Rn中积分中值点取值范围的改进

将Rn中积分中值定理的中值点取值范围,由积分区域D上缩小到D的内部DaD上取到.     

Sn中Moebius形式为零的曲面

本文证明了S     

Cn中点乘子集M(Dp)的一个刻划

本文用Dp-Carlesor测度对Cn中的单位球B上的Dirichlet型空间Dp的点乘子集M(Dp)就p＜2k的情形进行一种刻划.     

On the functional equationS(m,n)F[n/(m,n)] =F(n)h[n/(m,n)]

In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σ _d|n ?(d)h(n/d) andS(m n) = Σd|(m, n) ?(d)h(n/d). The well-known Hölder's identity is a special case of this functional equation (?(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions.     

Die Gleichungax n −by n =c

Ohne ZusammenfassungGeschrieben in Dankbarkeit und Verehrung für Edmund Landau zu seinem 60. Geburtstag am 14. Februar 1937     

n2-sets in a projective plane which determine exactly n2+n lines

In this paper n²-sets, in a projective plane, determining exactly n²+n lines are completely characterized.Research partially supported by G.N.S.A.G.A. (C.N.R.).Paper written while the second author was at the University of Rome supported by a grant within the Italian Hungarian cultural exchange program.     

The log-behavior of $$\root n \of {p(n)}$$ and $$\root n \of {p(n)/n}$$p(n)/nn

Let p(n) denote the partition function and let \(\Delta \) be the difference operator with respect to n. In this paper, we obtain a lower bound for \(\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}\), leading to a proof of a conjecture of Sun on the log-convexity of \(\{\root n \of {p(n)/n}\}_{n\ge 60}\). Using the same argument, it can be shown that for any real number \(\alpha \), there exists an integer \(n(\alpha )\) such that the sequence \(\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}\) is log-convex. Moreover, we show that \(\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}\). Finally, by finding an upper bound for \(\Delta ^2 \log \root n-1 \of {p(n-1)}\), we establish an inequality on the ratio \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\).     

Existence of 2 n Solutions of a System of n Nonlinear Equations in n Unknowns

It is demonstrated that, under certain conditions, a system of n nonlinear equations with n unknowns has at least 2 ⁿ solutions. Bibliography: 2 titles.     

Manifolds homotopy equivalent to P n # P n

We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space. Partially supported by a grant from the National Science Foundation.     

Stopping rules forx n/n and related problems

The following results illustrate the problems with which this note deals. Letx _n (n=1, 2, ...) be non-negative, independent, identically distributed random variables, letβ>1 andEx ₁ ^β <∞. Then there exists a stopping ruleτ withP{τ<∞}=1, which maximizesE x _t/t among all stopping rulest. Moreover, the same rule maximizesE max (x ₁, ...,x _t)/t andE max (x ₁,..,x _τ)/τ=Ex _τ/τ Research supported in part by Grant GP-5705 of the National Science Foundation, USA.     

Subspace Arrangements of Type B n and D n

Let ${\mathcal{D}}_{n,k} $ be the family of linear subspaces of ?ⁿ given by all equations of the form $\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$ for 1 ≤ < ? ? ? < i _k ≤ i and $\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $ Also let ${\mathcal{B}}_{n,k,h} $ be ${\mathcal{D}}_{n,k} $ enlarged by the subspaces $x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$ for 1 ≤. The special cases ${\mathcal{B}}_{n,2,1} $ and ${\mathcal{D}}_{n,2} $ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B _nand D _n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of ${\mathcal{B}}_{n,k,h,} 1 \leqslant h < k$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $\begin{gathered} {\mathcal{D}}_{n,2} \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \\ \end{gathered} $ . For instance, it is shown that $H^d \left( {M_{n,k,k - 1} } \right)$ is torsion-free and is nonzero if and only if d = t(k ? 2) for some $t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-0em} k}} \right]$ . Torsion-free cohomology follows also for the complement in ?ⁿof the complexification ${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h < k$ .