Related papers: The Arithmetical Complexity of Dimension and Rando…

The base-$k$ {\em Copeland-Erd\"os sequence} given by an infinite set $A$ of positive integers is the infinite sequence $\CE_k(A)$ formed by concatenating the base-$k$ representations of the elements of $A$ in numerical order. This paper…

The Hausdorff dimension of a product XxY can be strictly greater than that of Y, even when the Hausdorff dimension of X is zero. But when X is countable, the Hausdorff dimensions of Y and XxY are the same. Diagonalizations of covers define…

The two most important notions of fractal dimension are {\it Hausdorff dimension}, developed by Hausdorff (1919), and {\it packing dimension}, developed by Tricot (1982). Lutz (2000) has recently proven a simple characterization of…

Let $A=kQ/I$ be a finite dimensional basic algebra over an algebraically closed field $k$ which is a gentle algebra with the marked ribbon surface $(\mathcal{S}_A,\mathcal{M}_A,\Gamma_A)$. It is known that $\mathcal{S}_A$ can be divided…

The aim of this paper is to study the dominant dimension of two important classes of finite dimensional algebras, namely, hereditary algebras and tree algebras. We derive an explicit formula for the dominant dimension of each class.

While every polyadic algebra ($\PA$) of dimension 2 is representable, we show that not every atomic polyadic algebra of dimension two is completely representable; though the class is elementary. Using higly involved constructions of Hirsch…

Let $\mathcal K=\langle\mathcal R, \delta\rangle$ be a closed ordered differential field, in the sense of M. Singer, and $C$ its field of constants. In this note, we prove that, for sets definable in the pair $\mathcal M=\langle \mathcal R,…

This paper investigates the Hausdorff dimension properties of chains and antichains in Turing degrees and hyperarithmetic degrees. Our main contributions are threefold: First, for antichains in hyperarithmetic degrees, we prove that every…

Given a compact set of real numbers, a random $C^{m + \alpha}$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has…

The strong global dimension of a ring is the supremum of the length of perfect complexes that are indecomposable in the derived category. In this note we characterize the noetherian commutative rings that have finite strong global…

Kifer, Peres, and Weiss proved that there exists $c_0>0,$ such that $\dim \mu\leq 1-c_0$ for any probability measure $\mu$ which makes the digits of the continued fraction expansion i.i.d. random variables. In this paper we prove that…

Fix a positive integer number $r$. A class of $r$-dim Lie conformal superalgebras named $r$-dim $i$-linear Lie conformal superalgebras are studied for $1\leq i \leq r$. We present an equivalent characterization of this class of Lie…

We introduce appropriate definitions of dimensions in order to characterize the fractal properties of complex networks. We compute these dimensions in a hierarchically structured network of particular interest. In spite of the nontrivial…

In this article a collection of random self-similar fractal dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have relied on geometrical…

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with…

We locate winning strategies for various Sigma^0_3-games in the L-hierarchy in order to prove that Sigma^0_3 Determinacy is intermediate between Pi^1_3-CA_0 (even Pi^1_2-CA_0 (lightface) with Pi^1_3-lightface definable parameters allowed)…

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…

Let $\mathbb{Z}$ be the integer numbers, $\mathbb{K}$ an algebraically closed field, $\Lambda$ a finite dimensional $\mathbb{K}$-algebra, mod$\Lambda$ the category of finitely generated right modules, proj$\Lambda$ the full subcategory of…

Let $k$ be a natural number. We consider $k$-times continuously-differentiable real-valued functions $f:E\to\mathbb{R}$, where $E$ is some interval on the line having positive length. For $0<\alpha<1$ let $I_\alpha(f)$ denote the set of…