ALBERT

Beschreibung: The problem of exponential stability for the uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations is investigated in this paper. This study starts from the corresponding nominal systems with known and partially unknown transition rates, respectively. By constructing a novel augmented Lyapunov functional which contains triple-integral terms and fully utilizes the bound of the delay, the delay-range-dependent and rate-dependent exponential stability criteria are developed by the Lyapunov theory, reciprocally convex lemma, and free weighting matrices. Then, the results about nominal systems are extended to the uncertain case. Finally, numerical examples are given to demonstrate the effectiveness of the proposed methods.

Beschreibung: The local fractional Schrödinger equations in the one-dimensional Cantorian system are investigated. The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show that the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative.

Beschreibung: The problem of robust fault tolerant control for continuous-time fractional-order (FO) systems with interval parameters and sensor faults of has been investigated. By establishing sensor fault model and state observer, an observer-based FO output feedback controller is developed such that the closed-loop FO system is asymptotically stable, not only when all sensor components are working well, but also in the presence of sensor components failures. Finally, numerical simulation examples are given to illustrate the application of the proposed design method.

Beschreibung: This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is where is a -dimensional unit torus with a flat metric. The phase space of spins is , the subspace of formed by functions symmetric under the permutations of the arguments. The Fock space yields the phase space of a system of a varying (but finite) number of particles. We associate a space with each vertex of a graph satisfying a special bidimensionality property. (Physically, vertex represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i) , the minus a half of the Laplace operator on , responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials , , describing a field generated by a heavy atom, (b) two-body potentials , , showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials , , scaled along the graph distance between vertices , which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group acts on , represented by a Euclidean space or torus of dimension , preserving the metric and the volume in . Furthermore, we suppose that the potentials , , and are -invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is -invariant, provided that the thermodynamic variables (the fugacity and the inverse temperature ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.

Beschreibung: The periodic solution of fractional oscillation equation with periodic input is considered in this work. The fractional derivative operator is taken as, where the initial time is ; hence, initial conditions are not needed in the model of the present fractional oscillation equation. With the input of the harmonic oscillation, the solution is derived to be a periodic function of time t with the same circular frequency as the input, and the frequency of the solution is not affected by the system frequency c as is affected in the integer-order case. These results are similar to the case of a damped oscillation with a periodic input in the integer-order case. Properties of the periodic solution are discussed, and the fractional resonance frequency is introduced.

Beschreibung: Nonlinear integrable couplings of super Broer-Kaup-Kupershmidt hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.

Beschreibung: We employ the complex method to research the integrality of the Fisher equations with degree three. We obtain the sufficient and necessary condition of the integrable of the Fisher equations with degree three and the general meromorphic solutions of the integrable Fisher equations with degree three, which improves the corresponding results obtained by Feng and Li (2006), Guo and Chen (1991), and Ağırseven and Öziş (2010). Moreover, all are new general meromorphic solutions of the Fisher equations with degree three for . Our results show that the complex method provides a powerful mathematical tool for solving a large number of nonlinear partial differential equations in mathematical physics.

Beschreibung: Recently, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. In this paper we introduce a novel class of fractional chaotic systems in the pseudo state space and propose an adaptive sliding mode control scheme to stabilize the chaotic systems in the presence of uncertainties and external disturbances whose bounds are unknown. To verify the effectiveness of the proposed adaptive sliding mode control technique, numerical simulations of control design of fractional Lorenz's system and Chen's system are presented.

Beschreibung: This paper is concerned with a viscous shallow water equation, which includes both the viscous Camassa-Holm equation and the viscous Degasperis-Procesi equation as its special cases. The optimal control under boundary conditions is given, and the existence of optimal solution to the equation is proved.

Beschreibung: Recently, we introduced a type of autocorrelation function (ACF) to describe a long-range dependent (LRD) process indexed with two parameters, which takes standard fractional Gaussian noise (fGn for short) as a special case. For simplicity, we call it the generalized fGn (GfGn). This short paper gives the power spectrum density function (PSD) of GfGn.

Beschreibung: This paper describes an approach to measure the entrepreneurship orientation of online P2P lending platforms. The limitations of existing methods for calculating entropy of interval-valued intuitionistic fuzzy sets (IVIFSs) are significantly improved by a new entropy measure of IVIFS considered in this paper, and then the essential properties of the proposed entropy are introduced. Moreover, an evaluation procedure is proposed to measure entrepreneurship orientation of online P2P lending platforms. Finally, a case is used to demonstrate the effectiveness of this method.

Beschreibung: We give new characterizations of Riesz-type frames, on equivalent conditions for a continuous frame to be a Riesz-type frame and on equivalency relations between Riesz-type frames and continuous frames. We characterize also the Riesz-type frames by using a bounded linear operator . Finally, we study the stability of alternate duals of continuous frames and we prove that if two continuous frames are close to each other, then we can find alternate duals of them which are close to each other.

