The ascending central series of nilpotent Lie algebras with complex structure

We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra $\mathfrak g$ under the presence of a complex structure $J$. In particular, we find a bound for the dimension of the center of $\mathfrak g$ when it does not contain any non-trivial $J$-invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras $\mathfrak g$ admitting this particular type of complex structures $J$. Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs $(\mathfrak g, J)$.Comment: 28 pages, 1 figure. To appear in Trans. Amer. Math. So

An Entourage Approach to the Contraction Principle in Uniform Spaces Endowed with a Graph

In this paper, we study Banach contractions in uniform spaces endowed with a graph and give some sufficient conditions for a mapping to be a Picard operator. Our main results generalize some results of [J. Jachymski, "The contraction principle for mappings on a metric space with a graph", Proc. Amer. Math. Soc. 136 (2008) 1359-1373] employing the basic entourages of the uniform space.Comment: 20 page

On the real linear polarization constant problem

The present paper deals with lower bounds for the norm of products of linear forms. It has been proved by J. Arias-de-Reyna [2], that the so-called n(th) linear polarization constant c(n)(C-n) is n(n/2), for arbitrary n is an element of N. The same value for c(n) (R-n) is only conjectured. In a recent work A. Pappas and S. Revesz prove that c(n) (R-n) = n(n/2) for n <= 5. Moreover, they show that if the linear forms are given as f(j)(x) = [x, a(j)),for some unit vectors a(j) (1 <= j <= n), then the product of the f(j)'s attains at least the value n(-n/2) at the normalized signed sum of the vectors having maximal length. Thus they asked whether this phenomenon remains true for arbitrary n is an element of N. We show that for vector systems {a(j)}(j=1)(n) close to an orthonormal system, the Pappas-Revesz estimate does hold true. Furthermore, among these vector systems the only system giving n(-n/2) as the norm of the product is the orthonormal system. On the other hand, for arbitrary vector systems we answer the question of A. Pappas and S. Revesz in the negative when n is an element of N is large enough. We also discuss various further examples and counterexamples that may be instructive for further research towards the determination of c(n)(R-n)