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- estimation of regression (1)
- exact simulation method (1)
- exakte Simulation (1)
- existence (1)
- exponential function (1)
- exponential stability (1)
- exterior tensor product (1)
- fibre coordinates (1)
- filter (1)
- filtering (1)
- finiteness theorem (1)
- finsler distance (1)
- first boundary value problem (1)
- first variation (1)
- fixational eye movements (1)
- fixed point formula (1)
- flocking (1)
- foliated diffusion (1)
- force unification (1)
- free algebra (1)
- fully non-linear degenerate parabolic equations (1)
- functor geometry (1)
- fundamental solution (1)
- gauge group (1)
- generalized Laplace operator (1)
- geodesic distance (1)
- geodätischer Abstand (1)
- geomagnetism (1)
- geometric optics approximation (1)
- geordnete Gruppen von Conrad-Typ (1)
- global exact boundary controllability (1)
- global solution (1)
- global solutions (1)
- globally hyperbolic spacetime (1)
- good-inner function (1)
- goodness of fit (1)
- goodness-of-fit (1)
- goodness-of-fit testing (1)
- graphs (1)
- gravitation (1)
- gravitational wave (1)
- hard core interaction (1)
- heat asymptotics (1)
- heat kernel (1)
- heavy-tailed distributions (1)
- heterogeneity (1)
- high dimensional (1)
- higher operations (1)
- higher order rectifiability (1)
- higher singularities (1)
- higher-order Sturm–Liouville problems (1)
- hitting times (1)
- host-parasite stochastic particle system (1)
- hyperbolic dynamical system (1)
- hyperbolic operators (1)
- hyperbolic tilings (1)
- hyperequational theory (1)
- hypoelliptic estimate (1)
- höhere Operationen (1)
- höhere Singularitäten (1)
- ill-posed (1)
- ill-posed problems (1)
- illposed problem (1)
- indecomposable varifold (1)
- index formula (1)
- index of elliptic operator (1)
- index of stability (1)
- infinite-dimensional diffusion (1)
- initial boundary value problem (1)
- integral Fourier operators (1)
- integral representation method (1)
- integration by parts formula (1)
- integration by parts on path space (1)
- interacting particle systems (1)
- interaction matrix (1)
- interassociativity (1)
- interfaces with conical singularities (1)
- intrinsic diameter (1)
- intrinsischer Diameter (1)
- invariant (1)
- inverse Probleme (1)
- inverse Sturm–Liouville problems (1)
- inverse ill-posed problem (1)
- inverse problems (1)
- isoperimetric inequality (1)
- isoperimetrische Ungleichung (1)
- isotopic tiling theory (1)
- jump process (1)
- jump processes (1)
- kanten- und ecken-entartete Symbole (1)
- kernel estimator of the hazard rate (1)
- kernel method (1)
- kernel methods (1)
- kernel-based Bayesian inference (1)
- kernel-basierte Bayes'sche Inferenz (1)
- kleine Parameter (1)
- knots (1)
- komplexe Systeme (1)
- komplexe mechanistische Systeme (1)
- konstitutive Gleichungen (1)
- large-scale mechanistic systems (1)
- lattice packing and covering (1)
- lattice point (1)
- least squares estimator (1)
- left ordered groups (1)
- lidar (1)
- lifespan (1)
- limit theorem for integrated squared difference (1)
- linksgeordnete Gruppen (1)
- localisation (1)
- locality principle (1)
- locally indicable (1)
- logarithmic residue (1)
- logarithmic source condition (1)
- logistic regression analysis (1)
- logistische Regression (1)
- lokal indizierbar (1)
- low-lying eignvalues (1)
- lumping (1)
- macromolecular decay (1)
- magnetic (1)
- magnetic field modeling (1)
- magnetisch (1)
- makromolekularer Zerfall (1)
- manifold (1)
- manifold with boundary (1)
- manifold with edge (1)
- manifolds with cusps (1)
- manifolds with edge (1)
- mapping class groups (1)
- mapping degree (1)
- maps on surfaces (1)
- marked Gibbs point processes (1)
- matching of asymptotic expansions (1)
- mathematical modeling (1)
- mathematical modelling (1)
- mathematical physics (1)
- mathematics (1)
- mathematics education (1)
- mathematische Physik (1)
- mean curvature (1)
- mechanistic modeling (1)
- mechanistische Modellierung (1)
- meromorphe Fortsetzung (1)
- meromorphic continuation (1)
