36,027 research outputs found
Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory
Quantum theory (QT), namely in terms of Schr\"odinger's 1926 wave functions
in general requires complex numbers to be formulated. However, it soon turned
out to even require some hypercomplex algebra. Incorporating Special Relativity
leads to an equation (Dirac 1928) requiring pairwise anti-commuting
coefficients, usually matrices. A unitary ring of square matrices
is an associative hypercomplex algebra by definition. Since only the algebraic
properties and relations of the elements matter, we replace the matrices by
biquaternions. In this paper, we first consider the basics of non-relativistic
and relativistic QT. Then we introduce general hypercomplex algebras and also
show how a relativistic quantum equation like Dirac's one can be formulated
using biquaternions. Subsequently, some algebraic preconditions for operations
within hypercomplex algebras and their subalgebras will be examined. For our
purpose equations akin to Schr\"odinger's should be able to be set up and
solved. Functions of complementary variables should be Fourier transforms of
each other. This should hold within a purely non-real subspace which must hence
be a subalgebra. Furthermore, it is an ideal denoted by . It must
be isomorphic to , hence containing an internal identity element.
The bicomplex numbers will turn out to fulfil these preconditions, and
therefore, the formalism of QT can be developed within its subalgebras. We also
show that bicomplex numbers encourage the definition of several different kinds
of conjugates. One of these treats the elements of like the usual
conjugate treats complex numbers. This defines a quantity what we call a
modulus which, in contrast to the complex absolute square, remains non-real
(but may be called `pseudo-real'). However, we do not conduct an explicit
physical interpretation here but we leave this to future examinations.Comment: 21 pages (without titlepage), 14 without titlepage and appendi
DNA electrophoresis studied with the cage model
The cage model for polymer reptation, proposed by Evans and Edwards, and its
recent extension to model DNA electrophoresis, are studied by numerically exact
computation of the drift velocities for polymers with a length L of up to 15
monomers. The computations show the Nernst-Einstein regime (v ~ E) followed by
a regime where the velocity decreases exponentially with the applied electric
field strength. In agreement with de Gennes' reptation arguments, we find that
asymptotically for large polymers the diffusion coefficient D decreases
quadratically with polymer length; for the cage model, the proportionality
coefficient is DL^2=0.175(2). Additionally we find that the leading correction
term for finite polymer lengths scales as N^{-1/2}, where N=L-1 is the number
of bonds.Comment: LaTeX (cjour.cls), 15 pages, 6 figures, added correctness proof of
kink representation approac
Finite Lorentz Transformations, Automorphisms, and Division Algebras
We give an explicit algebraic description of finite Lorentz transformations
of vectors in 10-dimensional Minkowski space by means of a parameterization in
terms of the octonions. The possible utility of these results for superstring
theory is mentioned. Along the way we describe automorphisms of the two highest
dimensional normed division algebras, namely the quaternions and the octonions,
in terms of conjugation maps. We use similar techniques to define and
via conjugation, via symmetric multiplication, and via
both symmetric multiplication and one-sided multiplication. The
non-commutativity and non-associativity of these division algebras plays a
crucial role in our constructions.Comment: 24 pages, Plain TeX, 2 figures on 1 page submitted separately as
uuencoded compressed tar fil
BioEM: GPU-accelerated computing of Bayesian inference of electron microscopy images
In cryo-electron microscopy (EM), molecular structures are determined from
large numbers of projection images of individual particles. To harness the full
power of this single-molecule information, we use the Bayesian inference of EM
(BioEM) formalism. By ranking structural models using posterior probabilities
calculated for individual images, BioEM in principle addresses the challenge of
working with highly dynamic or heterogeneous systems not easily handled in
traditional EM reconstruction. However, the calculation of these posteriors for
large numbers of particles and models is computationally demanding. Here we
present highly parallelized, GPU-accelerated computer software that performs
this task efficiently. Our flexible formulation employs CUDA, OpenMP, and MPI
parallelization combined with both CPU and GPU computing. The resulting BioEM
software scales nearly ideally both on pure CPU and on CPU+GPU architectures,
thus enabling Bayesian analysis of tens of thousands of images in a reasonable
time. The general mathematical framework and robust algorithms are not limited
to cryo-electron microscopy but can be generalized for electron tomography and
other imaging experiments
Octonic Electrodynamics
In this paper we present eight-component values "octons", generating
associative noncommutative algebra. It is shown that the electromagnetic field
in a vacuum can be described by a generalized octonic equation, which leads
both to the wave equations for potentials and fields and to the system of
Maxwell's equations. The octonic algebra allows one to perform compact combined
calculations simultaneously with scalars, vectors, pseudoscalars and
pseudovectors. Examples of such calculations are demonstrated by deriving the
relations for energy, momentum and Lorentz invariants of the electromagnetic
field. The generalized octonic equation for electromagnetic field in a matter
is formulated.Comment: 12 pages, 1 figur
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