We provide a framework for studying randomly coloured point sets in a locally compact second-countable space on which a metrizable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical system for uniformly discrete uncoloured point sets. For point sets of finite local complexity, we characterize ergodicity geometrically in terms of pattern frequencies. The general framework allows us to incorporate a random colouring of the point sets. We derive an ergodic theorem for randomly coloured point sets with finite-range dependencies. Special attention is paid to the exclusion of exceptional instances for uniquely ergodic systems. The setup allows for a straightforward application to randomly coloured graphs

Differentiability properties of optimal value functions associated with perturbed optimization problems require strong assumptions. We consider such a set of assumptions which does not use compactness hypothesis but which involves a kind of coherence property. Moreover, a strict differentiability property is obtained by using techniques of Ekeland and Lebourg and a result of Preiss. Such a strengthening is required in order to obtain genericity results.

Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such that

The smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,

We present a new approach to graph limit theory that unifies and generalizes the two most well-developed directions, namely dense graph limits (even the more general $L^p$ limits) and Benjamini–Schramm limits (even in the stronger local-global setting). We illustrate by examples that this new framework provides a rich limit theory with natural limit objects for graphs of intermediate density. Moreover, it provides a limit theory for bounded operators (called P-operators) of the form $L^\infty (\Omega )\to L^1(\Omega )$ for probability spaces $\Omega $ . We introduce a metric to compare P-operators (for example, finite matrices) even if they act on different spaces. We prove a compactness result, which implies that, in appropriate norms, limits of uniformly bounded P-operators can again be represented by P-operators. We show that limits of operators, representing graphs, are self-adjoint, positivity-preserving P-operators called graphops. Graphons, $L^p$ graphons, and graphings (known from graph limit theory) are special examples of graphops. We describe a new point of view on random matrix theory using our operator limit framework.

Suppose we have n events Ai, … , An and let p(Aj1 … , Ajk) be the probability that the events Ah, . . . , Ajk occur jointly. The probability P0 that none of Ai, . . . , An occur is given by Poincaré's formula, the probabilistic version of the principle of inclusion and exclusion:

Introduction. The present note began from the observation that the arguments produced by J-M. Maranda in developing his very interesting theory of representations of groups by automorphisms of modules over Dedekind rings (4, 5) were applicable without essential change to arbitrary orders, instead of just group rings, provided that a suitable generalization of Theorem 1 of (4) could be supplied.

For two nodes x and y of a graph G, the distance δG(x,y) is the smallest integer k such that k edges form a path from x to y; δG(x, x) = 0, and δG(x,y) = ∞ when x ≠ y and there is no path from x to y. The diameter δG is the maximum of δG(x, y) as x and y range over the nodes of G. When G is connected, δ(G) is the smallest integer k such that any two nodes of G can be joined by a path formed from at most k edges. When G is not connected, δ(G) = ∞ and there is interest in δc(G), the maximum of δ(G) over the components C of G.

For 2 ≧ n < ∞ and 0 ≧ E ≧ n(n − l)/2, let denote the set of all loopless undirected graphs with the node-set {1, …, n} and exactly E edges.

The definition and some of the properties of what may be called a Perron second integral (P2-integral) were given in a previous paper [4]. This integral starts with a function f(x) defined in an interval (a, c) and goes directly to a second primitive F(x) with the property that the generalized second derivative D2F is equal to f(x) for almost all x in (a, c). In the present paper the definition is changed slightly and further properties are deduced.

We present a series of conjectures for immanants, together with the supporting evidence we possess for them. The conjectures are loosely organized into three families. The first concerns inequalities involving the immanants of totally positive matrices (Le.,real matrices with nonnegative minors). This includes, for example, the conjecture that immanants of totally positive matrices are nonnegative. The second family involves the immanants of Jacobi-Trudi matrices. These conjectures were suggested by a previous conjecture of Goulden and Jackson (recently proved by C. Greene) that the immanants of Jacobi-Trudi matrices are polynomials with nonnegative coefficients. The third family involves geometric and combinatorial structures associated with total positivity and paths in acyclic digraphs.

Let E be a Borel set of RN. The α-outer Hausdorff measure of E has been defined to be

where

and each Bi is a closed ball. d(Bi) denotes the diameter of Bi.