- meromorphic family (1)
- metaplectic operators (1)
- metastability (1)
- microdialysis (1)
- microlocal analysis (1)
- microlokale Analysis (1)
- microphysics (1)
- microsaccades (1)
- middle school (1)
- mild solution (1)
- minimax convergence rates (1)
- minimax optimality (1)
- minimax rate (1)
- mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung (1)
- mittlere Krümmung (1)
- mixed elliptic problems (1)
- mixed problems (1)
- mixture of bridges (1)
- mod k index (1)
- model order reduction (1)
- model selection (1)
- model-informed precision dosing (1)
- modeling (1)
- modellinformierte Präzisionsdosierung (1)
- modified Landweber method (1)
- moduli space of flat connections (1)
- modulo n index (1)
- molecular weaving (1)
- mollifier method (1)
- moment map (1)
- monotone method (1)
- monotonicity (1)
- multi-change point detection (1)
- multilayered coated and absorbing aerosol (1)
- multiple characteristics (1)
- multiplicative Lévy noise (1)
- multiplicative noise (1)
- multitype measure-valued branching processes (1)
- multiwavelength Lidar (1)
- multiwavelength lidar (1)
- multizeta functions (1)
- networks (1)
- new recursive algorithm (1)
- nicht-lineare gemischte Modelle (NLME) (1)
- nichtlineare partielle Differentialgleichung (1)
- non-coercive boundary conditions (1)
- non-linear integro-differential equations (1)
- non-regular drift (1)
- non-uniqueness (1)
- nondegenerate condition (1)
- nonhomogeneous boundary value problems (1)
- nonlinear (1)
- nonlinear equations (1)
- nonlinear invers problem (1)
- nonlinear optimization (1)
- nonlinear partial differential equations (1)
- nonlinear semigroup (1)
- nonlocal problem (1)
- nonparametric regression (1)
- nonparametric regression estimation (1)
- nonsmooth curves (1)
- norm estimates with respect to a parameter (1)
- normal bundle (1)
- normal reflection (1)
- numerical approximation (1)
- offene Wissenschaft (1)
- oncology (1)
- open science (1)
- operator algebras on manifolds with singularities (1)
- operators on manifolds with conical and edge singularities (1)
- operators on manifolds with edges (1)
- operators on manifolds with singularities (1)
- optimal rate (1)
- order filtration (1)
- order reduction (1)
- p-Branen (1)
- p-Laplace Operator (1)
- p-Laplace-Operator (1)
- p-Laplacian (1)
- p-branes (1)
- parallelizable spheres (1)
- parameter-dependent cone operators (1)
- parameter-dependent ellipticity (1)
- parameter-dependent pseudodifferential operators (1)
- parametrices (1)
- parity condition (1)
- parity conditions (1)
- partial algebras (1)
- partial least squares (1)
- partielle Integration (1)
- partielle Integration auf dem Pfadraum (1)
- periodic Gaussian process (1)
- periodic Ornstein-Uhlenbeck process (1)
- periodic entanglement (1)
- permanental- (1)
- pharmacokinetics (1)
- pharmacometrics (1)
- physics (1)
- physiologie-basierte Pharmacokinetic (PBPK) (1)
- point process (1)
- polydisc (1)
- polyhedra and polytopes (1)
- polymer (1)
- popPBPK (1)
- popPK (1)
- population analysis (1)
- porous medium equation (1)
- poset (1)
- principal symbolic hierarchies (1)
- problem of classification (1)
- profile likelihood (1)
- propor-tional hazard mode (1)
- pseudo-diferential operators (1)
- pseudo-differential equation (1)
- pseudo-differential operators (1)
- pseudo-differentialboundary value problems (1)
- pseudo-differentielle Gleichungen (1)
- pseudodifferential boundary value problems (1)
- pseudodifferential subspace (1)
- pseudodifferential subspaces (1)
- pseudodifferentiale Operatoren (1)
- quantizer (1)
- quantum field theory (1)
- quasi-linear potential theory (1)
- quasilinear Fredholm operator (1)
- quasilinear equation (1)
- quasilineare Potentialtheorie (1)
- random walk (1)
- random walk on Abelian group (1)
- rectifiable varifold (1)
- reflecting boundary (1)
- regular figures (1)
- regularisation (1)
- regularization methods (1)
- reinforcement learning (1)
- rejection sampling (1)
- rektifizierbare Varifaltigkeit (1)
