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Mathematical Proceedings of the Cambridge Philosophical Society
Journal Prestige (SJR): 1.086  Citation Impact (citeScore): 1 Number of Followers: 2  Subscription journalISSN (Print) 0305-0041 - ISSN (Online) 1469-8064 Published by Cambridge University Press  [388 journals]
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- PSP volume 169 issue 1 Cover and Front matter
- PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004120000092
Issue No: Vol. 169, No. 1 (2020)
- PubDate: 2020-07-01T00:00:00.000Z
- PSP volume 169 issue 1 Cover and Back matter
- PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004120000109
Issue No: Vol. 169, No. 1 (2020)
- PubDate: 2020-07-01T00:00:00.000Z
- Multisets in type theory
- Authors: HÅKON ROBBESTAD GYLTERUD
Pages: 1 - 18
Abstract: A multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative multisets, where multisets are iteratively built up from other multisets, in the context Martin–Löf Type Theory, in the presence of Voevodsky’s Univalence Axiom.In his 1978 paper, “the type theoretic interpretation of constructive set theory” Aczel introduced a model of constructive set theory in type theory, using a W-type quantifying over a universe, and an inductively defined equivalence relation on it. Our investigation takes this W-type and instead considers the identity type on it, which can be computed from the univalence axiom. Our thesis is that this gives a model of multisets. In order to demonstrate this, we adapt axioms of constructive set theory to multisets, and show that they hold for our model.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004119000045
Issue No: Vol. 169, No. 1 (2020)
- Authors: HÅKON ROBBESTAD GYLTERUD
- An origami of genus 3 with arithmetic Kontsevich–Zorich monodromy
- Authors: PASCAL HUBERT; CARLOS MATHEUS SANTOS
Pages: 19 - 30
Abstract: In this we exploit the arithmeticity criterion of Oh and Benoist–Miquel to exhibit an origami in the principal stratum of the moduli space of translation surfaces of genus three whose Kontsevich–Zorich monodromy is not thin in the sense of Sarnak.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004119000057
Issue No: Vol. 169, No. 1 (2020)
- Authors: PASCAL HUBERT; CARLOS MATHEUS SANTOS
- Topos points of quasi-coherent sheaves over monoid schemes
- Authors: ILIA PIRASHVILI
Pages: 31 - 74
Abstract: Let X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos
of quasi-coherent sheaves over X. As it turns out, every topos point of 
is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if 
and 
are equivalent.The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ∞, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004119000069
Issue No: Vol. 169, No. 1 (2020)
- Authors: ILIA PIRASHVILI
- Entiers friables dans des progressions arithmétiques de grand module
- Authors: R. DE LA BRETÈCHE; D. FIORILLI
Pages: 75 - 102
Abstract: We study the average error term in the usual approximation to the number of y-friable integers congruent to a modulo q, where a ≠ 0 is a fixed integer. We show that in the range exp{(log log x)5/3+ɛ} ⩽ y ⩽ x and on average over q ⩽ x/M with M → ∞ of moderate size, this average error term is asymptotic to −|a| Ψ(x/|a|, y)/2x. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of y-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004119000094
Issue No: Vol. 169, No. 1 (2020)
- Authors: R. DE LA BRETÈCHE; D. FIORILLI
- L-functions&rft.title=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.issn=0305-0041&rft.date=2020&rft.volume=169&rft.spage=103&rft.epage=140&rft.aulast=DEVIN&rft.aufirst=LUCILE&rft.au=LUCILE+DEVIN&rft_id=info:doi/10.1017/S0305004119000100">Chebyshev’s bias for analytic
- Authors: LUCILE DEVIN
Pages: 103 - 140
Abstract: We discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set
$ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$ for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004119000100
Issue No: Vol. 169, No. 1 (2020)
- Authors: LUCILE DEVIN
- Symmetric quotients of knot groups and a filtration of the Gordian graph
- Authors: SEBASTIAN BAADER; ALEXANDRA KJUCHUKOVA
Pages: 141 - 148
Abstract: We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004119000136
Issue No: Vol. 169, No. 1 (2020)
- Authors: SEBASTIAN BAADER; ALEXANDRA KJUCHUKOVA
- A support theorem for the X-ray transform on manifolds with plane covers
- Authors: NORBERT PEYERIMHOFF; EVANGELIA SAMIOU
Pages: 149 - 158
Abstract: This paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004119000148
Issue No: Vol. 169, No. 1 (2020)
- Authors: NORBERT PEYERIMHOFF; EVANGELIA SAMIOU
- Semantics of higher inductive types
- Authors: PETER LEFANU LUMSDAINE; MICHAEL SHULMAN
Pages: 159 - 208
Abstract: Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings.We introduce the notion of cell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has weakly stable typal initial algebras for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations.Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S030500411900015X
Issue No: Vol. 169, No. 1 (2020)
- Authors: PETER LEFANU LUMSDAINE; MICHAEL SHULMAN
- Triforce and corners
- Authors: JACOB FOX; ASHWIN SAH, MEHTAAB SAWHNEY, DAVID STONER, YUFEI ZHAO
Pages: 209 - 223
Abstract: May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and
$G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1017/S0305004119000173
Issue No: Vol. 169, No. 1 (2020)
- Authors: JACOB FOX; ASHWIN SAH, MEHTAAB SAWHNEY, DAVID STONER, YUFEI ZHAO
