Search results for: hamiltonian

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 [1]. Furthermore, based on computer experimental results in [1], 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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Authors: Dongqin Cheng

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The n-dimensional balanced hypercube BHn (n ≥ 1) has been proved to be a bipartite graph. Let P be a set of edges whose induced subgraph consists of pairwise vertex-disjoint paths. For any two vertices u, v from different partite sets of V (BHn). In this paper, we prove that if |P| ≤ 2n − 2 and the subgraph induced by P has neither u nor v as internal vertices, or both of u and v as end-vertices, then BHn contains a Hamiltonian path joining u and v passing through P. As a corollary, if |P| ≤ 2n−1, then the BHn contains a Hamiltonian cycle passing through P.

Keywords: interconnection network, balanced hypercube, Hamiltonian cycle, prescribed edges

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Authors: Salah Menouar, Sara Hassoul

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The dynamic of time-dependent three coupled oscillators is studied through an approach based on decoupling of them using the unitary transformation method. From a first unitary transformation, the Hamiltonian of the complicated original system is transformed to an equal but a simple one associated with the three coupled oscillators of which masses are unity. Finally, we diagonalize the matrix representation of the transformed hamiltonian by using a unitary matrix. The diagonalized Hamiltonian is just the same as the Hamiltonian of three simple oscillators. Through these procedures, the coupled oscillatory subsystems are completely decoupled. From this uncouplement, we can develop complete dynamics of the whole system in an easy way by just examining each oscillator independently. Such a development of the mechanical theory can be done regardless of the complication of the parameters' variations.

Keywords: schrödinger equation, hamiltonian, time-dependent three coupled oscillators, unitary transformation

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Authors: Ahmed Salam, Haithem Benkahla

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Recently, an efficient backward-stable algorithm for computing eigenvalues and vectors of a symmetric and Hamiltonian matrix has been proposed. The method preserves the symmetric and Hamiltonian structures of the original matrix, during the whole process. In this paper, we revisit the method. We derive a way for implementing the reduction of the matrix to the appropriate condensed form. Then, we construct a novel version of the implicit QR-algorithm for computing the eigenvalues and vectors.

Keywords: block implicit QR algorithm, preservation of a double structure, QR algorithm, symmetric and Hamiltonian structures

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Authors: Kazem Ghanbari, Yousef Gholami

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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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Authors: E. R. Swart, S. J. Gismondi, N. R. Swart, C. E. Bell

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We present an heuristic algorithm that decides graph non-Hamiltonicity. All graphs are directed, each undirected edge regarded as a pair of counter directed arcs. Each of the n! Hamilton cycles in a complete graph on n+1 vertices is mapped to an n-permutation matrix P where p(u,i)=1 if and only if the ith arc in a cycle enters vertex u, starting and ending at vertex n+1. We first create exclusion set E by noting all arcs (u, v) not in G, sufficient to code precisely all cycles excluded from G i.e. cycles not in G use at least one arc not in G. Members are pairs of components of P, {p(u,i),p(v,i+1)}, i=1, n-1. A doubly stochastic-like relaxed LP formulation of the Hamilton cycle decision problem is constructed. Each {p(u,i),p(v,i+1)} in E is coded as variable q(u,i,v,i+1)=0 i.e. shrinks the feasible region. We then implement the Weak Closure Algorithm (WCA) that tests necessary conditions of a matching, together with Boolean closure to decide 0/1 variable assignments. Each {p(u,i),p(v,j)} not in E is tested for membership in E, and if possible, added to E (q(u,i,v,j)=0) to iteratively maximize |E|. If the WCA constructs E to be maximal, the set of all {p(u,i),p(v,j)}, then G is decided non-Hamiltonian. Only non-Hamiltonian G share this maximal property. Ten non-Hamiltonian graphs (10 through 104 vertices) and 2000 randomized 31 vertex non-Hamiltonian graphs are tested and correctly decided non-Hamiltonian. For Hamiltonian G, the complement of E covers a matching, perhaps useful in searching for cycles. We also present an example where the WCA fails.