Beschreibung: Discrete neural models are of great importance in numerical simulations and practical implementations. In the current paper, a discrete model of continuous-time neural networks with variable and distributed delays is investigated. By Lyapunov stability theory and techniques such as linear matrix inequalities, sufficient conditions guaranteeing the existence and global exponential stability of the unique equilibrium point are obtained. Introduction of LMIs enables one to take into consideration the sign of connection weights. To show the effectiveness of the method, an illustrative example, along with numerical simulation, is presented.

Beschreibung: We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality.

Beschreibung: Quark matter is believed to exist in the center of neutron stars. A combined model consisting of quark matter and ordinary matter is used to show that the extreme conditions existing in the center could result in a topology change, that is, in the formation of wormholes.

Beschreibung: We consider a class of nonlinear two-dimensional dynamic systems of the neutral type We obtain sufficient conditions for all solutions of the system to be oscillatory. Our oscillation results when improve the oscillation results for dynamic systems on time scales that have been established by Fu and Lin (2010), since our results do not restrict to the case where . Also, as a special case when , our results do not require to be a positive real sequence. Some examples are given to illustrate the main results.

Beschreibung: We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.

Beschreibung: The chaos in a new system with order 3 is studied. We have shown that this chaotic system again will be chaotic when the order of system is less than 3. Generalized Adams-Bashforth algorithm has been used for investigating in stability of fixed points and existence of chaos.

Beschreibung: The Lie symmetry analysis is performed on the Rosenau equation which arises in modeling many physical phenomena. The similarity reductions and exact solutions are presented. Then the exact analytic solutions are considered by the power series method.

Beschreibung: Necessity for the periodic fundamental solutions arises when the periodic boundary value problems should be analyzed. The latter are naturally related to problems of finding the homogenized properties of the dispersed composites, porous media, and media with uniformly distributed microcracks or dislocations. Construction of the periodic fundamental solutions is done in terms of the convergent series in harmonic polynomials. An example of the periodic fundamental solution for the anisotropic porous medium is constructed, along with the simplified lower bound estimate.

Beschreibung: This paper presents results of an initial investigation into vehicle roll model and control strategies suitable for preventing vehicle untripped rollovers. For vehicles that are deemed to be susceptible to wheel-liftoff, various control strategies are implemented in simulation. In this study, the authors propose a method for rollover prevention that does not require such accurate contact information. The validity of the stability margin is shown, and it is used to realize rollover prevention in the direction of the roll. The primary assumption in their implementation is that the vehicle in question is equipped with a conventional controller system.

Beschreibung: We study the generalized quantum isotonic oscillator Hamiltonian given by H=−d2/dr2+l(l+1)/r2+w2r2+2g(r2−a2)/(r2+a2)2, g>0. Two approaches are explored. A method for finding the quasipolynomial solutions is presented, and explicit expressions for these polynomials are given, along with the conditions on the potential parameters. By using the asymptotic iteration method, we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters g, w, and a may be found to high accuracy.

Beschreibung: We study mixed geodesic GCR-lightlike submanifolds of indefinite Sasakian manifolds and obtain some necessary and sufficient conditions for a GCR-lightlike submanifold to be a GCR-lightlike product.

Beschreibung: A family of generalized binomial probability distributions attached to Landau levels on the Riemann sphere is introduced by constructing a kind of generalized coherent states. Their main statistical parameters are obtained explicitly. As an application, photon number statistics related to coherent states under consideration are discussed.

Beschreibung: Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). It is shown that multiplicative nature of the noise is the main reason for the successful application of the logarithmic gaps transforming the multiplicative noise into an additive one. A relation of this phenomenon to spontaneous neuron activity and to chaotic brain computations has been discussed.

Beschreibung: Bonome et al., 1997, provided an algebraic characterization for an indefinite Sasakian manifold to reduce to a space of constant ϕ-holomorphic sectional curvature. In this present paper, we generalize the same characterization for indefinite g⋅f⋅f-space forms.

Beschreibung: We consider N point vortices whose positions satisfy a stochastic ordinary differential equation on ℝ2N perturbed by spatially correlated Brownian noise. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) with a state-dependent stochastic term. As the number of vortices tends to infinity, we obtain a smooth solution to the SNSE, and we prove the conservation of total vorticity in this continuum limit.

Beschreibung: We make explicit in terms of categories a number of statements from the theory of partial inner product spaces (PIP-spaces) and operators on them. In particular, we construct sheaves and cosheaves of operators on certain PIP-spaces of practical interest.

Beschreibung: Based on a well-known Lie algebra, the multicomponent Guo hierarchy with self-consistent sources is proposed. With the help of a set of non-semisimple Lie algebra, the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources are obtained. It enriches the content of the integrable couplings of hierarchies with self-consistent sources. Finally, the Hamiltonian structures are worked out by employing the variational identity.

