Abstract: Periodic solutions for a second order semilinear Volterra equation metric and generalized projection operators in Banach spaces - properties and applications the rate of asymptotic regularity is 0(1/square root of n) global existence for second order functional differential equations iterative process for finding common fixed points of nonlinear mappings regularity for semilinear abstract Cauchy problems the KdV equation via semigroups a degree for maximal monotone operators on subjectivity of perturbed nonlinear m-accretive operators the fixed point property and mappings which are eventually nonexpansive approximation-solvability of semilinear equations and applications on the approximation of zeros for locally accretive operators quasimonotonicity and the Leray-Lions condition on nonlinear ill-posed problems II - monotone operator equations and monotone variational inequalities a classical hypergeometric proof of a transformation found by Ronald Bruck periodic solutions for nonlinear 2-D wave equations the existence of resolvents of holomorphic generators in Banach spaces existence of solutions to partial functional differential equations with delay zeros of weakly inward accretive mappings via A-proper maps nonlinear wave equations with asymptotically monotone damping.

Abstract: 1 The Spectrum of Linear Operators and Hilbert Spaces.- 2 The Geometry of a Hilbert Space and Its Subspaces.- 3 Exponential Decay of Eigenfunctions.- 4 Operators on Hilbert Spaces.- 5 Self-Adjoint Operators.- 6 Riesz Projections and Isolated Points of the Spectrum.- 7 The Essential Spectrum: Weyl's Criterion.- 8 Self-Adjointness: Part 1. The Kato Inequality.- 9 Compact Operators.- 10 Locally Compact Operators and Their Application to Schrodinger Operators.- 11 Semiclassical Analysis of Schrodinger Operators I: The Harmonic Approximation.- 12 Semiclassical Analysis of Schrodinger Operators II: The Splitting of Eigenvalues.- 13 Self-Adjointness: Part 2. The Kato-Rellich Theorem 131.- 14 Relatively Compact Operators and the Weyl Theorem.- 15 Perturbation Theory: Relatively Bounded Perturbations.- 16 Theory of Quantum Resonances I: The Aguilar-Balslev-Combes-Simon Theorem.- 17 Spectral Deformation Theory.- 18 Spectral Deformation of Schrodinger Operators.- 19 The General Theory of Spectral Stability.- 20 Theory of Quantum Resonances II: The Shape Resonance Model.- 21 Quantum Nontrapping Estimates.- 22 Theory of Quantum Resonances III: Resonance Width.- 23 Other Topics in the Theory of Quantum Resonances.- Appendix 1. Introduction to Banach Spaces.- A1.1 Linear Vector Spaces and Norms.- A1.2 Elementary Topology in Normed Vector Spaces.- A1.3 Banach Spaces.- A1.4 Compactness.- 1. Density results.- 2. The Holder Inequality.- 3. The Minkowski Inequality.- 4. Lebesgue Dominated Convergence.- Appendix 3. Linear Operators on Banach Spaces.- A3.1 Linear Operators.- A3.2 Continuity and Boundedness of Linear Operators.- A3.3 The Graph of an Operator and Closure.- A3.4 Inverses of Linear Operators.- A3.5 Different Topologies on L(X).- Appendix 4. The Fourier Transform, Sobolev Spaces, and Convolutions.- A4.1 Fourier Transform.- A4.2 Sobolev Spaces.- A4.3 Convolutions.- References.

Abstract: The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.

Abstract: A generalized “measure of distance” defined by . is generated from any member f of the class of Bregman functions. Although it is not. technically speaking. a distance function. it has been used in the past to define and study projection operators. In this paper we give new definitions of paracont ractions. convex combinations. and firmly nonexpansivc operators. based on Df (x,y), and study sequential and simultaneous iterative algorithms employing them for the solution of the problem of finding a common asymptotic fixed point of a family of operators. Applications to the convex feasibility problem. to optimization and to monotone operator theory are also included.

