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This appendix provides a quick summary (without proofs) of the standard technical results of holomorphic curves that are needed in the rest of the book, including the basic facts on simple and multiply covered curves, moduli spaces and their dimensions, Fredholm regularity, and compactifications. The treatment covers both closed and punctured (asymptotically cylindrical) holomorphic curves.
We prove a comparison isomorphism between certain moduli spaces of
$p$
-divisible groups and strict
${\mathcal{O}}_{K}$
-modules (RZ-spaces). Both moduli problems are of PEL-type (polarization, endomorphism, level structure) and the difficulty lies in relating polarized
$p$
-divisible groups and polarized strict
${\mathcal{O}}_{K}$
-modules. We use the theory of relative displays and frames, as developed by Ahsendorf, Lau and Zink, to translate this into a problem in linear algebra. As an application of these results, we verify new cases of the arithmetic fundamental lemma (AFL) of Wei Zhang: The comparison isomorphism yields an explicit description of certain cycles that play a role in the AFL. This allows, under certain conditions, to reduce the AFL identity in question to an AFL identity in lower dimension.
We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime
$p$
, extending previous work of Shparlinski [‘Multiplicative orders in orbits of polynomials over finite fields’, Glasg. Math. J.60(2) (2018), 487–493].
Under sufficiently strong assumptions about the first prime in an arithmetic progression, we prove that the number of Carmichael numbers up to
$X$
is
$\gg X^{1-R}$
, where
$R=(2+o(1))\log \log \log \log X/\text{log}\log \log X$
. This is close to Pomerance’s conjectured density of
$X^{1-R}$
with
$R=(1+o(1))\log \log \log X/\text{log}\log X$
.
Zacharias [‘Proof of a conjecture of Merca on an average of square roots’, College Math. J.49 (2018), 342–345] proved Merca’s conjecture that the arithmetic means
$(1/n)\sum _{k=1}^{n}\sqrt{k}$
of the square roots of the first
$n$
integers have the same floor values as a simple approximating sequence. We prove a similar result for the arithmetic means
$(1/n)\sum _{k=1}^{n}\sqrt[3]{k}$
of the cube roots of the first
$n$
integers.
This paper presents a flexible family which we call the
$\alpha$
-mixture of survival functions. This family includes the survival mixture, failure rate mixture, models that are stochastically closer to each of these conventional mixtures, and many other models. The
$\alpha$
-mixture is endowed by the stochastic order and uniquely possesses a mathematical property known in economics as the constant elasticity of substitution, which provides an interpretation for
$\alpha$
. We study failure rate properties of this family and establish closures under monotone failure rates of the mixture’s components. Examples include potential applications for comparing systems.
A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.
We introduce coset progressions and Bohr sets, and show that the two notions are roughly equivalent up to Freiman homomorphism. To facilitate the proof of this we introduce lattices and convex bodies and their basic properties, and prove Minkowski’s second theorem from the geometry of numbers.
We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related.
(T.1) A basic property of Cantor space
$2^ $
is Heine–Borel compactness: for any open covering of
$2^ $
, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any
$G:2^ \to $
, output a finite sequence
$\langle f_0 , \ldots ,f_n \rangle $
in
$2^ $
such that the neighbourhoods defined from
$\overline {f_i } G\left( {f_i } \right)$
for
$i \le n$
form a covering of
$2^ $
.
(T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.
Our study of (T.1) yields exotic objects in computability theory, while (T.2) leads to surprising results in Reverse Mathematics. We stress that (T.1) and (T.2) are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa.
Quasi-Sturmian words, which are infinite words with factor complexity eventually
$n+c$
share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.
Given a function f on the n-dimensional polydisc, the Bohr radius (recall Chapter 8) looks for the best r for which the supremum of ∑ | c_α z^α| for || z ||_∞ <r is less than or equal to the supremum of |f(z)| for || z ||_∞ <1. Here an analogous problem is considered, replacing the sup-norm by another p-norm. The corresponding Bohr radius for l_p-balls is defined, and its asymptotic behaviour is computed. This is done in three steps. First, an m-homogeneous version (where only m-homogeneous polynomials are considered) is defined, and it is shown how these m-homogeneous radii determine the general Bohr radius. In the second step, this homogenous radius is related to the unconditional basis constant of the monomials in the space of homogeneous polynomials on l_p. Finally, this unconditional basis constant is computed.
