8,140 research outputs found

    The Consistent Ethic of Life and Health Care Systems

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    Symmetric matrices related to the Mertens function

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    In this paper we explore a family of congruences over N\N^\ast from which one builds a sequence of symmetric matrices related to the Mertens function. From the results of numerical experiments, we formulate a conjecture about the growth of the quadratic norm of these matrices, which implies the Riemann hypothesis. This suggests that matrix analysis methods may come to play a more important role in this classical and difficult problem.Comment: Version submitted to LAA; some new reference

    Newman\u27s Idea of a University and Its Relevance to Catholic Higher Education

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    Minimum Entropy Orientations

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    We study graph orientations that minimize the entropy of the in-degree sequence. The problem of finding such an orientation is an interesting special case of the minimum entropy set cover problem previously studied by Halperin and Karp [Theoret. Comput. Sci., 2005] and by the current authors [Algorithmica, to appear]. We prove that the minimum entropy orientation problem is NP-hard even if the graph is planar, and that there exists a simple linear-time algorithm that returns an approximate solution with an additive error guarantee of 1 bit. This improves on the only previously known algorithm which has an additive error guarantee of log_2 e bits (approx. 1.4427 bits).Comment: Referees' comments incorporate

    The Complexity of Simultaneous Geometric Graph Embedding

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    Given a collection of planar graphs G1,,GkG_1,\dots,G_k on the same set VV of nn vertices, the simultaneous geometric embedding (with mapping) problem, or simply kk-SGE, is to find a set PP of nn points in the plane and a bijection ϕ:VP\phi: V \to P such that the induced straight-line drawings of G1,,GkG_1,\dots,G_k under ϕ\phi are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for R\exists\mathbb{R}, the existential theory of the reals. Hence the problem kk-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to kk-SGE, with the property that both numbers kk and nn are linear in the number of pseudolines. This implies not only the R\exists\mathbb{R}-hardness result, but also a 22Ω(n)2^{2^{\Omega (n)}} lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only 22Ω(n)2^{2^{\Omega (\sqrt{n})}}

    On Universal Point Sets for Planar Graphs

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    A set P of points in R^2 is n-universal, if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n>=15. Conversely, we use a computer program to show that there exist universal point sets for all n<=10 and to enumerate all corresponding order types. Finally, we describe a collection G of 7'393 planar graphs on 35 vertices that do not admit a simultaneous geometric embedding without mapping, that is, no set of 35 points in the plane supports a plane straight-line embedding of all graphs in G.Comment: Fixed incorrect numbers of universal point sets in the last par

    Solving kk-SUM using few linear queries

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    The kk-SUM problem is given nn input real numbers to determine whether any kk of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within PP, and it is in particular open whether it admits an algorithm of complexity O(nc)O(n^c) with c<k2c<\lceil \frac{k}{2} \rceil. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n3log3n)O(n^3\log^3 n) solving kk-SUM. Furthermore, we show that there exists a randomized algorithm that runs in O~(nk2+8)\tilde{O}(n^{\lceil \frac{k}{2} \rceil+8}) time, and performs O(n3log3n)O(n^3\log^3 n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8+8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of kk. The O(n3log3n)O(n^3\log^3 n) bound on the number of linear queries is also a tighter bound than any known algorithm solving kk-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-PP. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)o(n)-linear decision trees of depth o(n4)o(n^4)
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