In a free group no nontrivial commutator is a square. And in the free group F2 = F (x1, x2) freely generated by x1, x2 the commutator [x1, x2], is never the product of two squares in F2, although it is always the product of three squares. Let F2,3 = 〈x1, x2〉 be a free nilpotent group of rank 2 and class 3 freely generated by x1, x2. We prove that in F2,3 = 〈x1, x2〉, it is possible to write cert...

For countably generated ideals, J , of B(H), geometric stability is necessary for the canonical spectral characterization of sums of (J ; B(H)){commutators to hold. This answers a question raised by Dykema, Figiel, Weiss and Wodzicki. There are some ideals, J , having quasi{nilpotent elements that are not sums of (J ; B(H)){commutators. Also, every trace on every geometrically stable ideal is a...

In this paper, we obtain bounded criteria on certain weighted Morrey spaces for the commutators generalized by some sublinear integral operators and weighted Lipschitz functions. We also present bounded criteria for commutators generalized by such sublinear integral operators and weighted BMO function on the weighted Morrey spaces. As applications, our results yield the same bounded criteria fo...

Let T be the Calderón-Zygmund operator, Coifman, Rochberg and Weiss (see [4]) proves that the commutator [b, T ](f) = bT (f) − T (bf)(where b ∈ BMO(R)) is bounded on L(R) for 1 < p <∞. Chanillo (see [2]) proves a similar result when T is replaced by the fractional operators. In [8, 16], Janson and Paluszynski study these results for the Triebel-Lizorkin spaces and the case b ∈ Lipβ(R), where Li...

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups U (n). Every word w in the free group Fr on r generators determines a word map from U (n) to U (n), defined by ...

a gyrogroup is a nonassociative group-like structure modelled on the space of relativistically admissible velocities with a binary operation given by einstein&apos;s velocity addition law. in this article, we present a few of groups sitting inside a gyrogroup $g$, including the commutator subgyrogroup, the left nucleus, and the radical of $g$. the normal closure of the commutator subgyrogroup, ...

This paper proposes a low power commutator architecture based on triple port RAMs rather than dual port RAMs or conventional FIFO.for the radix-4 pipelined FFT processor implementation. The triple port RAM based commutator consumes less power than the other two for the first and second stages of a 64-point radix-4 pipelined FFT processor. This commutator is attractive for shorter FFT’s but can ...