Keywords: Hamilton cycle decision problem, computational complexity theory, graph theory, theoretical computer science

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Authors: Matthaios Katsanikas

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We study the structure of the invariant manifolds of complex unstable periodic orbits of a family of periodic orbits, in a 3D autonomous Hamiltonian system of galactic type, after a transition of this family from stability to complex instability (Hamiltonian Hopf bifurcation). We consider the case of a supercritical Hamiltonian Hopf bifurcation. The invariant manifolds of complex unstable periodic orbits have two kinds of structures. The first kind is represented by a disk confined structure on the 4D space of section. The second kind is represented by a complicated central tube structure that is associated with an extended network of tube structures, strips and flat structures of sheet type on the 4D space of section.

Keywords: dynamical systems, galactic dynamics, chaos, phase space

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Authors: Ouardi Okkacha, Kaarour Abedlkrim, Meskine Mohamed

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The effective Hamiltonians are widely used in molecular spectroscopy for the interpretation of the vibration-rotation spectra. Their construction is an ambiguous procedure due to the existence of unitary transformations that change the effective Hamiltonian but do not change its eigenvalues. As a consequence of this ambiguity, it may happen that some parameters of effective Hamiltonians cannot be recovered from experimental data in a unique way. The type of admissible transformations which keeps the operator form of the effective Hamiltonian unaltered and the number of empirically determinable parameters strongly depend on the symmetry type of a molecule (asymmetric top, spherical top, and so on) and on the degeneracy of the vibrational state. In this work, we report the study of the ambiguity of effective Hamiltonian for the fundamental degenerate states v3 of the Molecule 12CD4.

Keywords: 12CD4, high-resolution infrared spectra, tetrahedral tensorial formalism, vibrational states, rovibrational line position analysis, XTDS, SPVIEW

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Authors: Ahmet Kaya

Abstract:

Incoming cash flows and outgoing claims play an important role to determine how is companies’ profit or loss. In this matter, ruin probability provides to describe vulnerability of the companies against ruin. Quantum mechanism is one of the significant approaches to model ruin probability as stochastically. Using the Hamiltonian method, we have performed formalisation of quantum mechanics < x|e-ᵗᴴ|x' > and obtained the transition probability of 2x2 and 3x3 matrix as traditional and eigenvector basis where A is a ruin operator and H|x' > is a Schroedinger equation. This operator A and Schroedinger equation are defined by a Hamiltonian matrix H. As a result, probability of not to be in ruin can be simulated and calculated as stochastically.

Keywords: ruin probability, quantum mechanics, Hamiltonian technique, operator approach

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

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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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Authors: Hosein Falahaty, Hitoshi Gotoh, Abbas Khayyer

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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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Authors: Rosangliana Chawngthu, Ramkumar K. Thapa

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A model calculation of photoassisted field emission current (PFEC) by using transfer Hamiltonian method will be present here. When the photon energy is incident on the surface of the metals, such that the energy of a photon is usually less than the work function of the metal under investigation. The incident radiation photo excites the electrons to a final state which lies below the vacuum level; the electrons are confined within the metal surface. A strong static electric field is then applied to the surface of the metal which causes the photoexcited electrons to tunnel through the surface potential barrier into the vacuum region and constitutes the considerable current called photoassisted field emission current. The incident radiation is usually a laser beam, causes the transition of electrons from the initial state to the final state and the matrix element for this transition will be written. For the calculation of PFEC, transfer Hamiltonian method is used. The initial state wavefunction is calculated by using Kronig-Penney potential model. The effect of the matrix element will also be studied. An appropriate dielectric model for the surface region of the metal will be used for the evaluation of vector potential. FORTRAN programme is used for the calculation of PFEC. The results will be checked with experimental data and the theoretical results.

Keywords: photoassisted field emission, transfer Hamiltonian, vector potential, wavefunction

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Authors: Ahmet Kaya

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Quantum mechanism is one of the most important approaches to calculating non-ruin probability. We apply standard Dirac notation to model given Hamiltonians. By using the traditional method and eigenvector basis, non-ruin probability is found for several examples. Also, non-ruin probability is calculated for two different Hamiltonian by using the tensor product. Finally, the path integral method is applied to the examples and comparison is made for stochastic simulations and path integral calculation.

