ACTA SCIENTIARUM MATHEMATICARUM最新文献

In this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital (C^*)-algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the (C^*)-algebra of operators on a Hilbert space, even to general unital (C^*)-algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Various multivariable means have been defined for positive definite matrices, such as the Cartan mean, Wasserstein mean, and Rényi power mean. These multivariable means have corresponding matrix equations. In this paper, we consider the following non-linear matrix equation:

where (t in (0,1]). We prove that this equation has a unique solution and define a new mean, which we denote as (G_{t}(omega ; mathbb {A})). We explore important properties of the mean (G_{t}(omega ; mathbb {A})) including the relationship with matrix power mean, and show that the mean (G_{t}(omega ; mathbb {A})) is monotone in the parameter t. Finally, we connect the mean (G_{t}(omega ; mathbb {A})) to a barycenter for the log-determinant divergence.

Power difference means and Heron means are well-known numerical means with parameters. Their comparison in the positive definite sense is studied. More precisely, for a power difference mean M with each fixed parameter we try to determine the exact parameter range for Heron means majorizing M (in the positive definite sense). Since this order is known to determine validity of (unitarily invariant) norm inequalities between corresponding matrix power difference means and matrix Heron means, we obtain an abundance of very precise norm inequalities between these two matrix means.

We introduce a weaker form of continuity which is a necessary and sufficient condition for the existence of fixed points. The obtained theorems exhibit interesting fixed point – eventual fixed point patterns. If we slightly weaken the conditions then the mappings admit periodic points besides fixed points and such mappings possess interesting combinations of fixed and periodic points. Our results are applicable to contractive type as well as non-expansive type mappings. Our theorems are independent of almost all the existing results for contractive type mappings. The last theorem of Sect. 2 is applicable to mappings having various geometric patterns as their domain and is perhaps the first result of its type that also opens up scope for the study of periodic points and periodic point structures. We also give an application of our theorem to obtain the solutions of a nonlinear Diophantine equation; and also show that various well-known fixed point theorems are not applicable in solving this equation.

In this paper, we introduce and investigate a new class of operators known as almost unbounded L-weakly compact (in shortly, (_{au}L)-weakly compact) and almost unbounded M-weakly compact (in shortly, (_{au}M)-weakly compact) operators. We explore the lattice properties related to this class and examine their relationships with other established operator classes, such as L-weakly compact operators and almost L-weakly compact operators. We demonstrate that every L-weakly compact operator is an (_{au}L)-weakly compact operator, but the reverse implication does not necessarily hold in all cases.

Let (G=(V,E)) be a finite graph together with an initial assignment (Vrightarrow {0,1}) that represents the opinion of each vertex. Then discordant push voting is a discrete, non-deterministic protocol that alters the opinion of one vertex at a time until a consensus is reached. More precisely, at each round a discordant vertex u (i.e., one that has a neighbor with a different opinion) is chosen uniformly at random, and then we choose a neighbor v with different vote uniformly at random, and force v to change its opinion to that of u. In case of the discordant pull protocol we simply choose a discordant vertex uniformly at random and change its opinion. In this paper, we give asymptotically sharp estimations for the worst expected runtime of the discordant push and pull protocols on the star graph.

In this article we study log-majorizations related to the spectral geometric and Rényi means. Our goal is to establish certain geometric properties for them with respect to the Thompson metric and Kim’s semi-metric on the cone of positive definite matrices. We also study geodesic in-betweenness type results for these two means and some Audenaert-type in-betweenness inequalities.