Beschreibung: It is shown that the spanning set for provided by the eigenfunctions of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to , where and are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in turns out to be appropriate also for problems that are softly confined by U-shaped potentials, including those with strong singularities at . Specific examples are discussed in detail, along with some bound -boson systems.

Beschreibung: The method of approximate transformation groups, which was proposed by Baikov et al. (1988 and 1996), is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended procedure is applied in order to compute the approximate conservation laws and approximate recursion operators corresponding to these types of equations. In particular, as an application, a comprehensive analysis of the problem of approximate conservation laws and approximate recursion operators associated to the Gardner equation with the small parameters is presented.

Beschreibung: Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

Beschreibung: The method of approximate transformation groups, which was proposed by Baikov et al. (1988 and 1996), is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended procedure is applied in order to compute the approximate conservation laws and approximate recursion operators corresponding to these types of equations. In particular, as an application, a comprehensive analysis of the problem of approximate conservation laws and approximate recursion operators associated to the Gardner equation with the small parameters is presented.

Beschreibung: It is shown that the spanning set for provided by the eigenfunctions of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to , where and are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in turns out to be appropriate also for problems that are softly confined by U-shaped potentials, including those with strong singularities at . Specific examples are discussed in detail, along with some bound -boson systems.

Beschreibung: For a class of multiparameter statistical models based on

Beschreibung: We discuss how to obtain black hole quasinormal modes (QNMs) using the asymptotic iteration method (AIM), initially developed to solve second-order ordinary differential equations. We introduce the standard version of this method and present an improvement more suitable for numerical implementation. We demonstrate that the AIM can be used to find radial QNMs for Schwarzschild, Reissner-Nordström (RN), and Kerr black holes in a unified way. We discuss some advantages of the AIM over the continued fractions method (CFM). This paper presents for the first time the spin 0, 1/2 and 2 QNMs of a Kerr black hole and the gravitational and electromagnetic QNMs of the RN black hole calculated via the AIM and confirms results previously obtained using the CFM. We also present some new results comparing the AIM to the WKB method. Finally we emphasize that the AIM is well suited to higher-dimensional generalizations and we give an example of doubly rotating black holes.

Beschreibung: This paper is mainly a review concerning singular perturbation methods by means of Lie group analysis which has been presented by the author. We make use of a particular type of approximate Lie symmetries in those methods in order to construct reduced systems which describe the long-time behavior of the original dynamical system. Those methods can be used in analyzing not only ordinary differential equations but also difference equations. Although this method has been mainly used in order to derive asymptotic behavior, when we can find exact Lie symmetries, we succeed in construction of exact solutions.

Beschreibung: The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation for small- but finite-amplitude electrostatic ion-acoustic waves in weakly relativistic plasma consisting of warm ions and isothermal electrons. An algebraic method with computerized symbolic computation is applied in obtaining a series of exact solutions of the KdV equation. Numerical studies have been made using plasma parameters which reveal different solutions, that is, bell-shaped solitary pulses, rational pulses, and solutions with singularity at finite points, which called “blowup” solutions in addition to the propagation of an explosive pulses. The weakly relativistic effect is found to significantly change the basic properties (namely, the amplitude and the width) of the ion-acoustic waves. The result of the present investigation may be applicable to some plasma environments, such as ionosphere region.

Beschreibung: Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.

Beschreibung: An analysis is performed to investigate the effect of MHD and thermal radiation on the two-dimensional steady flow of an incompressible, upper-convected Maxwells (UCM) fluid in presence of external magnetic field. The governing system of partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations and is solved numerically by efficient shooting technique. Velocity and temperature fields have been computed and shown graphically for various values of physical parameters. For a Maxwell fluid, a thinning of the boundary layer and a drop in wall skin friction coefficient is predicted to occur for the higher elastic number which agrees with the results of Hayat et al. 2007 and Sadeghy et al. 2006. The objective of the present work is to investigate the effect of elastic parameter β, magnetic parameter Mn, Eckert number Ec, Radiation parameter N, and Prandtl number Pr on flow and heat transfer charecteristics.

Beschreibung: The Locally Rotationally Symmetric (LRS) Bianchi type I cosmological model with variable modified Chaplygin gas having the equation of state p=Aρ−B/ρα, where 0≤α≤1, A is a positive constant, and B is a positive function of the average-scale factor a(t) of the universe (i.e., B=B(a)) has been studied. It is shown that the equation of state of the variable modified Chaplygin gas interpolates from radiation-dominated era to quintessence-dominated era. The statefinder diagnostic pair (i.e., {r,s} parameter) is adopted to characterize different phases of the universe.

Beschreibung: We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite Kaehler manifold. We discuss the boundedness of holomorphic sectional curvature of GCR-lightlike submanifolds of an indefinite complex space form. We establish a condition for a GCR-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems for holomorphic sectional and holomorphic bisectional curvature.

