A close connection between the strict convexity of the Day norm to the concept of the Gruenhage compacta is shown. As a byproduct we give an elementary characterisation of Gul'ko compacta in the sigma-product of lines and a more elementary proof of Mercourakis' renorming result for Vašák spaces.

Generalised second-order derivatives introduced by Rockafellar in the finite dimensional setting are extended to convex functions defined on reflexive Banach spaces. Our approach is based on the characterisation of convex generalised quadratic forms defined in reflexive Banach spaces, from the graph of the associated subdifferentials. The main result which is obtained is the exhibition of a particular generalised Hessian when the function admits a generalised second derivative. Some properties of the generalised second derivative are pointed out along with further justifications of the concept.

The Blaschke–Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on ℝn. Here we discuss another decomposition called polar decomposition by considering ℝn×⋯×ℝn as ℳn×k and using its polar decomposition. This is a generalisation of the Blaschke–Petkantschin formula and may be useful when one needs to integrate a function g:ℝn×⋯×ℝn→ℝ with rotational symmetry, that is, for each orthogonal transformation O,g(O(x1),…,O(xk))=g(x1,…xk). As an application we compute the moments of a Gaussian determinant.

In this paper, we prove some value distribution results which lead to normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family, and improve some recent results. In particular, the main result of this paper leads to a counterexample to the converse of Bloch’s principle.

We obtain a basic B,-Y. Chen's inequality for a C-totally real submanifold in a (κ,μ)-contact space form involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and κ-Ricci curvature are also obtained. These results are applied to get corresponding results for C-totally real submanifolds in a Sasakian space form.

A submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

The existence of a certain type of Rudin-Shapiro sequence of functions is shown for all infinite compact groups which are not Lie, thus extending a recent result of the authors to all infinite compact groups.

A new linear expression in σ(ν), ν = l, 2, …, n, which vanishes identically is established. A linear expression in σ(ν)'s has been found for σ3(n). A similar expression in σ3(ν)'s has been proved for σ7(n) also, Ramanujan's τ(n) = P24(n-1) is given in three different ways as linear expressions in σ2k+1(n) and σK(ν)'s with k = 1, 3, 5 respectively. Again, the coefficient p48(n-2) is expressed as a linear expression in σ11(v)'S and σ5(ν)'s. In establishing these results advantage is taken of the general theorem, also established, that the coefficients of the square of a power series whose coefficients satisfy a certain functional equation are expressible as linear functions of the latter coefficients.

It is shown that two polycyclic-by-finite groups G and H, satisfying the same sentences with one alternation of quantifiers, are commensurable. In fact we show something stronger: given n > 1 there is a subgroup Hn of H and a subgroup Gn of G such that Hn≃G, Gn≃H and the indices [G: Gn] and [H: Hn] are finite and prime to n.

We study the k-Galois linear complementary dual (LCD) codes over the finite chain ring $R=\mathbb {F}_q+u\mathbb {F}_q$ with $u^2=0$ , where $q=p^e$ and p is a prime number. We give a sufficient condition on the generator matrix for the existence of k-Galois LCD codes over R. Finally, we show that a linear code over R (for $k=0, q> 3$ ) is equivalent to a Euclidean LCD code, and a linear code over R (for $0<k<e$ , $(p^{e-k}+1)\mid (p^e-1)$ and ${(p^e-1)}/{(p^{e-k}+1)}>1$ ) is equivalent to a k-Galois LCD code.

Let Bk denote the Euclidean unit ball in ℝκ equipped with the k-dimensional Lebesgue measure and let φ: ℝ+ → ℝ+ be a convex function satisfying φ(0) = 0, φ(t) > 0 for some t > 0. Denote by Eφ = Eφ(Bk) the Orlicz space of finite elements (see (1.6)) generated by φ. The aim of this paper is to show that there exists a retraction of the closed unit ball in Eφ onto the unit sphere in Eφ being a (2 + ɛ)γφ;-set contraction (Theorem 3.6), which generalises [9, Corollary 6] proved for the case of Lp[−1, 1], 1 ≤ p < ∞. Here γφ, denote the Hausdorff measure of noncompactness. This theorem is proved both for the Amemiya and the Luxemburg norms. Also some related results concerning the case of s-convex (0 < s ≤ 1) functions are presented.

We determine a bound for the valency in a family of dihedrants of twice odd prime orders which guarantees that the Cayley graphs are Ramanujan graphs. We take two families of Cayley graphs with the underlying dihedral group of order $2p$ : one is the family of all Cayley graphs and the other is the family of normal ones. In the normal case, which is easier, we discuss the problem for a wider class of groups, the Frobenius groups. The result for the family of all Cayley graphs is similar to that for circulants: the prime $p$ is ‘exceptional’ if and only if it is represented by one of six specific quadratic polynomials.

An investigation of representation of Abel type infinite sums in closed form will be presented in this paper. An arbitrary order forced differential-difference equation will be analysed from which sums will be generated. Identities of resulting infinite sums will be proved.

Let G/K be a noncompact symmetric space, Gc/K its compact dual, 𝔤=𝔨⊕𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian subspace of 𝔭, H be an element of 𝔞, a=exp (H) , and ac =exp (iH) . In this paper, we prove that if for some positive integer r, νrac is absolutely continuous with respect to the Haar measure on Gc, then νra is absolutely continuous with respect to the left Haar measure on G, where νac (respectively νa) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare [‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

A formula is given for the Möbius function of the poset Hom(P, Q) of all order-preserving maps between two finite posets P and Q. Two applications of the formula are presented.

The minimum area a(v) of a v–sided convex lattice polygon is known to satisfy . We conjecture that a(v) = cv3 + o(v3), for c a constant; we prove that , and that for some positive constant c, .

In this paper we prove two results relating the cardinalities of discrete subsets to those of dense subsets of a topological space. In particular, we show that

(1) closed discrete subsets of separable normal spaces are countable, and

(2) discrete subsets of separable regular spaces have cardinalities at most that of the continuum.

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta $ such that any preassigned sequence of $\ell $ digits occurs in the $q$-ary expansion of $\eta $ at the expected frequency, namely $1/q^\ell $. In a recent paper we constructed a large family of normal numbers, showing in particular that, if $P(n)$ stands for the largest prime factor of $n$, then the number $0.P(2)P(3)P(4)\ldots ,$ the concatenation of the numbers $P(2), P(3), P(4), \ldots ,$ each represented in base $q$, is a $q$-normal number, thereby answering in the affirmative a question raised by Igor Shparlinski. We also showed that $0.P(2+1)P(3+1)P(5+1) \ldots P(p+1)\ldots ,$ where $p$ runs through the sequence of primes, is a $q$-normal number. Here, we show that, given any fixed integer $k\ge 2$, the numbers $0.P_k(2)P_k(3)P_k(4)\ldots $ and $0. P_k(2+1)P_k(3+1)P_k(5+1) \ldots P_k(p+1)\ldots ,$ where $P_k(n)$ stands for the $k{\rm th}$ largest prime factor of $n$, are $q$-normal numbers. These results are part of more general statements.

Motivated by considerations of the quadratic orthogonal bisectional curvature, we address the question of when a weighted graph (with possibly negative weights) has nonnegative Dirichlet energy.