Thompson’s conjecture for simple groups with connected prime graph

We deal with finite simple groups G with the property π(G) ⊆ {2, 3, 5, 7, 11, 13, 17}, where π(G) is the set of all prime divisors of the order of a group G. The set of all such groups is denoted by ζ ₁₇. Thompson’s conjecture in [1, Question 12.38] is proved valid for all groups in ζ ₁₇ whose prime graph is connected.     

On Brouwer’s continuity principle

We reconsider Ishihara (1991, 1992) within a weak formalised framework of constructive reverse mathematics, focusing on Brouwer’s continuity principle for a mapping from the Baire space into the natural numbers.     

Weyl–Pedersen calculus for some semidirect products of nilpotent Lie groups

For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl–Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.     

Newton’s method on Lie groups

The purpose of this paper is to study the strong convergence of fixed points for a family of demi-continuous pseudo-contractions by hybrid projection algorithms in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others.     

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

A generalizaton of Hardy’s uncertainty principle on compact extensions of \mathbb R ^n

Let \(K\) be a compact subgroup of automorphisms of \(\mathbb R ^n\) . We prove in this paper a generalization of Hardy’s uncertainty principle on the semi-direct product \(K\ltimes \mathbb R ^n\) , and we solve the sharpness problem. As a consequence, a complete analogue of classical Hardy’s theorem is obtained. The representation theory and the Plancherel formula play an important role in the proofs.     

Harnack’s principle for quasiminimizers

Abstract We study Harnack type properties of quasiminimizers of the -Dirichlet integral on metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that an increasing sequence of quasiminimizers converges locally uniformly to a quasiminimizer, provided the limit function is finite at some point, even if the quasiminimizing constant and the boundary values are allowed to vary in a bounded way. If the quasiminimizing constants converge to one, then the limit function is the unique minimizer of the -Dirichlet integral. In the Euclidean case with the Lebesgue measure we obtain convergence also in the Sobolev norm. Keywords: Metric space, doubling measure, Poincaré inequality, Newtonian space, Harnack inequality, Harnack convergence theorem Mathematics Subject Classification (2000): 49J52, 35J60, 49J27     

On Hardy’s Uncertainty Principle for Solvable Locally Compact Groups

We establish analogues of Hardy’s uncertainty principle for the Fourier transform on ? for locally compact Abelian groups and for some classes of solvable Lie groups, such as diamond groups and exponential Lie groups with non-trivial centre.     

D’Alembert’s and Wilson’s functional equations on step 2 nilpotent groups

Summary. We describe the set of solutions of Wilsons functional equation on any step 2 nilpotent group and how the set of classical solutions in certain cases must be supplemented by 4-dimensional spaces of solutions.     

Heisenberg’s uncertainty principle in the sense of Beurling

We shed new light on Heisenberg??s uncertainty principle in the sense of Beurling, by offering a fundamentally different proof which allows us to weaken the assumptions rather substantially. The new formulation is pretty much optimal, as can be seen from examples. Our arguments involve Fourier and Mellin transforms. We also introduce a version which applies to two given functions. Finally, we show how our approach applies in the higher dimensional setting.     

Thompson’s conjecture for Lie type groups E 7(q)

In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type E ₇(q). More precisely, we show that every finite group G with the properties Z(G) = 1 and cs(G) = cs(E ₇(q)) is necessarily isomorphic to E ₇(q), where cs(G) and Z(G) are the set of lengths of conjugacy classes of G and the center of G respectively.     

Kantorovich’s majorants principle for Newton’s method

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s method, the relationship of the majorant function and the non-linear operator under consideration. This approach enables us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results. The research of O.P. Ferreira was supported in part by FUNAPE/UFG, CNPq Grant 475647/2006-8, CNPq Grant 302618/2005-8, PRONEX–Optimization(FAPERJ/CNPq) and IMPA. The research of B.F. Svaiter was supported in part by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization(FAPERJ/CNPq).     

On some problems involving Hardy’s function

Some problems involving the classical Hardy function

Up–down representations and ergodic theory in nilpotent Lie groups

Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let Γ be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up–down representation . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/Γ in the case when Γ is a lattice subgroup of G and A is a one-parameter subgroup of G.     

Tauvel’s height formula for quantum nilpotent algebras

Tauvel’s height formula, which provides a link between the height of a prime ideal and the Gelfand–Kirillov dimension of the corresponding factor algebra, is verified for quantum nilpotent algebras.