We derive a bilinear expansion expressing elements of lattice Kadomtsev-Petviashvili (KP) τ-functions, labeled by partitions, as sum over products pairs an associated BKP strict partitions. This generalizes earlier results relating determinants and Pfaffians minors skew symmetric matrices, with applications to Schur functions Q-functions. It is deduced using the representations KP τ-functions v...

The following conjecture generalizing the Contraction Mapping Theorem was made by Stein in [9]: Let (X, ρ) be a complete metric space and let F = {T1, . . . , Tn} be a finite family of self-maps of X. Suppose there is a constant γ ∈ (0, 1) such that for any x, y ∈ X there exists T ∈ F with ρ(T (x), T (y)) ≤ γρ(x, y). Then some composition of members of F has a fixed point. In this paper we disp...

It is shown that for $n \le 3$ the joint numerical range of a family commuting $n\times n$ complex matrices always convex; \ge 4$ there are two whose not convex.

BACKGROUND In most European origin populations measures of socioeconomic position are positively associated with leisure time physical activity (LTPA), this is unclear for active commuting. In addition, these associations have scarcely been studied in ethnic minority groups, who often have a high cardiovascular disease risk. Because of the expected public health potential, we assessed the relat...

We define the commuting algebra determinant of a finite group action on a finite set, a notion dual to the group determinant of Dedekind. We show that the following combinatorial example is a commuting algebra determinant. Let Bq(n) denote the set of all subspaces of an n-dimensional vector space over Fq. The type of an ordered pair (U, V ) of subspaces, where U, V ∈ Bq(n), is the ordered tripl...

BACKGROUND Route environments can positively influence people's active commuting and thereby contribute to public health. The Active Commuting Route Environment Scale (ACRES) was developed to study active commuters' perceptions of their route environments. However, bicycle commuters represent a small portion of the population in many cities and thus are difficult to study using population-based...

Let G be a group. A subset X of G is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in X. If |X| ≥ |Y | for any other set of pairwise non-commuting elements Y in G, then X is said to be a maximal subset of pairwise non-commuting elements. In this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

An operator in a Banach space is called upper (lower) semi-Browder if it is upper (lower) semi-Fredholm and has a finite ascent (descent). We extend this notion to n-tuples of commuting operators and show that this notion defines a joint spectrum. Further we study relations between semi-Browder and (essentially) semiregular operators. Denote by L(X) the algebra of all bounded linear operators i...