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1.
Joanna Janczewska 《Central European Journal of Mathematics》2004,2(4):561-572
In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R
k
→Y is a C
2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R
1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R
k
and we describe the solution set of the studied equation in a small neighbourhood of this point. 相似文献
2.
Song Li 《中国科学A辑(英文版)》2003,46(3):364-375
The purpose of this paper is to investigate the refinement equations of the form
where the vector of functions ϕ=(ϕ
1..., ϕ
r
)
T
is in (L
p
(ℝ
s
))
r
, 1⩽p⩽∞, a(α), α∈ℤ
s
is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vector of compactly supported functions φ
0∈(L
p
(ℝ
s
))
r
and use the iteration schemes f
n
:=Q
a
n
φ
0, n=1,2,..., where Q
n
is the linear operator defined on (L
p
(ℝ
s
))
r
given by
This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of some linear operators
determined by the sequence a and the set B restricted to a certain invariant subspace, where the set B is a complete set of representatives of the distinct cosets of the quotient group ℤs/Mℤs containing 0. 相似文献
3.
Domenico Perrone 《Mathematische Zeitschrift》2009,263(1):125-147
It is well known that a Hopf vector field on the unit sphere S
2n+1 is the Reeb vector field of a natural Sasakian structure on S
2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector
field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.
Supported by funds of the University of Lecce and M.I.U.R.(PRIN). 相似文献
4.
Thomas Vougiouklis 《数学学报(英文版)》2008,24(7):1067-1078
The hyperoperations, called theta-operations (δ), are motivated from the usual property, which the derivative has on the derivation of a product of functions. Using any map on a set, one can define δ-operations. In this paper, we continue our study on the δ-operations on groupoids, rings, fields and vector spaces or on the corresponding hyperstructures. Using δ-operations one obtains, mainly, Hwstructures, which form the largest class of the hyperstructures. For representation theory of hyperstructures, by hypermatrices, one needs special Hv-rings or Hy-fields, so these hyperstructures can be used. Moreover, we study the relation of these δ-structures with other classes of hyperstructures, especially with the Hv-structures. 相似文献
5.
The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function
of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p
−1. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable
function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there
exist infinity many different 2-adic orthonormal wavelet bases in ℒ2(ℚ2) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ2(ℚ2
n
) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral
analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also
that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary
prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators
develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).
相似文献
6.
The main focus in this paper is on homogenization of the parabolic problem ∂
t
uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ
r
)∇u
ɛ
) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made. 相似文献
7.
We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional
and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed
into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting
of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give
sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are
applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise.
Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001 相似文献
8.
The Evens-Lu-Weinstein representation (Q
A
, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly
modified to (Q
A
or
, Dor) by tensoring by orientation flat line bundle, Q
A
or
=QA⊗or (M) and D
or=D⊗∂
A
or
. It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q
A
or
, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial.
In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂
A
or
) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential
forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization
of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the
orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre
spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces:
assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which
can be infinite dimensional) implies the same for cohomology spaces. 相似文献
9.
In Krylov (Journal of the Juliusz Schauder Center 4 (1994), 355–364), a parabolic Littlewood–Paley inequality and its application
to an L
p
-estimate of the gradient of the heat kernel are proved. These estimates are crucial tools in the development of a theory
of parabolic stochastic partial differential equations (Krylov, Mathematical Surveys and Monographs vol. 64 (1999), 185–242).
We generalize these inequalities so that they can be applied to stochastic integrodifferential equations.
相似文献
10.
Multivariate Refinement Equations and Convergence of Cascade Algorithms in Lp(0〈p〈1)Spaces 总被引:1,自引:0,他引:1
SongLI 《数学学报(英文版)》2003,19(1):97-106
We consider the solutions of refinement equations written in the form
where the vector of functions ϕ = (ϕ
1, ..., ϕ
r
)
T
is unknown, g is a given vector of compactly supported functions on ℝ
s
, a is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s dilation matrix with m = |detM|. Inhomogeneous refinement equations appear in the construction of multiwavelets and the constructions of wavelets on a finite
interval. The cascade algorithm with mask a, g, and dilation M generates a sequence ϕ
n
, n = 1, 2, ..., by the iterative process
from a starting vector of function ϕ
0. We characterize the L
p
-convergence (0 < p < 1) of the cascade algorithm in terms of the p-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness
property of the solutions of the refinement equations associated with the homogeneous refinement equation.
This project is supported by the NSF of China under Grant No. 10071071 相似文献
11.
