TY - INPR A1 - Lauter, Robert A1 - Seiler, Jörg T1 - Pseudodifferential analysis on manifolds with boundary - a comparison of b-calculus and cone algebra N2 - We establish a relation between two different approaches to a complete pseudodifferential analysis of totally characteristic or Fuchs type operators on compact manifolds with boundary respectively conical singularities: Melrose's (overblown) b-calculus and Schulze's cone algebra. Though quite different in their definition, we show that these two pseudodifferential calculi basically contain the same operators. T3 - Preprint - (1999) 27 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25611 ER - TY - INPR A1 - Kytmanov, Alexander A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Analytic representation of CR Functions on hypersurfaces with singularities N2 - We prove a theorem on analytic representation of integrable CR functions on hypersurfaces with singular points. Moreover, the behaviour of representing analytic functions near singular points is investigated. We are aimed at explaining the new effect caused by the presence of a singularity rather than at treating the problem in full generality. T3 - Preprint - (1999) 29 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25631 ER - TY - INPR A1 - Böckmann, Christine A1 - Sarközi, Janos T1 - The ill-posed inversion of multiwavelength lidar data by a hybrid method of variable projection N2 - 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. T3 - NLD Preprints - 53 KW - Ill-posed problem KW - inversion KW - variable projection method KW - multiwavelength Lidar KW - aerosol distribution Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14847 ER - TY - INPR A1 - Schrohe, Elmar A1 - Seiler, Jörg T1 - Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces N2 - 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). T3 - Preprint - (1999) 28 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25621 ER - TY - INPR A1 - Fedosov, Boris T1 - Pseudo-differential operators and deformation quantization N2 - Using the Riemannian connection on a compact manifold X, we show that the algebra of classical pseudo-differential operators on X generates a canonical deformation quantization on the cotangent manifold T*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization. T3 - Preprint - (1999) 32 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25651 ER - TY - INPR A1 - Sadykov, Timour T1 - Hypergeometric systems of differential equations and amoebas of rational functions N2 - 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]. T3 - Preprint - (1999) 33 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25665 ER - TY - INPR A1 - Schrohe, Elmar T1 - Noncommutative residues, Dixmier's Trace, and heat trace expansions on manifolds with boundary N2 - 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. T3 - Preprint - (1999) 13 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25486 ER - TY - INPR A1 - Nazaikinskii, Vladimir E. A1 - Sternin, Boris T1 - Surgery and the relative index in elliptic theory N2 - 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. T3 - Preprint - (1999) 17 KW - elliptic operators KW - index theory KW - surgery KW - relative index KW - manifold with singularities KW - boundary value problems Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25538 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Elliptic operators in even subspaces N2 - 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. T3 - Preprint - (1999) 10 KW - index of elliptic operators in subspaces KW - K-theory KW - eta invariant KW - Atiyah-Patodi-Singer theory KW - boundary value problems Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25461 ER - TY - INPR A1 - Witt, Ingo T1 - On the factorization of meromorphic Mellin symbols N2 - It is prooved that mermorphic, parameter-dependet elliptic Mellin symbols can be factorized in a particular way. The proof depends on the availability of logarithms of pseudodifferential operators. As a byproduct, we obtain a characterization of the group generated by pseudodifferential operators admitting a logarithm. The factorization has applications to the theory os pseudodifferential operators on spaces with conical singularities, e.g., to the index theory and the construction of various sub-calculi of the cone calculus. T3 - Preprint - (1999) 05 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25427 ER - TY - INPR A1 - Aizenberg, Lev A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Bohr phenomenon for elliptic equations N2 - In 1914 Bohr proved that there is an r ∈ (0, 1) such that if a power series converges in the unit disk and its sum has modulus less than 1 then, for |z| < r, the sum of absolute values of its terms is again less than 1. Recently analogous results were obtained for functions of several variables. The aim of this paper is to comprehend the theorem of Bohr in the context of solutions to second order elliptic equations meeting the maximum principle. T3 - Preprint - (1999) 18 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25547 ER - TY - INPR A1 - Duduchava, Roland T1 - The Green formula and layer potentials N2 - 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. T3 - Preprint - (1999) 26 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25601 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Nazaikinskii, Vladimir E. A1 - Sternin, Boris T1 - On the homotopy classification of elliptic operators on manifolds with singularities N2 - 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. T3 - Preprint - (1999) 21 KW - elliptic operators KW - homotopy classification KW - manifold with singularities KW - Atiyah-Singer theorem Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25574 ER - TY - INPR A1 - Shlapunov, Alexander T1 - Iterations of self-adjoint operators and their applications to elliptic systems N2 - Let Hsub(0), Hsub(1) be Hilbert spaces and L : Hsub(0) -> Hsub(1) be a linear bounded operator with ||L|| ≤ 1. Then L*L is a bounded linear self-adjoint non-negative operator in the Hilbert space Hsub(0) and one can use the Neumann series ∑∞sub(v=0)(I - L*L)v L*f in order to study solvability of the operator equation Lu = f. In particular, applying this method to the ill-posed Cauchy problem for solutions to an elliptic system Pu = 0 of linear PDE's of order p with smooth coefficients we obtain solvability conditions and representation formulae for solutions of the problem in Hardy spaces whenever these solutions exist. For the Cauchy-Riemann system in C the summands of the Neumann series are iterations of the Cauchy type integral. We also obtain similar results 1) for the equation Pu = f in Sobolev spaces, 2) for the Dirichlet problem and 3) for the Neumann problem related to operator P*P if P is a homogeneous first order operator and its coefficients are constant. In these cases the representations involve sums of series whose terms are iterations of integro-differential operators, while the solvability conditions consist of convergence of the series together with trivial necessary conditions. T3 - Preprint - (1999) 03 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25401 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Elliptic operators in odd subspaces N2 - 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. T3 - Preprint - (1999) 11 KW - index of elliptic operators in subspaces KW - K-theory KW - eta invariant KW - Atiyah-Patodi-Singer theory KW - boundary value problems Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25478 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization and the wave packet transform T3 - Preprint - (1999) 08 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25447 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Regularisation of mixed boundary problems N2 - We show an application of the spectral theorem in constructing approximate solutions of mixed boundary value problems for elliptic equations. T3 - Preprint - (1999) 09 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25454 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Savin, Anton A1 - Sternin, Boris T1 - Elliptic operators in subspaces and the eta invariant N2 - 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. T3 - Preprint - (1999) 14 KW - index of elliptic operators in subspaces KW - K-theory KW - eta-invariant KW - mod k index KW - Atiyah-Patodi-Singer theory Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25496 ER - TY - INPR A1 - Schrohe, Elmar A1 - Schulze, Bert-Wolfgang T1 - Edge-degenerate boundary value problems on cones N2 - 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. T3 - Preprint - (1999) 06 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25436 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A general index formula on tropic manifolds with conical points N2 - We solve the index problem for general elliptic pseudodifferential operators on toric manifolds with conical points. T3 - Preprint - (1999) 15 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25501 ER -