In this paper various conditions are given under which the strict singularity (respectively, strict cosingularity) of a linear relation implies the strict singularity (respectively, strict cosingularity) of its conjugate.

Let κ ≥ 2 be an integer. We show that there exist infinitely many positive integers N such that the number of κ-full integers in the interval (Nκ, (N + 1)κ) is at least (log N)1/3+ο(1). We also show that the ABC-conjecture implies that for any fixed δ > 0 and sufficiently large N, the interval (N, N + N1−(2+δ)/κ) contains at most one κ-full number.

We prove a Seidel-Walsh-type theorem about the universality of a sequence of derivation-composition operators generated by automorphisms of the unit disk in the setting of the higher order Hardy spaces. Moreover, some related positive or negative assertions involving interpolating sequences and sequences between two tangent circles are established for the class of bounded functions in the unit disk. Our statements improve earlier ones due to Herzog and to the first and third authors.

Let SL(2, Γn) be the n-dimensional Clifford matrix group and G ⊂ SL(2, Γn) be a non-elementary subgroup. We show that G is the extension of a subgroup of SL(2, ℂ) if and only if G is conjugate in SL(2, Γn) to a group G′ which satisfies the following properties:

(1) there exist loxodromic elements g0, h ∈ G′ such that fix(g0) = {0, ∞}, fix(g0) ∩ fix(h) = ∅ and fix(h) ∩ ℂ ≠ ∅;

(2) tr(g) ∈ ℂ for each loxodromic element g ∈ G′.

Further G is the extension of a subgroup of SL(2, ℝ) if and only if G is conjugate in SL(2, Γn) to a group G′ which satisfies the following properties:

(1) there exists a loxodromic element g0 ∈ G′ such that fix(g0) ∩ {0, ∞} ≠ ∅;

(2) tr(g) ∈ ℝ for each loxodromic element g ∈ G′.

The discreteness of subgroups of SL(2, Γn) is also discussed.

Let Ω1 and Ω2 be two domains in the complex plane with a nonempty intersection. Suppose that μj are locally extremal Beltrami coefficients in Ωj (j = 1, 2) respectively. In 1980, Sheretov posed the problem: Will the coefficient μ defined by the condition μ(z) = μj(z) for z ∈ Ωj, j = 1, 2, be locally extremal in Ω1 ∪ Ω2? We give a counterexample to show that μ may not be locally extremal and not even be extremal.

Let f: [0, 1] × ℝ2 → ℝ be a function satisfying the Carathéodory conditions and t (1 − t) e (t) ∈ L1(0, 1). Let ai ∈ ℝ and ξi ∈ (0, 1) for i = 1, …, m − 2 where 0 < ξ1 < ξ2 < … < ξm−2 < 1. In this paper we study the existence of C[0, 1] solutions for the m-point boundary value problem The proof of our main result is based on the Leray-Schauder continuation theorem.

In this paper we prove the following result: Let R be a 2-torsion free semiprime ring. Suppose there exists an additive mapping T: R → R such that T(xyx) = T(x)yx − xT(y)x + xyT(x) holds for all pairs x, y ∈ R. Then T is of the form 2T(x) = qx + xq, where q is a fixed element in the symmetric Martindale ring of quotients of R.

Let G be a graph, and f: G → G be a continuous map with periodic points. In this paper we show that the following five statements are equivalent.

(1) f is equicontinuous.

(2) There exists some positive integer N such that fN is uniformly convergent.

(3) f is S-equicontinuous for some positive integer sequence S = {n1 < n2 < …}.

(4) Ω(x, f) = ω(x, f) for every x ∈ G.

(5) is a periodic map.

The index and period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element with index m and period 1 is called an m-potent element. For an element α of a finite full transformation semigroup with index m and period r, a unique factorisation α = σβ such that Shift(σ) ∩ Shift(β) = ∅ is obtained, where σ is a permutation of order r and β is an m-potent. Some applications of this factorisation are given.

In this paper, we show that Jacobi's two-square and two-triangular number theorems are immediate consequence of Ramanujan's continued fraction identities.

This paper studies the problem of numerical integration over the unit sphere S2 ⊆ ℝ3 for functions in the Sobolev space H3/2(S2). We consider sequences Qm(n), n ∈ ℕ, of cubature (or numerical integration) rules, where Qm(n) is assumed to integrate exactly all (spherical) polynomials of degree ≤ n, and to use m = m(n) values of f. The cubature weights of all rules Qm(n) are assumed to be positive, or alternatively the sequence Qm(n), n ∈ ℕ, is assumed to have a certain local regularity property which involves the weights and the points of the rules Qm(n), n ∈ ℕ. Under these conditions it is shown that the worst-case (cubature) error, denoted by E3/2 (Qm(n)), for all functions in the unit ball of the Hilbert space H3/2 (S2) satisfies the estimate E3/2 (Qm(n)) ≤ c n−3/2, where the constant c is a universal constant for all sequences of positive weight cubature rules. For a sequence Qm(n), n ∈ ℕ, of cubature rules that satisfies the alternative local regularity property the constant c may depend on the sequence Qm(n), n ∈ ℕ. Examples of cubature rules that satisfy the assumptions are discussed.

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

A ring is called Bezout if every finitely generated ideal is principal. We show that every ideal of a commutative Bezout ring R is flat if and only if every submodule of a free R-module is flat. Using this theorem we obtain Neville's theorem.

Soit (M, g0) une variété riemannienne compacte de courbure sectionnelle négative. Soit g1 une autre métrique riemannienne sur M de rang 1. On montre que l'égalité des spectres marqués des longueurs de g0 et g1 implique que le flot géodésique de g0 est un facteur de celui de g1.

We restrict primes and prime powers to sets . Let . Then the error in θH(x) has, unconditionally, the expected order of magnitude . However, if then ψH (x) = x log 2 + O (log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the “harmonic prime number theorem”, πH (x)/π (x) → log 2.

Interior point methods are not only the most effective methods for solving optimisation problems in practice but they also have polynomial time complexity. However, there is still a gap between the practical behavior of the interior point method algorithms and their theoretical complexity results. In this paper, by focusing on linear programming problems, we introduce a new family of kernel functions that have some simple and easy to check properties. We present a simplified analysis to obtain the complexity of generic interior point methods based on the proximity functions induced by these kernel functions. Finally, we prove that this family of kernel functions leads to improved iteration bounds of the large-update interior point methods.

This paper develops a one term Edgeworth expansion under minimal conditions for elementary symmetric polynomials of any degree based on trimmed samples. These statistics are special cases of trimmed U-statistics and natural extensions of the trimmed mean.

We consider the problem of determining all power integral bases for the maximal real subfield Q (ζ + ζ−1) of the p-th cyclotomic field Q (ζ), where p ≥ 5 is prime and ζ is a primitive p-th root of unity. The ring of integers is Z[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove that if or one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators for p ≤ 11, and it is plausible that there are no additional generators for p > 13 as well. For p = 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 < p < 1000.