Abstract: Principles of Construction of Adjoint Operators in Non-Linear Problems Dual Spaces and Adjoint Operators Construction of Adjoint Operators Based on Using the Lagrange Identity Definition of Adjoint Operators Based on Using Taylor's Formula Operators of the Class D and their Adjoint Operators Properties of Adjoint Operators Constructed on the Basis of Various Principles General Properties of Main and Adjoint Operators Corresponding to Non-Linear Operators Properties of Operators of the Class D Properties of Adjoint Operators Constructed with the Use of the Taylor Formula Solvability of Main and Adjoint Equations in Non-Linear Problems Main and Adjoint Equations. Problems Solvability of the Equation F(u) = y Solvability of the Equation A(u)v = y Solvability of the Equation A(u)v = y Solvability of the Equation A*(u)w = p Solvability of the Equation A*(u)w = p Transformation Groups, Conservation Laws and Construction of the Adjoint Operators in Non-Linear Problems Definitions. Non-Linear Equations and Operators. Conservation Laws Transformation of Equations Adjoint Equations Relationship between Different Adjoint Operators General Remarks on Constructing the Adjoint Equations with the Use of the Lie Groups and Conservation Laws Construction of Adjoint Operators with Prescribed Properties The Noether Theorem, Conservation Laws and Adjoint Operators On Some Applications of Adjoint Equations Perturbation Algorithms in Non-Linear Problems Perturbation Algorithms for Original Non-Linear Equations and Equations Involving Adjoint Operators Perturbation Algorithms for Non-Linear Functionals Based on Using Main and Adjoint Equations Spectral Method in Perturbation Algorithms Justification of the N-th Order Perturbation Algorithms Convergence Rate Estimates for Perturbation Algorithms. Comparison with the Successive Approximation Method Justification of Perturbation Algorithms in Quasi-Linear Elliptic Problems Adjoint Equations and the N-th Order Perturbation Algorithms in Non-Linear Problems of Transport Theory Some Problems of Transport Theory The N-th Order Perturbation Algorithms for an Eigenvalue Problem A Problem of Control and its Approximate Solution with the Use of Perturbation Algorithms Investigation and Approximate Solution of a Non-Linear Problem for the Transport Equation Adjoint Equations and Perturbation Algorithms for a Quasilinear Equation of Motion Statement of the Problem. Basic Assumptions. Operator Formulation Transformation of the Problem. Properties of the Non-Linear Operator Adjoint Equation An Algorithm for Computing the Functional The Problem on Chemical Exchange Processes Adjoint Equations and Perturbation Algorithms for a Non-Linear Mathematical Model of Mass Transfer in Soil Mathematical Models of Mass Transfer in Soil Formulation of a Non-Linear Mathematical Model Transformation of the Problem. Properties of the Non-Linear Operator Perturbation Algorithm. Adjoint Equation Approximate Solution of the Problem on Finding an Effective Dispersion Coefficient An Algorithm for Solving the Problem Applications of Adjoint Equations in Science and Technology Adjoint Equations in Data Assimilation Problems Application of Adjoint Equations for Solving the Problem of Liquid Boundary Conditions in Hydrodynamics Shape Optimization Using Adjoint Equation Approaches Global Transport of Pollutants Problems of Climate Change Sensitivity in Various Regions of the World

Abstract: in all dimensions, again under the assumption Q e L log L. It is conceivable that a variant of the arguments in [3] for the maximal operator could also work for the singular integral operator; in fact, in unpublished work, the authors of [3] obtained a weak type (1, 1) inequality in dimension d < 7. However their arguments if applied to the singular integral operator lead to substantial technical difficulties and no

Abstract: Let k[D] be the ring of differential operators with coefficients in a differential fieldk. We say that an elementL ofk[D] isreducible ifL=L 1·L 2 forL 1,L 2gEk[D],L 1,L 2∉k. We show that for a certain class of differential operators (completely reducible operators) there exists a Berlekamp-style algorithm for factorization. Furthermore, we show that operators outside this class can never be irreducible and give an algorithm to test if an operator belongs to the above class. This yields a new reducibility test for linear differential operators. We also give applications of our algorithm to the question of determining Galois groups of linear differential equations.

Abstract: The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, corresponding to the two filtrations by interior order and vanishing degree at the boundary, together with the exterior derivations of the algebra. This leads to an index formula which is a pseudodifferential extension of that of Atiyah, Patodi and Singer for Dirac operators; together with a symbolic term it involves the `eta' invariant on the suspended algebra over the boundary previously introduced by the first author.

