Chasing the Critical Wetting Transition. An Effective Interface Potential Method

Wettablity is one of the important characteristics defining a given surface. Here we show that the effective interface potential method of determining the wetting temperature, originally proposed by MacDowell and Müller for the surfaces exhibiting the first order wetting transition, can also be used to estimate the wetting temperature of the second order (continuous) wetting transition. Some selected other methods of determination of the wetting temperature are also discussed.

Orthogonally Biadditive Operators

In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator T : E × F ⟶ W defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W is orthogonally biadditive if all partial operators T y : E ⟶ W and T x : F ⟶ W are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators O B A r E , F ; W is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in O B A r E , F ; W . In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator.

On Bilinear Narrow Operators

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×F→W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators Tx and Ty are narrow for all x∈E,y∈F. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E×F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E×F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if |T| is.

Hahn-Banach-Type Theorems and Subdifferentials for Invariant and Equivariant Order Continuous Vector Lattice-Valued Operators with Applications to Optimization

We give some versions of Hahn-Banach, sandwich, duality, Moreau--Rockafellar-type theorems, optimality conditions and a formula for the subdifferential of composite functions for order continuous vector lattice-valued operators, invariant or equivariant with respect to a fixed group G of homomorphisms. As applications to optimization problems with both convex and linear constraints, we present some Farkas and Kuhn-Tucker-type results.

Global Adaptive Consensus Control for Multiagent Systems with Predefined Accuracy

This paper addresses a consensus problem for uncertain nonlinear multiagent systems with predefined precision under disturbance. By employing the neural networks method and backstepping technique, adaptive controllers for each agent are created. In contrast to the exiting global control methods for multiagent systems, global precision consensus control scheme is first put forward. Moreover, by using three n th-order continuous differentiable functions, adaptive tuning laws and virtual controllers and the real controller are designed. It is proved that the presented method can ensure that all signals are globally bounded and systems can be consistent with a given accuracy under disturbance. Finally, a practical simulation verifies the correctness for the devised control protocol.

A quantitative approach to disjointly non-singular operators

We introduce and study some operational quantities which characterize the disjointly non-singular operators from a Banach lattice E to a Banach space Y when E is order continuous, and some other quantities which characterize the disjointly strictly singular operators for arbitrary E.

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.