共查询到20条相似文献,搜索用时 31 毫秒
1.
Janusz Matkowski 《Central European Journal of Mathematics》2012,10(6):2215-2228
Let (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n ) n∈? of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit $$\mu _T (x) = \mathop {\lim }\limits_{n \to \infty } T^n (x)$$ is a fixed point of T. The problem of determining the form of µ T leads to the invariance equation µ T ○ T = µ T , which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p , where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M 1,...,M p ) of I p . In this paper we give the explicit forms of µM for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive. 相似文献
2.
An increasing triangular mapping T on the n-dimensional cube Θ = [0, 1] n transforming a measure μ to a measure ν is considered, where μ and ν are absolutely continuous Borel probability measures having densities ρ μ and ρ ν . It is shown that if there exist positive constants ? and M such that ? < ρ ν < M, ? < ρ ν < M, there exist numbers α, β > 1 such that p = αβ(n ? 1)?1 (α + β)?1 > 1 and ρ μ ∈ W 1,α (Θ), ρ ν ∈ W 1,β ) (Θ), where W 1,q denotes a Sobolev class, then the mapping T belongs to the class W 1,p (Θ). 相似文献
3.
A. V. Kolesnikov 《Mathematical Notes》2010,88(5-6):678-695
We generalize the well-known result due to Caffarelli concerning Lipschitz estimates for the optimal transportation T of logarithmically concave probability measures. Suppose that T: ? d → ? d is the optimal transportation mapping µ = e ?V dx to ν = e ?W dx. Suppose that the second difference-differential V is estimated from above by a power function and that the modulus of convexity W is estimated from below by the function A q |x|1+q , q ≥ 1. We prove that, under these assumptions, the mapping T is globally Hölder with the Hölder constant independent of the dimension. In addition, we study the optimal mapping T of a measure µ to Lebesgue measure on a convex bounded set K ? ? d . We obtain estimates of the Lipschitz constant of the mapping T in terms of d, diam(K), and DV, D 2 V. 相似文献
4.
Let M be a compact Riemannian manifold and let μ, d be the associated measure and distance on M. Robert McCann, generalizing results for the Euclidean case by Yann Brenier, obtained the polar factorization of Borel maps S: M → M pushing forward μ to a measure ν: each S factors uniquely a.e. into the composition S = T ? U, where U: M → M is volume preserving and T: M → M is the optimal map transporting μ to ν with respect to the cost function d2/2.In this article we study the polar factorization of conformal and projective maps of the sphere S n . For conformal maps, which may be identified with elements of Oo(1, n+1), we prove that the polar factorization in the sense of optimal mass transport coincides with the algebraic polar factorization (Cartan decomposition) of this Lie group. For the projective case, where the group GL+(n + 1) is involved, we find necessary and sufficient conditions for these two factorizations to agree. 相似文献
5.
Z. Buczolich 《Acta Mathematica Hungarica》2007,117(1-2):91-140
We construct a sequence (n
k
) such that n
k + 1 − n
k
→ ∞ and for any ergodic dynamical system (X, Σ, μ, T) and f ε L
1(μ) the averages converge to ∝
X
f
dμ for μ almost every x. Since the above sequence is of zero Banach density this disproves a conjecture of J. Rosenblatt and M. Wierdl about the
nonexistence of such sequences.
Research supported by the Hungarian National Foundation for Scientific research T049727. 相似文献
6.
Let P1,…,Pn be generic homogeneous polynomials in n variables of degrees d1,…,dn respectively. We prove that if ν is an integer satisfying ∑i=1ndi?n+1?min{di}<ν, then all multivariate subresultants associated to the family P1,…,Pn in degree ν are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of smooth isolated points in To cite this article: L. Busé, C. D'Andrea, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 相似文献
7.
