Buckminster Fuller has coined the name tetrahelix for a column of regular tetrahedra, each sharing two faces with neighbours, one 'below' and one 'above' [A. H. Boerdijk, Philips Research Reports 7 (1952), p. 309]. Such a column could well be employed in architecture, because it is both strong and attractive. The (n — 1)-dimensional analogue is based on a skew polygon such that every n consecutive vertices belong to a regular simplex. The generalized twist which shifts this polygon one step along itself is found to have the characteristic equation

(λ - 1)2{(n - 1)λn-2 + 2(n - 2)λn-3 + 3(n - 3)λn-4 + . . . + (n - 2)2λ + (n - 1)} = 0,

which can be derived from tan nθ = n tan θ by setting λ = exp (2θi).

For suitable polynomials f(x) ∊ ℤ[x] in n variables, of total degree d, it is shown that

This is, formally, a precise analogue of a theorem of Deligne [1] on exponential sums (mod p). However the proof uses no more than elementary algebraic geometry.

The projective special linear group PSL(V) is generated by dations. Among all factorizations of p ∈ PSL(V) into dations there will be one (or more) with the least number of factors. We determine this number, i.e. we solve the length problem for the projective special linear group. We solve a similar problem for the projective unimodular group which is generated by harmonic homologies. The projective special linear group and the projective unimodular group are the most important special cases of projective hyperreflection groups. We also solve the length problem for the general case.

Let T be a linear transformation acting on the space of n x n complex matrices. Let G(k) be the set of all hermitian matrices with k positive and n — k negative eigenvalues. Let T map some indefinite inertia class G(k) onto itself. We classify all such T. The possibilities are congruence, congruence followed by transposition, and, if n = 2k, it is possible that — T can be a congruence or a congruence followed by transposing. In other words, negation is an admissible transformation when n = 2k.

The exact form for the gamma factor for the L-function corresponding to the m-th symmetric power of a cuspidal automorphic representation of PGL(2) is given. This information is used to obtain, via a theorem of Landau, bounds for the eigenvalues of Hecke operators.

Regular C-fractions f(α) = 1 + a1α/1 + a2α/1 + . .. with an = an2 + bn + c + Vn, |Vn| sufficiently small are examined. In the case Vn = 0, exact expressions are obtained which reveal a two sheeted Riemann structure for f(α). If Vn ≠ 0 analytic properties are obtained by means of perturbation theory applied to the associated difference equation. A conjecture that f(α) is the ratio of two entire functions of for an even larger class of C-fractions is proved for the case .

The Dedekind zeta function of an algebraic number field E determines the rational equivalence class of the trace form of E. The Hasse symbols of the trace form are related to the local Artin root numbers of the zeta function by formulas of Serre and Deligne. This is used to settle the question of which families of complex numbers appear as the local Artin root numbers of a continuous real representation of the absolute Galois group of ℚ.

If χ is a Dirichlet character to a prime-power modulus pα, then the problem of estimating an incomplete character sum of the form ∑1≤x≤h χ (x) by the method of D. A. Burgess leads to a consideration of congruences of the type

f(x)g'(x) - f'(x)g(x) ≡ 0(pα),

where fg(x) ≢ 0(p) and f, g are monic polynomials of equal degree with coefficients in Ζ. Here, a characterization of the solution-set for cubics is given in terms of explicit arithmetic progressions.

This paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

The paper investigates the relationship between Perron - Stieltjes integrability and Lebesgue-Stieltjes integrability within the generalized Riemann approach. The main result states that with certain restrictions a Perron-Stieltjes integrable function is locally Lebesgue-Stieltjes integrable on an open dense set. This is then applied to show that a nonnegative Perron-Stieltjes integrable function is Lebesgue-Stieltjes integrable. Finally, measure theory is invoked to remove the restrictions in the main result.

Let P be a translation plane containing a net N which is replaceable by . Let P' denote the transposed plane. We note that N' is replaceable by ()'. This result shows how to relate the various constructions of the two translation planes of order 16 that admit PSL(2, 7).

In this note, we construct a model with a normal measure U over a measurable cardinal κ so that the cardinal structures of V and Vκ/U are the same ≤2κ. We then show that it is possible to construct a model where this is not true.

The homogeneous Besov-Lipschitz spaces, usually defined by difference operators or Fourier transform, are studied in terms of mean oscillation, and several equivalent characterisations are given.

If n is a natural number, let A(n) denote the number of integers, k, such that 0 < k < n and n divides . Let ϕ(n) denote Euler's totient function. Necessary and sufficient conditions are given so that A(n) = ϕ(n) when n is square-free.

Finiteness criteria are established for the negative spectra of higher order elliptic operators on Rn. The results are obtained by establishing isomorphism theorems for elliptic operators and applying the abstract finiteness criterion of Konno-Kuroda.

An example of the type described in the title is given.

We first note that each graph (x,A(x)) of a function A(x) with BMO derivative is a chord-arc curve. Using this, Muckenhoupt's Ap theory, and the theory of Calderón-Zygmund operators, we shall derive weighted norm inequalities for the Cauchy integral on such graphs from a recent theorem of G. David on the L2-boundedness of Cauchy integral on almost-lipschitzian curves.

Using the method of non-well-founded generic ultra-powers, we shall prove a generalization of a theorem of Taylor that certain values of completeness and saturatedness of a uniform ideal rule out certain sizes of the underlying index set.

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