- relative cohomology (1)
- relative index formulas (1)
- relative η-invariant (1)
- removable set (1)
- removable sets (1)
- reproducing kernel Hilbert space (1)
- rescaled lattice (1)
- residue (1)
- retrieval (1)
- reziproke Invarianten (1)
- saccade detection (1)
- sampling (1)
- scaled lattice (1)
- scattering amplitude (1)
- scattering theory (1)
- schlecht gestellt (1)
- seismic source inversion (1)
- seismische Quellinversion (1)
- self-assembly (1)
- semi-classical difference operator (1)
- semi-classical spectral estimates (1)
- semiclassical spectral asymptotics (1)
- semiclassics (1)
- semiconductors (1)
- semigroup (1)
- semipermeable barriers (1)
- semiprocess (1)
- sequences of microsaccades (1)
- series representation (1)
- shock wave (1)
- singular drifts (1)
- singular integral equations (1)
- singular manifolds (1)
- singular point (1)
- singuläre Mannigfaltigkeiten (1)
- skew diffusion (1)
- skew diffusions (1)
- skew field of fraction (1)
- small noise asymptotic (1)
- soft matter (1)
- space-time Gibbs field (1)
- spacetimes with timelike boundary (1)
- specific entropy (1)
- spectral boundary value problems (1)
- spectral independence (1)
- spectral kernel function (1)
- spectral resolution (1)
- spin Hall effect (1)
- spirallike function (1)
- stability and accuracy (1)
- stable limit cycle (1)
- stark Hughes-frei (1)
- state estimation (1)
- statistical inference (1)
- statistical inverse problem (1)
- statistical machine learning (1)
- statistical model selection (1)
- statistische Inferenz (1)
- statistisches maschinelles Lernen (1)
- step process (1)
- stochastic Burgers equations (1)
- stochastic bridges (1)
- stochastic interacting particles (1)
- stochastic mechanics (1)
- stochastische Anordnung (1)
- stochastische Differentialgleichungen (1)
- stochastische Mechanik (1)
- stochastische Zellulare Automaten (1)
- stochastisches interagierendes System (1)
- stopping rules (1)
- strongly Hughes-free (1)
- strongly pseudoconvex domains (1)
- structure formation (1)
- structured numbers (1)
- strukturierte Zahlen (1)
- subRiemannian geometry (1)
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- survival analysis (1)
- symmetry group (1)
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- symplectic manifold (1)
- symplectic methods (1)
- symplectic reduction (1)
- system Lame (1)
- systems of partial differential equations (1)
- systems pharmacology (1)
- tangles (1)
- teaching methods (1)
- terrigener Staub (1)
- terrigenous dust (1)
- tetration (1)
- the Dirichlet problem (1)
- the Goursat problem (1)
- the characteristic Cauchy problem (1)
- the first boundary value problem (1)
- the linearised Einstein equation (1)
- time reversal (1)
- time series (1)
- time series with heavy tails (1)
- time symmetry (1)
- tomogrphy (1)
- trace (1)
- transition path theory (1)
- travel time tomography (1)
- two-level interacting processes (1)
- ultracontractivity (1)
- unendlich teilbare Punktprozesse (1)
- unendliche Teilbarkeit (1)
- uniform compact attractor (1)
- unzerlegbare Varifaltigkeit (1)
- variable projection method (1)
- variational calculus (1)
- variational principle (1)
- variational stability (1)
- varifold (1)
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- weiche Materie (1)
- weighted Hölder spaces (1)
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- weighted Sobolev spaces with discrete saymptotics (1)
- weighted spaces with asymptotics (1)
- zero-noise limit (1)
- η-invariant (1)
- ∂-operator (1)
Institute
- Institut für Mathematik (476) (remove)
Given a system of entire functions in Cn with at most countable set of common zeros, we introduce the concept of zeta-function associated with the system. Under reasonable assumptions on the system, the zeta-function is well defined for all s ∈ Zn with sufficiently large components. Using residue theory we get an integral representation for the zeta-function which allows us to construct an analytic extension of the zeta-function to an infinite cone in Cn.
