A short proof of the following result of Bernstein and Ganea is given:

“Let X be a topological space which is homotopy dominated by a closed connected n-dimensional manifold M. If Hn (X; Z2 ) ≠ 0 then X has the homotopy type of M”.

It is also shown that the manifold in this theorem can be replaced by a Poincaré complex.

Almost tangent structures have been studied by Eliopoulos [1] and certain Riemannian structures subordinate to almost tangent structures have been studied by Closs [2]. In this paper we investigate those subordinate Riemannian structures for which the underlying almost tangent structure is without torsion and those for which the fundamental form is closed.

Similar studies have been carried out with respect to Riemannian structures subordinate to almost complex structures by Lichnerowicz [4] and with respect to Riemannian structures subordinate to almost product structures by Legrand [5].

Let Un be an n-dimensional subspace of C[0, 1]. We prove that if n ≥ 2, and Un contains a function which is strictly positive on (0, 1), then there exists an f ∈ C[0, 1] which has more than one best one-sided L '-approximation from Un. We also characterize those Un with the property that each f ∈ C[0, 1] has a unique best one-sided L1(w)-approximation from Un with respect to every strictly positive continuous weight function w.

The exact probability of making an error at the first stage of a stepwise regression is found, both when the independent variables are fixed and when they are random. The probabilities differ in the two cases because the random variables involved are pairwise independent but not jointly independent.

Let f(t) be integrable L(-π, π) and periodic with period 2π, and let

1.1

be its Fourier series. The series

1.2

obtained by the term by term differentiation of the series (1. 1), is called the Derived Fourier series of (1. 1). The series conjugate to (1. 2),

1.3

is called the Derived conjugate series of the Fourier series of f.

We characterize two important notions of amenability and compactness of a locally compact quantum group $\mathbb{G}$ in terms of certain homological properties. For this, we show that $\mathbb{G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra ${{L}^{1}}\left( \mathbb{G} \right)$. In particular, we improve an interesting result by Hu, Neufang, and Ruan.

Given a smooth compact hypersurface $M$ with boundary $\unicode[STIX]{x1D6F4}=\unicode[STIX]{x2202}M$, we prove the existence of a sequence $M_{j}$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\unicode[STIX]{x1D70E}_{k}(M_{j})$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_{j}$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.

Let A be a semisimple, unital, and complex Banach algebra. It is well known and easy to prove that A is commutative if and only $e^xe^y=e^{x+y}$ for all $x,y\in A$ . Elaborating on the spectral theory of commutativity developed by Aupetit, Zemánek, and Zemánek and Pták, we derive, in this paper, commutativity results via a spectral comparison of $e^xe^y$ and $e^{x+y}$ .

Set-theoretic solutions to the Yang–Baxter equation can be classified by their universal coverings and their fundamental groupoids. Extending previous results, universal coverings of irreducible involutive solutions are classified in the degenerate case. These solutions are described in terms of a group with a distinguished self-map. The classification in the nondegenerate case is simplified and compared with the description in the degenerate case.

We prove irregularities in the distribution of prime numbers in any Beatty sequence ${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ is a positive real irrational number of finite type.

Among the eight-dimensional stable planes, the compact connected generalized Hughes planes and the geometries induced on the outer points are characterized by the property that these planes admit an effective action of the group SL3 ℂ.

We construct a one-parameter generic polynomial for $\text{PSL}\left( 2,{{2}^{n}} \right)$ over ${{\mathbb{F}}_{{{2}^{n}}}}$.

A wavefront set is defined to be an image of Legendrian mapping. In this note we prove that generic wavefront sets have stable local topological structures by using Mather's stratification theory.

Let $X$ be a reduced nonsingular quasiprojective scheme over $\mathbb{R}$ such that the set of real rational points $X\left( \mathbb{R} \right)$ is dense in $X$ and compact. Then $X\left( \mathbb{R} \right)$ is a real algebraic variety. Denote by $H_{k}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ the group of homology classes represented by Zariski closed $k$-dimensional subvarieties of $X\left( \mathbb{R} \right)$. In this note we show that $H_{1}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ is a proper subgroup of ${{H}_{1}}\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ for a nonorientable hyperelliptic surface $X$. We also determine all possible groups $H_{1}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ for a real ruled surface $X$ in connection with the previously known description of all possible topological configurations of $X$.

In this paper, we prove a noncommutative (nc) analog of Schwarz lemma for the nc Schur–Agler class and prove that the regular nc Schur–Agler class and the regular free Herglotz–Agler class are homeomorphic. Moreover, we give a characterization of regular free Herglotz–Agler functions. As an application, we will show that any regular free Herglotz–Agler functions can uniformly be approximated by regular Herglotz–Agler free polynomials.

The notion of sequential topological algebra was introduced by this author and Ng [3], Among a number of results concerning these algebras, we showed that each multiplicative linear functional on a sequentially complete, sequential, locally convex algebra is bounded ([3], Theorem 1). From this it follows that every multiplicative linear functional on a sequential F-algebra (complete metrizable) is continuous ([3], Corollary 2).

We generalize a result of E. E. Floyd on unicoherent Peano continua with involution to unicoherent locally connected regular hereditarily Lindelöf spaces. The result has an application in the theory of connectivity functions.

There are a number of theories which assign to a function defined on the real line a measure that reflects somehow the variation of that function. The most familiar of these is, of course, the Lebesgue-Stieltjes measure associated with any monotonie function. The problem in general is to provide a construction of a measure from a completely arbitrary function in such a way that the values of this measure provide information about the total variation of the function over sets of real numbers and from which useful inferences can be drawn.

If $A$ is a set of $M$ positive integers, let $G(A)$ be the maximum of ${{a}_{i}}/\,\gcd ({{a}_{i}},\,\,{{a}_{j}}\,)$ over ${{a}_{i}},\,{{a}_{j}}\,\,\in \,\,A.$ We show that if $G(A)$ is not too much larger than $M$, then $A$ must have a special structure.

Let Zm denote the ring of integers modulo an integer m > 1 and let G(Zm) be the group under composition of all Zm-automorphisms of Zm[X]. In this paper we determine Zm[X]G(zm) the subring of Zm[X] left fixed by elements of G(Zm).