@unpublished{MartinSchulze2005, author = {Martin, C.-I. and Schulze, Bert-Wolfgang}, title = {The quantisation of edge symbols}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29959}, year = {2005}, abstract = {We investigate operators on manifolds with edges from the point of view of the symbolic calculus induced by the singularities. We discuss new aspects of the quantisation of edge-degenerate symbols which lead to continuous operators in weighted edge spaces.}, language = {en} } @unpublished{MeloNestSchrohe2001, author = {Melo, S. T. and Nest, R. and Schrohe, Elmar}, title = {C*-structure and K-theory of Boutet de Monvel's algebra}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26166}, year = {2001}, abstract = {We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the K-groups of A are topologically determined. In case the manifold, its boundary, and the cotangent space of its interior have torsion free K-theory, we get Ki(A,k) congruent Ki(C(X))⊕Ksub(1-i)(Csub(0)(T*X)),i = 0,1, with k denoting the compact ideal, and T*X denoting the cotangent bundle of the interior. Using Boutet de Monvel's index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis. For the case of orientable, two-dimensional X, Ksub(0)(A) congruent Z up(2g+m) and Ksub(1)(A) congruent Z up(2g+m-1), where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ k ⊂ G ⊂ A, with A/G commutative and G/k isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L²(R+).}, language = {en} } @unpublished{Messina2001, author = {Messina, Francesca}, title = {Local solvability for semilinear Fuchsian equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26124}, year = {2001}, abstract = {Contents: 1 Introduction 2 Differential operators on manifolds with conical singularities 3 When H up(s,y) (B) is an algebra 4 Statement of the main result 5 Proof of the theorem}, language = {en} } @unpublished{Murr2011, author = {Murr, R{\"u}diger}, title = {Characterization of L{\´e}vy Processes by a duality formula and related results}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-43538}, year = {2011}, abstract = {Processes with independent increments are characterized via a duality formula, including Malliavin derivative and difference operators. This result is based on a characterization of infinitely divisible random vectors by a functional equation. A construction of the difference operator by a variational method is introduced and compared to approaches used by other authors for L´evy processes involving the chaos decomposition. Finally we extend our method to characterize infinitely divisible random measures.}, language = {en} } @unpublished{Myslivets2000, author = {Myslivets, Simona}, title = {On the boundary behaviour of the logarithmic residue integral}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25733}, year = {2000}, abstract = {A formula of multidimensional logarithmic residue is proved for holomorphic maps with zeroes on the boundary of a bounded domain in Cn.}, language = {en} } @unpublished{NacinovichSchulzeTarkhanov1998, author = {Nacinovich, Mauro and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {On carleman formulas for the dolbeault cohomology}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25224}, year = {1998}, abstract = {We discuss the Cauchy problem for the Dolbeault cohomology in a domain of C n with data on a part of the boundary. In this setting we introduce the concept of a Carleman function which proves useful in the study of uniqueness. Apart from an abstract framework we show explicit Carleman formulas for the Dolbeault cohomology.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2004, author = {Nazaikinskii, Vladimir E. and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {On the homotopy classification of elliptic operators on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26769}, year = {2004}, abstract = {We obtain a stable homotopy classification of elliptic operators on manifolds with edges.}, language = {en} } @unpublished{NazaikinskiiSchulzeSternin1999, author = {Nazaikinskii, Vladimir E. and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Quantization methods in differential equations : Chapter 2: Quantization of Lagrangian modules}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25582}, year = {1999}, abstract = {In this chapter we use the wave packet transform described in Chapter 1 to quantize extended classical states represented by so-called Lagrangian sumbanifolds of the phase space. Functions on a Lagrangian manifold form a module over the ring of classical Hamiltonian functions on the phase space (with respect to pointwise multiplication). The quantization procedure intertwines this multiplication with the action of the corresponding quantum Hamiltonians; hence we speak of quantization of Lagrangian modules. The semiclassical states obtained by this quantization procedure provide asymptotic solutions to differential equations with a small parameter. Locally, such solutions can be represented by WKB elements. Global solutions are given by Maslov's canonical operator [2]; also see, e.g., [3] and the references therein. Here the canonical operator is obtained in the framework of the universal quantization procedure provided by the wave packet transform. This procedure was suggested in [4] (see also the references there) and further developed in [5]; our exposition is in the spirit of these papers. Some further bibliographical remarks can be found in the beginning of Chapter 1.}, language = {en} } @unpublished{NazaikinskiiSternin1999, author = {Nazaikinskii, Vladimir E. and Sternin, Boris}, title = {Surgery and the relative index in elliptic theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25538}, year = {1999}, abstract = {We prove a general theorem on the local property of the relative index for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions) this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities as well as for elliptic boundary value problems with a symmetry condition for the conormal symbol.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2003, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Differential operators on manifolds with singularities : analysis and topology : Chapter 3: Eta invariant and the spectral flow}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26595}, year = {2003}, abstract = {Contents: Chapter 3: Eta Invariant and the Spectral Flow 3.1. Introduction 3.2. The Classical Spectral Flow 3.2.1. Definition and main properties 3.2.2. The spectral flow formula for periodic families 3.3. The Atiyah-Patodi-Singer Eta Invariant 3.3.1. Definition of the eta invariant 3.3.2. Variation under deformations of the operator 3.3.3. Homotopy invariance. Examples 3.4. The Eta Invariant of Families with Parameter (Melrose's Theory) 3.4.1. A trace on the algebra of parameter-dependent operators 3.4.2. Definition of the Melrose eta invariant 3.4.3. Relationship with the Atiyah-Patodi-Singer eta invariant 3.4.4. Locality of the derivative of the eta invariant. Examples 3.5. The Spectral Flow of Families of Parameter-Dependent Operators 3.5.1. Meromorphic operator functions. Multiplicities of singular points 3.5.2. Definition of the spectral flow 3.6. Higher Spectral Flows 3.6.1. Spectral sections 3.6.2. Spectral flow of homotopies of families of self-adjoint operators 3.6.3. Spectral flow of homotopies of families of parameter-dependent operators 3.7. Bibliographical Remarks}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2003, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Differential operators on manifolds with singularities : analysis and topology : Chapter 4: Pseudodifferential operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26587}, year = {2003}, abstract = {Contents: Chapter 4: Pseudodifferential Operators 4.1. Preliminary Remarks 4.1.1. Why are pseudodifferential operators needed? 4.1.2. What is a pseudodifferential operator? 4.1.3. What properties should the pseudodifferential calculus possess? 