Let L(H) denote the C∗-algebra of adjointable operators on a Hilbert C∗-module H. In this paper, we introduce generalized Cauchy–Schwarz inequality for in L(H). More precisely, an operator A∈L(H) is said to satisfy if there exists ν∈(0,1) such that ‖⟨Ax,y⟩‖≤(‖Ax‖‖y‖)ν(‖Ay‖‖x‖)1−ν(x,y∈H).We investigate various properties which inequality. particular, prove A satisfies has polar decomposition, th...

We demonstrate the generation of high-quality narrowband biphotons from a Doppler-broadened hot rubidium atomic vapor cell. Choosing a double-K atomic energy level scheme for optimizing both spontaneous four-wave mixing nonlinear parametric interaction and electromagnetically induced transparency (EIT), we achieve a biphoton spectral brightness as high as 14 000 s MHz . Meanwhile, we apply a sp...

This paper follows the recent discussion on the sparse solution recovery with quasi-norms lq, q ∈ (0, 1) when the sensing matrix possesses a Restricted Isometry Constant δ2k (RIC). Our key tool is an improvement on a version of “the converse of a generalized Cauchy-Schwarz inequality” extended to the setting of quasi-norm. We show that, if δ2k ≤ 1/2, any minimizer of the lq minimization, at lea...

We present optimal order preconditioners for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems. A specific assembling process is proposed which allows us to use the hierarchy of geometrically nested meshes. We consider two variants of hierarchical splittings and study the angle between the resulting subspaces. Applying the corresponding two-l...

Every 3-graph in which no four vertices are independent and no four vertices span precisely three edges must have edge density ≥ 4/9(1 − o(1)). This bound is tight. The proof is a rather elaborate application of Cauchy–Schwarz-type arguments presented in the framework of flag algebras. We include further demonstrations of this method by re-proving a few known tight results about hypergraph Turá...

et strictement positive : x 6= 0⇒ 〈x | x〉 > 0 (4) Finalement, H, muni de la norme ‖x‖ := 〈x | x〉 doit être un espace de Banach, i.e., être complet. La condition (3) est équivalente à 〈x | x〉 ∈ R, en effet 4〈x | y〉 = 〈x+ y | x+ y〉 − 〈x− y | x− y〉 + i〈x+ iy | x+ iy〉 − i〈x− iy | x− iy〉. L’équation (3) est donc conséquence de (4). Venons-en à la norme ; le point de départ est l’inégalité de Cauchy-...

Single photons are generated from an ensemble of cold Cs atoms via the protocol of Duan et al. [Nature (London), ()]]. Conditioned upon an initial detection from field 1 at 852 nm, a photon in field 2 at 894 nm is produced in a controlled fashion from excitation stored within the atomic ensemble. The single-quantum character of field 2 is demonstrated by the violation of a Cauchy-Schwarz inequa...

1. INTRODUCTION. Hypergraphs. Information Theory. Cauchy-Schwarz. It seems reasonable to assume that most mathematicians would be puzzled to find these three terms as, say, key words for the same mathematical paper. (Just in case this puzzlement is a result of being unfamiliar with the term " hypergraph " : a hypergraph is nothing other than a family of sets, and will be defined formally later....