Biorthogonal system

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

Biorthogonal valence bond descriptions of electronic structure

Abstract: A valence bond approach based on nonorthogonal orbitals in a biorthogonal representation is presented. While the scheme suffers from the lack of a variational bound on the energy functional, it is shown that with a suitable optimization of the orbitals reliable molecular wavefunctions can be obtained. A review of the background theory is given emphasizing the similarity of this approach to the familiar spin-free unitary group formulation of quantum chemistry. The details of the computer implementation are discussed and the method is illustrated with model calculations on HF, H2O and F2O2.

Equilateral sets in infinite dimensional Banach spaces

Abstract: We show that every Banach space $X$ containing an isomorphic copy of $c_0$ has an infinite equilateral set and also that if $X$ has a bounded biorthogonal system of size $\alpha$ then it can be renormed so as to admit an equilateral set of equal size. If $K$ is any compact non metrizable space, then under a certain combinatorial condition on $K$ the Banach space $C(K)$ has an uncountable equilateral set.

RF simplex spread spectrum receiver and method with symbol deinterleaving prior to bit estimating

Abstract: A method and system are provided for the reception of a biorthogonally-modulated RF signal in a simplex free-space system. The method is found to substantially diminish the coupling between the error probabilities of those bits associated with signal selection and those bits associated with signal polarity in prior art biorthogonal receivers. By decoupling the error probabilities, the polarity information error rate is reduced and consequently the overall error rate of the receiver system is improved. The receiver system preserves soft-decision reliability information for the signal selection bits in a manner so that the information is insensitive to uncompensated amplitude fluctuations in the received signal. A symbol deinterleaver in the receiver is located between the correlators and the bit estimators. The method and system are applicable to all known biorthogonal signal sets but requires a specific interleaving structure in the corresponding transmitter system.

Linear independence of pseudo-splines

Abstract: In this paper, we show that the shifts of a pseudo-spline are lin- early independent. This is stronger than the (more obvious) statement that the shifts of a pseudo-spline form a Riesz system. In fact, the linear independence of a compactly supported (reflnable) function and its shifts has been studied in several areas of approximation and wavelet theory (see e.g. (4), (9), (10), (13), (14), (16) and (19)). Furthermore, the linear independence of the shifts of a pseudo-spline is a necessary and su-cient condition for the existence of a compactly supported function whose shifts form a biorthogonal dual system of the shifts of the pseudo-spline.

Wavelets from the Loop Scheme

Abstract: Anewwavelet-based geometric mesh compression algorithm was developed recently in the area of computer graphics by Khodakovsky, Schroder, and Sweldens in their interesting article [23]. The new wavelets used in [23] were designed from the Loop scheme by using ideas and methods of [26, 27], where orthogonal wavelets with exponential decay and pre-wavelets with compact support were constructed. The wavelets have the same smoothness order as that of the basis function of the Loop scheme around the regular vertices which has a continuous second derivative; the wavelets also have smaller supports than those wavelets obtained by constructions in [26, 27] or any other compactly supported biorthogonal wavelets derived from the Loop scheme (e.g., [11, 12]). Hence, the wavelets used in [23] have a good time frequency localization. This leads to a very efficient geometric mesh compression algorithm as proposed in [23]. As a result, the algorithm in [23] outperforms several available geometric mesh compression schemes used in the area of computer graphics. However, it remains open whether the shifts and dilations of the wavelets form a Riesz basis of L2(ℝ2). Riesz property plays an important role in any wavelet-based compression algorithm and is critical for the stability of any wavelet-based numerical algorithms. We confirm here that the shifts and dilations of the wavelets used in [23] for the regular mesh, as expected, do indeed form a Riesz basis of L2(ℝ2) by applying the more general theory established in this article.