The Möbius function for a group G was introduced in 1936 by Hall in order to count ordered generating sets of G. In this paper we determine the Möbius function of the simple small Ree groups, R(q) = G2(q) where q = 3 for m > 0, using their 2-transitive permutation representation of degree q + 1. We also describe their maximal subgroups in terms of this representation. We use this to enumerate s...

We report results on two physics topics from the SLD experiment at the SLAC Linear Collider, using our full sample of 550,000 events of the type e + e − → Z 0 → q ¯ q. The electron beam was polarized, enabling the quark and antiquark hemispheres to be tagged in each event. One physics topic is the first study of rapidities signed such that positive rapidity is along the quark rather than antiqu...

In this note we give a description of the representation of the wreath product Sn[G] of the symmetric group Sn and a finite group G on the homology of the product of n copies of a partially ordered set (poset for short) P on which G acts as a group of automorphisms. In the sequel all posets will be finite. We will consider three types of product constructions. The direct product P ×Q of two pos...

We study the set S nc(p, q) of annular non-crossing permutations of type B, and we introduce a corresponding set NC(p, q) of annular non-crossing partitions of type B, where p and q are two positive integers. We prove that the natural bijection between S nc(p, q) and NC (p, q) is a poset isomorphism, where the partial order on S nc(p, q) is induced from the hyperoctahedral group Bp+q, while NC ...

This is a report for my presentation at the upcoming meeting on Berechenbarkeitstheorie (“Computability Theory”), Oberwolfach, January 21–27, 2001. We use 2 to denote the space of infinite sequences of 0’s and 1’s. For X, Y ∈ 2, X ≤T Y means that X is Turing reducible to Y . For P,Q ⊆ 2 we say that P is Muchnik reducible to Q, abbreviated P ≤w Q, if for all Y ∈ Q there exists X ∈ P such that X ...

We give de nitions for fuzzy subalgebras, for fuzzy relations on fuzzy sets, and for fuzzy compatibility, which generalize, improve and correct the existing ones. The unit interval is replaced here by a partially ordered algebra. Then we study their connection with the corresponding crisp concepts through their newly de ned Q-cuts.

Pearson’s rank correlation coefficient (r) between the vectors of counts and residual values over all tuples was found to be very close to zero (−0.035). Kendall rank correlation coefficient τ between the ranked lists when (w, q) tuples are ordered by frequency and residual value, was found to be −4.65 × 10−9 . This indicates almost no correlation between counts and residuals. Problem Statement

Let i ≥ 1 be arbitrary. We wish to extend the results of [2, 2, 14] to measurable, linear, ordered points. We show that Q is isomorphic to Ĥ. Next, the groundbreaking work of F. Gupta on paths was a major advance. U. Sasaki [14] improved upon the results of E. Moore by extending non-linearly meager functionals.

Existence and multiplicity of positive solutions for the fourth order equation u′′′′ − M( ∫ 1 0 |u ′|2 dx)u′′ = q(x)f(x, u, u′), which models simply supported extensible beams, are considered using fixed point theorems in cones of ordered Banach spaces.

We study the q-state clock models on heptagonal lattices assigned on a negatively curved surface. We show that the system exhibits three classes of equilibrium phases; in between ordered and disordered phases, an intermediate phase characterized by a diverging susceptibility with no magnetic order is observed at every q>or=2. The persistence of the third phase for all q is in contrast with the ...