K- and L-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by Z/4

We compute the group homology, the topological K-theory of the reduced C^*-algebra, the algebraic K-theory and the algebraic L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by Z/4. These computations will follow from the more general treatment of a certain class of groups G which occur as extensions 1-->K-->G-->Q-->1 of a torsionfree group K by a group Q which satisfies certain assumptions. The key ingredients are the Baum-Connes and Farrell-Jones Conjectures and methods from equivariant algebraic topology.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper37.abs.htm

How does object fatness impact the complexity of packing in d dimensions?

Packing is a classical problem where one is given a set of subsets of Euclidean space called objects, and the goal is to find a maximum size subset of objects that are pairwise non-intersecting. The problem is also known as the Independent Set problem on the intersection graph defined by the objects. Although the problem is NP-complete, there are several subexponential algorithms in the literature. One of the key assumptions of such algorithms has been that the objects are fat, with a few exceptions in two dimensions; for example, the packing problem of a set of polygons in the plane surprisingly admits a subexponential algorithm. In this paper we give tight running time bounds for packing similarly-sized non-fat objects in higher dimensions. We propose an alternative and very weak measure of fatness called the stabbing number, and show that the packing problem in Euclidean space of constant dimension $d \geq 3$ for a family of similarly sized objects with stabbing number $\alpha$ can be solved in $2^{O(n^{1-1/d}\alpha)}$ time. We prove that even in the case of axis-parallel boxes of fixed shape, there is no $2^{o(n^{1-1/d}\alpha)}$ algorithm under ETH. This result smoothly bridges the whole range of having constant-fat objects on one extreme ($\alpha=1$) and a subexponential algorithm of the usual running time, and having very "skinny" objects on the other extreme ($\alpha=n^{1/d}$), where we cannot hope to improve upon the brute force running time of $2^{O(n)}$, and thereby characterizes the impact of fatness on the complexity of packing in case of similarly sized objects. We also study the same problem when parameterized by the solution size $k$, and give a $n^{O(k^{1-1/d}\alpha)}$ algorithm, with an almost matching lower bound.Comment: Short version appears in ISAAC 201

Smith equivalence and finite Oliver groups with Laitinen number 0 or 1

In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules at the two points are always isomorphic? We focus on the case G is an Oliver group and we present a classification of finite Oliver groups G with Laitinen number a_G = 0 or 1. Then we show that the Smith Isomorphism Question has a negative answer and a_G > 1 for any finite Oliver group G of odd order, and for any finite Oliver group G with a cyclic quotient of order pq for two distinct odd primes p and q. We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group G with a_G > 1. Moreover, we deduce that for a finite nonabelian simple group G, the answer to the Smith Isomorphism Question is affirmative if and only if a_G = 0 or 1.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-35.abs.htm

A Framework for Exponential-Time-Hypothesis--Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

We give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs of similarly-sized fat objects, yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixed dimension $d\ge 2$ for many well-known graph problems, including Independent Set, $r$-Dominating Set for constant $r$, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms are representation-agnostic, i.e., they work on the graph itself and do not require the geometric representation. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into $d$-dimensional grids, and it allows us to derive matching $2^{\Omega(n^{1-1/d})}$ lower bounds under the exponential time hypothesis even in the much more restricted class of $d$-dimensional induced grid graphs

Study protocol for a randomized controlled trial to explore the effects of personalized lifestyle advices and tandem skydives on pleasure in anhedonic young adults

Background:  Anhedonia is generally defined as the inability to feel pleasure in response to experiences that are usually enjoyable. Anhedonia is one of the two core symptoms of depression and is a major public health concern. Anhedonia has proven particularly difficult to counteract and predicts poor treatment response generally. It has often been hypothesized that anhedonia can be deterred by a healthy lifestyle. However, it is quite unlikely that a one-size-fits-all approach will be effective for everyone. In this study the effects of personalized lifestyle advice based on observed individual patterns of lifestyle behaviors and experienced pleasure will be examined. Further, we will explore whether a tandem skydive following the personalized lifestyle advice positively influences anhedonic young adults' abilities to carry out the recommended lifestyle changes, and whether this ultimately improves their self-reported pleasure. Methods:  Our study design is an exploratory intervention study, preceded by a cross-sectional survey as a screening instrument. For the survey, 2000 young adults (18-24 years old) will be selected from the general population. Based on survey outcomes, 72 individuals (36 males and 36 females) with persistent anhedonia (i.e., more than two months) and 60 individuals (30 males and 30 females) without anhedonia (non-anhedonic control group) will be selected for the intervention study. The non-anhedonic control group will fill out momentary assessments of pleasure and lifestyle behaviors three times a day, for one month. The anhedonic individuals will fill out momentary assessments for three consecutive months. After the first month, the anhedonic individuals will be randomly assigned to (1) no intervention, (2) lifestyle advice only, (3) lifestyle advice plus tandem skydive. The personalized lifestyle advice is based on patterns observed in the first month. Discussion:  The present study is the first to examine the effects of a personalized lifestyle advice and tandem skydive on pleasure in anhedonic young adults. Results of the present study may improve treatment for anhedonia, if the interventions are found to be effective