Search results for: Hamiltonian system
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 8362

Search results for: Hamiltonian system

8362 Hamiltonian Factors in Hamiltonian Graphs

Authors: Sizhong Zhou, Bingyuan Pu

Abstract:

Let G be a Hamiltonian graph. A factor F of G is called a Hamiltonian factor if F contains a Hamiltonian cycle. In this paper, two sufficient conditions are given, which are two neighborhood conditions for a Hamiltonian graph G to have a Hamiltonian factor.

Keywords: graph, neighborhood, factor, Hamiltonian factor.

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8361 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.

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8360 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

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8359 The Panpositionable Hamiltonicity of k-ary n-cubes

Authors: Chia-Jung Tsai, Shin-Shin Kao

Abstract:

The hypercube Qn is one of the most well-known and popular interconnection networks and the k-ary n-cube Qk n is an enlarged family from Qn that keeps many pleasing properties from hypercubes. In this article, we study the panpositionable hamiltonicity of Qk n for k ≥ 3 and n ≥ 2. Let x, y of V (Qk n) be two arbitrary vertices and C be a hamiltonian cycle of Qk n. We use dC(x, y) to denote the distance between x and y on the hamiltonian cycle C. Define l as an integer satisfying d(x, y) ≤ l ≤ 1 2 |V (Qk n)|. We prove the followings: • When k = 3 and n ≥ 2, there exists a hamiltonian cycle C of Qk n such that dC(x, y) = l. • When k ≥ 5 is odd and n ≥ 2, we request that l /∈ S where S is a set of specific integers. Then there exists a hamiltonian cycle C of Qk n such that dC(x, y) = l. • When k ≥ 4 is even and n ≥ 2, we request l-d(x, y) to be even. Then there exists a hamiltonian cycle C of Qk n such that dC(x, y) = l. The result is optimal since the restrictions on l is due to the structure of Qk n by definition.

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

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8358 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.

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8357 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.

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8356 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.

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8355 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.

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8354 Lyapunov Type Inequalities for Fractional Impulsive Hamiltonian Systems

Authors: Kazem Ghanbari, Yousef Gholami

Abstract:

This paper deals with study about fractional order impulsive Hamiltonian systems and fractional impulsive Sturm-Liouville type problems derived from these systems. The main purpose of this paper devotes to obtain so called Lyapunov type inequalities for mentioned problems. Also, in view point on applicability of obtained inequalities, some qualitative properties such as stability, disconjugacy, nonexistence and oscillatory behaviour of fractional Hamiltonian systems and fractional Sturm-Liouville type problems under impulsive conditions will be derived. At the end, we want to point out that for studying fractional order Hamiltonian systems, we will apply recently introduced fractional Conformable operators.

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

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8353 The Frequency Graph for the Traveling Salesman Problem

Authors: Y. Wang

Abstract:

Traveling salesman problem (TSP) is hard to resolve when the number of cities and routes become large. The frequency graph is constructed to tackle the problem. A frequency graph maintains the topological relationships of the original weighted graph. The numbers on the edges are the frequencies of the edges emulated from the local optimal Hamiltonian paths. The simplest kind of local optimal Hamiltonian paths are computed based on the four vertices and three lines inequality. The search algorithm is given to find the optimal Hamiltonian circuit based on the frequency graph. The experiments show that the method can find the optimal Hamiltonian circuit within several trials.

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

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8352 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

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8351 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.

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8350 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.

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8349 An Efficient Hamiltonian for Discrete Fractional Fourier Transform

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

Abstract:

Fractional Fourier Transform, which is a generalization of the classical Fourier Transform, is a powerful tool for the analysis of transient signals. The discrete Fractional Fourier Transform Hamiltonians have been proposed in the past with varying degrees of correlation between their eigenvectors and Hermite Gaussian functions. In this paper, we propose a new Hamiltonian for the discrete Fractional Fourier Transform and show that the eigenvectors of the proposed matrix has a higher degree of correlation with the Hermite Gaussian functions. Also, the proposed matrix is shown to give better Fractional Fourier responses with various transform orders for different signals.

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

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8348 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.

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8347 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.

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8346 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.

