This paper studies labeled sample compression for multi-label concept classes. For a specific extension of the notion of VC-dimension to multi-label classes, we prove that every maximum multilabel class of dimension d has a sample compression scheme in which every sample is compressed to a subset of size at most d. We further show that every multi-label class of dimension 1 has a sample compres...

Let A, B, be finite subsets of an abelian group, and let G ⊂ A×B be such that #A,#B,#{a + b : (a, b) ∈ G} ≤ N . We consider the question of estimating the quantity #{a − b : (a, b) ∈ G}. In [2] Bourgain improved the trivial upper bound of N to N 1 13 , and applied this to the Kakeya conjecture. We improve Bourgain’s estimate further to N 1 6 , and conclude that Besicovitch sets in Rn have Hausd...

The study aims to identify the possibilities offered by academic studies in Cuba for the training of information professionals as marketing managers. A theoretical analysis is conducted of the basic functions of marketing and its information dimension, the specific features of the work of the marketing manager, and the Revista Cubana de Información en Ciencias de la Salud 2014;25(2):234-248 htt...

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrands projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorf...

Minkowski dimension is one of the most widely used dimensions. Its popularity is largely due to its relative ease of mathematical calculation. The definition goes back at least to the 1930’s and it has been variously termed Kolmogorov entropy, metric dimension, information dimension, ... etc. Let F be any non-empty bounded subset of R and let Nγ(F ) be the smallest number of sets of diameter at...

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...

Let $(R, m)$ be a commutative noetherian local ring and let $Gamma$ be a finite group. It is proved that if $R$ admits a dualizing module, then the group ring $Rga$ has a dualizing bimodule as well. Moreover, it is shown that a finitely generated $Rga$-module $M$ has generalized Gorenstein dimension zero if and only if it has generalized Gorenstein dimension zero as an $R$-module.

In this paper we investigate the algebraic properties of dimension of fuzzy hypervector spaces. Also, we prove that two isomorphic fuzzy hypervector spaces have the same dimension.