**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**25

# Search results for: Hamiltonian

##### 25 Hamiltonian Factors in Hamiltonian Graphs

**Authors:**
Sizhong Zhou,
Bingyuan Pu

**Abstract:**

**Keywords:**
graph,
neighborhood,
factor,
Hamiltonian factor.

##### 24 A Sufficient Condition for Graphs to Have Hamiltonian [a, b]-Factors

**Authors:**
Sizhong Zhou

**Abstract:**

Let a and b be nonnegative integers with 2 ≤ a < b, and let G be a Hamiltonian graph of order n with n ≥ (a+b−4)(a+b−2) b−2 . An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. In this paper, it is proved that G has a Hamiltonian [a, b]-factor if |NG(X)| > (a−1)n+|X|−1 a+b−3 for every nonempty independent subset X of V (G) and δ(G) > (a−1)n+a+b−4 a+b−3 .

**Keywords:**
graph,
minimum degree,
neighborhood,
[a,
b]-factor,
Hamiltonian [a,
b]-factor.

##### 23 The Balanced Hamiltonian Cycle on the Toroidal Mesh Graphs

**Authors:**
Wen-Fang Peng,
Justie Su-Tzu Juan

**Abstract:**

The balanced Hamiltonian cycle problemis a quiet new topic of graph theorem. Given a graph G = (V, E), whose edge set can be partitioned into k dimensions, for positive integer k and a Hamiltonian cycle C on G. The set of all i-dimensional edge of C, which is a subset by E(C), is denoted as Ei(C).

**Keywords:**
Hamiltonian cycle,
balanced,
Cartesian product.

##### 22 Mutually Independent Hamiltonian Cycles of Cn x Cn

**Authors:**
Kai-Siou Wu,
Justie Su-Tzu Juan

**Abstract:**

In a graph G, a cycle is Hamiltonian cycle if it contain all vertices of G. Two Hamiltonian cycles C_1 = ⟨u_0, u_1, u_2, ..., u_{n−1}, u_0⟩ and C_2 = ⟨v_0, v_1, v_2, ..., v_{n−1}, v_0⟩ in G are independent if u_0 = v_0, u_i = ̸ v_i for all 1 ≤ i ≤ n−1. In G, a set of Hamiltonian cycles C = {C_1, C_2, ..., C_k} is mutually independent if any two Hamiltonian cycles of C are independent. The mutually independent Hamiltonicity IHC(G), = k means there exist a maximum integer k such that there exists k-mutually independent Hamiltonian cycles start from any vertex of G. In this paper, we prove that IHC(C_n × C_n) = 4, for n ≥ 3.

**Keywords:**
Hamiltonian,
independent,
cycle,
Cartesian product,
mutually independent Hamiltonicity

##### 21 Lyapunov Type Inequalities for Fractional Impulsive Hamiltonian Systems

**Authors:**
Kazem Ghanbari,
Yousef Gholami

**Abstract:**

**Keywords:**
Fractional derivatives and integrals,
Hamiltonian
system,
Lyapunov type inequalities,
stability,
disconjugacy.

##### 20 A Systematic Approach for Finding Hamiltonian Cycles with a Prescribed Edge in Crossed Cubes

**Authors:**
Jheng-Cheng Chen,
Chia-Jui Lai,
Chang-Hsiung Tsai,

**Abstract:**

The crossed cube is one of the most notable variations of hypercube, but some properties of the former are superior to those of the latter. For example, the diameter of the crossed cube is almost the half of that of the hypercube. In this paper, we focus on the problem embedding a Hamiltonian cycle through an arbitrary given edge in the crossed cube. We give necessary and sufficient condition for determining whether a given permutation with n elements over Zn generates a Hamiltonian cycle pattern of the crossed cube. Moreover, we obtain a lower bound for the number of different Hamiltonian cycles passing through a given edge in an n-dimensional crossed cube. Our work extends some recently obtained results.

