A sufficient condition is derived for a finite-time $L_2 $ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $ \ \lim_{ t \uparrow T_*} \sup \|\frac{ D \omega} { Dt}\|_{L_2(\Omega)} = \infty $, where $\Omega \subset \mathbb{R}$ moves with the fluid. In particular, $| \omega | $, $| \S_{ij} | $, and $| \P_{ij} | $ all become unbounded at one point $(x_1, T_1) $, $T_1 $ being the first blow-up time in $L_2 $.

Let k be a field, X a set, F = k 〈X〉 the free associative k-algebra, and b an element of F that is homogeneous with respect to the grading of F induced by some map . We show that the idealizer of b in F, S = {f∈F|fb∈bF}, is a free algebra.

Let Bk, Mn, Np be manifolds in the category C = Top, Duff or PL. Define Mn and Np to be h-enclosable in Bk if (1) k = n + p + 1, (2) there are C-imbeddings i:Mn ⊆ Bk and j: Np ⊆ Bk with disjoint images and (3) there are de formation retractions of Bk − i(Mn) onto j(Np) and of Bk − j(NP) onto i(Mn). This is expressed as Bk = [Mn, NP/i, j] (mod C). The manifolds are trivially h-enclosable in Bk if, in addition, each manifold has a product tubular neighbourhood in the category C.

In a Noetherian commutative ring with identity, every ideal can be expressed (not necessarily uniquely) as a finite intersection of primary ideals (called a primary decomposition). This note is concerned with powers of ideals generated by subsets of an R-sequence in a local ring R (i.e. a Noetherian commutative ring R with identity possessing a unique maximal ideal m) and with a decomposition of such ideals.

Let M denote a compact topological 3-manifold, without boundary, foliated by topological circles in the sense of Seifert's gefaserter Räume[8]. This will be called a Seifert structure, or Seifert fibration on M; the leaves of the foliation are called (regular or exceptional) fibres. Our main result is the following theorem, reminiscent of the theorem of Borsuk—Ulam [1] stating that every continuous function from S3 to S2 takes at least one pair of antipodal points to the same value.

Let denote the Fourier series expansion of a function . Féver's celebrated theorem states that the series is summable (C, 1) at each point of continuity of f, where (C, 1) denotes the Cesàro method of summability of order 1. Riesz extended this result to (C, α) for each α > 0. Since then many authors have established sufficient conditions on various methods of summability to guarantee similar results. Over the years the pattern has been to strive for weaker conditions on the matrix, and to replace the condition of continuity on the function by a less stringent one. Theorems have been proved not only for the summability of f, but the summability of the derived series, and other series, related to f. Beginning with the work of Hille and Tamarkin[13], many of the theorems have been extended to absolute summability.

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincaré series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincaré series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency relations. We also give a general formula for the Poincaré series of the arc filtration for isolated surface singularities which are stabilizations of plane curve ones.

Let S(N(t)) be defined by

where {N(t), t ≥ 0} is a Poisson process with intensity parameter 1/μ > 0 and {Xi i ≥ 1} is a family of independent random variables which are also independent of {N(t), t ≥ 0}.

We consider the problem of finding the maximum possible size of a family of d-dimensional subcubes of the n-cube {0, 1}n, none of which is contained in the union of the others. (We call such a family ‘irredundant’). Aharoni and Holzman [1] conjectured that for d > n/2, the answer is (which is attained by the family of all d-subcubes containing a fixed point). We prove this in the case where all the subcubes go through either (0, 0, . . . , 0) or (1, 1, . . . , 1). We also give a new proof of an upper bound of Meshulam [6]. Finally, we consider the case d ≤ n/2. We give a probabilistic construction showing that in this case, Meshulam's upper bound is correct to within a constant factor. When n = 2d, we construct an irredundant family of size .

We show that the dynamical degree of an (i.i.d) random sequence of dominant, rational self-maps on projective space is almost surely constant. We then apply this result to height growth and height counting problems in random orbits.

