Search results for "basis"
showing 10 items of 760 documents
Non-discrimination on the basis of nationality
2022
In this publication, I discuss non-discrimination on the basis of nationality
Riesz-like bases in rigged Hilbert spaces
2015
The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space $\D[t] \subset \H \subset \D^\times[t^\times]$. A Riesz-like basis, in particular, is obtained by considering a sequence $\{\xi_n\}\subset \D$ which is mapped by a one-to-one continuous operator $T:\D[t]\to\H[\|\cdot\|]$ into an orthonormal basis of the central Hilbert space $\H$ of the triplet. The operator $T$ is, in general, an unbounded operator in $\H$. If $T$ has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces.
Evolution semigroups and time operators on Banach spaces
2010
AbstractWe present new general methods to obtain shift representation of evolution semigroups defined on Banach spaces. We introduce the notion of time operator associated with a generalized shift on a Banach space and give some conditions under which time operators can be defined on an arbitrary Banach space. We also tackle the problem of scaling of time operators and obtain a general result about the existence of time operators on Banach spaces satisfying some geometric conditions. The last part of the paper contains some examples of explicit constructions of time operators on function spaces.
Strongly extreme points and approximation properties
2017
We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but …
Polyhedrality and decomposition
2018
Abstract The aim of this note is to present two results that make the task of finding equivalent polyhedral norms on certain Banach spaces, having either a Schauder basis or an uncountable unconditional basis, easier and more transparent. The hypotheses of both results are based on decomposing the unit sphere of a Banach space into countably many pieces, such that each one satisfies certain properties. Some examples of spaces having equivalent polyhedral norms are given.
Forward Kinematic Modelling with Radial Basis Function Neural Network Tuned with a Novel Meta-Heuristic Algorithm for Robotic Manipulators
2022
The complexity of forward kinematic modelling increases with the increase in the degrees of freedom for a manipulator. To reduce the computational weight and time lag for desired output transformation, this paper proposes a forward kinematic model mapped with the help of the Radial Basis Function Neural Network (RBFNN) architecture tuned by a novel meta-heuristic algorithm, namely, the Cooperative Search Optimisation Algorithm (CSOA). The architecture presented is able to automatically learn the kinematic properties of the manipulator. Learning is accomplished iteratively based only on the observation of the input–output relationship. Related simulations are carried out on a 3-Degrees…
Full configuration interaction calculation of the low lying valence and Rydberg states of BeH
2007
The all-electron full configuration interaction (FCI) vertical excitation energies for some low lying valence and Rydberg excited states of BeH are presented in this article. A basis set of valence atomic natural orbitals has been augmented with a series of Rydberg orbitals that have been generated as centered onto the Be atom. The resulting basis set can be described as 4s2p1d/2s1p (Be/H) + 4s4p3d. It allows to calculate Rydberg states up to n= {3,4,5} of the s, p, and d series of Rydberg states. The FCI vertical ionization potential for the same basis set and geometry amounts to 8.298 eV. Other properties such as FCI electric dipole and quadrupole moments and FCI transition dipole and qua…
Multiconfigurational second-order perturbation calculation of the electronic absorption spectrum of trisilane, Si3H8
2002
The low-lying singlet excited states of trisilane have been computed using the complete active space self-consistent field (CASSCF) method, second-order perturbation theory (CASPT2), and the extended multistate CASPT2 (MS-CASPT2) approach, and a generally contracted basis set of atomic natural orbitals (ANOs) including Rydberg functions. The ground state structure was obtained from an ab initio optimization at the second-order M⊘ller-Plesset perturbation theory (MP2) level using Dunning's correlation-consistent triple-zeta basis set (cc-pVTZ) and agrees well with experiment. The calculation of the electronic transitions included the lower valence excited states and two Rydberg series conver…
A theoretical study of the electronic spectrum of bithiophene
1995
The electronic spectrum of bithiophene in the energy range up to 6.0 eV has been studied using multiconfigurational second order perturbation theory (CASPT2) and a basis set of ANO type, with split valence quality and including polarization functions on all heavy atoms. Calculations were performed at a planar (trans) and twisted geometry. The calculated ordering of the excited singlet states is 1Bu, 1Bu, 1Ag, 1Ag, and 1Bu with 0–0 transition energies: 3.88, 4.15, 4.40, 4.71, and 5.53 eV, respectively. The first Rydberg transition (3s) has been found at 5.27 eV. The results have been used in aiding the interpretation of the experimental spectra, and in cases where a direct comparison is poss…
On the dissociation energy of Cu2 and CuH using a differential correlation approach
1990
The energy contributions involved in the “dissociation-consistent configuration interaction” (DCCI) scheme proposed by Goddard and co-workers plus the dispersion effects associated with the valence MOs adjacent to the bond (DISP) are evaluated for the Cu2 and CuH systems, following the “configuration interaction by perturbation with multiconfigurational zeroth-order wavefunction selected by iterative process” (CIPSI) algorithm. Using a relativistic pseudopotential to represent the neon core of copper, and a flexible basis set for the valence shell, the results obtained with the CIPSI/DCCI (+ DISP) approach shows a good agreement with previous theoretical results employing basis sets of simi…