Proof. Suppose that π ∈ Sn is such that the minimal j ∈ {1, . . . , n} for which aπ(j) ∈ BX(x, r+dX(x, y)) actually satisfies aπ(j) ∈ BX(x, r−dX(x, y)). Hence j r (x) = j and therefore aπ(j) = ar (x). Also, dX(aπ(j), y) 6 dX(aπ(j), x) + dX(x, y) 6 r, so j r (y) 6 j. But dX(x, ajπ r (y)) 6 dX(y, ajπ r (y)) + dX(x, y) 6 r+ dX(x, y), so by the definition of j we must have j r (y) > j. Thus j π r (...

The Political Parties, Elections and Referendums Act 2000 marked the most comprehensive and radical overhaul of British party finance for over 100 years. It instituted reforms in many areas with the notable exception of the comprehensive extension of state funding for political parties. However, despite the radicalism of the Act, questions have already arisen as to whether the further state fun...

k-plexes are cohesive subgraphs which were introduced to relax the structure of cliques. A co-k-plex is the complement of a k-plex and is therefore similar to a stable set. This paper derives the co-2-plex analogue for certain properties of the stable set polytope. We also describe a class of 0-1 matrices A for which the polytope {x ∈ R+ | Ax ≤ 2, x ≤ 1} is integral.

Direct observations provide constraints on the first two derivatives of the inflaton potential in slow-roll models. We discuss how present-day observations, combined with the flow equations in slow-roll parameter space, provide a non-trivial constraint on the third derivative of the inflaton potential. We find a lower bound on the third derivative of the inflaton potential V ′′′/V > −0.2. We al...

In this paper, we establish the following result: Let M be an n-dimensional complete totally real minimal submanifold immersed in CPn with Ricci curvature bounded from below. Then either M is totally geodesic or infr ≤ (3n+1)(n−2)/3, where r is the scalar curvature of M .

We study Yukawa unification, including the effects of a physical neutrino mass consistent with the Superkamiokande observations, in a string/D-brane inspired SU(4) ⊗ SU(2) L ⊗ SU(2) R model which allows the most general non-universal scalar and gaugino masses, including the usual D-term contributions which arise in SO(10). We investigate how the tight constraints from rare decays such as b → sγ...