共查询到20条相似文献,搜索用时 46 毫秒
1.
A. K. Steiner undE. F. Steiner described the socalled natural topology κ on spacesA B of transfinite sequences (a β), β∈B,a β∈A [J. Math. Anal. Appl.19, 174–178 (1967)]. These spaces generalize Baire's zerodimensional sequence-spaces. Using these spaces (A B, κ), we generalize two well known theorems of F. Hausdorff, W. Hurewicz, C. Kuratowski and K. Morita on metric spaces and their Lebesgue-dimension respectively, both involving Baire's sequence spaces. Thus we obtain a topological characterization of uniform spaces \((X,\mathfrak{U})\) with a linearly ordered base \(\mathfrak{B}\) of \(\mathfrak{U}\) . 相似文献
2.
T. H. Mark 《manuscripta mathematica》1974,11(3):211-220
Let (X, ) be a complex space and \(\mathfrak{F}\) a coherent -module. In analogy to the reduction red one can define a reduction \(\mathfrak{F}\) red= \(\mathfrak{F}\) / \(\mathfrak{F}\) ′, where \(\mathfrak{F}\) ′ ? \(\mathfrak{F}\) is the subsheaf of “nilvalent” elements of \(\mathfrak{F}\) . (Even if X is reduced, we may have \(\mathfrak{F}\) ′ ≠ 0.) We prove that \(\mathfrak{F}\) ′ is coherent. Therefore we can construct the sheaf \(\mathfrak{F}\) (2)=( \(\mathfrak{F}\) ′)′ of nilvalent elements with respect to \(\mathfrak{F}\) ′. Iterating this process, we get a sequence ( \(\mathfrak{F}\) (n))n∈N of subsheaves of \(\mathfrak{F}\) . We show that on every compact subset of X the sheaves \(\mathfrak{F}\) (n) vanish for n sufficiently large (Satz 2). 相似文献
3.
Eberhard M. Schröder 《Journal of Geometry》1995,53(1-2):167-178
Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques. 相似文献
4.
5.
P. P. Kargaev 《Journal of Mathematical Sciences》1997,85(2):1802-1807
Let F be a compact subset of ? let μ be a Borel measure on F, and let ρ(z) be the distance of z to F. Denote $$A_K (f)(z) = \int\limits_F {K(\varsigma ,z)f(\varsigma ) dm(\varsigma ), z \in \mathbb{C}\backslash F}$$ where K (ζ, z) is either (ζ-z)2 or (|ζ-z|(ζ-z))-1 and m is the Lebesgue measure. Let ψ be a monotone nondecreasing positive function on (0, ∞) and let Φ(z)=Ψ(ρ(z))ρ(z), z ε ?/F. Under some additional assumptions on μ, it is proved that AK is bounded from L2 (μ) to L2 (Φm) if and only if $$\int\limits_0^{ + 1} {\tfrac{{\psi (t)}}{t} + \int\limits_1^\infty {\tfrac{{\psi (t)}}{{t^2 }}dt< \infty } }$$ Thus, no interference of values of K of various signs is observed in such a situation. Bibliography: 4 titles. 相似文献
6.
André Gomes 《Results in Mathematics》2004,46(1-2):31-36
In this work we study a codimension-one C∞-foliation ${cal F}$ of a complete Riemannian manifold M. We assume that ${cal F}$ is transversely orientable. Under this hypothesis, we show that the mean curvature function of ${cal F}$ has a superior limit. Using this result, we find a necessary and sufficient condition for the foliation ${cal F}$ to be totally geodesic. 相似文献
7.
Richárd Balka 《Monatshefte für Mathematik》2014,174(1):1-28
In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$ . The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\mathbb {R}^n$ . In order to do so, we define the $n$ th inductive topological Hausdorff dimension, $\dim _{t^nH} K$ . Let $\dim _H K,\,\dim _t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\mathbb {R}^n$ . We show that $\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n$ for the generic $f \in C_n(K)$ , provided that $\dim _t K\ge n$ , otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces $K$ for which $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$ . We also generalize a result of Kirchheim by showing that if $K$ is self-similar and $\dim _t K\ge n$ then $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in {{\mathrm{int}}}f(K)$ . 相似文献
8.