**Keywords:**
Interconnection network,
Hamiltonian,
crossed cubes,
prescribed edge.

##### 19 An Efficient Hamiltonian for Discrete Fractional Fourier Transform

**Authors:**
Sukrit Shankar,
Pardha Saradhi K.,
Chetana Shanta Patsa,
Jaydev Sharma

**Abstract:**

**Keywords:**
Fractional Fourier Transform,
Hamiltonian,
Eigen
Vectors,
Discrete Hermite Gaussians.

##### 18 An Augmented Automatic Choosing Control Designed by Extremizing a Combination of Hamiltonian and Lyapunov Functions for Nonlinear Systems with Constrained Input

**Authors:**
Toshinori Nawata,
Hitoshi Takata

**Abstract:**

In this paper we consider a nonlinear feedback control called augmented automatic choosing control (AACC) for nonlinear systems with constrained input. Constant terms which arise from section wise linearization of a given nonlinear system are treated as coefficients of a stable zero dynamics.Parameters included in the control are suboptimally selectedby extremizing a combination of Hamiltonian and Lyapunov functions with the aid of the genetic algorithm. This approach is applied to a field excitation control problem of power system to demonstrate the splendidness of the AACC. Simulation results show that the new controller can improve performance remarkably well.

**Keywords:**
Augmented Automatic Choosing Control,
NonlinearControl,
Genetic Algorithm,
Hamiltonian,
Lyapunovfunction

##### 17 Hamiltonian Related Properties with and without Faults of the Dual-Cube Interconnection Network and Their Variations

**Authors:**
Shih-Yan Chen,
Shin-Shin Kao

**Abstract:**

**Keywords:**
Hypercubes,
dual-cubes,
fault-tolerant
hamiltonian property,
dual-cube extensive networks,
dual-cube-like
networks.

##### 16 A Hamiltonian Decomposition of 5-star

**Authors:**
Walter Hussak,
Heiko Schröder

**Abstract:**

Star graphs are Cayley graphs of symmetric groups of permutations, with transpositions as the generating sets. A star graph is a preferred interconnection network topology to a hypercube for its ability to connect a greater number of nodes with lower degree. However, an attractive property of the hypercube is that it has a Hamiltonian decomposition, i.e. its edges can be partitioned into disjoint Hamiltonian cycles, and therefore a simple routing can be found in the case of an edge failure. The existence of Hamiltonian cycles in Cayley graphs has been known for some time. So far, there are no published results on the much stronger condition of the existence of Hamiltonian decompositions. In this paper, we give a construction of a Hamiltonian decomposition of the star graph 5-star of degree 4, by defining an automorphism for 5-star and a Hamiltonian cycle which is edge-disjoint with its image under the automorphism.

**Keywords:**
interconnection networks,
paths and cycles,
graphs andgroups.

##### 15 The Spanning Laceability of k-ary n-cubes when k is Even

**Authors:**
Yuan-Kang Shih,
Shu-Li Chang,
Shin-Shin Kao

**Abstract:**

**Keywords:**
container,
Hamiltonian,
k-ary n-cube,
m*-connected.

##### 14 The Panpositionable Hamiltonicity of k-ary n-cubes

**Authors:**
Chia-Jung Tsai,
Shin-Shin Kao

**Abstract:**

**Keywords:**
Hamiltonian,
panpositionable,
bipanpositionable,
k-ary n-cube.

##### 13 On Chvátal’s Conjecture for the Hamiltonicity of 1-Tough Graphs and Their Complements

**Authors:**
Shin-Shin Kao,
Yuan-Kang Shih,
Hsun Su

**Abstract:**

In this paper, we show that the conjecture of Chv tal, which states that any 1-tough graph is either a Hamiltonian graph or its complement contains a specific graph denoted by F, does not hold in general. More precisely, it is true only for graphs with six or seven vertices, and is false for graphs with eight or more vertices. A theorem is derived as a correction for the conjecture.

**Keywords:**
Complement,
degree sum,
Hamiltonian,
tough.

##### 12 Isospectral Hulthén Potential

**Authors:**
Anil Kumar

**Abstract:**

Supersymmetric Quantum Mechanics is an interesting framework to analyze nonrelativistic quantal problems. Using these techniques, we construct a family of strictly isospectral Hulth´en potentials. Isospectral wave functions are generated and plotted for different values of the deformation parameter.

