We introduce a Morse theory for posets of Bestvina–Brady type combining matchings and height functions. This theory generalizes Forman's discrete Morse theory for regular CW-complexes and extends previous results on Morse theory for $h$-regular posets to all finite posets. We also develop a relative version of Morse theory which allows us to compare the topology of a poset with that of a given subposet.

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb{Z}$ with finitely generated kernels. The proof uses Bestvina–Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb{H}^{4}$ with fundamental domain the 120-cell or the 24-cell.

In this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:

$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill & {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill & {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$

$M:{\open R}_0^ + \to {\open R}^ + $

A compact set K ⊂ ℝn is Whitney 1-regular if the geodesic distance in K is equivalent to the Euclidean distance. Let P be the Chevalley map defined by an integrity basis of the algebra of polynomials invariant by a reflection group. This paper gives the Whitney 1-regularity of the image by P of any closed ball centred at the origin. The proof uses the works of Givental', Kostov and Arnol'd on the symmetric group. It needs a generalization of a property of the Vandermonde determinants to the Jacobian of the Chevalley mappings.

Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyse the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kähler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.

In Part I we describe the class of models under study and present our results in the case of one-dimensional (quantum mechanical) models, which is quite representative and at the same time simple enough to analyse explicitly. Part II will be devoted to supersymmetric two-dimensional sigma models and four-dimensional Yang–Mills theory. In Part III we will discuss non-supersymmetric models.

We establish the existence of two nontrivial solution for some elliptic systems. In the proofs we apply variational methods and Morse theory.

Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.