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The Schwarz methods are the oldest domain decomposition methods, but over the last decade several interesting developments happened. This mini-symposium gives an overview of recent results and new directions of research for Schwarz methods. It includes Schwarz methods with new transmission condi- tions, optimized Schwarz methods, restricted Schwarz methods, Schwarz waveform relaxation methods for evolution and delay differential equations and Schwarz methods for systems of equations

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In classical literature, invariants of families of differentialequations were considered for linear equations only, e.g. the renownedLaplace invariants for linear hyperbolic partial differential equationsand invariants of linear ordinary differential equations with variablecoefficients. The restriction to linear equations was essential inpioneering works of Cockle, Laguerre, Halphen, andForsyth ... [Show full abstract] for tackling the problem of invariants of differentialequations. Lie regretted that these authors did not use advantagesprovided by his theory of infinite continuous groups, but he himself didnot undertake further developments in this direction. Recently, the present author considered the possibility hinted byLie's remark and introduced the infinitesimal technique in thetheory of invariants of families of differential equations thatwas lacking in old methods. In consequence, a simple unifiedapproach was developed for calculation of invariants of algebraicand differential equations independent on the assumption oflinearity of equations. It was employed recently for calculationof Laplace type invariants for parabolic equations. Here, themethod is applied to calculation of invariants for the family ofnonlinear equations appearing in the problem on linearization ofnonlinear ordinary differential equations.

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l_1(t)<x<l_2(t)

,

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, where

l_1(t)

and

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The Riccati equation method is used to establish some new oscillatory criteria for the hamiltonian systems in a new direction, which is to break the positive definiteness restriction imposed on one of coefficients of the hamiltonian system. The obtained results are compared with some known oscillatory criteria. non negative (positive) definiteness, the least eigenvalue of the hermitian matrix. ... [Show full abstract] §1. Introduction. Let A(t), B(t) and C(t) be complex valued continuous matrix-functions of dimension n × n on [t 0 ; +∞) and let B(t) and C(t) be hermitian matrices, i.e. B(t) = B * (t), C(t) = C * (t), t ≥ t 0 , where * is the conjugation sign. Consider the hamiltonian system    Φ ′ = A(t)Φ + B(t)Ψ; Ψ ′ = C(t)Φ − A * (t)Ψ, t ≥ t 0. (1.1) Here Φ and Ψ are the unknown continuously differentiable on [t 0 ; +∞) matrix-functions of dimension n × n. Definition 1.1. A solution (Φ(t), Ψ(t)) of the system (1.1) is called prepared (or preferred), if Φ * (t)Ψ(t) = Ψ * (t)Φ(t), t ≥ t 0. Definition 1.2. A prepared solution (Φ(t), Ψ(t)) of the system (1.1) is called oscillatory if det Φ(t) has arbitrary large zeroes. Definition 1.3. The system (1.1) is called oscillatory if its every prepared solution is oscillatory.

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Differential operators are usually used to determine the rate of change and the direction of change of a signal modeled by a function in some appropriately selected function space. Gibbs derivatives are introduced as operators permitting differentiation of piecewise constant functions. Being initially intended for applications in Walsh dyadic analysis, they are defined as operators having Walsh ... [Show full abstract] functions as eigenfunctions. This feature was used in different generalizations and extensions of the concept firstly defined for functions on finite dyadic groups. In this paper, we provide a brief overview of the evolution of this concept into a particlar class of differential operators for functions on various groups.

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