4.2. Classical Pseudodifferential Operators on Smooth Manifolds 4.2.1. Definition of pseudodifferential operators on a manifold 4.2.2. H{\"o}rmander's definition of pseudodifferential operators 4.2.3. Basic properties of pseudodifferential operators 4.3. Pseudodifferential Operators in Sections of Hilbert Bundles 4.3.1. Hilbert bundles 4.3.2. Operator-valued symbols. Specific features of the infinite-dimensional case 4.3.3. Symbols of compact fiber variation 4.3.4. Definition of pseudodifferential operators 4.3.5. The composition theorem 4.3.6. Ellipticity 4.3.7. The finiteness theorem 4.4. The Index Theorem 4.4.1. The Atiyah-Singer index theorem 4.4.2. The index theorem for pseudodifferential operators in sections of Hilbert bundles 4.4.3. Proof of the index theorem 4.5. Bibliographical Remarks}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2004, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Differential operators on manifolds with singularities : analysis and topology : Chapter 6: Elliptic theory on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26757}, year = {2004}, abstract = {Contents: Chapter 6: Elliptic Theory on Manifolds with Edges Introduction 6.1. Motivation and Main Constructions 6.1.1. Manifolds with edges 6.1.2. Edge-degenerate differential operators 6.1.3. Symbols 6.1.4. Elliptic problems 6.2. Pseudodifferential Operators 6.2.1. Edge symbols 6.2.2. Pseudodifferential operators 6.2.3. Quantization 6.3. Elliptic Morphisms and the Finiteness Theorem 6.3.1. Matrix Green operators 6.3.2. General morphisms 6.3.3. Ellipticity, Fredholm property, and smoothness Appendix A. Fiber Bundles and Direct Integrals A.1. Local theory A.2. Globalization A.3. Versions of the Definition of the Norm}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2003, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Elliptic theory on manifolds with nonisolated singularities : V. Index formulas for elliptic problems on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26500}, year = {2003}, abstract = {For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2003, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Differential operators on manifolds with singularities : analysis and topology : Chapter 1: Localization (surgery) in elliptic theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26546}, year = {2003}, abstract = {Contents: Chapter 1: Localization (Surgery) in Elliptic Theory 1.1. The Index Locality Principle 1.1.1. What is locality? 1.1.2. A pilot example 1.1.3. Collar spaces 1.1.4. Elliptic operators 1.1.5. Surgery and the relative index theorem 1.2. Surgery in Index Theory on Smooth Manifolds 1.2.1. The Booß-Wojciechowski theorem 1.2.2. The Gromov-Lawson theorem 1.3. Surgery for Boundary Value Problems 1.3.1. Notation 1.3.2. General boundary value problems 1.3.3. A model boundary value problem on a cylinder 1.3.4. The Agranovich-Dynin theorem 1.3.5. The Agranovich theorem 1.3.6. Bojarski's theorem and its generalizations 1.4. (Micro)localization in Lefschetz theory 1.4.1. The Lefschetz number 1.4.2. Localization and the contributions of singular points 1.4.3. The semiclassical method and microlocalization 1.4.4. The classical Atiyah-Bott-Lefschetz theorem}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2002, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Elliptic theory on manifolds with nonisolated singularities : I. The index of families of cone-degenerate operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26327}, year = {2002}, abstract = {We study the index problem for families of elliptic operators on manifolds with conical singularities. The relative index theorem concerning changes of the weight line is obtained. AN index theorem for families whose conormal symbols satisfy some symmetry conditions is derived.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2002, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Elliptic theory on manifolds with nonisolated singularities : III. The spectral flow of families of conormal symbols}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26386}, year = {2002}, abstract = {When studyind elliptic operators on manifolds with nonisolated singularities one naturally encounters families of conormal symbols (i.e. operators elliptic with parameter p ∈ IR in the sense of Agranovich-Vishik) parametrized by the set of singular points. For homotopies of such families we define the notion of spectral flow, which in this case is an element of the K-group of the parameter space. We prove that the spectral flow is equal to the index of some family of operators on the infinite cone.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2002, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Elliptic theory on manifolds with nonisolated singularities : IV. Obstructions to elliptic problems on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26415}, year = {2002}, abstract = {The obstruction to the existence of Fredholm problems for elliptic differentail operators on manifolds with edges is a topological invariant of the operator. We give an explicit general formula for this invariant. As an application we compute this obstruction for geometric operators.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2002, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Elliptic theory on manifolds with nonisolated singularities : II. Products in elliptic theory on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26335}, year = {2002}, abstract = {Exterior tensor products of elliptic operators on smooth manifolds and manifolds with conical singularities are used to obtain examples of elliptic operators on manifolds with edges that do not admit well-posed edge boundary and coboundary conditions.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2003, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Differential operators on manifolds with singularities : analysis and topology : Chapter 5: Manifolds with isolated singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26659}, year = {2003}, abstract = {Contents: Chapter 5: Manifolds with Isolated Singularities 5.1. Differential Operators and the Geometry of Singularities 5.1.1. How do isolated singularities arise? Examples 5.1.2. Definition and methods for the description of manifolds with isolated singularities 5.1.3. Bundles. The cotangent bundle 5.2. Asymptotics of Solutions, Function Spaces,Conormal Symbols 5.2.1. Conical singularities 5.2.2. Cuspidal singularities 5.3. A Universal Representation of Degenerate Operators and the Finiteness Theorem 5.3.1. The cylindrical representation 5.3.2. Continuity and compactness 5.3.3. Ellipticity and the finiteness theorem 5.4. Calculus of ΨDO 5.4.1. General ΨDO 5.4.2. The subalgebra of stabilizing ΨDO 5.4.3. Ellipticity and the finiteness theorem}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2004, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Differential operators on manifolds with singularities : analysis and topology : Chapter 7: The index problem on manifolds with singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26700}, year = {2004}, abstract = {Contents: Chapter 7: The Index Problemon Manifolds with Singularities Preface 7.1. The Simplest Index Formulas 7.1.1. General properties of the index 7.1.2. The index of invariant operators on the cylinder 7.1.3. Relative index formulas 7.1.4. The index of general operators on the cylinder 7.1.5. The index of operators of the form 1 + G with a Green operator G 7.1.6. The index of operators of the form 1 + G on manifolds with edges 7.1.7. The index on bundles with smooth base and fiber having conical points 7.2. The Index Problem for Manifolds with Isolated Singularities 7.2.1. Statement of the index splitting problem 7.2.2. The obstruction to the index splitting 7.2.3. Computation of the obstruction in topological terms 7.2.4. Examples. Operators with symmetries 7.3. The Index Problem for Manifolds with Edges 7.3.1. The index excision property 7.3.2. The obstruction to the index splitting 7.4. Bibliographical Remarks}, language = {en} } @unpublished{NazaikinskiiSchulzeSternin2002, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Surgery and the relative index theorem for families of elliptic operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26300}, year = {2002}, abstract = {We prove a theorem describing the behaviour of the relative index of families of Fredholm operators under surgery performed on spaces where the operators act. In connection with additional conditions (like symmetry conditions) this theorem results in index formulas for given operator families. By way of an example, we give an application to index theory of families of boundary value problems.