Beschreibung: The covariant canonical analysis for Yang-Mills theory expressed as a BF-like action is performed. We study a BF-like action, that in spite of being the coupling of two topological terms, yield, on shell to Yang-Mills action. In addition, by using the results obtained in the covariant canonical approach we study the symmetries of the action, in particular we calculate its energy-momentum tensor obtaining the same tensor found for Yang-Mills theory; then we confirm those results by using Noether's theorem.

Beschreibung: We consider an open quantum system of N not directly interacting spins (qubits) in contact with both local and collective thermal environments. The qubit-environment interactions are energy conserving. We trace out the variables of the thermal environments and N−2 qubits to obtain the time-dependent reduced density matrix for two arbitrary qubits. We numerically simulate the reduced dynamics and the creation of entanglement (concurrence) as a function of the parameters of the thermal environments and the number of qubits, N. Our results demonstrate that the two-qubit entanglement generally decreases as N increases. We show analytically that, in the limit N→∞, no entanglement can be created. This indicates that collective thermal environments cannot create two-qubit entanglement when many qubits are located within a region of the size of the environment coherence length. We discuss possible relevance of our consideration to recent quantum information devices and biosystems.

Beschreibung: This paper presents the study of momentum and heat transfer characteristics in a hydromagnetic flow of dusty fluid over an inclined stretching sheet with non-uniform heat source/sink, where the flow is generated due to a linear stretching of the sheet. Using a similarity transformation, the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations and are solved numerically by Runge-Kutta-Fehlberg fourth-fifth-order method using symbolic software Maple. Our numerical solutions are shown to agree with the available results in the literature and then employ the numerical results to bring out the effects of the fluid-particle interaction parameter, local Grashof number, angle of inclination, heat source/sink parameter, Chandrasekhar number, and the Prandtl number on the flow and heat transfer characteristics. The results have possible technological applications in liquid-based systems involving stretchable materials.

Beschreibung: We construct a toy model of spacetime deformation-induced inertia effects, in which we prescribe to each and every particle individually a new fundamental constituent of hypothetical 2D, so-called master space (MS), subject to certain rules. The MS, embedded in the background 4D-spacetime, is an indispensable companion to the particle of interest, without relation to every other particle. The MS is not measurable directly, but we argue that a deformation (distortion of local internal properties) of MS is the origin of inertia effects that can be observed by us. With this perspective in sight, we construct the alternative relativistic theory of inertia. We go beyond the hypothesis of locality with special emphasis on distortion of MS, which allows to improve essentially the standard metric and other relevant geometrical structures referred to a noninertial frame in Minkowski spacetime for an arbitrary velocities and characteristic acceleration lengths. Despite the totally different and independent physical sources of gravitation and inertia, this approach furnishes justification for the introduction of the weak principle of equivalence (WPE), that is, the universality of free fall. Consequently, we relate the inertia effects to the more general post-Riemannian geometry.

Beschreibung: We present the relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. Classical equations for the spin tensor are the same as those of the Barut-Zanghi model of spinning particle.

Beschreibung: We prove that the Lie superalgebra of regular differential operators on the superspace ℂM|N[t,t−1] has an essentially unique non-trivial central extension.

Beschreibung: We show that a “dynamical” interaction for arbitrary spin can be constructed in a straightforward way if gauge and Lorentz transformations are placed on the same foundation. As Lorentz transformations act on space-time coordinates, gauge transformations are applied to the gauge field. Placing these two transformations on the same ground means that all quantized field like spin-1/2 and spin-3/2 spinors are functions not only of the coordinates but also of the gauge field components. As a consequence, on this stage the (electromagnetic) gauge field has to be considered as classical field. Therefore, standard quantum field theory cannot be applied. Despite this inconvenience, such a common ground is consistent with an old dream of physicists almost a century ago. Our approach, therefore, indicates a straightforward way to realize this dream.

Beschreibung: We prove a version of the Jacobs-de Leeuw-Glicksberg splitting theorem for weak* continuous one-parameter semigroups on dual Banach spaces. This result is applied to give sufficient conditions for a quantum dynamical semigroup to display decoherence. The underlying notion of decoherence is that introduced by Blanchard and Olkiewicz (2003). We discuss this notion in some detail.

Beschreibung: The goal of this paper is to provide estimates leading to a direct proof of the analyticity of the pressure for certain classical unbounded models. We use our new formula (Lo, 2007) to establish the analyticity of the pressure in the thermodynamic limit for a wide class of classical unbounded models in statistical mechanics.

Beschreibung: The paper presents a special class of embedding problems whoes solutions are important for the explicit solution of nonlinear partial differential equations. It is shown that these embedding problems are solvable and explicit solutions are given. Not only are the solutions new but also the mathematical framework of their construction which is defined by a nonstandard function theory built over nonstandard algebraical structures, denoted as “GAPs”. These GAPs must not be neither associative nor division algebras, but the corresponding function theories built over them preserve the most important symmetries from the classical complex function theory in a generalized form: a generalization of the Cauchy-Riemannian differential equations exists as well as a generalization of the classical Cauchy Integral Theorem.