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8345 Exterior Calculus: Economic Profit Dynamics

Authors: Troy L. Story

Abstract:

A mathematical model for the Dynamics of Economic Profit is constructed by proposing a characteristic differential oneform for this dynamics (analogous to the action in Hamiltonian dynamics). After processing this form with exterior calculus, a pair of characteristic differential equations is generated and solved for the rate of change of profit P as a function of revenue R (t) and cost C (t). By contracting the characteristic differential one-form with a vortex vector, the Lagrangian is obtained for the Dynamics of Economic Profit.

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

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8344 Hamiltonian Related Properties with and without Faults of the Dual-Cube Interconnection Network and Their Variations

Authors: Shih-Yan Chen, Shin-Shin Kao

Abstract:

In this paper, a thorough review about dual-cubes, DCn, the related studies and their variations are given. DCn was introduced to be a network which retains the pleasing properties of hypercube Qn but has a much smaller diameter. In fact, it is so constructed that the number of vertices of DCn is equal to the number of vertices of Q2n +1. However, each vertex in DCn is adjacent to n + 1 neighbors and so DCn has (n + 1) × 2^2n edges in total, which is roughly half the number of edges of Q2n+1. In addition, the diameter of any DCn is 2n +2, which is of the same order of that of Q2n+1. For selfcompleteness, basic definitions, construction rules and symbols are provided. We chronicle the results, where eleven significant theorems are presented, and include some open problems at the end.

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

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8343 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.

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8342 Regional Stability Analysis of Rotor-Ball Bearing and Rotor- Roller Bearing Systems Considering Switching Phenomena

Authors: Jafar Abbaszadeh Chekan, Kaveh Merat, Hassan Zohoor

Abstract:

In this study the regional stability of a rotor system which is supported on rolling bearings with radial clearance is studied. The rotor is assumed to be rigid. Due to radial clearance of bearings and dynamic configuration of system, each rolling elements of bearings has the possibility to be in contact with both of the races (under compression) or lose its contact. As a result, this change in dynamic of the system makes it to be known as switching system which is a type of Hybrid systems. In this investigation by adopting Multiple Lyapunov Function theorem and using Hamiltonian function as a candidate Lyapunov function, the stability of the system is studied. The purpose of this study is to inspect the regional stability of rotor-roller bearing and rotor-ball bearing systems.

Keywords: Stability analysis, Rotor-rolling bearing systems, Switching systems, Multiple Lyapunov Function Method

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8341 Computer Simulations of an Augmented Automatic Choosing Control Using Automatic Choosing Functions of Gradient Optimization Type

Authors: Toshinori Nawata

Abstract:

In this paper we consider a nonlinear feedback control called augmented automatic choosing control (AACC) using the automatic choosing functions of gradient optimization type for nonlinear systems. Constant terms which arise from sectionwise linearization of a given nonlinear system are treated as coefficients of a stable zero dynamics. Parameters included in the control are suboptimally selected by minimizing the Hamiltonian 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, nonlinear control, genetic algorithm, zero dynamics.

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8340 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.

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8339 Numerical Calculation of the Ionization Energy of Donors in a Cubic Quantum well and Wire

Authors: Sara Sedaghat, Mahmood Barati, Iraj Kazeminezhad

Abstract:

The ionization energy in semiconductor systems in nano scale was investigated by using effective mass approximation. By introducing the Hamiltonian of the system, the variational technique was employed to calculate the ground state and the ionization energy of a donor at the center and in the case that the impurities are randomly distributed inside a cubic quantum well. The numerical results for GaAs/GaAlAs show that the ionization energy strongly depends on the well width for both cases and it decreases as the well width increases. The ionization energy of a quantum wire was also calculated and compared with the results for the well.

Keywords: quantum well, quantum wire, quantum dot, impuritystate

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8338 The Spanning Laceability of k-ary n-cubes when k is Even

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

Abstract:

Qk n has been shown as an alternative to the hypercube family. For any even integer k ≥ 4 and any integer n ≥ 2, Qk n is a bipartite graph. In this paper, we will prove that given any pair of vertices, w and b, from different partite sets of Qk n, there exist 2n internally disjoint paths between w and b, denoted by {Pi | 0 ≤ i ≤ 2n-1}, such that 2n-1 i=0 Pi covers all vertices of Qk n. The result is optimal since each vertex of Qk n has exactly 2n neighbors.

