Halving Balls in Deterministic Linear Time

Let \D be a set of $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the set of their center points. A hyperplane \Hy is an \emph{$m$-separator} for \D if each closed halfspace bounded by \Hy contains at least $m$ points from $P$. This generalizes the notion of halving hyperplanes, which correspond to $n/2$-separators. The analogous notion for point sets has been well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only. We present three deterministic algorithms to bisect or approximately bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present a simple linear-time algorithm to construct an $\alpha n$-separator for balls in $\R^d$, for any $0<\alpha<1/2$, that intersects at most $cn^{(d-1)/d}$ balls, for some constant $c$ that depends on $d$ and $\alpha$. The number of intersected balls is best possible up to the constant $c$. Secondly, we present a near-linear time algorithm to construct an $(n/2-o(n))$-separator in $\R^d$ that intersects $o(n)$ balls. Finally, we give a linear-time algorithm to construct a halving line in $\R^2$ that intersects $O(n^{(5/6)+\epsilon})$ disks. Our results improve the runtime of a disk sliding algorithm by Bereg, Dumitrescu and Pach. In addition, our results improve and derandomize an algorithm to construct a space decomposition used by L{\"o}ffler and Mulzer to construct an onion (convex layer) decomposition for imprecise points (any point resides at an unknown location within a given disk)

Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed

Molekyylinmallinnusohjelma

Työn tavoitteena on suunnitella ja toteuttaa käyttökelpoinen molekyylinmallinnus-ohjelma, jota voisi mahdollisesti hyödyntää koulujen kemianopetuksessa tavanomaisen oppimisen ohella. Projektin idea lähti tekijän omasta mielenkiinnosta aiheeseen, ja käyttötarpeen selvittäminen sekä mahdollinen käyttöönotto on tarkoitus aloittaa vasta työn päättymisen jälkeen. Työn etenemiseen riitti teorian kannalta lukiossa opitut kemian sekä pitkän matematiikan taidot, joita jouduttiin kuitenkin kertaamaan internetlähteitä hyödyntäen. Ohjelma toteutettiin Unity-pelimoottorilla, joka on suunniteltu erilaisten ohjelmien helppoon tuottamiseen ja jonka käyttöönottokynnys on suhteellisen matala. Työn ohjelmointipuolen hankaluudesta johtuen sen jää kesken, joten jatkokehitystä on tehtävä, ennen kuin ohjelman käyttöä voidaan harkita opetuksessa.The aim of the thesis was to design and implement a 3D molecule modelling application that could be used in schools to help visualize molecules alongside with the traditional teaching methods. It remains to be seen whether the program will actually see any use, as the marketing side of the project was intentionally left out of the thesis work and will be conducted on a later date. For this thesis it was enough to know the basics of molecular chemistry and maths learned in high school, even though some rehearsing was necessary. The software was made using the Unity-engine, which is designed for easy developing of software. Its deployment has also been made simple, which helped in choosing it for this work. Due to the difficulties encountered during the development process the application was not made to the point originally planned. Further development is however planned for it and is needed before it can be considered to be used for teaching in schools

The Supremum Norm of the Discrepancy Function: Recent Results and Connections

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.Comment: 15 pages, 3 Figures. Reports on talks presented by the authors at the 10th international conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Sydney Australia, February 2011. v2: Comments of the referee are incorporate

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

Search for direct production of charginos and neutralinos in events with three leptons and missing transverse momentum in √s = 7 TeV pp collisions with the ATLAS detector

A search for the direct production of charginos and neutralinos in final states with three electrons or muons and missing transverse momentum is presented. The analysis is based on 4.7 fb−1 of proton–proton collision data delivered by the Large Hadron Collider and recorded with the ATLAS detector. Observations are consistent with Standard Model expectations in three signal regions that are either depleted or enriched in Z-boson decays. Upper limits at 95% confidence level are set in R-parity conserving phenomenological minimal supersymmetric models and in simplified models, significantly extending previous results

Quasi-Monte Carlo rules for numerical integration over the unit sphere S2\mathbb{S}^2

We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$

Elevated NSD3 histone methylation activity drives squamous cell lung cancer

Amplification of chromosomal region 8p11-12 is a common genetic alteration that has been implicated in the aetiology of lung squamous cell carcinoma (LUSC)(1-3). The FGFR1 gene is the main candidate driver of tumorigenesis within this region(4). However, clinical trials evaluating FGFR1 inhibition as a targeted therapy have been unsuccessful(5). Here we identify the histone H3 lysine 36 (H3K36) methyltransferase NSD3, the gene for which is located in the 8p11-12 amplicon, as a key regulator of LUSC tumorigenesis. In contrast to other 8p11-12 candidate LUSC drivers, increased expression of NSD3 correlated strongly with its gene amplification. Ablation of NSD3, but not of FGFR1, attenuated tumour growth and extended survival in a mouse model of LUSC. We identify an LUSC-associated variant NSD3(T1232A) that shows increased catalytic activity for dimethylation of H3K36 (H3K36me2) in vitro and in vivo. Structural dynamic analyses revealed that the T1232A substitution elicited localized mobility changes throughout the catalytic domain of NSD3 to relieve auto-inhibition and to increase accessibility of the H3 substrate. Expression of NSD3(T1232A) in vivo accelerated tumorigenesis and decreased overall survival in mouse models of LUSC. Pathological generation of H3K36me2 by NSD3(T1232A) reprograms the chromatin landscape to promote oncogenic gene expression signatures. Furthermore, NSD3, in a manner dependent on its catalytic activity, promoted transformation in human tracheobronchial cells and growth of xenografted human LUSC cell lines with amplification of 8p11-12. Depletion of NSD3 in patient-derived xenografts from primary LUSCs containing NSD3 amplification or the NSD3(T1232A)-encoding variant attenuated neoplastic growth in mice. Finally, NSD3-regulated LUSC-derived xenografts were hypersensitive to bromodomain inhibition. Thus, our work identifies NSD3 as a principal 8p11-12 amplicon-associated oncogenic driver in LUSC, and suggests that NSD3-dependency renders LUSC therapeutically vulnerable to bromodomain inhibition

Search for displaced muonic lepton jets from light Higgs boson decay in proton-proton collisions at root s=7 TeV with the ATLAS detector

A search is performed for collimated muon pairs displaced from the primary vertex produced in the decay of long-lived neutral particles in proton–proton collisions at root s=7 TeV centre-of-mass energy, with the ATLAS detector at the LHC. In a 1.9 fb−1event sample collected during 2011, the observed data are consistent with the Standard Model background expectations. Limits on the product of the production cross section and the branching ratio of a Higgs boson decaying to hidden-sector neutral long-lived particles are derived as a function of the particlesʼ mean lifetime