scholarly journals Path (or cycle)-trees with Graph Equations involving Line and Split Graphs

2017 ◽  
Vol 16 (3) ◽  
pp. 1-12
Author(s):  
H P Patil ◽  
V Raja
Keyword(s):  
Line Graphs ◽  
Chromatic Numbers ◽  
Higher Dimensional ◽  
Split Graphs ◽  
Hamiltonian Property ◽  
The Relationship

H-trees generalizes the existing notions of trees, higher dimensional trees and k-ctrees. The characterizations and properties of both Pk-trees for k at least 4 and Cn-trees for n at least 5 and their hamiltonian property, dominations, planarity, chromatic and b-chromatic numbers are established. The conditions under which Pk-trees for k at least 3 (resp. Cn-trees for n at least 4), are the line graphs are determined. The relationship between path-trees and split graphs are developed.

scholarly journals Algorithmic Aspects of Some Variations of Clique Transversal and Clique Independent Sets on Graphs

Algorithms ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 22
Author(s):  
Chuan-Min Lee
Keyword(s):  
Planar Graphs ◽  
Maximum Clique ◽  
Chordal Graphs ◽  
Perfect Graphs ◽  
Line Graphs ◽  
Np Completeness ◽  
Split Graphs ◽  
Transversal Numbers ◽  
The Relationship ◽  
Independence Numbers

This paper studies the maximum-clique independence problem and some variations of the clique transversal problem such as the {k}-clique, maximum-clique, minus clique, signed clique, and k-fold clique transversal problems from algorithmic aspects for k-trees, suns, planar graphs, doubly chordal graphs, clique perfect graphs, total graphs, split graphs, line graphs, and dually chordal graphs. We give equations to compute the {k}-clique, minus clique, signed clique, and k-fold clique transversal numbers for suns, and show that the {k}-clique transversal problem is polynomial-time solvable for graphs whose clique transversal numbers equal their clique independence numbers. We also show the relationship between the signed and generalization clique problems and present NP-completeness results for the considered problems on k-trees with unbounded k, planar graphs, doubly chordal graphs, total graphs, split graphs, line graphs, and dually chordal graphs.

scholarly journals Composing effective prediction at five points

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
John Joseph M. Carrasco ◽  
Laurentiu Rodina ◽  
Suna Zekioğlu
Keyword(s):  
Gauge Theory ◽  
Scattering Amplitude ◽  
Building Blocks ◽  
Tree Level ◽  
Critical Aspect ◽  
Higher Derivative ◽  
Higher Dimensional ◽  
Single Trace ◽  
The Relationship ◽  
Double Copy

Abstract Color-kinematics duality in the adjoint has proven key to the relationship between gauge and gravity theory scattering amplitude predictions. In recent work, we demonstrated that at four-point tree-level, a small number of color-dual EFT building blocks could encode all higher-derivative single-trace massless corrections to gauge and gravity theories compatible with adjoint double-copy. One critical aspect was the trivialization of building higher-derivative color-weights — indeed, it is the mixing of kinematics with non-adjoint-type color-weights (like the permutation-invariant d4) which permits description via adjoint double-copy. Here we find that such ideas clarify the predictions of local five-point higher-dimensional operators as well. We demonstrate how a single scalar building block can be combined with color structures to build higher-derivative color factors that generate, through double copy, the amplitudes associated with higher-derivative gauge-theory operators. These may then be suitably mapped, through another double-copy, to higher-derivative corrections in gravity.

