Covering dimension and finite-to-one maps

Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most 2n-to-1 map

On the State Hierarchy of Exploding Automata

A recently revisited question in finite automata theory considers the possible numbers n and d for which there exists an n-state minimal NFA with a minimal equivalent DFA of d states. We present a new class of finite automata, the NFA En of n states, which in a sense contains half of the state hierarchy [n, 2n]; that is, by making small modifications to En, we can create a minimal equivalent DFA of d states for any d ∈ (2n−1, 2n]. Although this is not stronger than the most recent of work that has been done on the problem, the value of this result lies in the systematic and intuitive method by which we, given the parameter d, construct the appropriate NFA from En. Specifically, the construction from En is a direct reflection of the binary representation of 2n−d, each 1-bit of which indicates a single modification to make to En. We conclude the thesis with a discussion of computational results to suggest that these methods can be extended to reach the entire state hierarchy, that is, to answer the question for any d ∈ [n, 2n]

Singularities of affine equidistants: projections and contacts

Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine $\lambda$-equidistants of $n$-dimensional closed submanifolds of $\mathbb R^q$, for $q\leq 2n$, whenever $(2n,q)$ is a pair of nice dimensions.Comment: 18 pages, v2 (minimal changes) agrees with version to appear in Journal of Singularitie

Holomorphicity of real Kaehler submanifolds

Let $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ denote an isometric immersion of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with codimension $p$. If $2p\leq 2n-1$, we show that generic rank conditions on the second fundamental form of the submanifold imply that $f$ has to be a minimal submanifold. In fact, for codimension $p\leq 11$ we prove that $f$ must be holomorphic with respect to some complex structure in the ambient space.Comment: 18 page