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

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8337 Single-qubit Quantum Gates using Magneto-optic Kerr Effect

Authors: Pradeep Kumar K

Abstract:

We propose the use of magneto-optic Kerr effect (MOKE) to realize single-qubit quantum gates. We consider longitudinal and polar MOKE in reflection geometry in which the magnetic field is parallel to both the plane of incidence and surface of the film. MOKE couples incident TE and TM polarized photons and the Hamiltonian that represents this interaction is isomorphic to that of a canonical two-level quantum system. By varying the phase and amplitude of the magnetic field, we can realize Hadamard, NOT, and arbitrary phase-shift single-qubit quantum gates. The principal advantage is operation with magnetically non-transparent materials.

Keywords: Quantum computing, qubit, magneto-optic kerr effect (MOKE), magneto-optical interactions, continuous variables.

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8336 Torque Ripple Minimization in Switched Reluctance Motor Using Passivity-Based Robust Adaptive Control

Authors: M.M. Namazi, S.M. Saghaiannejad, A. Rashidi

Abstract:

In this paper by using the port-controlled Hamiltonian (PCH) systems theory, a full-order nonlinear controlled model is first developed. Then a nonlinear passivity-based robust adaptive control (PBRAC) of switched reluctance motor in the presence of external disturbances for the purpose of torque ripple reduction and characteristic improvement is presented. The proposed controller design is separated into the inner loop and the outer loop controller. In the inner loop, passivity-based control is employed by using energy shaping techniques to produce the proper switching function. The outer loop control is employed by robust adaptive controller to determine the appropriate Torque command. It can also overcome the inherent nonlinear characteristics of the system and make the whole system robust to uncertainties and bounded disturbances. A 4KW 8/6 SRM with experimental characteristics that takes magnetic saturation into account is modeled, simulation results show that the proposed scheme has good performance and practical application prospects.

Keywords: Switched Reluctance Motor, Port HamiltonianSystem, Passivity-Based Control, Torque Ripple Minimization

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8335 Modeling and Simulations of Complex Low- Dimensional systems: Testing the Efficiency of Parallelization

Authors: Ryszard Matysiak, Grzegorz Kamieniarz

Abstract:

The deterministic quantum transfer-matrix (QTM) technique and its mathematical background are presented. This important tool in computational physics can be applied to a class of the real physical low-dimensional magnetic systems described by the Heisenberg hamiltonian which includes the macroscopic molecularbased spin chains, small size magnetic clusters embedded in some supramolecules and other interesting compounds. Using QTM, the spin degrees of freedom are accurately taken into account, yielding the thermodynamical functions at finite temperatures. In order to test the application for the susceptibility calculations to run in the parallel environment, the speed-up and efficiency of parallelization are analyzed on our platform SGI Origin 3800 with p = 128 processor units. Using Message Parallel Interface (MPI) system libraries we find the efficiency of the code of 94% for p = 128 that makes our application highly scalable.

Keywords: Deterministic simulations, low-dimensional magnets, modeling of complex systems, parallelization.

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8334 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.

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8333 Describing the Fine Electronic Structure and Predicting Properties of Materials with ATOMIC MATTERS Computation System

Authors: Rafal Michalski, Jakub Zygadlo

Abstract:

We present the concept and scientific methods and algorithms of our computation system called ATOMIC MATTERS. This is the first presentation of the new computer package, that allows its user to describe physical properties of atomic localized electron systems subject to electromagnetic interactions. Our solution applies to situations where an unclosed electron 2p/3p/3d/4d/5d/4f/5f subshell interacts with an electrostatic potential of definable symmetry and external magnetic field. Our methods are based on Crystal Electric Field (CEF) approach, which takes into consideration the electrostatic ligands field as well as the magnetic Zeeman effect. The application allowed us to predict macroscopic properties of materials such as: Magnetic, spectral and calorimetric as a result of physical properties of their fine electronic structure. We emphasize the importance of symmetry of charge surroundings of atom/ion, spin-orbit interactions (spin-orbit coupling) and the use of complex number matrices in the definition of the Hamiltonian. Calculation methods, algorithms and convention recalculation tools collected in ATOMIC MATTERS were chosen to permit the prediction of magnetic and spectral properties of materials in isostructural series.

Keywords: Atomic matters, crystal electric field, spin-orbit coupling, localized states, electron subshell, fine electronic structure.

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