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Abstract: Random Matrix Theory (RMT) is a powerful statistical tool to model spectral fluctuations. This approach has also found fruitful application in Quantum Chromodynamics (QCD). Importantly, RMT provides very efficient means to separate different scales in the spectral fluctuations. We try to identify the equivalent of a Thouless energy in complete spectra of the QCD Dirac operator for staggered fermions from SU(2) lattice gauge theory for different lattice size and gauge couplings. We focus on the bulk of the spectrum. In disordered systems, the Thouless energy sets the universal scale for which RMT applies. This relates to recent theoretical studies which suggest a strong analogy between QCD and disordered systems. The wealth of data allows us to analyze several statistical measures in the bulk of the spectrum with high quality. We find deviations which allows us to give an estimate for this universal scale. Other deviations than these are seen whose possible origin is discussed. Moreover, we work out higher order correlators as well, in particular three-point correlation functions.
Abstract: We show that the physical mechanism of population transfer in a 3-level system with a closed loop of coherent couplings (loop-STIRAP) is not equivalent to an adiabatic rotation of the dark-state of the Hamiltonian but coresponds to a rotation of a higher-order trapping state in a generalized adiabatic basis. The concept of generalized adiabatic basis sets is used as a constructive toolto design pulse sequences for stimulated Raman adiabatic passage (STIRAP) which give maximum population transfer also under conditions when the usual condition of adiabaticty is only poorly fulfilled. Under certain conditions for the pulses (generalized matched pulses) there exists a higher-order trapping state, which is an exact constant of motion and analytic solutions for the atomic dynamics can be derived.
Abstract: We analyze the long-time quantum dynamics of degenerate parametric down-conversion from an initial sub-harmonic vacuum (spontaenous down-conversion). Standard linearization of the Heisenberg equations of motions fails in this case, since it is based on an expansion around an unstable classical solution and neglects pump depletion. Introducing a mean-field approximation we find a periodic exchange of energy between the pump and subharmonic mode goverened by an anharmonic pendulum equation. From this equation the optimum interaction time or crystal length for maximum conversion can be determined. A numerical integration of the 2-mode Schrödinger equation using a dynamically optimized basis of displaced and squeezed number states verifies the characteristic times predicted by the mean-field approximation. In contrast to semiclassical and mean-field predictions it is found that quantum uctuations of the pump mode lead to a substantial limitation of the efficiency of parametric down-conversion.
Abstract: Generalized single-atom Maxwell-Bloch equations for optically dense media are derived taking into account non-cooperative radiative atom-atom interactions. Applying a Gaussian approximation and formally eliminating the degrees of freedom of the quantized radiation field and of all but a probe atom leads to an effective time-evolution operator for the probe atom. The mean coherent amplitude of the local field seen by the atom is shown to be given by the classical Lorentz-Lorenz relation. The second-order correlations of the field lead to terms that describe relaxation or pump processes and level shifts due to multiple scattering or reabsorption of spontaneously emitted photons. In the Markov limit a non-linear and nonlocal single-atom density matrix equation is derived. To illustrate the effects of the quantum corrections we discuss amplified spontaneous emission and radiation trapping in a dense ensemble of initially inverted two-level atoms and the effects of radiative interactions on intrinsic optical bistability in coherently driven systems.
Abstract: We predict the possibility of sharp, high-contrast resonances in the optical response of a broad class of systems, wherein interference effects are generated by coherent perturbation or interaction of dark states. The properties of these resonances can be manipulated to design a desired atomic response.
Thermal Properties of Interacting Bose Fields and Imaginary-Time Stochastic Differential Equations
(1998)
Abstract: Matsubara Green's functions for interacting bosons are expressed as classical statistical averages corresponding to a linear imaginary-time stochastic differential equation. This makes direct numerical simulations applicable to the study of equilibrium quantum properties of bosons in the non-perturbative regime. To verify our results we discuss an oscillator with quartic anharmonicity as a prototype model for an interacting Bose gas. An analytic expression for the characteristic function in a thermal state is derived and a Higgs-type phase transition discussed, which occurs when the oscillator frequency becomes negative.
