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.