Using a technique due to Macbeath (Jack and Macbeath 1959) this paper gives what the author hopes is a shorter and easier presentation of the evaluation of certain Jacobians of matrix transformations which have occurred in statistics and the theory of quadratic forms.

An asymptotic theory is developed for linear differential equations of odd order. Theory is applied with large coefficients. The forms of the asymptotic solutions are given under general conditions on the coefficients.

A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

In this paper, we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

The acyl groups in aromatic ketones with hydroxyl or methoxyl groups in the ortho or para positions are readily removed by hydrobromic acid in acetic acid. The results are correlated with the fine structure of the parent hydrocarbons. 5,8-Dimethoxy-I-tetralone and 1,3-dihydroxyxanthone fail to undergo ring-fission with the reagent. The mechanism is discussed and it seems probable that fission occurs by direct proton attack at nucleophilic carbon atoms bearing the acyl groups.

Asymptotic formulae for the Titchmarsh—Weyl m-coefficient on rays in the complex λ-plane for the equation − y ″ + qy = λy whenthe potential is limit circle and non-oscillatory at x = 0 are obtained under assumptions slightly more general than xq(x) ∈ L1(0,c). The behaviour of q at the right end-point is arbitrary and may fall in either the limit-point or limit-circle case. A method of regularization of the equation is given that can be made to depend either on a solution of the equation for λ = 0 or more directly on an approximation to the solution in terms of q. This enables equivalent definitions of the m-coefficient to be given for the singular Sturm—Liouville problem associated with a singular limit-circle boundary condition, and its associatedregular Sturm—Liouville problem. As a consequence, it becomes possible to apply asymptotic results obtained by Atkinson for the regular problem in order to give asymptoticresults for the singular problem. Potentials of the form q(x) = C/xj, 1 ≤ j < 2, are included. In the case j = 1, an independent calculation of the limit-point m-coefficient over the range (0,∞), relying on Whittaker functions, verifies the main result.

This paper shows that if S is a sphere in a Banach space and f: S → S is an α-contraction, then f has a fixed point. The paper generalizes a result of R. D. Nussbaum which holds for α-contractions only. The proof uses the Browder nonrepulsive fixed point theorem and is motivated by recent work of M. Martelli and G. D. Cooperman.

Geometric Langlands duality relates a representation of a simple Lie group Gv to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of Gv makes an unexpected appearance. This can be explained using gauge theory, as this paper will show, with the help of the equations of Nahm and Bogomolny.

The solution of the dual series relations

where {λn) is the sequence of positive zeros of the Bessel function Jν(αλ), arranged in order of increasing magnitude, þ and ν are real numbers (−1 <þ < 1, ν >0), the functions, f1(ρ), f2(ρ) being prescribed, is obtained by giving an integral representation of {αn} in terms of a single function g(t). The problem is reduced to that of solving a Fredholm integral equation of the second kind for the auxiliary function g(t).

We consider the nonlinear eigenvalue problem posed by a parameter-dependent semilinear second-order elliptic equation on a bounded domain with the Dirichlet boundary condition. The coefficients of the elliptic operator are bounded measurable functions and the boundary of the domain is only required to be regular in the sense of Wiener. The main results establish the existence of an unbounded branch of positive weak solutions.

For $N\geq 2$, a bounded smooth domain $\Omega$ in $\mathbb {R}^{N}$, and $g_0,\, V_0 \in L^{1}_{loc}(\Omega )$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem:

\[ -\Delta_p \phi + V |\phi|^{p-2}\phi = \lambda g |\phi|^{p-2}\phi \text{ in } \Omega, \quad \phi=0 \text{ on } \partial \Omega, \]

$g$

$V$

$g_0$

$V_0$

$(\underline {g},\,\underline {V})$

$(\overline {g},\,\overline {V})$

$g_0$

$V_0$

$p=2$

We study the asymptotic behaviour, as $p\to 1^+$, of the solutions of the following inhomogeneous Robin boundary value problem:P

\begin{equation*} \begin{cases} \displaystyle -\Delta_p u_p = f & \text{ in }\Omega,\\ \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p = g & \text{ on } \partial\Omega, \end{cases} \end{equation*}

