In the theory of enumeration, the part devoted to the counting of planar maps gives rather surprising results. Of special interest to the combinatorialists is the conspicuous feature of counting numbers associated with families of maps as discussed in the papers of Tutte, Brown and Mullin. These formulas are by no means easy to prove; this is also a part of their charm. We are convinced therefore that these maps possess deep combinatorial properties. We discuss such properties in this paper.

The main result of this article is the construction of a bisection between planar maps and a special family of trees: their vertices are labeled with numbers that differ by at most 1 on adjacent vertices. These trees are said to be well labeled. Combinatorialists usually claim that theorems specifying the “geometry” of objects lead to enumerating formulas. Our result obeys this general rule: we can indeed obtain a new proof of the formula counting the number of rooted planar maps with m edges. Four different proofs of this formula are known to date.

An orthogonal design in order n of type (u1, …, ut) on the commuting variables x1, …, xt is an n × n matrix X with entries 0, ±x1, …, ±xt such that

In [5] Geramita and Wallis show that if n = 24a+b·n0, where n0 is odd and 0 ≦ b > 4, then t ≧ ρ(n) = 8a + 2b. The result is essentially Radon's limit on the number of anti-commuting, real, anti-symmetric, orthogonal matrices in order n. Garamita and Pullman show that this limit is sharp for orthogonal designs: i.e., given n, there exists an orthogonal design in order n with ρ(n) variables [6].

Two orthogonal designs, X and F, are called amicable if XYt = YXt.

Such pairs of orthogonal designs are especially useful in generating new orthogonal designs [5] or [6].

In [1, b.2, VIII, 128] Pólya and Szegö give the following interesting result of A. Cohn:

THEOREM 1. If a prime p is expressed in the decimal system as

then the polynomial irreducible inZ[x].

The proof of this result rests on the following theorem of Pólya and Szegö [1, b.2, VIII, 127] which essentially states that a polynomial f(x) is irreducible if it takes on a prime value at an integer which is sufficiently far from the zeros of f(x).

THEOREM 2. Let f(x) ∊ Z[x] be a polynomial with the zeros α1, α2, …, αn. If there is an integer b for which f(b) is a prime, f(b – 1) ≠ 0, and for 1 ≦ i ≦ n, then f(x) is irreducible inZ[x].

In [2], Dempwolff constructs three translation planes of order 16 using sharply 2-transitive sets of permutations in S16. That is, if acting on Λ is a sharply 2-transitive set of permutations then an affine plane of order n may be defined as follows: The set of points = {(x, y)|x, y ∊ Λ} and the lines = {(x, y)|y = xg for fixed }, {(x, y)|x = c}, {(x, y)|y = c} for c ∊ Λ.

Let V be a vector space of dimension k over F ≅ GF(pr). A translation plane may be defined by finding a set M of pkr – 1 linear transformations such that xy–l is fixed point free on for all x ≠ y in M.

Notice that if we allow V to act on itself then MV is a sharply 2-transitive set on V if and only if xy–1 is fixed point free on for all x ≠ y in M.

Let K be a field of characteristic zero. The Schur subgroup S(K) of Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. It is well known that the classes in S(K) are represented by cyclotomic algebras, (see [16]). However it is not necessarily the case that the division algebra representatives of these classes are themselves cyclotomic. The main result of this paper is to provide necessary and sufficient conditions for the latter to occur when K is any algebraic number field.

Next we provide necessary and sufficient conditions for the Schur group of a local field to be induced from the Schur group of an arbitrary subfield. We obtain a corollary from this result which links it to the main result. Finally we link the concept of the stufe of a number field to the existence of certain quaternion division algebras in S(K).

Let L be a finitely generated extension of a field K of characteristic p ≠ 0. If L/K is algebraic, then there is a unique intermediate field S such that

S is just the maximal separable extension of K in L. If L/K is not algebraic, then Dieudonne [4] showed there exist maximal separable extensions D of K in L such that L ⊆ Kp–∞⊗KD. In general, not every maximal separable extension of K in L has the property. Those which do have the property are called distinguished. Kraft [7] established that a maximal separable extension D of K in L is distinguished if and only if [L:D] is as small as possible. If the minimum of the [L:D] is pr, r is called the order of inseparability of L/K, denoted inor (L/K).

Let L1 be an intermediate field of L/K. If L/K is algebraic, then the maximal separable extension S1 of K in L1 is contained in the maximal separable extension S of K in L, and moreover S is separable over S1.

