11,000 research outputs found
Remarks on Morphisms of Spectral Geometries
Having in view the study of a version of Gel'fand-Neumark duality adapted to
the context of Alain Connes' spectral triples, in this very preliminary review,
we first present a description of the relevant categories of geometrical
spaces, namely compact Hausdorff smooth finite-dimensional orientable
Riemannian manifolds (or more generally Hermitian bundles of Clifford modules
over them); we give some tentative definitions of the relevant categories of
algebraic structures, namely "propagators" and "spectral correspondences" of
commutative Riemannian spectral triples; and we provide a construction of
functors that associate a naive morphism of spectral triples to every smooth
(totally geodesic) map. The full construction of spectrum functors
(reconstruction theorem for morphisms) and a proof of duality between the
previous "geometrical' and "algebraic" categories are postponed to subsequent
works, but we provide here some hints in this direction. We also show how the
previous categories of "propagators" of commutative C*-algebras embed in the
mildly non-commutative environments of categories of suitable Hilbert
C*-bimodules, factorizable over commutative C*-algebras, with composition given
by internal tensor product.Comment: 9 pages, AMS-LaTeX2e. Reformatted, heavily revised and corrected
version, only for arXiv, of a previous review paper published in East-West
Journal of Mathematics. The main results presented in this review are now
part of F.Jaffrennou PhD thesis "Morphisms of Spectral Geometries" (Mahidol
University, June 2014
Improved Chebyshev series ephemeris generation capability of GTDS
An improved implementation of the Chebyshev ephemeris generation capability in the operational version of the Goddard Trajectory Determination System (GTDS) is described. Preliminary results of an evaluation of this orbit propagation method for three satellites of widely different orbit eccentricities are also discussed in terms of accuracy and computing efficiency with respect to the Cowell integration method. An empirical formula is deduced for determining an optimal fitting span which would give reasonable accuracy in the ephemeris with a reasonable consumption of computing resources
Matrix product states, geometry, and invariant theory
Matrix product states play an important role in quantum information theory to
represent states of many-body systems. They can be seen as low-dimensional
subvarieties of a high-dimensional tensor space. In these notes, we consider
two variants: homogeneous matrix product states and uniform matrix product
states. Studying the linear spans of these varieties leads to a natural
connection with invariant theory of matrices. For homogeneous matrix product
states, a classical result on polynomial identities of matrices leads to a
formula for the dimension of the linear span, in the case of 2x2 matrices.
These notes are based partially on a talk given by the author at the
University of Warsaw during the thematic semester "AGATES: Algebraic Geometry
with Applications to TEnsors and Secants", and partially on further research
done during the semester. This is still a preliminary version; an updated
version will be uploaded over the course of 2023.Comment: 10 pages; comments welcome
Monodromy of Projective Curves
The uniform position principle states that, given an irreducible
nondegenerate curve C in the projective r-space , a general (r-2)-plane L
is uniform, that is, projection from L induces a rational map from C to
whose monodromy group is the full symmetric group. In this paper we show the
locus of non-uniform (r-2)-planes has codimension at least two in the
Grassmannian for a curve C with arbitrary singularities. This result is optimal
in . For a smooth curve C in that is not a rational curve of degree
three, four or six, we show any irreducible surface of non-uniform lines is a
Schubert cycle of lines through a point , such that projection from is
not a birational map of onto its image.Comment: corrected typo in first paragraph of introduction, 23 pages, AMSLaTe
Brane actions, Categorification of Gromov-Witten theory and Quantum K-theory
Let X be a smooth projective variety. Using the idea of brane actions
discovered by To\"en, we construct a lax associative action of the operad of
stable curves of genus zero on the variety X seen as an object in
correspondences in derived stacks. This action encodes the Gromov-Witten theory
of X in purely geometrical terms and induces an action on the derived category
Qcoh(X) which allows us to recover the Quantum K-theory of Givental-Lee.Comment: final version, 64 pages, accepted for publication in Geometry &
Topolog
The Convex Hull of a Variety
We present a characterization, in terms of projective biduality, for the
hypersurfaces appearing in the boundary of the convex hull of a compact real
algebraic variety.Comment: 12 pages, 2 figure
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
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