A note on a conjecture concerning rank one perturbations of singular M-matrices

On a Conjecture concerning Doubly Stochastic Matrices

On A Conjecture Concerning Doubly Stochastic Matrices

In a 2002 paper, Kirkland showed that if T ∈ Rn×n is an irreducible stochastic matrix with stationary distribution vector πT , then for A = I − T , maxj=1,...,n πj‖A j ‖∞ ≥ n−1 n , where Aj , j = 1, . . . , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the j–th row and column of A. He also conjectured that equality holds in that lower bound if and only if either T...

A Note on a Conjecture Concerning Symmetric Resilient Functions

In 1985, Chor et al 2] conjectured that the only 1-resilient symmetric functions are the exclusive-or of all n variables and its negation. In this note the existence of symmetric resilient functions is shown to be equivalent to the existence of a solution to a simultaneous subset sum problem. Then, using arithmetic properties of certain binomial coeecients, an innnite class of counterexamples t...

On low rank perturbations of matrices

The article is devoted to different aspects of the question: ”What can be done with a matrix by a low rank perturbation?” It is proved that one can change a geometrically simple spectrum drastically by a rank 1 perturbation, but the situation is quite different if one restricts oneself to normal matrices. Also the Jordan normal form of a perturbed matrix is discussed. It is proved that with res...

A Note on Alberti’s Rank–one Theorem

The aim of these notes is to illustrate a proof of the following remarkable Theorem of Alberti (first proved in [1]). Here, when μ is a Radon measure on Ω ⊂ R, we denote by μ its absolutely continuous part (with respect to the Lebesgue measure L ), by μ := μ− μ its singular part, and by |μ| its total variation measure. Clearly, |μ|a = |μa| and |μ|s = μ. When μ = Du for some u ∈ BV (Ω,R), we wil...