**Keywords:**
Hulth´en potential,
Isospectral Hamiltonian.

##### 11 Remarks on Energy Based Control of a Nonlinear, Underactuated, MIMO and Unstable Benchmark

**Authors:**
Guangyu Liu

**Abstract:**

In the last decade, energy based control theory has undergone a significant breakthrough in dealing with underactated mechanical systems with two successful and similar tools, controlled Lagrangians and controlled Hamiltanians (IDA-PBC). However, because of the complexity of these tools, successful case studies are lacking, in particular, MIMO cases. The seminal theoretical paper of controlled Lagrangians proposed by Bloch and his colleagues presented a benchmark example–a 4 d.o.f underactuated pendulum on a cart but a detailed and completed design is neglected. To compensate this ignorance, the note revisit their design idea by addressing explicit control functions for a similar device motivated by a vector thrust body hovering in the air. To the best of our knowledge, this system is the first MIMO, underactuated example that is stabilized by using energy based tools at the courtesy of the original design idea. Some observations are given based on computer simulation.

**Keywords:**
Controlled Lagrangian,
Energy Shaping,
Spherical Inverted Pendulum,
Controlled Hamiltonian.

##### 10 Exterior Calculus: Economic Growth Dynamics

**Authors:**
Troy L. Story

**Abstract:**

Mathematical models of dynamics employing exterior calculus are mathematical representations of the same unifying principle; namely, the description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus, to a set of differential equations and a characteristic tangent vector (vortex vector) which define transformations of the system. Using this principle, a mathematical model for economic growth is constructed by proposing a characteristic differential one-form for economic growth dynamics (analogous to the action in Hamiltonian dynamics), then generating a pair of characteristic differential equations and solving these equations for the rate of economic growth as a function of labor and capital. By contracting the characteristic differential one-form with the vortex vector, the Lagrangian for economic growth dynamics is obtained.

**Keywords:**
Differential geometry,
exterior calculus,
Hamiltonian geometry,
mathematical economics.

##### 9 A Further Study on the 4-Ordered Property of Some Chordal Ring Networks

**Authors:**
Shin-Shin Kao,
Hsiu-Chunj Pan

**Abstract:**

Given a graph G. A cycle of G is a sequence of vertices of G such that the first and the last vertices are the same. A hamiltonian cycle of G is a cycle containing all vertices of G. The graph G is k-ordered (resp. k-ordered hamiltonian) if for any sequence of k distinct vertices of G, there exists a cycle (resp. hamiltonian cycle) in G containing these k vertices in the specified order. Obviously, any cycle in a graph is 1-ordered, 2-ordered and 3- ordered. Thus the study of any graph being k-ordered (resp. k-ordered hamiltonian) always starts with k = 4. Most studies about this topic work on graphs with no real applications. To our knowledge, the chordal ring families were the first one utilized as the underlying topology in interconnection networks and shown to be 4-ordered. Furthermore, based on our computer experimental results, it was conjectured that some of them are 4-ordered hamiltonian. In this paper, we intend to give some possible directions in proving the conjecture.

**Keywords:**
Hamiltonian cycle,
4-ordered,
Chordal rings,
3-regular.

##### 8 Quantum Localization of Vibrational Mirror in Cavity Optomechanics

**Authors:**
Madiha Tariq,
Hena Rabbani

**Abstract:**

Recently, cavity-optomechanics becomes an extensive research field that has manipulated the mechanical effects of light for coupling of the optical field with other physical objects specifically with regards to dynamical localization. We investigate the dynamical localization (both in momentum and position space) for a vibrational mirror in a Fabry-Pérot cavity driven by a single mode optical field and a transverse probe field. The weak probe field phenomenon results in classical chaos in phase space and spatio temporal dynamics in position |ψ(x)²| and momentum space |ψ(p)²| versus time show quantum localization in both momentum and position space. Also, we discuss the parametric dependencies of dynamical localization for a designated set of parameters to be experimentally feasible. Our work opens an avenue to manipulate the other optical phenomena and applicability of proposed work can be prolonged to turn-able laser sources in the future.