Abstract: We investigate the quantum properties of fields generated by resonantly enhanced wave mixing based on atomic coherence in Raman systems. We show that such a process can be used for generation of pairs of Stokes and anti-Stokes fields with nearly perfect quantum correlations, yielding almost complete (i.e. 100%) squeezing without the use of a cavity. We discuss the extension of the wave mixing interactions into the domain of a few interacting light quanta.
Abstract: Resonant optical pumping in dense atomic media is discussed, where the absorption length is less than the smallest characteristic dimension of the sample. It is shown that reabsorption and multiple scattering of spontaneous photons (radiation trapping) can substantially slow down the rate of optical pumping. A very slow relaxation out of the target state of the pump process is then sufficient to make optical pumping impossible. As model systems an inhomogeneously and a radiatively broadened 3-level system resonantly driven with a strong broad-band pump field are considered.
Abstract: The effect of intracavity Electromagnetically Induced Transparency on the properties of optical resonators and active laser devices is discussed theoretically. A pronounced frequency pulling and cavity linewidth narrowing are predicted. The effect can be used to substantially reduce classical and quantum phase noise of the beat-note of optical oscillators. Fundamental limits of this stabilization mechanism are discussed as well as its potential application to high-resolution spectroscopy.
Abstract: We develop a constructive method to derive exactly solvable quantum mechanical models of rational (Calogero) and trigonometric (Sutherland) type. This method starts from a linear algebra problem: finding eigenvectors of triangular finite matrices. These eigenvectors are transcribed into eigenfunctions of a selfadjoint Schrödinger operator. We prove the feasibility of our method by constructing an " AG_3 model" of trigonometric type (the rational case was known before from Wolfes 1975). Applying a Coxeter group analysis we prove its equivalence with the B_3 model. In order to better understand features of our construction we exhibit the F_4 rational model with our method.
Verbale Sacherschließung
(1998)
Das Skript gibt eine Einführung in die Geschichte, die Terminologie und die Verfahren der verbalen Sacherschließung. Im deutschsprachigen und englischsprachigen Raum etablierte Verfahren, wie die "Regeln für den Schlagwortkatalog (RSWK)" und die "Library of Congress Subject Headings (LCSH)", werden eingehend beschrieben und Aspekte der Kooperation und Tauglichkeit für Online-Kataloge diskutiert. Charakteristika sowie Vor- und Nachteile der automatischen Indexierung werden anhand des Verfahrens "Maschinelle Indexierung zur verbesserten Literaturerschließung in Online Systemen (MILOS)" dargestellt.
Superselection rules induced by the interaction with the environment are investigated with the help of exactly soluble Hamiltonian models. Starting from the examples of Araki and of Zurek more general models with scattering are presented for which the projection operators onto the induced superselection sectors do no longer commute with the Hamiltonian. The example of an environment given by a free quantum field indicates that infrared divergence plays an essential role for the emergence of induced superselection sectors. For all models the induced superselection sectors are uniquely determined by the Hamiltonian, whereas the time scale of the decoherence depends crucially on the initial state of the total system.
In this paper we derive nonparametric stochastic volatility models in discrete time. These models generalize parametric autoregressive random variance models, which have been applied quite successfully to nancial time series. For the proposed models we investigate nonparametric kernel smoothers. It is seen that so-called nonparametric deconvolution estimators could be applied in this situation and that consistency results known for nonparametric errors- in-variables models carry over to the situation considered herein.
We present a particle method for the numerical simulation of boundary value problems for the steady-state Boltzmann equation. Referring to some recent results concerning steady-state schemes, the current approach may be used for multi-dimensional problems, where the collision scattering kernel is not restricted to Maxwellian molecules. The efficiency of the new approach is demonstrated by some numerical results obtained from simulations for the (two-dimensional) BEnard's instability in a rarefied gas flow.
