660 research outputs found
On the equivalence of modes of convergence for log-concave measures
An important theme in recent work in asymptotic geometric analysis is that
many classical implications between different types of geometric or functional
inequalities can be reversed in the presence of convexity assumptions. In this
note, we explore the extent to which different notions of distance between
probability measures are comparable for log-concave distributions. Our results
imply that weak convergence of isotropic log-concave distributions is
equivalent to convergence in total variation, and is further equivalent to
convergence in relative entropy when the limit measure is Gaussian.Comment: v3: Minor tweak in exposition. To appear in GAFA seminar note
Technology Needs Assessment of an Atmospheric Observation System for Multidisciplinary Air Quality/Meteorology Missions, Part 2
The technology advancements that will be necessary to implement the atmospheric observation systems are considered. Upper and lower atmospheric air quality and meteorological parameters necessary to support the air quality investigations were included. The technology needs were found predominantly in areas related to sensors and measurements of air quality and meteorological measurements
Remarks on the Central Limit Theorem for Non-Convex Bodies
In this note, we study possible extensions of the Central Limit Theorem for
non-convex bodies. First, we prove a Berry-Esseen type theorem for a certain
class of unconditional bodies that are not necessarily convex. Then, we
consider a widely-known class of non-convex bodies, the so-called p-convex
bodies, and construct a counter-example for this class
Triangulations and Severi varieties
We consider the problem of constructing triangulations of projective planes
over Hurwitz algebras with minimal numbers of vertices. We observe that the
numbers of faces of each dimension must be equal to the dimensions of certain
representations of the automorphism groups of the corresponding Severi
varieties. We construct a complex involving these representations, which should
be considered as a geometric version of the (putative) triangulations
The central limit problem for random vectors with symmetries
Motivated by the central limit problem for convex bodies, we study normal
approximation of linear functionals of high-dimensional random vectors with
various types of symmetries. In particular, we obtain results for distributions
which are coordinatewise symmetric, uniform in a regular simplex, or
spherically symmetric. Our proofs are based on Stein's method of exchangeable
pairs; as far as we know, this approach has not previously been used in convex
geometry and we give a brief introduction to the classical method. The
spherically symmetric case is treated by a variation of Stein's method which is
adapted for continuous symmetries.Comment: AMS-LaTeX, uses xy-pic, 23 pages; v3: added new corollary to Theorem
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
Effects of mesenchymal stromal cells versus serum on tendon healing in a controlled experimental trial in an equine model
Abstract Background Mesenchymal stromal cells (MSC) have shown promising results in the treatment of tendinopathy in equine medicine, making this therapeutic approach seem favorable for translation to human medicine. Having demonstrated that MSC engraft within the tendon lesions after local injection in an equine model, we hypothesized that they would improve tendon healing superior to serum injection alone. Methods Quadrilateral tendon lesions were induced in six horses by mechanical tissue disruption combined with collagenase application 3 weeks before treatment. Adipose-derived MSC suspended in serum or serum alone were then injected intralesionally. Clinical examinations, ultrasound and magnetic resonance imaging were performed over 24 weeks. Tendon biopsies for histological assessment were taken from the hindlimbs 3 weeks after treatment. Horses were sacrificed after 24 weeks and forelimb tendons were subjected to macroscopic and histological examination as well as analysis of musculoskeletal marker expression. Results Tendons injected with MSC showed a transient increase in inflammation and lesion size, as indicated by clinical and imaging parameters between week 3 and 6 (p < 0.05). Thereafter, symptoms decreased in both groups and, except that in MSC-treated tendons, mean lesion signal intensity as seen in T2w magnetic resonance imaging and cellularity as seen in the histology (p < 0.05) were lower, no major differences could be found at week 24. Conclusions These data suggest that MSC have influenced the inflammatory reaction in a way not described in tendinopathy studies before. However, at the endpoint of the current study, 24 weeks after treatment, no distinct improvement was observed in MSC-treated tendons compared to the serum-injected controls. Future studies are necessary to elucidate whether and under which conditions MSC are beneficial for tendon healing before translation into human medicine
Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces
This paper contains a thorough study of the trigonometry of the homogeneous
symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex
Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and
hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and
some non-compact symmetric spaces associated to SL(N+1,R) are the generic
members in this family. The method encapsulates trigonometry for this whole
family of spaces into a single "basic trigonometric group equation", and has
'universality' and '(self)-duality' as its distinctive traits. All previously
known results on the trigonometry of CP^N and CH^N follow as particular cases
of our general equations. The physical Quantum Space of States of any quantum
system belongs, as the complex Hermitian space member, to this parametrised
family; hence its trigonometry appears as a rather particular case of the
equations we obtain.Comment: 46 pages, LaTe
There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems
There is no 5,7-triangulation of the torus, that is, no triangulation with
exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no
3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be
bicolored. Similar statements hold for 4,8-triangulations and
2,6-quadrangulations. We prove these results, of which the first two are known
and the others seem to be new, as corollaries of a theorem on the holonomy
group of a euclidean cone metric on the torus with just two cone points. We
provide two proofs of this theorem: One argument is metric in nature, the other
relies on the induced conformal structure and proceeds by invoking the residue
theorem. Similar methods can be used to prove a theorem of Dress on infinite
triangulations of the plane with exactly two irregular vertices. The
non-existence results for torus decompositions provide infinite families of
graphs which cannot be embedded in the torus.Comment: 14 pages, 11 figures, only minor changes from first version, to
appear in Geometriae Dedicat
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