We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then there exists at least a piecewise smooth horizontal curve in this hypersurface connecting any two given points in it. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry: there exists a unique nonholonomic connection which is completely determined by the sub-Riemannian structure and is “symmetric” and compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.

In this paper, we show that the direct of infinite finitely presented groups is always properly 3-realisable. We also show that classical hyperbolic groups are properly 3-realisable. We recall that a finitely presented group G is said to be properly 3-realisable if there exists a compact 2-polyhedron K with π1 (K) ≅ G and whose universal cover has the proper homotopy type of a (p.1.) 3-manifold with boundary. The question whether or not every finitely presented is properly 3-realisable remains open.

Let be a group homomorphism between free products of groups such that GλΘ = Bλ for all λ ∈ Λ. Let H ⊆ G be a subgroup such that HΘ = B. Then such that HλΘ = Bλ and where Fλ is free.

In this paper, we give some evaluation formulas for the values of double L-series of Tornheim's type, in terms of the Dirichlet L-values and the Riemann zeta values at positive integers. As special cases, these give the formulas for double L-values given by Terhune.

Let R be a ring with 1 and (R) its Jacobson radical. Then 1 + (R) is a normal subgroup of the group of units, G(R). The existence of a complement to this subgroup was explored in a paper by Coleman and Easdown; in particular the ring R = Matn(ℤpk) was considered. We prove the remaining cases to determine for which n, P and k a complement exists in this ring.

We study the fixed point property with respect to general vector topologies in L-embedded Banach spaces. Considering a class of topologies in l1 such that the standard basis is convergent, we characterise those of them for which the fixed point property holds. We show that in c0-sums of some Banach spaces the weak topology is in a sense the coarsest topology for which the fixed point property holds.

A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. Here the controllability of rolling bodies is investigated with a global approach. By using simple geometric facts, this problem has been completely solved in the special case where one of them is a plane or a sphere.

A uniform tree is a tree that covers a finite connected graph. Let X be any locally finite tree. Then G = Aut(X) is a locally compact group. We show that if X is uniform, and if the restriction of G to the unique minimal G-invariant subtree X0 ⊆ X is not discrete then G contains non-uniform lattices; that is, discrete subgroups Γ for which Γ/G is not compact, yet carries a finite G-invariant measure. This proves a conjecture of Bass and Lubotzky for the existence of non-uniform lattices on uniform trees.

Assuming the Generalised Riemann Hypothesis we determine all imaginary Abelian number fields N whose Galois group G(N/ℚ) is isomorphic to (ℤ/2ℤ)n for some integers n ≥ 1 and the square of every ideal of N is principal.

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. In 1999 Nicholson asked whether every semiperfect ring is strongly clean and whether the matrix ring of a strongly clean ring is strongly clean. In this paper, we prove that if R = {m/n ∈ ℚ: n is odd}, then M2(R) is a semiperfect ring but not strongly clean. Thus, we give negative answers to both questions. It is also proved that every upper triangular matrix ring over the ring R is strongly clean.

Let A, B ⊆ {1, …, n}. For m ∈ Z, let rA,B(m) be the cardinality of the set of ordered pairs (a, b) ∈ A × B such that a + b = m. For t ≥ 1, denote by (A + B)t the set of the elements m for which rA, B(m) ≥ t. In this paper we prove that for any subsets A, B ⊆ {1, …, n} such that |A| + |B| ≥ (4n + 4t − 3)/3, the sumset (A + B)t contains a block of consecutive integers with the length at least |A| + |B| − 2t + 1, and that (a) for any two subsets A and B of {1, …, n} such that |A| + |B| ≥ (4n)/3, there exists an arithmetic progression of length n in A + B; (b) for any 2 ≤ r ≤ (4n − 1)/3, there exist two subsets A and B of {1, …, n} with |A| + |B| = r such that any arithmetic progression in A + B has the length at most (2n − 1)/3 + 1.

The author estimates the semi-norm and norm of composition operators on the Block space; and obtains several necessary conditions for a composition operator to be isometric.

A Riemannian manifold Mn is called IP, if, at every point x ∈ Mn, the eigenvalues of its skew-symmetric curvature operator R(X, Y) are the same, for every pair of orthonormal vectors X, Y ∈ TxMn. In [5, 6, 12] it was shown that for all n ≥ 4, except n = 7, an IP manifold either has constant curvature, or is a warped product, with some specific function, of an interval and a space of constant curvature. We prove that the same result is still valid in the last remaining case n = 7, and also study 3-dimensional IP manifolds.

We give a proof based on an integral equation for an explicit prediction formula for fractional Brownian motion with Hurst index less than 1/2.

We give sufficient conditions so that the distance of a point to the intersection of two sets agrees with the maximum of the distances to each of them. The results are established in several settings: complete metric spaces, Banach spaces and spaces of subsets of Banach spaces.

Suppose G is a complex Lie group and H is a closed complex subgroup of G. Let G′ denote the commutator subgroup of G. If there are no nonconstant holomorphic functions on G/H and H is not contained in any proper parabolic subgroup of G, then Akhiezer [2] asked whether every analytic hypersurface in G which is invariant under the right action of H is also invariant under the right action of G′. In this paper we answer a related question in two settings. Under the assumptions stated above we show that the orbits of the radical of G in G/H cannot be Cousin groups, provided G/H is Kähler. We also introduce an intermediate fibration of G/H induced by the holomorphic reduction of the radical orbits and resolve the related question in a situation arising from this fibration.

In a recent referee's report reviewing [2], it was pointed out to me that Theorem 2.2 in [1] was already known in the literature, and is (originally) due to Mammone and Moresi in [3]. Also in [3], the authors establish the excellence of inseparable quadratic extensions in a shorter way than I presented in [1]. As the paper [3] was unknown to me at the time of submitting [1], I hope by this note to acknowledge and credit [3] for the result(s).