Introduction 2 1. Normed Rings 4 1.1. Elementwise Norms 4 1.2. Ideal norms 5 1.3. Finite Quotient Domains 5 1.4. Euclidean norms 6 2. Euclidean quadratic forms and ADC forms 7 2.1. Euclidean quadratic forms 7 2.2. Euclideanity 7 2.3. ADC-forms 8 2.4. Euclidean Forms are ADC Forms 8 2.5. The Generalized Cassels-Pfister Theorem 9 2.6. Maximal Lattices 10 3. Localization and Completion 11 3.1. Loc...

Let k be a number field. Let Ω denote the set of places of k and for v ∈ Ω, let kv denote the completion of k at v. A classical theorem of Hasse-Minkowski states that a quadratic form q over k represents zero non-trivially provided it represents zero non-trivially over kv for all v ∈ Ω; in particular, two quadratic forms over k are isomorphic if they are isomorphic over kv for all v ∈ Ω. Anothe...

Let NSymm be the Hopf algebra of non-commutative symmetric functions (in an infinity of indeterminates): NSymm=Z 1 2 , ,... Z Z   . It is shown that an associative algebra A with a Hasse-Schmidt derivation 1 2 ( , , ,...) d id d d  on it is exactly the same as an NSymm module algebra. The primitives of NSymm act as ordinary derivations. There are many formulas for the generators Zi in terms ...

We apply theorems of Fischer, Morris and Shapiro on affine semigroup rings to show that if a certain affine semigroup ring defined by a poset is a complete intersection, then the poset is either unicyclic or contains a chain, the removal of which increases the number of connected components of the Hasse diagram. This is the converse of a theorem of Boussicault, Feray, Lascoux and Reiner [2]. We...

We describe a technique for user interaction with the interim results of Formal Concept Analysis which we hypothesise will expedite user comprehension of the resultant concept lattice. Given any algorithm which enumerates the concepts of a formal context, this technique incrementally updates the set of formal concepts generated so far, the transitive reduction of the ordering relation between t...

Transient precursor states are often experimentally observed for molecules adsorbing on surfaces. However, such precursor states are typically rather short-lived, quickly yielding to more stable adsorption configurations. Here we employ first-principles calculations to systematically explore the interaction mechanism for benzene derivatives on metal surfaces, enabling us to selectively tune the...

We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.

We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n2) features, in an O(n)×O(n) grid in O(n2) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) features in an O(n)×O(n) grid i...

The edges in the Hasse diagram of the partitions of an integer ordered by reverse refinement are studied. A recursive count of the edges between ranks is developed. Let P, be the poset of partitions of n, ordered by refinement. It contains p(n; n-t) elements of rank t, where p(n; k) denotes the number of partitions of n into h-parts, p(n) will denote the number of partitions of n. The number NE...

An< accurate, stable and very fast corner finder (for feature based vision) has been developed', based on a novel definition of corners, using:no image derivatives. This note describes the algorithm and the results obtained from its use.