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1.
In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra
with unit
over the fields
or
, the infimum of its norms with respect to all scalar products in this algebra (with
) is either infinite or at most
. Sufficient conditions for this bound to be not less than
are obtained. The finiteness of this bound for infinite-dimensional Grassmann algebras was first proved by Kupsh and Smolyanov (this was used for constructing a functional representation for Fock superalgebras). 相似文献
2.
For a given homogeneous elliptic partial differential operator
with constant complex coefficients, two Banach spaces
and
of distributions in
, and compact sets
and
in
, we study joint approximations in the norms of the spaces
and
(the spaces of Whitney jet-distributions) by the solutions of the equation
in neighborhoods of the set
. We obtain a localization theorem, which, under certain conditions, allows one to reduce the above-cited approximation problem to the corresponding separate problems in each of the spaces. 相似文献
3.
We construct the trajectory attractor
of a three-dimensional Navier--Stokes system with exciting force
. The set
consists of a class of solutions to this system which are bounded in
, defined on the positive semi-infinite interval
of the time axis, and can be extended to the entire time axis
so that they still remain bounded-in-
solutions of the Navier--Stokes system. In this case any family of bounded-in-
solutions of this system comes arbitrary close to the trajectory attractor
. We prove that the solutions
are continuous in t if they are treated in the space of functions ranging in
. The restriction of the trajectory attractor
to
,
, is called the global attractor of the Navier--Stokes system. We prove that the global attractor
thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as
the trajectory attractors
and the global attractors
of the
-order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors
and
, respectively. Similar problems are studied for the cases of an exciting force of the form
depending on time
and of an external force
rapidly oscillating with respect to the spatial variables or with respect to time
. 相似文献
4.
We prove the absolute continuity of the spectrum of the Schrödinger operator in
,
, with periodic (with a common period lattice
) scalar
and vector
potentials for which either
,
, or the Fourier series of the vector potential
converges absolutely,
, where
is an elementary cell of the lattice
,
for
, and
for
, and the value of
is sufficiently small, where
and
otherwise,
, and
. 相似文献
5.
It is proved that if a normal semifinite weight on a von Neumann algebra
satisfies the inequality
for any selfadjoint operators
in
, then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality
is refined and reinforced. 相似文献
6.
In this paper, we study the Banach algebra
generated by multidimensional pair integral operators with homogeneous kernels. We describe necessary and sufficient conditions for operators from the algebra
to be Fredholm and present a formula for calculating the index. 相似文献
7.
We study a version of the Gauss map
for a surface
immersed in
and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff
has a constant mean curvature. As a corollary, we conclude that an embedded flat torus
with constant mean curvature is a spherical Delonay surface. 相似文献
8.
We consider the series
and
whose coefficients satisfy the condition
for
, where the sequence
can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If
as
, then the series
is uniformly convergent. If
for all
, then the sequence of partial sums of this series is uniformly bounded. If the series
is convergent for
and
as
, then this series is uniformly convergent. If the sequence of partial sums of the series
for
is bounded and
for all
, then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence
is lacunary. In the general case, they are not necessary. 相似文献
9.
Let
be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x
1,x
2),x
3), ((x
4,x
5),x
6)), (x
7,x
8)) = 0 (((x
1
x
2 · x
3)(x
4
x
5 · x
6))(x
7
x
8) = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety
(
, respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety
has the cardinality of the continuum and that there are algebras in
with undecidable word problem. 相似文献
10.
We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order
for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian
-matrix with a given
-block
in block
-representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian
-matrix
of rank not greater than
so that the obtained matrix, being added to a given Hermitian
-matrix
, will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems. 相似文献
11.
A nondegenerate null-pair of the real projective space
consists of a point and of a hyperplane nonincident to this point. The manifold of all nondegenerate null-pairs
carries a natural Kählerian structure of hyperbolic type and of constant nonzero holomorphic sectional curvature. In particular,
is a symplectic manifold. We prove that
is endowed with the structure of a fiber bundle over the projective space
, whose typical fiber is an affine space. The vector space associated to a fiber of the bundle is naturally isomorphic to the cotangent space to
. We also construct a global section of this bundle; this allows us to construct a diffeomorphism
between the manifold of nondegenerate null-pairs and the cotangent bundle over the projective space. The main statement of the paper asserts that the explicit diffeomorphism
is a symplectomorphism of the natural symplectic structure on
to the canonical symplectic structure on
. 相似文献
12.
Suppose that
is an arbitrary finite complex Borel measure on the interval
is its Poisson integral,
and
are the conjugate harmonics of
, and
is the nontangential limiting value of the analytic function
as
. In this paper, we consider the problem of representing the analytic function
in terms of its boundary values
. 相似文献
13.
Two numerical characteristics of a nonrectifiable arc
generalizing the notion of length are introduced. Geometrically, this notion can naturally be generalized as the least upper bound of the sums
, where
are the lengths of segments of a polygonal line inscribed in the curve
and
is a given function. On the other hand, the length of
is the norm of the functional
in the space
; its norms in other spaces can be considered as analytical generalizations of length. In this paper, we establish conditions under which the generalized geometric rectifiability of a curve
implies its generalized analytic rectifiability. 相似文献
14.
We establish an analog of the Banach—Mazur theorem for real separable linear spaces with asymmetric norm: every such space can be linearly and isometrically embedded in the space of continuous functions f on the interval [0,1] equipped with the asymmetric norm
. This assertion is used to obtain nontrivial representations of an arbitrary convex closed body
, an arbitrary compact set
, and an arbitrary continuous function
. 相似文献
15.
In the open disk
of the complex plane, we consider the following spaces of functions: the Bloch space
; the Hardy--Sobolev space
; and the Hardy--Besov space
. It is shown that if all the poles of the rational function R of degree n,
, lie in the domain
, then
, where
and
depends only on
. The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions. 相似文献
16.
We study nonlinear elliptic systems of the form
,
even,
, with the natural energy space
. We establish that for
solutions from
belong to the Morrey space and the Morrey exponent does not tend to zero under the degeneration of ellipticity. In the case
, a similar result is obtained under an additional structure condition on the system. 相似文献
17.
Quasiconcave functions
and
belong to the same scale if there exist quasiconcave functions
and
and numbers
such that
and
. We establish a criterion for such functions to belong to the same scale up to equivalence. This criterion is obtained in terms of nodes of the corresponding linear-constant step-functions. It turns out that nodes must be equivalent to sequences. 相似文献
18.
Let
be the free product of two Abelian torsion-free groups, let
and
, where
is the Cartesian subgroup of the group
, and let
F contain no zero divisors. In the paper it is proved that, in this case, any automorphism of the group
is inner. This result generalized the well-known result of Bachmuth, Formanek, and Mochizuki on the automorphisms of groups of the form
,
,
, where
is a free group of rank two. 相似文献
19.
In the Sobolev space
, where is a bounded domain in n with a Lipschitzian boundary, for an arbitrarily given
, we construct a basis such that the error of approximation of a function
the Nth partial sum of its expansion with respect to this basis can be estimated in terms of the modulus of smoothness
of order
. 相似文献
20.
For a polynomial algebra
in several variables over a commutative ring R with a Hopf algebra structure
the existence of the dual Hopf algebra
is proved. 相似文献