The Gibbs–Bogoliubov inequality states that the free energy of a system is always lower than calculated by trial function. In this study, we show counterpart holds on Nishimori line for Ising spin-glass models with Gaussian randomness. Our quenched using key component proof convexity pressure function \(\mathbb{E}[\log Z]\) respect to parameters along line, which differs from conventional inver...

We consider two different notions of convexity of metric spaces, namely (strict/uniform) ball convexity and (strict/uniform) distance convexity. Our main theorem states that (strict/uniform) distance convexity is preserved under a fairly general product construction, whereas we provide an example which shows that the same does not hold for (strict/uniform) ball convexity, not even when consider...

Abstract We study the sparse phase retrieval problem, recovering an s -sparse length- n signal from m magnitude-only measurements. Two-stage non-convex approaches have drawn much attention in recent studies. Despite non-convexity, many two-stage algorithms provably converge to underlying solution linearly when appropriately initialized. However, terms of sample complexity, bottleneck those with...

Problem of defining convexity of a digital region is considered. Definition of DL− (digital line) convexity is proposed, and it is shown to be stronger than the other two definitions, T−(triangle) convexity and L−(line) convexity. In attempt to connect the convexity of digital sets, to the fact that digital set can be a fuzzy set, the notion of convexity of the membership function is introduced...

In this paper we extend the result of [6] on the characteristic of convexity of Orlicz spaces to the more general case of Musielak-Orlicz spaces over a non-atomic measure space. Namely, the characteristic of convexity of these spaces is computed whenever the Musielak-Orlicz functions are strictly convex.

This note investigates the convexity of the indefinite joint numerical range of a tuple of Hermitian matrices in the setting of Krein spaces. Its main result is a necessary and sufficient condition for convexity of this set. A new notion of “quasi-convexity” is introduced as a refinement of pseudo-convexity.

This paper presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A → Tr eK+logA is concave, ...