In this paper we present a domain decomposition approach for the coupling of Boltzmann and Euler equations. Particle methods are used for both equations. This leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains. Adaptive time and space discretizations and a direct coupling procedure leads to considerable gains in CPU time compared to a solution of the full Boltzmann equation. Several test cases involving a large range of Knudsen numbers are numerically investigated.
Application of Moment Realizability Criteria for Coupling of the Boltzmann and Euler Equations
(1998)
The moment realizability criteria have been used to test the domains of validity of the Boltzmann and Euler Equations. With the help of this criteria teh coupling of the Boltzmann and Euler equations have been performed in two dimensional spatial space. The time evolution of domain decompositions for such equations have been presented in different time steps. The numerical resulta obtained from the coupling code have been compared with those from the pure Boltzmann one.
We develop a constructive method to derive exactly solvable quantum mechanical models of rational (Calogero) and trigonometric (Sutherland) type. This method starts from a linear algebra problem: finding eigenvectors of triangular finite matrices. These eigenvectors are transcribed into eigenfunctions of a selfadjoint Schrödinger operator. We prove the feasibility of our method by constructing a new "\(AG_3\) model" of trigonometric type (the rational case was known before from Wolfes 1975). Applying a Coxeter group analysis we prove its equivalence with the \(B_3\) model. In order to better understand features of our construction we exhibit the \(F_4\) rational model with our method.
The critical points of the continuous series are characterized by two complex numbers l_1,l_2 (Re(l_1,l_2)< 0), and a natural number n (n>=3) which enters the string susceptibility constant through gamma = -2/(n-1). The critical potentials are analytic functions with a convergence radius depending on l_1 or l_2. We use the orthogonal polynomial method and solve the Schwinger-Dyson equations with a technique borrowed from conformal field theory.
Monomial representations and operations for Gröbner bases computations are investigated from an implementation point of view. The technique ofvectorized monomial operations is introduced and it is shown how it expedites computations of Gröbner bases. Furthermore, a rank-based monomialrepresentation and comparison technique is examined and it is concluded that this technique does not yield an additional speedup over vectorizedcomparisons. Extensive benchmark tests with the Computer Algebra System SINGULAR are used to evaluate these concepts.
Groups can be studied using methods from different fields such as combinatorial group theory or string rewriting. Recently techniques from Gröbner basis theory for free monoid rings (non-commutative polynomial rings) respectively free group rings have been added to the set of methods due to the fact that monoid and group presentations (in terms of string rewriting systems) can be linked to special polynomials called binomials. In the same mood, the aim of this paper is to discuss the relation between Nielsen reduced sets of generators and the Todd-Coxeter coset enumeration procedure on the one side and the Gröbner basis theory for free group rings on the other. While it is well-known that there is a strong relationship between Buchberger's algorithm and the Knuth-Bendix completion procedure, and there are interpretations of the Todd-Coxeter coset enumeration procedure using the Knuth-Bendix procedure for special cases, our aim is to show how a verbatim interpretation of the Todd-Coxeter procedure can be obtained by linking recent Gröbner techniques like prefix Gröbner bases and the FGLM algorithm as a tool to study the duality of ideals. As a side product our procedure computes Nielsen reduced generating sets for subgroups in finitely generated free groups.
This paper describes a tableau-based higher-order theorem prover HOT and an application to natural language semantics. In this application, HOT is used to prove equivalences using world knowledge during higher-order unification (HOU). This extended form of HOU is used to compute the licensing conditions for corrections.
Simultaneous quantifier elimination in sequent calculus is an improvement over the well-known skolemization. It allows a lazy handling of instantiations as well as of the order of certain reductions. We prove the soundness of a sequent calculus which incorporates a rule for simultaneous quantifier elimination. The proof is performed by semantical arguments and provides some insights into the dependencies between various formulas in a sequent.