V. S. Kal’nitskii 《Journal of Mathematical Sciences》1998,91(6):3476-3491
We study the structure of those vector fields on the tangent bundle of an arbitrary smooth manifold which commute with the
geodesic vector field defined by an affine connection. The study is restricted to polylinear fields generated by a pair of
symmetric pseudotensor fields of type (k, 1) and (k+1,1), k≥0, defined on the manifold. We establish an isomorphism between
the space of infinitesimal automorphisms of fixed type and the space ℌk of the solutions of a partial differential equation generalizing the Jacobi equation for the infinitesimal automorphisms
of the connection. It is shown that the spaces ℌk are finite-dimensional and form a graduated Lie algebra ℌ=⊕
k=0
∞
ℌk. These algebras are classified in the case of one-dimensional manifolds. It is proved that if the geodesic vector field is
complete, then so are the automorphisms corresponding to covariant constant fields of type (1, 1). Bibliography: 5 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 222–244.
Translated by V. S. Kal’nitskii. 相似文献
12.
A variational theory for monotone vector fields 总被引:1,自引:0,他引:1
Nassif Ghoussoub 《Journal of Fixed Point Theory and Applications》2008,4(1):107-135
Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators,
but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials
in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We
describe here how the selfdual variational calculus, developed recently by the author, provides a variational approach to
PDEs and evolution equations driven by maximal monotone operators. To any such vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian L
T
on the phase space X × X
* that can be seen as a “potential” for T, in the sense that the problem of inverting T reduces to minimizing a convex energy functional derived from L
T
. This variational approach to maximal monotone operators allows their theory to be analyzed with the full range of methods—computational
or not—that are available for variational settings. Standard convex analysis (on phase space) can then be used to establish
many old and new results concerned with the identification, superposition, and resolution of such vector fields.
Dedicated to Felix Browder on his 80th birthday 相似文献
13.
Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed>1 and {0} if d>1.Let Bcd be a cd-homogeneous C^*-algebra over S^2×T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2×Z^m-2) in Bcd ×C^*(Z^m-3) by a totally skew multiplier ρ on T^2×Z^m-2。It is shown that Sρ^cd×Mρ∞ is isomorphic to C(S^2)×C^*(T^2×Z^m-2,ρ)× Mcd(C)×Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p. 相似文献
14.
15.
Andrzej Rozkosz 《Probability Theory and Related Fields》2003,125(3):393-407
We extend the definition of solutions of backward stochastic differential equations to the case where the driving process
is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of
solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic
partial differential equations in Sobolev spaces.
Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002
Research supported by KBN Grant 0253 P03 2000 19.
Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55
Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution 相似文献
16.
Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝ
an
K
, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū\∑.
Partially supported by the European RTN Network RAAG (contract no. HPRN-CT-00271) 相似文献
17.
Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(
n
n+x
). Here we prove that the order x Veronese embedding ofP
n
is not weakly (k−1)-defective, i.e. for a general S⊃P
n
such that #(S) = k+1 the projective space | I
2S
(x)| of all degree t hypersurfaces ofP
n
singular at each point of S has dimension (
n
/n+x
)−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I
2S
(x)| has an ordinary double point at each P∈ S and Sing (F)=S.
The author was partially supported by MIUR and GNSAGA of INdAM (Italy). 相似文献
18.
We consider the one-dimensional stochastic differential equation dX
t=b(t, Xt−) dZ
t, whereZ is a symmetric α-stable Lévy process with α ε (1, 2] andb is a Borel function. We give sufficient conditions under which the equation has a weak nonexploding solution.
Partially supported by Programma Professori Visitatori of G. N. A. F. A. (Italy).
Partially supported by MURST (Italy). The present research was completed while the second author was visiting the Institute
of Mathematics and Informatics (Vilnius, Lithuania) in spring of 1999.
Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 361–385, July–September, 2000.
Translated by H. Pragarauskas 相似文献
19.
We study the existence theory for parabolic variational inequalities in weighted L
2 spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal
stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality.
The weighted L
2 setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion
coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we
treat the cases of assets with stochastic volatility and with path-dependent payoffs. 相似文献
20.
Albert G. Buckley 《Mathematical Programming》1986,36(3):256-275
This work concerns the derivation of formulae for updating quasi-Newton matrices used in algorithms for computing approximate
minima of smooth unconstrained functions. The paper concentrates strictly on the techniques used to derive update formulae.
It demonstrates a technique in which problems of finding matrices in ℝ
n ×n
of minimum Frobenius norm are converted to equivalent problems, using vector representations in ℝ
n2
and ℝ
n(n+1)/2 of these matrices, and then solvingl
2-minimization problems. These problems are more directly dealt with, and indeed, the paper demonstrates how this technique
may be used to handle weighted sparse updates. 相似文献