Beschreibung: We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially time correlated (Ornstein-Uhlenbeck noise). Different methods of achieving the marginal densities for correlated and uncorrelated noise are discussed. Analytical results are presented for a deterministic linear friction force and a stochastic force that is uncorrelated or exponentially correlated.

Beschreibung: We study spectral properties of a family of quasiperiodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp (2005).

Beschreibung: Recent developments are reviewed and some new results are presented in the study of time in quantum mechanics and quantum electrodynamics as an observable, canonically conjugate to energy. This paper deals with the maximal Hermitian (but nonself-adjoint) operator for time which appears in nonrelativistic quantum mechanics and in quantum electrodynamics for systems with continuous energy spectra and also, briefly, with the four-momentum and four-position operators, for relativistic spin-zero particles. Two measures of averaging over time and connection between them are analyzed. The results of the study of time as a quantum observable in the cases of the discrete energy spectra are also presented, and in this case the quasi-self-adjoint time operator appears. Then, the general foundations of time analysis of quantum processes (collisions and decays) are developed on the base of time operator with the proper measures of averaging over time. Finally, some applications of time analysis of quantum processes (concretely, tunneling phenomena and nuclear processes) are reviewed.

Beschreibung: We consider a Hamiltonian with cutoffs describing the weak decay of spin 1 massive bosons into the full family of leptons. The Hamiltonian is a self-adjoint operator in an appropriate Fock space with a unique ground state. We prove a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold for a sufficiently small coupling constant. As a corollary, we prove the absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval.

Beschreibung: A series of compact implicit schemes of fourth and sixth orders are developed for solving differential equations involved in geodynamics simulations. Three illustrative examples are described to demonstrate that high-order convergence rates are achieved while good efficiency in terms of fewer grid points is maintained. This study shows that high-order compact implicit difference methods provide high flexibility and good convergence in solving some special differential equations on nonuniform grids.

Beschreibung: The geometrical application of split octonions is considered. The new representation of products of the basis units of split octonionic having David's star shape (instead of the Fano triangle) is presented. It is shown that active and passive transformations of coordinates in octonionic “eight-space” are not equivalent. The group of passive transformations that leave invariant the pseudonorm of split octonions is SO(4,4), while active rotations are done by the direct product of O(3,4)-boosts and real noncompact form of the exceptional group G2. In classical limit, these transformations reduce to the standard Lorentz group.

Beschreibung: The results of minimizing the action for string-like systems on a simply connected world sheet are shown to encode the Cartesian components of real null momentum four-vectors into coordinates on the world sheet. This identification arises consistently from different approaches to the problem.

Beschreibung: Quantum Hall systems are a suitable theme for a case study in the general area of nanotechnology. In particular, it is a good framework to search for universal principles relevant to nanosystem modeling and nanosystem-specific signal processing. Recently, we have been able to construct a partial differential equations-based model of a quantum Hall system, which consists of the Schrödinger equation supplemented with a special-type nonlinear feedback loop. This result stems from a novel theoretical approach, which in particular brings to the fore the notion of quantum information. Here we undertake to modify the original model by substituting the dynamics based on the Dirac operator. This leads to a model that consists of a system of three nonlinearly coupled first-order elliptic equations in the plane.

Beschreibung: In this paper, we apply to (almost) all the “named” polynomials of the Askey scheme, as defined by their standard three-term recursion relations, the machinery developed in previous papers. For each of these polynomials we identify at least one additional recursion relation involving a shift in some of the parameters they feature, and for several of these polynomials characterized by special values of their parameters, factorizations are identified yielding some or all of their zeros—generally given by simple expressions in terms of integers (Diophantine relations). The factorization findings generally are applicable for values of the Askey polynomials that extend beyond those for which the standard orthogonality relations hold. Most of these results are not (yet) reported in the standard compilations.

Beschreibung: An algebraic scheme for constructing deformations of structure constants for associative algebras generated by deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable systems are studied.

Beschreibung: We discuss the concept of “hydrodynamic” stochastic theory, which is not based on the traditional Markovian concept. A Wigner function developed for friction is used for the study of operators in quantum physics, and for the construction of a quantum equation with friction. We compare this theory with the quantum theory, the Liouville process, and the Ornstein-Uhlenbeck process. Analytical and numerical examples are presented and compared.

Beschreibung: There are two types of master equations in describing nonequilibrium phenomena with memory effect: (i) the memory function type and (ii) the nonstationary type. A generalized Polya process is studied within the framework of a non-stationary type master equation approach. For a transition-rate with an arbitrary time-dependent relaxation function, the exact solution of a generalized Polya process is obtained. The characteristic features of temporal variation of the solution are displayed for some typical time-dependent relaxation functions reflecting memory in the systems.

Beschreibung: Constructive field theory can be considered as a reorganization of perturbation theory in a convergent way. In this pedagogical note we propose to wander through five different methods to compute the number of connected graphs of the zero-dimensional ϕ4 field theory, in increasing order of sophistication and power.

