Refine
Year of publication
- 1999 (397) (remove)
Document Type
- Preprint (397) (remove)
Keywords
- Case-Based Reasoning (10)
- Fallbasiertes Schliessen (5)
- Location Theory (5)
- case-based problem solving (5)
- Abstraction (4)
- Fallbasiertes Schließen (4)
- Knowledge Acquisition (4)
- Internet (3)
- Knowledge acquisition (3)
- Maschinelles Lernen (3)
Faculty / Organisational entity
- Kaiserslautern - Fachbereich Informatik (206)
- Kaiserslautern - Fachbereich Mathematik (121)
- Kaiserslautern - Fachbereich Physik (51)
- Kaiserslautern - Fachbereich Elektrotechnik und Informationstechnik (9)
- Fraunhofer (ITWM) (6)
- Kaiserslautern - Fachbereich Wirtschaftswissenschaften (2)
- Kaiserslautern - Fachbereich Maschinenbau und Verfahrenstechnik (1)
- Universitätsbibliothek (1)
Abstract: Recently, the contributions of chiral logarithms predicted by quenched chiral perturbation theory have been extracted from lattice calculations of hadron masses. We argue that a detailed comparison of random matrix theory and lattice calculations allows for a precise determination of such corrections. We estimate the relative size of the m log(m), m, and m^2 corrections to the chiral condensate for quenched SU(2).
Abstract: Random matrix theory (RMT) is a powerful statistical tool to model spectral fluctuations. In addition, RMT provides efficient means to separate different scales in spectra. Recently RMT has found application in quantum chromodynamics (QCD). In mesoscopic physics, the Thouless energy sets the universal scale for which RMT applies. We try to identify the equivalent of a Thouless energy in complete spectra of the QCD Dirac operator with staggered fermions and SU_(2) lattice gauge fields. Comparing lattice data with RMT predictions we find deviations which allow us to give an estimate for this scale.
Beyond the Thouless energy
(1999)
Abstract: The distribution and the correlations of the small eigenvalues of the Dirac operator are described by random matrix theory (RMT) up to the Thouless energy E_= 1 / sqrt (V), where V is the physical volume. For somewhat larger energies, the same quantities can be described by chiral perturbation theory (chPT). For most quantities there is an intermediate energy regime, roughly 1/V < E < 1/sqrt (V), where the results of RMT and chPT agree with each other. We test these predictions by constructing the connected and disconnected scalar susceptibilities from Dirac spectra obtained in quenched SU(2) and SU(3) simulations with staggered fermions for a variety of lattice sizes and coupling constants. In deriving the predictions of chPT, it is important totake into account only those symmetries which are exactly realized on the lattice.
Abstract: Recently, the chiral logarithms predicted by quenched chiral perturbation theory have been extracted from lattice calculations of hadron masses. We argue that the deviations of lattice results from random matrix theory starting around the so-called Thouless energy can be understood in terms of chiral perturbation theory as well. Comparison of lattice data with chiral perturbation theory formulae allows us to compute the pion decay constant. We present results from a calculation for quenched SU(2) with Kogut-Susskind fermions at ß = 2.0 and 2.2.
Abstract: We describe a general technique that allows for an ideal transfer of quantum correlations between light fields and metastable states of matter. The technique is based on trapping quantum states of photons in coherently driven atomic media, in which the group velocity is adiabatically reduced to zero. We discuss possible applications such as quantum state memories, generation of squeezed atomic states, preparation of entangled atomic ensembles and quantum information processing.
Abstract: We show that it is possible to "store" quantum states of single-photon fields by mapping them onto collective meta-stable states of an optically dense, coherently driven medium inside an optical resonator. An adiabatic technique is suggested which allows to transfer non-classical correlations from traveling-wave single-photon wave-packets into atomic states and vise versa with nearly 100% efficiency. In contrast to previous approaches involving single atoms, the present technique does not require the strong coupling regime corresponding to high-Q micro-cavities. Instead, intracavity Electromagnetically Induced Transparency is used to achieve a strong coupling between the cavity mode and the atoms.
Mirrorless oscillation based on resonantly enhanced 4-wave mixing: All-order analytic solutions
(1999)
Abstract: The phase transition to mirrorless oscillation in resonantly enhanced four-wave mixing in double-A systems are studied analytically for the ideal case of infinite lifetimes of ground-state coherences. The stationary susceptibilities are obtained in all orders of the generated fields and analytic solutions of the coupled nonlinear differential equations for the field amplitudes are derived and discussed.
Abstract: We utilize the generation of large atomic coherence to enhance the resonant nonlinear magneto-optic effect by several orders of magnitude, thereby eliminating power broadening and improving the fundamental signal-to-noise ratio. A proof-of-principle experiment is carried out in a dense vapor of Rb atoms. Detailed numerical calculations are in good agreement with the experimental results. Applications such as optical magnetometry or the search for violations of parity and time reversal symmetry are feasible.
Abstract: Spontaneous emission and Lamb shift of atoms in absorbing dielectrics are discussed. A Green's-function approach is used based on the multipolar interaction Hamiltonian of a collection of atomic dipoles with the quantised radiation field. The rate of decay and level shifts are determined by the retarded Green's-function of the interacting electric displacement field, which is calculated from a Dyson equation describing multiple scattering. The positions of the atomic dipoles forming the dielectrics are assumed to be uncorrelated and a continuum approximation is used. The associated unphysical interactions between different atoms at the same location is eliminated by removing the point-interaction term from the free-space Green's-function (local field correction). For the case of an atom in a purely dispersive medium the spontaneous emission rate is altered by the well-known Lorentz local-field factor. In the presence of absorption a result different from previously suggested expressions is found and nearest-neighbour interactions are shown to be important.
Abstract: We aim to establish a link between path-integral formulations of quantum and classical field theories via diagram expansions. This link should result in an independent constructive characterisation of the measure in Feynman path integrals in terms of a stochastic differential equation (SDE) and also in the possibility of applying methods of quantum field theory to classical stochastic problems. As a first step we derive in the present paper a formal solution to an arbitrary c-number SDE in a form which coincides with that of Wick's theorem for interacting bosonic quantum fields. We show that the choice of stochastic calculus in the SDE may be regarded as a result of regularisation, which in turn removes ultraviolet divergences from the corresponding diagram series.
