIMODIFIKASI DALAM PENENTUAN AKAR GANDA PERSAMAAN NON LINEAR

2019 ◽  
Vol 21 (1) ◽  
pp. 21-31
Author(s):  
Patrisius Batarius ◽  
Alfri Aristo SinLae
Keyword(s):  
Numerical Methods ◽  
Linear Equations ◽  
Secant Method ◽  
Multiple Roots ◽  
Initial Value ◽  
Single Root ◽  
Non Linear ◽  
Mathematical Equations ◽  
Raphson Method ◽  
Non Linear Equations

Determining the root of an equation means making the equation equal zero, (f (f) = 0). In engineering, there are often complex mathematical equations. With the numerical method approach, the equation can be searching for the value of the equation root. However, to find a double root approach with several numerical methods such as the bisection method, regulatory method, Newton-Raphson method, and Secant method, it is not efficient in determining multiple roots. This study aims to determine the roots of non-linear equations that have multiple roots using the modified Secant method. Besides knowing the effect of determining the initial value for the Secant method that is modifying in determining the non-linear root of persistence that has multiple roots. Comparisons were also make to other numerical methods in determining twin roots with the modified Secant method. A comparison is done to determine the initial value used. Simulations are performing on equations that have one single root and two or more double roots.

scholarly journals Kempe-Locking Configurations

2018 ◽  
Author(s):  
James A. Tilley
Keyword(s):  
Bipartite Graph ◽  
Systematic Search ◽  
Single Edge ◽  
Heuristic Argument ◽  
Bottom Boundary ◽  
Single Vertex ◽  
Planar Triangulation ◽  
Isomorphism Classes ◽  
Connectivity Property ◽  
Proper Subgraph

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

scholarly journals Hogyan befolyásolja a villamosenergia-hálózatról rendelkezésre álló információ a fizikai támadások által okozott sérülékenységről alkotott képet?

2021 ◽  
Vol 2 (2) ◽  
pp. 155-163
Author(s):  
Bálint Hartmann
Keyword(s):  
Power Grid ◽  
Deep Understanding ◽  
Graph Representation ◽  
Single Edge ◽  
Weighted Network ◽  
Single Vertex ◽  
Targeted Attacks ◽  
Edge Removal ◽  
Element Removal ◽  
Vertex Removal

Összefoglaló. A villamosenergia-rendszerek fizikai támadásokkal szembeni ellenálló képessége a közelmúltban világszerte történt események ismeretében egyre nagyobb hangsúlyt kap a tématerület kutatásaiban. Az ilyen eseményekre való megfelelő felkészüléshez elengedhetetlen az üzemeltetett infrastruktúrának, elsősorban annak gyengeségeinek pontos ismerete. A cikkben Magyarország villamosenergia-hálózatának adatai alapján készített súlyozatlan és súlyozott gráfokon végzünk vizsgálatokat, hogy megértsük a különböző stratégia mentén kiválasztott célpontok elleni támadások milyen mértékben csökkentik a topológiai hatékonyságot. A cikk célja egyben a magyar hálózat sérülékenységének általános bemutatása is, mely hasznos bemeneti információ lehet a kockázati tervek elkészítésekor. Summary. Tolerance of the power grid against physical intrusions has gained importance in the light of various attacks that have taken place around the world. To adequately prepare for such events, grid operators have to possess a deep understanding of their infrastructure, more specifically, of its weaknesses. A graph representation of the Hungarian power grid was created in a way that the vertices are generators, transformers, and substations and the edges are high-voltage transmission lines. All transmission and sub-transmission elements were considered, including the 132 kV network as well. The network is subjected to various types of single and double element attacks, objects of which are selected according to different aspects. The vulnerability of the network is measured as a relative drop in efficiency when a vertex or an edge is removed from the network. Efficiency is a measure of the network’s performance, assuming that the efficiency for transmitting electricity between vertices i and j is proportional to the reciprocal of their distance. In this paper, simultaneous removals were considered, arranged into two scenarios (single or double element removal) and a total of 5 cases were carried out (single vertex removal, single edge removal, double vertex removal, double edge removal, single vertex and single edge removal). During the examinations, all possible removal combinations were simulated, thus the 5 cases represent 385, 504, 73920, 128271 and 193797 runs, respectively. After all runs were performed, damage values were determined for random and targeted attacks, and attacks causing maximal damage were also identified. In all cases, damage was calculated for unweighted and weighted networks as well, to enable the comparison of those two models. The aims of this paper are threefold: to perform a general assessment on the vulnerability of the Hungarian power grid against random and targeted attacks; to compare the damage caused by different attack strategies; and to highlight the differences between using unweighted and weighted graphs representations. Random removal of a single vertex or a single edge caused 0.3–0.4% drop in efficiency, respectively, which indicates a high tolerance against such attacks. Damage for random double attacks was still only in the range of 0.6–0.8%, which is acceptable. It was shown that if targets are selected by the attacker based on the betweenness rank of the element, damage would be below the maximal possible values. Comparison of the damage measured in the unweighted and the weighted network representations has shown that damage to the weighted network tends to be bigger for vertex attacks, but the contrary is observed for edge attacks. Numerical differences between the two representations do not show any trend that could be generalised, but in the case of the most vulnerable elements significant differences were found in damage measures, which underlines the importance of using weighted models.

