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Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.


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TL;DR: In this article, a systematic and comprehensive analysis of the excited-state coupled-cluster (CC) methods for electronic excitation is given, extending and generalizing previous such studies.
Abstract: Single-reference coupled-cluster (CC) methods for electronic excitation are based on a biorthogonal representation (bCC) of the (shifted) Hamiltonian in terms of excited CC states, also referred to as correlated excited (CE) states, and an associated set of states biorthogonal to the CE states, the latter being essentially configuration interaction (CI) configurations. The bCC representation generates a non-hermitian secular matrix, the eigenvalues representing excitation energies, while the corresponding spectral intensities are to be derived from both the left and right eigenvectors. Using the perspective of the bCC representation, a systematic and comprehensive analysis of the excited-state CC methods is given, extending and generalizing previous such studies. Here, the essential topics are the truncation error characteristics and the separability properties, the latter being crucial for designing size-consistent approximation schemes. Based on the general order relations for the bCC secular matrix and the (left and right) eigenvector matrices, formulas for the perturbation-theoretical order of the truncation errors (TEO) are derived for energies, transition moments, and property matrix elements of arbitrary excitation classes and truncation levels. In the analysis of the separability properties of the transition moments, the decisive role of the so-called dual ground state is revealed. Due to the use of CE states, the bCC approach can be compared to so-called intermediate state representation (ISR) methods based exclusively on suitably orthonormalized CE states. As the present analysis shows, the bCC approach has decisive advantages over the conventional CI treatment, but also distinctly weaker TEO and separability properties in comparison to a full (and hermitian) ISR method.

38 citations

Journal ArticleDOI
TL;DR: The main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required by Dahmen and Schneider.
Abstract: A key ingredient of the construction of biorthogonal wavelet bases for Sobolev spaces on manifolds, which is based on topological isomorphisms is the Hestenes extension operator. Here we firstly investigate whether this particular extension operator can be replaced by another extension operator. Our main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required by Dahmen and Schneider. In the second part of this paper, we investigate and optimize the Hestenes extension operator. The results of the optimization process allow us to implement the construction of biorthogonal wavelets from Dahmen and Schneider. As an example, we illustrate a wavelet basis on the 2-sphere.

37 citations

Journal ArticleDOI
TL;DR: An algorithm for computing biorthogonal compactly supported dyadic wavelets related to the Walsh functions on the positive half-line ℝ+.
Abstract: In this paper, we describe an algorithm for computing biorthogonal compactly supported dyadic wavelets related to the Walsh functions on the positive half-line ℝ+. It is noted that a similar technique can be applied in very general situations, e.g., in the case of Cantor and Vilenkin groups. Using the feedback-based approach, some numerical experiments comparing orthogonal and biorthogonal dyadic wavelets with the Haar, Daubechies, and biorthogonal 9/7 wavelets are prepared.

37 citations

Journal ArticleDOI
TL;DR: This paper presents a 2-dimensional biorthogonal DWT processor design based on the residue number system that is able to fit into a 1,000,000-gate FPGA device and be able to complete a first level 2-D DWT decomposition of a 32/spl times/32-pixel image in 205 /spl mu/s.
Abstract: Discrete wavelet transform has been incorporated as part of the JPEG2000 image compression standard and is used in many consumer imaging products. This paper presents a 2-dimensional biorthogonal DWT processor design based on the residue number system. The symmetric extension scheme is employed to reduce distortion at image boundaries. Hardware complexity reduction and utilization improvement are achieved by hardware sharing. Our implementation results show that the design is able to fit into a 1,000,000-gate FPGA device and is able to complete a first level 2-D DWT decomposition of a 32/spl times/32-pixel image in 205 /spl mu/s.

37 citations

Posted Content
TL;DR: In this article, the authors use generating functions to express orthogonality relations in the form of $q$-beta integrals, which are then used as a weight function for a new set of orthogonal or biorthogonal relations.
Abstract: We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal

37 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202329
2022105
202155
202058
201960