A well-known theorem of Minkowski on the product of two linear forms states that if

are two linear forms with real coefficients and determinant Δ = |αδ − βγ| ≠ 0, then given any real numbers c1, c2 we can find integers x, y such that

In 1914 Carathéodory defined m–dimensional measure in n–dimensional space. He considered one-dimensional measure as a generalization of length and he proved that the length of a rectifiable curve coincides with its one-dimensional measure.

The purpose of this note is to calculate the specific heat and paramagnetic susceptibility of an electron gas obeying the Fermi-Dirac statistics for all temperatures, including those temperatures for which the gas is partially degenerate. The results are applicable to the electrons in a metal, whether free or moving in a periodic field, provided only that the number of electronic states per gram atom with energy between E and E + dE can be expressed in the form

as for free electrons.

In this paper we calculate part of the integral cohomology ring of the sporadic simple group J4; this group has order 221.33.5.7. 113.23.29.31.37.43. More precisely, we obtain all of the cohomology ring except for the 2-primary part. As the cohomology has already been written down [9] at the primes which divide the group order only once, we concentrate here on the primes 3 and 11. In both of these cases the Sylow p-subgroups are extraspecial of order p3 and exponent p. We use the method which identifies the p-primary cohomology with the ring of stable classes in the cohomology of a Sylow p-subgroup. The stable classes are all invariant under the action of the Sylow p-normalizer; and some time is spent finding invariant classes in the cohomology ring of , the extraspecial group. Section 2 studies the prime 11: the invariant classes are the stable classes, because the Sylow 11-subgroups have the Trivial Intersection (T.I.) property. In Section 3 we study the prime 3, and see that all conditions for invariant classes to be stable reduce to one condition.

This paper continues our study started in [5] of the group G defined by the presentation

formula here

where m [ges ] 2, n [ges ] 2 and ∈i = ±1 for 1 [les ] i [les ] 2. The problem under consideration can be stated as: precisely when does both a have order m and b have order n in the group G? This question arose as a result of our investigation into equations over groups and the connection is explained in Section 2.

This note describes measurements of the velocity of sound in liquid nitrogen by a method which has previously been used in the case of liquid oxygen(1). The method depends on the diffraction of light by suprasonic waves.

Flips occur in the theory of minimal models of algebraic varieties. For an introduction and references see [1, lecture no. 5]. For varieties X− and X+, I denote the canonical class by K− and respectively. A flip is a diagram X−→X←X+ of normal complex quasiprojective 3-folds satisfying the conditions:

1. both morphisms are birational and projective, contracting only finitely many curves C±⊂X± to an isolated singular point P∈X;

2. the divisors −K− and K+ are relatively ample, that is, −K−Γ>0 for any curve Γ contracted by the morphism X−→X and similarly for K+;

3. the two varieties X− and X+ have only terminal singularities.

A diagram satisfying condition 1 is called a flip diagram. It is said to be directed by the canonical class if it also satisfies condition 2. Notice that condition 3 is overstated since under all the other conditions X+ will automatically have terminal singularities (see [4, 5-1-11(2)]).

In this paper, we study involutions of Godeaux surfaces whose bicanonical system has no base components. We classify all possible fixed loci and find corresponding examples to the classification.

Motivated by the work of Farber, Tabachnikov and Yuzvinsky on the motion planning problem for projective spaces, we give an estimate for the topological complexity (TC) of lens spaces in terms of certain generalized “skew” maps between spheres. This last concept turns out to be closely related to that for a generalized axial map developed by Astey, Davis and the author to characterize the smallest Euclidean dimension where (2-torsion) lens spaces can be immersed. As a result, this suggests an alternative simpler “TC-approach” to the classical immersion problem for real projective spaces, whose initial stages we settle by means of techniques in obstruction theory.

In Problem 3·39 (B) and (C) of Kirby's collection [10], Giffen and Thurston asked whether, for a closed 3-manifold M, the order of finite subgroups of Diff M is bounded, so that it contains no infinite torsion subgroups unless M admits a circle action. In this paper, we answer this question affirmatively for homotopy geometric manifolds, and then discuss some hyperbolic 3-manifolds with only a few actions as examples showing poor symmetry of 3-manifolds in general.

The pressure fluctuations in a compressible fluid in statistically steady isotropic motion are studied. These depend crucially on the scale of the turbulent region. If the region is small enough, the mean square acoustic pressure varies as the linear scale of the region, and as the fourth power of the turbulence Mach number. In the limit of infinite scale, however, diffusive effects limit the otherwise infinite pressure fluctuations as first shown by Lighthill(6). The mean square acoustic pressure then varies as the Mach number and as the turbulence Reynolds number. The pressure fluctuations above a sheet composed of statistically uniformly fluctuating sources are also examined. Here the mean square pressure diverges, initially, in proportion to the logarithm of the scale of the sheet, until viscous effects again become significant in limiting the pressure.

