Volume 8, Issue 4

Necessary and sufficient conditions and effective sufficient conditions are established for the so-called ξ -exponentially asymptotic stability of the linear system 𝑑𝑥(𝑡) = 𝑑𝘈(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are respectively matrix- and vector-functions with bounded variation components, on every closed interval from [0, +∞[ and ξ : [0, +∞[ → [0, +∞[ is a nondecreasing function such that .

Let 𝑋 be a smooth projective curve of genus 𝑔 ≥ 2 and 𝑆(𝑟, 𝑑) the moduli scheme of all rank 𝑟 stable vector bundles of degree 𝑑 on 𝑋. Fix an integer 𝑘 with 0 < 𝑘 < 𝑟. H. Lange introduced a natural stratification of 𝑆(𝑟, 𝑑) using the degree of a rank 𝑘 subbundle of any 𝐸 ∈ 𝑆(𝑟, 𝑑) with maximal degree. Every non-dense stratum, say 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎), has in a natural way a fiber structure ℎ : 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎) → Pic 𝑎 (𝑋) × Pic 𝑏 (𝑋) with ℎ dominant. Here we study the rationality or the unirationality of the generic fiber of ℎ.

Sufficient conditions are given, under which the equation 𝑦 (𝑛) = ƒ(𝑡, 𝑦, 𝑦′, . . . , 𝑦 (𝑙) )𝑔(𝑦 (𝑛 – 1) ) has a singular solution 𝑦[𝑇, τ ) → 𝐑, τ < ∞ satisfying , 𝑖 = 0, 1, . . . , 𝑙 and for 𝑗 = 𝑙 + 1, . . . , 𝑛 – 1 where 𝑙 ∈ {0, 1, . . . , 𝑛 – 2}.

In the general setting of groupoid enriched categories, notions of suspender and looper of a map are introduced, formalizing a generalization of the classical homotopy-theoretic notions of suspension and loop space. The formalism enables subtle analysis of these constructs. In particular, it is shown that the suspender of a principal coaction splits as a coproduct. This result leads to the notion of theories with suspension and to the cohomological classification of certain groupoid enriched categories.

We introduce theories in which suspension, resp. loop objects, are defined and we describe the cohomology of such theories. Cohomology classes in degree 3 classify track theories.

We discuss some global properties of an abstract geometric model for Riemann–Hilbert problems introduced by the first author. In particular, we compute the homotopy groups of elliptic Riemann–Hilbert problems and describe some connections with the theory of Fredholm structures which enable one to introduce more subtle geometrical and topological invariants for families of such problems.

The equivalence of the decreasing rearrangement of the ergodic maximal function and the maximal function of the decreasing rearrangement is proved. Exact constants are obtained in the corresponding inequalities.

We generalize some results shown by J. E. Hutchinson in [Indiana Univ. Math. J. 30: 713–747, 1981]. Let be finite systems of contractions on a complete metric space; then, under some conditions on (𝔉 𝑛 ), there exists a unique non-empty compact set 𝐾 such that the sequence defined by ((𝔉 1 ○ 𝔉 2 ○ ⋯ ○ 𝔉 𝑛 )(𝐶)) converges to 𝐾 in the Hausdorff metric for every non-empty closed and bounded set 𝐶. If the metric space is also separable and for every there are real numbers strictly between 0 and 1, satisfying the condition , then there exists a unique probability Radon measure μ 𝐾 such that the sequence weakly converges to μ 𝐾 for every probability Borel regular measure ν with bounded support (where by we denote the image measure of ν under a contraction 𝑓). Moreover, 𝐾 is the support of μ 𝐾 .

Boundary contact problems of statics of the moment (couple-stress) theory of elasticity are studied in the case of a unilateral contact of two elastic anisotropic nonhomogeneous media. A problem, in which during deformation the contact zone lies within the boundaries of some domain, and a problem, in which the contact zone can extend, are given a separate treatment. Concrete problems suitable for numerical realizations are considered.

An effective solution of a number of boundary value and boundary contact problems of thermoelastic equilibrium is constructed for a homogeneous isotropic rectangular parallelepiped in terms of asymmetric and pseudo-asymmetric elasticity (Cosserat's continuum and pseudo- continuum). Two opposite faces of a parallelepiped are affected by arbitrary surface disturbances and a stationary thermal field, while for the four remaining faces symmetry or anti-symmetry conditions (for a multilayer rectangular parallelepiped nonhomogeneous contact conditions are also defined) are given. The solutions are constructed in trigonometric series using the method of separation of variables.

Some properties of the dimension function dim on the class of separable metric spaces are studied by means of geometric construction which can be realized in Euclidean spaces. In particular, we prove that if dim(𝑋 × 𝙔) = dim 𝑋 + dim 𝙔 for separable metric spaces 𝑋 and 𝙔, then there exists a pair of maps , , 𝑠 = dim 𝑋 + dim 𝙔, with stable intersections.

Sufficient conditions are established for the solvability of the boundary value problem 𝑥 (𝑛) (𝑡) = 𝑓(𝑥)(𝑡), ℎ 𝑖 (𝑥) = 0 (𝑖 = 1, . . . , 𝑛), where 𝑓 is an operator (ℎ 𝑖 (𝑖 = 1, . . . , 𝑛) are operators) acting from some subspace of the space of (𝑛 – 1)-times differentiable on the interval ]𝑎, 𝑏[ 𝑚-dimensional vector functions into the space of locally integrable on ]𝑎, 𝑏[ 𝑚-dimensional vector functions (into the space ).

Given a quasisymmetric map , let 𝑓 0 be an extremal quasiconformal extension of ℎ onto the upper half-plane 𝘜 = {𝑧 ∈ ℂ : 𝔍𝑧 > 0} whose dilatation 𝑘(𝑓 0 ) = inf{𝑘(𝑓) : 𝑓| ∂ 𝘜 = ℎ 0 } ≕ 𝑘(ℎ). Let 𝑘 𝑛 be the minimal dilatation of polygonal quasiconformal maps 𝑓 : 𝘜 → 𝘜 satisfying 𝑓(𝑥 𝑗 ) = ℎ(𝑥 𝑗 ), 𝑗 = 1, 2, . . . , 𝑛, for any 𝑛 points of (vertices of 𝑛-gons). Already a long time ago, the question was posed whether 𝑘(ℎ) = sup 𝑘 𝑛 , where the supremum is taken over all possible 𝑛-gons of such kind. The answer is obtained (in the negative) for the case of quadrilaterals (𝑛 = 4). We show that in the case of pentagons (𝑛 = 5) the answer is also negative, i.e., there are quasisymmetric ℎ with 𝑘(ℎ) > sup 𝑘 5 .

The well-known A. Grothendieck's theorem on a homomorphism between locally convex spaces is generalized to the case of topologies which are incompatible with dualities. On the basis of this theorem, necessary and sufficient conditions are obtained for a weak homomorphism (resp. its adjoint operator, resp. its double adjoint operator) to be again a homomorphism in various topologies of the initial (resp. dual, resp. bidual) spaces. Some new classes of pairs of locally convex spaces satisfying these conditions are established. The results obtained have enabled us to reveal new properties of frequently encountered homomorphisms and weakly open operators, as well as to strengthen and generalize some well-known results.