We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To save this undefined to your undefined account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your undefined account.
Find out more about saving content to .
To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
The generating degree $\text{g}\deg \left( A \right)$ of a topological commutative ring $A$ with char $A\,=\,0$ is the cardinality of the smallest subset $M$ of $A$ for which the subring $\mathbb{Z}\left[ M \right]$ is dense in $A$. For a prime number $p$, ${{\mathbb{C}}_{p}}$ denotes the topological completion of an algebraic closure of the field ${{\mathbb{Q}}_{p}}$ of $p$-adic numbers. We prove that $\text{g}\deg \left( {{\mathbb{C}}_{p}} \right)\,=\,1$, i.e., there exists $t$ in ${{\mathbb{C}}_{p}}$ such that $\mathbb{Z}\left[ t \right]$ is dense in ${{\mathbb{C}}_{p}}$. We also compute $\text{gdeg}\left( A\left( U \right) \right)$ where $A\left( U \right)$ is the ring of rigid analytic functions defined on a ball $U$ in ${{\mathbb{C}}_{p}}$. If $U$ is a closed ball then $\text{gdeg}\left( A\left( U \right) \right)\,=\,2$ while if $U$ is an open ball then $\text{gdeg}\left( A\left( U \right) \right)$ is infinite. We show more generally that $\text{gdeg}\left( A\left( U \right) \right)$ is finite for any affinoid$U$ in ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$ and $\text{gdeg}\left( A\left( U \right) \right)$ is infinite for any wide open subset $U$ of ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$.
This paper is concernedwith the structure of the arithmetic sum of a finite number of central Cantor sets. The technique used to study this consists of a duality between central Cantor sets and sets of subsums of certain infinite series. One consequence is that the sum of a finite number of central Cantor sets is one of the following: a finite union of closed intervals, homeomorphic to the Cantor ternary set or an $M$-Cantorval.
We show that an $\text{RA}$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\text{RA}$ loop can be normal in its unit loop. Over fields, this can never happen.
The Hamiltonian potentials of the bending deformations of $n$-gons in ${{\mathbb{E}}^{3}}$ studied in $\left[ \text{KM} \right]$ and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra ${{P}_{n}}$ of the pure braid group ${{P}_{n}}$ on the moduli space ${{M}_{r}}$ of $n$-gon linkages with the side-lengths $r\,=\,\left( {{r}_{1}},\ldots ,{{r}_{n}} \right)$ in ${{\mathbb{E}}^{3}}$. If $e\,\in \,{{M}_{r}}$ is a singular point we may linearize the vector fields in ${{P}_{n}}$ at $e$. This linearization yields a flat connection $\nabla$ on the space $\mathbb{C}_{*}^{n}$ of $n$ distinct points on $\mathbb{C}$. We show that the monodromy of $\nabla$ is the dual of a quotient of a specialized reduced Gassner representation.
The paper is dealing with determination of the integral ${{\int }_{\gamma }}\,f$ along the fractal arc $\gamma $ on the complex plane by terms of polynomial approximations of the function $f$. We obtain inequalities for polynomials and conditions of integrability for functions from the Hölder, Besov and Slobodetskii spaces.
Motivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complexmanifold with an almost complex structure. We describe various properties of this structure.
A challenge by R. Padmanabhan to prove by group theory the commutativity of cancellative semigroups satisfying a particular law has led to the proof of more general semigroup laws being equivalent to quite simple ones.
The singular spectrum of $u$ in a convolution equation $\mu *u\,=\,f$, where $\mu$ and $f$ are tempered ultra distributions of Beurling or Roumieau type is estimated by
The purpose of this paper is to show the limitations of the conjectures of algebraic approximation. For this, we construct points of ${{\mathbf{C}}^{m}}$ which do not admit good algebraic approximations of bounded degree and height, when the bounds on the degree and the height are taken from specific sequences. The coordinates of these points are Liouville numbers.
Simple necessary conditions for weak type (1, 1) of invariant operators on $L\left( {{\mathbb{R}}^{d}} \right)$ and their applications to rational Fourier multiplier are given.
Around 1995, Professors Lupacciolu, Chirka and Stout showed that a closed subset of ${{\mathbb{C}}^{N}}\left( N\ge 2 \right)$ is removable for holomorphic functions, if its topological dimension is less than or equal to $N\,-\,2$. Besides, they asked whether closed subsets of ${{\mathbb{C}}^{2}}$ homeomorphic to the real line (the simplest 1-dimensional sets) are removable for holomorphic functions. In this paper we propose a positive answer to that question.