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.     

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.     

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.     

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     

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.     

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 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     

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.     

Compact operators which factor through subspaces of lp

Let 1 ≤ p ≤ ∞. A subset K of a Banach space X is said to be relatively p ‐compact if there is an 〈x_n 〉 ∈ l^s _p (X) such that for every k ∈ K there is an 〈α_n 〉 ∈ l_{p ′} such that k = σ^∞_n=1 α_n x_n . A linear operator T: X → Y is said to be p ‐compact if T (Ball (X)) is relatively p ‐compact in Y. The set of all p ‐compact operators K_p (X, Y) from X to Y is a Banach space with a suitable factorization norm κ_p and (K_p , κ_p ) is a Banach operator ideal. In this paper we investigate the dual operator ideal (K^d _p , κ^d _p ). It is shown that κ^d _p (T) = π_p (T) for all T ∈ B (X, Y) if either X or Y is finite‐dimensional. As a consequence it is proved that the adjoint ideal of K^d _p is I_{p ′}, the ideal of p ′‐integral operators. Further, a composition/decomposition theorem K^d _p = Π_p K is proved which also yields that (Π^min _p )^inj = K^d _p . Finally, we discuss the density of finite rank operators in K^d _p and give some examples for different values of p in this respect. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Bounding an Index by the Largest Character Degree of a p-Solvable Group

In this article, we show that if p is a prime and G is a p-solvable group, then |G: O _p (G)| _p  ≤ (b(G) ^p /p)^1/(p?1), where b(G) is the largest character degree of G. If p is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow p-subgroup of G is at most p, then |G: O _p (G)| _p  ≤ b(G).     

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)     

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.     

Some probabilistic and extremal results on subdivisions and odd subdivisions of graphs

A probabilistic result of Bollobás and Catlin concerning the largest integer p so that a subdivision of K_p is contained in a random graph is generalized to a result concerning the largest integer p so that a subdivision of A_p is contained in a random graph for some sequence A₁, A₂,… of graphs such that A_i+1 contains a subdivision of A_i. A similar result is proved for subdivisions with odd paths or cycles. The result is applied to disprove a conjecture of Chartrand, Geller, and Hedetniemi. The maximum number of edges in a graph without a subdivision of K_p, p = 4, 5, with odd paths or cycles is determined.     

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.     

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.     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

The Permutations 123p 4…p m and 321p 4…p m are Wilf-Equivalent

Write p ₁, p ₂…p _m for the permutation matrix δ _{pi, j} . Let S _n (M) be the set of n×n permutation matrices which do not contain the m×m permutation matrix M as a submatrix. In [7] Simion and Schmidt show bijectively that |S _n (123) |=|S _n (213) |. In [9] this was generalised to a bijection between S _n (12 p ₃…p _m ) and S _n (21 p ₃…p _m ). In the present paper we obtain a bijection between S _n (123 p ₄…p _m ) and S _n (321 p ₄…p _m ). Revised: March 24, 1999