This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.
We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein{Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for hyperbolic and elliptic equations in these spaces. The well posedness is obtained in these spaces.
We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.
Parabolic equations on manifolds with singularities require a new calculus of anisotropic pseudo-differential operators with operator-valued symbols. The paper develops this theory along the lines of sn abstract wedge calculus with strongly continuous groups of isomorphisms on the involved Banach spaces. The corresponding pseodo-diferential operators are continuous in anisotropic wedge Sobolev spaces, and they form an alegbra. There is then introduced the concept of anisotropic parameter-dependent ellipticity, based on an order reduction variant of the pseudo-differential calculus. The theory is appled to a class of parabolic differential operators, and it is proved the invertibility in Sobolev spaces with exponential weights at infinity in time direction.
In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].
In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields.
The International Project for the Evaluation of Educational Achievement (IEA) was formed in the 1950s (Postlethwaite, 1967). Since that time, the IEA has conducted many studies in the area of mathematics, such as the First International Mathematics Study (FIMS) in 1964, the Second International Mathematics Study (SIMS) in 1980-1982, and a series of studies beginning with the Third International Mathematics and Science Study (TIMSS) which has been conducted every 4 years since 1995. According to Stigler et al. (1999), in the FIMS and the SIMS, U.S. students achieved low scores in comparison with students in other countries (p. 1). The TIMSS 1995 “Videotape Classroom Study” was therefore a complement to the earlier studies conducted to learn “more about the instructional and cultural processes that are associated with achievement” (Stigler et al., 1999, p. 1). The TIMSS Videotape Classroom Study is known today as the TIMSS Video Study. From the findings of the TIMSS 1995 Video Study, Stigler and Hiebert (1999) likened teaching to “mountain ranges poking above the surface of the water,” whereby they implied that we might see the mountaintops, but we do not see the hidden parts underneath these mountain ranges (pp. 73-78). By watching the videotaped lessons from Germany, Japan, and the United States again and again, they discovered that “the systems of teaching within each country look similar from lesson to lesson. At least, there are certain recurring features [or patterns] that typify many of the lessons within a country and distinguish the lessons among countries” (pp. 77-78). They also discovered that “teaching is a cultural activity,” so the systems of teaching “must be understood in relation to the cultural beliefs and assumptions that surround them” (pp. 85, 88). From this viewpoint, one of the purposes of this dissertation was to study some cultural aspects of mathematics teaching and relate the results to mathematics teaching and learning in Vietnam. Another research purpose was to carry out a video study in Vietnam to find out the characteristics of Vietnamese mathematics teaching and compare these characteristics with those of other countries. In particular, this dissertation carried out the following research tasks: - Studying the characteristics of teaching and learning in different cultures and relating the results to mathematics teaching and learning in Vietnam - Introducing the TIMSS, the TIMSS Video Study and the advantages of using video study in investigating mathematics teaching and learning - Carrying out the video study in Vietnam to identify the image, scripts and patterns, and the lesson signature of eighth-grade mathematics teaching in Vietnam - Comparing some aspects of mathematics teaching in Vietnam and other countries and identifying the similarities and differences across countries - Studying the demands and challenges of innovating mathematics teaching methods in Vietnam – lessons from the video studies Hopefully, this dissertation will be a useful reference material for pre-service teachers at education universities to understand the nature of teaching and develop their teaching career.
In this paper we establish the regularity, exponential stability of global (weak) solutions and existence of uniform compact attractors of semiprocesses, which are generated by the global solutions, of a two-parameter family of operators for the nonlinear 1-d non-autonomous viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.
We analyze a general class of difference operators containing a multi-well potential and a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we treat the eigenvalue problem as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix similar to the analysis for the Schrödinger operator, and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.
This article assesses the distance between the laws of stochastic differential equations with multiplicative Lévy noise on path space in terms of their characteristics. The notion of transportation distance on the set of Lévy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate.
