62 research outputs found
On the minima and convexity of Epstein Zeta function
Let be the Epstein zeta function defined as the
meromorphic continuation of the function
\sum_{k\in\Z^n\setminus\{0\}}(\sum_{i=1}^n [a_i k_i]^2)^{-s},
\text{Re} s>\frac{n}{2}
to the complex plane. We show that for fixed , the function
, as a function of with
fixed , has a unique minimum at the point .
When is fixed, the function can be shown to be a convex function of any of
the variables . These results are then applied to the study of
the sign of when is in the critical range . It is shown that when , as a
function of , can be both positive and negative for
every . When , there are some open subsets of
, where is positive for all . By regarding as a function of , we
find that when , the generalized Riemann hypothesis is false for all
.Comment: 27 page
Division Algebras and Quantum Theory
Quantum theory may be formulated using Hilbert spaces over any of the three
associative normed division algebras: the real numbers, the complex numbers and
the quaternions. Indeed, these three choices appear naturally in a number of
axiomatic approaches. However, there are internal problems with real or
quaternionic quantum theory. Here we argue that these problems can be resolved
if we treat real, complex and quaternionic quantum theory as part of a unified
structure. Dyson called this structure the "three-fold way". It is perhaps
easiest to see it in the study of irreducible unitary representations of groups
on complex Hilbert spaces. These representations come in three kinds: those
that are not isomorphic to their own dual (the truly "complex"
representations), those that are self-dual thanks to a symmetric bilinear
pairing (which are "real", in that they are the complexifications of
representations on real Hilbert spaces), and those that are self-dual thanks to
an antisymmetric bilinear pairing (which are "quaternionic", in that they are
the underlying complex representations of representations on quaternionic
Hilbert spaces). This three-fold classification sheds light on the physics of
time reversal symmetry, and it already plays an important role in particle
physics. More generally, Hilbert spaces of any one of the three kinds - real,
complex and quaternionic - can be seen as Hilbert spaces of the other kinds,
equipped with extra structure.Comment: 30 pages, 3 encapsulated Postscript figure
Unified Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Five Dimensions
Unified N=2 Maxwell-Einstein supergravity theories (MESGTs) are supergravity
theories in which all the vector fields, including the graviphoton, transform
in an irreducible representation of a simple global symmetry group of the
Lagrangian. As was established long time ago, in five dimensions there exist
only four unified Maxwell-Einstein supergravity theories whose target manifolds
are symmetric spaces. These theories are defined by the four simple Euclidean
Jordan algebras of degree three. In this paper, we show that, in addition to
these four unified MESGTs with symmetric target spaces, there exist three
infinite families of unified MESGTs as well as another exceptional one. These
novel unified MESGTs are defined by non-compact (Minkowskian) Jordan algebras,
and their target spaces are in general neither symmetric nor homogeneous. The
members of one of these three infinite families can be gauged in such a way as
to obtain an infinite family of unified N=2 Yang-Mills-Einstein supergravity
theories, in which all vector fields transform in the adjoint representation of
a simple gauge group of the type SU(N,1). The corresponding gaugings in the
other two infinite families lead to Yang-Mills-Einstein supergravity theories
coupled to tensor multiplets.Comment: Latex 2e, 28 pages. v2: reference added, footnote 14 enlarge
In-source decay and fragmentation characteristics of peptides using 5-aminosalicylic acid as a matrix in matrix-assisted laser desorption/ionization mass spectrometry
Journal of BIOPHOTONICS R E P R I N T Comparison of two methods for noninvasive determination of carotenoids in human and animal skin: Raman spectroscopy versus reflection spectroscopy
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