Abstract: 0. Introductory Material.- 1. Functional Operators.- 1. Weighted Shift Operators. Boundedness Conditions.- 2. The Operator Approach. Model Examples.- 3. Discussion of the Model Examples Discontinuity of the Spectral Radius.- 4. Operators Generated by a Finite Transformation Group.- 5. Spectral Radius of a Weighted Shift Operator.- 2. Banach Algebras Generated by Functional Operators.- 6. Extension of Operator Algebras by Operators that Generate Automorphisms.- 7. The Isomorphism Theorem for C* -Algebras Generated by Dynamical Systems.- 8. Corollaries of the Isomorphism Theorem.- 3. Invertibility Conditions for Functional Operators. L2-Theory.- 9. Characterization of Invertible Operators by Means of Hyperbolic Linear Extensions.- 10. Discretization as an Orbital Approach to Study Invertibility.- 11. Operators with a Convex Rationally Independent System of Shifts.- 12. Algebras of Type C*(A, G, Tg) in the Case of an Arbitrary Group Action.- 4. Functional Operators in Some Special Function Spaces.- 13. Weighted Shift Operators in Spaces of Smooth Functions.- 14. Functional Equations in a Space of Periodic Distributions.- 5. Applications to Some Classes of Equations and Boundary Value Problems.- 15. Functional-Differential Equations of Neutral Type.- 16. The Symbol of a Functional-Partial Differential Operator.- 17. Nonlocal Boundary Value Problems for Elliptic Equations.- 18. Equations with Periodic and Almost Periodic Coefficients.- 19. Boundary Value Problems with Data on the Entire Boundary for the Equation of a Vibrating String.- 20. Index Formulas.- Comments and Bibliographic Information.- References.

Abstract: We analyse the structure of the first-order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules, thereby generalizing several proposals.

Abstract: Distribution theory for discontinuous test functions and differential operators with generalized coefficients

Abstract: Local intrinsic dimensionality is shown to be an elementary structural property of multidimensional signals that cannot be evaluated using linear filters We derive a class of polynomial operators for the detection of intrinsically 2-D image features like curved edges and lines, junctions, line ends, etc Although it is a deterministic concept, intrinsic dimensionality is closely related to signal redundancy since it measures how many of the degrees of freedom provided by a signal domain are in fact used by an actual signal Furthermore, there is an intimate connection to multidimensional surface geometry and to the concept of 'Gaussian curvature' Nonlinear operators are inevitably required for the processing of intrinsic dimensionality since linear operators are, by the superposition principle, restricted to OR-combinations of their intrinsically 1-D eigenfunctions The essential new feature provided by polynomial operators is their potential to act on multiplicative relations between frequency components Therefore, such operators can provide the AND-combination of complex exponentials, which is required for the exploitation of intrinsic dimensionality Using frequency design methods, we obtain a generalized class of quadratic Volterra operators that are selective to intrinsically 2-D signals These operators can be adapted to the requirements of the signal processing task For example, one can control the "curvature tuning" by adjusting the width of the stopband for intrinsically 1-D signals, or the operators can be provided in isotropic and in orientation-selective versions We first derive the quadratic Volterra kernel involved in the computation of Gaussian curvature and then present examples of operators with other arrangements of stop and passbands Some of the resulting operators show a close relationship to the end-stopped and dot-responsive neurons of the mammalian visual cortex

Abstract: Local quasinilpotence, cycles and invariant subspaces invariant mean value and harmonicity in Cartan and Siegel domains generalized de Leeuw theorems and extension theorems for weak type multipliers transference couples and their applications to convolution operators and maximal operators generalized radial limits associated with representing measures functional calculus for Hilbert space operators with bounded imaginary powers Machado's theorem and the abstract Bicho for some

Abstract: The aim of this paper is to introduce a new approach to the modal operators of necessity and possibility. This approach is based on the existence of two negations in certain lattices that we call bi-Heyting algebras. Modal operators are obtained by iterating certain combinations of these negations and going to the limit. Examples of these operators are given by means of graphs.

Abstract: Invited lecture at the XIV-th workshop on geometric methods in physics, Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global Toeplitz operators, approximation results for the quantum operators are shown. From them it follows that the quantum operators have the correct classical limit. A star product deformation of the Poisson algebra is constructed.

Abstract: The question of seminormality of tensor products of nonzero bounded linear operators on Hilbert spaces is investigated. It is shown that A 09 B is subnormal if and only if so are A and B.

Abstract: This second in the series of three volumes builds upon the basic theory of linear PDE given in volume 1, and pursues more advanced topics. Analytical tools introduced here include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. The book also develops basic differential geometrical concepts, centred about curvature. Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion.

Abstract: Sufficient conditions for the local null controllability of non-linear functional differential systems with unbounded linear operators in Banach space are established. The results are obtained using the semigroup of linear operators, fractional powers of operators, and the Schauder fixed-point theorem. Applications to parabolic differential systems are given.