Toeplitz operators, and ellipticity of boundary value problems with global projection conditions
(2003)
Ellipticity of (pseudo-) differential operators A on a compact manifold X with boundary (or with edges) Y is connected with boundary (or edge) conditions of trace and potential type, formulated in terms of global projections on Y together with an additional symbolic structure. This gives rise to operator block matrices A with A in the upper left corner. We study an algebra of such operators, where ellipticity is equivalent to the Fredhom property in suitable scales of spaces: Sobolev spaces on X plus closed subspaces of Sobolev spaces on Y which are the range of corresponding pseudo-differential projections. Moreover, we express parametrices of elliptic elements within our algebra and discuss spectral boundary value problems for differential operators.
This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These operators are bounded in these (Lipshitz and Djrbashian) spaces. As an application, we show a theorem about the division by good-inner functions in the mentioned classes is proved.
Studying the influence of the updating scheme for MCMC algorithm on spatially extended models is a well known problem. For discrete-time interacting particle systems we study through simulations the effectiveness of a synchronous updating scheme versus the usual sequential one. We compare the speed of convergence of the associated Markov chains from the point of view of the time-to-coalescence arising in the coupling-from-the-past algorithm. Unlike the intuition, the synchronous updating scheme is not always the best one. The distribution of the time-to-coalescence for these spatially extended models is studied too.
Data assimilation has been an active area of research in recent years, owing to its wide utility. At the core of data assimilation are filtering, prediction, and smoothing procedures. Filtering entails incorporation of measurements' information into the model to gain more insight into a given state governed by a noisy state space model. Most natural laws are governed by time-continuous nonlinear models. For the most part, the knowledge available about a model is incomplete; and hence uncertainties are approximated by means of probabilities. Time-continuous filtering, therefore, holds promise for wider usefulness, for it offers a means of combining noisy measurements with imperfect model to provide more insight on a given state.
The solution to time-continuous nonlinear Gaussian filtering problem is provided for by the Kushner-Stratonovich equation. Unfortunately, the Kushner-Stratonovich equation lacks a closed-form solution. Moreover, the numerical approximations based on Taylor expansion above third order are fraught with computational complications. For this reason, numerical methods based on Monte Carlo methods have been resorted to. Chief among these methods are sequential Monte-Carlo methods (or particle filters), for they allow for online assimilation of data. Particle filters are not without challenges: they suffer from particle degeneracy, sample impoverishment, and computational costs arising from resampling.
The goal of this thesis is to:— i) Review the derivation of Kushner-Stratonovich equation from first principles and its extant numerical approximation methods, ii) Study the feedback particle filters as a way of avoiding resampling in particle filters, iii) Study joint state and parameter estimation in time-continuous settings, iv) Apply the notions studied to linear hyperbolic stochastic differential equations.
The interconnection between Itô integrals and stochastic partial differential equations and those of Stratonovich is introduced in anticipation of feedback particle filters. With these ideas and motivated by the variants of ensemble Kalman-Bucy filters founded on the structure of the innovation process, a feedback particle filter with randomly perturbed innovation is proposed. Moreover, feedback particle filters based on coupling of prediction and analysis measures are proposed. They register a better performance than the bootstrap particle filter at lower ensemble sizes.
We study joint state and parameter estimation, both by means of extended state spaces and by use of dual filters. Feedback particle filters seem to perform well in both cases. Finally, we apply joint state and parameter estimation in the advection and wave equation, whose velocity is spatially varying. Two methods are employed: Metropolis Hastings with filter likelihood and a dual filter comprising of Kalman-Bucy filter and ensemble Kalman-Bucy filter. The former performs better than the latter.
We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y , i.e., Au = f in int X, T±u = g± on int Y±, where Y is subdivided into subsets Y± with an interface Z and boundary conditions T± on Y± that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z ⊂ Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T− Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in [3]. With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z \ {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.
Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge Sobolev spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. Our methods from the calculus of boundary value problems on a manifold with edges will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator.
We study an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted Lp-space. Under suitable conditions we show that the resolvent (λ - A )-¹ exists in a sector of the complex plane and decays like 1/|λ| as |λ| -> ∞. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with striaght conical degeneracy and establish maximal regularity for the Cauchy problem u - Δu = f, u(0) = 0.
