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430
Weierstrass and Approximation Theory
"... We discuss and examine Weierstrass' main contributions to approximation theory. ..."
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Cited by 190 (9 self)
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We discuss and examine Weierstrass' main contributions to approximation theory.
Deformable Kernels for Early Vision
 IEEE Trans. Pattern Anal. Mach. Intell
, 1995
"... AbstractEarly vision algorithms often have a first stage of linearfiltering that ‘extracts ’ from the image information at multiple scales of resolution and multiple orientations. A common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsel ..."
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Cited by 145 (11 self)
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AbstractEarly vision algorithms often have a first stage of linearfiltering that ‘extracts ’ from the image information at multiple scales of resolution and multiple orientations. A common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs. This discretization produces anisotropies due to a loss of translation, rotation, and scalinginvariance that makes early vision algorithms less precise and more difficult to design. This need not be so: one can compute and store efficiently the response of families of linear filters defined on a continuum of orientations and scales. A technique is presented that allows 1) computing the best approximation of a given family using linear combinations of a small number of ‘basis ’ functions; 2) describing all finitedimensional families, i.e., the families of filters for which a finite dimensional representation is possible with no error. The technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations; the relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated. Experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multiscale 2D edgedetection kernels. The implementation issues are also discussed. Index TermsSteerable filters, wavelets, early vision, multiresolution image analysis, multirate filtering, deformable filters, scalespace I.
Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
 J. Amer. Math. Soc
, 1996
"... Recall that a subset of R n is called semialgebraic if it can be represented as a (finite) boolean combination of sets of the form {�α ∈ R n: p(�α) =0}, {�α ∈R n: q(�α)>0}where p(�x), q(�x) arenvariable polynomials with real coefficients. A map from R n to R m is called semialgebraic if its gr ..."
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Cited by 128 (4 self)
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Recall that a subset of R n is called semialgebraic if it can be represented as a (finite) boolean combination of sets of the form {�α ∈ R n: p(�α) =0}, {�α ∈R n: q(�α)>0}where p(�x), q(�x) arenvariable polynomials with real coefficients. A map from R n to R m is called semialgebraic if its graph, considered
Group and field definitions
 Journal of Formalized Mathematics
, 1989
"... Summary. The article contains exactly the same definitions of group and field as those in [4]. These definitions were prepared without the help of the definitions and properties of Nat and Real modes included in the MML. This is the first of a series of articles in which we are going to introduce th ..."
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Cited by 102 (1 self)
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Summary. The article contains exactly the same definitions of group and field as those in [4]. These definitions were prepared without the help of the definitions and properties of Nat and Real modes included in the MML. This is the first of a series of articles in which we are going to introduce the concept of the set of real numbers in a elementary axiomatic way.
Fading Memory and the Problem of Approximating Nonlinear Operators with Volterra Series
, 1985
"... Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any timeinvariant (TZ) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finitedimensional l ..."
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Cited by 97 (0 self)
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Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any timeinvariant (TZ) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finitedimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that any TZ operator with fading memory can be approximated (in our strong sense) by a nonlinear movingaverage operator. Some further discussion of the notion of fading memory is given.
Affine processes and applications in finance
 Annals of Applied Probability
, 2003
"... Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and OrnsteinUhlenbeck type processes. We show, and provide foundations for, a wide range of financial ap ..."
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Cited by 95 (5 self)
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Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and OrnsteinUhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.
SteerableScalable Kernels for Edge Detection and Junction Analysis
 Image and Vision Computing
, 1992
"... Families of kernels that are useful in a variety of early vision algorithms may be obtained by rotating and scaling in a continuum a `template' kernel. These multiscale multiorientation family may be approximated by linear interpolation of a discrete finite set of appropriate `basis' ker ..."
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Cited by 91 (1 self)
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Families of kernels that are useful in a variety of early vision algorithms may be obtained by rotating and scaling in a continuum a `template' kernel. These multiscale multiorientation family may be approximated by linear interpolation of a discrete finite set of appropriate `basis' kernels. A scheme for generating such a basis together with the appropriate interpolation weights is described. Unlike previous schemes by Perona, and Simoncelli et al. it is guaranteed to generate the most parsimonious one. Additionally, it is shown how to exploit two symmetries in edgedetection kernels for reducing storage and computational costs and generating simultaneously endstop and junctiontuned filters for free.
Global conservative solutions of the CamassaHolm equation
"... Abstract. This paper develops a new approach in the analysis of the CamassaHolm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new va ..."
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Cited by 90 (7 self)
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Abstract. This paper develops a new approach in the analysis of the CamassaHolm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time. The nonlinear partial differential equation ut − utxx + 3uux = 2uxuxx + uuxxx, t> 0, x ∈ IR, was derived by Camassa and Holm [CH] as a model for the propagation of shallow water waves, with
2005 Madden–Julian oscillation
 Rev. Geophys
"... Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This eoffprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to selfarchive your work, please use the accepted author’s versio ..."
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Cited by 77 (0 self)
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Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This eoffprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to selfarchive your work, please use the accepted author’s version for posting to your own website or your institution’s repository. You may further deposit the accepted author’s version on a funder’s repository at a funder’s request, provided it is not made publicly available until 12 months after publication.
Richardson: Cohomology and deformations in graded Lie algebras
 Bull. Amer. Math. Soc
, 1966
"... authors gave an outline of the similarities between the deformations of complexanalytic structures on compact manifolds on one hand, and the deformations of associative algebras on the other. The first theory had been stimulated in 1957 by a paper [7] by Nijenhuis ..."
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Cited by 74 (0 self)
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authors gave an outline of the similarities between the deformations of complexanalytic structures on compact manifolds on one hand, and the deformations of associative algebras on the other. The first theory had been stimulated in 1957 by a paper [7] by Nijenhuis