This paper is an appendix to “Finite unitary reflection groups” by J. A. Todd and the author (3) in which the irreducible finite groups generated by reflections in unitary space (2, p. 364) are enumerated and abstract definitions are given for all those n-dimensional groups which may be generated by n reflections.

Let G be an almost simple complex algebraic group defined over R, and let G(R) be the group of real points of G. We enumerate the G(R)-conjugacy classes of maximal R-tori of G. Each of these conjugacy classes is also a single G(R)˚-conjugacy class, where G(R)˚ is the identity component of G(R), viewed as a real Lie group. As a consequence we also obtain a new and short proof of the Kostant-Sugiura's theorem on conjugacy classes of Cartan subalgebras in simple real Lie algebras.

A connected real Lie group P is said to be weakly exponential (w.e.) if the image of its exponential map is dense in P. This concept was introduced in [HM] where also the question of identifying all w.e. almost simple real Lie groups was raised. By using a theorem of A. Borel and our classification of maximal R-tori we answer the above question when P is of the form G(R)˚.

We show that quasimap Floer cohomology for varying symplectic quotients resolves several puzzles regarding displaceability of toric moment fibers. For example, we present a compact Hamiltonian torus action containing an open subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and we determine displaceability for most of the moment fibers of a symplectic ellipsoid.

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by “spread modules,” which are sometimes called “interval modules” in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the “single-source” spread modules gives rise to a new invariant which is finer than the rank invariant.

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X,Y)$ has the pointwise Bishop–Phelps–Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X,Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X,Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_{p}(\unicode[STIX]{x1D707})$ spaces fail to have this property when $p>2$. No universal pointwise BPB range space can be simultaneously uniformly convex and uniformly smooth unless its dimension is one. We also discuss a version of the pointwise BPB property for compact operators.

A matrix H of order n = 4t with all entries from the set ﹛1, —1﹜ is Hadamard if HHt = 4tI. The set of Hadamard matrices is . A matrix is of type I or is skew-Hadamard if H = S — I where St = —S (some authors also use H = S + I). The set of type I members is . A matrix P is a signed permutation matrix if each row and each column has exactly one non-zero entry, and that entry is from the set ﹛1, —1﹜.

According to a classical theorem originally proved by L. Autonne (1; 3) in 1915, every m × n matrix of rank r with entries from the complex field can be decomposed as

where U1 and U2 are unitary matrices of order m and n respectively and D is an m × n matrix having the form

1

where Δ is a non-singular diagonal matrix whose rank is r. If r = m, then the row of zero matrices of (1) does not actually appear. If r = n, then the column of zero matrices of (1) does not appear. The main purpose of this paper is to give a necessary and sufficient condition under which both U1 and U2 may be chosen to be real orthogonal matrices.

The object of this paper is to prove the following:

Theorem. Suppose that λ, μ are real non-zero numbers, not both negative, λ is irrational, and k is a positive integer. Then there exist infinitely many primes p and pairs of primes p1, p2 such that

In particular [λp1 + μp2] represents infinitely many primes.

Here [x] denotes the greatest integer not exceeding x.

McCoy, following Frobenius, studied a problem which can be described as follows. Let k be an arbitrary field, kc its algebraic closure, and any algebra of n ⨯ n matrices over k which contains the identity I . Define a canonical ordering to be a set of n mappings λ of , or of a subset of , into kc such that the sequence λ1(A),λ2(A), …, λn(A), for each A ∈ , consists of the characteristic values (roots of det(A — xI) = 0) of A, each with the right multiplicity. Define a canonical ordering to be a Frobenius ordering if, for all non-commutative polynomials f(x1, x2, … , xm) and all finite subsets A1, A2, …, Am of ,

A Prüfer domain is an integral domain D with the property that for every proper prime ideal P of D the quotient ring DP is a valuation ring. Examples of such domains are valuation rings and Dedekind domains, a Dedekind domain being merely a noetherian Prüfer domain. The integral closure of the integers in an infinite algebraic extension of the rationals is another example of a Prüfer domain (5, p. 555, Theorem 8). This third example has been studied initially by Krull (4) and then by Nakano (8).

This paper is concerned with two things. The first is a (primarily) geometric axiomatic description for the systems of real roots of Lie algebras arising from (generalized) Cartan matrices. The description is base free and is a natural extension of the well-known axiomatic description of finite root systems. The primary component of our description is an open convex cone which, following Looijenga [3], we call the Tits cone. In fact it was Looijenga's paper that led to this axiomatic formulation. Unlike his construction, the dimension of the Tits cone is not tightly connected to the dimension of the Cartan matrix which it eventually yields. This leads us to the second part of the paper which concerns the construction of Cartan matrices of low row rank. We can show that if we have an l × l Cartan matrix of row rank n, then we can model an axiomatic description of it with a cone of dimension n + 1.

