An α-disconnected space has no proper monic preimage

AbstractAll spaces are compact Hausdorff. α is an uncountable cardinal or the symbol ∞. A continuous map τ:X→Y is called an α-SpFi morphism if τ-1(G) is dense in X whenever G is a dense α-cozero set of Y. We thus have a category α-SpFi (spaces with the α-filter) which, like any category, has its monomorphisms; these need not be one-to-one. For general α, we cannot say what the α-SpFi monics are, but we show, and R.G. Woods showed, that ∞-SpFi monic means range-irreducible. The main theorem here is: X has no proper α-SpFi monic preimage if and only if X is α-disconnected. This generalizes (by putting in α = ∞) the well-known fact: X has no proper irreducible preimage if and only if X is extremally disconnected. If, in our theorem, we restrict to Boolean spaces and apply Stone duality, we have the theorem of R. Lagrange, that in Boolean α-algebras, epimorphisms are surjective.The theory of spaces with filters has a lot of connections with ordered algebra—Boolean algebras of course, but also lattice-ordered groups and frames. This paper is a contribution to the development of this topological theory

A Better Bouncers Algorithm

Suppose we have a set of materials - e.g., drugs or genes - some combinations of which react badly together. We can experiment to see whether subsets contain any bad combinations and we want to find a maximal subset that does not. This problem is equivalent to finding a maximal independent set (or minimal vertex cover) in a hypergraph using group tests on the vertices. Consider the simple greedy algorithm that adds vertices one by one; after adding each vertex, the algorithm tests whether the subset now contains any edges and, if so, removes that vertex and discards it. We call this the "bouncer's algorithm" because it is reminiscent of how order is maintained as patrons are admitted to some bars. If this algorithm processes the vertices according to a given total preference order, then its solution is the unique optimum with respect to that order. Our main contribution is another algorithm that produces the same solution but uses fewer tests when few vertices are discarded: if the bouncer's algorithm discards d of the n vertices in the hypergraph, then our algorithm uses at most d(⌈log2 n⌉+1)+1 tests. It follows that, given black-box access to a monotone Boolean formula on n variables, we can find a minimal satisfying truth assignment using at most d(⌈log 2 n⌉+1)+1 tests, where d is the number of variables set to true. We also prove some bounds for partially adaptive algorithms

A Better Bouncer's Algorithm

Suppose we have a set of materials - e.g., drugs or genes - some combinations of which react badly together. We can experiment to see whether subsets contain any bad combinations and we want to find a maximal subset that does not. This problem is equivalent to finding a maximal independent set (or minimal vertex cover) in a hypergraph using group tests on the vertices. Consider the simple greedy algorithm that adds vertices one by one; after adding each vertex, the algorithm tests whether the subset now contains any edges and, if so, removes that vertex and discards it. We call this the "bouncer's algorithm" because it is reminiscent of how order is maintained as patrons are admitted to some bars. If this algorithm processes the vertices according to a given total preference order, then its solution is the unique optimum with respect to that order. Our main contribution is another algorithm that produces the same solution but uses fewer tests when few vertices are discarded: if the bouncer's algorithm discards d of the n vertices in the hypergraph, then our algorithm uses at most d(\u2308log2 n\u2309+1)+1 tests. It follows that, given black-box access to a monotone Boolean formula on n variables, we can find a minimal satisfying truth assignment using at most d(\u2308log 2 n\u2309+1)+1 tests, where d is the number of variables set to true. We also prove some bounds for partially adaptive algorithms

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Exordium for DNA Codes

We describe how deletion-correcting codes may be enhanced to yield codes with double-strand DNA-sequence codewords. This enhancement involves abstractions of the pertinent aspects of DNA; it nevertheless ensures specificity of binding for all pairs of single strands derived from its codewords-the key desideratum of DNA codes- i.e. with binding feasible only between reverse complementary strands. We defer discussing the combinatorial-optimization superincumbencies of code construction. Generalization of deletion similarity to an optimal sequence-alignment score could readily effect advantageous improvements (Kaderali, Master's Thesis, Informatics, U. Köln, 2001) but would render the combinatorics opaque. We mention motivating applications of DNA codes