2,103 research outputs found

    Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties

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    These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology.Comment: To appear in the proceedings of the 2008 Srni Winter School on Geometry and Physic

    Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley

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    We answer a question of K. Mulmuley: In [Efremenko-Landsberg-Schenck-Weyman] it was shown that the method of shifted partial derivatives cannot be used to separate the padded permanent from the determinant. Mulmuley asked if this "no-go" result could be extended to a model without padding. We prove this is indeed the case using the iterated matrix multiplication polynomial. We also provide several examples of polynomials with maximal space of partial derivatives, including the complete symmetric polynomials. We apply Koszul flattenings to these polynomials to have the first explicit sequence of polynomials with symmetric border rank lower bounds higher than the bounds attainable via partial derivatives.Comment: 18 pages - final version to appear in Theory of Computin

    Permanent v. determinant: an exponential lower bound assumingsymmetry and a potential path towards Valiant's conjecture

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    International audienceWe initiate a study of determinantal representations with symmetry. We show that Grenet's determinantal representation for the permanent is optimal among determinantal representations respecting left multiplication by permutation and diagonal matrices (roughly half the symmetry group of the permanent). In particular, if any optimal determinantal representation of the permanent must be polynomially related to one with such symmetry, then Valiant's conjecture on permanent v. determinant is true

    Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture

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    We make a first geometric study of three varieties in Cm⊗Cm⊗Cm\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m (for each mm), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen's Asymptotic Rank Conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.Comment: Final version. Revisions in Section 1 and Section

    Thermodynamics of a black hole in a cavity

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    We present a unified thermodynamical description of the configurations consisting on self-gravitating radiation with or without a black hole. We compute the thermal fluctuations and evaluate where will they induce a transition from metastable configurations towards stable ones. We show that the probability of finding such a transition is exponentially small. This indicates that, in a sequence of quasi equilibrium configurations, the system will remain in the metastable states till it approaches very closely the critical point beyond which no metastable configuration exists. Near that point, we relate the divergence of the local temperature fluctuations to the approach of the instability of the whole system, thereby generalizing the usual fluctuations analysis in the cases where long range forces are present. When angular momentum is added to the cavity, the above picture is slightly modified. Nevertheless, at high angular momentum, the black hole loses most of its mass before it reaches the critical point at which it evaporates completely.Comment: 27 pages, latex file, contains 3 figures available on request at [email protected]

    Comparative analysis of three brevetoxin-associated bottlenose dolphin (Tursiops truncatus) mortality events in the Florida Panhandle region (USA)

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    In the Florida Panhandle region, bottlenose dolphins (Tursiops truncatus) have been highly susceptible to large-scale unusual mortality events (UMEs) that may have been the result of exposure to blooms of the dinoflagellate Karenia brevis and its neurotoxin, brevetoxin (PbTx). Between 1999 and 2006, three bottlenose dolphin UMEs occurred in the Florida Panhandle region. The primary objective of this study was to determine if these mortality events were due to brevetoxicosis. Analysis of over 850 samples from 105 bottlenose dolphins and associated prey items were analyzed for algal toxins and have provided details on tissue distribution, pathways of trophic transfer, and spatial-temporal trends for each mortality event. In 1999/2000, 152 dolphins died following extensive K. brevis blooms and brevetoxin was detected in 52% of animals tested at concentrations up to 500 ng/g. In 2004, 105 bottlenose dolphins died in the absence of an identifiable K. brevis bloom; however, 100% of the tested animals were positive for brevetoxin at concentrations up to 29,126 ng/mL. Dolphin stomach contents frequently consisted of brevetoxin-contaminated menhaden. In addition, another potentially toxigenic algal species, Pseudo-nitzschia, was present and low levels of the neurotoxin domoic acid (DA) were detected in nearly all tested animals (89%). In 2005/2006, 90 bottlenose dolphins died that were initially coincident with high densities of K. brevis. Most (93%) of the tested animals were positive for brevetoxin at concentrations up to 2,724 ng/mL. No DA was detected in these animals despite the presence of an intense DA-producing Pseudo-nitzschia bloom. In contrast to the absence or very low levels of brevetoxins measured in live dolphins, and those stranding in the absence of a K. brevis bloom, these data, taken together with the absence of any other obvious pathology, provide strong evidence that brevetoxin was the causative agent involved in these bottlenose dolphin mortality events
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