$\Omega$

$\mathbb {R}^{N}$

$\nu$

$\Delta _p v$

$p$

$p>1$

$f\in L^{N,\infty }(\Omega )$

$\lambda,\,g$

$\partial \Omega$

$\lambda \ge 0$

$p$

$1$

$p\to 1^+$

In this paper, we consider the Cauchy problem for an inviscid compressible Oldroyd-B model in three dimensions. The global well posedness of strong solutions and the associated time-decay estimates in Sobolev spaces are established near an equilibrium state. The vanishing of viscosity is the main challenge compared with [47] where the viscosity coefficients are included and the decay rates for the highest-order derivatives of the solutions seem not optimal. One of the main objectives of this paper is to develop some new dissipative estimates such that the smallness of the initial data and decay rates are independent of the viscosity. Moreover, we prove that the decay rates for the highest-order derivatives of the solutions are optimal, which is of independent interest. Our proof relies on Fourier theory and delicate energy method.

If a cord be knotted, any number of times, according to the pattern below

it is obviously perverted by simple inversion. Hence, when the free ends are joined it is an amphicheiral knot. Its simplest form is that of 4-fold knottiness. All its forms have knottiness expressible as 4n.

Any reasonable generalisation on possibly fundamental rules of specific and generic differentiation would appear to be worth consideration. Defining our main terms briefly, we may here state that the genera and species of quantitative evolution are the genera and species recognised as such by responsible systematists who have considered entire groups in detail: the relatively few “critical” genera do not yield comparable data.

1. When the univalent anions (of potassium salts) are arranged according to their precipitation concentration for a ferric hydroxide sol the sequence is lyotrope:

A, CNS, Cl, Br, NO3, ClO3, I.

The concentrations range from 0·01 for acetate to 0·093 for iodide; thiocyanate alone is in an unusual position.

The order for kations is also lyotrope:

Li, Na, Mg, K, Rb, NH4, Ca. The concentration range is much smaller, from 0·037 for lithium to 0·055 for ammonium.

2. The pairs of univalent-bivalent kations—Li. and Mg¨, K. and Ca¨—show no trace of antagonistic action in the precipitation of Fe(OH)3 sol—the results being strictly ADDITIVE.

3. The pair of univalent-bivalent anions—ClO3′ and SO4″—on the contrary, show a well-marked ADJUVANT action, which amounts to 50 per cent.

4. The effect of salts on the opalescence temperature of a phenol-water system (10 to 11 per cent, phenol) is lyotrope, and is very marked at a concentration of 0·04 n. The order is—

Anions … C1, Br, NO3, I, CNS

Kations ‥ Na, Li, K, NH4

The effect of the valency of the ion is slight, or non-existent.

5. A preliminary account is given of a method which promises to provide a simple and reliable means of determining the amount of “free” and “bound” water in a solution (applicable to electrolytes and nonelectrolytes, and apparently to colloidal solutions, as e.g. gum arabic).

6. An explanation of the Lyotrope Effect is put forward.

The value of the continued-fraction, as a means of representing an analytic function f(x), is now fully recognised. Other representations, such as power-series or Fourier series or Dirichlet series or series of inverse factorials, converge in general only over limited regions of the x-plane, and fail to converge over the rest of the plane, whereas the representation of the function by a continued-fraction converges (in a large class of cases at any rate) over the whole x-plane, with the exception of certain singular curves.

§ 1. The theory of group-velocity has been developed by Stokes, Osborne Reynolds, Lord Rayleigh, and later by Professor Lamb. The application of the theory to light-waves was made by Gouy and Lord Rayleigh, and its importance in this connection was emphasised by Professor Schuster in his paper “On Interference Phenomena,” Phil. Mag., vol. xxxvii., 1894.

Writers on mammalian anatomy are united in stating that one of the characteristics of the Cetacean family Physeteridæ is the presence of functional teeth in the lower jaw only. Typical statements are those of Flower and Lydekker (1891): “No functional teeth in the upper jaw”; Beddard (1900): “Teeth found in both jaws, but those of lower jaw alone functional; often very reduced in number”; Turner (1912): “Functional teeth in lower, but not in upper jaw.” A possible exception is the Pigmy Whale, Kogia, in which the upper jaw may altogether lack teeth or may possess an anterior rudimentary pair (as in K. simus, Owen) the utility of which is a matter of some doubt.

In 1849 William Swan, subsequently Professor of Natural Philosophy in the University of St Andrews, read a paper on the “Gradual Production of Luminous Impressions on the Eye and other Phenomena of Vision” before the Royal Society of Edinburgh (see Transactions, Vol. XVI.). This paper contains some results of high interest, but I have no recollection of ever having seen it referred to in modern literature on the subject.