Let Tn, n = 1,2,… be a sequence of linear contractions on the space where is a finite measure space. Let M be the subspace of L1 for which Tng → g weakly in L1 for g?M. If Tn1 → 1 strongly, then Tnf → f strongly for all f in the closed vector sublattice in L1 generated by M.This result can be applied to the determination of Korovkin sets and shadows in L1. Given a set G ? L1, its shadow S(G) is the set of all f?L1 with the property that Tnf → f strongly for any sequence of contractions Tn, n = 1, 2,… which converges strongly to the identity on G; and G is said to be a Korovkin set if S(G) = L1. For instance, if 1 ?G, then, where M is the linear hull of G and M is the sub-σ-algebra of generated by {x?X: g(x) > 0} for g?M. If the measure algebra is separable, has Korovkin sets consisting of two elements. 相似文献
8.
G. A. Kalyabin 《Proceedings of the Steklov Institute of Mathematics》2014,284(1):161-167
Explicit upper and lower estimates are given for the norms of the operators of embedding of , n ∈ ?, in L q (dµ), 0 < q < ∞. Conditions on the measure µ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ1, n, β , β > 0, of the boundary value problems (?d 2/dx 2) n u(x) = λ|x| β?1, x ∈ (?1, 1), u (k)(±1) = 0, k ∈ {0, 1, ..., n ? 1}. 相似文献
9.
Elifalet López-González 《Bulletin of the Brazilian Mathematical Society》2013,44(3):393-412
We consider Riccati foliations ?ρ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π 1(S) → PSL(2,?) in a quasi-Fuchsian group. The foliated geodesic flow has a repeller-attractor dynamic with generic statistics µ+ and µ? for positive and negative times, respectively. These measures have a common projection to a harmonic measure μρ for the Riccati foliation. We describe μ ρ + , μ ρ - and μρ in terms of the Patterson-Sullivan construction, and we show that the measures μρ provide examples of the conformal harmonic measures introduced by M. Brunella. 相似文献
10.
Hermann König Vitali Milman 《Proceedings of the Steklov Institute of Mathematics》2013,280(1):191-207
We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T 1, T 2,A: C k (?) → C(?) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ C k (?), namely, V (f · g) = (T 1 f) · g + f · (T 2 g) for k = 1, V (f · g) = (T 1 f) · g + f · (T 2 g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T 1 f) ○ g · (T 2 g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T 1 f) · (Ag) + (Af) · (T 2 g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T 1 and T 2 must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · Tg + B(f, g), and show under suitable conditions on B in the first case that B = 0 and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation. 相似文献
11.
A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20. 相似文献
12.
E.B Dynkin 《Journal of Functional Analysis》1985,62(3):397-434
Let Xt be the Brownian motion in d. The random set Γ = {(t1,…, tn, z): Xtl = ··· = Xtn = z} in d + n is empty a.s. except in the following cases: (a) n = 1, d = 1, 2,…; (b) d = 2, n = 2, 3,…; (c) d = 3, n = 2. In each of these cases, a family of random measures Mλ concentrated on Γ is constructed (λ takes values in a certain class of measures on d). Measures Mλ characterize the time-space location of self-intersections for Brownian paths. If n = d = 1, then Mλ(dt, dz) = λ(dz) Nz(dt) where N2 is the local time at z. In the case n = 2, the set Γ can be identified with the set of Brownian loops. The measure Mλ “explodes” on the diagonal {t1 = t2} and, to study small loops, a random distribution which regularizes Mλ is constructed. 相似文献
13.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1998,326(3):347-352
Let Td : L2([0, 1]d) → C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k−1 (log k)d− 1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(−Cɛ−2¦ log ɛ¦2d−1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms. 相似文献
14.
