This note gives a “Zorn’s Lemma” style proof that any two bases in a vector space have the same cardinality. One of the most fundamental notions in linear algebra is that of a basis: A subset B of a vector space V is a basis if every element of V is a unique linear combination of elements of B. The following theorem of Georg Hamel expresses one of the most important aspects of this definition. ...

Probably the two best-known exact string matching algorithms are the linear-time algorithm of Knuth, Morris and Pratt (KMP), and the fast on average algorithm of Boyer and Moore (BM). The efficiency of these algorithms is based on using a suitable failure function. When a mismatch occurs in the currently inspected text position, the purpose of a failure function is to tell how many positions th...

For graphs with maximum degree A and diameter D, an upper bound on the number of vertices is 1 + AxF:-,‘(A 1)‘. This bound is called the Moore bound for graphs and the graphs that attain it are called Moore graphs. Similar bounds for directed graphs and for hypergraphs have been defined and the existence of directed Moore graphs and of Moore hypergraphs has been studied. In this article, we def...

Sometimes the bad character heuristics fails. In the following situation the comparison a-b causes a mismatch. An alignment of the rightmost occurence of the pattern symbol a with the text symbol a would produce a negative shift. Instead, a shift by 1 would be possible. However, in this case it is better to derive the maximum possible shift distance from the structure of the pattern. This metho...

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore-like bound in terms of its diameter k and the maximum out-degrees (d1; d2) of its partite sets of vertices. It has been proved that, when d1d2 > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist w...

For the relevant literature and most of the definitions and conventions the reader is referred to [1]. The next section contains an explicit description of the (reduced) characteristic polynomial of L4(s, t), and it is shown that this polynomial admits a decomposition into two closely related factors of degree 2 each. In Section 3 it is first shown that at least one of these factors must be red...

A simple language is defined, a probabilistic semantics and a partial correctness logic is proposed in terms of probabilistic relations. It is shown how a derandomized semantics can be constructed through the Eilenberg-Moore algebras. We investigate the category of EilenbergMoore algebras for the Giry monad associated with stochastic relations over Polish spaces with continuous maps as morphism...

In this paper we introduce some notions of stabilizability and detectability for discrete time-varying systems. These concepts are introduced by making state-space decompositions of the system. This allows an intuitive interpretation and shows the difficulties which occur if one tries to derive general stabilizability and detectability properties. Moreover, we show by means of a counterexample ...

Let ΩX be the space of Moore loops on a finite, qconnected, n-dimensional CW complex X , and let p be an odd prime. We prove the theorem: If p ≥ n/q, then the Hurewicz homomorphism induces isomorphisms U(Fπ∗(ΩX(p))) = −→ FH∗(ΩX(p)) and UE ∞ π (ΩX) = −→ E∞ H (ΩX) of Hopf algebras, where E ∞ π and E ∞ H are the limit terms of the homotopy and homology Bockstein spectral sequences, respectively, a...