共查询到20条相似文献,搜索用时 15 毫秒
1.
Yann Bugeaud 《Journal d'Analyse Mathématique》2004,92(1):327-349
We compute the generalized Hausdorff measure of sets of points in R
s
which satisfy an inhomogeneous system of Diophantine inequalities infinitely often. This provides an inhomogeneous analogue
of a classical result of Jarník on simultaneous Diophantine approximation. 相似文献
2.
R. Lucchetti P. Shunmugaraj Y. Sonntag 《Numerical Functional Analysis & Optimization》2013,34(1-2):103-115
Given a continuous function f: X→R, sufficient conditions are offered for the continuity of the value function v(A):=inf{f{x): x ε A} and of the level set multifunction Lev(A, α) := {x ε A: f(x)?α}, with respect to recently defined topologies on the closed sets of a metric space. 相似文献
3.
We introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces. The latter are functions f○T:Rn×Q→Q with f○T(x,ω)=f(T(x)ω) where Q is a compact (Hausdorff topological) space, f∈C(Q) and T(x):Q→Q, x∈Rn, is an n-dimensional continuous dynamical system endowed with an invariant Radon probability measure μ. It can be easily shown that for almost all ω∈Q the realization f(T(x)ω) belongs to an algebra with mean value, that is, an algebra of functions in BUC(Rn) containing all translates of its elements and such that each of its elements possesses a mean value. This notion was introduced by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators, Mat. Zametki 33 (1983) 571-582, English transl. in Math. Notes 33 (1983) 294-300]. We then establish the existence of multiscale Young measures in the setting of algebras with mean value, where the compactifications of Rn provided by such algebras plays an important role. These parametrized measures are useful in connection with the existence of correctors in homogenization problems. We apply this framework to the homogenization of a porous medium type equation in Rn with a stationary continuous process as a stiff oscillatory external source. This application seems to be new even in the classical context of periodic homogenization. 相似文献
4.
Strong commutativity preserving maps on Lie ideals 总被引:2,自引:0,他引:2
Jer-Shyong Lin 《Linear algebra and its applications》2008,428(7):1601-1609
Let A be a prime ring and let R be a noncentral Lie ideal of A. An additive map f:R→A is called strong commutativity preserving (SCP) on R if [f(x),f(y)]=[x,y] for all x,y∈R. In this paper we show that if f is SCP on R, then there exist λ∈C, λ2=1 and an additive map μ:R→Z(A) such that f(x)=λx+μ(x) for all x∈R where C is the extended centroid of A, unless charA=2 and A satisfies the standard identity of degree 4. 相似文献
5.
Tord Sjödin 《manuscripta mathematica》1997,94(1):169-186
We extend a recent result of A. Jonsson about mutual absolute continuity of twoD
s
-measures on ans-setF ⊂R
n
to the homogeneous spaces (X, d, μ) of Coifman, Weiss. Here we define Hausdorff measure, Hausdorff dimension,D
s
-set andd-set relative to the measureμ. Our main result holds for so called (s, d)-sets,d ≥s, and is stronger than Jonssons result even inR
n
. As applications we interpret this Hausdorff dimension as a relative dimension for very regular sets and show that it in
general depends strongly onμ. For this purpose we construct a strictly increasing functionf :R →R, whose measure is doubling and concentrated on a set of arbitrary small Hausdorff dimension. The extension off to a quasiconformal map of the half plane onto itself sharpens a classical example of Ahlfors-Beurling. 相似文献
6.
Let G be a graph and f:G→G be a continuous map. Denote by P(f), R(f) and Ω(f) the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If L=(x,y) is an open arc contained in an edge of G such that {fm(x),fk(y)}⊂(x,y) for some m,k∈N, then R(f)∩(x,y)≠∅; (2) Any isolated point of P(f) is also an isolated point of Ω(f); (3) If x∈Ω(f)−Ω(fn) for some n∈N, then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) [9] and J. Xiong (1983, 1986) [17] and [19] on interval maps or tree maps. 相似文献
7.
A.G. Ramm 《Journal of Computational and Applied Mathematics》2010,234(12):3326-3331
A new understanding of the notion of the stable solution to ill-posed problems is proposed. The new notion is more realistic than the old one and better fits the practical computational needs. A method for constructing stable solutions in the new sense is proposed and justified. The basic point is: in the traditional definition of the stable solution to an ill-posed problem Au=f, where A is a linear or nonlinear operator in a Hilbert space H, it is assumed that the noisy data {fδ,δ} are given, ‖f−fδ‖≤δ, and a stable solution uδ:=Rδfδ is defined by the relation limδ→0‖Rδfδ−y‖=0, where y solves the equation Au=f, i.e., Ay=f. In this definition y and f are unknown. Any f∈B(fδ,δ) can be the exact data, where B(fδ,δ):={f:‖f−fδ‖≤δ}.The new notion of the stable solution excludes the unknown y and f from the definition of the solution. The solution is defined only in terms of the noisy data, noise level, and an a priori information about a compactum to which the solution belongs. 相似文献
8.
In 1998, Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation (and indeed
the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine
exponent w
0(x) = 1/n for almost every point x on a nondegenerate submanifold M \mathcal{M} of
\mathbbRn {\mathbb{R}^n} . In this paper, the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any
“inhomogeneous” vector θ ∈
\mathbbRn {\mathbb{R}^n} we prove that the simultaneous inhomogeneous Diophantine exponent w
0(x
,
θ) is 1/n for almost every point x on M \mathcal{M} . The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w
0(x) is 1/n for almost all x ∈ M \mathcal{M} if and only if, for any θ ∈
\mathbbRn {\mathbb{R}^n} , the inhomogeneous exponent w
0(x
,
θ) = 1/n for almost all x ∈ M \mathcal{M} . The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered
by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront
the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference
principle in all its glory. 相似文献
9.
