3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Using 3-Sasakian reduction techniques we obtain infinite familiesof new 3-Sasakian manifolds M (p ₁,p ₂, p ₃) andM (p ₁,p ₂, p ₃, p ₄) in dimension 11 and 15 respectively. The metric cone on (p ₁,p ₂, p ₃) is a generalization ofthe Kronheimer hyperkähler metric on the regular maximalnilpotent orbit of sl (3, C)whereas the cone on M (p ₁,p ₂, p ₃, p ₄)generalizes the hyperkähler metric onthe 16-dimensional orbit of so(6, C).These are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)-reductions of M(p ₁,p ₂, p ₃).These yield examples of nontoric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families O() of compact self-dual positivescalar curvature Einstein metrics with orbifoldsingularities and with only one Killing vector field.     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups III

Let p₁, p₂, p₃ be primes. This is the final paper in a series of three on the (p₁, p₂, p₃)-generation of the finite projective special unitary and linear groups PSU ₃(pⁿ), PSL ₃(pⁿ), where we say a noncyclic group is (p₁, p₂, p₃)-generated if it is a homomorphic image of the triangle group T_{p1, p2, p3} . This article is concerned with the case where p₁ = 2 and p₂ ≠ p₃. We determine for any primes p₂ ≠ p₃ the prime powers pⁿ such that PSU ₃(pⁿ) (respectively, PSL ₃(pⁿ)) is a quotient of T = T_{2, p2, p3} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU ₃(pⁿ)) (respectively, Hom(T, PSL ₃(pⁿ))) is surjective as pⁿ tends to infinity.     

Cp

The spacec _p is the class of operators on a Hilbert space for which thec _p norm |T| _p =[trace(T*T) ^p/2]^1/p is finite. We prove many of the known results concerningc _p in an elementary fashion, together with the result (new for 1<p<2) thatc _p is as uniformly convex a Banach space asl _p. In spite of the remarkable parallel of norm inequalities in the spacesc _p andl _p, we show thatp ≠ 2, noc _p built on an infinite dimensional Hilbert space is equivalent to any subspace of anyl _p orL _p space. The author was supported by National Science Foundation Grant GP-5707.     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups II

Let p ₁, p ₂, p ₃ be primes. This is the second article in a series of three on the (p ₁, p ₂, p ₃)-generation of the finite projective special unitary and linear groups PSU₃(p ⁿ ), PSL₃(p ⁿ ), where we say a noncyclic group is (p ₁, p ₂, p ₃)-generated if it is a homomorphic image of the triangle group T _{p 1, p 2, p 3} . This paper is concerned with the case where p ₁ = 2 and p ₂ = p ₃. We determine for any prime p ₂ the prime powers p ⁿ such that PSU₃(p ⁿ ) (respectively, PSL₃(p ⁿ )) is a quotient of T = T _{2, p 2, p 2} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU₃(p ⁿ )) (respectively, Hom(T, PSL₃(p ⁿ ))) is surjective as p ⁿ tends to infinity.     

Embedding of C q and R q into noncommutative L p -spaces, 1≤p<q≤2

We prove that a quotient of a subspace of C_p⊕_pR_p (1≤p<2) embeds completely isomorphically into a noncommutative L_p -space, where C_p and R_p are respectively the p-column and p-row Hilbertian operator spaces. We also represent C_q and R_q (p<q≤2) as quotients of subspaces of C_p⊕_pR_p. Consequently, C_q and R_q embed completely isomorphically into a noncommutative L_p (M). We further show that the underlying von Neumann algebra M cannot be semifinite.     

4-valent graphs

Let {p_k}k≥2, k≠4 be a sequence of non-negative integers which satisfies 8 + Σ_k≥3(k — 4)p_k = 0. Then there exists an integer p₄ such that there exists a 2-connected planar graph with exactly p_k k-gons as faces for all k ≥ 2. This paper determines all such p₄ when p_k = 0 for k ≥ 5 and determines that there is a constant C ≥ 1 such that for some m ≤ p₂ + 1/4p₃ + C, there exists a 2-connected planar graph with exactly p_k faces for each p₄ = m + 2w, w a positive integer. When there exists at least one odd k ≥ 3 for which p_k ≠ 0, the coefficient 2 of w in the above equation may be replaced by 1. These conclusions do not hold if the coefficients of p₂ and p₃ are any smaller than 1 and 1/4, respectively.     

Modes and moments of unimodal distributions

Summary For a unimodal distribution relations of its modea with its absolute momentβ _p and central absolute momentγ _p of orderp are considered. The best constantA _p andB _p are given for the inequalities |a|≦A _p β _p ^1/p (p>0) and |a−m|≦B _p γ _p ^1/p (p≧1) wherem is the mean. the results follow from discussion of more general moments.     

A mean value theorem on the binary Goldbach problem and its application

We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers n satisfying n ≢ 2(mod 6), the equation p ₁ + p ₂ = n is solvable in prime variables p ₁, p ₂ such that p ₁ + 2 = P ₃, and for every sufficiently large odd integer [`(n)]{\bar n} satisfying [`(n)]{\bar n} ≢ 1(mod 6), the equation p ₁ + p ₂ + p ₃ = [`(n)]{\bar n} is solvable in prime variables p ₁, p ₂, p ₃ such that p ₁ + 2 = P ₂, p ₂ + 2 = P ₃. Here P _k denotes any integer with no more than k prime factors, counted according to multiplicity.     

p-Banach Spaces and p-Totally Convex Spaces

We introduce the categories Vec _p of p-normed vector spaces, Ban _p of _p -Banach spaces, AC _p of _p -absolutely and TC _p of _p -totally convex spaces (0 < p 1). It will be shown that TC _p (AC _p ) is the Eilenberg–Moore category of Ban _p (Vec _p ). Then congruence relations on TC _p (AC _p )-spaces are studied. There are many differences between TC _p (AC _p )-spaces and totally (absolutely) convex spaces (i.e. p = 1) (Pumplün and Röhrl, 1984, 1985), which will become apparent in Section 4.     

