1,208 research outputs found
A new graph parameter related to bounded rank positive semidefinite matrix completions
The Gram dimension \gd(G) of a graph is the smallest integer
such that any partial real symmetric matrix, whose entries are specified on the
diagonal and at the off-diagonal positions corresponding to edges of , can
be completed to a positive semidefinite matrix of rank at most (assuming a
positive semidefinite completion exists). For any fixed the class of graphs
satisfying \gd(G) \le k is minor closed, hence it can characterized by a
finite list of forbidden minors. We show that the only minimal forbidden minor
is for and that there are two minimal forbidden minors:
and for . We also show some close connections to
Euclidean realizations of graphs and to the graph parameter of
\cite{H03}. In particular, our characterization of the graphs with \gd(G)\le
4 implies the forbidden minor characterization of the 3-realizable graphs of
Belk and Connelly \cite{Belk,BC} and of the graphs with of van
der Holst \cite{H03}.Comment: 31 pages, 6 Figures. arXiv admin note: substantial text overlap with
arXiv:1112.596
Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra
We study a C*-dynamical system arising from the ring inclusion of the 2\times
2 integer matrices in the rational ones. The orientation preserving affine
groups of these rings form a Hecke pair that is closely related to a recent
construction of Connes and Marcolli; our dynamical system consists of the
associated reduced Hecke C*-algebra endowed with a canonical dynamics defined
in terms of the determinant function. We show that the Schlichting completion
also consists of affine groups of matrices, over the finite adeles, and we
obtain results about the structure and induced representations of the Hecke
C*-algebra. In a somewhat unexpected parallel with the one dimensional case
studied by Bost and Connes, there is a group of symmetries given by an action
of the finite integral ideles, and the corresponding fixed point algebra
decomposes as a tensor product over the primes. This decomposition allows us to
obtain a complete description of a natural class of equilibrium states which
conjecturally includes all KMS_\beta-states for \beta\ne 0,1.Comment: 30 pages; minor corrections, final versio
Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces
We define symmetric spaces in arbitrary dimension and over arbitrary
non-discrete topological fields \K, and we construct manifolds and symmetric
spaces associated to topological continuous quasi-inverse Jordan pairs and
-triple systems. This class of spaces, called smooth generalized projective
geometries, generalizes the well-known (finite or infinite-dimensional) bounded
symmetric domains as well as their ``compact-like'' duals. An interpretation of
such geometries as models of Quantum Mechanics is proposed, and particular
attention is paid to geometries that might be considered as "standard models"
-- they are associated to associative continuous inverse algebras and to Jordan
algebras of hermitian elements in such an algebra
Frame completions for optimally robust reconstruction
In information fusion, one is often confronted with the following problem:
given a preexisting set of measurements about an unknown quantity, what new
measurements should one collect in order to accomplish a given fusion task with
optimal accuracy and efficiency. We illustrate just how difficult this problem
can become by considering one of its more simple forms: when the unknown
quantity is a vector in a Hilbert space, the task itself is vector
reconstruction, and the measurements are linear functionals, that is, inner
products of the unknown vector with given measurement vectors. Such
reconstruction problems are the subject of frame theory. Here, we can measure
the quality of a given frame by the average reconstruction error induced by
noisy measurements; the mean square error is known to be the trace of the
inverse of the frame operator. We discuss preliminary results which help
indicate how to add new vectors to a given frame in order to reduce this mean
square error as much as possible
From p-adic to real Grassmannians via the quantum
Let F be a local field. The action of GL(n,F) on the Grassmann variety
Gr(m,n,F) induces a continuous representation of the maximal compact subgroup
of GL(n,F) on the space of L^2-functions on Gr(m,n,F). The irreducible
constituents of this representation are parameterized by the same underlying
set both for Archimedean and non-Archimedean fields.
This paper connects the Archimedean and non-Archimedean theories using the
quantum Grassmannian. In particular, idempotents in the Hecke algebra
associated to this representation are the image of the quantum zonal spherical
functions after taking appropriate limits. Consequently, a correspondence is
established between some irreducible representations with Archimedean and
non-Archimedean origin.Comment: 24 pages, final version, to appear in Advances in Mathematic
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