The spectral function ρα(μ) (−∞<μ<∞) associated with the Sturm–Liouville equation

and a boundary condition

is a non-decreasing function of μ which is defined in terms of the Titchmarsh–Weyl function mα(λ) for (1.1) and (1.2). Thus, taking into account a standardization of the sign attached to mα(λ), we have

[4, Chapter 9, Theorem 3.1; 9, Section 2.3; 21, Sections 3.3 and 6.7].

We give a characterization for isoperimetric invariants, including the Cheeger constant and the isoperimetric number of a graph. This leads to an isoperimetric inequality for the Cartesian products of graphs.

In this note we give an answer to the following problem of Todorcevic: Find out the combinatorial essence behind the fact that the family ℋ of the ground-model infinite sets of integers in a Perfect-set forcing extension has the property that for any Borel f: [ℕ]ω → {0, 1} there exists an A ∈ ℋ such that f is constant on [A]ω (see [7], [13]). In other words, one needs to capture the combinatorial properties of the family ℋ of ground-model subsets of ℕ which assure that it diagonalizes all Borel partitions. It turns out that the notion which results from our analysis of this problem is a bit more optimal than the older notion of a “happy family” (or selective coideal) introduced by A.R.D. Mathias [16] long ago in order to extend the well-known theorems of Galvin–Prikry [6] and Silver [25] (see Theorems 3.1 and 4.1 below). We should remark that these Mathias-style extensions can indeed be as useful in the applications as the original partition theorems.

Let M be a loopless matroid with rank r and c components. Let P(M, t) be the characteristic polynomial of M. We shall show that (−1)rP(M, t)[ges ](1−t)r for t∈(−∞, 1), that the multiplicity of the zeros of P(M, t) at t=1 is equal to c, and that (−1)r+cP(M, t)[ges ](t−1)r for t∈(1, 32/27]. Using a result of C. Thomassen we deduce that the maximal zero-free intervals for characteristic polynomials of loopless matroids are precisely (−∞, 1) and (1, 32/27].

It was conjectured by Bartels and Welsh [1] that removing an edge from an n-vertex graph G will not increase the average number of colours in the proper colourings of G if k=n colours are available. This paper shows that for all n>3, and for each k∈{3, 4, …, [lfloor](3n−6)/2[rfloor]}, there is a graph on n vertices with an edge whose removal increases the average number of colours in the k-colourings, thus disproving the conjecture.

L'objet de cet article est d'étudier un procédé de summation associé á certaines séries. Notant P(n) le plus grand facteur premier d'un entier générique n, nous rappellons les définitions de P-convergence et de P-régularité d'une série, introduites dans [7].

Deuber's theorem states that, given any m, p, c, r in IN, there exist n, q, μ in IN such that, whenever an (n, q, cμ)-set is r-coloured, there is a monochrome (m, p, c)-set. This theorem has been used in conjunction with the algebraic structure of the Stone–Čech compactification βIN of IN to derive several strengthenings of itself. We present here an algebraic proof of the main results in βIN and derive Deuber's theorem as a consequence.

We study the concept of list total colourings and prove that every multigraph of maximum degree 3 is 5-total-choosable. We also show that the total choice number of a graph of maximum degree 2 is equal to its total chromatic number.

Let kp(G) denote the number of complete subgraphs of order p in the graph G. Bollobás proved that any real linear combination of the form [sum ]apkp(G) attains its maximum on a complete multipartite graph. We show that the same is true for a linear combination of the form [sum ]apkp(G)+bpkp(G¯), provided bp[ges ]0 for every p.

We consider Riemannian orbifolds with Ricci curvature nonnegative outside a compact set and prove that the number of ends is finite. We also show that if that compact set is small then the Riemannian orbifolds have only two ends. A version of splitting theorem for orbifolds also follows as an easy consequence.

The boundary-layer spots involved here come from large-time theory and related computations on the Euler equations to cover the majority of the global properties of the spot disturbances, which are nonlinear, three-dimensional, and transitional rather than turbulent. The amplitude levels investigated are higher than those examined in detail previously and produce a new near-wall momentum contribution in the mean flow, initially close to the wingtips of the spot. This enables the amplitude levels in the analysis to be raised successively, a process which gradually causes the wing-tip region to spread inwards. The process is accompanied by subtle increases in the induced phase variations. Among other things the work finds the details of how nonlinear effects grow from the wing-tips to eventually alter the entire trailing edge, and then the centre of the spot, in a strongly nonlinear fashion. Comparisons with earlier suggestions and with experiments are described at the end.

There is an error in our paper “Bounds on the covering radius of a lattice” published in this journal in 1996 (vol. 43, pp. 159–164). Theorem 1 of the paper should be corrected as follows.

Let f be a complex valued function from the open upper halfplane E of the complex plane. We study the set of all z∈∂E such that there exist two Stoltz angles V1, V2 in E with vertices in z (i.e., Vi is a closed angle with vertex at z and Vi\{z} ⊂ E, i = 1, 2) such that the function f has different cluster sets with respect to these angles at z. E. P. Dolzhenko showed that this set of singular points is G∂σ and σ-porous for every f. He posed the question of whether each G∂σ σ-porous set is a set of such singular points for some f. We answer this question negatively. Namely, we construct a G∂ porous set, which is a set of such singular points for no function f.

Let X be a vertex-transitive graph and let S be an arbitrary finite subset of its vertices. Denote by @∂S the set of vertices adjacent to S but not in S. Babai and Szegedy proved that for an infinite, connected, locally finite X with subexponential growth we have

formula here

where d(S) is the diameter of S. The aim of this note is to provide a slightly better, tight lower bound on this quantity. We prove that

formula here

under the same conditions.

Let m and n be integers with 0<m<n and let μ be a Radon measure on ℝn with compact support. For the Hausdorff dimension, dimH, of sections of measures we have the following equality: for almost all (n − m)-dimensional linear subspaces V

provided that dimH μ > m. Here μv,a is the sliced measure and V⊥ is the orthogonal complement of V. If the (m + d)-energy of the measure μ is finite for some d>0, then for almost all (n − m)-dimensional linear subspaces V we have

Let K be an arbitrary compact space and C(K) the space of continuous functions on K endowed with its natural supremum norm. We show that for any subset B of the unit sphere of C(K)* on which every function of C(K) attains its norm, a bounded subset A of C(K) is weakly compact if, and only if, it is compact for the topology tp(B) of pointwise convergence on B. It is also shown that this result can be extended to a large class of Banach spaces, which contains, for instance, all uniform algebras. Moreover we prove that the space (C(K), tp(B)) is an angelic space in the sense of D. H. Fremlin.

In the present paper we show that a non-zero, continuous, locally finite Borel measure on R cannot be b-concentrated for any 1 <b≤2.

We study three quantities that can each be viewed as the time needed for a finite irreducible Markov chain to ‘forget’ where it started. One of these is the mixing time, the minimum mean length of a stopping rule that yields the stationary distribution from the worst starting state. A second is the forget time, the minimum mean length of any stopping rule that yields the same distribution from any starting state. The third is the reset time, the minimum expected time between independent samples from the stationary distribution.

Our main results state that the mixing time of a chain is equal to the mixing time of the time-reversed chain, while the forget time of a chain is equal to the reset time of the reverse chain. In particular, the forget time and the reset time of a time-reversible chain are equal. Moreover, the mixing time lies between absolute constant multiples of the sum of the forget time and the reset time.

We also derive an explicit formula for the forget time, in terms of the 'access times' introduced in [11]. This enables us to relate the forget and reset times to other mixing measures of the chain.