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  • On the iteration of quasimeromorphic mappings
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-07-06
    LUKE WARREN

    The Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.

    更新日期:2020-01-04
  • Krull's principal ideal theorem in non-Noetherian settings
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-08-08
    BRUCE OLBERDING

    Let P be a finitely generated ideal of a commutative ring R. Krull's principal ideal theorem states that if R is Noetherian and P is minimal over a principal ideal of R, then P has height at most one. Straightforward examples show that this assertion fails if R is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.

    更新日期:2020-01-04
  • Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-12
    JOONAS ILMAVIRTA; JERE LEHTONEN; MIKKO SALO

    We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

    更新日期:2020-01-04
  • Median and injective metric spaces
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-07-27
    BRIAN H. BOWDITCH

    We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other facts about median metrics, and the geometry of CAT(0) cube complexes. One motivation for the study of such metrics is that they arise as asymptotic cones of certain naturally occurring spaces.

    更新日期:2020-01-04
  • Small sets containing any pattern
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-07-31
    URSULA MOLTER; ALEXIA YAVICOLI

    Given any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.

    更新日期:2020-01-04
  • Regulator constants of integral representations of finite groups
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-05
    ALEX TORZEWSKI

    Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$ [G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$ [G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$ , and a cohomological invariant of Yakovlev.

    更新日期:2020-01-04
  • Interior of sums of planar sets and curves
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-05
    KÁROLY SIMON; KRYSTAL TAYLOR

    Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise ${\mathcal C}^2$ curve and A ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S1)° ≠ ∅. If, however, we assume that A is a kind of generalised product of two reasonably large sets, then (A + Γ)° ≠ ∅ whenever Γ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of C := Cγ × Cγ, γ ⩾ 1/3, pinned at any point of C has non-empty interior, where Cγ (see (1.1)) is the middle 1 − 2γ Cantor set (including the usual middle-third Cantor set, C1/3). Our proof for the middle-third Cantor set requires a separate method. We also prove that C + S1 has non-empty interior.

    更新日期:2020-01-04
  • The distribution of consecutive prime biases and sums of sawtooth random variables
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-08-02
    ROBERT J. LEMKE OLIVER; KANNAN SOUNDARARAJAN

    In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

    更新日期:2020-01-04
  • Supercongruences for truncated hypergeometric series and p-adic gamma function
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-08-10
    RUPAM BARMAN; NEELAM SAIKIA

    We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

    更新日期:2020-01-04
  • On the weak forms of the 2-part of Birch and Swinnerton-Dyer conjecture
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-05
    SHUAI ZHAI

    In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$ .

    更新日期:2020-01-04
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  • 更新日期:2020-01-04
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