128 research outputs found
Halving Balls in Deterministic Linear Time
Let \D be a set of pairwise disjoint unit balls in and the
set of their center points. A hyperplane \Hy is an \emph{-separator} for
\D if each closed halfspace bounded by \Hy contains at least points
from . This generalizes the notion of halving hyperplanes, which correspond
to -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 -separator for balls
in , for any , that intersects at most
balls, for some constant that depends on and . The number of
intersected balls is best possible up to the constant . Secondly, we present
a near-linear time algorithm to construct an -separator in
that intersects balls. Finally, we give a linear-time algorithm to
construct a halving line in that intersects
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)
Molekyylinmallinnusohjelma
Työn tavoitteena on suunnitella ja toteuttaa kÀyttökelpoinen molekyylinmallinnus-ohjelma, jota voisi mahdollisesti hyödyntÀÀ koulujen kemianopetuksessa tavanomaisen oppimisen ohella. Projektin idea lÀhti tekijÀn omasta mielenkiinnosta aiheeseen, ja kÀyttötarpeen selvittÀminen sekÀ mahdollinen kÀyttöönotto on tarkoitus aloittaa vasta työn pÀÀttymisen jÀlkeen.
Työn etenemiseen riitti teorian kannalta lukiossa opitut kemian sekÀ pitkÀn matematiikan taidot, joita jouduttiin kuitenkin kertaamaan internetlÀhteitÀ hyödyntÀen. Ohjelma toteutettiin Unity-pelimoottorilla, joka on suunniteltu erilaisten ohjelmien helppoon tuottamiseen ja jonka kÀyttöönottokynnys on suhteellisen matala.
Työn ohjelmointipuolen hankaluudesta johtuen sen jÀÀ kesken, joten jatkokehitystÀ on tehtÀvÀ, ennen kuin ohjelman kÀyttöÀ 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
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
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
A quantitative version of the non-abelian idempotent theorem
Suppose that G is a finite group and A is a subset of G such that 1_A has
algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of
subgroups of G, and L can be taken to be triply tower in O(M). This is a
quantitative version of the non-abelian idempotent theorem.Comment: 82 pp. Changed the title from `Indicator functions in the
Fourier-Eymard algebra'. Corrected the proof of Lemma 19.1. Expanded the
introduction. Corrected typo
Quasi-Monte Carlo rules for numerical integration over the unit sphere
We study numerical integration on the unit sphere using equal weight quadrature rules, where the weights are such
that constant functions are integrated exactly.
The quadrature points are constructed by lifting a -net given in the
unit square to the sphere 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
. And finally, we prove an upper bound on the spherical cap
-discrepancy of order (where denotes the
number of points). This slightly improves upon the bound on the spherical cap
-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm.
Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the
-nets lifted to the sphere have spherical cap
-discrepancy converging with the optimal order of
Search for direct production of charginos and neutralinos in events with three leptons and missing transverse momentum in âs = 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 fbâ1 of protonâproton 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
Jet size dependence of single jet suppression in lead-lead collisions at sqrt(s(NN)) = 2.76 TeV with the ATLAS detector at the LHC
Measurements of inclusive jet suppression in heavy ion collisions at the LHC
provide direct sensitivity to the physics of jet quenching. In a sample of
lead-lead collisions at sqrt(s) = 2.76 TeV corresponding to an integrated
luminosity of approximately 7 inverse microbarns, ATLAS has measured jets with
a calorimeter over the pseudorapidity interval |eta| < 2.1 and over the
transverse momentum range 38 < pT < 210 GeV. Jets were reconstructed using the
anti-kt algorithm with values for the distance parameter that determines the
nominal jet radius of R = 0.2, 0.3, 0.4 and 0.5. The centrality dependence of
the jet yield is characterized by the jet "central-to-peripheral ratio," Rcp.
Jet production is found to be suppressed by approximately a factor of two in
the 10% most central collisions relative to peripheral collisions. Rcp varies
smoothly with centrality as characterized by the number of participating
nucleons. The observed suppression is only weakly dependent on jet radius and
transverse momentum. These results provide the first direct measurement of
inclusive jet suppression in heavy ion collisions and complement previous
measurements of dijet transverse energy imbalance at the LHC.Comment: 15 pages plus author list (30 pages total), 8 figures, 2 tables,
submitted to Physics Letters B. All figures including auxiliary figures are
available at
http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/HION-2011-02
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