scholarly journals Experimental and Numerical Simulation Study on the Shear Strength Characteristics of Magnolia multiflora Root-Soil Composites

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
ZiFan Sui ◽  
Wen Yi ◽  
YunGang Lu ◽  
Liang Deng
Keyword(s):  
Shear Strength ◽  
Inclination Angle ◽  
Numerical Models ◽  
Shear Resistance ◽  
Enhancement Effect ◽  
Shear Direction ◽  
Multiple Roots ◽  
Shear Tests ◽  
Single Root ◽  
Laboratory Test Results

The shear strength of the soil refers to the ultimate strength of the soil against shear failure, which is one of the important indicators used to measure slope stability. This paper presents a simulation of direct shear tests on root-soil composites with different root embedding angles under different stress conditions. By comparing and analyzing the simulation results of ABAQUS software and the laboratory test results, the enhancement effect of plant roots on soil shear strength was explored. Conclusions can be drawn as follows: the excellent agreement between numerical models and laboratory shear tests suggested that the developed model can quickly and conveniently predict the shear strength of the root-soil composites. The shear strength was related to the rooting arrangement. For a single root system, when the inclination angle of the root was about 64° to the shear direction, the shear resistance of soil was much improved, while the root reinforcement had less effect when the inclination angle was greater than 90°. In the case of multiple roots, the hybrid rooting method can more effectively improve the shear resistance of the root-soil composite. Therefore, in the practical application of using the root to strengthen the soil, the angle of a single root and arrangement of multiple roots should be comprehensively considered.

scholarly journals On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Mudassir Shams ◽  
Nazir Ahmad Mir ◽  
Naila Rafiq ◽  
A. Othman Almatroud ◽  
Saima Akram
Keyword(s):  
Nonlinear Equation ◽  
Numerical Test ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Optimal Order ◽  
Multiple Roots ◽  
Optimal Order Of Convergence ◽  
Single Root ◽  
Root Finding

In this article, we construct an optimal family of iterative methods for finding the single root and then extend this family for determining all the distinct as well as multiple roots of single-variable nonlinear equations simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in the case of single root finding methods and 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The computational cost, basins of attraction, efficiency, log of residual, and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in the literature.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals A Microcomputed Tomographic Evaluation of Root Canal Morphology of Maxillary Second Premolars in a Pakistani Cohort

2021 ◽  
Vol 11 (11) ◽  
pp. 5086
Author(s):  
Mazen F. Alkahtany ◽  
Saqib Ali ◽  
Abdul Khabeer ◽  
Shafqat A. Shah ◽  
Khalid H. Almadi ◽  
...  
Keyword(s):  
Root Canal ◽  
Three Dimensional ◽  
Ct Scanning ◽  
Micro Ct ◽  
Single Root ◽  
Root Canal Morphology ◽  
Spss Software ◽  
Variable Morphology ◽  
Canal Morphology

This study aimed to investigate variations in the root canal morphology of maxillary second premolar (MSP) teeth using microcomputed tomography (micro-CT). Sixty (N = 60) human extracted MSPs were collected and prepared for micro-CT scanning. The duration for scanning a single sample ranged between 30 and 40 min and a three-dimensional (3-D) image was obtained for all the MSPs. The images were evaluated by a single observer who recorded the canal morphology type, number of roots, canal orifices, apical foramina(s), apical delta(s), and accessory canals. The root canal configuration was categorized in agreement with Vertucci’s classification, and any configuration not in agreement with Vertucci’s classification was reported as an “additional canal configuration”. Descriptive statistics (such as mean percentages) were calculated using SPSS software. The most common types agreeing with Vertucci’s classification (in order of highest to lowest incidence) were types I, III, V, VII, II, and VI. The teeth also exhibited four additional configurations that were different from Vertucci’s classification: types 2-3, 1-2-3, 2-1-2-1, and 1-2-1-3. A single root was found in 96.7% and the majority of the samples demonstrated two canals (73.3%). Further, 80% of the teeth showed one canal orifice. The number of apical foramina’s in the teeth was variable, with 56.7% having solitary apical foramen. The accessory canal was found in 33.3%, and apical delta was found in only 20% of the samples. Variable morphology of the MSPs was detected in our study. The canal configuration most prevalent was type 1; however, the results also revealed some additional canal types.

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