Related papers: Substituting the typical compact sets into a power…

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…

The (projective) convergence set of a divergent formal power series $f(x_{1},...,x_{n})$ is defined to be the image in $\PP^{n-1}$ of the set of all $x\in \mathbb{C}^{n}$ such that $f(x_{1}t,...,x_{n}t)$, as a series in $t$, converges…

Let $K_1, K_2$ be two compact convex sets in $\mathit{C}$. Their Minkowski product is the set $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$. We show that the set $K_1K_2$ is star-shaped if $K_1$ is a line segment or a circular disk. Examples for…

The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…

Minkowski sums are of theoretical interest and have applications in fields related to industrial backgrounds. In this paper we focus on the specific case of summing polytopes as we want to solve the tolerance analysis problem described in…

We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski…

One can think of power series or polynomials in one variable, such as $P(x)=2x^3+x+5$, as functors from the category $\mathsf{Set}$ of sets to itself; these are known as polynomial functors. Denote by $\mathsf{Poly}_{\mathsf{Set}}$ the…

It is shown that for every $\e\in (0,1)$, every compact metric space $(X,d)$ has a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\e)$, and $$\dim_H(S)\ge (1-\e)\dim_H(X),$$ where $\dim_H(\cdot)$…

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the…

We describe the norming sets for the space of global holomorphic sections to a $k$-power of a positive holomorphic line bundle on a compact complex manifold $X$. We characterize in metric terms the sequence of measurable subsets…

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes…

This paper studies the equivalence between generalized holomorphic functions (GHF) and complex analytic functions in the framework of Robinson-Colombeau generalized numbers. In every non-Archimedean ring, the use of ordinary series is…

The conformal compactification is considered in a hierarchy of hypercomplex projective spaces with relevance in physics including Minkowski and Anti-de Sitter space. The geometries are expressed in terms of bicomplex Vahlen matrices and…

Given a $k$-point configuration $x\in (\mathbb{R}^d)^k$, we consider the $\binom{k}{d}$-vector of volumes determined by choosing any $d$ points of $x$. We prove that a compact set $E\subset \R^d$ determines a positive measure of such volume…

We construct holomorphically varying families of Fatou-Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $\mathbb C^n$ for $n>1$. This provides a simple proof of the recent result of…

We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of…

Given a compact set $E \subset \mathbb{R}^{d - 1}$, $d \geq 1$, write $K_{E} := [0,1] \times E \subset \mathbb{R}^{d}$. A theorem of C. Bishop and J. Tyson states that any set of the form $K_{E}$ is minimal for conformal dimension: if…

Consider a random power series of the form $P(z) = \sum_{n\ge 1} \varepsilon_n a_n z^{n}$ where $a_n \in \mathbb{C}$ are deterministic and $\varepsilon_n$ are chosen independently and uniformly at random from $\{\pm 1\}$. Kolmogorov's…

In this paper, the Beatty multiple shift is introduced, which is a generalization of the multiplicative shift of finite type (multiple SFT) [Kenyon, Peres and Solomyak, Ergodic Theory and Dynamical Systems, 2012] and the affine multiple…

In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity. Working in the hyperspace $\mathbf{K}(X)$ of compact subsets of $X$, with $X=[0,1]^d$ or…