}, language = {en} } @unpublished{NazaikinskiiSchulzeSternin1999, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Quantization and the wave packet transform}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25447}, year = {1999}, language = {en} } @unpublished{NazaikinskiiSchulzeSternin2000, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Quantization methods in differential equations : Chapter 3: Applications of noncommutative analysis to operator algebras on singular manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25801}, year = {2000}, abstract = {Content: Chapter 3: Applications of Noncommutative Analysis to Operator Algebras on Singular Manifolds 3.1 Statement of the problem 3.2 Operators on the Model Cone 3.3 Operators on the Model Cusp of Order k 3.4 An Application to the Construction of Regularizers and Proof of the Finiteness Theorem}, language = {en} } @unpublished{NazaikinskiiSchulzeSternin2000, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Quantization methods in differential equations : Chapter 2: Exactly soluble commutation relations (The simplest class of classical mechanics)}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25796}, year = {2000}, abstract = {Content: Chapter 2: Exactly SolubleCommutation Relations (The Simplest Class of Classical Mechanics) 2.1 Some examples 2.2 Lie commutation relations 2.3 Non-Lie (nonlinear) commutation relations}, language = {en} } @unpublished{NazaikinskiiSchulzeSternin2000, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Quantization methods in differential equations : Chapter 11: Noncommutative analysis and high-frequency asymptotics}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25857}, year = {2000}, abstract = {Content: Chapter 11: Noncommutative Analysis and High-Frequency Asymptotics 11.1 Statement of the Problem 11.2 Mixed Asymptotics: the General Scheme 11.3 The Asymptotic Solution of Main Problem 11.4 Analysis of the Asymptotic Solution}, language = {en} } @unpublished{NazaikinskiiSchulzeSternin2000, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Quantization methods in differential equations : Part II: Quantization by the method of ordered operators (Noncommutative Analysis) : Chapter 1: Noncommutative Analysis: Main Ideas, Definitions, and Theorems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25762}, year = {2000}, abstract = {Content: 0.1 Preliminary Remarks Chapter 1: Noncommutative Analysis: Main Ideas, Definitions, and Theorems 1.1 Functions of One Operator (Functional Calculi) 1.2 Functions of Several Operators 1.3 Main Formulas of Operator Calculus 1.4 Main Tools of Noncommutative Analysis 1.5 Composition Laws and Ordered Representations}, language = {en} } @unpublished{NazaikinskiiSchulzeSternin2001, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Localization problem in index theory of elliptic operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26175}, year = {2001}, abstract = {This is a survey of recent results concerning the general index locality principle, associated surgery, and their applications to elliptic operators on smooth manifolds and manifolds with singularities as well as boundary value problems. The full version of the paper is submitted for publication in Russian Mathematical Surveys.}, language = {en} } @unpublished{NazaikinskiiSchulzeSterninetal.1997, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris and Shatalov, Victor}, title = {A Lefschetz fixed point theorem for manifolds with conical singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25073}, year = {1997}, abstract = {We establish an Atiyah-Bott-Lefschetz formula for elliptic operators on manifolds with conical singular points.}, language = {en} } @unpublished{NazaikinskiiSchulzeSterninetal.1997, author = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris and Shatalov, Victor}, title = {Quantization of symplectic transformations on manifolds with conical singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25084}, year = {1997}, abstract = {The structure of symplectic (canonical) transformations on manifolds with conical singularities is established. The operators associated with these transformations are defined in the weight spaces and their properties investigated.}, language = {en} } @unpublished{NazaikinskiiSternin2000, author = {Nazaikinskii, Vladimir and Sternin, Boris}, title = {On surgery in elliptic theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25873}, year = {2000}, abstract = {We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.}, language = {en} } @unpublished{NazaikinskiiSternin2002, author = {Nazaikinskii, Vladimir and Sternin, Boris}, title = {Relative elliptic theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26400}, year = {2002}, abstract = {This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms.}, language = {en} } @unpublished{NazaikinskiiSternin2001, author = {Nazaikinskii, Vladimir and Sternin, Boris}, title = {Some problems of control of semiclassical states for the Schr{\"o}dinger equation}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26130}, year = {2001}, abstract = {Contents: Introduction Controlled Quantum Systems The Asymptotic Controllability Problem The Stabilization Problem Unitarily Nonlinear Equations The Quantum Problem The Stabilization Problem for the Schr{\"o}dinger Equation with a Unitarily Non-linear Control}, language = {en} } @unpublished{NehringPoghosyanZessin2013, author = {Nehring, Benjamin and Poghosyan, Suren and Zessin, Hans}, title = {On the construction of point processes in statistical mechanics}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64080}, year = {2013}, abstract = {By means of the cluster expansion method we show that a recent result of Poghosyan and Ueltschi (2009) combined with a result of Nehring (2012) yields a construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in R^d of Ginibre's Fermi-Dirac gas of such loops. The latter will be identified as a Gibbs perturbation of the ideal Fermi gas. On generalizing these considerations we will obtain the existence of a large class of Gibbs perturbations of the so-called KMM-processes as they were introduced by Nehring (2012). Moreover, it is shown that certain "limiting Gibbs processes" are Gibbs in the sense of Dobrushin, Lanford and Ruelle if the underlying potential is positive. And finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive.}, language = {en} } @unpublished{NehringZessin2010, author = {Nehring, Benjamin and Zessin, Hans}, title = {A path integral representation of the moment measures of the general ideal Bose gas}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49635}, year = {2010}, abstract = {We reconsider the fundamental work of Fichtner ([2]) and exhibit the permanental structure of the ideal Bose gas again, using another approach which combines a characterization of infinitely divisible random measures (due to Kerstan,Kummer and Matthes [5, 6] and Mecke [8, 9]) with a decomposition of the moment measures into its factorial measures due to Krickeberg [4]. To be more precise, we exhibit the moment measures of all orders of the general ideal Bose gas in terms of certain path integrals. This representation can be considered as a point process analogue of the old idea of Symanzik [11] that local times and self-crossings of the Brownian motion can be used as a tool in quantum field theory. Behind the notion of a general ideal Bose gas there is a class of infinitely divisible point processes of all orders with a Levy-measure belonging to some large class of measures containing the one of the classical ideal Bose gas considered by Fichtner. It is well known that the calculation of moments of higher order of point processes are notoriously complicated. See for instance Krickeberg's calculations for the Poisson or the Cox process in [4].}, language = {en} } @unpublished{OliaroSchulze2002, author = {Oliaro, Alessandro and Schulze, Bert-Wolfgang}, title = {Parameter-dependent boundary value problems on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26424}, year = {2002}, abstract = {As is known from Kondratyev's work, boundary value problems for elliptic operators on a manifold with conical singularities and boundary are controlled by a principal symbolic hierarchy, where the conormal symbols belong to the typical new components, compared with the smooth case, with interior and boundary symbols. A similar picture may be expected on manifolds with corners when the base of the cone itself is a manifold with conical or edge singularities. This is a natural situation in a number of applications, though with essential new difficulties. We investigate here corresponding conormal symbols in terms of a calculus of holomorphic parameter-dependent edge boundary value problems on the base. We show that a certain kernel cut-off procedure generates all such holomorphic families, modulo smoothing elements, and we establish conormal symbols as an algebra as is necessary for a parametrix constructions in the elliptic case.}, language = {en} } @unpublished{Paneah2005, author = {Paneah, B.}, title = {On the general theory of the cauchy type functional equations with applications in analysis}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30004}, year = {2005}, abstract = {Contents: 1 The main notations and definitions. 2 Statement of the problems and main results. 2.1 The case of a Z-configuration. 