Keywords: quantum physics, Hamiltonian system, path integral, tensor product, ruin probability

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Authors: Shih-Yan Chen, Shin-Shin Kao

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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: dual-cubes, dual-cube extensive networks, dual-cube-like networks, hypercubes, fault-tolerant hamiltonian property

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Authors: Suleiman Bashir Adamu, Lawan Sani Taura

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We study the role of boundary conditions via -symmetric quantum mechanics, where denotes parity operator and denotes time reversal operator. Using the one-dimensional Schrödinger Hamiltonian for a free particle in an infinite square well, we introduce symmetric boundary conditions. We find solutions of the 1D Klein-Gordon equation for a free particle in an infinite square well with Hermitian boundary and symmetry boundary conditions, where in both cases the energy eigenvalues and eigenfunction, respectively, are obtained.

Keywords: Eigenvalues, Eigenfunction, Hamiltonian, Klein- Gordon equation, PT-symmetric quantum mechanics

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Authors: O. D. Arbeláez-Echeverri, E. Restrepo-Parra, J. C. Riano-Rojas

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A two dimensional geometric model of Bit Patterned Media is proposed, the model is based on the crystal structure of the materials commonly used to produce the nano islands in bit patterned materials and the possible defects that may arise from the interaction between the nano islands and the matrix material. The dynamic magnetic properties of the material are then computed using time aware integration methods for the multi spin Hamiltonian. The Hamiltonian takes into account both the spatial and topological disorder of the sample as well as the high perpendicular anisotropy that is pursued when building bit patterned media. The main finding of the research was the possibility of replicating the results of previous experiments on similar materials and the ability of computing the switching field distribution given the geometry of the material and the parameters required by the model.

Keywords: nanostructures, Monte Carlo, pattern media, magnetic properties

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Authors: Yaw-Ling Lin

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A de Bruijn graph of order k is a graph whose vertices representing all length-k sequences with edges joining pairs of vertices whose sequences have maximum possible overlap (length k−1). Every Hamiltonian cycle of this graph defines a distinct, minimum length de Bruijn sequence containing all k-mers exactly once. A Kautz sequence is the minimal generating sequence so as the sequence of minimal length that produces all possible length-k sequences with the restriction that every two consecutive alphabets in the sequences must be different. A collection of de Bruijn/Kautz sequences are orthogonal if any two sequences are of maximally differ in sequence composition; that is, the maximum length of their common substring is k. In this paper, we discuss how such a collection of (maximal) orthogonal de Bruijn/Kautz sequences can be made and use the algorithm to build up a web application service for the synthesized DNA and other related biomolecular sequences.

Keywords: biomolecular sequence synthesis, de Bruijn sequences, Eulerian cycle, Hamiltonian cycle, Kautz sequences, orthogonal sequences

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Authors: Sahil Imtiyaz

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One of the biggest challenges has emerged from the ever-expanding, dynamic, and instantaneously changing space-Big Data; and to find a data point and inherit wisdom to this space is a hard task. In this paper, we reduce the space of big data in Hamiltonian formalism that is in concordance with Ising Model. For this formulation, we simulate the system using dye laser in FORTRAN and analyse the dynamics of the data point in energy well of rhodium atom. After mapping the photon intensity and pulse width with energy and potential we concluded that as we increase the energy there is also increase in probability of tunnelling up to some point and then it starts decreasing and then shows a randomizing behaviour. It is due to decoherence with the environment and hence there is a loss of ‘quantumness’. This interprets the efficiency parameter and the extent of quantum evolution. The results are strongly encouraging in favour of the use of ‘Topological Property’ as a source of information instead of the qubit.

Keywords: big data, optimization, quantum evolution, hamiltonian, dye laser, fermionic computations

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Authors: Wenjun Hou, Marek Perkowski

