Abstract We establish general “collapse to the mean” principles that provide conditions under which a law-invariant functional reduces an expectation. In convex setting, we retrieve and sharpen known results from literature. However, our also apply beyond setting. illustrate this by providing complete account of for quasiconvex functionals. special cases consistent risk measures Choquet integra...

In this work we consider the moduli space of all noncommutative metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over $\mathbb C$ and R$ up to dimension 5 algebras in 6. We introduce cyclic reduced cohomology identify their deformations.

Let V be an n-dimensional complex linear space and L(V) the algebra of all linear transformations on V . We prove that every linear map on L(V), which maps every operator into an operator with isomorphic lattice of invariant subspaces, is an inner automorphism or an inner antiautomorphism multiplied by a nonzero constant and additively perturbed by a scalar type operator. The same result holds ...

This paper focuses on the invariance control problem for discrete-time switched nonlinear systems. The proposed approach computes controlled invariant sets in a finite number of iterations and directly yields a partitionbased invariance controller using the information recorded during the computation. In contrast with Lyapunov-based control methods, this method does not require the subsystems t...

Correctness: The first step is to choose a useful loop invariant. Consider the following invariant: At the end of the k iteration of the outer for loop, the array A[1 · · k] is sorted and A[k] is the k smallest element in A. Let us focus on the first execution of the outer for loop. In this case, the inner for loop executes (n− 1) times. After the inner for loop has executed for the first time,...

Certain interesting classes of functions on a real inner product space are invariant under an associated group of orthogonal linear transformations. This invariance can be made explicit via a simple decomposition. For example, rotationally invariant functions on R2 are just even functions of the Euclidean norm, and functions on the Hermitian matrices (with trace inner product) which are invaria...

In this article, we introduce a notion of relative mean metric dimension with potential for factor map \(\pi : (X,d, T)\rightarrow (Y, S)\) between two topological dynamical systems. To link it ergodic theory, establish four variational principles in terms entropy partitions, Shapira’s entropy, Katok’s and Brin–Katok local respectively. Some results on respect to fixed open cover are obtained t...

We study divisibilities between bounded holomorphic quasi-inner functions in H∞ and operator-valued bounded holomorphic quasi-inner functions in H∞(Ω, L(K)) where Ω is a bounded finitely connected region. Furthermore, we characterize these divisibilities by using rationally-invariant subspaces.