We show that the solution to an arbitrary c-number stochastic differential equation (SDE) can be represented as a diagram series. Both the diagram rules and the properties of the graphical elements reflect causality properties of the SDE and this series is therefore called a causal diagram series. We also discuss the converse problem, i.e. how to construct an SDE of which a formal solution is a given causal diagram series. This then allows for a nonperturbative summation of the diagram series by solving this SDE, numerically or analytically.
Abstract: We propose a simple method for measuring the populations and the relative phase in a coherent superposition of two atomic states. The method is based on coupling the two states to a third common (excited) state by means of two laser pulses, and measuring the total fluorescence from the third state for several choices of the excitation pulses.
Abstract: We present experimental and theoretical results of a detailed study of laser-induced continuum structures (LICS) in the photoionization continuum of helium out of the metastable state 2s^1 S_0. The continuum dressing with a 1064 nm laser, couples the same region of the continuum to the 4s^1 S_0 state. The experimental data, presented for a range of intensities, show pronounced ionization suppression (by asmuch as 70% with respect to the far-from-resonance value) as well as enhancement, in a Beutler-Fano resonance profile. This ionization suppression is a clear indication of population trapping mediated by coupling to a contiuum. We present experimental results demonstrating the effect of pulse delay upon the LICS, and for the behavior of LICS for both weak and strong probe pulses. Simulations based upon numerical solution of the Schrödinger equation model the experimental results. The atomic parameters (Rabi frequencies and Stark shifts) are calculated using a simple model-potential method for the computation of the needed wavefunctions. The simulations of the LICS profiles are in excellent agreement with experiment. We also present an analytic formulation of pulsed LICS. We show that in the case of a probe pulse shorter than the dressing one the LICS profile is the convolution of the power spectra of the probe pulse with the usual Fano profile of stationary LICS. We discuss some consequences of deviation from steady-state theory.
We present results from a study of the coherence properties of a system involving three discrete states coupled to each other by two-photon processes via a common continuum. This tripod linkage is an extension of the standard laser-induced continuum structure (LICS) which involves two discrete states and two lasers. We show that in the tripod scheme, there exist two population trapping conditions; in some cases these conditions are easier to satisfy than the single trapping condition in two-state LICS. Depending on the pulse timing, various effects can be observed. We derive some basic properties of the tripod scheme, such as the solution for coincident pulses, the behaviour of the system in the adiabatic limit for delayed pulses, the conditions for no ionization and for maximal ionization, and the optimal conditions for population transfer between the discrete states via the continuum. In the case when one of the discrete states is strongly coupled to the continuum, the population dynamics reduces to a standard two-state LICS problem (involving the other two states) with modified parameters; this provides the opportunity to customize the parameters of a given two-state LICS system.
Abstract: In this paper we present a renormalizability proof for spontaneously broken SU (2) gauge theory. It is based on Flow Equations, i.e. on the Wilson renormalization group adapted to perturbation theory. The power counting part of the proof, which is conceptually and technically simple, follows the same lines as that for any other renormalizable theory. The main difficulty stems from the fact that the regularization violates gauge invariance. We prove that there exists a class of renormalization conditions such that the renormalized Green functions satisfy the Slavnov-Taylor identities of SU (2) Yang-Mills theory on which the gauge invariance of the renormalized theory is based.
Spektralsequenzen
(1999)
Magnetic anisotropies of MBE-grown fcc Co(110)-films on Cu(110) single crystal substrates have been determined by using Brillouin light scattering(BLS) and have been correlated with the structural properties determined by low energy electron diffraction (LEED) and scanning tunneling microscopy (STM). Three regimes of film growth and associated anisotropy behavior are identified: coherent growth in the Co film thickness regime of up to 13 Å, in-plane anisotropic strain relaxation between 13 Å and about 50 Å and inplane isotropic strain relaxation above 50 Å. The structural origin of the transition between anisotropic and isotropic strain relaxation was studied using STM. In the regime of anisotropic strain relaxation long Co stripes with a preferential [ 110 ]-orientation are observed, which in the isotropic strain relaxation regime are interrupted in the perpendicular in-plane direction to form isotropic islands. In the Co film thickness regime below 50 Å an unexpected suppression of the magnetocrystalline anisotropy contribution is observed. A model calculation based on a crystal field formalism and discussed within the context of band theory, which explicitly takes tetragonal misfit strains into account, reproduces the experimentally observed anomalies despite the fact that the thick Co films are quite rough.
Absract: We report on measurements of the two-dimensional intensity distribtion of linear and non-linear spin wave excitations in a LuBiFeO film. The spin wave intensity was detected with a high-resolution Brillouinlight scatteringspectroscopy setup. The observed snake-like structure of the spin wave intensity distribution is understood as a mode beating between modes with different lateral spin wave intensity distributions. The theoretical treatment of the linear regime is performed analytically, whereas the propagation of non-linear spin waves is simulated by a numerical solution of a non-linear Schrödinger equation with suitable boundary conditions.
Das Skript vermittelt die für eine Beschlagwortung nach den Regeln für den Schlagwortkatalog (RSWK) notwendigen Grundkenntnisse. Darüber hinaus wird beschrieben, wie die Schlagwortdaten in der Verbunddatenbank des Südwestdeutschen Bibliotheksverbund (SWB) strukturiert sind, welche Prinzipien bei der kooperativen Beschlagwortung im SWB einzuhalten sind und wie die Daten erfasst werden müssen . Des weiteren werden die in der Datenbank des SWB realisierten Suchmöglichkeiten aufgezeigt und aufgelistet, wie die dazugehörigen Suchbefehle lauten. Für Fragen der Organisation des Geschäftsgang der Teilnehmerbibliotheken wird exemplarisch der Arbeitsablauf an der UB Kaiserslautern dargestellt.
Abstract: The periodic bounce configurations responsible for quantum tunneling are obtained explicitly and are extended to the finite energy case for minisuperspace models of the Universe. As a common feature of the tunneling models at finite energy considered here we observe that the period of the bounce increases with energy monotonically. The periodic bounces do not have bifurcations and make no contribution to the nucleation rate except the one with zero energy. The sharp first order phase transition from quantum tunneling to thermal activation is verified with the general criterions.
