This paper follows the recent discussion on the sparse solution recovery with quasi-norms `q, 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...

Quantum photonics offers much promise for the development of new technologies. The ability to control the interaction of light and matter at the level of single quantum excitations is a prerequisite for the construction of potentially powerful devices. Here we use the rotational levels of a room temperature ensemble of hydrogen molecules to couple two distinct optical modes at the single photon...

Exercise [2.1.5] Let > 0 and pick K = K( ) finite such that if k ≥ K then r(k) ≤ . Applying the Cauchy-Schwarz inequality for Xi −EXi and Xj −EXj we have that Cov(Xi, Xj) ≤ [Var(Xi)Var(Xj)] ≤ r(0) <∞ for all i, j. Thus, breaking the double sum in Var(Sn) = ∑n i,j=1 Cov(Xi, Xj) into {(i, j) : |i − j| < K} and {(i, j) : |i− j| ≥ K} gives the bound Var(Sn) ≤ 2Knr(0) + n . Dividing by n we see that...

This paper aims to present some inequalities for unitarily invariant norms. In section 2, we give a refinement of the Cauchy-Schwarz inequality for matrices. In section 3, we obtain an improvement for the result of Bhatia and Kittaneh [Linear Algebra Appl. 308 (2000) 203-211]. In section 4, we establish an improved Heinz inequality for the Hilbert-Schmidt norm. Finally, we present an inequality...

paper addresses the high dimensionality problem in blind source separation (BSS), where the number of sources is greater than two. Two pairwise iterative schemes are proposed to tackle this high dimensionality problem. The two pairwise schemesrealize non-parametric independent component analysis (ICA) algorithms based on a new high-performance Convex Cauchy–Schwarz Divergence (CCS-DIV). These t...

We prove one-sided universal bounds on coarsening rates for two kinds of mean field models of phase transitions, one with a coarsening rate l ∼ t and the other with l ∼ t. Here l is a characteristic length scale. These bounds are both proved by following a strategy developed by Kohn and Otto (Comm. Math. Phys. 229 (2002), 375-395). The l ∼ t rate is proved using a new dissipation relation which...

In a 1918 paper [Sch18], Schur proved a remarkable inequality that related group representations, Hermitian forms and determinants. He also gave concise necessary and sufficient conditions for equality. In [Mar64], Marcus gave a beautiful short proof of Schur’s inequality by applying the Cauchy-Schwarz inequality to symmetric tensors, but he did not discuss the case of equality. In [Wil69], Wil...

Exercise [2.1.5] Let > 0 and pick K = K( ) finite such that if k ≥ K then r(k) ≤ . Applying the Cauchy-Schwarz inequality for Xi −EXi and Xj −EXj we have that Cov(Xi, Xj) ≤ [Var(Xi)Var(Xj)] ≤ r(0) <∞ for all i, j. Thus, breaking the double sum in Var(Sn) = ∑n i,j=1 Cov(Xi, Xj) into {(i, j) : |i − j| < K} and {(i, j) : |i− j| ≥ K} gives the bound Var(Sn) ≤ 2Knr(0) + n . Dividing by n we see that...