Abstract: IfA i i=1, 2 are quasi-similarp-hyponormal operators such thatUi is unitary in the polar decompositionA i =U i |A i |, then σ(A1)=σ(A2) andσc(A1) = σe(A2). Also a Putnam-Fuglede type commutativity theorem holds for p-hyponomral operators.

Abstract: Let X be a real Banach space and G a bounded, open and convex subset of X. The solvability of the fixed point problem (*) Tx + Cx 3 x in D(T) n G is considered, where T . X D D(T) -* 2X is a possibly discontinuous m-dissipative operator and C: G -X is completely continuous. It is assumed that X is uniformly convex, D(T) n G $& 0 and (T + C)(D(T) n OG) C G. A result of Browder, concerning single-valued operators T that are either uniformly continuous or continuous with X* uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let IF {= 7Z+ -+ R-; /3(r) -O 0 as r -* oo}. The effect of a weak boundary condition of the type (u + Cx, x) > -/3(11xfl)flxl12 on the range of operators T + C is studied for m-accretive and maximal monotone operators T. Here, ,B E F, x C D(T) with sufficiently large norm and u C Tx. Various new eigenvalue results are given involving the solvability of Tx + ACx 3 0 with respect to (A, x) E (0, oo) x D(T). Several results do not require the continuity of the operator C. Four open problems are also given, the solution of which would improve upon certain results of the paper.

Abstract: In this work the connections between the fuzzy closure operators and the graded consequence relations are examined Namely, as it is well known, in the crisp case there is a complete equivalence between the notion of closure operator and the one of consequence relation. We extend this result by proving that the graded consequence relations are related to a particular class of fuzzy closure operators, namely the class of fuzzy closure operators that can be obtained by a chain of classical closure operators.

Abstract: If the scattering interaction in linear particle transport problems is highly peaked about zero momentum transfer, a common and often useful approximation is the replacement of the integral scattering operator with the differential Fokker-Planck operator. This operator involves a first derivative in energy and second derivatives in angle. In this paper, higher order Fokker-Planck scattering operators are derived, involving higher derivatives in both energy and angle. The applicability of these higher order differential operators to representative scattering kernels is discussed. It is shown that, depending upon the details of the scattering kernel in the integral operator, higher order Fokker-Planck approximations may or may not be valid. Even the classic low-order Fokker-Planck operator fails as an approximation for certain highly peaked scattering kernels. In particular, no Fokker-Planck operator is a valid approximation for scattering involving the widely used Henyey-Greenstein scattering kernel.

Abstract: This volume illuminates the depth and variety of analysis of partial differential equations. It begins with a survey on the use of semi-classical analysis and operator theory, before going on to present the perturbation theory for generators of Markov semigroups acting on Lp. The third section provides a self-contained introduction to continuous wavelet analysis, including its relations to function spaces and microlocal regularity. Finally, the text explores pseudo-differential analysis on singular configurations, with special emphasis on C-algebra techniques, Merlin operators, and analytical index formulas.

Abstract: The median operator is a nonlinear image transformation celebrated for its noise cleaning capacities. It treats the foreground and background of an image identically, i.e., it is self-dual. Unfortunately, the median operator has one major drawback: it is not idempotent. Even worse, subsequent iterations of a given image may lead to oscillations. This paper describes a general method for the construction of morphological operators which are self-dual. This construction is based upon the concept of a switch operator. Subsequently, the paper treats a class of operators, the so-called activity-extensive operators, which have the intriguing property that every sequence of iterates of a given image is pointwise monotone and therefore convergent. The underlying concept is that of the activity ordering. Every increasing, self-dual operator can be modified in such a way that it becomes activity-extensive. The sequence of iterates of this modification converges to a self-dual morphological filter.

Abstract: Foreword. 1. Nonstandard Theory of Vector Lattices A.G. Kusraev, S.S. Kutateladze. 2. Operator Classes Determined by Order Conditions A.V. Bukhvalov. 3. Stably Dominated and Stably Regulator Operators B.M. Makarov. 4. Integral Operators A.V. Bukhvalov, V.B. Korotkov, B.M. Makarov. 5. Disjointness Preserving Operators A.E. Gutman. Index.

Abstract: Systems of equations and matrices vector spaces linear operators inner product spaces diagnosable linear operators the structure of normal operators bilinear and quadradic forms small oscillations factorizations and canonical forms.