15,944 research outputs found
Commutators and Anti-Commutators of Idempotents in Rings
We show that a ring has two idempotents with an invertible
commutator if and only if for a
ring in which is a sum of two units. In this case, the
"anti-commutator" is automatically invertible, so we study also
the broader class of rings having such an invertible anti-commutator. Simple
artinian rings (along with other related classes of matrix rings) with
one of the above properties are completely determined. In this study, we also
arrive at various new criteria for {\it general\} matrix rings.
For instance, is such a matrix ring if and only if it has an invertible
commutator where .Comment: 21 page
On vanishing sums of th roots of unity in finite fields
In an earlier work, the authors have determined all possible weights for
which there exists a vanishing sum of th roots
of unity in characteristic 0. In this paper, the same problem is
studied in finite fields of characteristic . For given and , results
are obtained on integers such that all integers are in the
``weight set'' . The main result in this paper guarantees,
under suitable conditions, the existence of solutions of
with all coordinates not equal to zero over a finite field
Elementary Proofs Of Two Theorems Involving Arguments Of Eigenvalues Of A Product Of Two Unitary Matrices
We give elementary proofs of two theorems concerning bounds on the maximum
argument of the eigenvalues of a product of two unitary matrices --- one by
Childs \emph{et al.} [J. Mod. Phys., \textbf{47}, 155 (2000)] and the other one
by Chau [arXiv:1006.3614]. Our proofs have the advantages that the necessary
and sufficient conditions for equalities are apparent and that they can be
readily generalized to the case of infinite-dimensional unitary operators.Comment: 8 pages in Revtex 4.1 preprint format, to appear in Journal of
Inequalities and Application
Quasi-dark Mode in a Metamaterial for Analogous Electromagnetically-induced Transparency
We study a planar metamaterial supporting electromagnetically-induced
transparency (EIT)-like effect by exploiting the coupling between bright and
quasi-dark eigenmodes. The specific design of such a metamaterial consists of a
cut-wire (CW) and a single-gap split-ring resonator (SRR). From the numerical
and the analytical results we demonstrate that the response of SRR, which is
weakly excited by external electric field, is mitigated to be a quasi-dark
eigenmode in the presence of strongly radiative CW. This result suggests more
relaxed conditions for the realization of devices utilizing the EIT-like
effects in metamaterial, and thereby widens the possibilities for many
different structural implementations.Comment: 11 pages, 4 figure
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