The projective 3-annihilating-ideal hypergraphs

In this paper, we investigate the crosscap of 3-annihilating-ideal hypergraph [Formula: see text] of a commutative ring [Formula: see text] and the topological embedding of [Formula: see text] to the nonorientable compact surfaces. Furthermore, we determine all Artinian commutative non-local rings [Formula: see text] (up to isomorphism) such that [Formula: see text] is a projective graph.

The Brouwer invariance theorems in reverse mathematics

In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

Shadows of a Closed Curve

A shadow of a geometric object A in a given direction v is the orthogonal projection of A on the hyperplane orthogonal to v. We show that any topological embedding of a circle into Euclidean d-space can have at most two shadows that are simple paths in linearly independent directions. The proof is topological and uses an analog of basic properties of degree of maps on a circle to relations on a circle. This extends a previous result that dealt with the case d = 3.

A closer look at the stable completion

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

The geometry of Niggli reduction:SAUC– search of alternative unit cells

A database of lattices using theG6representation of the Niggli-reduced cell as the search key provides a more robust and complete search than older techniques. Searching is implemented by finding the distance from the probe cell to other cells using a topological embedding of the Niggli reduction inG6, so that all cells representing similar lattices will be found. The embedding provides the first fully linear measure of distances between unit cells. Comparison of results with those from older cell-based search algorithms suggests significant value in the new approach.

A topological embedding of the lexicon for semantic distance computation

We show how a quantitative context may be established for what is essentially qualitative in nature by topologically embedding a lexicon (here, WordNet) in a complete metric space. This novel transformation establishes a natural connection between the order relation in the lexicon (e.g., hyponymy) and the notion of distance in the metric space, giving rise to effective word-level and document-level lexical semantic distance measures. We provide a formal account of the topological transformation and demonstrate the value of our metrics on several experiments involving information retrieval and document clustering tasks.