Bekanntlich gibt es keinen befriedigenden unendlich dimensionalen Ersatz für das Lebesgue-Mass. Andererseits lassen sich viele Techniken klassischer Analysis auch auf unendlich dimensionale Situationen übertragen. Eine Möglichkeit hierzu gibt die Theorie differenzierbarer Masse. Man definiert Richtungsableitungen für Masse ähnlich wie für Funktionen. Eines der zentralen Beispiele ist das Wiener-Mass. Stochastische Integration bezüglich der Brownschen Bewegung, insbesondere das Skorokhod-Integral ergeben sich in natürlicher Weise durch diesen Ansatz und auch die Grundideen des MalliavinKalküls lassen sich in diesem Rahmen einfach erläutern. Die Vorträge geben die meisten Beweise.
On a family F of probability measures on a measure space we consider the Hellinger and Kullback-Leibler distances. We show that under suitable regulari ty conditions Jeffreys' prior is proportional to the k-dimensional Hausdorff measure w.r.t. Hellinger dis tance respectively to the k2 -dimensional Hausdorff measure w.r.t. Kullback-Leibler distance. The proof i s based on an area-formula for the Hausdorff measure w.r.t. to generalized distances.
The paper studies differential and related properties of functions of a real variable with values in the space of signed measures. In particular the connections between different definitions of differentiability are described corresponding to different topologies on the measures. Some conditions are given for the equivalence of the measures in the range of such a function. These conditions are in terms of socalled logarithmic derivatives and yield a generalization of the Cameron-Martin-Maruyama-Girsanov formula. Questions of this kind appear both in the theory of differentiable measures on infinite-dimensional spaces and in the theory of statistical experiments.
The notion of formal description techniques for timed systems (T-FDTs) has been introduced in [EDK98a] to provide a unifying framework for description techniques that are formal and that allow to describe the ongoing behavior of systems. In this paper we show that three well known temporal logics, MTL, MTL-R , and CTL*, can be embedded in this framework. Moreover, we provide evidence that a large number of dioeerent kinds of temporal logics can be considered as T-FDTs.
Das Problem der Integration heterogener Softwaresysteme stellt sich auch auf dem Gebiet der CAx-Systeme, wie sie in vielfältigen Ausprägungen etwa in der Automobilbranche für die Fahrzeugentwicklung eingesetzt werden. Zunächst werden die heute in diesem Bereich
praktizierten Lösungen und die dabei auftretenden Probleme kurz dargestellt. Danach werden der neue Standard für Produktdaten, STEP, und der Standard für die Interoperabilität heterogener Softwaresysteme, CORBA, sowie einige CORBA-Entwurfsmuster erläutert. Als nächstes wird eine auf diesen beiden Standards basierende CAx-Integrationsarchitektur, die im Projekt ANICA entwickelt wurde, vorgestellt und die prinzipielle Vorgehensweise bei
ihrer Realisierung beschrieben. Daran anschließend wird über eine erste Umsetzung dieser Architektur in die Praxis berichtet. Zum Abschluß wird kurz auf die gewonnenen Erfahrungen eingegangen und ein Ausblick auf zukünftige Entwicklungen gegeben.