The rank of a completion of a dedekind domain

Let D be a Dedekind domain with quotient field K. Let C_p be the completion of the localisationD_p , of D at a nonzero prime idealp, of D. Let r_p be the rank of C_p as a D-module, ier_p , is the dimension of the K-vector space K_p , = K? _DC_p . The following results on r_p are deduced from well-known theorems: if r_p is finite for at least one prime ideal p, then D is a discrete valuation ring; and D = C_p if _p = 1. If D is a discrete valuation ring, then r_p = dimExt(K, D) + 1. A module M is extensionless if every extension of M by M splits. The D-module r_C is an estensionless indecomposable module. If r_C is infinite for every nonzero prime ideal, it is shown that an estensionless D-module of finite rank is a direct sum or certain rank one modulcs.     

A P‐Completeness Result for Visibility Graphs of Simple Polygons

For each vertex of a simple polygon P an integer valued weight is given. We consider the path p₁, p₂, ..., p_k in P which is created according to the following strategy: p₁ is a designated start vertex s and p_i+1 is obtained by choosing the vertex with smallest weight among all vertices visible from p_i and different from p₁, p₂, ..., p_i. If there is no such vertex the path is finished. This path is called geometric lexicographic dead end path. We shall prove the problem of determining whether a distinguished vertex t of P is on the geometric lexicographic dead end path or not to be P‐complete.     

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p ₂>p ₁>p _c , each infinite p ₂-cluster contains an infinite p ₁-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ _v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p _c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group. Received: 22 December 1997 / Revised version: 1 July 1998     

On some algebra of continuous linear operators

We present a criterion for an operator on L _p to belong to the set I _p of all sums of integral operators on L _p and multiplication operators by functions in L _∞. We describe the closure of I _p in the operator norm. We prove that the set L _p,1 of all sums of multiplication operators and operators on L _p mapping the unit ball of L _p into compact subsets of L ₁ is a Banach algebra.     

On the complemented subspaces ofX p

In this paper we prove some results related to the problem of isomorphically classifying the complemented subspaces ofX _p.We characterize the complemented subspaces ofX _pwhich are isomorphic toX _pby showing that such a space must contain a canonical complemented subspace isomorphic toX _p.We also give some characterizations of complemented subspaces ofX _pisomorphic tol _p⊕l ₂. Research supported in part by NSF grant DMS 890237.     

Some analogues of eberhard’s theorem on convex polytopes

Letp _k denote the number ofk-gonal faces of a simple 3-polytope. Euler’s relation leads to an equation between thep _k ’s which does not involvep ₆. Eberhard proved in 1891 that every sequence of non-negative integers (p ₃,p ₄,…) satisfying this equation corresponds to a polytope for suitable values ofp ₆. In the present paper it is established that ifp ₃=p ₄=0 then every valuep ₆≧8 is suitable. Research supported in part by the National Science Foundation under grant GP-7536     

Carleson measures and the BMO space on the p ‐adic vector space

For a prime number p, let Q _p be the p ‐adic field and let Q _p ^d denote a vector space over Q _p which consists of all d ‐tuples of Q _p . Then we study the p ‐adic version of the Calderón–Zygmund decomposition, Carleson measures on the vector space Q _p ^{d +1} and the space BMO ( Q _p ^d ) of functions of bounded mean oscillation on Q _p ^d . In particular, it turns out that the operator norms of various oncoming operators are independent of the dimension d and the prime number p, which is one of the big differences from that of the Euclidean case. Interestingly, the independence of the dimension d and p makes it possible to develop Harmonic Analysis on the infinite dimensional p ‐adic vector space as the importance had already been pointed out in the Euclidean case (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ON Sums of Products of Polynomials

Abstract

Given a polynomial P(t₁ ,…, t _n) = σ a_a t_a ^a1 …t_n ^an in several variables, we consider the p-norms |P|_p = (σ |a_a | ^p )^1/p (1≥ p < ∞) and |p|_∞ = max |a_a |. Our goal is to establish a generalization to the p-norms (1 ≥ p ≥ ∞) of a theorem originally obtained by P. Enflo for the l-norm.     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

The eigen functions of the complex projective space

In the complexn-dimensional projective spaceCP ⁿ , let λ _p (=4p(p+n)) be the eigen value of the Laplace-Beltrami operator andH _p be the space of all eigen functions of eigen value λ _p . The reproducing kernelh _p (z, w) ofH _p is constructed explicitly in this paper, and a system of complete orthogohal functions ofH _p is constructed fromh _p (z,w)(p=1,2, …). Partially supported by NSF of China     

Automorphism Groups of Finite Abelian p-groups and Transvections

We define a particular type of automorphisms called transvections on a finite finite abelian p-group H_p. It is proved that the subgroup E of the automorphism group Aut(H_p) of H_p generated by those transvections is normal in it, and that Aut(H_p) can be written as the product of E and some abelian subgroup K. The center of Aut(H_p) is also determined.