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The Traveling Salesman problem is famous in computing and graph theory. In short, it asks for the Hamiltonian cycle of the least total weight in a given graph with N nodes. All variations on this problem, such as those with K-bounded-degree nodes, are classified as NP-complete in classical computing. Although several papers propose theoretical high-level designs of quantum algorithms for the Traveling Salesman Problem, no quantum circuit implementation of these algorithms has been created up to our best knowledge. In contrast to previous papers, the goal of this paper is not to optimize some abstract complexity measures based on the number of oracle iterations, but to be able to evaluate the real circuit and time costs of the quantum computer. Using the emerging quantum programming language Q# developed by Microsoft, which runs quantum circuits in a quantum computer simulation, an implementation of the bounded-degree problem and its respective quantum circuit were created. To apply Grover’s algorithm to this problem, a quantum oracle was designed, evaluating the cost of a particular set of edges in the graph as well as its validity as a Hamiltonian cycle. Repeating the Grover algorithm with an oracle that finds successively lower cost each time allows to transform the decision problem to an optimization problem, finding the minimum cost of Hamiltonian cycles. N log₂ K qubits are put into an equiprobablistic superposition by applying the Hadamard gate on each qubit. Within these N log₂ K qubits, the method uses an encoding in which every node is mapped to a set of its encoded edges. The oracle consists of several blocks of circuits: a custom-written edge weight adder, node index calculator, uniqueness checker, and comparator, which were all created using only quantum Toffoli gates, including its special forms, which are Feynman and Pauli X. The oracle begins by using the edge encodings specified by the qubits to calculate each node that this path visits and adding up the edge weights along the way. Next, the oracle uses the calculated nodes from the previous step and check that all the nodes are unique. Finally, the oracle checks that the calculated cost is less than the previously-calculated cost. By performing the oracle an optimal number of times, a correct answer can be generated with very high probability. The oracle of the Grover Algorithm is modified using the recalculated minimum cost value, and this procedure is repeated until the cost cannot be further reduced. This algorithm and circuit design have been verified, using several datasets, to generate correct outputs.

Keywords: quantum computing, quantum circuit optimization, quantum algorithms, hybrid quantum algorithms, quantum programming, Grover’s algorithm, traveling salesman problem, bounded-degree TSP, minimal cost, Q# language

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Authors: Haileyesus Tessema Alemneh, Abiyu Enyew Molla, Oluwole Daniel Makinde

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Introduction: Faba bean is one of the most important grown plants worldwide for humans and animals. Several biotic and abiotic elements have limited the output of faba beans, irrespective of their diverse significance. Many faba bean pathogens have been reported so far, of which the most important yield-limiting disease is chocolate spot disease (Botrytis fabae). The dynamics of disease transmission and decision-making processes for intervention programs for disease control are now better understood through the use of mathematical modeling. Currently, a lot of mathematical modeling researchers are interested in plant disease modeling. Objective: In this paper, a deterministic mathematical model for chocolate spot disease (CSD) on faba bean plant with an optimal control model was developed and analyzed to examine the best strategy for controlling CSD. Methodology: Three control interventions, quarantine (u2), chemical control (u3), and prevention (u1), are employed that would establish the optimal control model. The optimality system, characterization of controls, the adjoint variables, and the Hamiltonian are all generated employing Pontryagin’s maximum principle. A cost-effective approach is chosen from a set of possible integrated strategies using the incremental cost-effectiveness ratio (ICER). The forward-backward sweep iterative approach is used to run numerical simulations. Results: The Hamiltonian, the optimality system, the characterization of the controls, and the adjoint variables were established. The numerical results demonstrate that each integrated strategy can reduce the diseases within the specified period. However, due to limited resources, an integrated strategy of prevention and uprooting was found to be the best cost-effective strategy to combat CSD. Conclusion: Therefore, attention should be given to the integrated cost-effective and environmentally eco-friendly strategy by stakeholders and policymakers to control CSD and disseminate the integrated intervention to the farmers in order to fight the spread of CSD in the Faba bean population and produce the expected yield from the field.

Keywords: CSD, optimal control theory, Pontryagin’s maximum principle, numerical simulation, cost-effectiveness analysis

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Authors: S. Behnia, S. Fathizadeh, A. Akhshani

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In the recent decades, DNA has increasingly interested in the potential technological applications that not directly related to the coding for functional proteins that is the expressed in form of genetic information. One of the most interesting applications of DNA is related to the construction of nanostructures of high complexity, design of functional nanostructures in nanoelectronical devices, nanosensors and nanocercuits. In this field, DNA is of fundamental interest to the development of DNA-based molecular technologies, as it possesses ideal structural and molecular recognition properties for use in self-assembling nanodevices with a definite molecular architecture. Also, the robust, one-dimensional flexible structure of DNA can be used to design electronic devices, serving as a wire, transistor switch, or rectifier depending on its electronic properties. In order to understand the mechanism of the charge transport along DNA sequences, numerous studies have been carried out. In this regard, conductivity properties of DNA molecule could be investigated in a simple, but chemically specific approach that is intimately related to the Su-Schrieffer-Heeger (SSH) model. In SSH model, the non-diagonal matrix element dependence on intersite displacements is considered. In this approach, the coupling between the charge and lattice deformation is along the helix. This model is a tight-binding linear nanoscale chain established to describe conductivity phenomena in doped polyethylene. It is based on the assumption of a classical harmonic interaction between sites, which is linearly coupled to a tight-binding Hamiltonian. In this work, the Hamiltonian and corresponding motion equations are nonlinear and have high sensitivity to initial conditions. Then, we have tried to move toward the nonlinear dynamics and phase space analysis. Nonlinear dynamics and chaos theory, regardless of any approximation, could open new horizons to understand the conductivity mechanism in DNA. For a detailed study, we have tried to study the current flowing in DNA and investigated the characteristic I-V diagram. As a result, It is shown that there are the (quasi-) ohmic areas in I-V diagram. On the other hand, the regions with a negative differential resistance (NDR) are detectable in diagram.