1. Changes in the relative intensities of the lines in the fluorescent L-spectrum of Cerium excited by radiation of various wave-lengths have been observed.

2. These results imply a change in the relative absorbing powers of the three L-levels as the wave-length of the absorbed radiation diminishes from a value just below the absorption wave-length of the L-levels to a value considerably below. The absorbing power of the LI-level becomes increased relative to the absorbing powers of the other L-levels as the wave-length diminishes. The results agree with those published by H. Robinson in a recent paper.

3. These results imply a breakdown of the law that μ/λ3 is a constant (where μ is the absorption coefficient of X-rays of wave-length λ) as applied to the individual L-levels of an element.

4. A comparison is made between the above results, and some results on the relative absorbing process of the L-levels obtained by Ellis and Skinner from β-ray spectra.

We improve the best known sum-product estimates over the reals. We prove that

\[\max(|A+A|,|A+A|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,,\]

$A\subset \mathbb {R}$

Furthermore,

\[|AA+AA|\geq |A|^{\frac{127}{80} - o(1)}\,.\]

\[|A+A|\geq |A|^{\frac{30}{19}-o(1)}\,.\]

The division of ideal theory into the additive and multiplicative branches of the subject is a marked one of long standing. It is the object of this paper to carry the distinction a stage further, by showing how the group-theoretic methods of the additive branch can be extended to modules, that is, to Abelian groups, provided the ring of operators is commutative.

In a series of recent papers Verma and Upadhyay (7,8,9) developed the theory of basic hypergeometric series with two bases q and q½. These investigations were made in an attempt to discover a summation formula for a bilateral basic hypergeometric series 2Ψ2 analogous to that for a 2H2 (cf. Bailey (2,3)) and in finding relations between certain q-infinite products. In one of their papers they mentioned that it did not seem possible to develop the corresponding general theory for basic series with two unconnected bases q and q1. A recent paper by Andrews (1) indicates that transformations between basic hypergeometric series with two unconnected bases can be very interesting and useful in the study of ‘mock’ theta functions and their extensions. Besides this interest, such a theory also enables one to extend the entire existing transformation theory of the generalized basic hypergeometric series.

Let (R, m) be a 2-dimensional regular local ring and I an m-primary ideal. The aim of this paper is to find conditions on I so that the associated graded ring of I,

and the Rees ring of I,

where t is an indeterminate, are Cohen–Macaulay (resp. Gorenstein). To this end, we use the results and techniques from Zariski's theory of complete ideals ([14], appendix 5) and its later generalizations and refinements due to Huneke [7] and Lipman[8]. The main result is an application of three deep theorems: (i) a generalization of Macaulay's classical theorem on Hilbert series of Gorenstein graded rings [13], (ii) a generalization of the Briançon–Skoda theorem due to Lipman and Sathaye [9], and (iii) a formula for the length of R/I, where I is a complete m-primary ideal, due to Hoskin[4] and Deligne[1].

We study local singularities of ruled surfaces in 3. We show that any map germ f : (2, 0) → (3, 0) with a simple singularity is -equivalent to a ruled surface. Moreover, we give a topological classification of -finitely determined singularities of ruled surfaces and show that there are just eleven topological classes.

This paper relates the concept of n-connection for graphs to Tutte's theory of n-connection for matroids (12). In particular, we show how Tutte's definition may be modified to give a matroid concept directly generalizing the graph-theoretic notion of n-connection.

1. The half-life of Th C′ was determined by a new method of coincidence technique.

2. Previous investigators have recognized difficulties in regard to this investigation. These were: (a) random delays in the counters; (b) coincidences due to other disintegrations; (c) possibility of difference in delays in recording α- and β-particles. In this investigation the difficulties arising from these effects have been overcome.

3. From this type of experiment it is possible to determine the order in which the particles are emitted.

4. It would appear that the time of breakdown of a counter depends slightly upon the magnitude of the initialionization.

5. The value for the half-life of Th C′ was found to be 3.0 ± 0.15 × 10−7 sec.

Let (Xt: t ≥ 0) be the Markov branching process (MBP) with a density independent catastrophe component. It is denned to be the Feller process on the non-negative integers having the generator

Here {pj} is the offspring distribution which satisfies p1 = 0 and p0 < 1, ρ is the per capita birth rate, κ is the rate of occurrence of catastrophe events, {δj: j ≥ 0} is the decrement distribution and . Thus Xt can be interpreted as the size of a population in which individuals reproduce according to the rules of a MBP – see Athreya and Ney[1], chap, III – and where there is an external and independent Poisson process of catastrophe events, κ per unit time, and if j < i each such event reduces the population size by j with probability δj. Usually we assume that δ0 = 0 on the basis that a catastrophe always reduces the population size. Let f(s) = Σpjsj and assume that . This ensures that the MBP obtained by setting κ = 0 is regular ([1], p. 105) and hence (Xt) is the unique Markov process corresponding to the above generator when κ > 0.