D.L. Berman has proved several divergence theorems about “extended” Hermite-Fejér interpolation on the Chebyshev nodes of the first kind. These are surprising in light of the classical convergence theorem of L. Fejér concerning ordinary Hermite-Fejér interpolation on these nodes. However there is one case which has been neglected so far: the case of quasi-Hermite-Fejér interpolation on these nodes. In this paper it is proved that quasi-Hermite-Fejér interpolation polynomials on the Chebyshev nodes converge uniformly to the continuous function being interpolated. In addition, an estimate for the rate of convergence is established.

Let ${\mathcal{D}}$ be a Schauder decomposition on some Banach space $X$. We prove that if ${\mathcal{D}}$ is not $R$-Schauder, then there exists a Ritt operator $T\in B(X)$ which is a multiplier with respect to ${\mathcal{D}}$ such that the set $\{T^{n}:n\geq 0\}$ is not $R$-bounded. Likewise, we prove that there exists a bounded sectorial operator $A$ of type $0$ on $X$ which is a multiplier with respect to ${\mathcal{D}}$ such that the set $\{e^{-tA}:t\geq 0\}$ is not $R$-bounded.

In this paper, a generalisation of the Andersen, Hilton, Rodger Theorem for embedding partial idempotent latin squares is proved. This result is then used to prove that a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order (2n + 1)m if m is odd, of order 2nm if m ≥ 8 is even, 12n + 1 if m = 6 and approximately 2n + √2n if m = 4.

We show that a certain class of Fibonacci groups can not be right ordered. A question remaining is: Are the torsion-free members of this class unique products groups?

It is proved that every metrizable topological space without isolated points is the union of a continuum or fewer nowhere dense subsets.

Let $\ell >0$ be arbitrary. We introduce the extremal quantities

$$\begin{eqnarray}G(\ell ):=\sup _{f}\int _{-\ell }^{\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\quad C(\ell ):=\sup _{f}\sup _{a\in \mathbb{R}}\int _{a-\ell }^{a+\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\end{eqnarray}$$

$\ell =2$

$\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}G(k+\unicode[STIX]{x1D700})$

$\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}C(k+\unicode[STIX]{x1D700})$

$(k\in \mathbb{N})$

$\ell$

Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

D.P. Baturov proved in ‘Subspaces of function spaces’ Vestnik Moskov University Series I (1987) that Lindelöf degree equals extent for subspaces of Cp(Χ) when Χ is a Lindelöf Σ-space. We prove that if the Lindelöf degree of the subspace is “big enough” the equality is true for a topological space Χ not necessarily Lindelöf Σ.

We describe a type of Lie colour algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. Moreover, it is an iterated Ore extension. We provide an application and show Gröbner basis methods can be used to study universal enveloping algebras of factors of generic Lie colour algebras.

It has been shown by W.F. LaMartin that the Pontryagin dual of a k-space product of Hausdorff abelian k-groups is the coproduct of their duals. Here we offer a different proof, based partly on LaMartin's, that the dual of a product is the coproduct of the duals in a more general setting.

Sargent and later Bullen and Mukhopadhyay obtained a definition of absolutely continuous functions, functions, that is related to kth Peano derivatives. The generalised notions of ACkG*, [ACkG*], ACkG* above, etcetera functions led Bullen and Mukhopadhyay to define certain general integrals of the kth order.

The present work is concerned with a further simplification of the definitions of such functions by the use of divided differences but still retaining similar fundamental properties. These concepts lead to the introduction of Denjoy and Ridder type integrals which are shown to be equivalent to a Perron type integral that corresponds to kth Riemann* derivatives. All three of these integrals are shown to be equivalent to the three integrals of Bullen and Mukhopadhyay.

We determine the irreducible subgroups G of GL (r,ℂ), where r is prime and all proper subgroups of G are reducible.

We give single laws for two subvarieties of Steiner quasigroups (squags); the subvariety of squags whose underlying triple systems are the affine spaces over GF[3], and the subvariety of squags whose underlying triple systems are the affine triple systems.

In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups G1 and G2 of an infinite co-volume Kleinian group G⊂Isom(H3) having Λ(G1)=Λ(G2) are commensurable. In particular, we prove that any finitely generated subgroup H of a Kleinian group G⊂Isom(H3) with Λ(H)=Λ(G) is of finite index if and only if H is not a virtually fibered subgroup.

It is proved that if a special Hermitian manifold complies with the 1-cosymplectic hypersurfaces axiom, then it is a Kählerian manifold.

Let $p(z)= z{f}^{\prime } (z)/ f(z)$ for a function $f(z)$ analytic on the unit disc $\mid z\mid \lt 1$ in the complex plane and normalised by $f(0)= 0, {f}^{\prime } (0)= 1$. We provide lower and upper bounds for the best constants ${\delta }_{0} $ and ${\delta }_{1} $ such that the conditions ${e}^{- {\delta }_{0} / 2} \lt \mid p(z)\mid \lt {e}^{{\delta }_{0} / 2} $ and $\mid p(w)/ p(z)\mid \lt {e}^{{\delta }_{1} } $ for $\mid z\mid , \mid w\mid \lt 1$ respectively imply univalence of $f$ on the unit disc.

Let G be a graph, and k a positive integer. Let h:E(G)→[0,1] be a function. If ∑ e∋xh(e)=k holds for each x∈V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh={e∈E(G)∣h(e)>0}. In this paper we use neighbourhoods to obtain a new sufficient condition for a graph to have a fractional k-factor. Furthermore, this result is shown to be best possible in some sense.

Let B be the class of functions ω(z) regular in |z| < 1 and satisfying ω(0) = 0, |ω(z)|<1 in |z|<1. We denote by P(A, B), −1 ≤ B < A ≤1, the class of functions p(z) = l+p1z+… regular in |z| < 1 and such that p(z) = [1+Aω(z)]/[1+Bω(z)] for some ω(z) ∈ Β. This paper establishes sharp lower and upper bounds on |z| = r<1 for the functional

where p(z) varies in P(A, B). The results are then used to study certain geometric properties of the corresponding class of meromorphic starlike univalent functions