We consider a (2 + 1)-dimensional mechanical system with the Lagrangian linear in the torsion of a light-like curve. We give Hamiltonian formulation of this system and show that its mass and spin spectra are defined by one-dimensional nonrelativistic mechanics with a cubic potential. Consequently, this system possesses the properties typical of resonance-like particles.
Starting from the Hamiltonian operator of the noncompensated two-sublattice model of a small antiferromagnetic particle, we derive the e effective Lagrangian of a biaxial antiferromagnetic particle in an external magnetic field with the help of spin-coherent-state path integrals. Two unequal level-shifts induced by tunneling through two types of barriers are obtained using the instanton method. The energy spectrum is found from Bloch theory regarding the periodic potential as a superlattice. The external magnetic field indeed removes Kramers' degeneracy, however a new quenching of the energy splitting depending on the applied magnetic field is observed for both integer and half-integer spins due to the quantum interference between transitions through two types of barriers.
Die Entwicklung des Zusammenlebens der Menschen geht immer mehr den Weg zur Informations- und Mediengesellschaft. Nicht zuletzt aufgrund der weltweiten Vernetzung ist es uns in minutenschnelle möglich, fast alle erdenklichen Informationen zu Hause auf den Bildschirm geliefert zu bekommen. Es findet sich so jeder zwar in einer gewissen schützenden Anonymität, aber dennoch einer genauso gewollten, wie erschreckenden Transparenz wieder. Jeder klassifiziert in gewisser Weise Informationen, die er preisgibt etwa in öffentliche, persönliche und vertrauliche Nachrichten. Gerade hier müssen Techniken und Methoden bereitstehen, um in dieser anonymen Transparenz Informationen, die nur für spezielle Empfänger gedacht sind vor unbefugtem Zugriff zu schützen und nur denjenigen zugänglich zu machen, die dazu berechtigt sind. Diesen Wunsch hat nicht nur allgemein die Gesellschaft, sondern im speziellen wird die Entwicklung auf diesem Gebiet gerade von staatlichen und militärischen Einrichtungen gefordert und gefördert. So sind häufig eingesetzte Werkzeuge die Methoden der Kryptologie, aber solange es geheime Nachrichten gibt, wird es Angreifer geben, die versuchen, sich unberechtigten Zugang zu diesen Informationen zu verschaffen. Da die ständig wachsende Leistung von EDV-Anlagen das "Knacken" von Verschlüsselungsmethoden begünstigt, muß zu immer sichereren Chiffrierverfahren übergegangen werden. Dieser Umstand macht das Thema Kryptologie für den Moment hochaktuell und auf lange Sicht zu einem zeitlosen Forschungsgebiet der Mathematik und Informatik.
We consider three applications of impulse control in financial mathematics, a cash management problem, optimal control of an exchange rate, and portfolio optimisation under transaction costs. We sketch the different ways of solving these problems with the help of quasi-variational inequalities. Further, some viscosity solution results are presented.
Continuous and discrete superselection rules induced by the interaction with the environment are investigated for a class of exactly soluble Hamiltonian models. The environment is given by a Boson field. Stable superselection sectors can only emerge if the low frequences dominate and the ground state of the Boson field disappears due to infrared divergence. The models allow uniform estimates of all transition matrix elements between different superselection sectors.
In a discrete-time financial market setting, the paper relates various concepts introduced for dynamic portfolios (both in discrete and in continuous time). These concepts are: value preserving portfolios, numeraire portfolios, interest oriented portfolios, and growth optimal portfolios. It will turn out that these concepts are all associated with a unique martingale measure which agrees with the minimal martingale measure only for complete markets.
Facility Location Problems are concerned with the optimal location of one or several new facilities, with respect to a set of existing ones. The objectives involve the distance between new and existing facilities, usually a weighted sum or weighted maximum. Since the various stakeholders (decision makers) will have different opinions of the importance of the existing facilities, a multicriteria problem with several sets of weights, and thus several objectives, arises. In our approach, we assume the decision makers to make only fuzzy comparisons of the different existing facilities. A geometric mean method is used to obtain the fuzzy weights for each facility and each decision maker. The resulting multicriteria facility location problem is solved using fuzzy techniques again. We prove that the final compromise solution is weakly Pareto optimal and Pareto optimal, if it is unique, or under certain assumptions on the estimates of the Nadir point. A numerical example is considered to illustrate the methodology.
In this paper we deal with the determination of the whole set of Pareto-solutions of location problems with respect to Q general criteria.These criteria include median, center or cent-dian objective functions as particular instances.The paper characterizes the set of Pareto-solutions of a these multicriteria problems. An efficient algorithm for the planar case is developed and its complexity is established. Extensions to higher dimensions as well as to the non-convexcase are also considered.The proposed approach is more general than the previously published approaches to multi-criteria location problems and includes almost all of them as particular instances.
Value Preserving Strategies and a General Framework for Local Approaches to Optimal Portfolios
(1999)
We present some new general results on the existence and form of value preserving portfolio strategies in a general semimartingale setting. The concept of value preservation will be derived via a mean-variance argument. It will also be embedded into a framework for local approaches to the problem of portfolio optimisation.
Discretizations for the Incompressible Navier-Stokes Equations based on the Lattice Boltzmann Method
(1999)
A discrete velocity model with spatial and velocity discretization based on a lattice Boltzmann method is considered in the low Mach number limit. A uniform numerical scheme for this model is investigated. In the limit, the scheme reduces to a finite difference scheme for the incompressible Navier-Stokes equation which is a projection method with a second order spatial discretization on a regular grid. The discretization is analyzed and the method is compared to Chorin's original spatial discretization. Numerical results supporting the analytical statements are presented.
In this paper we derive fluid dynamic equations byperforming asymptotic analysis for the generalized Boltzmann equationfor polyatomic gases. In particular, we consider the steady state,one-dimensional Boltzmann equation with one additional internal energyand different relaxation times. Moreover, we present a new approachto define coupling procedures for the Boltzmann equation and Navier-Stokesequations based on the 14-moments expansion of Levermore. These coupledmodels are validated by numerical simulations.
