1,190 research outputs found
Products, coproducts and singular value decomposition
Products and coproducts may be recognized as morphisms in a monoidal tensor
category of vector spaces. To gain invariant data of these morphisms, we can
use singular value decomposition which attaches singular values, ie generalized
eigenvalues, to these maps. We show, for the case of Grassmann and Clifford
products, that twist maps significantly alter these data reducing degeneracies.
Since non group like coproducts give rise to non classical behavior of the
algebra of functions, ie make them noncommutative, we hope to be able to learn
more about such geometries. Remarkably the coproduct for positive singular
values of eigenvectors in yields directly corresponding eigenvectors in
A\otimes A.Comment: 17 pages, three eps-figure
Hopf algebras and characters of classical groups
Schur functions provide an integral basis of the ring of symmetric functions.
It is shown that this ring has a natural Hopf algebra structure by identifying
the appropriate product, coproduct, unit, counit and antipode, and their
properties. Characters of covariant tensor irreducible representations of the
classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur
functions, and the Hopf algebra is exploited in the determination of
group-subgroup branching rules and the decomposition of tensor products. The
analysis is carried out in terms of n-independent universal characters. The
corresponding rings, CharGL, CharO and CharSp, of universal characters each
have their own natural Hopf algebra structure. The appropriate product,
coproduct, unit, counit and antipode are identified in each case.Comment: 9 pages. Uses jpconf.cls and jpconf11.clo. Presented by RCK at
SSPCM'07, Myczkowce, Poland, Sept 200
On an easy transition from operator dynamics to generating functionals by Clifford algebras
Clifford geometric algebras of multivectors are treated in detail. These
algebras are build over a graded space and exhibit a grading or multivector
structure. The careful study of the endomorphisms of this space makes it clear,
that opposite Clifford algebras have to be used also. Based on this
mathematics, we give a fully Clifford algebraic account on generating
functionals, which is thereby geometric. The field operators are shown to be
Clifford and opposite Clifford maps. This picture relying on geometry does not
need positivity in principle. Furthermore, we propose a transition from
operator dynamics to corresponding generating functionals, which is based on
the algebraic techniques. As a calculational benefit, this transition is
considerable short compared to standard ones. The transition is not injective
(unique) and depends additionally on the choice of an ordering. We obtain a
direct and constructive connection between orderings and the explicit form of
the functional Hamiltonian. These orderings depend on the propagator of the
theory and thus on the ground state. This is invisible in path integral
formulations. The method is demonstrated within two examples, a non-linear
spinor field theory and spinor QED. Antisymmetrized and normal-ordered
functional equations are derived in both cases.Comment: 23p., 76kB, plain LaTeX, [email protected]
A Hopf laboratory for symmetric functions
An analysis of symmetric function theory is given from the perspective of the
underlying Hopf and bi-algebraic structures. These are presented explicitly in
terms of standard symmetric function operations. Particular attention is
focussed on Laplace pairing, Sweedler cohomology for 1- and 2-cochains, and
twisted products (Rota cliffordizations) induced by branching operators in the
symmetric function context. The latter are shown to include the algebras of
symmetric functions of orthogonal and symplectic type. A commentary on related
issues in the combinatorial approach to quantum field theory is given.Comment: 29 pages, LaTeX, uses amsmat
Z_2-gradings of Clifford algebras and multivector structures
Let Cl(V,g) be the real Clifford algebra associated to the real vector space
V, endowed with a nondegenerate metric g. In this paper, we study the class of
Z_2-gradings of Cl(V,g) which are somehow compatible with the multivector
structure of the Grassmann algebra over V. A complete characterization for such
Z_2-gradings is obtained by classifying all the even subalgebras coming from
them. An expression relating such subalgebras to the usual even part of Cl(V,g)
is also obtained. Finally, we employ this framework to define spinor spaces,
and to parametrize all the possible signature changes on Cl(V,g) by
Z_2-gradings of this algebra.Comment: 10 pages, LaTeX; v2 accepted for publication in J. Phys.
On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
Clifford algebras are naturally associated with quadratic forms. These
algebras are Z_2-graded by construction. However, only a Z_n-gradation induced
by a choice of a basis, or even better, by a Chevalley vector space isomorphism
Cl(V) \bigwedge V and an ordering, guarantees a multi-vector decomposition
into scalars, vectors, tensors, and so on, mandatory in physics. We show that
the Chevalley isomorphism theorem cannot be generalized to algebras if the
Z_n-grading or other structures are added, e.g., a linear form. We work with
pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which
we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford
algebras'. It turns out, that in this sense, all multi-vector Clifford algebras
of the same quadratic but different bilinear forms are non-isomorphic. The
usefulness of such algebras in quantum field theory and superconductivity was
shown elsewhere. Allowing for arbitrary bilinear forms however spoils their
diagonalizability which has a considerable effect on the tensor decomposition
of the Clifford algebras governed by the periodicity theorems, including the
Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which
can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1}
\otimes Cl_{1,1}. The general case used in quantum field theory lacks this
feature. Theories with non-symmetric bilinear forms are however needed in the
analysis of multi-particle states in interacting theories. A connection to
q-deformed structures through nontrivial vacuum states in quantum theories is
outlined.Comment: 25 pages, 1 figure, LaTeX, {Paper presented at the 5th International
Conference on Clifford Algebras and their Applications in Mathematical
Physics, Ixtapa, Mexico, June 27 - July 4, 199
Early transitions and tertiary enrolment: The cumulative impact of primary and secondary effects on entering university in Germany
Our aim is to assess how the number of working class students entering German universities can effectively be increased. Therefore, we estimate the proportion of students from the working class that would successfully enter university if certain policy interventions were in place to eliminate primary effects (performance differentials between social classes) and/or secondary effects (choice differentials net of performance) at different transition points. We extend previous research by analysing the sequence of transitions between elementary school enrolment and university enrolment and by accounting for the impact that manipulations at earlier transitions have on the performance distribution and size of the student ‘risk-set’ at subsequent transitions. To this end, we develop a novel simulation procedure which also seeks to find viable solutions to the shortcomings in the German data landscape. Our findings show that interventions are most effective if they take place early in the educational career. Neutralizing secondary effects at the transition to upper secondary school proves to be the single most effective means to increase participation rates in tertiary education among working class students. However, this comes at the expense of lower average performance levels. (DIPF/author
Next steps towards AMH as a robust biomarker for assessing ovarian aging in individual women
No abstract available
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