Beschreibung: The 2×2 Schlesinger system for the case of four regular singularities is equivalent to the Painlevé VI equation. The Painlevé VI equation can in turn be rewritten in the symmetric form of Okamoto's equation; the dependent variable in Okamoto's form of the PVI equation is the (slightly transformed) logarithmic derivative of the Jimbo-Miwa tau-function of the Schlesinger system. The goal of this note is twofold. First, we find a universal formulation of an arbitrary Schlesinger system with regular singularities in terms of appropriately defined Virasoro generators. Second, we find analogues of Okamoto's equation for the case of the 2×2 Schlesinger system with an arbitrary number of poles. A new set of scalar equations for the logarithmic derivatives of the Jimbo-Miwa tau-function is derived in terms of generators of the Virasoro algebra; these generators are expressed in terms of derivatives with respect to singularities of the Schlesinger system.

Beschreibung: This work is based on the field of reference frames based on quantum representations of real and complex numbers described in other work. Here frame domains are expanded to include space and time lattices. Strings of qukits are described as hybrid systems as they are both mathematical and physical systems. As mathematical systems they represent numbers. As physical systems in each frame the strings have a discrete Schrodinger dynamics on the lattices. The frame field has an iterative structure such that the contents of a stage j frame have images in a stage j-1 (parent) frame. A discussion of parent frame images includes the proposal that points of stage j frame lattices have images as hybrid systems in parent frames. The resulting association of energy with images of lattice point locations, as hybrid systems states, is discussed. Representations and images of other physical systems in the different frames are also described.

Beschreibung: We study the initial value problem for the quadratic nonlinear Klein-Gordon equation Lu=〈i∂x〉-1u¯2, (t,x)∈R×R, u(0,x)=u0(x), x∈R, where L=∂t+i〈i∂x〉 and 〈i∂x〉=1-∂x2̅. Using the Shatah normal forms method, we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data which was assumed in the previous works.

Beschreibung: Since the discussion of Kapteyn series occurrences in astronomical problems the wealth of mathematical physics problems in which such series play dominant roles has burgeoned massively. One of the major concerns is the ability to sum such series in closed form so that one can better understand the structural and functional behavior of the basic physics problems. The purpose of this review article is to present some of the recent methods for providing such series in closed form with applications to: (i) the summation of Kapteyn series for radiation from pulsars; (ii) the summation of other Kapteyn series in radiation problems; (iii) Kapteyn series arising in terahertz sideband spectra of quantum systems modulated by an alternating electromagnetic field; and (iv) some plasma problems involving sums of Bessel functions and their closed form summation using variations of the techniques developed for Kapteyn series. In addition, a short review is given of some other Kapteyn series to illustrate the ongoing deep interest and involvement of scientists in such problems and to provide further techniques for attempting to sum divers Kapteyn series.

Beschreibung: For a system of n interacting particles moving in the background of a “homogeneous” potential, we show that if the single particle Hamiltonian admits a density of states, so does the interacting n-particle Hamiltonian. Moreover, this integrated density of states coincides with that of the free particle Hamiltonian. For the interacting n-particle Anderson model, we prove regularity properties of the integrated density of states by establishing a Wegner estimate.

Beschreibung: We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in ℝ2d, d≥1. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We give asymptotic formulas for the rate of accumulation of eigenvalues in these clusters. When the compact is a Reinhardt domain we are able to show a more precise asymptotic formula.

Beschreibung: We construct a new supertwistor space suited for establishing a Penrose-Ward transform between certain bundles over this space and solutions to the 𝒩=8 super-Yang-Mills equations in three dimensions. This mini-superambitwistor space is obtained by dimensional reduction of the superambitwistor space, the standard superextension of the ambitwistor space. We discuss in detail the construction of this space and its geometry before presenting the Penrose-Ward transform. We also comment on a further such transform for purely bosonic Yang-Mills-Higgs theory in three dimensions by considering third-order formal “subneighborhoods” of a miniambitwistor space.

Beschreibung: From the integration of nonsymmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions (pdfs). A very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. We obtain, as particular cases, the generalized error function, the Zipf-Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.

Beschreibung: We show that a nonlinear equation that represents third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids is integrable for a particular choice of its parameters, since in this case it is equivalent with an integrable equation which has recently appeared in the literature. We also discuss the integrability of both second- and third-order approximations of additional cases.

Beschreibung: We present a review of the basics of supermanifold theory (in the sense of Berezin-Kostant-Leites-Manin) from a physicist's point of view. By considering a detailed example of what does it mean the expression “to integrate an ordinary superdifferential equation” we show how the appearance of anticommuting parameters playing the role of time is very natural in this context. We conclude that in dynamical theories formulated whithin the category of supermanifolds, the space that classically parametrizes time (the real line ℝ) must be replaced by the simplest linear supermanifold ℝ1|1. This supermanifold admits several different Lie supergroup structures, and we analyze from a group-theoretic point of view what is the meaning of the usual covariant superderivatives, relating them to a change in the underlying group law. This result is extended to the case of N-supersymmetry.

