8,340 research outputs found

    Hyperelliptic jacobians with real multiplication

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    Let KK be a field of characteristic p≠2p \neq 2, and let f(x)f(x) be a sextic polynomial irreducible over KK with no repeated roots, whose Galois group is isomorphic to \A_5. If the jacobian J(C)J(C) of the hyperelliptic curve C:y2=f(x)C:y^2=f(x) admits real multiplication over the ground field from an order of a real quadratic field DD, then either its endomorphism algebra is isomorphic to DD, or p>0p > 0 and J(C)J(C) is a supersingular abelian variety. The supersingular outcome cannot occur when pp splits in DD.Comment: Corrected typos; clarified proofs; added more examples in positive characteristi

    Artificial in its own right

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    Artificial Cells, , Artificial Ecologies, Artificial Intelligence, Bio-Inspired Hardware Systems, Computational Autopoiesis, Computational Biology, Computational Embryology, Computational Evolution, Morphogenesis, Cyborgization, Digital Evolution, Evolvable Hardware, Cyborgs, Mathematical Biology, Nanotechnology, Posthuman, Transhuman

    Optimal Euclidean spanners: really short, thin and lanky

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    In a seminal STOC'95 paper, titled "Euclidean spanners: short, thin and lanky", Arya et al. devised a construction of Euclidean (1+\eps)-spanners that achieves constant degree, diameter O(log⁑n)O(\log n), and weight O(log⁑2n)β‹…Ο‰(MST)O(\log^2 n) \cdot \omega(MST), and has running time O(nβ‹…log⁑n)O(n \cdot \log n). This construction applies to nn-point constant-dimensional Euclidean spaces. Moreover, Arya et al. conjectured that the weight bound can be improved by a logarithmic factor, without increasing the degree and the diameter of the spanner, and within the same running time. This conjecture of Arya et al. became a central open problem in the area of Euclidean spanners. In this paper we resolve the long-standing conjecture of Arya et al. in the affirmative. Specifically, we present a construction of spanners with the same stretch, degree, diameter, and running time, as in Arya et al.'s result, but with optimal weight O(log⁑n)β‹…Ο‰(MST)O(\log n) \cdot \omega(MST). Moreover, our result is more general in three ways. First, we demonstrate that the conjecture holds true not only in constant-dimensional Euclidean spaces, but also in doubling metrics. Second, we provide a general tradeoff between the three involved parameters, which is tight in the entire range. Third, we devise a transformation that decreases the lightness of spanners in general metrics, while keeping all their other parameters in check. Our main result is obtained as a corollary of this transformation.Comment: A technical report of this paper was available online from April 4, 201

    On Efficient Distributed Construction of Near Optimal Routing Schemes

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    Given a distributed network represented by a weighted undirected graph G=(V,E)G=(V,E) on nn vertices, and a parameter kk, we devise a distributed algorithm that computes a routing scheme in (n1/2+1/k+D)β‹…no(1)(n^{1/2+1/k}+D)\cdot n^{o(1)} rounds, where DD is the hop-diameter of the network. The running time matches the lower bound of Ξ©~(n1/2+D)\tilde{\Omega}(n^{1/2}+D) rounds (which holds for any scheme with polynomial stretch), up to lower order terms. The routing tables are of size O~(n1/k)\tilde{O}(n^{1/k}), the labels are of size O(klog⁑2n)O(k\log^2n), and every packet is routed on a path suffering stretch at most 4kβˆ’5+o(1)4k-5+o(1). Our construction nearly matches the state-of-the-art for routing schemes built in a centralized sequential manner. The previous best algorithms for building routing tables in a distributed small messages model were by \cite[STOC 2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but suffers from substantially larger routing tables of size O(n1/2+1/k)O(n^{1/2+1/k}), while the latter has sub-optimal running time of O~(min⁑{(nD)1/2β‹…n1/k,n2/3+2/(3k)+D})\tilde{O}(\min\{(nD)^{1/2}\cdot n^{1/k},n^{2/3+2/(3k)}+D\})

    Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence

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    We study the {edge-coloring} problem in the message-passing model of distributed computing. This is one of the most fundamental and well-studied problems in this area. Currently, the best-known deterministic algorithms for (2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01}, where Delta is the maximum degree of the input graph. Also, recent results of \cite{BE10} for vertex-coloring imply that one can get an O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an arbitrarily small constant epsilon > 0. In this paper we devise a drastically faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves the previous state-of-the-art {exponentially} in a wide range of Delta, specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In addition, for small values of Delta our deterministic algorithm outperforms all the existing {randomized} algorithms for this problem. On our way to these results we study the {vertex-coloring} problem on the family of graphs with bounded {neighborhood independence}. This is a large family, which strictly includes line graphs of r-hypergraphs for any r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to {general} graphs. Our main technical contribution is a subroutine that computes an O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq Delta
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