We show that for every smooth curve in Rn, there is a quadrilateral with equal sides and equal diagonals whose vertices lie on the curve. In the case of a smooth plane curve, this implies that the curve admits an inscribed square, strengthening a theorem of Schnirelmann and Guggenheimer. “Smooth” means having a continuously turning tangent. We give a weaker condition which is still sufficient for the existence of an inscribed square in a plane curve, and which is satisfied if the curve is convex, if it is a polygon, or (with certain restrictions) if it is piecewise of class C1. For other curves, the question remains open.

§1. Introduction. In this paper, as in [4], we are concerned with integer points (m, n) lying close to the curve

in the sense that

where ║t║ denotes the distance of the real number t from the nearest integer. We shall always suppose that

and that F(x) is at least twice continuously differentiable. Let R be the number of solutions of (1.1) with m an integer, O ≤ m ≤ L. The obvious method of estimating R uses the row-of-teeth or rounding error function

with

Oler's lattice-point theorem gives a sharp upper bound for the lattice-point enumerator GΛ of a certain class of lattices in the plane. We give a sharp lower bound for GΛ of the corresponding class of lattices in all dimensions. This result is closely related to the Blichfeldt-van der Corputgeneralization of Minkowski's fundamental lattice-point theorem.

Introduction. Polyhedra in 3-dimensional hyperbolic space which give rise to discrete groups generated by reflections in their faces have been investigated in [14], [17], [29] and in the case of tetrahedra there are precisely nine compact non-congruent ones with dihedral angles integral submultiples of π [14]. These polyhedral groups give rise to hyperbolic 3-orbifolds and examples of these have been studied, for example, in [3], [15], [18], [24], [25].

The usual definition for vertex-criticality with respect to the chromatic index is that a multigraph G is vertex-critical if G is Class 2, connected, and χ'(G\υ) <χ'(G) for all υ ε V(G). We consider here an allied notion, that of vertex-criticality with respect to the chromatic class–in this case G is vertex critical if G is Class 2 and connected, but G\υ is Class 1 for all υ ε V(G). We also investigate the analogues of these two notions for edge-criticality.

§1. Introduction. Let two probability spaces (X, , μ,) and (Y, ℬ, ν) be given. For a subset D of X × Y and a real number d ≥ 0 we consider the following problem

(MP) Does there exist a measure » on X × Y having μ and ν as marginals and such that λ (D) ≥ 1 − d?

This problem comes from Strassen's paper [12], where Borel probabilities on Polish spaces were treated. Further, it was investigated by many authors in more general settings (cf. [2], [4]-[7], [11]-[13]).

§1. Introduction. In [1], Odoni discusses the iterates of the polynomial x2 +1 and their Galois groups over the rationals (a problem initially proposed by J. McKay). Setting f1,(x) = x2+1 and fn(x) =f1(fn-1(X)) for n ≥ 2, write Kn for the splitting field of fn(x) over and Ωn = Gal (Kn/). Then Odoni proves that Ωn is isomorphic to a subgroup of [C2]n, the nth wreath power of the cyclic group C2 of order 2, and gives a simple rational criterion for Ωn [C2]n to hold. In this note we describe a computer implementation of Odoni's criterion, and state the result that Ωn [C2]n for n ≤ 5 × 107.

The nonlinear interactions that evolve between a planar or nearly planar Tollmien-Schlichting (TS) wave and the associated longitudinal vortices are considered theoretically, for a boundary layer at high Reynolds numbers. The vortex flow is either induced by the TS nonlinear forcing or is input upstream, and similarly for the nonlinear wave development. Three major kinds of nonlinear spatial evolution, Types I-III, are found. Each can start from secondary instability and then becomes nonlinear, Type I proving to be relatively benign but able to act as a pre-cursor to the Types II, III which turn out to be very powerful nonlinear interactions. Type II involves faster streamwise dependence and leads to a finite-distance blow-up in the amplitudes, which then triggers the full nonlinear three-dimensional triple-deck response, thus entirely altering the mean-flow profile locally. In contrast, Type III involves slower streamwise dependence but a faster spanwise response, with a small TS amplitude thereby causing an enhanced vortex effect which, again, is substantial enough to entirely alter the mean-flow profile, on a more global scale. Concentrated spanwise formations in which there is localized focussing of streamwise vorticity and/or wave amplitude can appear, and certain of the nonlinear features also suggest by-pass processes for transition and significant change in the flow structure downstream. The powerful nonlinear 3D interactions II, III seem potentially very relevant to experimental and computational findings in fully fledged transition; in particular, it is suggested in an appendix that the Type-Ill interaction can terminate in a form of 3D boundary-layer separation which appears possibly connected with the formation of lambda vortices in practice.

§1. Introduction. In this note we shall discuss a certain dichotomy concerning the pointwise convergence of sequences of analytic sets in completely metrizable separable spaces (Proposition 2.1). The dichotomy is closely related to some reasoning due to W. Szlenk [Sz]; we comment on this in Section 5.1.

Let X be a completely regular Hausdorff topological space and let C(X) (the set of all real-valued bounded and continuous in X functions) be endowed with the sup-norm. Let ßX, as usual, denotes the Stone-Čech compactification of X. We give a characterization of those X for which the set

contains a dense -subset of C(X). These are just the spaces X which contain a dense Čech complete subspace. We call such spaces almost Čech complete. We also prove that X contains a dense completely metrizable subspace, if, and only if, C(X) contains a dense -subset of functions which determine Tykhonov well-posed optimization problems over X. For a compact Hausdorff topological space X the latter result was proved by Čoban and Kenderov [CK1.CK2]. Relations between the well-posedness and Gâteaux and Fréchet differentiability of convex functionals in C(X) are investigated. In particular it is shown that the sup-norm in C(X) is Frechet differentiable at the points of a dense -subset of C(X), if, and only if, the set of isolated points of X is dense in X. Conditions and examples are given when the set of points of Gateaux differentiability of the sup-norm in C(X) is a dense and Baire subspace of C(X) but does not contain a dense -subset of C(X).

In this note, we investigate those Hausdorff measures which obey a simple scaling law. Consider a continuous increasing function θ defined on with θ(0)= 0 and let be the corresponding Hausdorff measure. We say that obeys an order α scaling law provided whenever K⊂ and c> 0, then

or, equivalently, if T is a similarity map of with similarity ratio c:

Introduction. This paper describes a natural way to associate fractal setsto a certain class of absolutely convergent series in In Theorem 1 we give sufficient conditions for such series. Theorem 2 shows that each analytic function gives a different fractal series for each number in a certain open set. Theorem 3 gives the Hausdorff dimension of the associated sets to fractal series, under suitable conditions on the series. This theorem can be applied to some standard series in analysis, such as the binomial, exponential and trigonometrical complex series. The associated sets to geometrical complex series are selfsimilar sets previously studied by M. F. Barnsley from a different (dynamical) point of view (see refs. [5], [6]).

We consider a metric space (X, ρ) of a certain class studied by H. Federer in “Geometric Measure Theory”. Let Ф be any derivation basis on X, which is formed by open balls and is ρ-fine. We show that Ф allows mutual derivation of two arbitrary Borel regular measures on X, which are σ-finite and finite-valued on bounded sets. The proof is based on the so-called De Giorgi property studied in a previous paper.