Die virtuelle Produktentwicklung in verteilter Umgebung erfordert eine intensive Kommunika-tion zwischen den beteiligten CAx-Systemen. Diese findet bisher in Form des dateibasierten Datenaustausches mit Hilfe von Direktkonvertern oder neutralen Schnittstellen statt. Der Datenaustausch wird hierbei meist in mehreren Iterationsschleifen durchgeführt und ist oft mit Datenverlusten sowie Unterbrechungen der Entwicklungsaktivitäten verbunden. Demgegenüber steht als neuer Ansatz für die Interoperabilität zwischen CAx-Systemen das Konzept eines CAx-Objektbusses auf Basis von CORBA und STEP. Dieser Ansatz ermög-licht eine plattformübergreifende Online-Kopplung heterogener CAx-Systeme. Im Gegensatz zum dateibasierten Datenaustausch ist hierbei ein transparenter Zugriff sowohl auf Daten als auch auf Funktionen der angebundenen Systeme möglich. Dadurch kann die Durchgängigkeit der Produktdaten in der Prozeßkette deutlich erhöht werden. Zur Beurteilung der Praxistauglichkeit wird dieser neue Ansatz dem dateibasierten Daten-austausch am Beispiel virtueller Einbauuntersuchungen gegenübergestellt. Dabei werden für unterschiedliche praxisrelevante Modellgrößen die für die Übertragung von Geometrie und Topologie erforderlichen Zeiten analysiert und verglichen. Weiterhin werden die generellen Vor- und Nachteile der beiden Lösungen dargestellt. Abschließend wird auf die Potentiale des neuen Ansatzes für den Einsatz in anderen Bereichen eingegangen.
The paper addresses two problems of comprehensible proof presentation, the hierarchically structured presentation at the level of proof methods and different presentation styles of construction proofs. It provides solutions for these problems that can make use of proof plans generated by an automated proof planner.
In this paper we study a particular class of \(n\)-node recurrent neural networks (RNNs).In the \(3\)-node case we use monotone dynamical systems theory to show,for a well-defined set of parameters, that,generically, every orbit of the RNN is asymptotic to a periodic orbit.Then, within the usual 'learning' context of NeuralNetworks, we investigate whether RNNs of this class can adapt their internal parameters soas to 'learn' and then replicate autonomously certain external periodic signals.Our learning algorithm is similar to identification algorithms in adaptivecontrol theory. The main feature of the adaptation algorithm is that global exponential convergenceof parameters is guaranteed. We also obtain partial convergence results in the \(n\)-node case.
Wavelets on closed surfaces in Euclidean space R3 are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of functional values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at infinity. Furthermore, an exact fully discrete wavelet approximation is developed in case of band-limited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem.
For the determination of the earth" s gravity field many types of observations are available nowadays, e.g., terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gradiometry etc. The mathematical connection between these observables on the one hand and gravity field and shape of the earth on the other hand, is called the integrated concept of physical geodesy. In this paper harmonic wavelets are introduced by which the gravitational part of the gravity field can be approximated progressively better and better, reflecting an increasing flow of observations. An integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are integration formulas relating an integral over an internal sphere to suitable linear combinations of observation functionals, i.e., linear functionals representing the geodetic observables. A scale discrete version of multiresolution is described for approximating the gravitational potential outside and on the earth" s surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of band-limited wavelets. A method for combined global outer harmonic and local harmonic wavelet modelling is proposed corresponding to realistic earth" s models. As examples, the role of wavelets is discussed for the classical Stokes problem, the oblique derivative problem, satellite-to-satellite tracking, satellite gravity gradiometry, and combined satellite-to-satellite tracking and gradiometry.