2.2 The case of a P-configuration. 3 Proofs of Theorems 1-7. 4 Applications. 4.1 Multiplicative Cauchy type functional equation. 4.2 On some integral equations relating to a geometric problem 4.3 On the solvability of boundary problem for hyperbolic differential equations.}, language = {en} } @unpublished{Paneah2002, author = {Paneah, Boris}, title = {Dynamic methods in the general theory of cauchy type functional equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26295}, year = {2002}, abstract = {Contents: 1 Introduction. Denfitions and Discussions 2 Solvability of the Cauchy Type Functional Equations 2.1 The Case of a P-configuration 2.2 The Case of a Z-configuration 2.3 Multiplicative Cauchy type functional equations 3 Problems in Analysis Reducing to Cauchy Type Functional Equations 3.1 Some problems in Integral Geometry and Cauchy Functional Equations 3.2 First Boundary Problem for Hyperbolic Differential Equations and Cauchy Type Functional Equations 4 Functional Equations Determining Polynomials}, language = {en} } @unpublished{Paneah2006, author = {Paneah, Boris}, title = {Another approach to the stability of linear functional operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30139}, year = {2006}, abstract = {Contents: 1 Introduction. 2 The main notations and notions. 3 Statement of results 4 Proofs of the main results.}, language = {en} } @unpublished{Paneah2001, author = {Paneah, Boris}, title = {On a new problem in integral geometry related to boundary problems for partial differential equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26089}, year = {2001}, abstract = {Contents: 1 Introduction 2 Statement of the problem and definitions 3 The main results 4 Proof of theorem 2 4.1 Reduction of problem (2) to functional - integral equations 4.2 The uniqueness of a solution of equation (2) 4.3 The existence of a solution of equation (2) 5 Proof of theorem 1 6 Proof of theorem 3 7 First boundary problem for hyperbolic differential equations 7.1 Statement of the problem 7.2 The formulation of the result and a sketch of the proof}, language = {en} } @unpublished{PaneahSchulze1998, author = {Paneah, Boris and Schulze, Bert-Wolfgang}, title = {On the existence of smooth solutions of the dirichlet problem for hyperbolic : differential equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25179}, year = {1998}, language = {en} } @unpublished{Popivanov2004, author = {Popivanov, Petăr R.}, title = {Lorenz transformations and creation of logarithmic singularities to the solutions of some nonstrictly hyperbolic semilinear systems with two space variables}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26710}, year = {2004}, language = {en} } @unpublished{PrasseGrubenMachlikaetal.2016, author = {Prasse, Paul and Gruben, Gerrit and Machlika, Lukas and Pevny, Tomas and Sofka, Michal and Scheffer, Tobias}, title = {Malware Detection by HTTPS Traffic Analysis}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-100942}, pages = {10}, year = {2016}, abstract = {In order to evade detection by network-traffic analysis, a growing proportion of malware uses the encrypted HTTPS protocol. We explore the problem of detecting malware on client computers based on HTTPS traffic analysis. In this setting, malware has to be detected based on the host IP address, ports, timestamp, and data volume information of TCP/IP packets that are sent and received by all the applications on the client. We develop a scalable protocol that allows us to collect network flows of known malicious and benign applications as training data and derive a malware-detection method based on a neural networks and sequence classification. We study the method's ability to detect known and new, unknown malware in a large-scale empirical study.}, language = {en} } @unpublished{PrenovTarkhanov2001, author = {Prenov, B. and Tarkhanov, Nikolai Nikolaevich}, title = {Kernel spikes of singular problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26195}, year = {2001}, abstract = {Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This paper is aimed at describing refined asymptotics in the Dirichlet problem in a ball. The beauty of explicit formulas highlights the structure of asymptotic expansions in the calculi on singular varieties.}, language = {en} } @unpublished{Penisson2010, author = {P{\´e}nisson, Sophie}, title = {Estimation of the infection parameter in the different phases of an epidemic modeled by a branching process}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49607}, year = {2010}, abstract = {The aim of this paper is to build and compare estimators of the infection parameter in the different phases of an epidemic (growth and extinction phases). The epidemic is modeled by a Markovian process of order d > 1 (allowing non-Markovian life spans), and can be written as a multitype branching process. We propose three estimators suitable for the different classes of criticality of the process, in particular for the subcritical case corresponding to the extinction phase. We prove their consistency and asymptotic normality for two asymptotics, when the number of ancestors (resp. number of generations) tends to infinity. We illustrate the asymptotic properties with simulated examples, and finally use our estimators to study the infection intensity in the extinction phase of the BSE epidemic in Great-Britain.}, language = {en} } @unpublished{Penisson2010, author = {P{\´e}nisson, Sophie}, title = {Conditional Limit Theorems for Multitype Branching Processes and Illustration in Epidemiological Risk Analysis}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49589}, year = {2010}, abstract = {This thesis is concerned with the issue of extinction of populations composed of different types of individuals, and their behavior before extinction and in case of a very late extinction. We approach this question firstly from a strictly probabilistic viewpoint, and secondly from the standpoint of risk analysis related to the extinction of a particular model of population dynamics. In this context we propose several statistical tools. The population size is modeled by a branching process, which is either a continuous-time multitype Bienaym{\´e}-Galton-Watson process (BGWc), or its continuous-state counterpart, the multitype Feller diffsion process. We are interested in different kinds of conditioning on nonextinction, and in the associated equilibrium states. These ways of conditioning have been widely studied in the monotype case. However the literature on multitype processes is much less extensive, and there is no systematic work establishing connections between the results for BGWc processes and those for Feller diffusion processes. In the first part of this thesis, we investigate the behavior of the population before its extinction by conditioning the associated branching process Xt on non-extinction (Xt 6= 0), or more generally on non-extinction in a near future 0 < 1 (Xt+ 0 = 0), and by letting t tend to infinity. We prove the result, new in the multitype framework and for 0 > 0, that this limit exists and is nondegenerate. This re ects a stationary behavior for the dynamics of the population conditioned on non-extinction, and provides a generalization of the so-called Yaglom limit, corresponding to the case 0 = 0. In a second step we study the behavior of the population in case of a very late extinction, obtained as the limit when 0 tends to infinity of the process conditioned by Xt+ 0 = 0. The resulting conditioned process is a known object in the monotype case (sometimes referred to as Q-process), and has also been studied when Xt is a multitype Feller diffusion process. We investigate the not yet considered case where Xt is a multitype BGWc process and prove the existence of the associated Q-process. In addition, we examine its properties, including the asymptotic ones, and propose several interpretations of the process. Finally, we are interested in interchanging the limits in t and 0, as well as in the not yet studied commutativity of these limits with respect to the high-density-type relationship between BGWc processes and Feller processes. We prove an original and exhaustive list of all possible exchanges of limit (long-time limit in t, increasing delay of extinction 0, diffusion limit). The second part of this work is devoted to the risk analysis related both to the extinction of a population and to its very late extinction. We consider a branching population model (arising notably in the epidemiological context) for which a parameter related to the first moments of the offspring distribution is unknown. We build several estimators adapted to different stages of evolution of the population (phase growth, decay phase, and decay phase when extinction is expected very late), and prove moreover their asymptotic properties (consistency, normality). In particular, we build a least squares estimator adapted to the Q-process, allowing a prediction of the population development in the case of a very late extinction. This would correspond to the best or to the worst-case scenario, depending on whether the population is threatened or invasive. These tools enable us to study the extinction phase of the Bovine Spongiform Encephalopathy epidemic in Great Britain, for which we estimate the infection parameter corresponding to a possible source of horizontal infection persisting after the removal in 1988 of the major route of infection (meat and bone meal). This allows us to predict the evolution of the spread of the disease, including the year of extinction, the number of future cases and the number of infected animals. In particular, we produce a very fine analysis of the evolution of the epidemic in the unlikely event of a very late extinction.