Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N . One calls (K,N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G := K nN which meet the annihilator k⊥ of the Lie algebra k of K do so in single K-orbits. This generalizes a result of the authors and R....

There have been a number of mathematical results recently identifying algebras over certain operads [4, 25, 17, 28, 16, 10, 14, 11]. See [1, 26] for expository surveys of the basics of operad theory. Before citing any of these results, let us mention some trivial classical examples. Let A denote one of the three words: “commutative”, “associative” and “Lie”. In each of these cases, consider the...

This paper proposes the implicative rating measure developed on the typicality measure. The paper also proposes a new recommendation model presenting the top N items to the active users. The proposed model is based on the user-based collaborative filtering approach using the implicative intensity measure to find the nearest neighbors of the active users, and the proposed measure to predict user...

Given two k-graphs H and F , a perfect F -packing in H is a collection of vertexdisjoint copies of F in H which together cover all the vertices in H. In the case when F is a single edge, a perfect F -packing is simply a perfect matching. For a given fixed F , it is often the case that the decision problem whether an n-vertex k-graph H contains a perfect F -packing is NP-complete. Indeed, if k ≥...

Let k be an algebraically closed field. We list the finitely many isomorphism types of rank n commutative k-algebras for n ≤ 6. There are infinitely many types for each n ≥ 7. All algebras are assumed to be commutative, associative, and with 1 (except briefly in Remark 1.1). We assume that k is an algebraically closed field, except in Section 2. By the rank of a k-algebra, we mean its dimension...

It was known that the quaternion group and the octic group could not be generated by the squares of any group [5, pp. 193-194]. A natural question is which groups are generated by the squares of some groups. Clearly, groups of odd order and simple groups are generated by their own squares. In this paper, we show in a concrete manner that abelian groups are generated by the squares of some group...

We investigate examples of quasi-spectral triples over two-dimensional commutative sphere, which are obtained by modifying the order-one condition. We find quasi-Dirac operators and calculate the index paring with a representant of K-theory class to prove that the quasispectral triples are mutually inequivalent. MSC 2000: 58B34, 46L87, 34L40

The setting of this research is Bishop’s constructive mathematics. Following ideas of Chan and Shum, exposed in their famous paper “Homomorphisms of implicative semigroups”, we discuss the structure of implicative semigroups on sets with tight apartness. Moreover, we use anti-orders instead of partial orders. We study concomitant issues induced by existence of apartness and anti-orders giving s...