Der vorliegende Artikel setzt die Beitragsreihe zur Vorstellung der Ergebnisse der FEMEX fort, die mit der Präsentation einer allgemeinen Feature-Definition in [BWE-96] begonnen wurde. FEMEX (Feature Modelling Experts) ist eine internationale und interdisziplinäre Gruppe von Forschern, Entwicklern und Anwendern aus Universitäten, Forschungsinstituten und Industrie, die sich zum Ziel gesetzt haben, Grundlagen für eine Feature-basierte Produktentwicklung zu erarbeiten. Der Anwender steht dabei im Mittelpunkt der Bemühungen: die Feature-Technologie hat die Aufgabe, ihm Methoden und Werkzeuge an die Hand zu geben, mit denen er in den unterschiedlichen Phasen einer komplexen Prozeßkette effizient arbeiten kann. Vier Arbeitsgruppen wurden gebildet, die sich mit unterschiedlichen Aspekten der Feature-Technologie beschäftigen. In diesem Beitrag werden die Ergebnisse der Arbeitsgruppe II Feature Modelling Methods and Application Areas" vorgestellt. Ihre Aufgabe ist es, die Modellierungsmethoden und Anwendungsgebiete der Feature-Technologie im Kontext des Produktentwicklungs- prozesses zu untersuchen. Ausgangspunkt für die Arbeiten ist neben den benutzerspezifischen Anforderungen die Feature-Definition der Arbeitsgruppe I [BWE-96]. An dieser Definition ist hervorzuheben, daß Features keine physikalischen Elemente sind und auch keine physikalischen Entsprechungen haben müssen, sondern nur in der Welt der informationstechnischen Modelle existieren. Desweiteren sind die für den Anwender relevanten Eigenschaften der bearbeiteten Objekte, welcher Art sie auch sein mögen (beispielsweise die Funktion des Bauteils), die eigentliche Grundlage der Definition. Keiner Eigenschaft wird von vorneherein eine höhere Priorität gegeben, wodurch die Bauteilgeometrie ihre tragende Rolle bei der Modellierung verliert (bei den meisten der heute angebotenen CAD/CAM-Systemen wird dagegen üblicherweise davon ausgegangen, daß die in einem System verarbeitete Produktgeometrie die Basis für das gesamte Produktmodell darstellt).
Aesthetic Design bzw. Styling ist mehr und mehr ein zentrales Merkmal für den Erfolg von Automobilen auf dem Weltmarkt. Entsprechend den firmenspezifischen Vorstellungen werden diese Eigenschaften der Karosserien in komplexen Abläufen herausgearbeitet. Computer Aided Styling (CAS), Computer Aided Aesthetic Design (CAAD) sind die Werkzeuge zur Schaffung optimaler Karosserieformen. Die Abläufe sind von Unternehmen zu Unternehmen unterschiedlich, haben aber ähnliche Strukturen: es wird die Form der Karosserie erstellt, anschließend wird mit Hilfe geeigneter Werkzeuge die Qualität der Flächen beurteilt. In einem nächsten Schritt werden die Flächen entsprechend dieser Beurteilung wieder verändert. Diese Schleifen werden wiederholt, bis das Ergebnis die Verantwortlichen zufriedenstellt. Im Brite-EuRam-Projekt FIORES von 12 Partnern aus 6 Ländern, mit Automobilunternehmen (BMW, Saab), Design-Firmen (Eiger, Formtech, Pininfarina, Taurus), Systemherstelllern und Forschungsinstituten wird jetzt versucht, Methoden zu entwickeln, die den Design-Ablauf verbessern könnten: Die Bewertungskriterien für ästhetische Flächen sollen formalisiert werden und dann direkt zur Modifikation der Freiformflächen benutzt werden im Sinne einer zielgesteuerten Modellierung (Engineering in Reverse, EiR). Dieser Artikel stellt die Ergebnisse des Projekts innerhalb des ersten Jahres dar: der Design-Prozeß in verschiedenen Unternehmen wird analysiert, die sich daraus ergebenden Beurteilungskriterien für ästhetische Formen werden formalisiert und der zielgesteuerten Modellierung zugeführt. Ausblicke auf weitere Ziele des Projekts werden gegeben. Die vorgestellten Arbeiten sind das gemeinsame Ergebnis des Projekt-Konsortiums.
On the one hand, in the world of Product Data Technology (PDT), the ISO standard STEP (STandard for the Exchange of Product model data) gains more and more importance. STEP includes the information model specification language EXPRESS and its graphical notation EXPRESS-G. On the other hand, in the Software Engineering world in general, mainly other modelling languages are in use - particularly the Unified Modeling Language (UML), recently adopted to become a standard by the Object Management Group, will probably achieve broad acceptance. Despite a strong interconnection of PDT with the Software Engineering area, there is a lack of bridging elements concerning the modelling language level. This paper introduces a mapping between EXPRESS-G and UML in order to define a linking bridge and bring the best of both worlds together. Hereby the feasibility of a mapping is shown with representative examples; several problematic cases are discussed as well as possible solutions presented.