We consider a scale discrete wavelet approach on the sphere based on spherical radial basis functions. If the generators of the wavelets have a compact support, the scale and detail spaces are finite-dimensional, so that the detail information of a function is determined by only finitely many wavelet coefficients for each scale. We describe a pyramid scheme for the recursive determination of the wavelet coefficients from level to level, starting from an initial approximation of a given function. Basic tools are integration formulas which are exact for functions up to a given polynomial degree and spherical convolutions.
Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems
(1999)
Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtained simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved. .....
The paper shows that characterizing the causal relationship between significant events is an important but non-trivial aspect for understanding the behavior of distributed programs. An introduction to the notion of causality and its relation to logical time is given; some fundamental results concerning the characterization of causality are pre- sented. Recent work on the detection of causal relationships in distributed computations is surveyed. The relative merits and limitations of the different approaches are discussed, and their general feasibility is analyzed.
The Hamiltonian of the \(N\)-particle Calogero model can be expressed in terms of generators of a Lie algebra for a definite class of representations. Maintaining this Lie algebra, its representations, and the flatness of the Riemannian metric belonging to the second order differential operator, the set of all possible quadratic Lie algebra forms is investigated. For \(N = 3\) and \(N = 4\) such forms are constructed explicitly and shown to correspond to exactly solvable Sutherland models. The results can be carried over easily to all \(N\).
Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the coroot lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The invariants of the basis can be used as coordinates in any cell of the coroot space and lead to an exactly solvable model of Sutherland type. We apply this construction to the \(F_4\) case.
Mechanised reasoning systems and computer algebra systems have apparentlydifferent objectives. Their integration is, however, highly desirable, since in manyformal proofs both of the two different tasks, proving and calculating, have to beperformed. Even more importantly, proof and computation are often interwoven andnot easily separable. In the context of producing reliable proofs, the question howto ensure correctness when integrating a computer algebra system into a mechanisedreasoning system is crucial. In this contribution, we discuss the correctness prob-lems that arise from such an integration and advocate an approach in which thecalculations of the computer algebra system are checked at the calculus level of themechanised reasoning system. This can be achieved by adding a verbose mode to thecomputer algebra system which produces high-level protocol information that can beprocessed by an interface to derive proof plans. Such a proof plan in turn can beexpanded to proofs at different levels of abstraction, so the approach is well-suited forproducing a high-level verbalised explication as well as for a low-level machine check-able calculus-level proof. We present an implementation of our ideas and exemplifythem using an automatically solved extended example.
We propose a specification language for the formalization of data types with par-tial or non-terminating operations as part of a rewrite-based logical frameworkfor inductive theorem proving. The language requires constructors for designat-ing data items and admits positive/negative conditional equations as axioms inspecifications. The (total algebra) semantics for such specifications is based onso-called data models. We present admissibility conditions that guarantee theunique existence of a distinguished data model with properties similar to thoseof the initial model of a usual equational specification. Since admissibility of aspecification requires confluence of the induced rewrite relation, we provide aneffectively testable confluence criterion which does not presuppose termination.
To prove difficult theorems in a mathematical field requires substantial know-ledge of that field. In this paper a frame-based knowledge representation formalismis presented, which supports a conceptual representation and to a large extent guar-antees the consistency of the built-up knowledge bases. We define a semantics ofthe representation by giving a translation into the underlaying logic.
The amount of user interaction is the prime cause of costs in interactiveprogram verification. This paper describes an internal analogy techniquethat reuses subproofs in the verification of state-based specifications. Itidentifies common patterns of subproofs and their justifications in orderto reuse these subproofs; thus significant savings on the number of userinteractions in a verification proof are achievable.
We present an empirical study of mathematical proofs by diagonalization, the aim istheir mechanization based on proof planning techniques. We show that these proofs canbe constructed according to a strategy that (i) finds an indexing relation, (ii) constructsa diagonal element, and (iii) makes the implicit contradiction of the diagonal elementexplicit. Moreover we suggest how diagonal elements can be represented.
The problem of providing connectivity for a collection of applications is largely one of data integration: the communicating parties must agree on thesemantics and syntax of the data being exchanged. In earlier papers [#!mp:jsc1!#,#!sg:BSG1!#], it was proposed that dictionaries of definitions foroperators, functions, and symbolic constants can effectively address the problem of semantic data integration. In this paper we extend that earlier work todiscuss the important issues in data integration at the syntactic level and propose a set of solutions that are both general, supporting a wide range of dataobjects with typing information, and efficient, supporting fast transmission and parsing.
Chains of Recurrences (CRs) are a tool for expediting the evaluation of elementary expressions over regular grids. CR based evaluations of elementaryexpressions consist of 3 major stages: CR construction, simplification, and evaluation. This paper addresses CR simplifications. The goal of CRsimplifications is to manipulate a CR such that the resulting expression is more efficiently to evaluate. We develop CR simplification strategies which takethe computational context of CR evaluations into account. Realizing that it is infeasible to always optimally simplify a CR expression, we give heuristicstrategies which, in most cases, result in a optimal, or close-to-optimal expressions. The motivations behind our proposed strategies are discussed and theresults are illustrated by various examples.
Algorithms in Singular
(1999)
Top-down and bottom-up theorem proving approaches have each specific ad-vantages and disadvantages. Bottom-up provers profit from strong redundancycontrol and suffer from the lack of goal-orientation, whereas top-down provers aregoal-oriented but have weak calculi when their proof lengths are considered. Inorder to integrate both approaches our method is to achieve cooperation betweena top-down and a bottom-up prover: The top-down prover generates subgoalclauses, then they are processed by a bottom-up prover. We discuss theoreticaspects of this methodology and we introduce techniques for a relevancy-basedfiltering of generated subgoal clauses. Experiments with a model eliminationand a superposition-based prover reveal the high potential of our cooperation approach.The author was supported by the Deutsche Forschungsgemeinschaft (DFG).
