Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the k-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Thornton.Our results also show that Nikodym sets are typical among all sets which contain, for every x ∈ R n , a punctured hyperplane H \ {x} through x. With similar methods we also construct a Borel subset of R n of Lebesgue measure zero containing a hyperplane at every positive distance from every point.2010 Mathematics Subject Classification. 28A78, 54E52.
Abstract. De Bruijn and Newman introduced a deformation of the Riemann zeta function ζ(s), and found a real constant Λ which encodes the movement of the zeros of ζ(s) under the deformation. The Riemann hypothesis (RH) is equivalent to Λ ≤ 0. Newman made the conjecture that Λ ≥ 0 along with the remark that "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so."Newman's conjecture is still unsolved, and previous work could only handle the Riemann zeta function and quadratic Dirichlet L-functions, obtaining lower bounds very close to zero (for example, for ζ(s) the bound is at least −1.14541 · 10 −11 , and for quadratic Dirichlet L-functions it is at least −1.17·10 −7 ). We generalize the techniques to apply to automorphic L-functions as well as function field L-functions. We further determine the limit of these techniques by studying linear combinations of L-functions, proving that these methods are insufficient.We explicitly determine the Newman constants in various function field settings, which has strong implications for Newman's quantitative version of RH. In particular, let D ∈ Z[T ] be a square-free polynomial of degree 3. Let D p be the polynomial in √ p ; by Sato-Tate (if the curve is non-CM) there exists a sequence of primes such that lim n→∞ Λ Dp n = 0. We end by discussing connections with random matrix theory.
This paper looks at the problem of finding the minimum cost curvatureconstrained path between two directed points where the cost at every point along the path depends on the instantaneous direction. This generalises the results obtained by Dubins for curvature-constrained paths of minimum length, commonly referred to as Dubins paths. We conclude that if the reciprocal of the directional-cost function is strictly polarly convex, then the forms of the optimal paths are of the same forms as Dubins paths. If we relax the strict polar convexity to weak polar convexity, then we show that there exists a Dubins path which is optimal. The results obtained can be applied to optimising the development of underground mine networks, where the paths need to satisfy a curvature constraint and the cost of development of the tunnel depends on the direction due to the geological characteristics of the ground.
This paper studies the problem of finding the minimum cost curvature-constrained path between two directed points where the cost at every point along the path depends on the instantaneous direction. This generalises the results obtained by Dubins for curvatureconstrained paths of minimum length, commonly referred to as Dubins paths. We show that there always exists a path of the form CSCSC or a degeneracy which is optimal, where C represents an arc of maximum curvature, and S represents a straight line. This result is also extended to the case where there is not only a directional-cost, but the cost of curved sections are scaled up by a factor w C ≥ 1. The results obtained can be applied to optimising the development of underground mine networks, where the paths need to satisfy a curvature constraint, the cost of development of the tunnel depends on the direction due to the geological characteristics of the ground, and curved sections may incur more cost due to additional support and ventilation.
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