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We consider a class of parabolic-type equations with double nonlinearity and derive sufficient conditions for finite time blow-up of its solutions in a bounded domain under the homogeneous Dirichlet condition. To prove the solution blow-up we use a modification of Levine's method. Bibliography: 13 titles.
We classify Q-Fano threefolds of Fano index >2 and sufficiently big degree. Bibliography: 20 titles.
The paper is concerned with the existence of a Lipschitz selection for the operator T{sub C} (the Chebyshev-centre map) that assigns to any bounded subset M of a Banach space X the set T{sub C}(M) of its Chebyshev centres. It is proved that if the unit sphere S(X) of X has an exposed smooth point, then T{sub C} does not have a Lipschitz selection. It is also proved that if X is finite dimensional the operator T{sub C} has a Lipschitz selection if and only if X is polyhedral. The operator T{sub C} is also shown to have a Lipschitz selection in the space c{sub 0}(K) and c-spaces. Bibliography: 4 titles.
Let G be a zero-dimensional locally compact Abelian group all of whose elements are compact, and let C(G) be the space of all complex-valued continuous functions on G. A closed linear subspace H⊆C(G) is said to be an invariant subspace if it is invariant with respect to the translations τ{sub y}:f(x)↦f(x+y), y∈G. In the paper, it is proved that any invariant subspace H admits spectral synthesis, that is, H coincides with the closed linear span of the characters of G belonging to H. Bibliography: 25 titles.
This paper is concerned with describing the spectrum of the difference operator K:l{sub α}{sup p}(Z,X)→l{sub α}{sup p}(Z......athscrKx)(n)=Bx(n−1), n∈Z, x∈l{sub α}{sup p}(Z,X), with a constant operator coefficient B, which is a bounded linear operator in a Banach space X. It is assumed that K acts in the weighted space l{sub α}{sup p}(Z,X), 1≤p≤∞, of two-sided sequences of vectors from X. The main results are obtained in terms of the spectrum σ(B) of the operator coefficient B and properties of the weight function. Applications to the study of the spectrum of a differential operator with an unbounded operator coefficient (the generator of a strongly continuous semigroup of operators) in weighted function spaces are given. Bibliography: 23 titles.
We provide a complete list of real Lie algebras with generic orbits of dimension 2 in the coadjoint representation. Bibliography: 20 titles.
In the paper, a new description of the generalized Lions-Peetre method of means is found, which enables one to evaluate the interpolation orbits of spaces constructed by this method. The list of these spaces includes all Lorentz spaces with functional parameters, Orlicz spaces, and spaces close to them. This leads in turn to new optimal embedding theorems for Sobolev spaces produced using the Lions-Peetre construction in rearrangement invariant spaces. It turns out that the optimal space of the embedding is also a generalized Lions-Peetre space whose parameters are explicitly evaluated. Bibliography: 18 titles.
The problem of a sphere rolling on the plane, with twisting but without slipping, is considered. It is required to roll the sphere from one configuration to another in such a way that the minimum of the action is attained. We obtain a complete parametrization of the extremal trajectories and analyse the natural symmetries of the Hamiltonian system of the Pontryagin maximum principle (rotations and reflections) and their fixed points. Based on the estimates obtained for the fixed points we prove upper estimates for the cut time, that is, the moment of time when an extremal trajectory loses optimality. We consider the problem of re-orienting the sphere in more detail; in particular, we find diffeomorphic domains in the pre-image and image of the exponential map which are used to construct the optimal synthesis. Bibliography: 15 titles.
We obtain sharp constants in multiplicative inequalities that characterize embeddings of the Sobolev spaces into the space of bounded continuous functions. Bibliography: 2 titles.
The Hodge conjecture on algebraic cycles is proved for a smooth projective model X of a fibre product X{sub 1}×{sub C}X{sub 2} of nonisotrivial 1-parameter families of K3 surfaces (possibly with degeneracies) X{sub k}→C (k=1,2) over a smooth projective curve C under the assumption that, for generic geometric fibres X{sub 1s} and X{sub 2s}, the ring End{sub Hg(X{sub 1{sub s)}}}NS{sub Q}(X{sub 1s}){sup ⊥} is an imaginary quadratic field, rankNS(X{sub 1s})≠18, and End{sub Hg(X{sub 2{sub s)}}}NS{sub Q}(X{sub 2s}){sup ⊥} is a totally real field or else rankNS(X{sub 1s})<rankNS(X{sub 2s}). Bibliography: 10 titles.
