368,554 research outputs found
On the Proof Theory of Regular Fixed Points
International audienceWe consider encoding finite automata as least fixed points in a proof theoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to proof-theory. This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic proofs in particular
Singularities of Algebraic Differential Equations
We combine algebraic and geometric approaches to general systems of algebraic
ordinary or partial differential equations to provide a unified framework for
the definition and detection of singularities of a given system at a fixed
order. Our three main results are firstly a proof that even in the case of
partial differential equations regular points are generic. Secondly, we present
an algorithm for the effective detection of all singularities at a given order
or, more precisely, for the determination of a regularity decomposition.
Finally, we give a rigorous definition of a regular differential equation, a
notion that is ubiquitous in the geometric theory of differential equations,
and show that our algorithm extracts from each prime component a regular
differential equation. Our main algorithmic tools are on the one hand the
algebraic resp. differential Thomas decomposition and on the other hand the
Vessiot theory of differential equations.Comment: 45 pages, 5 figure
Geometric singularities and a flow tangent to the Ricci flow
We consider a geometric flow introduced by Gigli and Mantegazza which, in the
case of smooth compact manifolds with smooth metrics, is tangen- tial to the
Ricci flow almost-everywhere along geodesics. To study spaces with geometric
singularities, we consider this flow in the context of smooth manifolds with
rough metrics with sufficiently regular heat kernels. On an appropriate non-
singular open region, we provide a family of metric tensors evolving in time
and provide a regularity theory for this flow in terms of the regularity of the
heat kernel.
When the rough metric induces a metric measure space satisfying a Riemannian
Curvature Dimension condition, we demonstrate that the distance induced by the
flow is identical to the evolving distance metric defined by Gigli and
Mantegazza on appropriate admissible points. Consequently, we demonstrate that
a smooth compact manifold with a finite number of geometric conical
singularities remains a smooth manifold with a smooth metric away from the cone
points for all future times. Moreover, we show that the distance induced by the
evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by
Gigli-Mantegazza.Comment: Fixed proof of Lemma 5.4, updated references to published work
Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular
Because of Minty’s classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or “almost have” fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors
(nearest point mappings). The proof resides in a Hilbert product space and it relies upon the Brezis-Haraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex.Natural Sciences and Engineering Research Council of CanadaCanada Research Chair ProgramDirección General de Enseñanza SuperiorJunta de Andaluci
- …