This paper devotes to studying uncertainty principles of Heisenberg type for signals defined on Rn taking values in a Clifford algebra. For real-para-vector-valued signals possessing all first-order ...partial derivatives we obtain two uncertainty principles of which both correspond to the strongest form of the Heisenberg type uncertainty principles for the one-dimensional space. The lower-bounds of the new uncertainty principles are in terms of a scalar-valued phase derivative. Through Hardy spaces decomposition we also obtain two forms of uncertainty principles for real-valued signals of finite energy with the first order Sobolev type smoothness.
We introduce a useful approach to find asymptotically explicit expressions for the Casimir free energy at large temperature. The resulting expressions contain the classical terms as well as the few ...first terms of the corresponding heat-kernel expansion, as expected. This technique works well for many familiar configurations in Euclidean as well as non-Euclidean spaces. By utilizing this approach, we provide some new numerically considerable results for the Casimir pressure in some rectangular ideal-metal cavities. For instance, we show that at sufficiently large temperature, the Casimir pressure acting on the sidewalls of a rectangular tube can be up to twice that of the two parallel planes. We also apply this technique for calculating the Casimir free energy on a 3-torus as well as a 3-sphere. We show that a nonzero mass term for both scalar and spinor fields as well on the torus as on the sphere, violates the third law of thermodynamics. We obtain some negative values for the Casimir entropy on the 3-torus as well as on the 3-sphere. We speculate that these negative Casimir entropies can be interpreted thermodynamically as an instability of the vacuum state at finite temperatures.
In this paper, we obtain a classification theorem for generalized Yamabe solitons, called conformal solitons, on pseudo–Riemannian hypersurfaces in pseudo–Euclidean spaces by taking the vector field ...as the tangential part of the position vector field. By using the theorem, we also obtain complete classification of Yamabe solitons, almost Yamabe solitons,
k
–Yamabe solitons and
h
–almost Yamabe solitons on pseudo–Riemannian hypersurfaces in pseudo–Euclidean space under the same assumption.
The Banach space
P
(
2
X
)
of 2-homogeneous polynomials on the Banach space
X
can be naturally embedded in the Banach space
Lip
0
(
B
X
)
of real-valued Lipschitz functions on
B
X
that vanish at 0. ...We investigate whether
P
(
2
X
)
is a complemented subspace of
Lip
0
(
B
X
)
. This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that
P
(
1
X
)
=
X
∗
is complemented in
Lip
0
(
B
X
)
for every Banach space
X
. Our main result asserts that
P
(
2
X
)
is not complemented in
Lip
0
(
B
X
)
for every Banach space
X
with non-trivial type.
The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space R n and, in a more general setup, on a ...connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups.
PDF
