Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry

An action minimizing path between two given configurations, spatial or planar, of the $n$-body problem is always a true -- collision-free -- solution. Based on a remarkable idea of Christian Marchal, this theorem implies the existence of new "simple" symmetric periodic solutions, among which the Eight for 3 bodies, the Hip-Hop for 4 bodies and their generalizations

Ind--varieties of generalized flags as homogeneous spaces for classical ind--groups

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind--groups $SL(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in terms of generalized flags. Generalized flags in $V$ are chains of subspaces which in general cannot be enumerated by integers. Given a basis $E$ of $V$, we define a notion of $E$--commensurability for generalized flags, and prove that the set \cFl (\cF, E) of generalized flags E$--commensurable with a fixed generalized flag$\cF$in$V$has a natural structure of an ind--variety. In the case when$V$is the standard representation of$G = SL(\infty)$, all homogeneous ind--spaces$G/P$for parabolic subgroups$P$containing a fixed splitting Cartan subgroup of$G$, are of the form$\cFl (\cF, E)$. We also consider isotropic generalized flags. The corresponding ind--spaces are homogeneous spaces for$SO(\infty)$and$Sp(\infty)$. As an application of the construction, we compute the Picard group of$\cFl (\cF, E)$(and of its isotropic analogs) and show that$\cFl (\cF, E)$is a projective ind--variety if and only if$\cF$is a usual, possibly infinite, flag of subspaces in$V$

The management of segmental tibial shaft fractures: A systematic review.

INTRODUCTION: Segmental tibial fractures are complex injuries associated with significant soft tissue damage that are difficult to treat. This study aimed to identify the most effective method of treating segmental tibial fractures. METHOD: A PRISMA compliant systematic review was conducted. Studies investigating the management of segmental tibial fractures with intramedullary nail fixation (IMN), open reduction and internal fixation (ORIF) or circular external fixation (CEF) were included for review. The primary outcome measure was time to fracture union. Secondary outcomes were complications and functional outcome. A narrative analysis was undertaken as meta-analysis was inappropriate due to heterogeneity of the data. RESULTS: Thirteen studies were eligible and included. No randomised controlled trials were identified. Fixation with an intramedullary nail provided the fastest time to union, followed by open reduction and internal fixation and then CEF. The rate of deep infection was highest after IMN (5/162 [3%]), followed by open reduction and internal fixation (2/78 [2.5%]) and CEF (1/54 [2%]). However, some studies reported particularly high rates of infection following IMN for open segmental tibial fractures. There was limited reporting of postoperative deformities. From the studies that did include such data, there was a higher rate of deformity following ORIF (8/53 [15%]), compared to IMN (13/138 [9%]), and CEF (4/44 [9%]). Three studies, not including IMN, described patient reported outcome measures with results ranging from 'excellent' to 'fair'. DISCUSSION: The available evidence was of poor quality, dominated by retrospective case series. This prevented statistical analysis, and precludes firm conclusions being drawn from the results available. CONCLUSION: IMN has the fastest time to fracture union, however there are concerns regarding an increased deep infection rate in open segmental tibial fractures. In this subgroup, the data suggests CEF provides the most satisfactory results. However, the available literature does not provide sufficient detail to make this statement with certainty. We recommend a randomised controlled study to further investigate this challenging problem

Grothendieck ring of pretriangulated categories

We consider the abelian group $PT$ generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of "multiplication" $\bullet$ on the collection of DG categories which makes this abelian group into a commutative ring. A few applications are considered: representability of "standard" functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.Comment: 33 page

NGC1052 - A study of the pc-scale twin jet

We present results of a VLBA multi-frequency study of the pc-scale twin jet in NGC1052. We observed this object at epoch 1998.99 with the VLBA at 5, 8.4, 22 and 43 GHz both in total and linearly polarized intensity. The spectral analysis confirms the necessity of a free-free absorbing medium, obscuring the innermost part of both jets. At 5 GHz we found a compact linearly polarized emission region at the base of the eastern jet with a degree of polarization of 1.5%. At higher frequencies there is no evidence for polarization in our data. A core shift analysis constrains the position of the central engine to ~0.03 pc. The shift rates of the apparent core position with frequency confirm the strong influence of free-free absorption in conjunction with steep pressure gradients at the bases of both jets.Comment: Proceedings of the conference "SRT: the impact of large antennas on Radio Astronomy and Space Science", Cagliari, Italy, 7-10 November 2001, in press; 6 pages, 5 figures, needs srt_style.st

Accelerating Incremental Gradient Optimization with Curvature Information

This paper studies an acceleration technique for incremental aggregated gradient ({\sf IAG}) method through the use of \emph{curvature} information for solving strongly convex finite sum optimization problems. These optimization problems of interest arise in large-scale learning applications. Our technique utilizes a curvature-aided gradient tracking step to produce accurate gradient estimates incrementally using Hessian information. We propose and analyze two methods utilizing the new technique, the curvature-aided IAG ({\sf CIAG}) method and the accelerated CIAG ({\sf A-CIAG}) method, which are analogous to gradient method and Nesterov's accelerated gradient method, respectively. Setting $\kappa$ to be the condition number of the objective function, we prove the $R$ linear convergence rates of $1 - \frac{4c_0 \kappa}{(\kappa+1)^2}$ for the {\sf CIAG} method, and $1 - \sqrt{\frac{c_1}{2\kappa}}$ for the {\sf A-CIAG} method, where $c_0,c_1 \leq 1$ are constants inversely proportional to the distance between the initial point and the optimal solution. When the initial iterate is close to the optimal solution, the $R$ linear convergence rates match with the gradient and accelerated gradient method, albeit {\sf CIAG} and {\sf A-CIAG} operate in an incremental setting with strictly lower computation complexity. Numerical experiments confirm our findings. The source codes used for this paper can be found on \url{http://github.com/hoitowai/ciag/}.Comment: 22 pages, 3 figures, 3 tables. Accepted by Computational Optimization and Applications, to appea

Scaling by 5 on a 1/4-Cantor Measure

Each Cantor measure (\mu) with scaling factor 1/(2n) has at least one associated orthonormal basis of exponential functions (ONB) for L^2(\mu). In the particular case where the scaling constant for the Cantor measure is 1/4 and two specific ONBs are selected for L^2(\mu), there is a unitary operator U defined by mapping one ONB to the other. This paper focuses on the case in which one ONB (\Gamma) is the original Jorgensen-Pedersen ONB for the Cantor measure (\mu) and the other ONB is is 5\Gamma. The main theorem of the paper states that the corresponding operator U is ergodic in the sense that only the constant functions are fixed by U.Comment: 34 page

A Lane to a Road

This is a blog about all things food. From sharing my own kitchen creations and recipes, to pondering the inspiration I get from others. But mainly it is about the journey I have recently embarked upon, in establishing my own food business. When I was really young, about two or three years old, my dad used to ask me what I wanted to be when I grew up. Because my name is Elaine, my reply – which he had taught me to say – was ‘a road’. Cheesey eh? But I think it’s a good analogy for where I’m (hopefully) going :

On the crystalline period map

The paper contains a proof of the Fontaine-Jannsen conjecture based on a crystalline version of the p-adic Poincar'e lemma (different proofs were found earlier by Faltings, Niziol and Tsuji).Comment: 38 pages. The previous version of the paper will appear in March 2013 issue of Cambridge Journal of Mathematics. In present version the exposition of Hyodo-Kato theory (sect. 1.15-1.16) is simplifie