It is known that the box dimension of any Martin-Löf random closed set of ${\{0,1\}^\mathbb{N}}$ is ${\log_2(\frac{4}{3})}$ . Barmpalias et al. [J Logic Comput 17(6):1041–1062, 2007] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of ${\{0,1\}^\mathbb{N}}$ , and propose a general method for random closed sets in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets. 相似文献
9.
Parseval frames have particularly useful properties, and in some cases, they can be used to reconstruct signals which were analyzed by a non-Parseval frame. In this paper, we completely describe the degree to which such reconstruction is feasible. Indeed, notice that for fixed frames \({\mathcal{F}}\) and \({\mathcal{X}}\) with synthesis operators F and X, the operator norm of FX ??I measures the (normalized) worst-case error in the reconstruction of vectors when analyzed with \({\mathcal{X}}\) and synthesized with \({\mathcal{F}}\) . Hence, for any given frame \({\mathcal{F}}\) , we compute explicitly the infimum of the operator norm of FX ??I, where \({\mathcal{X}}\) is any Parseval frame. The \({\mathcal{X}}\) ’s that minimize this quantity are called Parseval quasi-dual frames of \({\mathcal{F}}\) . Our treatment considers both finite and infinite Parseval quasi-dual frames. 相似文献
10.
In this paper we study the arithmetic of Artin–Schreier extensions of $\mathbb {F}_{q}(T)$ . We determine the integral closure of $\mathbb {F}_{q}[T]$ in Artin–Schreier extension of $\mathbb {F}_{q}(T)$ . We also investigate the average values of the $L$ -functions of orders of Artin–Schreier extensions and study the average values of ideal class numbers when $p=3$ in detail. 相似文献
11.
Let \(B\) be an \(n\times n\) real expanding matrix and \(\mathcal {D}\) be a finite subset of \(\mathbb {R}^n\) with \(0\in \mathcal {D}\) . The self-affine set \(K=K(B,\mathcal {D})\) is the unique compact set satisfying the set-valued equation \(BK=\bigcup _{d\in \mathcal {D}}(K+d)\) . In the case where \(\#\mathcal D=|\det B|,\) we relate the Lebesgue measure of \(K(B,\mathcal {D})\) to the upper Beurling density of the associated measure \(\mu =\lim _{s\rightarrow \infty }\sum _{\ell _0, \ldots ,\ell _{s-1}\in \mathcal {D}}\delta _{\ell _0+B\ell _1+\cdots +B^{s-1}\ell _{s-1}}.\) If, on the other hand, \(\#\mathcal D<|\det B|\) and \(B\) is a similarity matrix, we relate the Hausdorff measure \(\mathcal {H}^s(K)\) , where \(s\) is the similarity dimension of \(K\) , to a corresponding notion of upper density for the measure \(\mu \) . 相似文献
12.
Henrik Shahgholian 《Arkiv f?r Matematik》1994,32(2):475-492
This paper deals with a free boundary porblem connected with the concept “quadrature surface”. Let Ω?R n be a bounded domain with aC 2 boundary and μ a measure compactly supported in Ω. Then we say ?Ω is a quadrature surface with respect to μ if the following overdetermined Cauchy problem has a solution. $$\Delta u = - \mu in \Omega ,u = 0 and \frac{{\partial u}}{{\partial v}} = - 1 on \partial \Omega .$$ Applying simple techniques, we derive basic inequalities and show uniform boundedness for the set of solutions. Distance estimates as well as uniqueness results are obtained in special cases, e.g. we show that if ?Ω and ?D are two quadrature surfaces for a fixed measure μ and Ω is convex, thenD?Ω. The main observation, however, is that if ?Ω is a quadrature surface for μ≥0 andxε?Ω, then the inward normal ray to ?Ω atx intersects the convex hull of supp μ. We also study relations between quadrature surfaces and quadrature domains.D is said to be a quadrature domain with respect to a mesure μ if there is a solution to the following overdetermined Cauchy problem: $$\Delta u = 1 - \mu in D, andu = |\nabla u| = 0 on \partial D.$$ Finally, we apply our results to a problem of electrochemical machining. 相似文献
13.