We consider smooth solutions of the Cauchy problem for the Navier-Stokes equations on the scale of smooth functions which are periodic with respect to x∈R{sup 3}. We obtain existence theorems for global (with respect to t>0) and local solutions of the Cauchy problem. The statements of these depend on the smoothness and the norm of the initial vector function. Upper bounds for the behaviour of solutions in both classes, which depend on t, are also obtained. Bibliography: 10 titles.
The method of partial stabilization of locally minimal networks, which was invented by Ivanov and Tuzhilin to construct examples of shortest trees with given topology, is developed. According to this method, boundary vertices of degree 2 are not added to all edges of the original locally minimal tree, but only to some of them. The problem of partial stabilization of locally minimal trees in a finite-dimensional Euclidean space is solved completely in the paper, that is, without any restrictions imposed on the number of edges remaining free of subdivision. A criterion for the realizability of such stabilization is established. In addition, the general problem of searching for the shortest forest connecting a finite family of boundary compact sets in an arbitrary metric space is formalized; it is shown that such forests exist for any family of compact sets if and only if for any finite subset of the ambient space there exists a shortest tree connecting it. The theory developed here allows us to establish further generalizations of the stabilization theorem both for arbitrary metric spaces and for metric spaces with some special properties. Bibliography: 10 titles.
It is proved that a real Banach space realizes minimal fillings for all its finite subsets (a shortest network spanning a fixed finite subset always exists and has the minimum possible length) if and only if it is a predual of L{sub 1}. The spaces L{sub 1} are characterized in terms of Steiner points (medians). Bibliography: 25 titles. (paper)
An analogue of a Lie theorem is obtained for (not necessarily continuous) finite-dimensional representations of soluble finite-dimensional locally compact groups with connected quotient group by the centre. As a corollary, the following automatic continuity proposition is obtained for locally bounded finite-dimensional representations of connected locally compact groups: if G is a connected locally compact group, N is a compact normal subgroup of G such that the quotient group G/N is a Lie group, N{sub 0} is the connected identity component in N, H is the family of elements of G commuting with every element of N{sub 0}, and π is a (not necessarily continuous) locally bounded finite-dimensional representation of G, then π is continuous on the commutator subgroup of H (in the intrinsic topology of the smallest analytic subgroup of G containing this commutator subgroup). Bibliography: 23 titles. (paper)
As an application of Kronrod's construction of the tree of a function to convex analysis, a characterization of quasiconvex functions is obtained. Namely, a function is quasiconvex if and only if the associated function defined on the tree of the original function is quasiconvex. A new proof of one of the existence lemmas in Kronrod's construction of the tree of a function (the original proof of which turned out to be incorrect) is given. Bibliography: 4 titles. (paper)
The paper investigates questions related to Borsuk's classical problem of partitioning a set in Euclidean space into subsets of smaller diameter, as well as to the well-known Nelson-Erdős-Hadwiger problem on the chromatic number of a Euclidean space. The results of the work are obtained using combinatorial and geometric methods alike. A new approach to the investigation of such problems is suggested; it leads to a collection of results which significantly improve all results known so far. Bibliography: 58 titles.
We consider the billiard dynamical system in a domain bounded by confocal parabolas. We describe such domains in which the billiard problem can be correctly stated. In each such domain we prove the integrability for the system, analyse the arising Liouville foliation, and calculate the invariant of Liouville equivalence--the so-called marked molecule. It turns out that billiard systems in certain parabolic domains have the same closures of solutions (integral trajectories) as the systems of Goryachev-Chaplygin-Sretenskii and Joukowski at suitable energy levels. We also describe the billiard motion in noncompact domains bounded by confocal parabolas, namely, we describe the topology of the Liouville foliation in terms of rough molecules. Bibliography: 16 titles.
The paper looks at problems concerning the recovery of operators from noisy information in non-Euclidean metrics. A number of general theorems are proved and applied to recovery problems for functions and their derivatives from the noisy Fourier transform. In some cases, a family of optimal methods is found, from which the methods requiring the least amount of original information are singled out. Bibliography: 25 titles.