In this note, we discuss the exceptional set E ⊆ [−1, 1] of points x0 satisfying the inequality where λ > 0, λ ≠ 2, 4, … and Ln(fλ,.) is the Lagrange interpolation polynomial of degree at most n to fλ(x):= |x|λ on the interval [−1, 1] associated with the equidistant nodes. It is known that E has Lebesgue measure zero. Here we show that E contains infinite families of rational and irrational numbers.

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.

The question of which Abelian groups can be the inner mapping group of a loop has been considered by Niemenmaa, Kepka and others. We give a construction which shows that many finite Abelian groups can be the inner mapping group of a loop.

Ramanujan recorded many beautiful continued fractions in his notebooks. In this paper, we derive several indentities involving the Ramanujan continued fraction c(q), including relations between c(q) and c(qn). We also obtain explicit evaluations of for various positive integers n.

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).

One of the most important examples of conditional Cauchy equations with the condition dependent on the unknown function is Dhombres' equation This paper is devoted to the stability of the above equation.

We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We givn an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a p-minimal surface to have no singular points. For general complete immersed p-minimal surfaces, we prove a half space theorem and give a criterion for the properness.

In this paper we consider inequalities of the form , Where M is a mean. The main results of the paper offer sufficient conditions on M so that the above inequality holds with a finite constant C. The results obtained extend Hardy's and Carleman's classical inequalities together with their various generalisations in a new dirction.