}, language = {en} } @unpublished{Penisson2009, author = {P{\´e}nisson, Sophie}, title = {Continuous-time multitype branching processes conditioned on very late extinction}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49548}, year = {2009}, abstract = {Multitype branching processes and Feller diffusion processes are conditioned on very late extinction. The conditioned laws are expressed as Doob h-transforms of the unconditioned laws, and an interpretation of the conditioned paths for the branching process is given, via the immortal particle. We study different limits for the conditioned process (increasing delay of extinction, long-time behavior, scaling limit) and provide an exhaustive list of exchangeability results.}, language = {en} } @unpublished{RabinovichSchulzeTarkhanov1998, author = {Rabinovich, Vladimir and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {Boundary value problems in cuspidal wedges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25363}, year = {1998}, abstract = {The paper is devoted to pseudodifferential boundary value problems in domains with cuspidal wedges. Concerning the geometry we even admit a more general behaviour, namely oscillating cuspidal wedges. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to edges.}, language = {en} } @unpublished{RabinovichSchulzeTarkhanov2000, author = {Rabinovich, Vladimir and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {C*-algebras of ISO's with oscillating symbols}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25847}, year = {2000}, abstract = {For a domain D subset of IRn with singular points on the boundary and a weight function ω infinitely differentiable away from the singularpoints in D, we consider a C*-algebra G (D; ω) of operators acting in the weighted space L² (D, ω). It is generated by the operators XD F-¹ σ F XD where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for G (D; ω). When combined with the local principle, this leads to describing the Fredholm operators in G (D; ω).}, language = {en} } @unpublished{RabinovichSchulzeTarkhanov1997, author = {Rabinovich, Vladimir and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {A calculus of boundary value problems in domains with Non-Lipschitz Singular Points}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-24957}, year = {1997}, abstract = {The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.}, language = {en} } @unpublished{RabinovichSchulzeTarkhanov1999, author = {Rabinovich, Vladimir and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {Boundary value problems in domains with corners}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25552}, year = {1999}, abstract = {We describe Fredholm boundary value problems for differential equations in domains with intersecting cuspidal edges on the boundary.}, language = {en} } @unpublished{Rafler2009, author = {Rafler, Mathias}, title = {Gaussian loop- and polya processes : a point process approach}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51638}, year = {2009}, abstract = {Zuf{\"a}llige Punktprozesse beschreiben eine (zuf{\"a}llige) zeitliche Abfolge von Ereignissen oder eine (zuf{\"a}llige) r{\"a}umliche Anordnung von Objekten. Deren wichtigster Vertreter ist der Poissonprozess. Der Poissonprozess zum Intensit{\"a}tsmaß, das Lebesgue-Maß ordnet jedem Gebiet sein Volumen zu, erzeugt lokal, d.h in einem beschr{\"a}nkten Gebiet B, gerade eine mit dem Volumen von B poissonverteilte Anzahl von Punkten, die identisch und unabh{\"a}ngig voneinander in B plaziert werden; im Mittel ist diese Anzahl (B). Ersetzt man durch ein Vielfaches a, so wird diese Anzahl mit dem a-fachen Mittelwert erzeugt. Poissonprozesse, die im gesamten Raum unendlich viele Punkte realisieren, enthalten bereits in einer einzigen Stichprobe gen{\"u}gend Informationen, um Statistik betreiben zu k{\"o}nnen: Bedingt man lokal bzgl. der Anzahl der Teilchen einer Stichprobe, so fragt man nach allen Punktprozessen, die eine solche Beobachtung h{\"a}tten liefern k{\"o}nnen. Diese sind Limespunktprozesse zu dieser Beobachtung. Kommt mehr als einer in Frage, spricht man von einem Phasen{\"u}bergang. Da die Menge dieser Limespunktprozesse konvex ist, fragt man nach deren Extremalpunkten, dem Rand. Im ersten Teil wird ein Poissonprozess f{\"u}r ein physikalisches Teilchenmodell f{\"u}r Bosonen konstruiert. Dieses erzeugt sogenannte Loops, das sind geschlossene Polygonz{\"u}ge, die dadurch charakterisiert sind, dass man an einem Ort mit einem Punkt startet, den mit einem normalverteilten Schritt l{\"a}uft und dabei nach einer gegebenen, aber zuf{\"a}lligen Anzahl von Schritten zum Ausgangspunkt zur{\"u}ckkehrt. F{\"u}r verschiedene Beobachtungen von Stichproben werden zugeh{\"o}rige Limespunktprozesse diskutiert. Diese Beobachtungen umfassen etwa das Z{\"a}hlen der Loops gem{\"a}aß ihrer L{\"a}nge, das Z{\"a}hlen der Loops insgesamt, oder das Z{\"a}hlen der von den Loops gemachten Schritte. Jede Wahl zieht eine charakteristische Struktur der invarianten Punktprozesse nach sich. In allen hiesigen F{\"a}llen wird ein charakteristischer Phasen{\"u}bergang gezeigt und Extremalpunkte werden als spezielle Poissonprozesse identifiziert. Insbesondere wird gezeigt, wie die Wahl der Beobachtung die L{\"a}nge der Loops beeinflusst. Geometrische Eigenschaften dieser Poissonprozesse sind der Gegenstand des zweiten Teils der Arbeit. Die Technik der Palmschen Verteilungen eines Punktprozesses erlaubt es, unter den unendlich vielen Loops einer Realisierung den typischen Loop herauszupicken, dessen Geometrie dann untersucht wird. Eigenschaften sind unter anderem die euklidische L{\"a}nge eines Schrittes oder, nimmt man mehrere aufeinander folgende Schritte, das Volumen des von ihnen definierten Simplex. Weiterhin wird gezeigt, dass der Schwerpunkt eines typischen Loops normalverteilt ist mit einer festen Varianz. Der dritte und letzte Teil befasst sich mit der Konstruktion, den Eigenschaften und der Statistik eines neuartigen Punktprozesses, der Polyascher Summenprozess genannt wird. Seine Konstruktion verallgemeinert das Prinzip der Polyaschen Urne: Im Gegensatz zum Poissonprozess, der alle Punkte unabh{\"a}ngig und vor allem identisch verteilt, werden hier die Punkte nacheinander derart verteilt, dass der Ort, an dem ein Punkt plaziert wird, eine Belohnung auf die Wahrscheinlichkeit bekommt, nach der nachfolgende Punkte verteilt werden. Auf diese Weise baut der Polyasche Summenprozess "T{\"u}rmchen", indem sich verschiedene Punkte am selben Ort stapeln. Es wird gezeigt, dass dennoch grundlegende Eigenschaften mit denjenigen des Poissonprozesses {\"u}bereinstimmen, dazu geh{\"o}ren unendliche Teilbarkeit sowie Unabh{\"a}ngigkeit der Zuw{\"a}chse. Zudem werden sein Laplace-Funktional sowie seine Palmsche Verteilung bestimmt. Letztere zeigt, dass die H{\"o}he der T{\"u}rmchen gerade geometrisch verteilt ist. Abschließend werden wiederum Statistiken, nun f{\"u}r den Summenprozess, diskutiert. Je nach Art der Beobachtung von der Stichprobe, etwa Anzahl, Gesamth{\"o}he der T{\"u}rmchen oder beides, gibt es in jedem der drei F{\"a}lle charakteristische Limespunktprozesse und es stellt sich heraus, dass die zugeh{\"o}rigen Extremalverteilungen wiederum Polyasche Summenprozesse sind.}, language = {en} } @unpublished{Rebahi1998, author = {Rebahi, Y.}, title = {Asymptotics of solutions of differential equations on manifolds with cusps}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25372}, year = {1998}, language = {en} } @unpublished{Roelly2013, author = {Roelly, Sylvie}, title = {Reciprocal processes : a stochastic analysis approach}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64588}, year = {2013}, abstract = {Reciprocal processes, whose concept can be traced back to E. Schr{\"o}dinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. L{\´e}onard.