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1.
Emilien Tarquini 《Monatshefte für Mathematik》2007,151(4):333-339
In this paper we consider the Gross-Pitaevskii equation iu
t
= Δu + u(1 − |u|2), where u is a complex-valued function defined on
, N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x
1 − ct, x
2, …, x
N
), where
is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence
result for non-constant travelling waves of fixed speed having small energy. 相似文献
2.
Let
be a univariate, separable polynomial of degree n with roots x
1,…,x
n
in some algebraic closure
of the ground field
. It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all
such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on
by permutation of the coordinates and each
defines a relation P − P(x
1,…,x
n
) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an
H-basis of G-invariant relations if and only if the algebra of coinvariants
has dimension ‖G‖ over
. To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite
group G has dimension ‖G‖ if and only if G = Σ
n
acting via the tautological permutation representation. 相似文献
3.
Anders Olofsson 《Monatshefte für Mathematik》2006,148(3):229-239
We consider the singular biharmonic equation
with Dirichlet boundary conditions u = f0 and ∂nu = f1 on
. In our setup the boundary values fj (j = 0,1) are elements in two homogeneous Banach spaces Bj (j = 0,1) on
. We give a sufficient condition on the spaces Bj (j = 0,1) to ensure that the solution u of this Dirichlet problem has the appropriate boundary values fj (j = 0,1) in the sense of convergence in spaces Bj (j = 0,1). Our results also apply in the unweighted case. 相似文献
4.
Michael Voit 《Monatshefte für Mathematik》2005,145(3):247-260
Let (Bt)t ≥ 0 be a Brownian motion on
with the corresponding Gaussian convolution semigroup (μt)t ≥ 0 and generator L. We show that algebraic relations between L and the generators of the matrix semigroups
lead to
for t → s, k ≥ 1, and all coordinates i,j. These relations will form the basis for a martingale characterization of (Bt)t ≥ 0 in terms of generalized heat polynomials. This characterization generalizes a corresponding result for the Brownian motion on
in terms of Hermite polynomials due to J. Wesolowski and may be regarded as a variant of the Lévy characterization without continuity assumptions. 相似文献
5.
Let M
n
be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere
, then M
n
is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M
n
under the M?bius transformation group of
. A classical theorem of M?bius geometry states that M
n
(n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All
the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter
are Dupin hypersurfaces.
In this paper, we determine all M?bius isoparametric hypersurfaces in
by proving the following classification theorem: If
is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in
; or (ii) the pre-image of the stereographic projection of the cone in
over the Cartan isoparametric hypersurface in
with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures
in
. The classification of hypersurfaces in
with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart
of the classification for Dupin hypersurfaces in
up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen.
Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM.
Partially supported by the Zhongdian grant No. 10531090 of NSFC.
Partially supported by RFDP, 973 Project and Jiechu grant of NSFC. 相似文献
6.
Let
be a simply connected domain in
, such that
is connected. If g is holomorphic in Ω and every derivative of g extends continuously on
, then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and
we denote
. We prove the existence of a function f ∈ A∞(Ω), such that the following hold:
相似文献
i) | There exists a strictly increasing sequence μn ∈ {0, 1, 2, …}, n = 1, 2, …, such that, for every pair of compact sets Γ, Δ ⊂ and every l ∈ {0, 1, 2, …} we have |
ii) | For every compact set with and Kc connected and every function continuous on K and holomorphic in K0, there exists a subsequence of , such that, for every compact set we have |
7.
Let X be a locally compact topological space and (X, E, Xω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set Xω ⊆ X, such that all internal subsets of Xω are relatively compact in the induced topology and X is homeomorphic to the quotient Xω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function
. The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient
, for certain external subspaces
of the hyperfinite dimensional Banach space
, with the norm ‖f‖1 = ∑x ∈ X |f(x)|. If additionally X = G is a hyperfinite group, Xω = Gω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G0 of Gω, and G is isomorphic to the locally compact group Gω/G0, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and
are isometrically isomorphic as Banach algebras.
Research of both authors supported by a grant by VEGA – Scientific Grant Agency of Slovak Republic. 相似文献
8.
Given two sets
, the set of d dimensional vectors over the finite field
with q elements, we show that the sumset
contains a geometric progression of length k of the form vΛ
j
, where j = 0,…, k − 1, with a nonzero vector
and a nonsingular d × d matrix Λ whenever
. We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic
varieties. 相似文献
9.
In this paper we deal with the following problem. Let (M
n
,〈,〉) be an n-dimensional Riemannian manifold and
an isometric immersion. Find all Riemannian metrics on M
n
that can be realized isometrically as immersed hypersurfaces in the Euclidean space
. More precisely, given another Riemannian metric
on M
n
, find necessary and sufficient conditions such that the Riemannian manifold
admits an isometric immersion
into the Euclidean space
. If such an isometric immersion exists, how can one describe
in terms of f?
Author’s address: Thomas Hasanis and Theodoros Vlachos, Department of Mathematics, University of Ioannina, 45110 Ioannina,
Greece 相似文献
10.
Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums
, as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form
, where
is a continuous function with
,
runs over
, the set of Farey fractions of order Q in the unit interval [0,1] and
are consecutive elements of
. We show that the limit lim
Q→∞
A
h
(Q) exists and is independent of h. 相似文献
11.
Let Ω and Π be two finitely connected hyperbolic domains in the complex plane
and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities
are finite for all
if and only if ∂Ω and ∂Π do not contain isolated points.