We examine an approach for demand-driven cooperative theorem proving.We briefly point out the problems arising from the use of common success-driven cooperation methods, and we propose the application of our approachof requirement-based cooperative theorem proving. This approach allows for abetter orientation on current needs of provers in comparison with conventional co-operation concepts. We introduce an abstract framework for requirement-basedcooperation and describe two instantiations of it: Requirement-based exchangeof facts and sub-problem division and transfer via requests. Finally, we reporton experimental studies conducted in the areas superposition and unfailing com-pletion.The author was supported by the Deutsche Forschungsgemeinschaft (DFG).
HOT is an automated higher-order theorem prover based on HTE, an extensional higher-order tableaux calculus (Kohlhase 95). The first part of the paper introduces a variant of the calculus which closely corresponds to the proof procedure implemented in HOT. The second part discusses HOT's design that can be characterized as a concurrent Blackboard architecture. We show the usefulness of the implementation by including benchmark results for over one hundred solved problems from logic and set theory.
Orderings on polynomial interpretations of operators represent a powerful technique for proving thetermination of rewriting systems. One of the main problems of polynomial orderings concerns thechoice of the right interpretation for a given rewriting system. It is very difficult to develop techniquesfor solving this problem. Here, we present three new heuristic approaches: (i) guidelines for dealingwith special classes of rewriting systems, (ii) an algorithm for choosing appropriate special polynomialsas well as (iii) an extension of the original polynomial ordering which supports the generation ofsuitable interpretations. All these heuristics will be applied to examples in order to illustrate theirpractical relevance.
We study families V of curves in P2(C) of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to be irreducible. For T-smoothness these conditions involve new invariants of curve singularities and are conjectured to be asymptotically proper, i.e., optimal up to a constant factor. To obtain the results, we study the Castelnuovo function, prove the irreducibility of the Hilbert scheme of zero-dimensional schemes associated to a cluster of infinitely near points of the singularities and deduce new vanishing theorems for ideal sheaves of zero-dimensional schemes in P2. Moreover, we give a series of examples of cuspidal curves where the family V is reducible, but where ss1(P2nC) coincides (and is abelian) for all C 2 V .
After the notion of Gröbner bases and an algorithm for constructing them was introduced by Buchberger [Bu1, Bu2] algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Nontrivial problems coming either from logic, mathematics or applications usually lead to nontrivial Gröbner basis computations, which is the reason why several improvements have been provided by many people and have been implemented in general purpose systems like Axiom, Maple, Mathematica, Reduce, etc., and systems specialized for use in algebraic geometry and commutative algebra like CoCoA, Macaulay and Singular. The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field. In the second chapter we introduce standard bases (generalization of Gr"obner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions. The last chapter describes a new algorithm for computing the normalization of a reduced affine ring and gives an elementary introduction to singularity theory. Then we describe algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities. It is impossible to list all papers where Gr"obner bases have been used in local and global algebraic geometry, and even more impossible to give an overview about these contributions. We have, therefore, included only references to papers mentioned in this tutorial paper. The interested reader will find many more in the other contributions of this volume and in the literature cited there.
Algorithmic ideal theory
(1999)
Algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Non-trivial mathematical problems usually lead to non-trivial Gröbner basis computations, which is the reason why several improvements and efficient implementations have been provided by algebraic geometers (for example, the systems CoCoA, Macaulay and SINGULAR). The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field. In the second chapter we introduce standard bases (generalization of Gröbner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions. In the third chapter several algorithms for primary decomposition of polynomial ideals are presented, together with a discussion of improvements and preferable choices. We also describe a newly invented algorithm for computing the normalization of a reduced affine ring. The last chapter gives an elementary introduction to singularity theory and then describes algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities. It is impossible to list all papers where Gröbner basis have been used in local and global algebraic geometry, and even more impossible to give an overview about these contributions. We have, therefore, included only a few references to papers which contain interesting applications and which are not mentioned in this tutorial paper. The interested reader will find many more in the other contributions of this volume and in the literature cited there.
Singular algebraic curves, their existence, deformation, families (from the local and global point of view) attract continuous attention of algebraic geometers since the last century. The aim of our paper is to give an account of results, new trends and bibliography related to the geometry of equisingular families of algebraic curves on smooth algebraic surfaces over an algebraically closed field of characteristic zero. This theory is founded in basic works of Plücker, Severi, Segre, Zariski, and has tight links and finds important applications in singularity theory, topology of complex algebraic curves and surfaces, and in real algebraic geometry.
If \(A\) generates a bounded cosine function on a Banach space \(X\) then the negative square root \(B\) of \(A\) generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by \(B\). This characterization relies on new results on the inversion of the vector-valued conjugate potential transform.
We consider regularizing iterative procedures for ill-possed problems with random and nonrandom additive errors. The rate of square-mean convergence for iterative procedures with random errors is studied. The comparison theorem is established for the convergence of procedures with and without additive errors.
The computational complexity of combinatorial multiple objective programming problems is investigated. NP-completeness and #P-completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The performance of the well known tree and Christofides' heuristics for the TSP is investigated in the multicriteria case with respect to the two definitions of approximability.
Convex Operators in Vector Optimization: Directional Derivatives and the Cone of Decrease Directions
(1999)
The paper is devoted to the investigation of directional derivatives and the cone of decrease directions for convex operators on Banach spaces. We prove a condition for the existence of directional derivatives which does not assume regularity of the ordering cone K. This result is then used to prove that for continuous convex operators the cone of decrease directions can be represented in terms of the directional derivatices . Decrease directions are those for which the directional derivative lies in the negative interior of the ordering cone K. Finally, we show that the continuity of the convex operator can be replaced by its K-boundedness.
The notion of the balance number introduced in [3,page 139] through a certain set contraction procedure for nonscalarized multiobjective global optimization is represented via a min-max operation on the data of the problem. This representation yields a different computational procedure for the calculation of the balance number and allows us to generalize the approach for problems with countably many performance criteria.
In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facility where a set of polygonal barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the Median and Center objective functions). An exact Algorithm and a heuristic solution procedure based on this reduction result are developed.