}, language = {en} } @unpublished{RoellyRuszel2013, author = {Roelly, Sylvie and Ruszel, Wioletta M.}, title = {Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69014}, year = {2013}, abstract = {We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.}, language = {en} } @unpublished{Rozenblum2000, author = {Rozenblum, G.}, title = {On some analytical index formulas related to operator-valued symbols}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25811}, year = {2000}, abstract = {For several classes of pseudodifferential operators with operator-valued symbol analytic index formulas are found. The common feature is that uasual index formulas are not valid for these operators. Applications are given to pseudodifferential operators on singular manifolds.}, language = {en} } @unpublished{RuedigerFeudelSeehafer1998, author = {R{\"u}diger, Sten and Feudel, Fred and Seehafer, Norbert}, title = {Dynamo bifurcations in an array of driven convection-like rolls}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14678}, year = {1998}, abstract = {The bifurcations in a three-dimensional incompressible, electrically conducting fluid with an external forcing of the Roberts type have been studied numerically. The corresponding flow can serve as a model for the convection in the outer core of the Earth and is realized in an ongoing laboratory experiment aimed at demonstrating a dynamo effect. The symmetry group of the problem has been determined and special attention has been paid to symmetry breaking by the bifurcations. The nonmagnetic, steady Roberts flow loses stability to a steady magnetic state, which in turn is subject to secondary bifurcations. The secondary solution branches have been traced until they end up in chaotic states.}, language = {en} } @unpublished{Sadykov1999, author = {Sadykov, Timour}, title = {Hypergeometric systems of differential equations and amoebas of rational functions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25665}, year = {1999}, abstract = {We study the approach to the theory of hypergeometric functions in several variables via a generalization of the Horn system of differential equations. A formula for the dimension of its solution space is given. Using this formula we construct an explicit basis in the space of holomorphic solutions to the generalized Horn system under some assumptions on its parameters. These results are applied to the problem of describing the complement of the amoeba of a rational function, which was posed in [12].}, language = {en} } @unpublished{SavinSchulzeSternin1998, author = {Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {On the invariant index formulas for spectral boundary value problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25285}, year = {1998}, abstract = {In the paper we study the possibility to represent the index formula for spectral boundary value problems as a sum of two terms, the first one being homotopy invariant of the principal symbol, while the second depends on the conormal symbol of the problem only. The answer is given in analytical, as well as in topological terms.}, language = {en} } @unpublished{SavinSchulzeSternin2000, author = {Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Elliptic operators in subspaces}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25701}, year = {2000}, abstract = {We construct elliptic theory in the subspaces, determined by pseudodifferential projections. The finiteness theorem as well as index formula are obtained for elliptic operators acting in the subspaces. Topological (K-theoretic) aspects of the theory are studied in detail.}, language = {en} } @unpublished{SavinSternin2000, author = {Savin, Anton and Sternin, Boris}, title = {Eta-invariant and Pontrjagin duality in K-theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25747}, year = {2000}, abstract = {The topological significance of the spectral Atiyah-Patodi-Singer η-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.}, language = {en} } @unpublished{SavinSternin2000, author = {Savin, Anton and Sternin, Boris}, title = {Eta invariant and parity conditions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25869}, year = {2000}, abstract = {We give a formula for the η-invariant of odd order operators on even-dimensional manifolds, and for even order operators on odd-dimensional manifolds. Geometric second order operators are found with nontrivial η-invariants. This solves a problem posed by P. Gilkey.}, language = {en} } @unpublished{SavinSternin1999, author = {Savin, Anton and Sternin, Boris}, title = {Elliptic operators in even subspaces}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25461}, year = {1999}, abstract = {An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.}, language = {en} } @unpublished{SavinSternin2005, author = {Savin, Anton and Sternin, Boris}, title = {Pseudodifferential subspaces and their applications in elliptic theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29937}, year = {2005}, abstract = {The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah-Patodi-Singer eta invariant, when it defines a homotopy invariant (Gilkey's problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.}, language = {en} } @unpublished{SavinSternin1999, author = {Savin, Anton and Sternin, Boris}, title = {Elliptic operators in odd subspaces}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25478}, year = {1999}, abstract = {An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.}, language = {en} } @unpublished{SavinSternin2001, author = {Savin, Anton and Sternin, Boris}, title = {Index defects in the theory of nonlocal boundary value problems and the η-invariant}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26146}, year = {2001}, abstract = {The paper deals with elliptic theory on manifolds with boundary represented as a covering space. We compute the index for a class of nonlocal boundary value problems. For a nontrivial covering, the index defect of the Atiyah-Patodi-Singer boundary value problem is computed. We obtain the Poincar{\´e} duality in the K-theory of the corresponding manifolds with singularities.}, language = {en} } @unpublished{ScheelSeehafer1997, author = {Scheel, Stefan and Seehafer, Norbert}, title = {Bifurcation to oscillations in three-dimensional Rayleigh-B{\´e}nard convection}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14370}, year = {1997}, abstract = {Three-dimensional bouyancy-driven convection in a horizontal fluid layer with stress-free boundary conditions at the top and bottom and periodic boundary conditions in the horizontal directions is investigated by means of numerical simulation and bifurcation-analysis techniques. The aspect ratio is fixed to a value of 2√2 and the Prandtl number to a value of 6.8. Two-dimensional convection rolls are found to be stable up to a Rayleigh number of 17 950, where a Hopf bifurcation leads to traveling waves. These are stable up to a Rayleigh number of 30 000, where a secondary Hopf bifurcation generates modulated traveling waves. We pay particular attention to the symmetries of the solutions and symmetry breaking by the bifurcations.}, language = {en} } @unpublished{SchmidtmannFeudelSeehafer1997, author = {Schmidtmann, Olaf and Feudel, Fred and Seehafer, Norbert}, title = {Nonlinear Galerkin methods for the 3D magnetohydrodynamic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14431}, year = {1997}, abstract = {The usage of nonlinear Galerkin methods for the numerical solution of partial differential equations is demonstrated by treating an example. We desribe the implementation of a nonlinear Galerkin method based on an approximate inertial manifold for the 3D magnetohydrodynamic equations and compare its efficiency with the linear Galerkin approximation. Special bifurcation points, time-averaged values of energy and enstrophy as well as Kaplan-Yorke dimensions are calculated for both schemes in order to estimate the number of modes necessary to correctly describe the behavior of the exact solutions.}, language = {en} } @unpublished{Schrohe2000, author = {Schrohe, Elmar}, title = {A short introduction to Boutet de Monvel's calculus}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25696}, year = {2000}, abstract = {This paper provides an introduction to Boutet de Monvel's calculus on the half-space IRn (positiv) in the framework of a pseudodifferential calculus with operator-valued symbols.