The logarithmic coefficients $\unicode[STIX]{x1D6FE}_{n}$ of an analytic and univalent function $f$ in the unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ with the normalisation $f(0)=0=f^{\prime }(0)-1$ are defined by $\log (f(z)/z)=2\sum _{n=1}^{\infty }\unicode[STIX]{x1D6FE}_{n}z^{n}$. In the present paper, we consider close-to-convex functions (with argument 0) with respect to odd starlike functions and determine the sharp upper bound of $|\unicode[STIX]{x1D6FE}_{n}|$, $n=1,2,3$, for such functions $f$.

Let be a nilpotent Lie algebra which is an exponential -module, being an exponential algebra of derivations of . Put = exp and = exp . If Ω is a closed orbit of * under the action of , then Ker is dense in Ker Ω for the topology of L1 () and the algebra Ker is nilpotent, where denotes the minimal closed ideal of L1() whose hull is Ω. Moreover, the -prime ideals of Ll() coincide with the kernels Ker Ω, where Ω denotes an arbitrary orbit (not necessarily closed) in *.

We show that an isometric immersion of an n−dimensional compact Riemannian manifold of non-negative Ricci curvature with scalar curvature always less than n(n−1)λ−2 into a Euclidean space of dimension n + 1 can never be contained in a ball of radius λ.

We prove an asymptotic formula for the sum $\sum _{n\leq N}d(n^{2}-1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our proof, we also furnish an asymptotic formula for the sum $\sum _{d\leq N}g(d)$, where $g(d)$ denotes the number of solutions $x$ in $\mathbb{Z}_{d}$ to the equation $x^{2}\equiv 1~(\text{mod}~d)$.

It is shown that if G is an arbitrary regular product of its subgroups Aλ, ∈ ϵ I, then the multiplicator, M(G), is the director product of the M(Aλ) together with a certain other group. This extends a calculation of M(A1 × A2) due to Schur. As an application, we find the multiplicator of a vertai wreath product A wrVB where A is abelian. A representing group for a finite regular product is also constructed.

We give an elementary and self-contained proof of the result that the sheaf M~ determined by an injective module M over a commutative noetherian ring is flasque.

Let 𝒯n(F) denote the monoid of all upper triangular n×n matrices over a finite field F. It has been shown by Volkov and Goldberg that 𝒯n(F) is nonfinitely based if ∣F∣>2 and n≥4, but the cases when ∣F∣>2 and n=2,3 or when ∣F∣=2 have remained open. In this paper, it is shown that the monoid 𝒯2 (F) is finitely based when ∣F∣=2 , and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety generated by 𝒯2 (F) with ∣F∣=2 are determined.

We describe a contact metric manifold whose Reeb vector field belongs to the (κ,μ)-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric (κ,μ)-spaces in terms of a canonical connection which can be naturally defined on them.

Let G be a nonabelian finite p-group of order pm. A long-standing conjecture asserts that G admits a noninner automorphism of order p. In this paper we prove the validity of the conjecture if exp (G)=pm−2. We also show that if G is a finite p-group of maximal class, then G has at least p(p−1) noninner automorphisms of order p which fix Φ(G) elementwise.

We describe all group gradings of the matrix algebra M2(k), where k is an arbitrary field. We prove that any such grading reduces to a grading of type C2, a grading of type C2 × C2, or to a good grading. We give new simple proofs for the description of C2-gradings and C2 × C2-gradings on M2(K).

The natural completion functor on the category of cλ-embedded Cauchy spaces induces a completion functor on a category of sequential Cauchy spaces which preserves sequential regularity.

A simple characterisation is given of compact sets of the space K(X), of nonempty compact subsets of a complete metric space X, with the Hausdorff metric dH. It is used to give a new proof of the Blaschke selection theorem for compact starshaped sets.

Let $G$ be a finite group, let ${\it\pi}(G)$ be the set of prime divisors of $|G|$ and let ${\rm\Gamma}(G)$ be the prime graph of $G$. This graph has vertex set ${\it\pi}(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

The skeleton of a variety of groups is defined to be the intersection of the section closed classes of groups which generate . If m is an integer, m > 1, is the variety of all abelian groups of exponent dividing m, and , is any locally finite variety, it is shown that the skeleton of the product variety is the section closure of the class of finite monolithic groups in . In particular, S) generates . The elements of S are described more explicitly and as a consequence it is shown that S consists of all finite groups in if and only if m is a power of some prime p and the centre of the countably infinite relatively free group of , is a p–group.

We improve upon earlier effective bounds for the magnitude of integer points on an elliptic curve ε defined over a number field K. We slightly refine the dependence on the discriminant of K. In most of the previous papers, the estimates obtained are exponential in the height of ε. In this work, taking also into consideration the prime ideals dividing the discriminant of ε, we provide a totally explicit bound which is only polynomial in the height.

A characterization of finite (weak) projectives in an equational class of distributive pseudocomplemented lattices is given. In the class of all such lattices, a finite lattice is projective if and only if its poset of join–irreducible elements forms a semilattice in which the minimal elements below the join of x and y are exactly the mininal elements below x or y. A similar condition works for any equational subclass.

We relate the kernel of the cup product of 1-dimensional cohomology classes for a group G acting trivially on a field R to Hom(G2/G3,R), the space of group homomorphisms of the second stage of the lower central series for G into R, by means of explicit computations with cocycles. The precise result depends on whether the characteristic of the field is 0, an odd prime or 2.

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilizers. However, beside a well-known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilizers, each significant for its own reason.

It is shown that an entire function having k distinct entire asymptotic functions of order less than ¼ is of lower order ½k, mean type at least; further that if f is of lower order ½k, mean type, then its order is ½k.