How Unsplittable-Flow-Covering helps Scheduling with Job-Dependent Cost Functions

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time $(1+\epsilon)$-approximation algorithm that yields an $(e+\epsilon)$-approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with \emph{optimal }cost at $1+\epsilon$ speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most $\log n$ many classes. This algorithm allows the jobs even to have up to $\log n$ many distinct release dates.Comment: 2 pages, 1 figur

Temporal Correlations and Persistence in the Kinetic Ising Model: the Role of Temperature

We study the statistical properties of the sum $S_t=\int_{0}^{t}dt' \sigma_{t'}$, that is the difference of time spent positive or negative by the spin $\sigma_{t}$, located at a given site of a $D$-dimensional Ising model evolving under Glauber dynamics from a random initial configuration. We investigate the distribution of $S_{t}$ and the first-passage statistics (persistence) of this quantity. We discuss successively the three regimes of high temperature ($T>T_{c}$), criticality ($T=T_c$), and low temperature ($T<T_{c}$). We discuss in particular the question of the temperature dependence of the persistence exponent $\theta$, as well as that of the spectrum of exponents $\theta(x)$, in the low temperature phase. The probability that the temporal mean $S_t/t$ was always larger than the equilibrium magnetization is found to decay as $t^{-\theta-\frac12}$. This yields a numerical determination of the persistence exponent $\theta$ in the whole low temperature phase, in two dimensions, and above the roughening transition, in the low-temperature phase of the three-dimensional Ising model.Comment: 21 pages, 11 PostScript figures included (1 color figure

The Discovery of Cherenkov Radiation and its use in the detection of extensive air showers

Cascades of charged particles are created when high-energy cosmic rays enter the earth's atmosphere: these 'extensive air-showers' are studied to gain information on the energy spectrum, arrival direction distribution and mass composition of the particles above 1014 eV where direct observations using instruments carried by balloons or satellites become impractical. Detection of light in the visible and ultra-violet ranges of the electromagnetic spectrum plays a key role in this work, the two processes involved being the emission of Cherenkov light and the production of fluorescence radiation. In this paper I will outline some of the history of the discovery of the Cherenkov process and describe the use to which it has been put in the study of extensive air-showers at ground level.Comment: To appear in Proceedings of CRIS2010: Cosmic Ray International Seminar on '100 years of Cosmic Rays: from Pioneering Experiments to Physics in Space

Stochastic Budget Optimization in Internet Advertising

Internet advertising is a sophisticated game in which the many advertisers "play" to optimize their return on investment. There are many "targets" for the advertisements, and each "target" has a collection of games with a potentially different set of players involved. In this paper, we study the problem of how advertisers allocate their budget across these "targets". In particular, we focus on formulating their best response strategy as an optimization problem. Advertisers have a set of keywords ("targets") and some stochastic information about the future, namely a probability distribution over scenarios of cost vs click combinations. This summarizes the potential states of the world assuming that the strategies of other players are fixed. Then, the best response can be abstracted as stochastic budget optimization problems to figure out how to spread a given budget across these keywords to maximize the expected number of clicks. We present the first known non-trivial poly-logarithmic approximation for these problems as well as the first known hardness results of getting better than logarithmic approximation ratios in the various parameters involved. We also identify several special cases of these problems of practical interest, such as with fixed number of scenarios or with polynomial-sized parameters related to cost, which are solvable either in polynomial time or with improved approximation ratios. Stochastic budget optimization with scenarios has sophisticated technical structure. Our approximation and hardness results come from relating these problems to a special type of (0/1, bipartite) quadratic programs inherent in them. Our research answers some open problems raised by the authors in (Stochastic Models for Budget Optimization in Search-Based Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio

Assessing microplastic exposure of large marine filter-feeders

Large filter-feeding animals are potential sentinels for understanding the extent of microplastic pollution, as their mode of foraging and prey mean they are continuously sampling the environment. However, there is considerable uncertainty about the total and mode of exposure (environmental vs trophic). Here, we explore microplastic exposure and ingestion by baleen whales feeding year-round in coastal Auckland waters, New Zealand. Plastic and DNA were extracted concurrently from whale scat, with 32 ± 24 (mean ± SD, n = 21) microplastics per 6 g scat sample detected. Using a novel stochastic simulation modeling incorporating new and previously published DNA diet information, we extrapolate this to total microplastic exposure levels of 24,028 (95% CI: 2119, 69,270) microplastics per mouthful of prey, or 3,408,002 microplastics (95% CI: 295,810, 10,031,370) per day, substantially higher than previous estimates for large filter-feeding animals. Critically, we find that the total exposure is four orders of magnitude more than expected from microplastic measurements of local coastal surface waters. This suggests that trophic transfer, rather than environmental exposure, is the predominant mode of exposure of large filter feeders for microplastic pollution. Measuring plastic concentration from the environment alone significantly underestimates exposure levels, an important consideration for future risk assessment studies.Environmental Biolog

On some algebraic identities and the exterior product of double forms

We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally generalize the previous results to higher multilinear forms namely to double forms. In particular, we show that the Cayley-Hamilton theorem once applied to the second fundamental form of a hypersurface of the Euclidean space is equivalent to a linearized version of the Gauss-Bonnet theorem, and once its generalization is applied to the Riemann curvature tensor (seen as a $(2,2)$ double form) is an infinitisimal version of the general Gauss-Bonnet-Chern theorem. In addition to that, the general Cayley-Hamilton theorems generate several universal curvature identities. The generalization of the classical Laplace expansion of the determinant to double forms is shown to lead to new general Avez type formulas for all Gauss-Bonnet curvatures.Comment: 32 pages, in this new version we added: an introduction to the exterior and composition products of double forms, a new section about hyperdeterminants and hyperpfaffians and reference