}, language = {en} } @unpublished{Schrohe1999, author = {Schrohe, Elmar}, title = {Noncommutative residues, Dixmier's Trace, and heat trace expansions on manifolds with boundary}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25486}, year = {1999}, abstract = {For manifolds with boundary, we define an extension of Wodzicki's noncommutative residue to boundary value problems in Boutet de Monvel's calculus. We show that this residue can be recovered with the help of heat kernel expansions and explore its relation to Dixmier's trace.}, language = {en} } @unpublished{SchroheSchulze1999, author = {Schrohe, Elmar and Schulze, Bert-Wolfgang}, title = {Edge-degenerate boundary value problems on cones}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25436}, year = {1999}, abstract = {We consider edge-degenerate families of pseudodifferential boundary value problems on a semi-infinite cylinder and study the behavior of their push-forwards as the cylinder is blown up to a cone near infinity. We show that the transformed symbols belong to a particularly convenient symbol class. This result has applications in the Fredholm theory of boundary value problems on manifolds with edges.}, language = {en} } @unpublished{SchroheSeiler1999, author = {Schrohe, Elmar and Seiler, J{\"o}rg}, title = {Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25621}, year = {1999}, abstract = {Given a manifold B with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale of Lp-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces, it turns out to be independent of the choice of p. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators on Lp(B).}, language = {en} } @unpublished{SchroheSeiler2002, author = {Schrohe, Elmar and Seiler, J{\"o}rg}, title = {The resolvent of closed extensions of cone differential operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26378}, year = {2002}, abstract = {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.}, language = {en} } @unpublished{Schulze2003, author = {Schulze, Bert-Wolfgang}, title = {Crack theory with singularties at the boundary}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26600}, year = {2003}, abstract = {We investigate crack problems, where the crack boundary has conical singularities. Elliptic operators with two-sided elliptc boundary conditions on the plus and minus sides of the crack will be interpreted as elements of a corner algebra of boundary value problems. The corresponding operators will be completed by extra edge conditions on the crack boundary to Fredholm operators in corner Sobolev spaces with double weights, and there are parametrices within the calculus.}, language = {en} } @unpublished{Schulze2006, author = {Schulze, Bert-Wolfgang}, title = {Pseudo-differential calculus on manifolds with geometric singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30204}, year = {2006}, abstract = {Differential and pseudo-differential operators on a manifold with (regular) geometric singularities can be studied within a calculus, inspired by the concept of classical pseudo-differential operators on a C1 manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy σ = (σj)0≤j≤k, with k being the order of the singularity and σk operator-valued for k ≥ 1. The symbols determine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by various quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give examples and also comment on new challenges and interesting problems of the recent development.}, language = {en} } @unpublished{Schulze2009, author = {Schulze, Bert-Wolfgang}, title = {Boundary value problems with the transmission property}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30377}, year = {2009}, abstract = {We give a survey on the calculus of (pseudo-differential) boundary value problems with the transmision property at the boundary, and ellipticity in the Shapiro-Lopatinskij sense. Apart from the original results of the work of Boutet de Monvel we present an approach based on the ideas of the edge calculus. In a final section we introduce symbols with the anti-transmission property.}, language = {en} } @unpublished{Schulze2008, author = {Schulze, Bert-Wolfgang}, title = {The iterative structure of corner operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30353}, year = {2008}, abstract = {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.}, language = {en} } @unpublished{Schulze2006, author = {Schulze, Bert-Wolfgang}, title = {Elliptic differential operators on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30188}, year = {2006}, abstract = {On a manifold with edge we construct a specific class of (edgedegenerate) elliptic differential operators. The ellipticity refers to the principal symbolic structure σ = (σψ, σ^) of the edge calculus consisting of the interior and edge symbol, denoted by σψ and σ^, respectively. For our choice of weights the ellipticity will not require additional edge conditions of trace or potential type, and the operators will induce isomorphisms between the respective edge spaces.}, language = {en} } @unpublished{Schulze2008, author = {Schulze, Bert-Wolfgang}, title = {On a paper of Krupchyk, Tarkhanov, and Tuomela}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30325}, year = {2008}, language = {en} } @unpublished{Schulze2006, author = {Schulze, Bert-Wolfgang}, title = {The structure of operators on manifolds with polyhedral singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30099}, year = {2006}, abstract = {We discuss intuitive ideas and historical background of concepts in the analysis on configurations with singularities, here in connection with our iterative approach for higher singularities.}, language = {en} } @unpublished{Schulze2001, author = {Schulze, Bert-Wolfgang}, title = {Operators with symbol hierarchies and iterated asymptotics}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25948}, year = {2001}, abstract = {Contents: Introduction 1 Edge calculus with parameters 1.1 Cone asymptotics and Green symbols 1.2 Mellin edge symbols 1.3 The edge symbol algebra 1.4 Operators on a manifold with edges 2 Corner symbols and iterated asymptotics 2.1 Holomorphic corner symbols 2.2 Meromorphic corner symbols and ellipicity 2.3 Weighted corner Sobolev spaces 2.4 Iterated asymptotics 3 The edge corner algebra with trace and potential conditions 3.1 Green corner operators 3.2 Smoothing Mellin corner operators 3.3 The edge corner algebra 3.4 Ellipicity and regularity with asymptotics 3.5 Examples and remarks}, language = {en} } @unpublished{Schulze2003, author = {Schulze, Bert-Wolfgang}, title = {Toeplitz operators, and ellipticity of boundary value problems with global projection conditions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26510}, year = {2003}, abstract = {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.}, language = {en} } @unpublished{Schulze1999, author = {Schulze, Bert-Wolfgang}, title = {Operator algebras with symbol hierarchies on manifolds with singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25647}, year = {1999}, abstract = {Problems for elliptic partial differential equations on manifolds M with singularities M' (here with piece-wise smooth geometry)are studied in terms of pseudo-differential algebras with hierarchies of symbols that consist of scalar and operator-valued components. Classical boundary value problems (with or without the transmission property) belong to the examples. They are a model for operator algebras on manifolds M with higher "polyhedral" singularities. The operators are block matrices that have upper left corners containing the pseudo-differential operators on the regular M\M' (plus certain Mellin and Green summands) and are degenerate (in streched coordinates) in a typical way near M'. By definition M' is again a manifold with singularities. The same is true of M'', and so on. The block matrices consist of trace, potential and Mellin and Green operators, acting between weighted Sobolev spaces on M(j) and M(k), with 0 ≤ j, k ≤ ord M; here M(0) denotes M, M(1) denotes M', etc. We generate these algebras, including their symbol hierarchies, by iterating so-called "edgifications" and "conifications" os algebras that have already been constructed, and we study ellipicity, parametrics and Fredholm property within these algebras.}, language = {en} } @unpublished{Schulze1999, author = {Schulze, Bert-Wolfgang}, title = {An algebra of boundary value problems not requiring Shapiro-Lopatinskil conditions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25596}, year = {1999}, abstract = {We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro-Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces.