Narrow Results

Let $(R,𝔪)$ be a local commutative Noetherian ring and $I$ an ideal of $R.$ Assume that $M$ is a finitely generated $R$-module of finite projective dimension $p$ and $N$ is a finitely generated $R$-module of dimension $d.$ We prove that $H_{𝔪}^1(H_I^{p+d-1}(M,N))$ is Artinian. Moreover, if ${\mathrm{\text{Hom}}}_R(R/I,H_I^{p+d-1}(M,N))$ is finitely generated, then $H_I^{p+d-1}(M,N)$ is an $I$-cofinite $R$-module.

A pair of Mond–Weir type nondifferentiable second-order symmetric primal and dual problems in mathematical programming is formulated. Weak duality, strong duality, and converse duality theorems are established under η-pseudobonvexity assumptions. Symmetric minimax mixed integer primal and dual problems are also investigated. Moreover, the self duality theorem is also discussed.

In this paper, we establish a strong duality theorem for a pair of Mond–Weir type second-order nondifferentiable symmetric dual problems. This removes certain inconsistencies in some of the earlier results.

Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. After reviewing the concept of minimax state estimation, we use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.