1,205 research outputs found
Shapely monads and analytic functors
In this paper, we give precise mathematical form to the idea of a structure
whose data and axioms are faithfully represented by a graphical calculus; some
prominent examples are operads, polycategories, properads, and PROPs. Building
on the established presentation of such structures as algebras for monads on
presheaf categories, we describe a characteristic property of the associated
monads---the shapeliness of the title---which says that "any two operations of
the same shape agree". An important part of this work is the study of analytic
functors between presheaf categories, which are a common generalisation of
Joyal's analytic endofunctors on sets and of the parametric right adjoint
functors on presheaf categories introduced by Diers and studied by
Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among
the analytic endofunctors, and may be characterised as the submonads of a
universal analytic monad with "exactly one operation of each shape". In fact,
shapeliness also gives a way to define the data and axioms of a structure
directly from its graphical calculus, by generating a free shapely monad on the
basic operations of the calculus. In this paper we do this for some of the
examples listed above; in future work, we intend to do so for graphical calculi
such as Milner's bigraphs, Lafont's interaction nets, or Girard's
multiplicative proof nets, thereby obtaining canonical notions of denotational
model
Equivariant chain complexes, twisted homology and relative minimality of arrangements
We show that the equivariant chain complex associated to a minimal
CW-structure X on the complement M(A) of a hyperplane arrangement A, is
independent of X.
When A is a sufficiently general linear section of an aspheric arrangement,
we explain a new way for computing the twisted homology of M(A).Comment: 22 page
Singular trajectories of control-affine systems
When applying methods of optimal control to motion planning or stabilization
problems, some theoretical or numerical difficulties may arise, due to the
presence of specific trajectories, namely, singular minimizing trajectories of
the underlying optimal control problem. In this article, we provide
characterizations for singular trajectories of control-affine systems. We prove
that, under generic assumptions, such trajectories share nice properties,
related to computational aspects; more precisely, we show that, for a generic
system -- with respect to the Whitney topology --, all nontrivial singular
trajectories are of minimal order and of corank one. These results, established
both for driftless and for control-affine systems, extend previous results. As
a consequence, for generic systems having more than two vector fields, and for
a fixed cost, there do not exist minimizing singular trajectories. We also
prove that, given a control system satisfying the LARC, singular trajectories
are strictly abnormal, generically with respect to the cost. We then show how
these results can be used to derive regularity results for the value function
and in the theory of Hamilton-Jacobi equations, which in turn have applications
for stabilization and motion planning, both from the theoretical and
implementation issues
Modal logics are coalgebraic
Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pick-and-choose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors' firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility
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