Let G be a finite group and let K and L be normal subgroups of G such that |K : L| and |G : K| are relatively prime, and assume that |K : L| is odd. Let H be a subgroup of G such that G = HK and H ∩ K = L. Let φ be an irreducible character of L that is invariant under the action of L and is fully ramified with respect to K/L. If χ ∈ Irr(G) is a constituent of φG, then we prove that χH has a unique irreducible constituent having odd multiplicity.

An explicit description of a hyperbolic canonical coordinate system for an expansive automorphism of a compact connected abelian group is given. These dynamical systems are factors of subshifts of finite type. Some properties of the associated crossed product C*-algebra are discussed. In these examples, the C* -algebras of Ruelle are crossed product algebras.

The concept of a mitotic group was introduced in [3] by Baumslag, Dyer and Heller who showed that mitotic groups were acyclic. In [8] one of the authors introduced the concept of a pseudo-mitotic group, a concept weaker than that of a mitotic group, and showed that pseudo-mitotic groups were acyclic and that the group Gnof homeomorphisms of Rn with compact support is pseudo-mitotic. In our present paper we develop techniques to prove pseudomitoticity of certain other homeomorphism groups. In [5] Kan and Thurston observed that the group of set theoretic bijections of Q with bounded support is acyclic. A natural question is to decide whether the group of homeomorphisms of Q (resp. the irrationals I ) with bounded support is acyclic or not. In the present paper we develop techniques to answer this question in the affirmative.

Let G be a complete normed abelian group with norm N1. Let S be an interval (bounded or otherwise) of real numbers. We propose to study the Stieltjes integral equation

where p is in G, a is in S, and each Fk is a function on S each value of which is a function from G to G. Our primary tools of investigation will be the works of J. S. MacNerney [6; 7] and their extensions by the author [4; 5]. Our main result, Theorem 4, will show that the equation above can be solved by a product integral of infinite products of solutions for the summands of the integrator. After obtaining our results, we shall specialize them to a linear situation and then show how this specialization allows us to obtain representations for analytic functions having only invertible values in a complex Banach algebra with identity.

Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.

In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the type

with r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

The classVk(p). We generalize the class Vk of analytic functions of bounded boundary rotation [8] by allowing critical points in the unit disc U.

Definition. Let f(z) = aqzq + . . . (q 1) be analytic in U. Then f(z) belongs to the class Vk(p) if for r sufficiently close to 1,

and

We note that (1.1) implies that / has precisely p — 1 critical points in U.

Berg's interchange technique is generalized to the context of certain new objects called pseudo-actions. This is used to find a more geometric proof of the Pimsner-Voiculescu theorem on the AF embedding of the irrational rotation algebras. Connections with Berg's original results are briefly examined.

Embedding diagrams are introduced to provide a uniform way of describing embeddings of transformation group C*-algebras C(X) ⋊ ℤ into AF algebras. Pimsner has classified the transformation group C* -algebras which can be AF embedded. We present a new proof of this result using embedding diagrams and pseudo-actions. The need to calculate the join of an open cover with its iterates under the transformation has been eliminated.

Let G′ be a quasi–split reductive group over a local field F, ƒ′ the characteristic function of a maximal compact subgroup K′ of G′, N′ a maximal unipotent subgroup of G′. We consider the orbits of maximal dimension for the action of N′ × N′ on G′ and the weighted orbital integral of f′ on such an orbit, the weight being a generic character. The resulting integral, we call a Kloosterman integral. A relative version of this construction is to consider a symmetric space S associated to a quasi-split group G, a maximal unipotent subgroup N of G, a maximal compact K of G and the orbits of maximal dimension for the action of N on S. The weighted orbital integral of the characteristic function f of K ∩ S on such an orbit is what we call a relative Kloosterman integral; the weight is an appropriate character of N. We conjecture that a relative Kloosterman integral is actually a Kloosterman integral for an appropriate group G′. We prove the conjecture in a simple case: E is an unramified quadratic extension of F,G is GL(3, E), S is the set of 3 × 3 matrices s such that the group G′ is then the quasi-split unitary group in three variables.

Blow up a sphere in one of the interstices of a lattice until it is held rigidly. There will be no lattice points in the interior and sufficiently many on the boundary so that their convex hull is a solid figure. Such a sphere was called an empty sphere by B. N. Delone in 1924 when he introduced his method for lattice coverings [3, 4]. The circumscribed polytope is called an L-polytope. Our interest in such matters stems from the following result [6, Theorems 2.1 and 2.3]: With a list of the L-polytopes for lattices of dimension ≦n one can give a geometrical description of the possible sets of integer solutions of

where f satisfies the following condition (in which Z denotes the integers):