We address the formulation of a scale-space theory for discrete signals. In one dimension it is possible to characterize the smoothing transformations completely and an exhaustive treatment is given, answering the following two main questions: 1. Which linear transformations remove structure in the sense that the number of local extrema (or zero-crossings) in the output signal does not exceed the number of local extrema (or zero-crossings) in the original signal? 2. How should one create a multi-resolution family of representations with the property that a signal at a coarser level of scale never contains more structure than a signal at a finer level of scale? We propose that there is only one reasonable way to define a scale-space for 1D discrete signals comprising a continuous scale parameter, namely by (discrete) convolution with the family of kernels T (n; t) = e I n (t), where I n are the modified Bessel functions of integer order. Similar arguments applied in the continuous case uniquely lead to the Gaussian kernel.

Abstract. The primary goal of this paper is to provide a rigorous theoretical justification of Cartan’s method of moving frames for arbitrary finite-dimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential operators on jet bundles, as well as equivalence, symmetry, and rigidity theorems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential invariants, explicit commutation formulae for the associated invariant differential operators, and a general classification theorem for syzygies of the higher order differentiated differential invariants. A variety of illustrative examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations.

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Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem. Fractional Cauchy problems are useful in physics to model anomalous di#usion.

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This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators A on a Banach space E with nonempty resolvent set for which the abstract Cauchy problem u'(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for every initial value x e D(A n). Investigating these operators we are led to the class of "integrated semigroups". Among others, this class contains the classes of strongly continuous semigroups and cosine families and the class of exponentially bounded distribution semigroups. The given characterizations of the generators of these integrated semigroups unify and generalize the classical characterizations of generators of strongly continuous semigroups, cosine families or exponentially bounded distribution semigroups. We indicate how integrated semigroups can be used studying second order Cauchy problems u"{t) — A\u'{t)- Aiu(t) = 0, operator valued equations U'(t) = A { U(t) + U(t)A2 and nonautonomous equations u'{t) = A(t)u(t). 1. Introduction. We

ABSTRACT. It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix V is concave (convex) with respect to V. Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from − ∞ to λ (from λ to +∞) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered.

It is well known that for infinite-dimensional systems, exponential stability is not necessarily determined by the location of spectrum.

Introduction Our object is to study domains which are the region of negative definiteness of an operator-valued Hermitian form defined on a space of operators and to investigate the biholomorphic linear fractional transformations between them. This is a unified setting in which to consider operator balls, operator half-planes, strictly J-contractive operators, strictly J-dissipative operators, etc., and the biholomorphic images of these domains under linear fractional transformations. Our approach is close in spirit to that of Potapov [28], Krein and Smuljan [27] and Smuljan [33]. At the same time, because we consider subspaces of operators, our circular domains include the matrix balls which E. Cartan [6] obtained as the classical bounded symmetric domains and they include the Siegel domains of genus 2 and 3 which Pyatetskii-Shapiro [29] associates with these domains as well as the infinite dimensional analogues of both types of domains given in [18] and [19]. Thus

Let X and Y be real normed linear spaces and let OE : X ! R be a non-negative function satisfying OE(x + y) OE(x) + kyk for all x; y 2 X . We show that there exist optimal constants c m;k such that if P : X ! Y is any polynomial satisfying kP (x)k OE(x) m for all x 2 X , then k D k P (x)k c m;k OE(x) m\Gammak whenever x 2 X and 0 k m. We obtain estimates for these constants and present applications to polynomials and multilinear mappings in normed spaces.