87,662 research outputs found
A Subdivision Approach to Planar Semi-algebraic Sets
International audienceSemi-algebraic sets occur naturally when dealing with implicit models and boolean operations between them. In this work we present an algorithm to efficiently and in a certified way compute the connected components of semi-algebraic sets given by intersection or union of conjunctions of bi-variate equalities and inequalities. For any given precision, this algorithm can also provide a polygonal and isotopic approximation of the exact set. The idea is to localize the boundary curves by subdividing the space and then deduce their shape within small enough cells using only boundary information. Then a systematic traversal of the boundary curve graph yields polygonal regions isotopic to the connected components of the semi-algebraic set. Space subdivision is supported by a kd-tree structure and localization is done using Bernstein representation. We conclude by demonstrating our C++ implementation in the CAS Mathemagix
Continuity argument revisited: geometry of root clustering via symmetric products
We study the spaces of polynomials stratified into the sets of polynomial
with fixed number of roots inside certain semialgebraic region , on its
border, and at the complement to its closure. Presented approach is a
generalisation, unification and development of several classical approaches to
stability problems in control theory: root clustering (-stability) developed
by R.E. Kalman, B.R. Barmish, S. Gutman et al., -decomposition(Yu.I.
Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A.
Fam, J. Meditch, J.Ackermann).
Our approach is based on the interpretation of correspondence between roots
and coefficients of a polynomial as a symmetric product morphism.
We describe the topology of strata up to homotopy equivalence and, for many
important cases, up to homeomorphism. Adjacencies between strata are also
described. Moreover, we provide an explanation for the special position of
classical stability problems: Hurwitz stability, Schur stability,
hyperbolicity.Comment: 45 pages, 4 figure
A regularizing commutant duality for a kinematically covariant partial ordered net of observables
We consider a net of *-algebras, locally around any point of observation,
equipped with a natural partial order related to the isotony property. Assuming
the underlying manifold of the net to be a differentiable, this net shall be
kinematically covariant under general diffeomorphisms. However, the dynamical
relations, induced by the physical state defining the related net of (von
Neumann) observables, are in general not covariant under all diffeomorphisms,
but only under the subgroup of dynamical symmetries.
We introduce algebraically both, IR and UV cutoffs, and assume that these are
related by a commutant duality. The latter, having strong implications on the
net, allows us to identify a 1-parameter group of the dynamical symmetries with
the group of outer modular automorphisms.
For thermal equilibrium states, the modular dilation parameter may be used
locally to define the notions of both, time and a causal structure.Comment: LaTeX, to appear in: Proc. XXI. Int. Sem. on Group Theor. Methods,
Goslar (1996), eds. Doebner et a
Frequency-based brain networks: From a multiplex framework to a full multilayer description
We explore how to study dynamical interactions between brain regions using
functional multilayer networks whose layers represent the different frequency
bands at which a brain operates. Specifically, we investigate the consequences
of considering the brain as a multilayer network in which all brain regions can
interact with each other at different frequency bands, instead of as a
multiplex network, in which interactions between different frequency bands are
only allowed within each brain region and not between them. We study the second
smallest eigenvalue of the combinatorial supra-Laplacian matrix of the
multilayer network in detail, and we thereby show that the heterogeneity of
interlayer edges and, especially, the fraction of missing edges crucially
modify the spectral properties of the multilayer network. We illustrate our
results with both synthetic network models and real data sets obtained from
resting state magnetoencephalography. Our work demonstrates an important issue
in the construction of frequency-based multilayer brain networks.Comment: 13 pages, 8 figure
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