This work was supported by a grant of the Deutsche Forschungsgemeinschaft for F. G. Avkhadiev. 相似文献
12.
It is proved that some velocity changes in flows on the torus determined by quasi-periodic Hamiltonians on :
where α1/α2 is an irrational number with bounded partial quotients, lead to singular flows on with an ergodic component having a minimal set of self-joinings.
Authors’ address: K. Frączek and M. Lemańczyk, Faculty of Mathematics and Computer Science, Nicolaus Copernicus University,
ul. Chopina 12/18, 87-100 Toruń, Poland
Research partially supported by KBN grant 1 P03A and by Marie Curie “Transfer of Knowledge” program, project MTKD-CT-2005-030042
(TODEQ) 03826. 相似文献
13.
In this paper, we establish several decidability results for pseudovariety joins of the form
, where
is a subpseudovariety of
or the pseudovariety
. Here,
(resp.
) denotes the pseudovariety of all
-trivial (resp.
-trivial) semigroups. In particular, we show that the pseudovariety
is (completely) κ-tame when
is a subpseudovariety of
with decidable κ-word problem and
is (completely) κ-tame. Moreover, if
is a κ-tame pseudovariety which satisfies the pseudoidentity x1 ⋯ xryω+1ztω = x1 ⋯ xryztω, then we prove that
is also κ-tame. In particular the joins
,
,
, and
are decidable.
Partial support by FCT, through the Centro de Matemática da Universidade do Porto, is also gratefully acknowledged.
Partial support by FCT, through the Centro de Matemática da Universidade do Minho, is also gratefully acknowledged. 相似文献
14.
Given a finite subset
of an additive group
such as
or
, we are interested in efficient covering of
by translates of
, and efficient packing of translates of
in
. A set
provides a covering if the translates
with
cover
(i.e., their union is
), and the covering will be efficient if
has small density in
. On the other hand, a set
will provide a packing if the translated sets
with
are mutually disjoint, and the packing is efficient if
has large density.
In the present part (I) we will derive some facts on these concepts when
, and give estimates for the minimal covering densities and maximal packing densities of finite sets
. In part (II) we will again deal with
, and study the behaviour of such densities under linear transformations. In part (III) we will turn to
.
Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395,
USA
The first author was partially supported by NSF DMS 0074531. 相似文献
15.
Godofredo Iommi 《Monatshefte für Mathematik》2007,150(2):91-95
We consider a class of countable Markov shifts
and a locally H?lder potential φ. We prove that the existence of φ-optimal measures is closely related to the behaviour of
the pressure function t → P(tφ). Using a Theorem by Sarig it is possible to prove that there exists a critical value t
c
∈ (0, ∞] such that for t < t
c
the pressure is analytic and for t > t
c
is linear. We prove that if t
c
is finite, then there are no φ-optimal measures, and if it is infinite, then φ-optimal measures do exist.
The author was partially supported by FCT/POCTI/FEDER and the grant SFRH/BPD/21927/2005. 相似文献
16.
Let H be an atomic monoid. For let denote the set of all with the following property: There exist atoms (irreducible elements) u
1, …, u
k
, v
1, …, v
m
∈ H with u
1· … · u
k
= v
1 · … · v
m
. We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number
fields). We show that, for every , max which settles Problem 38 in [4].
Authors’ addresses: W. Gao, Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China; A. Geroldinger, Institut
für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstra?e 36, 8010 Graz, Austria 相似文献
17.
For 1 ≤ i < j < d, a j-dimensional subspace L of
and a convex body K in
, we consider the projection K|L of K onto L. The directed projection function v
i,j
(K;L,u) is defined to be the i-dimensional size of the part of K|L which is illuminated in direction u ∈ L. This involves the i-th surface area measure of K|L and is motivated by Groemer’s [17] notion of semi-girth of bodies in
. It is well-known that centrally symmetric bodies are determined (up to translation) by their projection functions, we extend
this by showing that an arbitrary body is determined by any one of its directed projection functions. We also obtain a corresponding
stability result. Groemer [17] addressed the case i = 1, j = 2, d = 3. For j > 1, we then consider the average of v
1,j
(K;L,u) over all spaces L containing u and investigate whether the resulting function
determines K. We will find pairs (d,j) for which this is the case and some pairs for which it is false. The latter situation will be seen to be related to some
classical results from number theory. We will also consider more general averages for the case of centrally symmetric bodies.
The research of the first author was supported in part by NSF Grant DMS-9971202 and that of the second author by a grant from
the Volkswagen Foundation. 相似文献
18.
In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if
is a Gabor frame for
with frame bounds A and B, then the following two inequalities hold:
and
. In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form
, where Δ
k
and Λ
k
are arbitrary sequences of points in
and
, 1 ≤ k ≤ r.
Corresponding author for second author
Authors’ address: Lili Zang and Wenchang Sun, Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China 相似文献
19.
L. Olsen 《Monatshefte für Mathematik》2005,146(2):143-157
For a probability measure μ on a subset of
, the lower and upper Lq-dimensions of order
are defined by
We study the typical behaviour (in the sense of Baire’s category) of the Lq-dimensions
and
. We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower Lq-dimension
attains the smallest possible value and the upper Lq-dimension
attains the largest possible value. 相似文献
20.
Zhiting Xu 《Monatshefte für Mathematik》2007,150(2):157-171
Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation
of Emden-Fowler type
where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q
1,
. The results of this paper extend and improve some known results. In particular, two interesting examples that point out
the importance of our theorems are also included. 相似文献