Keywords: DNA conductivity, Landauer resistance, negative dierential resistance, Chaos theory, mean Lyapunov exponent

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Authors: Khaled I. Nawafleh

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In this work, we apply the method of Hamilton-Jacobi to obtain solutions of Hamiltonian systems in classical mechanics with two certain structures: the first structure plays a central role in the theory of time-dependent Hamiltonians, whilst the second is used to treat classical Hamiltonians, including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the nonstationary case can be obtained naturally in Hamilton-Jacobi formalism. Then, another expression of geometry of the Hamilton Jacobi equation is retrieved for Hamiltonians with time-dependent and frictional terms. Both approaches shall be applied to many physical examples.

Keywords: Hamilton-Jacobi, time dependent lagrangians, dissipative systems, variational principle

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Authors: Khawlh A. Alzubaidi, Khadijah B. Alziyadi, Amor M. Alsayari

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Now a days (Ga, Mn) is one of the most extensively studied and best understood diluted magnetic semiconductors. Also, the study of (Ga, Mn)As is a fervent research area since it allows to explore of a variety of novel functionalities and spintronics concepts that could be implemented in the future. In this work, we will calculate the energy gap of (Ga, Mn)As using the eight-band model. In the Hamiltonian, the effects of spin-orbit, spin-splitting, and strain will be considered. The dependence of the energy gap on Mn content, and the effect of the strain, which is varied continuously from tensile to compressive, will be studied. Finally, analytical expressions for the (Ga, Mn)As energy band gap, taking into account both parameters (Mn concentration and strain), will be provided.

Keywords: energy gap, diluted magnetic semiconductors, k.p method, strain

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Authors: Michael Danilov, Sergei Iskakov, Vladimir Mazurenko

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The present paper describes a high-performance parallel realization of an exact diagonalization solver for quantum-electron models in a shared memory computing system. The proposed algorithm contains a storage format for efficient computing eigenvalues and eigenvectors of a quantum electron Hamiltonian matrix. The results of the test calculations carried out for 15 sites Hubbard model demonstrate reduction in the required memory and good multiprocessor scalability, while maintaining performance of the same order as compressed row storage.

Keywords: sparse matrix, compressed format, Hubbard model, Anderson model

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Authors: Kotbi Lakhdar, Hachi Mostefa

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We show that a simple transformation between the regular lattices (the square, the triangular, and the honeycomb) belonging to the same dimensionality can explain in a natural way the universality of the critical exponents found in phase transitions and critical phenomena. It suffices that the Hamiltonian and the lattice present similar writing forms. In addition, it appears that if a property can be calculated for a given lattice then it can be extrapolated simply to any other lattice belonging to the same dimensionality. In this study, we have restricted ourselves on the spectral power amplification (SPA), we note that the SPA does not have an effect on the critical exponents but does have an effect by the criticality temperature of the lattice; the generalisation to other lattice could be shown according to the containment principle.

Keywords: ising model, phase transitions, critical temperature, critical exponent, spectral power amplification

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Authors: Elimhan N. Mahmudov

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The paper concerns the necessary and sufficient conditions of optimality for Cauchy problem of fourth order discrete (PD) and discrete-approximate (PDA) inclusions. The main problem is formulation of the fourth order adjoint discrete and discrete-approximate inclusions and transversality conditions, which are peculiar to problems including fourth order derivatives and approximate derivatives. Thus the necessary and sufficient conditions of optimality are obtained incorporating the Euler-Lagrange and Hamiltonian forms of inclusions. Derivation of optimality conditions are based on the apparatus of locally adjoint mapping (LAM). Moreover in the application of these results we consider the fourth order linear discrete and discrete-approximate inclusions.

