By Kevin Walker

This publication describes an invariant, l, of orientated rational homology 3-spheres that's a generalization of labor of Andrew Casson within the integer homology sphere case. enable R(X) denote the gap of conjugacy sessions of representations of p(X) into SU(2). permit (W, W, F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is said to be an adequately outlined intersection variety of R(W) and R(W) inside of R(F). The definition of this intersection quantity is a fragile job, because the areas concerned have singularities. A formulation describing how l transforms below Dehn surgical procedure is proved. The formulation contains Alexander polynomials and Dedekind sums, and will be used to offer a slightly straightforward facts of the lifestyles of l. it's also proven that once M is a Z-homology sphere, l(M) determines the Rochlin invariant of M

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1. Let (Ω, A, P ) be a probability space, (V, · ) be a normed space and S a linear space of V -valued random variables (possibly the entire space). 1) where P is a probability measure on Ω. The couple (S, ν) is called an E-normed space (EN space). Such spaces have been introduced by Sherwood (1969, 1979). The results that follow have been established in Lafuerza-Guill´en, Rodr´ıguez Lallena and Sempi (1977) and Lafuerza-Guill´en and Sempi (2003). For the sake of completeness the complete proof will be reproduced adapting it from Sherwood (1969, 1979) although the new part, which is necessary in order to extend the proof to PN spaces, is that regarding axiom (N4).

It is therefore no surprise that many results, mainly in the area of ﬁxed points, that hold in ordinary metric or normed spaces extend at once to Menger spaces under M . 7 Open Questions (1) Let (S, G) be an EN space with base (Ω, A, P ) and target (V, ·, · ). Then for every ω ∈ Ω, the function iω : S × S → R deﬁned by iω p, q := p(ω), q(ω) is a pseudo-inner product on S. Since distinct functions p and q may agree at a particular point ω ∈ Ω, so that p(ω) = q(ω) while p = q, iω need not be an inner product on S.

An EN space (S, ν) is a PPN space under the triangle functions τW and τM . Proof. Only the proof of the properties (N3) and (N4) is needed. For all p, q, and r in S and for every t > 0, let u and v in [0, +∞] be such that page 38 June 30, 2014 17:11 Probabilistic Normed Spaces 9in x 6in b1779-ch02 39 Probabilistic Normed Spaces u + v = t. Deﬁne the sets A, B, and C by A = {ω ∈ Ω : p(ω) < u} B = {ω ∈ Ω : q(ω) < u} C = {ω ∈ Ω : p(ω) + q(ω) < u}. Since the norm · A ∩ B ⊂ C, so that satisﬁes the triangle inequality, it follows that P (C) ≥ P (A ∩ B) ≥ W (P (A), P (B)).