}, language = {en} } @unpublished{SchulzeNazaikinskiiSternin1999, author = {Schulze, Bert-Wolfgang and Nazaikinskii, Vladimir E. and Sternin, Boris}, title = {On the homotopy classification of elliptic operators on manifolds with singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25574}, year = {1999}, abstract = {We study the homotopy classification of elliptic operators on manifolds with singularities and establish necessary and sufficient conditions under which the classification splits into terms corresponding to the principal symbol and the conormal symbol.}, language = {en} } @unpublished{SchulzeNazaikinskiiSternin1998, author = {Schulze, Bert-Wolfgang and Nazaikinskii, Vladimir and Sternin, Boris}, title = {The index of quantized contact transformations on manifolds with conical singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25276}, year = {1998}, abstract = {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.}, language = {en} } @unpublished{SchulzeNazaikinskiiSternin1998, author = {Schulze, Bert-Wolfgang and Nazaikinskii, Vladimir and Sternin, Boris}, title = {A semiclassical quantization on manifolds with singularities and the Lefschetz Formula for Elliptic Operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25296}, year = {1998}, abstract = {For general endomorphisms of elliptic complexes on manifolds with conical singularities, the semiclassical asymptotics of the Atiyah-Bott-Lefschetz number is calculated in terms of fixed points of the corresponding canonical transformation of the symplectic space.}, language = {en} } @unpublished{SchulzeNazaikinskiiSterninetal.1997, author = {Schulze, Bert-Wolfgang and Nazaikinskii, Vladimir and Sternin, Boris and Shatalov, Victor}, title = {Spectral boundary value problems and elliptic equations on singular manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25147}, year = {1997}, abstract = {For elliptic operators on manifolds with boundary, we define spectral boundary value problems, which generalize the Atiyah-Patodi-Singer problem to the case of nonhomogeneous boundary conditions, operators of arbitrary order, and nonself-adjoint conormal symbols. The Fredholm property is proved and equivalence with certain elliptic equations on manifolds with conical singularities is established.}, language = {en} } @unpublished{SchulzeQin2005, author = {Schulze, Bert-Wolfgang and Qin, Yuming}, title = {Uniform compact attractors for a nonlinear non-autonomous equation of viscoelasticity}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29892}, year = {2005}, abstract = {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.}, language = {en} } @unpublished{SchulzeSavinSternin1999, author = {Schulze, Bert-Wolfgang and Savin, Anton and Sternin, Boris}, title = {Elliptic operators in subspaces and the eta invariant}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25496}, year = {1999}, abstract = {The paper deals with the calculation of the fractional part of the η-invariant for elliptic self-adjoint operators in topological terms. The method used to obtain the corresponding formula is based on the index theorem for elliptic operators in subspaces obtained in [1], [2]. It also utilizes K-theory with coefficients Zsub(n). In particular, it is shown that the group K(T*M,Zsub(n)) is realized by elliptic operators (symbols) acting in appropriate subspaces.}, language = {en} } @unpublished{SchulzeSeiler2001, author = {Schulze, Bert-Wolfgang and Seiler, J{\"o}rg}, title = {The edge algebra structure of boundary value problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25955}, year = {2001}, abstract = {Boundary value problems for pseudodifferential operators (with or without the transmission property) are characterised as a substructure of the edge pseudodifferential calculus with constant discrete asymptotics. The boundary in this case is the edge and the inner normal the model cone of local wedges. Elliptic boundary value problems for non-integer powers of the Laplace symbol belong to the examples as well as problems for the identity in the interior with a prescribed number of trace and potential conditions. Transmission operators are characterised as smoothing Mellin and Green operators with meromorphic symbols.}, language = {en} } @unpublished{SchulzeSeiler2002, author = {Schulze, Bert-Wolfgang and Seiler, J{\"o}rg}, title = {Pseudodifferential boundary value problems with global projection conditions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26233}, year = {2002}, abstract = {Contents: Introduction 1 Operators with the transmission property 1.1 Operators on a manifold with boundary 1.2 Conditions with pseudodifferential projections 1.3 Projections and Fredholm families 2 Boundary value problems not requiring the transmission property 2.1 Interior operators 2.2 Edge amplitude functions 2.3 Boundary value problems 3 Operators with global projection conditions 3.1 Construction for boundary symbols 3.2 Ellipticity of boundary value problems with projection data 3.3 Operators of order zero}, language = {en} } @unpublished{SchulzeShlapunovTarkhanov1999, author = {Schulze, Bert-Wolfgang and Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {Regularisation of mixed boundary problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25454}, year = {1999}, abstract = {We show an application of the spectral theorem in constructing approximate solutions of mixed boundary value problems for elliptic equations.}, language = {en} } @unpublished{SchulzeShlapunovTarkhanov2000, author = {Schulze, Bert-Wolfgang and Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {Green integrals on manifolds with cracks}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25777}, year = {2000}, abstract = {We prove the existence of a limit in Hm(D) of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S ⊂ ∂D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchy problem in D with data on S, for an elliptic operator A of order m ≥ 1, whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of the double layer potential. A similar regularisation is constructed also for a mixed problem in D.}, language = {en} } @unpublished{SchulzeSterninSavin1999, author = {Schulze, Bert-Wolfgang and Sternin, Boris and Savin, Anton}, title = {The homotopy classification and the index of boundary value problems for general elliptic operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25568}, year = {1999}, abstract = {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.}, language = {en} } @unpublished{SchulzeSterninShatalov1997, author = {Schulze, Bert-Wolfgang and Sternin, Boris and Shatalov, Victor}, title = {Operator algebras on singular manifolds. I}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25011}, year = {1997}, language = {en} } @unpublished{SchulzeSterninShatalov1997, author = {Schulze, Bert-Wolfgang and Sternin, Boris and Shatalov, Victor}, title = {On the index of differential operators on manifolds with conical singularities}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-24965}, year = {1997}, abstract = {The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.}, language = {en} } @unpublished{SchulzeSterninShatalov1997, author = {Schulze, Bert-Wolfgang and Sternin, Boris and Shatalov, Victor}, title = {Nonstationary problems for equations of Borel-Fuchs type}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-24973}, year = {1997}, abstract = {In the paper, the nonstationary problems for equations of Borel-Fuchs type are investigated. The asymptotic expansion are obtained for different orders of degeneration of operators in question. The approach to nonstationary problems based on the asymptotic theory on abstract algebras is worked out.}, language = {en} } @unpublished{SchulzeSterninShatalov1997, author = {Schulze, Bert-Wolfgang and Sternin, Boris and Shatalov, Victor}, title = {Operator algebras on singular manifolds. IV, V}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25062}, year = {1997}, language = {en} } @unpublished{SchulzeSterninShatalov1997, author = {Schulze, Bert-Wolfgang and Sternin, Boris and Shatalov, Victor}, title = {On general boundary value problems for elliptic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25138}, year = {1997}, abstract = {We construct a theory of general boundary value problems for differential operators whose symbols do not necessarily satisfy the Atiyah-Bott condition [3] of vanishing of the corresponding obstruction. A condition for these problems to be Fredholm is introduced and the corresponding finiteness theorems are proved.}, language = {en} } @unpublished{SchulzeTarkhanov1999, author = {Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {Ellipticity and parametrices on manifolds with caspidal edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25411}, year = {1999}, language = {en} }