It is easily seen that the same value Hα(E) is obtained if we consider coverings of E by open balls or by balls which may be either open or closed.

By dim(E) we will mean the usual Hausdorff-Besicovitch dimension of E, where

Following the work of Kelly (8), Harary and Palmer (5), and Bondy (1) on the reconstruction of trees, and of Manvel (10) on the reconstruction of connected graphs with a single cycle, it was a natural step to attempt to solve the reconstruction problem for cacti. The solution of this problem, presented here, assumes both Kelly's Theorem and the result of Manvel in (10). Any definitions not given here can be found in (2).

Let graph G have point set V = {v1 v2, …, vp} and line set X = {x1, x2, …, xq}. For each vi ∈ V, Gi = G – vi is the maximal subgraph of G which does not contain vi and is formed by deleting vi and all lines incident with it from G.

In this paper, we shall be concerned with bounded, holomorphic functions of the form

where

(1)

(2)

and

(3)

B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).

According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

Let n be a positive integer. An f-ring A is said to satisfy the left nth-convexity property if for any u, v ∊ A such that v ≧ 0 and 0 ≧ u ≧ vn, there exists a w ∊ A such that u = wv. The right nth-convexity property is defined similarly and an f-ring is said to satisfy the nth-convexity property if it satisfies both the left and the right nth-convexity property. In this paper, we study arbitrary f-rings which satisfy one of the convexity properties.

Those f-rings which satisfy one or more of these properties have been studied by several authors. In [3, 1D], L. Gillman and M. Jerison note that any C(X) satisfies the nth-convexity property for all n ≧ 2, and in [3, 14.25], they give several properties that in C(X) are equivalent to the 1st-convexity property. M. Henriksen proves some results about the ideal theory of an f-ring satisfying the 2nd-convexity property in [5] and S. Steinberg studies left quotient rings of f-rings satisfying the left 1st-convexity property in [13].

The theory of minimal models, as developed by Sullivan [6; 8; 16] gives a method of computing the rational homotopy groups of a space X (that is, the homotopy groups of X tensored with the additive group of rationals Q). One associates to X a free, differential, graded-commutative lgebra , over Q, called the minimal model of X, from which one can read off the rational homotopy groups of X.

Our aim in this paper is to give a characterization of the order types of the countable initial segments of many-one degrees (m-degrees). The basic definitions and background information can be found in [2] from where we draw most of our notation and terminology. We expand the usual notion of m-reducibility by adopting the convention that R ≦m ∅ and R ≦mN for every recursive set R. This has the effect of giving all recursive sets the same m-degree; that m-degree will be denoted by 0. We shall denote by ≦ the partial ordering of m-degrees induced by ≦m, and shall denote by a ∪ b the least upper bound of the m-degrees a, b. We call a ∪ b the union of a and b.

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].

All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.

In this paper we describe reducibility of non-unitary generalized principal series for classical $p$-adic groups in terms of the classification of discrete series due to Mœglin and Tadić.

A famous theorem of Levitzki states that in a left Noetherian ring each nil left ideal is nilpotent. Lanski [5] has extended Levitzki's theorem by proving that in a left Goldie ring each nil subring is nilpotent. Another important theorem in this area which is due to Herstein and Small [3] states that if a ring satisfies the ascending chain condition on both left and right annihilators, then each nil subring is nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields both Lanski's theorem and Herstein- Small's theorem. We make use of the ascending chain condition on principal left annihilators in order to obtain, at an intermediate step, a theorem (Theorem 1.1) which produces sufficient conditions for a nil subring to be left T-nilpotent.

A combinatorial n-sphere is a simplicial n-complex whose body (i.e., the union of its members) is homeomorphic to the topological n-sphere Sn. A combinatorial n-manifold is a simplicial n-complex M such that M is connected, and for every vertex x in M the complex linker, M), the link of x in M, is a combinatorial (n — 1)-sphere. For more details the reader should consult Alexander [1] and Grünbaum [16]. All the spheres and manifolds to which we refer are combinatorial.

A fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fractiondense ƒ- rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fractiondense spaces are defined as those for which C(X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover. R-embeddings of Tychonoff spaces are re-introduced and examined in the context of fraction-density.