3,933 research outputs found
An operator extension of the parallelogram law and related norm inequalities
We establish a general operator parallelogram law concerning a
characterization of inner product spaces, get an operator extension of Bohr's
inequality and present several norm inequalities. More precisely, let
be a -algebra, be a locally compact Hausdorff space
equipped with a Radon measure and let be a continuous
field of operators in such that the function is
norm continuous on and the function is integrable. If
is a measurable function such that
for all , then we show that
\begin{align*} \int_T\int_T&\left|\alpha(t,s) A_t-\alpha(s,t)
A_s\right|^2d\mu(t)d\mu(s)+\int_T\int_T\left|\alpha(t,s) B_t-\alpha(s,t)
B_s\right|^2d\mu(t)d\mu(s) \nonumber &= 2\int_T\int_T\left|\alpha(t,s)
A_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) -
2\left|\int_T(A_t-B_t)d\mu(t)\right|^2\,. \end{align*}Comment: 9 pages; To appear in Math. Inequal. Appl. (MIA
Hamilton's Turns for the Lorentz Group
Hamilton in the course of his studies on quaternions came up with an elegant
geometric picture for the group SU(2). In this picture the group elements are
represented by ``turns'', which are equivalence classes of directed great
circle arcs on the unit sphere , in such a manner that the rule for
composition of group elements takes the form of the familiar parallelogram law
for the Euclidean translation group. It is only recently that this construction
has been generalized to the simplest noncompact group , the double cover of SO(2,1). The present work develops a theory of
turns for , the double and universal cover of SO(3,1) and ,
rendering a geometric representation in the spirit of Hamilton available for
all low dimensional semisimple Lie groups of interest in physics. The geometric
construction is illustrated through application to polar decomposition, and to
the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.Comment: 13 pages, Late
Polarimetric Control of Reflective Metasurfaces
This letter addresses the synthesis of reflective cells approaching a given
desired Floquet's scattering matrix. This work is motivated by the need to
obtain much finer control of reflective metasurfaces by controlling not only
their co-polarized reflection but also their cross-coupling behavior. The
demonstrated capability will enable more powerful design approaches -involving
all field components in phase and magnitude- and consequently better
performance in applications involving reflective metasurfaces. We first expose
some fundamental theoretical constraints on the cell scattering parameters.
Then, a successful procedure for controlling all four scattering parameters by
applying parallelogram and trapezoid transformations to square patches is
presented, considering both normal and oblique incidence
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