The source of light used as a background is an important factor in determining the convenience and accuracy of ultra-violet absorption work, etc. If a source of light of constant intensity is available, a direct comparison method can be used and it is only necessary to calibrate the plates. If the source of light is not constant in intensity, it is necessary to divide the light into two beams and use one to check the variations of intensity while the other goes through the absorbing substance or (during calibration) the reducing sector or wedge. This latter method requires much more complicated apparatus and if the variations in the source are at all large it becomes inaccurate. In addition to being constant in intensity a good background for ultra-violet absorption spectra should possess the following qualities:

(1) Most of the energy should be emitted in the form of a continuous spectrum.

(2) It is desirable to be able to use one photograph of the whole region to be investigated. For this purpose it is necessary that the variations of intensity in different parts of the spectrum should be small enough for it to be possible to arrange the exposure so that all parts of the spectrum are within the correct exposure range, i.e. it must not be necessary to over-expose any part in order to get a strong enough intensity at another wave-length.

The hydrogen continuous spectrum possesses both these qualities and is an excellent background for the region on the short wave-length side of 3200 A.U. It may be used for longer wavelengths, but the hydrogen secondary lines are apt to prove trouble-some unless a fairly large dispersion is used.

We determine the rate of growth of the expected number of real zeros of a class of random polynomials whose coefficients form a stationary uniformly mixing sequence.

To the conditions of reaction in homogeneous systems is added yet another condition, the adsorption of the reactants on the centres of activity of the surface. The surface is treated as if it were homogeneous, reaction taking place as if those centres on which the heat of activation is smallest were alone responsible for chemical change.

The general equation is worked out in terms of the rate of bombardment, and the mean lives of the molecules on the surface. It is incidentally shown that the same areas associated with the forward reaction must inevitablycatalyse the backward reaction.

The general solution is impracticable, so the special cases of irreversible synthesis and decomposition are considered and results are obtained.

Since the publication of a previous paper on chemical constants, some further experimental data upon the dissociation of chlorine by Wohl have appeared, from which a new and rather more satisfactory value of the chemical constant of diatomic chlorine can be calculated. Wohl concludes that Q0, the heat of dissociation at absolute zero, is – 57,000 calories. This is in good agreement with Henglein's value – 54,000 but differs from that of Trautz and Stackel, namely – 71,000, which is the value adopted in the previous paper. Q0 is necessarily an adjustable constant, and since Wold's value gives results which are a good deal more concordant with each other and with the theory, we now take Q0 = − 57,000. Wohl also uses hv0/k = 902 instead of 1093, but as the difference made by this change is very small compared with other disagreements, we retain the value 1093. The following table gives the new values of Γ (Cl2), calculated in the same way as before. The initials at the heads of the columns refer to the results of Henglein, Trautz and Wohl respectively.

This note is concerned with a result about spines of manifolds embedded in a sphere, which proved useful at one point in (2).

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over ℚ, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

1. Kepler's equation

has, when θ, x are real with 0 < x < 1, just one real root ξ = ξ*(θ, x). For this root there are the well-known formulae, dating from Lagrange and Bessel,

The objects of this paper are (i) to obtain analogous formulae for the unreal roots ξ of the equation, and (ii) to sum the conjugates (as Fourier series in θ) of the series on the right of (2) and (3).

Little is known about definite integrals for the Whittaker functions. The object of this paper is to establish such results and the following formulae are established

where the symbol means that in the expression following it i is to be replaced by –i and the two expressions are to be added.

We study the weak star accumulation points of sequences of probability measures of the form [XKn(x) ρn(x)dσ(x)]/ [∫Kn ρn(t)dσ(t)], where ρ(x)>0 is continuous on Rκ, σ denotes Lebesgue measure in Rκ and the sequence of compact sets Kn⊂Rκ converges in the sense of Hausdorff towards a compact set K. The motivation of our study was given by a result relating interval averages with the winding number. Similar probability measures are considered in partial differential equation problems and we extend our study to this case.

This investigation is to discuss some of the conditions which influence the absorption spectra of various “saturated” and “unsaturated” organic compounds. A condensed cadmium spark was used as the source of radiant energy; alcoholic solutions of the substances were employed, except for stearic acid, which was dissolved in ether. The vapours were examined in a quartz tube 200 mm. long, placed in a bath which could be heated to various temperatures.