Throughout this paper a surface is a 2-dimensional (not necessarily compact) complex manifold. A surface X is homogeneous if a complex Lie group G of holomorphic transformations acts holomorphically and transitively on it. Concisely, X is homogeneous if it can be identified with the left coset space G/H, where if is a closed complex Lie subgroup of G. We emphasize that the assumption that G is a complex Lie group is an essential part of the definition. For example, the 2-dimensional ball B2 is certainly “homogeneous” in the sense that its automorphism group acts transitively. But it is impossible to realize B2 as a homogeneous space in the above sense. The purpose of this paper is to give a detailed classification of the homogeneous surfaces. We give explicit descriptions of all possibilities.

Let (X, , μ) and (X, , μ′) be measure spaces with the measures μ and μ′ totally finite. Suppose {Uλ: λ > 0} is a family of positive (i.e., ϕ ≧ 0 ⇒ Uλϕ ≧ 0) continuous linear operators from L2(X, dμ′) to L2(X,dμ) with the following additional properties: if ϕ ≧ 0 then Uλϕ is non-decreasing as λ increases, while λ−1Uλϕ is nonincreasing.

A family {Mλ:λ > 0} of continuous linear operators from L2(X, dμ) to L2(X, dμ′) satisfies the “generalized resolvent equation” relative to {Uλ} if

(0.1)

for positive λ and v. If Uλ = λI, then (0.1) is just the well-known resolvent equation. The family {Mλ} is called submarkov if Mλ is a positive operator and

(0.2)

it is conservative if

(0.3)

It is well-known [5, 19] that every finite group can appear as a group of automorphisms of an algebraic Riemann surface. Hurwitz [9, 10] showed that the order of such a group can never exceed 84 (g – 1) provided that the genus g is ≧2. In fact, he showed that this bound is the best possible since groups of automorphisms of order 84 (g – 1) are obtainable for some surfaces of genus g. The problems considered by Hurwitz and others can be considered as particular cases of a more general question: Given a finite group G, what is the minimum genus of the surface for which it is a group of automorphisms? This question has been completely answered for cyclic groups by Harvey [7]. Wiman's bound 2(2g + 1), the best possible, materializes as a consequence. A further step was taken by Maclachlan who answered this question for non-cyclic Abelian groups.

1.1. The isometries of the Hardy spaces Hp(0 < p < ∞, p ≠ 2) of the unit disc were determined by Forelli in [2]. Generalizations to several variables: For the polydisc the isometries of Hp onto itself were characterized by Schneider [9]. For the unit ball the case p > 2 was then done by Forelli [3]; Rudin [8] removed the restriction p > 2 by proving a theorem on equimeasurability. Finally, Koranyi and Vagi [6] noted that the methods developed by Forelli, Rudin and Schneider applied to bounded symmetric domains.

In this note it will be shown that their methods also apply to the Bergman spaces over bounded Runge domains. The isometries which are onto are completely characterized; the special cases of the ball and polydisc are particularly nice and are given separately.

The fundamental theorem of this paper is stated in Section 8. In this theorem, the stochastic differential equation dX = a(X)dZ is studied when Z is a *-dominated (cf. [15]) Banach space valued process and a is a predictable functional which is continuous for the uniform norm.

For such an equation, the existence of a “weak solution” is stated; actually, the notion of weak solution here considered is more precise than this one introduced by Strook and Varadhan (cf. [30], [31], [23]).

Namely, this weak solution is a probability, so-called “rule,” defined on (DH × Ω), DH being the classical Skorohod space of all the cadlag sample paths and Ω is the initial space which Z is defined on: the marginal distribution of R on Ω is the given probability P on Ω. This concept of rule is defined in Section 3.

We let Z denote the ring of rational integers, Q the field of rational numbers, R the field of real numbers, and C the field of complex numbers.

For elements e and f of a Lie algebra, [e,f] denotes the bracket of e and f. A generalized Cartan matrix C = (cij) is a square matrix of integers satisfying cii = 2, cij ≦ 0 if i ≠ j, cij = 0 if and only if cji = 0. For any generalized Cartan matrix C = (cij) of size l × l and for any field F of characteristic zero, denotes the Lie algebra over F generated by 3l generators e1, …, el, h1, …, hl, f1, …, fl with the defining relations

for all i, j,

for distinct i, j.

