共查询到20条相似文献,搜索用时 875 毫秒
1.
Li Songying 《数学学报(英文版)》1988,4(2):97-110
In this paper, we shall prove the existence of the singular directions related to Hayman's problems[1]. The results are as follows.
- Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H: argz= θ0 (0≤θ0>2π) such that for every positive ε, every integer p(≠0, ?1) and every finite complex number b(≠0), we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' \cdot \{ f\} ^p = b)} \right\} = + \infty $$
- Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive ε, every integrer p(≥3) and any finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$
- Suppose that f(z) is a meromorphic function in the finite plane and satisfies the following condition $$\mathop {\lim }\limits_{r \to \infty } \frac{{T(r,f)}}{{(\log r)^3 }} = + \infty $$ then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive ε, every integer p(≥5) and every two finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$
2.
Liyuan Chen 《分析论及其应用》1994,10(4):56-71
In this paper,for the plane curve T=.we define an analytic family of maximal functions asso-ciated to T asM_2f(λ)=sup_n>oh~-1∫_R相似文献
3.
Frédéric Maire 《Graphs and Combinatorics》1994,10(2-4):263-268
A graph istriangulated if it has no chordless cycle with at least four vertices (?k ≥ 4,C k ?G). These graphs Jhave been generalized by R. Hayward with theweakly triangulated graphs $(\forall k \geqslant 5,C_{k,} \bar C_k \nsubseteq G)$ . In this note we propose a new generalization of triangulated graphs. A graph G isslightly triangulated if it satisfies the two following conditions;
- G contains no chordless cycle with at least 5 vertices.
- For every induced subgraphH of G, there is a vertex inH the neighbourhood of which inH contains no chordless path of 4 vertices.
4.
M. Pellegrini 《Discrete and Computational Geometry》1994,12(1):203-221
We show some combinatorial and algorithmic results concerning finite sets of lines and terrains in 3-space. Our main results include:
- An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of the set of lines that can be translated to infinity without intersecting a given finite set ofn lines, wherec is a suitable constant. This bound is almost tight.
- AnO(n 1.5+ε) randomized expected time algorithm that tests whether a directionv exists along which a set ofn red lines can be translated away from a set ofn blue lines without collisions. ε>0 is an arbitrary small but fixed constant.
- An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of theenvelope of lines above a terrain withn edges, wherec is a suitable constant.
- An algorithm for computing the intersection of two polyhedral terrains in 3-space withn total edges in timeO(n 4/3+ε+k 1/3 n 1+ε+klog2 n), wherek is the size of the output, and ε>0 is an arbitrary small but fixed constant. This algorithm improves on the best previous result of Chazelleet al. [5].
5.
Let $\mathcal{K}$ be the family of graphs on ω1 without cliques or independent subsets of sizew 1. We prove that
- it is consistent with CH that everyGε $\mathcal{K}$ has 2ω many pairwise non-isomorphic subgraphs,
- the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω1 either G?G[A] orG?G[B],
- the failure of (*) is consistent with ZFC.
6.
7.
Our main results are:
- Let α ≠ 0 be a real number. The function (Γ ? exp) α is convex on ${\mathbf{R}}$ if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x 0 = 1.4616... denotes the only positive zero of ${\psi = \Gamma'/\Gamma}$ .
- Assume that a function f: (0, ∞) → (0, ∞) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$
8.
Horst Herrlich 《Applied Categorical Structures》1996,4(1):1-14
In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:
- C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
- Equivalent are:
- the axiom of choice,
- A-compactness = D-compactness,
- B-compactness = D-compactness,
- C-compactness = D-compactness and complete regularity,
- products of spaces with finite topologies are A-compact,
- products of A-compact spaces are A-compact,
- products of D-compact spaces are D-compact,
- powers X k of 2-point discrete spaces are D-compact,
- finite products of D-compact spaces are D-compact,
- finite coproducts of D-compact spaces are D-compact,
- D-compact Hausdorff spaces form an epireflective subcategory of Haus,
- spaces with finite topologies are D-compact.
- Equivalent are:
- the Boolean prime ideal theorem,
- A-compactness = B-compactness,
- A-compactness and complete regularity = C-compactness,
- products of spaces with finite underlying sets are A-compact,
- products of A-compact Hausdorff spaces are A-compact,
- powers X k of 2-point discrete spaces are A-compact,
- A-compact Hausdorff spaces form an epireflective subcategory of Haus.
- Equivalent are:
- either the axiom of choice holds or every ultrafilter is fixed,
- products of B-compact spaces are B-compact.
- Equivalent are:
- Dedekind-finite sets are finite,
- every set carries some D-compact Hausdorff topology,
- every T 1-space has a T 1-D-compactification,
- Alexandroff-compactifications of discrete spaces and D-compact.
9.
V. N. Potapov 《Mathematical Notes》2013,93(3-4):479-486
Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1]. 相似文献
10.
