Search results for "basis"
showing 10 items of 760 documents
Isotropic stochastic flow of homeomorphisms on Rd associated with the critical Sobolev exponent
2008
Abstract We consider the critical Sobolev isotropic Brownian flow in R d ( d ≥ 2 ) . On the basis of the work of LeJan and Raimond [Y. LeJan, O. Raimond, Integration of Brownian vector fields, Ann. Probab. 30 (2002) 826–873], we prove that the corresponding flow is a flow of homeomorphisms. As an application, we construct an explicit solution, which is also unique in a certain space, to the stochastic transport equation when the associated Gaussian vector fields are divergence free.
Generalized Riesz systems and orthonormal sequences in Krein spaces
2018
We analyze special classes of bi-orthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with generalized Riesz systems. In this way, the notion of the first/second type sequences is introduced and studied. We also discuss their relevance in some concrete quantum mechanical system driven by manifestly non self-adjoint Hamiltonians.
Gabor-like systems in ${cal L}^2({bf R}^d)$ and extensions to wavelets
2008
In this paper we show how to construct a certain class of orthonormal bases in starting from one or more Gabor orthonormal bases in . Each such basis can be obtained acting on a single function with a set of unitary operators which operate as translation and modulation operators in suitable variables. The same procedure is also extended to frames and wavelets. Many examples are discussed.
Hamiltonians defined by biorthogonal sets
2017
In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not self-adjoint but still may have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed is some previous papers. However, in many physical models, one has to deal not with o.n. bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of $\mat…
Unitary Representations of Quantum Superpositions of two Coherent States and beyond
2013
The construction of a class of unitary operators generating linear superpositions of generalized coherent states from the ground state of a quantum harmonic oscillator is reported. Such a construction, based on the properties of a new ad hoc introduced set of hermitian operators, leads to the definition of new basis in the oscillator Hilbert space, extending in a natural way the displaced Fock states basis. The potential development of our method and our results are briefly outlined.
Quantum Walk Search on Johnson Graphs
2016
The Johnson graph $J(n,k)$ is defined by $n$ symbols, where vertices are $k$-element subsets of the symbols, and vertices are adjacent if they differ in exactly one symbol. In particular, $J(n,1)$ is the complete graph $K_n$, and $J(n,2)$ is the strongly regular triangular graph $T_n$, both of which are known to support fast spatial search by continuous-time quantum walk. In this paper, we prove that $J(n,3)$, which is the $n$-tetrahedral graph, also supports fast search. In the process, we show that a change of basis is needed for degenerate perturbation theory to accurately describe the dynamics. This method can also be applied to general Johnson graphs $J(n,k)$ with fixed $k$.
On the sign recovery by LASSO, thresholded LASSO and thresholded Basis Pursuit Denoising
2020
Basis Pursuit (BP), Basis Pursuit DeNoising (BPDN), and LASSO are popular methods for identifyingimportant predictors in the high-dimensional linear regression model Y = Xβ + ε. By definition, whenε = 0, BP uniquely recovers β when Xβ = Xb and β different than b implies L1 norm of β is smaller than the L1 norm of b (identifiability condition). Furthermore, LASSO can recover the sign of β only under a much stronger irrepresentability condition. Meanwhile, it is known that the model selection properties of LASSO can be improved by hard-thresholdingits estimates. This article supports these findings by proving that thresholded LASSO, thresholded BPDNand thresholded BP recover the sign of β in …
A new tetranuclear copper(I) complex based on allyl(5-phenyl-1,3,4-thiadiazol-2-yl)azanide ligand: Synthesis and structural characterization
2015
Abstract By means of alternating current electrochemical technique a new tetranuclear crystalline copper(I) complex [Cu I 4 ( L − ) 4 ] ( L − – allyl(5-phenyl-1,3,4-thiadiazol-2-yl)azanide ion) has been obtained and characterized by X-ray single crystal diffraction ( Sp. gr. I 4 1 / a ) and Raman spectroscopy. The metal center adopts linear arrangement, composed of one thiadiazole N atom from the one L − anion and one azanide N atom of the other L − ligand. A bridged Cu atoms stitch four L − ligands into the firstly observed tetranuclear copper(I) azanide complex with intramolecular Cu(I)⋯Cu(I) interactions at the distance of 2.7451(6) A. Molecular structure and Raman spectrum of the compo…
Gold(I) Complexes of Ferrocenyl Polyphosphines: Aurophilic Gold Chloride Formation and Phosphine-Concerted Shuttling of a Dinuclear [ClAu···AuCl] Fra…
2016
International audience; A smart steric control of the metallocene backbone in bis- and poly(phosphino)ferrocene ligands favors intramolecular aurophilic interactions between [AuCl] fragments in polynuclear gold(I) complexes. We synthesized and characterized by multinuclear NMR and X-ray diffraction analysis mono-, di-, and polynuclear gold complexes of constrained ferrocenyl diphosphines, which bear either bulky tert-butyl groups or more flexible siloxane substituents at the cyclopentadienyl rings. The complexes meso-1,1'-bis-(diphenylphosphino)-3,3'-di-tert-butylferrocene (4-m), rac-1,1'-bis[bis-(5-methy1-2-furyl)phosphino]-3,3'-di-tert-butyfferrocene (5-r), and rac-1,1'-bis ( diphenylphos…
Finitary formal topologies and Stone’s representation theorem
2008
AbstractWe study the concept of finitary formal topology, a point-free version of a topological space with a basis of compact open subsets. The notion of finitary formal topology is defined from the perspective of the Basic Picture (introduced by the second author) and thus it is endowed with a binary positivity relation. As an application, we prove a constructive version of Stone’s representation theorem for distributive lattices. We work within the framework of a minimalist foundation (as proposed by Maria Emilia Maietti and the second author). Both inductive and co-inductive methods are used in most proofs.