An example is given of a topological space that does not admit a strongly complete quasi-uniform structure.

This short note refines a noncommutative (nc) Oka–Weil theorem by using a characterization of free compact nc sets based on the notion of dilation hulls. A consequence of it is that any free holomorphic function can be represented as a free polynomial on each free compact nc set.

In this note we study the finite groups whose subgroup lattices are dismantlable.

Let $\omega ^*(n)$ be the number of primes p such that $p-1$ divides n. Recently, M. R. Murty and V. K. Murty proved that

$$ \begin{align*}x(\log\log x)^3\ll\sum_{n\le x}\omega^*(n)^2\ll x\log x.\end{align*} $$

They further conjectured that there is some positive constant C such that

$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\sim Cx\log x,\end{align*} $$

as $x\rightarrow \infty $ . In this short note, we give the correct order of the sum by showing that

$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\asymp x\log x.\end{align*} $$

We provide a novel proof of the existence of regulator indecomposables in the cycle group $C{{H}^{2}}\left( X,\,1 \right)$, where $X$ is a sufficiently general product of two elliptic curves. In particular, the nature of our proof provides an illustration of Beilinson rigidity.

Let $\Omega $ be a bounded Reinhardt domain in $\mathbb {C}^n$ and $\phi _1,\ldots ,\phi _m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{\phi _m}\cdots T_{\phi _1}=0$ on the Bergman space on $\Omega $ , then $\phi _j=0$ for some j.

The Hermite conjugate Poisson integral of a given f ∊ L1(μ), dμ(y)= exp(—y2) dy, was defined by Muckenhoupt [5, p. 247] as

where

If the Hermite conjugate function operator T is defined by (Tf) a.e., then one of the main results of [5] is that T is of weak-type (1, 1) and strongtype (p,p) for all p>l.

The Banach contraction principle has been generalized by Tan [6] to the mappings in separated locally convex spaces. We show that the result of Sehgal [5] and also of Holmes [3] can be generalized in the same way.

Throughout this note, we let X be a separated locally convex space, U a base for the closed absolutely convex neighborhoods of the origin O in X, K a nonempty subset of X, and Ta mapping from K to K.

This note determines the quadratic field generated by the square root of the discriminant of the modular equation satisfied by the special value j(α) of the modular function α for a an integer in an imaginary quadratic field.

This note presents a very simple proof for the integration by parts formula for the Perron integral.

In 1948 Gabor Szegő [9] gave four proofs of a remarkable inequality communicated to him by Paul Turán, who later published an original proof [10]. The Turán theorem states that if Pn(x) is the Le gendre polynomial, then

1.1

with equality holding only when |x| = 1.

Since then many similar inequalities have been found for various special functions, particularly for the Legendre and Hermite polynomials. Reference may be had to the recent work of Danese [2] and Chatterjea [1]. Danese gives an extensive bibliography.

The role of binomial distribution in the probability theory and its applications is well known. Recently M. P. Singh [4] has been discussing the socalled inflated binomial distribution. This distribution was introduced for situations which are described by simple binomial except for zero cell which is inflated, that is, there more observations than can be expected on the basis of simple binomial. The investigation of this inflated binomial distribution seems to be useful.

As an analog of a well-known theoremon the bilinear fractional integral on ${{\mathbb{R}}^{n}}$ by Kenig and Stein, we establish the similar boundedness property for a bilinear fractional integral on a compact Lie group. Our result is also a generalization of our recent theorem about the bilinear fractional integral on torus.

As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree 5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type ${{E}_{8}}$.

We consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors $({{x}_{\text{n}}})$ to be eigenvectors $T{{x}_{n}}={{\lambda }_{n}}{{x}_{n}}({{\lambda }_{n}}\ne {{\lambda }_{k}}\text{for}\,n\ne k)$ of a bounded operator $T$ (admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1) for the sequence ${{\left( {{x}_{n}}+{{\epsilon }_{n}}{{v}_{n}} \right)}_{n\ge k(\epsilon )}}$ to be admissible for every admissible $({{x}_{\text{n}}})$ and for a suitable choice of small numbers ${{\epsilon }_{n}}\,\ne \,0$ it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist ${{\text{ }\!\!\gamma\!\!\text{ }}_{n}}\,\in \,C$ such that ${{v}_{n}}\,=\,{{\gamma }_{n}}{{v}_{k}}\,\text{for}\,n\,\ge \,\text{k}$ (Theorem 2); (2) for a bounded operator $A$ to transform admissible families $({{x}_{\text{n}}})$ into admissible families $(A{{x}_{n}})$ it is necessary and sufficient that $A$ be left invertible (Theorem 4).

Dans [2] nous avons introduit, en nous inspirant des notions équivalentes pour les anneaux [3], les notions de monoïde maximal Q(S) et de monoïde complet B(S) des quotients d'un monoïde donné S, la premiére construction n'étant valide que lorsque S est commutatif, et nous avons montré que dans ce cas, on a toujours les inclusions (en général strictes) de monoïdes suivantes: S ⊂ Q(S) ⊂ B(S).

In this paper we answer an open question of C. J. Earle ([2] §3.3 remarks (a) and (b)) in several cases. We first give some definitions and state some results which are given in greater detail in [2].

We let X be a smooth surface of genus g ≥ 2 and let M (X) be the space of smooth complex structures with the C∞ topology. If μ∈M(X) let Xμ denote the Riemann surface determined by μ.

We use tk moves on links to prove that the first homology group with Z2-coefficients of an odd sheeted cyclic cover of S3 branched over a link is even dimensional.

The purpose of this note is to give an elementary geometric proof of the following result stated by Sullivan (see (4)).

Theorem 1 (Sullivan). Let K be a finite simplicial complex with vertices v1, …, vN and corresponding barycentric coordinates b1, …, bN. Then the algebra of rational PL forms on K