Yuri Bilu 《Israel Journal of Mathematics》1995,90(1-3):235-252
LetK be an algebraic number field,S?S \t8 a finite set of valuations andC a non-singular algebraic curve overK. Letx∈K(C) be non-constant. A pointP∈C(K) isS-integral if it is not a pole ofx and |x(P)| v >1 impliesv∈S. It is proved that allS-integral points can be effectively determined if the pair (C, x) satisfies certain conditions. In particular, this is the case if
- x:C→P 1 is a Galois covering andg(C)≥1;
- the integral closure of $\bar Q$ [x] in $\bar Q$ (C) has at least two units multiplicatively independent mod $\bar Q$ *.
11.
If φ: [0, 1) → (0,∞) is a non-decreasing unbounded function, then the φ-order of a meromorphic function f in the unit disc is defined as $$ \sigma _\phi (f) = \mathop {\lim \sup }\limits_{r \to 1^ - } \frac{{\log ^ + T(r,f)}} {{\log \phi (r)}}, $$ where T(r, f) is the Nevanlinna characteristic of f. In particular, $ \sigma _{\tfrac{1} {{1 - r}}} $ f is the order of f, and $ \sigma _{\log \tfrac{1} {{1 - r}}} $ f is the logarithmic order of f. Several results on the finiteness of the φ-order of solutions of $$ f^{(k)} + A_{k - 1} (z)f^{(k - 1)} + \cdots + A_1 (z)f' + A_0 (z)f = 0 $$ are obtained in the case when the coefficients A 0(z), ...,A k?1(z) are analytic functions in the unit disc. This paper completes some earlier results by various authors. 相似文献
12.
Ramon van Handel 《Probability Theory and Related Fields》2013,155(3-4):911-934
Let ${\mathcal{F}}$ be a separable uniformly bounded family of measurable functions on a standard measurable space ${(X, \mathcal{X})}$ , and let ${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$ be the smallest number of ${\varepsilon}$ -brackets in L 1(μ) needed to cover ${\mathcal{F}}$ . The following are equivalent:
- ${\mathcal{F}}$ is a universal Glivenko–Cantelli class.
- ${N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}$ for every ${\varepsilon > 0}$ and every probability measure μ.
- ${\mathcal{F}}$ is totally bounded in L 1(μ) for every probability measure μ.
- ${\mathcal{F}}$ does not contain a Boolean σ-independent sequence.
13.
V. A. Baskakov 《Analysis Mathematica》1983,9(1):9-22
Для линейных методов суммирования рядов Ф урье (1) $$L_n (f;x) = \frac{1}{\pi }\mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2} + \sum\limits_{k = 1}^n {\lambda _{k,n} } \cos kt} \right)dt$$ на классах $$C(\varepsilon ) = \{ f:E_n (f) \leqq \varepsilon _n ;\forall n \geqq 0\} ,\varepsilon = \{ \varepsilon _n \} _{n = 0.}^\infty \varepsilon _n \downarrow 0,$$ доказываются:
- оценки для порядка р оста норм ∥{Ln∥, если из вестен порядок приближения операторами (1) некоторого классаС (?) (при этом, если опера торы (1) приближают класс С(е) с наилучшим порядком, то находится точная а симптотика возрастания норм {∥ Ln∥);
- сравнительные оцен ки порядков приближе ния классовС(?) операторами (1), если известен порядок при ближения ими некотор ого более узкого класса С(?*).
14.
Letk n be the smallest constant such that for anyn-dimensional normed spaceX and any invertible linear operatorT ∈L(X) we have $|\det (T)| \cdot ||T^{ - 1} || \le k_n |||T|^{n - 1} $ . LetA + be the Banach space of all analytic functionsf(z)=Σ k≥0 a kzk on the unit diskD with absolutely convergent Taylor series, and let ‖f‖A +=σ k≥0 |ακ|; define ? n on $\overline D ^n $ by $ \begin{array}{l} \varphi _n \left( {\lambda _1 ,...,\lambda _n } \right) \\ = inf\left\{ {\left\| f \right\|_{A + } - \left| {f\left( 0 \right)} \right|; f\left( z \right) = g\left( z \right)\prod\limits_{i = 1}^n {\left( {\lambda _1 - z} \right), } g \in A_ + , g\left( 0 \right) = 1 } \right\} \\ \end{array} $ . We show thatk n=sup {? n(λ1,…, λ n ); (λ1,…, λ n )∈ $\overline D ^n $ }. Moreover, ifS is the left shift operator on the space ?∞:S(x 0,x 1, …,x p, …)=(x 1,…,x p,…) and if Jn(S) denotes the set of allS-invariantn-dimensional subspaces of ?∞ on whichS is invertible, we have $k_n = \sup \{ |\det (S|_E )|||(S|_E )^{ - 1} ||E \in J_n (S)\} $ . J. J. Schäffer (1970) proved thatk n≤√en and conjectured thatk n=2, forn≥2. In factk 3>2 and using the preceding results, we show that, up to a logarithmic factor,k n is of the order of √n whenn→+∞. 相似文献
15.
