Hadamard powers of rank two, doubly nonnegative matrices

The critical exponent conjecture for powers of doubly nonnegative matrices

Doubly non-negative matrices arise naturally in many setting including Markov random fields (positively banded graphical models) and in the convergence analysis of Markov chains. In this short note, we settle a recent conjecture by C.R. Johnson et al. [Linear Algebra Appl. 435 (2011)] by proving that the critical exponent beyond which all continuous conventional powers of $n$-by-$n$ doubly nonnegative matrices are doubly nonnegative is exactly $n-2$. We show that the conjecture follows immediately by applying a general characterization from the literature. We prove a stronger form of the conje…

Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t \ge r-2. We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices.

从分数平铺到集合平铺：格对的情形

From Fractional to Set Tilings for Pairs of Lattices

将Isbell定理推广到可数无限双随机矩阵，证明同时格平铺的支持约化定理：若函数被1控制且沿两个满秩格平移平铺，则存在非可测集的特征函数满足相同平铺恒等式。