The quantum cosmological wavefunction for a quadratic gravity theory derived from the heterotic string effective action is obtained near the inflationary epoch and during the initial Planck era. Neglecting derivatives with respect to the scalar field, the wavefunction would satisfy a third-order differential equation near the inflationary epoch which has a solution that is singular in the scale factor limit a(t) → 0. When scalar field derivatives are included, a sixth-order differential equation is obtained for the wavefunction and the solution by Mellin transform is regular in the a → 0 limit. It follows that inclusion of the scalar field in the quadratic gravity action is necessary for consistency of the quantum cosmology of the theory at very early times.
When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for
the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on
a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two.
The classical Navier-Stokes equations of hydrodynamics are usually written in terms of vector analysis. More promising is the formulation of these equations in the language of differential forms of degree one. In this way the study of Navier-Stokes equations includes the analysis of the de Rham complex. In particular, the Hodge theory for the de Rham complex enables one to eliminate the pressure from the equations. The Navier-Stokes equations constitute a parabolic system with a nonlinear term which makes sense only for one-forms. A simpler model of dynamics of incompressible viscous fluid is given by Burgers' equation. This work is aimed at the study of invariant structure of the Navier-Stokes equations which is closely related to the algebraic structure of the de Rham complex at step 1. To this end we introduce Navier-Stokes equations related to any elliptic quasicomplex of first order differential operators. These equations are quite similar to the classical Navier-Stokes equations including generalised velocity and pressure vectors. Elimination of the pressure from the generalised Navier-Stokes equations gives a good motivation for the study of the Neumann problem after Spencer for elliptic quasicomplexes. Such a study is also included in the work.We start this work by discussion of Lamé equations within the context of elliptic quasicomplexes on compact manifolds with boundary. The non-stationary Lamé equations form a hyperbolic system. However, the study of the first mixed problem for them gives a good experience to attack the linearised Navier-Stokes equations. On this base we describe a class of non-linear perturbations of the Navier-Stokes equations, for which the solvability results still hold.
We consider a mixed problem for a degenerate differentialoperator equation of higher order. We establish some embedding theorems in weighted Sobolev spaces and show existence and uniqueness of the generalized solution of this problem. We also give a description of the spectrum for the corresponding operator.
This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincaré-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions.
The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincaré-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincaré-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts.
The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.
We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.
For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number.
We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27 - 31, 2008, at the MSRI, University of Berkeley.
The quantization of contact transformations of the cosphere bundle over a manifold with conical singularities is described. The index of Fredholm operators given by this quantization is calculated. The answer is given in terms of the Epstein-Melrose contact degree and the conormal symbol of the corresponding operator.
The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol.
For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.
The ill-posed inversion of multiwavelength lidar data by a hybrid method of variable projection
(1999)
The ill-posed problem of aerosol distribution determination from a small number of backscatter and extinction lidar measurements was solved successfully via a hybrid method by a variable dimension of projection with B-Splines. Numerical simulation results with noisy data at different measurement situations show that it is possible to derive a reconstruction of the aerosol distribution only with 4 measurements.
The homotopy classification and the index of boundary value problems for general elliptic operators
(1999)
We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition.
The Green formula is proved for boundary value problems (BVPs), when "basic" operator is arbitrary partial differential operator with variable matrix coefficients and "boundary" operators are quasi-normal with vector-coeficients. If the system possesses the fundamental solution, representation formula for a solution is derived and boundedness properties of participating layer potentials from function spaces on the boundary (Besov, Zygmund spaces) into appropriate weighted function spaces on the inner and the outer domains are established. Some related problems are discussed in conclusion: traces of functions from weighted spaces, traces of potential-type functions, Plemelji formulae,Calderón projections, restricted smoothness of the underlying surface and coefficients. The results have essential applications in investigations of BVPs by the potential method, in apriori estimates and in asymptotics of solutions.
In this article we study the geometry associated with the sub-elliptic operator ½ (X²1 +X²2), where X1 = ∂x and X2 = x²/2 ∂y are vector fields on R². We show that any point can be connected with the origin by at least one geodesic and we provide an approximate formula for the number of the geodesics between the origin and the points situated outside of the y-axis. We show there are in¯nitely many geodesics between the origin and the points on the y-axis.
We construct a special asymptotic solution for the forced KdV equation. In the frame of the shallow water model this kind of the external driving force is related to the atmospheric disturbance. The perturbation slowly passes through a resonance and it leads to the solution exchange. The detailed asymptotic description of the process is presented.
This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case.