Let rC and rD be two convexdistance funtions in the plane with convex unit balls C and D. Given two points, p and q, we investigate the bisector, B(p,q), of p and q, where distance from p is measured by rC and distance from q by rD. We provide the following results. B(p,q) may consist of many connected components whose precise number can be derived from the intersection of the unit balls, C nd D. The bisector can contain bounded or unbounded 2-dimensional areas. Even more surprising, pieces of the bisector may appear inside the region of all points closer to p than to q. If C and D are convex polygons over m and m vertices, respectively, the bisector B(p,q) can consist of at most min(m,n) connected components which contain at most 2(m+n) vertices altogether. The former bound is tight, the latter is tight up to an additive constant. We also present an optimal O(m+n) time algorithm for computing the bisector.
In planar location problems with barriers one considers regions which are forbidden for the siting of new facilities as well as for trespassing. These problems areimportant since they reflect various real-world situations.The resulting mathematical models have a non-convex objectivefunction and are therefore difficult to tackle using standardmethods of location theory even in the case of simple barriershapes and distance funtions.For the case of center objectives with barrier distancesobtained from the rectilinear or Manhattan metric it is shown that the problem can be solved by identifying a finitedominating set (FDS) the cardinality of which is bounded bya polynomial in the size of the problem input. The resultinggenuinely polynomial algorithm can be combined with bound computations which are derived from solving closely connectedrestricted location and network location problems.It is shown that the results can be extended to barrier center problems with respect to arbitrary block norms having fourfundamental directions.
In this paper we deal with an NP-hard combinatorial optimization problem, the k-cardinality tree problem in node weighted graphs. This problem has several applications , which justify the need for efficient methods to obtain good solutions. We review existing literature on the problem. Then we prove that under the condition that the graph contains exactly one trough, the problem can be solved in ploynomial time. For the general NP-hard problem we implemented several local search methods to obtain heuristics solutions, which are qualitatively better than solutions found by constructive heuristics and which require significantly less time than needed to obtain optimal solutions. We used the well known concepts of genetic algorithms and tabu search with useful extensions. We show that all the methods find optimal solutions for the class of graphs containing exactly one trough. The general performance of our methods as compared to other heuristics is illustrated by numerical results.
Given a finite set of points in the plane and a forbidden region R, we want to find a point X not an element of int(R), such that the weighted sum to all given points is minimized. This location problem is a variant of the well-known Weber Problem, where we measure the distance by polyhedral gauges and allow each of the weights to be positive or negative. The unit ball of a polyhedral gauge may be any convex polyhedron containing the origin. This large class of distance functions allows very general (practical) settings - such as asymmetry - to be modeled. Each given point is allowed to have its own gauge and the forbidden region R enables us to include negative information in the model. Additionally the use of negative and positive weights allows to include the level of attraction or dislikeness of a new facility. Polynomial algorithms and structural properties for this global optimization problem (d.c. objective function and a non-convex feasible set) based on combinatorial and geometrical methods are presented.
In this paper a new trend is introduced into the field of multicriteria location problems. We combine the robustness approach using the minmax regret criterion together with Pareto-optimality. We consider the multicriteria Weber location problem which consists of simultaneously minimizing a number of weighted sum-distance functions and the set of Pareto-optimal locations as its solution concept. For this problem, we characterize the Pareto-optimal solutions within the set of robust locations for the original weighted sum-distance functions. These locations have both the properties of stability and non-domination which are required in robust and multicriteria programming.
Nonlinear dissipativity, asymptotical stability, and contractivity of (ordinary) stochastic differential equations (SDEs) with some dissipative structure and their discretizations are studied in terms of their moments in the spirit of Pliss (1977). For this purpose, we introduce the notions and discuss related concepts of dissipativity, growth- bounded and monotone coefficient systems, asymptotical stability and contractivity in wide and narrow sense, nonlinear A-stability, AN-stability, B-stability and BN-stability for stochastic dynamical systems - more or less as stochastic counterparts to deterministic concepts. The test class of in a broad sense interpreted dissipative SDEs as natural analogon to dissipative deterministic differential systems is suggested for stochastic-numerical methods. Then, in particular, a kind of mean square calculus is developed, although most of ideas and analysis can be carried over to general "stochastic Lp-case" (p * 1). By this natural restriction, the new stochastic concepts are theoretically meaningful, as in deterministic analysis. Since the choice of step sizes then plays no essential role in related proofs, we even obtain nonlinear A-stability, AN-stability, B-stability and BN-stability in the mean square sense for this implicit method with respect to appropriate test classes of moment-dissipative SDEs.
We consider the "representation type" of the classification problem of vector bundles on a projective curve. We prove that this problem is always either finite, or tame, or wild and we completely describe those curves which are of finite, resp. tame, vector bundle type. We also give a complete list of indecomposable vector bundles for the finite and tame cases.
Let P2r be the projective plane blown up at r generic points. Denote by E0; E1; : : : ; Er the strict transform of a generic straight line on P2 and the exceptional divisors of the blown-up points on P2r respectively. We consider the variety Virr of all irreducible curves C with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r ^ 9 we give the complete answer concerning the existence of nodal curves in Virr.
Many discrepancy principles are known for choosing the parameter \(\alpha\) in the regularized operator equation \((T^*T+ \alpha I)x_\alpha^\delta = T^*y^\delta\), \(||y-y^d||\leq \delta\), in order to approximate the minimal norm least-squares solution of the operator equation \(Tx=y\). In this paper we consider a class of discrepancy principles for choosing the regularization parameter when \(T^*T\) and \(T^*y^\delta\) are approximated by \(A_n\) and \(z_n^\delta\) respectively with \(A_n\) not necessarily self - adjoint. Thisprocedure generalizes the work of Engl and Neubauer (1985),and particular cases of the results are applicable to the regularized projection method as well as to a degenerate kernel method considered by Groetsch (1990).
A compact subset E of the complex plane is called removable if all bounded analytic functions on its complement are constant or, equivalently, i f its analytic capacity vanishes. The problem of finding a geometric characterization of the removable sets is more than a hundred years old and still not comp letely solved.
Let (Epsilon_k) be a sequence of experiments with the same finite parameter set. Suppose only that identification of the parameter is possible asymptotically. For large classes of information functionals we show that their exponential rates of convergence towards complete information coincide. As a special case we obtain the rate of the Shannon capacity of product experiments.
