In ‎this ‎paper, ‎we ‎study ‎the ‎relation ‎between ‎the ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎with ‎the ‎lattice theory. ‎We ‎introduce ‎the ‎concepts ‎of ‎the ‎lattice ‎of ‎soft ‎sets, ‎full ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎and next, we ‎verify ‎some ‎properties ‎of ‎them. We ‎prove ‎that ‎the ‎lattice ‎of ‎the ‎soft ‎sets ‎on ‎a fixed parameter set is isomorphic to the power set of a ...

Let S be a subset of a normed space X = (X ,‖ · ‖) and T and I self-mappings of X . Then T is called (1) nonexpansive on S if ‖Tx− Ty‖ ≤ ‖x− y‖ for all x, y ∈ S; (2) Inonexpansive on S if ‖Tx − Ty‖ ≤ ‖Ix − I y‖ for all x, y ∈ S; (3) I-contraction on S if there exists k ∈ [0,1) such that ‖Tx − Ty‖ ≤ k‖Ix − I y‖ for all x, y ∈ S. The set of fixed points of T (resp., I) is denoted by F(T) (resp., ...

We show that the graph of the reachable set of a control system given by a differential inclusion is uniquely characterized by a Hamilton-Jacobi equation involving proximal normals. Résumé On démontre que le graphe de l’ensemble des points accessibles d’un système de contrôle décrit par une inclusion différentielle est caractérisé par une équation Hamilton-Jacobi impliquant les normales proxima...

The reeectivity due to stimulated Brillouin backscattering (SBS) from an ensemble of independent laser speckles is investigated for diierent speckle statistics. Calculations are based on numerical simulations with a multidi-mensional code and a compact model describing the main features of speckle self-focusing. In particular, the simulations and the model are applied to speckle ensembles due t...

High-speed, tunable integrated silicon photonic delay lines are demonstrated by cascading complementary apodized silicon grating waveguides. The cascaded grating waveguides, with inward and outward super-Gaussian apodization profiles, compensate each other's dispersion and allow high-speed operation. Characterization of the compact delay lines shows that they have low loss, offer true time dela...

In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bis. Some examples with positive or zero Hausdorff metric entropy are given. Moreov...

A compact traveling-wave electroabsorption modulator-based ring oscillator is utilized to perform clock recovery and simultaneously reshaping for 40-Gb/s optical packets. The clock locks within 300 ns and has less than 0.6 ps timing jitter.

Schmüdgen proved a Positivstellensatz about the representation of positive definite polynomials on a basic closed semialgebraic set using methods from functional analysis in connection with Stengle's Positivstellensatz. Later Wörmann gave a purely algebraic proof. The present note contains yet another proof that uses classical results by Kadison about the representation of partially ordered alg...