We consider the case of hyperbolic basic sets $\Lambda $ of saddle type for holomorphic maps $f:{\mathbb{P }}^2{\mathbb{C }}\rightarrow {\mathbb{P }}^2{\mathbb{C }}$ . We study equilibrium measures $\mu _\phi $ associated to a class of Hölder potentials $\phi $ on $\Lambda $ , and find the measures $\mu _\phi $ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $\delta _{\mu _\phi }$ of $\mu _\phi $ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $\mu _\phi $ in the case when the preimage counting function is constant on $\Lambda $ . For terminal/minimal saddle sets we prove that an invariant measure $\nu $ obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the restriction $f|_\Lambda $ . This allows then to obtain formulas for the measure $\nu $ of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of $\nu $ . 相似文献
14.
Andrea Colesanti Daniel Hug Eugenia Saorín Gómez 《Journal of Geometric Analysis》2014,24(2):1064-1091
We consider a functional $\mathcal{F}$ on the space of convex bodies in ? n of the form $$ {\mathcal{F}}(K)=\int_{\mathbb{S}^{n-1}} f(u) \mathrm{S}_{n-1}(K,du), $$ where $f\in C(\mathbb{S}^{n-1})$ is a given continuous function on the unit sphere of ? n , K is a convex body in ? n , n≥3, and S n?1(K,?) is the area measure of K. We prove that $\mathcal{F}$ satisfies an inequality of Brunn–Minkowski type if and only if f is the support function of a convex body, i.e., $\mathcal{F}$ is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree n?1 and satisfy a Brunn–Minkowski type inequality. 相似文献
15.
We obtain fractal Lipschitz–Killing curvature-direction measures for a large class of self-similar sets $F$ in $\mathbb{R }^{d}$ . Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable sub-manifolds. They decouple as independent products of the unit Hausdorff measure on $F$ and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions. 相似文献
16.
Yurii Khomskii 《Archive for Mathematical Logic》2014,53(1-2):57-64
Todor?evi? (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$ . 相似文献
17.
David Ralston 《Monatshefte für Mathematik》2014,175(4):595-612
We investigate the set of \(x \in S^1\) such that for every positive integer \(N\) , the first \(N\) points in the orbit of \(x\) under rotation by irrational \(\theta \) contain at least as many values in the interval \([0,1/2]\) as in the complement. By using a renormalization procedure, we show both that the Hausdorff dimension of this set is the same constant (strictly between zero and one) for almost-every \(\theta \) , and that for every \(d \in [0,1]\) there is a dense set of \(\theta \) for which the Hausdorff dimension of this set is \(d\) . 相似文献
18.
Dan Liu 《Calculus of Variations and Partial Differential Equations》2011,41(1-2):179-202
In this paper, we study the critical set of a complex-valued solution to a Schr?dinger equation involving the magnetic field and with a nonlinear term, where the critical set is ${\{x\in\Omega:~\psi(x)=0, ~\nabla\psi(x)=0\}}$ . We consider this equation in a bounded domain of ${\mathbb{R}^3}$ with the boundary condition: ${\nabla _{\mathbf{A}}\psi\cdot \nu=0}$ , and we establish a global 1-dimensional Hausdorff measure estimate for the critical sets. From the proof of global estimates, we find that our methods work as well for more general equations with a magnetic potential. 相似文献
19.
In the present paper we introduce a new concept of $A$ -distributional convergence in an arbitrary Hausdorff topological space which is equivalent to $A$ -statistical convergence for a degenerate distribution function. We investigate $A$ -distributional convergence as a summability method in an arbitrary Hausdorff topological space. We also study the summability of spliced sequences, in particular, for metric spaces and give the Bochner integral representation of $A$ -limits of the spliced sequences for Banach spaces. 相似文献
20.
Diego Moreira Lihe Wang 《Calculus of Variations and Partial Differential Equations》2014,49(3-4):1237-1261
In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: $ F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) $ and $ \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)$ , where $\beta _{\varepsilon }$ approaches Dirac $\delta _{0}$ as $\varepsilon \rightarrow 0$ and $f_{\varepsilon }$ has a uniform control in $L^{q}, q>N.$ Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the $\varepsilon -$ level surfaces are established for these variational and nonvaritional solutions. Finally, letting $\varepsilon \rightarrow 0$ basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained. 相似文献