The uniqueness of the adjoint operation. II

Let H be a complex, finite-dimensional Hilbert space, and let L(H) denote the set of linear transformations mapping H into itself. For certain interesting subsets A(H) of L(H) [nonsingular transformations and L(H) are examples], the functions h: A(H) → L(H) which have the properties h(ST) = h(T)h(S) and h(S)S ⩾ 0 are characterized.     

The uniqueness of the core

Simple proofs are given for two theorems of Duffus and Rival: If a finite poset is dismantled by irreducibles as much as possible, the subposet one finally obtains is unique up to isomorphism. If one dismantles by doubly irreducibles, the subposet is unique.     

The complete matrix ring over an adjoint regular ring is adjoint regular

We prove that the ring of all n×n matrices over an adjoint regular ring is adjoint regular, thus confirming a longstanding conjecture in the theory of adjoint semigroups.     

The randomized Kaczmarz method with mismatched adjoint

This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exact—a situation we refer to as “mismatched adjoint”. We show that the method may still converge both in the over- and underdetermined consistent case under appropriate conditions, and we calculate the expected asymptotic rate of linear convergence. Moreover, we analyze the inconsistent case and obtain results for the method with mismatched adjoint as for the standard method. Finally, we derive a method to compute optimized probabilities for the choice of the rows and illustrate our findings with numerical examples.     

The computation of the adjoint generator for a singular neutral equation

We compute the adjoint generator for the infinitesimal generator A of a C₀-semigroup. The operator A is associated with a particular scalar nonatomic neutral equation.     

The uniqueness of multilinear PageRank vectors

The uniqueness of multilinear PageRank vectors is discussed, and the new uniqueness condition is given. The new results are better than the one given in the work of Gleich et al. published in SIAM J Matrix Anal Appl. 2015;36;1409‐1465. Numerical examples are given to demonstrate the new theoretical results.     

Herschel图的匹配唯一性

研究了H erschel图的一些性质,以及它的匹配唯一性.     

The uniqueness of certain automorphic graphs

It is shown that there is only one automorphic graph with intersection array {12, 10, 5; 1, 1, 8}, and only one automorphic graph with intersection array {12, 9, 9; 1, 1, 4}.The second author is grateful for financial support from the British Council.     

The uniqueness of polynomial crystallographic actions

Let be a polycyclic-by-finite group. It is proved in [8] that admits a polynomial action of bounded degree on which is properly discontinuous and such that the quotient is compact. We prove here that such an action is unique up to conjugation by a polynomial transformation of . Received: 30 October 2000 / Revised version: 12 July 2001 / Published online: 18 January 2002