Leandro Arosio Filippo Bracci Hidetaka Hamada Gabriela Kohr 《Journal d'Analyse Mathématique》2013,119(1):89-114
We present a new geometric construction of Loewner chains in one and several complex variables which holds on complete hyperbolic complex manifolds and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence, we obtain a univalent solution (f t : M → N) of any Loewner-Kufarev PDE. The problem of finding solutions given by univalent mappings (f t : M → ? n ) is reduced to investigating whether the complex manifold ∪ t≥0 f t (M) is biholomorphic to a domain in ? n . We apply such results to the study of univalent mappings f: B n → ? n . 相似文献
15.
Both of the following conditions are equivalent to the absoluteness of a norm ν in n: (1) for all n×n diagonal matrices D=(dk), the subordinate operator norm Nν(D)=maxk|dk|; (2) for all n×n matrices A, Nν(A) ?Nν(|A|). These conditions are modified for partitioned matrices by replacing absolute values with norms of blocks. A generalization of absoluteness is thus obtained. 相似文献
16.
Let ρ be a real-valued function on [0, T], and let LSI(ρ) be a class of Gaussian processes over time interval [0, T], which need not have stationary increments but their incremental variance σ(s, t) is close to the values ρ(|t ? s|) as t → s uniformly in s ∈ (0, T]. For a Gaussian processesGfrom LSI(ρ), we consider a power variation V n corresponding to a regular partition π n of [0, T] and weighted by values of ρ(·). Under suitable hypotheses on G, we prove that a central limit theorem holds for V n as the mesh of π n approaches zero. The proof is based on a general central limit theorem for random variables that admit a Wiener chaos representation. The present result extends the central limit theorem for a power variation of a class of Gaussian processes with stationary increments and for bifractional and subfractional Gaussian processes. 相似文献
17.
I. V. Shragin 《Mathematical Notes》2006,80(5-6):868-874
Given a measurable space (T, F), a set X, and a map ?: T → X, the σ-algebras N Ф = ??∈Φ N ?, and M Φ = ??∈Φ N ?, where G ?(t) = (t, ?(t)) and Φ ? X T , are considered. These σ-algebras are used to characterize the (F, B, ?)-measurability of the compositions g ○ ? and f о G ?, where g: X → Y, f: T × X → Y, and (Y, ?) is a measurable space. Their elements are described without using the operations ? ?1 and G ? ?1 . 相似文献
18.
Let M be an n-dimensional complete Riemannian manifold with Ricci curvature n- 1. By developing some new techniques, Colding(1996) proved that the following three conditions are equivalent: 1)dGH(M, S~n) → 0; 2) the volume of M Vol(M) → Vol(S~n); 3) the radius of M rad(M) →π. By developing a different technique, Petersen(1999) gave the 4th equivalent condition, namely he proved that the n + 1-th eigenvalue of M, λ_(n+1)(M) → n, is also equivalent to the radius of M, rad(M) →π, and hence the other two.In this paper, we use Colding's techniques to give a new proof of Petersen's theorem. We expect our estimates will have further applications. 相似文献
19.
V. S. Panferov 《Mathematical Notes》1973,14(5):936-942
Let ∥·∥ be a norm in R2 and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R2 rational if (x1 ? y1)/(x2 ? y2) or (x2 ? y2)/(x1 ? y1) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T2 of positive planar measure implies ∥T v ∥L4 (T2) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T v } and a set E ? T2, ¦E¦ > 0, such that T v (x) → 0(v → ∞) on E; however, ¦cn¦ ? 0 for ∥n∥ → ∞. 相似文献
20.
Xavier Dahan 《Combinatorica》2014,34(4):407-426
For every integer d≥10, we construct infinite families {G n } n∈? of d+1-regular graphs which have a large girth ≥log d |G n |, and for d large enough ≥1.33 · log d |G n |. These are Cayley graphs on PGL 2(F q ) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is a prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I n } n∈? of d + 1-regular graphs, realized as Cayley graphs on SL 2(F q ), and which are displaying a girth ≥0.48·log d |I n |. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M n } n∈N of 2 k +1-regular graphs were shown to have girth ≥2/3·log2 k |M n |. 相似文献