In 1979, J.M. Bernardo argued heuristically that in the case of regular product experiments his information theoretic reference prior is equal to Jeffreys' prior. In this context, B.S. Clarke and A.R. Barron showed in 1994, that in the same class of experiments Jeffreys' prior is asymptotically optimal in the sense of Shannon, or, in Bayesian terms, Jeffreys' prior is asymptotically least favorable under Kullback Leibler risk. In the present paper, we prove, based on Clarke and Barron's results, that every sequence of Shannon optimal priors on a sequence of regular iid product experiments converges weakly to Jeffreys' prior. This means that for increasing sample size Kullback Leibler least favorable priors tend to Jeffreys' prior.
We compare different notions of differentiability of a measure along a vector field on a locally convex space. We consider in the L2-space of a differ entiable measure the analoga of the classical concepts of gradient, divergence and Laplacian (which coincides with the OrnsteinUhlenbeck operator in the Gaussian case). We use these operators for the extension of the basic results of Malliavin and Stroock on the smoothness of finite dimensional image measures under certain nonsmooth mappings to the case of non-Gaussian measures. The proof of this extension is quite direct and does not use any Chaos-decomposition. Finally, the role of this Laplacian in the procedure of quantization of anharmonic oscillators is discussed.
Starting from the uniqueness question for mixtures of distributions this review centers around the question under which formally weaker assumptions one can prove the existence of SPLIFs, in other words perfect statistics and tests. We mention a couple of positive and negative results which complement the basic contribution of David Blackwell in 1980. Typically the answers depend on the choice of the set theoretic axioms and on the particular concepts of measurability.
A large set of criteria to evaluate formal methods for reactive systems is presented. To make this set more comprehensible, it is structured according to a Concept-Model of formal methods. It is made clear that it is necessary to make the catalogue more specific before applying it. Some of the steps needed to do so are explained. As an example the catalogue is applied within the context of the application domain building automation systems to three different formal methods: SDL, statecharts, and a temporallogic.
Im Bereich der Expertensysteme ist das Problemlösen auf der Basis von bekannten Fallbeispielen ein derzeit sehr aktuelles Thema. Auch für Diagnoseaufgaben gewinnt der fallbasierte Ansatz immer mehr an Bedeutung. In diesem Papier soll der im Rahmen des Moltke -Projektes1 an der Universität Kaiserslautern entwickelte fallbasierte Problemlöser Patdex/22 vorgestellt werden. Ein erster Prototyp, Patdex/1, wurde bereits 1988 entwickelt.
We present a mathematical knowledge base containing the factual know-ledge of the first of three parts of a textbook on semi-groups and automata,namely "P. Deussen: Halbgruppen und Automaten". Like almost all math-ematical textbooks this textbook is not self-contained, but there are somealgebraic and set-theoretical concepts not being explained. These concepts areadded to the knowledge base. Furthermore there is knowledge about the nat-ural numbers, which is formalized following the first paragraph of "E. Landau:Grundlagen der Analysis".The data base is written in a sorted higher-order logic, a variant of POST ,the working language of the proof development environment OmegaGamma mkrp. We dis-tinguish three different types of knowledge: axioms, definitions, and theorems.Up to now, there are only 2 axioms (natural numbers and cardinality), 149definitions (like that for a semi-group), and 165 theorems. The consistency ofsuch knowledge bases cannot be proved in general, but inconsistencies may beimported only by the axioms. Definitions and theorems should not lead to anyinconsistency since definitions form conservative extensions and theorems areproved to be consequences.
Das System ART (ASF RRL Translation) stellt im wesentlichen eine Umgebung dar,in welcher die Modularisierbarkeit von Beweisen (Induktionsbeweisen über Gleichungs-spezifikationen) untersucht werden kann. Es wurde die bereits bestehende Spezifikati-onsprache ASF (siehe [BeHeKl89]), in welcher modularisierte Spezifikationen möglichsind, so erweitert, daß zusätzlich auch Beweisaufgaben spezifiziert werden können. Imfolgenden wird diese erweiterte Spezifikationsprache auch ASF genannt. Als Bewei-ser für die Beweisaufgaben einer Spezifikation wurde RRL (siehe [KaZh89]) gewählt.RRL kann sowohl Kommandos aus einem File abarbeiten, wie auch Sitzungsprotokolleanfertigen, mit deren Hilfe sich die Beweisverläufe und Benutzereingaben der entspre-chenden RRL-Sitzung rekonstruieren lassen. In ART kann nun eine ASF-Spezifikation,die Beweisaufgaben umfassen kann, in ein File übersetzt werden, welches von RRLabgearbeitet werden kann. Dies wird im folgenden kurz mit 'Übersetzung von ASF nach RRL' bezeichnet. Bei der Abarbeitung eines solchen Files wird von RRL ein Sit-zungsprotokoll angelegt. ART kann dieses Sitzungsprotokoll dazu heranziehen, neueErgebnisse, wie etwa den erfolgreichen Beweis einer Beweisaufgabe, zu ermitteln, umdiese Ergebnisse der ursprüngliche Spezifikation hinzuzufügen. Dies wird im folgendenkurz mit 'Rückübersetzung von RRL nach ASF' bezeichnet. Im Kern besteht ART alsoaus einer Komponente zur Übersetzung von ASF nach RRL und aus einer Komponentezur Rückübersetzung von RRL nach ASF.