Beschreibung: We will show that a two-parameter extended entropy function is characterized by a functional equation. As a corollary of this result, we obtain that Tsallis entropy function is characterized by a functional equation, which is a different form that used by Suyari and Tsukada, 2009, that is, in a proposition 2.1 in the present paper. We give an interpretation of the functional equation in our main theorem.

Beschreibung: The homological Kähler-de Rham differential mechanism models the dynamical behavior of physical fields by purely algebraic means and independently of any background manifold substratum. This is of particular importance for the formulation of dynamics in the quantum regime, where the adherence to such a fixed substratum is problematic. In this context, we show that the functorial formulation of the Kähler-de Rham differential mechanism in categories of sheaves of commutative algebras, instantiating generalized localization environments of physical observables, induces a consistent functorial framework of dynamics in the quantum regime.

Beschreibung: The quantum Langevin equation has been studied for dissipative system using the approach of Ford et al. Here, we have considered the inverted harmonic oscillator potential and calculated the effect of dissipation on tunneling time, group delay, and the self-interference term. A critical value of the friction coefficient has been determined for which the self-interference term vanishes. This approach sheds new light on understanding the ion transport at nanoscale.

Beschreibung: We present an explicit construction of a unitary representation of the commutator algebra satisfied by position and momentum operators on the Moyal plane.

Beschreibung: Through the systematic use of the Minlos theorem on the support of cylindrical measures on R∞, we produce several mathematically rigorous finite-volume euclidean path integrals in interacting euclidean quantum fields with Gaussian free measures defined by generalized powers of finite-volume Laplacian operator.

Beschreibung: It is the purpose of this paper to give a simple proof of the fact that solutions of the KdV equation can be approximated via solutions of the NLS equation. The proof is based on an elimination of the quadratic terms of the KdV equation via the Miura transformation.

Beschreibung: The mechanism of differential geometric calculus is based on the fundamental notion of a connection on a module over a commutative and unital algebra of scalars defined together with the associated de Rham complex. In this communication, we demonstrate that the dynamical mechanism of physical fields can be formulated by purely algebraic means, in terms of the homological Kähler-De Rham differential schema, constructed by connection inducing functors and their associated curvatures, independently of any background substratum. In this context, we show explicitly that the application of this mechanism in General Relativity, instantiating the case of gravitational dynamics, is related with the absolute representability of the theory in the field of real numbers, a byproduct of which is the fixed background manifold construct of this theory. Furthermore, the background independence of the homological differential mechanism is of particular importance for the formulation of dynamics in quantum theory, where the adherence to a fixed manifold substratum is problematic due to singularities or other topological defects.

Beschreibung: We consider the initial value problem for the reduced fifth-order KdV-type equation: ∂tu-∂x5u-10∂x(u3)+10∂x(∂xu)2=0, t,x∈R, u(0,x)=ϕ(x), x∈R. This equation is obtained by removing the nonlinear term 10u∂x3u from the fifth-order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables if the initial data ϕ∈Hs(R)  (s>1/8) satisfies the condition ∑k=0∞(A0k/k!)∥(x∂x)kϕ∥Hs<∞, for some constant A0    (0<A0<1). Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the contraction principle and the bootstrap argument used in the third-order KdV equation (K. Kato and Ogawa 2000). The key of the proof is to obtain the estimate of ∂x(∂xu)2 on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).

Beschreibung: We extend the Hijazi type inequality, involving the energy-momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spinc manifolds without boundary and of finite volume. Under some additional assumptions, using the refined Kato inequality, we prove the Hijazi type inequality for elements of the essential spectrum. The limiting cases are also studied.

Beschreibung: Metzler et al. introduced a fractional Fokker-Planck equation (FFPE) describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath. In this paper, we present an interval Shannon wavelet numerical method for the FFPE. In this method, a new concept named “dynamic interval wavelet” is proposed to solve the problem that the numerical solution of the fractional PDE is usually sensitive to boundary conditions. Comparing with the traditional wavelet defined in the interval, the Newton interpolator is employed instead of the Lagrange interpolation operator, so, the extrapolation points in the interval wavelet can be chosen dynamically to restrict the boundary effect without increase of the calculation amount. In order to avoid unlimited increasing of the extrapolation points, both the error tolerance and the condition number are taken as indicators for the dynamic choice of the extrapolation points. Then, combining with the finite difference technology, a new numerical method for the time fractional partial differential equation is constructed. A simple Fokker-Planck equation is taken as an example to illustrate the effectiveness by comparing with the Grunwald-Letnikov central difference approximation (GL-CDA).