Keywords: difference, optimization, fourth, approximation, transversality

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Authors: S.I.Mukhin, S. Seidov, A. Mukherjee

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The Dicke model is a key tool for the description of correlated states of quantum atomic systems, excited by resonant photon absorption and subsequently emitting spontaneous coherent radiation in the superradiant state. The Dicke Hamiltonian (DH) is successfully used for the description of the dynamics of the Josephson Junction (JJ) array in a resonant cavity under applied current. In this work, we have investigated a generalized model, which is described by DH with a frustrating interaction term. This frustrating interaction term is explicitly the infinite coordinated interaction between all the spin half in the system. In this work, we consider an array of N superconducting islands, each divided into two sub-islands by a Josephson Junction, taken in a charged qubit / Cooper Pair Box (CPB) condition. The array is placed inside the resonant cavity. One important aspect of the problem lies in the dynamical nature of the physical observables involved in the system, such as condensed electric field and dipole moment. It is important to understand how these quantities behave with time to define the quantum phase of the system. The Dicke model without frustrating term is solved to find the dynamical solutions of the physical observables in analytic form. We have used Heisenberg’s dynamical equations for the operators and on applying newly developed Rotating Holstein Primakoff (HP) transformation and DH we have arrived at the four coupled nonlinear dynamical differential equations for the momentum and spin component operators. It is possible to solve the system analytically using two-time scales. The analytical solutions are expressed in terms of Jacobi's elliptic functions for the metastable ‘bound luminosity’ dynamic state with the periodic coherent beating of the dipoles that connect the two double degenerate dipolar ordered phases discovered previously. In this work, we have proceeded the analysis with the extended DH with a frustrating interaction term. Inclusion of the frustrating term involves complexity in the system of differential equations and it gets difficult to solve analytically. We have solved semi-classical dynamic equations using the perturbation technique for small values of Josephson energy EJ. Because the Hamiltonian contains parity symmetry, thus phase transition can be found if this symmetry is broken. Introducing spontaneous symmetry breaking term in the DH, we have derived the solutions which show the occurrence of finite condensate, showing quantum phase transition. Our obtained result matches with the existing results in this scientific field.

Keywords: Dicke Model, nonlinear dynamics, perturbation theory, superconductivity

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Authors: Francisco Bulnes

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Through the Fukaya conjecture and the wrapped Floer cohomology, the correspondences between paths in a loop space and states of a wrapping space of states in a Hamiltonian space (the ramification of field in this case is the connection to the operator that goes from TM to T*M) are demonstrated where these last states are corresponding to bosonic extensions of a spectrum of the space-time or direct image of the functor Spec, on space-time. This establishes a distinguished diffeomorphism defined by the mapping from the corresponding loops space to wrapping category of the Floer cohomology complex which furthermore relates in certain proportion D-branes (certain D-modules) with strings. This also gives to place to certain conjecture that establishes equivalences between moduli spaces that can be consigned in a moduli identity taking as space-time the Hitchin moduli space on G, whose dual can be expressed by a factor of a bosonic moduli spaces.

Keywords: Floer cohomology, Fukaya conjecture, Lagrangian submanifolds, quantum topological diffeomorphism

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Authors: Milad Zoghi, M. Zahangir Kabir

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Performance of armchair graphene nanoribbon (AGNR) resonant tunneling diodes (RTD) alter if they go under strain. This may happen due to either using stretchable substrates or real working conditions such as heat generation. Therefore, it is informative to understand how mechanical deformations such as uniaxial strain can impact the performance of AGNR RTDs. In this paper, two platforms of AGNR RTD consist of width-modified AGNR RTD and electric-field modified AGNR RTD are subjected to both compressive and tensile uniaxial strain ranging from -2% to +2%. It is found that characteristics of AGNR RTD markedly change under both compressive and tensile strain. In particular, peak to valley ratio (PVR) can be totally disappeared upon strong enough strain deformation. Numerical tight binding (TB) coupled with Non-Equilibrium Green's Function (NEGF) is derived for this study to calculate corresponding Hamiltonian matrices and transport properties.

Keywords: armchair graphene nanoribbon, resonant tunneling diode, uniaxial strain, peak to valley ratio

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Authors: Madiha Tariq, Hena Rabbani

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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

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