On Foundations of Estimation for Nonparametric Regression With Continuous Optimization

2020 ◽  
pp. 177-203
Author(s):  
Pakize Taylan
Keyword(s):  
Nonparametric Regression ◽  
Model Building ◽  
Continuous Optimization ◽  
Parametric Models ◽  
Independent Variables ◽  
Regression Parameters ◽  
Higher Dimensional ◽  
Finite Set ◽  
Regression Techniques ◽  
The Relationship

The aim of parametric regression models like linear regression and nonlinear regression are to produce a reasonable relationship between response and independent variables based on the assumption of linearity and predetermined nonlinearity in the regression parameters by finite set of parameters. Nonparametric regression techniques are widely-used statistical techniques, and they not only relax the assumption of linearity in the regression parameters, but they also do not need a predetermined functional form as nonlinearity for the relationship between response and independent variables. It is capable of handling higher dimensional problem and sizes of sample than regression that considers parametric models because the data should provide both the model building and the model estimates. For this purpose, firstly, PRSS problems for MARS, ADMs, and CR will be constructed. Secondly, the solution of the generated problems will be obtained with CQP, one of the famous methods of convex optimization, and these solutions will be called CMARS, CADMs, and CKR, respectively.

scholarly journals Quasi-Periodic Bifurcations of Higher-Dimensional Tori

2016 ◽  
Vol 26 (07) ◽  
pp. 1630016 ◽  
Author(s):  
Motomasa Komuro ◽  
Kyohei Kamiyama ◽  
Tetsuro Endo ◽  
Kazuyuki Aihara
Keyword(s):  
Periodic Solutions ◽  
Phase Locking ◽  
Period Doubling ◽  
Double Covering ◽  
Heteroclinic Cycle ◽  
One Dimensional ◽  
Local Bifurcations ◽  
Saddle Node ◽  
Higher Dimensional ◽  
The Relationship

We classify the local bifurcations of quasi-periodic [Formula: see text]-dimensional tori in maps (abbr. MT[Formula: see text]) and in flows (abbr. FT[Formula: see text]) for [Formula: see text]. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT[Formula: see text] can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FT[Formula: see text] can be classified into three classes: saddle-node, double covering, and Neimark–Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MT[Formula: see text] and FT[Formula: see text]: namely, saddle-node cycle and heteroclinic cycle bifurcations of the [Formula: see text]-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST[Formula: see text]) and zero-dimensional tori in sections (abbr. ST[Formula: see text]). The bifurcations of ST[Formula: see text] can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST[Formula: see text] can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark–Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST[Formula: see text]/ST[Formula: see text] and the bifurcations of MT[Formula: see text]/FT[Formula: see text]. We present examples of all of these bifurcations.

The Chaotic Hierarchy

1983 ◽  
Vol 38 (7) ◽  
pp. 788-801 ◽  
Author(s):  
Otto E. Rössler
Keyword(s):  
Chaotic Attractor ◽  
Dynamical Behavior ◽  
Electronic Systems ◽  
State Variables ◽  
Flip Flop ◽  
Lc Oscillator ◽  
Higher Dimensional ◽  
Dissipative Dynamical Systems ◽  
Open Question ◽  
The Relationship

Abstract The complexity of dynamical behavior possible in nonlinear (for example, electronic) systems depends only on the number of state variables involved. Single-variable dissipative dynamical systems (like the single-transistor flip-flop) can only possess point attractors. Two-variable systems (like an LC-oscillator) can possess a one-dimensional attractor (limit cycle). Three-variable systems admit two even more complicated types of behavior: a toroidal attractor (of doughnut shape) and a chaotic attractor (which looks like an infinitely often folded sheet). The latter is easier to obtain. In four variables, we analogously have the hyper-toroidal and the hyper-chaotic attractor, respectively; and so forth. In every higher-dimensional case, all of the lower forms are also possible as well as “mixed cases” (like a combined hypertoroidal and chaotic motion, for example). Ten simple ordinary differential equations, most of them easy to implement electronically, are presented to illustrate the hierarchical tree. A second tree, in which one more dimension is needed for every type, is called the weak hierarchy because the chaotic regimes contained cannot be detected physically and numerically. The relationship between the two hierarchies is posed as an open question. It may be approached empirically - using electronic systems, for example.