For the Heyting Arithmetic HA,
$HA^{\text{*}} $
is defined [14, 15] as the theory
$\left\{ {A|HA \vdash A^\square } \right\}$
, where
$A^\square $
is called the box translation of A (Definition 2.4). We characterize the
${\text{\Sigma }}_1 $
-provability logic of
$HA^{\text{*}} $
as a modal theory
$iH_\sigma ^{\text{*}} $
(Definition 3.17).
The aim of this paper is to find a broad family of means defined on a subinterval of
$I\subset [0,+\infty )$
such that
$$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }\mathscr{M}(a_{1},\ldots ,a_{n})<+\infty \quad \text{for all }a\in \ell _{1}(I).\end{eqnarray}$$
Equivalently, the averaging operator
$(a_{1},\,a_{2},a_{3}\,,\ldots )\mapsto (a_{1},\,\mathscr{M}(a_{1},a_{2}),\,\mathscr{M}(a_{1},a_{2},a_{3}),\ldots )$
is a selfmapping of
$\ell _{1}(I)$
. This property is closely related to the so-called Hardy inequality for means (which additionally requires boundedness of this operator). We prove that these two properties are equivalent in a broad family of so-called Gini means. Moreover, we show that this is not the case for quasi-arithmetic means, that is functions
$f^{-1}(\sum f(a_{i})/n)$
, where
$f:I\rightarrow \mathbb{R}$
is continuous and strictly monotone,
$n\in \mathbb{N}$
and
$a\in I^{n}$
. However, the weak Hardy property is localisable for this family.
The Tabula Alimentaria of Veleia records the details of two second-century a.d. imperial alimentary schemes at the northern Italian town of Veleia, providing a rare insight into the workings of these schemes. Imperial loans are made to local landowners in exchange for pledges of specified property. Interest paid by landowners is used to fund cash subsidies for the upbringing of selected local children. In the early twentieth century, the French scholar Félix de Pachtere came close to demonstrating a consistent arithmetical relationship between a landowner's declared property value and the loan received. However, anomalies remained. This article proposes a revised formula which establishes a precise and consistent linkage between loan amounts and property declarations. Based on this arithmetical dataset, the paper proposes some hypotheses about how these fractional computations might have been performed in second-century Rome.
We prove two results about the width of words in
$\operatorname{SL}_{n}(\mathbb{Z})$
. The first is that, for every
$n\geqslant 3$
, there is a constant
$C(n)$
such that the width of any word in
$\operatorname{SL}_{n}(\mathbb{Z})$
is less than
$C(n)$
. The second result is that, for any word
$w$
, if
$n$
is big enough, the width of
$w$
in
$\operatorname{SL}_{n}(\mathbb{Z})$
is at most 87.
Let
$p$
be a prime. If an integer
$g$
generates a subgroup of index
$t$
in
$(\mathbb{Z}/p\mathbb{Z})^{\ast },$
then we say that
$g$
is a
$t$
-near primitive root modulo
$p$
. We point out the easy result that each coprime residue class contains a subset of primes
$p$
of positive natural density which do not have
$g$
as a
$t$
-near primitive root and we prove a more difficult variant.
Let be a dominant rational self-map of a smooth projective variety defined over
$\overline{\mathbb{Q}}$
. For each point
$P\in X(\overline{\mathbb{Q}})$
whose forward
$f$
-orbit is well defined, Silverman introduced the arithmetic degree
$\unicode[STIX]{x1D6FC}_{f}(P)$
, which measures the growth rate of the heights of the points
$f^{n}(P)$
. Kawaguchi and Silverman conjectured that
$\unicode[STIX]{x1D6FC}_{f}(P)$
is well defined and that, as
$P$
varies, the set of values obtained by
$\unicode[STIX]{x1D6FC}_{f}(P)$
is finite. Based on constructions by Bedford and Kim and by McMullen, we give a counterexample to this conjecture when
$X=\mathbb{P}^{4}$
.