211,238 research outputs found
A note on the abelianizations of finite-index subgroups of the mapping class group
For some , let be a finite index subgroup of the mapping
class group of a genus surface (possibly with boundary components and
punctures). An old conjecture of Ivanov says that the abelianization of
should be finite. In this note, we prove two theorems supporting this
conjecture. For the first, let denote the Dehn twist about a simple
closed curve . For some , we have . We prove
that is torsion in the abelianization of . Our second result
shows that the abelianization of is finite if contains a
"large chunk" (in a certain technical sense) of the Johnson kernel, that is,
the subgroup of the mapping class group generated by twists about separating
curves. This generalizes work of Hain and Boggi.Comment: 6 pages, 1 figure; a few revisions; to appear in Proc. Amer. Math.
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Regularity for general functionals with double phase
We prove sharp regularity results for a general class of functionals of the
type featuring non-standard growth
conditions and non-uniform ellipticity properties. The model case is given by
the double phase integral with . This changes its ellipticity rate according to the geometry of the level set
of the modulating coefficient . We also present new
methods and proofs, that are suitable to build regularity theorems for larger
classes of non-autonomous functionals. Finally, we disclose some new
interpolation type effects that, as we conjecture, should draw a general
phenomenon in the setting of non-uniformly elliptic problems. Such effects
naturally connect with the Lavrentiev phenomenon
Intelligence via ultrafilters: structural properties of some intelligence comparators of deterministic Legg-Hutter agents
Legg and Hutter, as well as subsequent authors, considered intelligent agents through the lens of interaction with reward-giving environments, attempting to assign numeric intelligence measures to such agents, with the guiding principle that a more intelligent agent should gain higher rewards from environments in some aggregate sense. In this paper, we consider a related question: rather than measure numeric intelligence of one Legg- Hutter agent, how can we compare the relative intelligence of two Legg-Hutter agents? We propose an elegant answer based on the following insight: we can view Legg-Hutter agents as candidates in an election, whose voters are environments, letting each environment vote (via its rewards) which agent (if either) is more intelligent. This leads to an abstract family of comparators simple enough that we can prove some structural theorems about them. It is an open question whether these structural theorems apply to more practical intelligence measures
Massless spin-two field S-duality
We present a review of the homological algebra tools involved in the standard de Rham theory and their subsequent generalizations relevant for the understanding of free massless higher spin gauge structure. M-theory arguments suggest the existence of an extension of (Abelian) S-duality symmetry for non-Abelian gauge theories, like the four dimensional Yang-Mills or Einstein theories. Some no-go theorems prove that this extension, if it exists, should fall outside the scope of local perturbative field theory
Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry
This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages.
This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove useful for the control of underwater vehicles
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