Interoperability between different CAx systems involved in the development process of cars is presently one of the most critical issues in the automotive industry. None of the existing CAx systems meets all requirements of the very complex process network of the lifecycle of a car. With this background, industrial engineers have to use various CAx systems to get an optimal support for their daily work. Today, the communication between different CAx systems is done via data files using special direct converters or neutral system independent standards like IGES, VDAFS, and recently STEP, the international standard for product data description. To reduce the dependency on individual CAx s ystem vendors, the German automotive industry developed an open CAx system architecture based on STEP as guiding principle for CAx system development. The central component of this architecture is a common, system-independent access interface to CAx functions and data of all involved CAx systems, which is under development in the project ANICA. Within this project, a CAx object bus has been developed based on a STEP data description using CORBA as an integration platform. This new approach allows a transparent access to data and functions of the integrated CAx systems without file-based data exchange. The product development process with various CAx systems concerns objects from different CAx systems. Thus, mechanisms are needed to handle the persistent storage of the CAx objects distributed over the CAx object bus to give the developing engineers a consistent view of the data model of their product. The following paper discusses several possibilities to guarantee consistent data management and storage of distributed CAx models. One of the most promising approaches is the enhancement of the CAx object bus by a STEP-based object-oriented data server to realise a central data management.
We propose a new discretization scheme for solving ill-posed integral equations of the third kind. Combining this scheme with Morozov's discrepancy principle for Landweber iteration we show that for some classes of equations in such method a number of arithmetic operations of smaller order than in collocation method is required to appoximately solve an equation with the same accuracy.
In this paper we study the space-time asymptotic behavior of the solutions and derivatives to th incompressible Navier-Stokes equations. Using moment estimates we obtain that strong solutions to the Navier-Stokes equations which decay in \(L^2\) at the rate of \(||u(t)||_2 \leq C(t+1)^{-\mu}\) will have the following pointwise space-time decay \[|D^{\alpha}u(x,t)| \leq C_{k,m} \frac{1}{(t+1)^{ \rho_o}(1+|x|^2)^{k/2}} \]
where \( \rho_o = (1-2k/n)( m/2 + \mu) + 3/4(1-2k/n)\), and \(|a |= m\). The dimension n is \(2 \leq n \leq 5\) and \(0\leq k\leq n\) and \(\mu \geq n/4\)
The paper discusses the metastable states of a quantum particle in a periodic potential under a constant force (the model of a crystal electron in a homogeneous electric ,eld), which are known as the Wannier-Stark ladder of resonances. An ecient procedure to ,nd the positions and widths of resonances is suggested and illustrated by numerical calculation for a cosine potential.
We have computed ensembles of complete spectra of the staggered Dirac operator using four-dimensional SU(2) gauge fields, both in the quenched approximation and with dynamical fermions. To identify universal features in the Dirac spectrum, we compare the lattice data with predictions from chiral random matrix theory for the distribution of the low-lying eigenvalues. Good agreement is found up to some limiting energy, the so-called Thouless energy, above which random matrix theory no longer applies. We determine the dependence of the Thouless energy on the simulation parameters using the scalar susceptibility and the number variance.
The tunneling splitting of the energy levels of a ferromagnetic particle in the presence of an applied magnetic field - previously derived only for the ground state with the path integral method - is obtained in a simple way from Schr"odinger theory. The origin of the factors entering the result is clearly understood, in particular the effect of the asymmetry of the barriers of the potential. The method should appeal particularly to experimentalists searching for evidence of macroscopic spin tunneling.