A tight lower bound instance for k-means++ in constant dimension

The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the first center randomly from the given points. For $i > 1$, pick a point to be the $i^{th}$ center with probability proportional to the square of the Euclidean distance of this point to the closest previously $(i-1)$ chosen centers. The k-means++ seeding algorithm is not only simple and fast but also gives an $O(\log{k})$ approximation in expectation as shown by Arthur and Vassilvitskii. There are datasets on which this seeding algorithm gives an approximation factor of $\Omega(\log{k})$ in expectation. However, it is not clear from these results if the algorithm achieves good approximation factor with reasonably high probability (say $1/poly(k)$). Brunsch and R\"{o}glin gave a dataset where the k-means++ seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability that is exponentially small in $k$. However, this and all other known lower-bound examples are high dimensional. So, an open problem was to understand the behavior of the algorithm on low dimensional datasets. In this work, we give a simple two dimensional dataset on which the seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability exponentially small in $k$. This solves open problems posed by Mahajan et al. and by Brunsch and R\"{o}glin.Comment: To appear in TAMC 2014. arXiv admin note: text overlap with arXiv:1306.420

The Azimuthal Asymmetry at large p_t seem to be too large for a ``Jet Quenching''

We discuss simple generic model of ``jet quenching'' in which matter absorption is defined by one parameter. We show that as absorption grows, the azimuthal asymmetry v_2 grows as well, reaching the finite limit with a simple geometric interpretation. It turns out, that this limit is still below the experimental values for 6 > p_t > 2 GeV, according to preliminary data from STAR experiment at RHIC. We thus conclude that ``jet quenching'' models alone cannot account for the observed phenomenon, and speculate about alternative scenarios.Comment: 3 pages, 2 figs, 1 table. The final version contaning note added in proofs for PRC, which reflects experimental development which seem to suggest that the geometrical model for v2 is in fact correct description of data at pt=2-10 Ge

On Existence and Properties of Approximate Pure Nash Equilibria in Bandwidth Allocation Games

In \emph{bandwidth allocation games} (BAGs), the strategy of a player consists of various demands on different resources. The player's utility is at most the sum of these demands, provided they are fully satisfied. Every resource has a limited capacity and if it is exceeded by the total demand, it has to be split between the players. Since these games generally do not have pure Nash equilibria, we consider approximate pure Nash equilibria, in which no player can improve her utility by more than some fixed factor $\alpha$ through unilateral strategy changes. There is a threshold $\alpha_\delta$ (where $\delta$ is a parameter that limits the demand of each player on a specific resource) such that $\alpha$-approximate pure Nash equilibria always exist for $\alpha \geq \alpha_\delta$, but not for $\alpha < \alpha_\delta$. We give both upper and lower bounds on this threshold $\alpha_\delta$ and show that the corresponding decision problem is ${\sf NP}$-hard. We also show that the $\alpha$-approximate price of anarchy for BAGs is $\alpha+1$. For a restricted version of the game, where demands of players only differ slightly from each other (e.g. symmetric games), we show that approximate Nash equilibria can be reached (and thus also be computed) in polynomial time using the best-response dynamic. Finally, we show that a broader class of utility-maximization games (which includes BAGs) converges quickly towards states whose social welfare is close to the optimum

The Origin of the Extragalactic Gamma-Ray Background and Implications for Dark-Matter Annihilation

The origin of the extragalactic $\gamma$-ray background (EGB) has been debated for some time. { The EGB comprises the $\gamma$-ray emission from resolved and unresolved extragalactic sources, such as blazars, star-forming galaxies and radio galaxies, as well as radiation from truly diffuse processes.} This letter focuses on the blazar source class, the most numerous detected population, and presents an updated luminosity function and spectral energy distribution model consistent with the blazar observations performed by the {\it Fermi} Large Area Telescope (LAT). We show that blazars account for 50$^{+12}_{-11}$\,\% of the EGB photons ($>$0.1\,GeV), and that {\it Fermi}-LAT has already resolved $\sim$70\,\% of this contribution. Blazars, and in particular low-luminosity hard-spectrum nearby sources like BL Lacs, are responsible for most of the EGB emission above 100\,GeV. We find that the extragalactic background light, which attenuates blazars' high-energy emission, is responsible for the high-energy cut-off observed in the EGB spectrum. Finally, we show that blazars, star-forming galaxies and radio galaxies can naturally account for the amplitude and spectral shape of the background in the 0.1--820\,GeV range, leaving only modest room for other contributions. This allows us to set competitive constraints on the dark-matter annihilation cross section.Comment: On behalf of the Fermi-LAT collaboration. Contact authors: M. Ajello, D. Gasparrini, M. Sanchez-Conde, G. Zaharijas, M. Gustafsson. Accepted for publication on ApJ

The Epsilon Calculus and Herbrand Complexity

Hilbert's epsilon-calculus is based on an extension of the language of predicate logic by a term-forming operator $\epsilon_{x}$. Two fundamental results about the epsilon-calculus, the first and second epsilon theorem, play a role similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.Comment: 23 p

Gamma Ray Bursts: recent results and connections to very high energy Cosmic Rays and Neutrinos

Gamma-ray bursts are the most concentrated explosions in the Universe. They have been detected electromagnetically at energies up to tens of GeV, and it is suspected that they could be active at least up to TeV energies. It is also speculated that they could emit cosmic rays and neutrinos at energies reaching up to the $10^{18}-10^{20}$ eV range. Here we review the recent developments in the photon phenomenology in the light of \swift and \fermi satellite observations, as well as recent IceCube upper limits on their neutrino luminosity. We discuss some of the theoretical models developed to explain these observations and their possible contribution to a very high energy cosmic ray and neutrino background.Comment: 12 pages, 7 figures. Text of a plenary lecture at the PASCOS 12 conference, Merida, Yucatan, Mexico, June 2012; to appear in J.Phys. (Conf. Series

p34cdc2-mediated phosphorylation at T124 inhibits nuclear import of SV-40 T antigen proteins

The nuclear import of transcription regulatory proteins appears to be used by the cell to trigger transitions in cell cycle, morphogenesis, and transformation. We have previously observed that the rate at which SV-40 T antigen fusion proteins containing a functional nuclear localization sequence (NLS; residues 126-132) are imported into the nucleus is enhanced in the presence of the casein kinase II (CK-II) site S111/112. In this study purified p34cdc2 kinase was used to phosphorylate T antigen proteins specifically at T124 and kinetic measurements at the single-cell level performed to assess its effect on nuclear protein import. T124 phosphorylation, which could be functionally simulated by a T-to-D124 substitution, was found to reduce the maximal extent of nuclear accumulation whilst negligibly affecting the import rate. The inhibition of nuclear import depended on the stoichiometry of phosphorylation. T124 and S111/112 could be phosphorylated independently of one another. Two alterative mechanisms were considered to explain the inhibition of nuclear import by T124 phosphorylation: inactivation of the NLS and cytoplasmic retention, respectively. Furthermore, we speculate that in vivo T124 phosphorylation may regulate the small but functionally significant amount of cytoplasmic SV-40 T antigen. A sequence comparison showed that many transcription regulatory proteins contain domains comprising potential CK-II-sites, cdc2-sites, and NLS. This raises the possibility that the three elements represent a functional unit regulating nuclear protein import