Die systematische Verbesserung von Techniken zur Entwicklung und Betreuung von Software setzt eine explizite Darstellung der in einem Projekt ablaufenden Vorgnge (Prozesse) voraus. Diese Darstellungen (Prozemodelle) werden durch Software- Prozemodellierung gewonnen. Eine Sprache zur Beschreibung solcher Modelle ist MVP-L. Verschiedene Standard-Prozemodelle existieren bereits. Bisher gibt es jedoch kaum dokumentierte Software-Entwicklungsprozesse, die speziell fr die Entwicklung reaktiver Systeme entworfen worden sind, d. h. auf die besonderen Anfordernisse bei der Entwicklung reaktiver Systeme zugeschnitten sind. Auch ist bisher nur wenig Erfahrung dokumentiert, fr welche Art von Projektkontexten diese Prozesse gltig sind. Eine Software- Entwicklungsmethode, die - mit Einschrnkungen - zur Entwicklung reaktiver Systeme geeignet ist, ist SOMT (SDL-oriented Object Modeling Technique). Dieser Bericht beschreibt die erfahrungsbasierte Modellierung der Software-Entwicklungsprozesse von SOMT mit MVP-L. Zunchst werden inhaltliche Grundlagen der Software-Entwicklungsmethode SOMT beschrieben. Insbesondere wird auf die eingesetzten Techniken und deren Kombination eingegangen. Anschlieend werden mgliche Projektkontexte charakterisiert, in denen das SOMT-Modell im Sinne eines Erfahrungselements Gltigkeit hat. Darauf werden der Modellierungsvorgang sowie hierbei gemachte Erfahrungen dokumentiert. Eine vollstndige Darstellung des Modells in grafischer MVP-L-Notation befindet sich im Anhang. Die Darstellung des Modells in textueller Notation kann der SFB-Erfahrungsdatenbank entnommen werden.
In this paper we prove a reduction result for the number of criteria in convex multiobjective optimization. This result states that to decide wheter a point x in the decision space is pareto optimal it suffices to consider at most n? criteria at a time, where n is the dimension of the decision space. The main theorem is based on a geometric characterization of pareto, strict pareto and weak pareto solutions
Ramsey Numbers of K_m versus (n,k)-graphs and the Local Density of Graphs not Containing a K_m
(1999)
In this paper generalized Ramsey numbers of complete graphs K_m versus the set langle ,n,k angle of (n,k)-graphs are investigated. The value of r(K_m,langle n,k angle) is given in general for (relative to n) values of k small compared to n using a correlation with Turan numbers. These generalized Ramsey numbers con be used to determine the local densities of graphs not containing a subgraph K_m.
The Weber problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new facility X* in R^2 such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. In this paper we consider distances defined by polyhedral gauges. A variation of this problem is obtained if barriers are introduced which are convex polygonal subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers like lakes, military regions, national parks or mountains are frequently encountered in practice.From a mathematical point of view barrier problems are difficult, since the prensence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a descretization result: One of the grid points in the grid defined by the existing facilities and the fuundamental directions of the gauge distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm.
Kernel smoothing in nonparametric autoregressive schemes offers a powerful tool in modelling time series. In this paper it is shown that the bootstrap can be used for estimating the distribution of kernel smoothers. This can be done by mimicking the stochastic nature of the whole process in the bootstrap resampling or by generating a simple regression model. Consistency of these bootstrap procedures will be shown.
In this paper we consider generalizations of multifacility location problems in which as an additional constraint the new facilities are not allowed to be located in a presprcified region. We propose several different solution schemes for this non-convex optimization problem. These include a linear programming type approach, penalty approaches and barrier approaches. Moreover, structural results as well as illustratrive examples showing the difficulties of this problem are presented
To present the decision maker's (DM) preferences in multicriteria decision problems as a partially ordered set is an effective method to catch the DM's purpose and avoid misleading results. Since our paper is focused on minimal path problems, we regard the ordered set of edges (E,=). Minimal paths are defined in repect to power-ordered sets which provides an essential tool to solve such problems. An algorithm to detect minimal paths on a multicriteria minimal path problem is presented
Let P be a probability measure of the real line R such that each of the product measures P^{otimes n} assigns the value 1/2 to every half space in R^{n} having the origin as a boundary point. Then P is symmetric.Example: A strictly stable law on R is symmetric iff it has median zero. The treated symmetry problem is related to the problem of characterizing the distribution of X_1 by the distribution of (X_2 + X_1, ... ,X_n + X_1), with X_1, ... ,X_n being independent and identically distributed random variables.
In continous location problems we are given a set of existing facilities and we are looking for the location of one or several new facilities. In the classical approaches weights are assigned to existing facilities expressing the importance of the new facilities for the existing ones. In this paper, we consider a pointwise defined objective function where the weights are assigned to the existing facilities depending on the location of the new facility. This approach is shown to be a generalization of the median, center and centdian objective functions. In addition, this approach allows to formulate completely new location models. Efficient algorithms as well as structure results for this algebraic approach for location problems are presented. Extensions to the multifacility and restricted case are also considered.
In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.
In this paper we deal with the determination of the whole set of Pareto-solutions of location problems with respect to Q general criteria. These criteria include as particular instances median, center or cent-dian objective functions. The paper characterizes the set of Pareto-solutions of all these multicriteria problems. An efficient algorithm for the planar case is developed and its complexity is established. the proposed approach is more general than the previously published approaches to multicriteria location problems and includes almost all of them as particular instances.
The purpose of GPS-satellite-to-satellite tracking (GPS-SST) is to determine the gravitational potential at the earth's surface from measured ranges (geometrical distances) between a low-flying satellite and the high-flying satellites of the Global Posittioning System (GPS). In this paper GPS-satellite-to-satellite tracking is reformulated as the problem of determining the gravitational potential of the earth from given gradients at satellite altitude. Uniqueness and stability of the solution are investigated. The essential tool is to split the gradient field into a normal part (i.e. the first order radial derivative) and a tangential part (i.e. the surface gradient). Uniqueness is proved for polar, circular orbits corresponding to both types of data (first radial derivative and/or surface gradient). In both cases gravity recovery based on satellite-to-satellite tracking turns out to be an exponentially ill-posed problem. As an appropriate solution method regularization in terms of spherical wavelets is proposed based on the knowledge of the singular system. Finally, the extension of this method is generalized to a non-spherical earth and a non-spherical orbital surface based on combined terrestrial and satellite data material.
Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. it is shown that the process is an irreducible and aperiodic T-chain, regardless of the dimension of both data space and network and the special shape of the neighborhood function. Moreover the validity of Deoblin's condition is proved. These imply the convergence in distribution of the process to a finite invariant measure with a uniform geometric rate. In addition we show the process is positive Harris recurrent, which enables us to use statistical devices to measure its centrality and variability as the time goes to infinity.