624 research outputs found

    Are sketch-and-precondition least squares solvers numerically stable?

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    Sketch-and-precondition techniques are popular for solving large least squares (LS) problems of the form Ax=bAx=b with ARm×nA\in\mathbb{R}^{m\times n} and mnm\gg n. This is where AA is ``sketched" to a smaller matrix SASA with SRcn×mS\in\mathbb{R}^{\lceil cn\rceil\times m} for some constant c>1c>1 before an iterative LS solver computes the solution to Ax=bAx=b with a right preconditioner PP, where PP is constructed from SASA. Popular sketch-and-precondition LS solvers are Blendenpik and LSRN. We show that the sketch-and-precondition technique is not numerically stable for ill-conditioned LS problems. Instead, we propose using an unpreconditioned iterative LS solver on (AP)y=b(AP)y=b with x=Pyx=Py when accuracy is a concern. Provided the condition number of AA is smaller than the reciprocal of the unit round-off, we show that this modification ensures that the computed solution has a comparable backward error to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to provide a convincing argument that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems.Comment: 22 page

    A1_3 I believe in MIRACLs, where're you from?

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    The Mid-Infrared Advanced Chemical Laser (MIRACL) antimissile system is currently being developed by the US Army. Its aim is to destroy incoming missiles with lasers. This paper determines that there is a maximum of 10 minutes preparation time (assuming the missile is fired from at least 10 minutes flight time away), and it would take a 1 MW laser about 0.82 seconds to destroy a typical missile made of aluminium. It does not however cover the practical difficulties involved with implementing such a system

    Circadian gating of dark-induced increases in chloroplast- and cytosolic-free calcium in Arabidopsis.

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    Changes in the spatiotemporal concentration of free Ca2+ ([Ca2+ ]) in different organelles of the cell contribute to responses of plants to physiological and environmental stimuli. One example are [Ca2+ ] increases in the stroma of chloroplasts during light-to-dark transitions; however, the function and mechanisms responsible are unknown, in part because there is a disagreement in the literature concerning whether corresponding dark-induced changes in cytosolic [Ca2+ ] ([Ca2+ ]cyt ) can be detected. We have measured changes in [Ca2+ ]cyt upon darkness in addition to the already known dark-induced increases in [Ca2+ ]stroma in the aerial part of the Arabidopsis thaliana plant. These [Ca2+ ]cyt transients depend on the photoperiod and time of day, peaking at anticipated dusk, and are superimposed on daily 24 h oscillations in [Ca2+ ]cyt . We also find that the magnitude of the dark-induced increases in Ca2+ in both the cytosol and chloroplasts are gated by the nuclear circadian oscillator. The modulation of the magnitude of dark-induced increases in [Ca2+ ]stroma and [Ca2+ ]cyt by transcriptional regulators in the nucleus that are part of the circadian oscillator demonstrates a new role for the circadian system in subcellular Ca2+ signalling, in addition to its role in driving circadian oscillations of [Ca2+ ] in the cytosol and chloroplasts.Broadbank Awar

    A1_5 Atomising Death Ray

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    This report examines the weapons commonly found in science fiction movies which are able to completely atomise a human being, leaving nothing visible behind. To atomise a human body, this report uses the approximation that every bond connecting their atoms must be broken simultaneously. The energy required from such a device is found to be ~3.75 GJ and it is found that it would require 11.06 mg of Deuterium-Tritium to undergo fusion to provide this energy. It is also found that targets will glow deep violet in the visible spectrum shot

    A1_7 The Neutrino Problem

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    This report examines the premise from the movie "2012" (2009) that solar neutrinos start interacting with the core of the Earth, causing it to melt. A lower energy limit required for this interaction is found to be 9.55*10^4 GeV  and it is shown that the source of such energetic neutrinos is unlikely to be from within the solar system

    A1_2 Duck, duck... Spruce!

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    The H-4 Hercules was a prototype aircraft built in 1947 which was never properly flight tested. This report explores a theory that this aircraft would never have been able to fly more than a few hundred feet off of the ground due to the ground effect and that the plane was not viable for the purpose of transporting heavy military equipment. By exploring the pressure difference over the wing surfaces, the aircraft is shown to be able to reach a maximum altitude of 9,660 m and to require a minimum takeoff speed of 113.6 km/h. It is therefore shown that the aircraft would have been able to takeoff fully loaded and could have reached a high altitude successfully.Â