**Keywords:**
Dynamical localization,
cavity optomechanics,
hamiltonian chaos,
probe field.

##### 7 A Lagrangian Hamiltonian Computational Method for Hyper-Elastic Structural Dynamics

**Authors:**
Hosein Falahaty,
Hitoshi Gotoh,
Abbas Khayyer

**Abstract:**

Performance of a Hamiltonian based particle method in simulation of nonlinear structural dynamics is subjected to investigation in terms of stability and accuracy. The governing equation of motion is derived based on Hamilton's principle of least action, while the deformation gradient is obtained according to Weighted Least Square method. The hyper-elasticity models of Saint Venant-Kirchhoff and a compressible version similar to Mooney- Rivlin are engaged for the calculation of second Piola-Kirchhoff stress tensor, respectively. Stability along with accuracy of numerical model is verified by reproducing critical stress fields in static and dynamic responses. As the results, although performance of Hamiltonian based model is evaluated as being acceptable in dealing with intense extensional stress fields, however kinds of instabilities reveal in the case of violent collision which can be most likely attributed to zero energy singular modes.

**Keywords:**
Hamilton's principle of least action,
particle based method,
hyper-elasticity,
analysis of stability.

##### 6 Pseudo-almost Periodic Solutions of a Class Delayed Chaotic Neural Networks

**Authors:**
Farouk Cherif

**Abstract:**

This paper is concerned with the existence and unique¬ness of pseudo-almost periodic solutions to the chaotic delayed neural networks (t)= —Dx(t) ± A f (x (t)) B f (x (t — r)) C f (x(p))dp J (t) . t-o Under some suitable assumptions on A, B, C, D, J and f, the existence and uniqueness of a pseudo-almost periodic solution to equation above is obtained. The results of this paper are new and they complement previously known results.

**Keywords:**
Chaotic neural network,
Hamiltonian systems,
Pseudo almost periodic.

##### 5 An Improved Construction Method for MIHCs on Cycle Composition Networks

**Authors:**
Hsun Su,
Yuan-Kang Shih,
Shin-Shin Kao

**Abstract:**

Many well-known interconnection networks, such as kary n-cubes, recursive circulant graphs, generalized recursive circulant graphs, circulant graphs and so on, are shown to belong to the family of cycle composition networks. Recently, various studies about mutually independent hamiltonian cycles, abbreviated as MIHC-s, on interconnection networks are published. In this paper, using an improved construction method, we obtain MIHC-s on cycle composition networks with a much weaker condition than the known result. In fact, we established the existence of MIHC-s in the cycle composition networks and the result is optimal in the sense that the number of MIHC-s we constructed is maximal.

**Keywords:**
Hamiltonian cycle,
k-ary n-cube,
cycle composition networks,
mutually independent.

##### 4 Exterior Calculus: Economic Profit Dynamics

**Authors:**
Troy L. Story

**Abstract:**

**Keywords:**
Differential geometry,
exterior calculus,
Hamiltonian
geometry,
mathematical economics,
economic functions,
and
dynamics

##### 3 The Frequency Graph for the Traveling Salesman Problem

**Authors:**
Y. Wang

**Abstract:**

**Keywords:**
Traveling salesman problem,
frequency graph,
local
optimal Hamiltonian path,
four vertices and three lines inequality.

##### 2 An Optimal Control Problem for Rigid Body Motions on Lie Group SO(2, 1)

**Authors:**
Nemat Abazari,
Ilgin Sager

**Abstract:**

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.

**Keywords:**
Optimal control,
Hamiltonian vector field,
Darboux vector,
maximum principle,
lie group,
Rigid body motion,
Lorentz metric.

##### 1 Planning Rigid Body Motions and Optimal Control Problem on Lie Group SO(2, 1)

**Authors:**
Nemat Abazari,
Ilgin Sager

**Abstract:**

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimizes the integral of the Lorentz inner product of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.

**Keywords:**
Optimal control,
Hamiltonian vector field,
Darboux vector,
maximum principle,
lie group,
rigid body motion,
Lorentz metric.