Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.

Let K be a compact subset of the real Euclidean space En. We say that K has constant width if the distance between each pair of distinct parallel hyperplanes which support K is constant. The collection of all compact convex subsets of En which have constant width is denoted .

We write ω for the set of natural numbers (including zero) and for the set of all finite sequences of natural numbers. If n ∊ ω we write . If x is a function which takes its values in cu and whose domain of definition contains then we write for the element (x(0), …, x(n)) of .

In certain boundary value problems associated with two-parameter ordinary differential equations defined and having 2p, (p>1), turning points in a given interval, there arises certain equations in integers whose solutions determine the coefficients in the asymptotic expansions for the eigenvalues [1,2, 3, pp. 134-139].

The following theorem is well known [1, p. 66]: Suppose that, in a ballot, candidate P scores p votes and candidate Q scores q votes, where p > q. The probability that throughout the counting there are always more votes for P than for Q, equals (p-q)/(p+q).

In this note we present a simple alternative proof for the Bernstein problem in the three dimensional Heisenberg group $\text{Ni}{{\text{l}}_{3}}$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential.

The interesting question of the title was posed by J. Bergen and this note answers it in the negative. The main result characterizes rings which can be a finite union of proper right annihilators, and shows that any such commutative ring must have a total annihilator.

This is a continuation of the author's article [3], and it contains further problems connected with the theory of convex sets in En. To the list of general references in [3] may be added the recent book [2] on convex polyhedra.

1) Let A and B be two convex bodies in E2 and let a packing P = {B1, B2, …} be an infinite sequence of homothetic images of B sucn that:

1. a) each Bn is a subset of A,

2. b) no two of them share interior points,

3. c) Area (A) =

The existence of such packings is guaranteed by Vitali's Theorem. Let D(X) be the diameter of the set X and put .

Considering a mapping $g$ holomorphic on a neighbourhood of a rationally convex set $K\subset {{\mathbb{C}}^{n}}$, and range into the complex projective space $\mathbb{C}{{\mathbb{P}}^{m}}$, the main objective of this paper is to show that we can uniformly approximate $g$ on $K$ by rational mappings defined from ${{\mathbb{C}}^{n}}$ into $\mathbb{C}{{\mathbb{P}}^{m}}$. We only need to ask that the second Čech cohomology group ${{\overset{\scriptscriptstyle\smile}{H}}^{2}}\left( K,\mathbb{Z} \right)$ vanishes.

We give a criterion in order that an affine variety defined over any field has a complete intersection (ci.) embedding into some affine space. Moreover we give an example of a smooth real curve C all of whose embeddings into affine spaces are c.i.; nevertheless it has an embedding into ℝ3 which cannot be realized as a c.i. by polynomials.

This paper is devoted to obtaining sequence space representations of spaces of vector-valued Ck-functions defined on an open subset, Ω, of ℝn, whose kth derivatives satisfy a Lipschitz condition on compact subsets of Ω.

In a variety generated by a finite non-Abelian group, the commutator subgroup is not definable by a first-order formula.

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

In [6] B. Sz.-Nagy has proved that every operator on a Hilbert space such that

1

is similar to a unitary operator.

The following problem is an extension of this result: If T and S are two operators such that

1. 1. sup {‖Tn‖, ‖Sn‖}<∞ (n = 0, ±1, ±2,…)

2. 2. TS = ST

then there exists a selfadjoint operator Q such that QTQ-1, QSQ-1 are unitary operators?

Also, in [7] B. Sz.-Nagy has proved that every compact operator T such that

sup ‖Tn‖<∞ (n = 1, 2, 3,…)

is similar to a contraction.

In what follows small latin letters denote rational integers (whole numbers) and we write

Let us consider the system of k equations

1

that is

These equations are obviously satisfied if the b are a permutation of the a; such a solution we call trivial.

Let A, B, denote two non-void finite complexes (= subsets) of the torsion free abelian group G,

Let d(A),… denote the maximum number of linearly independent elements of A,… and let n = n(A, B) denote the number of elements of A + B whose representation in the form a + b is unique. In the preceding paper, Tarwater and Entringer [1] proved that n ≥ d(A).

In this note we study dependence in matroids as an exercise in combinational algebra. Because the work seems to have little connection with graph theory we will not use Tutte′s approach (1) which uses dual concepts. To define a matroid we use Edmunds (2).

We study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.