Link community detection based on line graphs with a novel link similarity measure

2016 ◽  
Vol 30 (06) ◽  
pp. 1650023 ◽  
Author(s):  
Guishen Wang ◽  
Lan Huang ◽  
Yan Wang ◽  
Wei Pang ◽  
Qin Ma
Keyword(s):  
Community Detection ◽  
Similarity Measure ◽  
Detection Algorithm ◽  
Line Graphs ◽  
The Novel ◽  
On Line ◽  
Community Detection Algorithm ◽  
Relationship Of ◽  
The Relationship ◽  
Link Similarity

Link community gradually unfolds its capacity in complex network research. In this paper, a novel link similarity measure on line graphs is proposed. This measure can be adapted to different types of networks with an adjustable parameter. We prove its value converges to a limit on line graphs with the relationship of the nonneighbor links taken into account. Based on this similarity measure, we propose a novel link community detection algorithm for link clustering on line graphs. The detection algorithm combines the novel link similarity measure with the classic Markov Cluster (MCL) Algorithm and determines the link community partitions by calculating an extended modularity measure. Extensive experiments on two types of complex networks demonstrate the effectiveness, reliability and rationality of our solution in contrast to the other two classical algorithms.

scholarly journals Annotations on the Relationship Among Discriminant Functions

2020 ◽  
pp. 161-165
Author(s):  
Awogbemi Clement Adeyeye
Keyword(s):  
Linear Relationship ◽  
Discriminant Function ◽  
Dimensional Space ◽  
Discriminant Functions ◽  
Classification Problems ◽  
Linear Discriminant ◽  
Equal Variance ◽  
Quadratic Discriminant Function ◽  
Higher Dimensional ◽  
The Relationship

Different forms of discriminant functions and the essence of their appearances were considered in this study. Various forms of classification problems were also considered, and in each of the cases mentioned, classification from simple functions of the observational vector rather than complicated regions in the higher-dimensional space of the original vector were made. Violation of condition of equal variance covariance matrix for Linear Discriminant Function (LDF) results to Quadratic Discriminant Function (QDF). The relationships among the classification statistics examined were established: The Anderson’s (W) and Rao’s (R) statistics are equivalent when the two sample sizes are equal, and when a constant is equal to 1, W, R and John-Kudo’s (Z) classification statistics are asymptotically comparable. A linear relationship is also established between W and Z classification statistics.

Reconfigurable Spatial Sub-Chain Evolution

2013 ◽  
Vol 389 ◽  
pp. 876-880
Author(s):  
Li Ping Zhang ◽  
Gui Bing Pang ◽  
Mao Jun Zhou ◽  
Zhi Yuan Jin
Keyword(s):  
Special Configuration ◽  
Novel Structure ◽  
Higher Dimensional ◽  
Evolutionary Variation ◽  
Variable Topology ◽  
Fixed Topology ◽  
Constraint Information ◽  
Set Up ◽  
Displacement Group ◽  
The Relationship

This paper investigates the built-in spatial modules extended with reconfigurable character based on reconfiguration modules in the form of spatial kinematic pairs and associated links. Known that reconfiguration modules are to be served to develop reconfiguration leading to novel structure expansion, the key issue is to assemble reconfiguration modules and to derive a reconfiguration mechanism as a self-reconfigurable set. The module exerts its reconfiguration through changing the number of mobility or type of its built-in kinematic pair and changing its combined components. Its reconfiguration characteristics come from its decomposition, transformation, degeneration and combination. It is clear that reconfiguration module extension serves as an effective category to set up the relationship and transformation categories between these reconfiguration modules. More often, there exists multiple module group solutions for a higher dimensional module and this is the key for topology reconfiguration and variation. Spatial reconfiguration process uses reconfiguration principles which is consistent with displacement group operations. The essence of reconfiguration is the reconfiguration mechanism characteristic which convert a mechanism from fixed topology to variable topology analogous to evolutionary variation. In fact, these can be the effective and available constraint information as geometrical ways to reach the special configuration states and then produce reconfigurations with special geometric and parametric dimension design.

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