Let (X, ) be a Steiner triple system on v = |X| points, and suppose that is a partial parallel class (transversal, clear set, set of pairwise disjoint blocks) of maximum size . We want to derive a bound on . (I conjecture that in fact r is bounded, e.g., r ≦ 4 – 4 is attained for the Fano plane, but all that has been proved so far (cf. [1], [2]) are bounds r < C.v for some C. Here we shall prove r < 5v2/3.)

Define a sequence of positive real numbers by q0 = Q · r2/v, , where l is determined by ql ≧ 6, , i.e.,

(The constant Q will be chosen later.) Define inductively sets Ai, Ki and collections as follows. Let

Let and denote infinite dimensional Hilbert spaces and let denote the space of all bounded linear operators from to . For A in and B in , let τAB denote the operator on defined by τAB(X) = AX – XB. The purpose of this note is to characterize the semi-Fredholm domain of τAB (Corollary 3.16). Section 3 also contains formulas for ind(τAB – λ). These results depend in part on a decomposition theorem for Hilbert space operators corresponding to certain “singular points” of the semi-Fredholm domain (Theorem 2.2). Section 4 contains a particularly simple formula for ind(τAB – λ) (in terms of spectral and algebraic invariants of A and B) for the case when τAB – λ is Fredholm (Theorem 4.2). This result is used to prove that (τBA) = –ind(τAB) (Corollary 4.3). We also prove that when A and B are bi-quasi-triangular, then the semi-Fredholm domain of τAB contains no points corresponding to nonzero indices.

Let o be a quasi semilocal semihereditary ring, i.e., o is a commutative ring with 1 which has finitely many maximal ideals {Ai|i ∊ I} and the localization oAi at any maximal ideal Ai is a valuation ring. We assume 2 is a unit in o. Furthermore * denotes an involution on o with the property that there exists a unit θ in o such that θ* = –θ. V is an n-ary free module over o with f : V × V → o a λ-Hermitian form. Thus λ is a fixed element of o with λλ* = 1 and f is a sesquilinear form satisfying f(x, y)* = λf(y, x) for all x, y in V. Assume the form is nonsingular; that is, the mapping M → Hom (M, A) given by x → f( , x) is an isomorphism. In this paper we shall write f(x, y) = xy for x, y in V.

It is a well known and useful fact [4] that every (partially) ordered set P has a linear extension L (that is, a totally ordered set (chain) on the same underlying set as P and satisfying a ≦ b in L whenever a ≦ b in P). It is just as well known that an ordered set P can be embedded in an ordered set P′ which, in turn, has a complete linear extension L′ (that is, a linear extension in which every subset has both a supremum and an infimum); just take L′ to be the “completion by cuts” of L. However, an arbitrary ordered set P need not, itself, have a complete linear extension (for example, if P is the chain of integers or, for that matter, if P is any noncomplete chain). It is natural to ask which ordered sets have a complete linear extension?

Let Δ denote the unit disk in the complex plane C. The space BMO has been extensively studied by many authors (see [3] for a good exposition of this topic). Recently, the subspace BMOA (Δ) has become a topic of interest. An analytic function f, in the Hardy class H2(A), belongs to BMOA (Δ) if

(1)

where

It is known (see [3, p. 96]) that (1) is equivalent to

(2)

In his work on the Dirichlet problem for the Heisenberg group Greiner [5] showed that each Lα-spherical harmonic is a unique linear combination of functions of the form

with k = 0, 1,2, … and n = 0, ±l, ±2 , …, where Hk(α, n)(θiθ) is defined by the generating function

Since Hk(0,0)(eiθ) = Pk(cos θ), where Pk(x) is the Legendre polynomial of degree k, and these functions satisfy the orthogonality relation

Greiner raised the question of whether the functions Hk(0,0)(eiθ) are orthogonal or biorthogonal with respect to some complex valued weight function.

If M is an associative algebra with product xy, M can be made into a Lie algebra by endowing M with a new multiplication [x, y] = xy – yx. The Poincare-Birkoff-Witt Theorem, in part, shows that every Lie algebra is (Lie) isomorphic to a Lie subalgebra of such an associative algebra M. A Lie ideal in M is a linear subspace U ⊆ M such that [x, u] ∊ U for all x £ M, u ∊ U. In [9], as a step in characterizing Lie mappings between von Neumann algebras, Lie ideals which are closed in the ultra-weak topology, and closed under the adjoint operation are characterized when If is a von Neumann algebra. However the restrictions of ultra-weak closure and adjoint closure seemed unnatural, and in this paper we characterize those uniformly closed linear subspaces which can occur as Lie ideals in von Neumann algebras.