K. -J. Wirths 《Analysis Mathematica》1975,1(4):313-318
Последовательность {itak} (n) k =1/∞ вещественных ч исел называется дважды мо нотонной, еслиa k -2a k+1 +a k+2 ≧0 дляk≧1. В работе доказываютс я следующие утвержде ния, являющиеся обобщени ем двух теорем Фейера:
- Если {itak — дважды моно тонная последовател ьность, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^n {a_\kappa z^\kappa } > 1/2$$ дляи≧ 1.
- Если О≦β<1 и последова тельность (k+1-2β)ak} дважд ы монотонна, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {ka_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } > \beta $$ , то есть $$\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } \varepsilon S_\beta ^\kappa $$ . При помощи 2) получены о бобщения и уточнения теорем из работы [1] о линейных комбинациях некотор ых однолистных функц ий.
16.
O. M. Fomenko 《Journal of Mathematical Sciences》2002,110(6):3143-3149
Let Sk(Γ0(N)Ψ) be the space of holomorphic Γ0(N)-cusp forms of integral weight k and of character Ψ(mod n), let f(z) be a newform of the space Sk(Γ0(N),Ψ), and let Lf(s) be the corresponding L-function. The following statements are proved. (1) Let $\mathcal{F}_0 $ be the set of all newforms of Sk(Γ0(p),1), let p be prime, and let k≥2 be a constant even number. Then $\sum\limits_{f \in \mathcal{F}_0 :L_f (k/2) \ne 0} {1 \gg \frac{p}{{\log ^2 p}}} {\text{ (}}p \to \infty ).$ (2) Let $\mathcal{F}_0 $ be the set of all Hecke eigenforms of the space Sk(Γ0(1),1) and let k≡0 (mod 4). Then $\sum\limits_{f \in \mathcal{F}_0 :L_f (k/2) \ne 0} {1 \gg \frac{k}{{\log ^2 p}}} {\text{ (}}k \to \infty ).$ Bibliography: 11 titles. 相似文献
17.
We characterize functional equations of the form ${f(zf(z))=f(z)^{k+1},z\in\mathbb {C}}$ , with ${k\in\mathbb N}$ , like those generalized Dhombres equations ${f(zf(z))=\varphi (f(z))}$ , ${z\in\mathbb C}$ , with given entire function ${\varphi}$ , which have a nonconstant polynomial solution f. 相似文献
18.
Denoting byS k (ξ k ) the set of solutions of the Cauchy problem $\dot x \in F_k (t,x),x(0) = \xi _k $ , fork∈N∪{∞}, we prove that, under appropriate assumptions, the sequence {S k (ξ k )} k ∈ N converges toS ∞(∈∞) in the Kuratowski sense as well as in the Mosco sense. This result together with some facts from Γ-convergence theory are used to prove a result concerning the existence and the asymptotic behavior of the minima to the optimization problem $$\min \int_0^T {[g_k (t,x(t)) + h_k (t,\dot x(t))]} dt + \psi _k (\xi ),x \in S_k (\xi ),\xi \in K$$ withK a compact subset ofR n . 相似文献
19.
Yu. Ya. Vymenets 《Journal of Mathematical Sciences》1997,87(5):3828-3858
The resutls of this paper show that the structure of sets mentioned in the title is not trivial. For example, it is shown that there exist countalbe sets of uniqueness for logarithmic potential, i.e., closed countable subsets E of the unit circle $\mathbb{T}$ such that $$f \in C(\mathbb{T}),f|_E = 0,U^f |_E = 0 \Rightarrow f \equiv 0.$$ Here $U^f (z) = \tfrac{1}{\pi }\int\limits_0^{2\pi } {f(e^{i\theta } )\log \tfrac{1}{{\left| {z - e^{i\theta } } \right|}}d\theta } $ . On the other hand, it is shown that every countable porous closed subset of $\mathbb{T}$ is a nonuniqueness set. Bibliography: 9 titles. 相似文献
20.
LetS be a locally compact (σ-compact) group or semi-group, and letT(t) be a continuous representation ofS by contractions in a Banach spaceX. For a regular probability μ onS, we study the convergence of the powers of the μ-averageUx=∫T(t)xdμ(t). Our main results for random walks on a groupG are:
- if μ is adapted and strictly aperiodic, and generates a recurrent random walk, thenU n (U-I) converges strongly to 0. In particular, the random walk is completely mixing.
- If μ×μ is ergodic onG×G, then for every unitary representationT(.) in a Hilbert space,U n converges strongly to the orthogonal projection on the space of common fixed points. These results are proved for semigroup representations, along with some other results (previously known only for groups) which do not assume ergodicity.
- If μ is spread-out with supportS, then $\left\| {\mu ^{n + K} - \mu ^n } \right\| \to 0$ if and only if e $ \in \overline { \cup _{j = 0}^\infty S^{ - j} S^{j + K} } .$ .