Let μ be a self-similar measure in Rd. A point x∈Rd for which the limit does not exist is called a divergence point. Very recently there has been an enormous interest in investigating the fractal structure of various sets of divergence points. However, all previous work has focused exclusively on the study of the Hausdorff dimension of sets of divergence points and nothing is known about the packing dimension of sets of divergence points. In this paper we will give a systematic and detailed account of the problem of determining the packing dimensions of sets of divergence points of self-similar measures. An interesting and surprising consequence of our results is that, except for certain trivial cases, many natural sets of divergence points have distinct Hausdorff and packing dimensions. 相似文献
10.
For a function f:{0,1}n→R and an invertible linear transformation L∈GLn(2), we consider the function Lf:{0,1}n→R defined by Lf(x)=f(Lx). We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I(Lf)≥I(f), where I(f) is the total influence of f. Second, we conjecture that if both f and L(f) are monotone, then f=L(f) (up to a permutation of the coordinates). We prove the second conjecture in the case where L is upper triangular. 相似文献
11.
In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear Schrödinger equations −Δu+V(x)u=f(x,u), x∈RN, where the primitive of the nonlinearity f is of superquadratic growth near infinity in u and the potential V is allowed to be sign-changing. 相似文献
12.
Adam P. Wójcik 《Monatshefte für Mathematik》1988,105(1):75-81
LetE be a compact subset of the complex planeC such that Leja's extremal functionL
E
forE is continuous. If almost all zeros of the polynomials of best approximation to a functionfC(E) are outside the setE
R
={zC:L
E
(z<R)}, for someR>1, thenf is extendible to a holomorphic function inE
R
. If the zeros ofn-th, polynomial of best approximation tof are outside
and the sequence {R
n
–n
} rapidly decreases to zero thenf can be extended to aC
function on 075-4}. 相似文献
13.
14.
Convex programs with an additional reverse convex constraint 总被引:2,自引:0,他引:2
H. Tuy 《Journal of Optimization Theory and Applications》1987,52(3):463-486
A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject toxD,g(x)0, whereD is a closed convex subset ofR
n
andf,g are convex finite functionsR
n
. Under suitable stability hypotheses, it is shown that a feasible point
is optimal if and only if 0=max{g(x):xD,f(x)f(
)}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ
k
,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS
k
. The method is similar to the outer approximation method for maximizing a convex function over a compact convex set. 相似文献
15.
Zachary Maddock 《Monatshefte für Mathematik》2010,121(3):167-186
The Takagi function τ(x) is a continuous non-differentiable function on the unit interval defined by Takagi in 1903. This paper studies level sets
L(y) = {x : τ(x) = y} of the Takagi function and bounds their Minkowski dimensions and Hausdorff dimensions above by 0.668. There exist level
sets with Minkowski dimension 1/2. The method of proof involves a multiscale analysis that relies upon the self-similarity
of τ(x) up to affine shifts. 相似文献
16.
Ifn>2 and iff is a locally quasiconformal mapping from the ballB
n= {x∈R
n:⋎x⋎<1} intoR
n ∪ {∞} thenf is injective inB
n (r)={x∈R
n:⋎x⋎ <r} wherer>0 depends only onn and the maximal dilatation off.
Supported in part by the Samuel Neaman Fund, Special Year in Complex Analysis, Technion, I.I.T., Haifa, Israel, 1975/76. 相似文献
17.
Let T : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J andf : J →R denote a Hölder continuous function satisfyingf(x)?log | T′(x vb for allx in J. Then for any pointz 0 in J define the set Dz 0(f) of “well-approximable” points to be the set of points in J which lie in the Euclidean ball $B(\gamma ,{\text{ exp(}} - \sum {_{i - 0}^{\mathfrak{n} - 1} } f(T^\ell x)))$ for infinitely many pairs (y, n) satisfying T n (y)=z0. We prove that the Hausdorff dimension of Dz 0(f) is the unique positive numbers(f) satisfying the equation P(T,?s(f).f)=0, where P is the pressure on the Julia set. This result is then shown to have consequences for the limsups of ergodic averages of Hölder continuous functions. We also obtain local counting results which are analogous to the orbital counting results in the theory of Kleinian groups. 相似文献
18.
Hong Li 《Journal of Mathematical Analysis and Applications》2010,365(1):338-333
Under the simple conditions on f and g, we show that entire positive radial solutions exist for the semilinear elliptic system Δu=p(|x|)f(v), Δv=q(|x|)g(u), x∈RN, N?3, where the functions are continuous. 相似文献
19.
José Luis Bravo Manuel Fernández 《Journal of Mathematical Analysis and Applications》2007,336(1):438-454
We study the structure induced by the number of periodic solutions on the set of differential equations x′=f(t,x) where f∈C3(R2) is T-periodic in t, fx3(t,x)<0 for every (t,x)∈R2, and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold. 相似文献
20.
Wioletta Karpińska 《Journal of Mathematical Analysis and Applications》2007,334(2):1462-1480
In the paper we study the existence and uniqueness of bounded solutions for differential equations of the form: x′−Ax=f(t,x), x″−Ax=f(t,x), where A∈L(Rm), is a Carathéodory function and the homogeneous equations x′−Ax=0, x″−Ax=0 have nontrivial solutions bounded on R. Using a perturbation of the equations, the Leray-Schauder Topological Degree and Fixed Point Theory, we overcome the difficulty that the linear problems are non-Fredholm in any reasonable Banach space. 相似文献