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Operator (computer programming)

About: Operator (computer programming) is a research topic. Over the lifetime, 40896 publications have been published within this topic receiving 671452 citations. The topic is also known as: operator symbol & operator name.


Papers
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Journal ArticleDOI
TL;DR: The regression problem in learning theory is investigated with least square Tikhonov regularization schemes in reproducing kernel Hilbert spaces (RKHS) and the sampling operator is applied to the error analysis in both the RKHS norm and the L2 norm.
Abstract: The regression problem in learning theory is investigated with least square Tikhonov regularization schemes in reproducing kernel Hilbert spaces (RKHS). We follow our previous work and apply the sampling operator to the error analysis in both the RKHS norm and the L2 norm. The tool for estimating the sample error is a Bennet inequality for random variables with values in Hilbert spaces. By taking the Hilbert space to be the one consisting of Hilbert-Schmidt operators in the RKHS, we improve the error bounds in the L2 metric, motivated by an idea of Caponnetto and de Vito. The error bounds we derive in the RKHS norm, together with a Tsybakov function we discuss here, yield interesting applications to the error analysis of the (binary) classification problem, since the RKHS metric controls the one for the uniform convergence.

598 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal, called a ''symmetric, informationally complete'' POVM (SIC-POVM).
Abstract: We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

597 citations

MonographDOI
10 Mar 2010
TL;DR: The main subject of this book is applications of methods of scattering theory to differential operators, primarily the Schrodinger operator as discussed by the authors, and it is based on graduate courses taught by the author at Saint-Petersburg (Russia) and Rennes (France) Universities.
Abstract: The main subject of this book is applications of methods of scattering theory to differential operators, primarily the Schrodinger operator There are two different trends in scattering theory for differential operators The first one relies on the abstract scattering theory The second one is almost independent of it In this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation In this book both of these trends are presented The first half of this book begins with the summary of the main results of the general scattering theory of the previous book by the author, ""Mathematical Scattering Theory: General Theory"", American Mathematical Society, 1992 The next three chapters illustrate basic theorems of abstract scattering theory, presenting, in particular their applications to scattering theory of perturbations of differential operators with constant coefficients and to the analysis of the trace class method In the second half of the book direct methods of scattering theory for differential operators are presented After considering the one-dimensional case, the author returns to the multi-dimensional problem and discusses various analytical methods and tools appropriate for the analysis of differential operators, including, among others, high- and low-energy asymptotics of the Green function, the scattering matrix, ray and eikonal explansions The book is based on graduate courses taught by the author at Saint-Petersburg (Russia) and Rennes (France) Universities and is oriented towards a reader interested in studying deep aspects of scattering theory (for example, a graduate student in mathematical physics)

597 citations

01 Jul 2004
TL;DR: In this paper, a dilatation operator for planar 4D conformal quantum field theories is presented, based on the symmetry algebra and structural properties of Feynman diagrams.
Abstract: In this work we review recent progress in four-dimensional conformal quantum field theories and scaling dimensions of local operators. Here we consider the example of maximally supersymmetric gauge theory and present techniques to derive, investigate and apply the dilatation operator which measures the scaling dimensions. We construct the dilatation operator by purely algebraic means: Relying on the symmetry algebra and structural properties of Feynman diagrams we are able to bypass involved, higher-loop field theory computations. In this way we obtain the complete one-loop dilatation operator and the planar, three-loop deformation in an interesting subsector. These results allow us to address the issue of integrability within a planar four-dimensional gauge theory: We prove that the complete dilatation generator is integrable at one loop and present the corresponding Bethe ansatz. We furthermore argue that integrability extends to three loops and beyond. Assuming that it holds indeed, we finally construct a novel spin chain model at five loops and propose a Bethe ansatz which might be valid at arbitrary loop order!

591 citations

Journal ArticleDOI
TL;DR: In this paper, it is shown that there are two natural regularization schemes, each of which leads to a well-defined operator, which can be completely specified by giving their action on states labelled by graphs.
Abstract: A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics.

589 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202236
20212,210
20202,380
20192,310
20182,164
20171,834