Beschreibung: An inverse source identification problem for a time fractional diffusion equation is discussed. The unknown heat source is supposed to be space dependent only. Based on the use of Green’s function, an effective numerical algorithm is developed to recover both the intensities and locations of unknown point sources from final measurements. Numerical results indicate that the proposed method is efficient and accurate.

Beschreibung: By summarizing some classical active contour models from the view of level set representation, a simple energy function expression with the Gaussian kernel of fractional order is proposed, and then a novel region-based geometric active contour model is established. In this proposed model, the energy function with value of [−1, 1] is built, the local mean and global mean of the inside and outside of the evolution curve are employed, and the segmentation results are obtained by controlling the expansion and contraction of the evolution curve. The model is simple and easy to implement; it can also protect weak edges because of considering more statistical information. Experimental results on synthetic and natural images show that the proposed model is much more effective in dealing with the images with weak or blurred edges, and it takes less time.

Beschreibung: This paper introduces two novel numerical algorithms for the efficient solution of coupled systems of nonlinear boundary value problems. The methods are benchmarked against existing methods by finding dual solutions of the highly nonlinear system of equations that model the flow of a viscoelastic liquid of Oldroyd-B type in a channel of infinite extent. The methods discussed here are the spectral relaxation method and spectral quasi-linearisation method. To verify the accuracy and efficiency of the proposed methods a comparative evaluation of the performance of the methods against established numerical techniques is given.

Beschreibung: This paper is concerned with dynamics of the solution to the system of two high-order nonlinear difference equations , , , , where , , and . Moreover the rate of convergence of a solution that converges to the equilibrium of the system is discussed. Finally, some numerical examples are considered to show the results obtained.

Beschreibung: The diffusion equation is discretized in spacial direction and transformed into the ordinary differential equations. The ordinary differential equations are solved by Lie group method and the explicit Runge-Kutta method. Numerical results showed that Lie group method is more stable than the corresponding explicit Runge-Kutta method.

Beschreibung: We study the necessary and sufficient conditions on the masses for the periodic solution of planar 4-body problems, where three particles locate at the vertices of an equilateral triangle and rotate with constant angular velocity about a resting particle. We prove that the above periodic motion is a solution of Newtonian 4-body problems if and only if the resting particle is at the origin and the masses of the other three particles are equal and their angular velocity satisfies a special condition.

Beschreibung: Based on solving numerically the generalized nonlinear Schrödinger equation describing the propagation of high order femtosecond soliton in silicon waveguide under certain parametric conditions by the split-step Fourier and Runge-Kutta integration methods, dispersive wave emission and supercontinuum generation in silicon waveguides are numerically investigated by propagating femtosecond solitons. The numerical results show that the efficient dispersive wave emission can be generated in silicon waveguide, which plays an important role in the process of the supercontinuum generation with the form of Cherenkov radiation, and it is also shown that the high order low-energy solitons and short waveguides are efficient for the dispersive wave emission.

Beschreibung: Fuzzy and fractional differential equations are used to model problems with uncertainty and memory. Using the fractional fuzzy Laplace transformation we have solved the fuzzy fractional eigenvalue differential equation. By illustrative examples we have shown the results.

Beschreibung: Heat transfer characteristics of a Berman flow of water based nanofluids containing copper (Cu) and alumina (Al2O3) as nanoparticles in a porous channel with Navier slip, viscous dissipation, and convective cooling are investigated. It is assumed that the exchange of heat with the ambient surrounding takes place at the channel walls following Newton’s law of cooling. The governing partial differential equations and boundary conditions are converted into a set of nonlinear ordinary differential equations using appropriate similarity transformations. These equations are solved analytically by regular perturbation methods with series improvement technique and numerically using an efficient Runge-Kutta Fehlberg integration technique coupled with shooting scheme. The effects of the governing parameters on the dimensionless velocity, temperature, skin friction, pressure drop, and Nusselt numbers are presented graphically and discussed quantitatively.

Beschreibung: We present applications of the representation theory of Lie groups to the analysis of structure and local unitary classification of Werner states, sometimes called the decoherence-free states, which are states of n quantum bits left unchanged by local transformations that are the same on each particle. We introduce a multiqubit generalization of the singlet state and a construction that assembles these qubits into Werner states.

Beschreibung: This paper is devoted to the study of the chaotic properties of some specific backward shift unbounded operators Hp=A*pAP+1; p=0,1,… realized as differential operators in Bargmann space, where A and A* are the standard Bose annihilation and creation operators such that [A,A*]=I.

Beschreibung: we propose a new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic partial differential equation of the form utt=A(x,y,t,u)uxx+B(x,y,t,u)uyy+g(x,y,t,u,ux,uy,ut), 0<x, y<1, t>0 subject to appropriate initial and Dirichlet boundary conditions , where k>0 and h>0 are mesh sizes in time and space directions, respectively. We use only five evaluations of the function g as compared to seven evaluations of the same function discussed by (Mohanty et al., 1996 and 2001). We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Some examples and their numerical results are provided to justify the usefulness of the proposed method.