The effect of the presence of singularities on the method of stationary phase for a certain integral of two variables is examined. The singularities lie on two straight lines whose positions depend upon parameters. For certain values of the parameters the straight lines can pass through the point of stationary phase.

The asymptotic development, which is uniform with respect to the parameters, is determined. It is found that it can be expressed in terms of a function which, in certain circumstances, reduces to a Fresnel integral. The main properties of this function are derived in an appendix.

Explicit formulae are given for the dominant asymptotic terms in the cases when the point of stationary phase is approached by (i) neither line, (ii) only one of the lines, (iii) both lines.

Let W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.

Let be the one-sided maximal operator and let Ф be a convex non-decreasing function on (0, ∞), Ф(0) = 0. We present necessary and sufficient conditions on a couple of weight functions (σ, ϱ) such that the integral inqualities of weak type

and of extra-weak type

hold. Our proofs do not refer to the theory of Orlicz spaces.

In this paper, we consider the equation

where A is a linear operator, N = ψ′ with ψ ∈ C1(E, R), and E is an Hilbert space. We suppose that N has a derivative at infinity N′(∞) and that 0 belongs to the spectrum of A–N′(∞). We prove an abstract theorem for multiplicity of solutions for the above equation. We then apply this theorem to the study of periodic solutions of Hamiltonian systems and of semilinear wave equations when the period is prescribed.

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀 denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1 , . . . , eαm, m ⩾ 2, where α0 = 0, α1, . . . , αm are m + 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies on m are improved.

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.

This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in $\mathbb {R}^{d}$ with $d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in $\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in $\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

We consider elliptic systems with discontinuous coefficients and prove that if the known term belongs to the Morrey space Lp,λ, then the highest-order derivatives of the local solution belong to the same space. We also obtain local Hölder continuity for lower-order derivatives.

This paper is concerned with the asymptotic behaviour of solutions to quasilinear hyperbolic equations with nonlinear damping on the quarter-plane (x, t) ∈ ℝ+ x ∈ ℝ+. We obtain the Lp (1 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

We consider the following equation:

where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up (p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up . Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x 1 only, and W satisfies

Using a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation

$$-\Delta u-k^2u = Q(x) \vert u \vert ^{2^*-2}u,\quad u\in W^{2,2^*}({\open R}^{N})$$

$Q \in L^{\infty }({\open R}^{N}){\setminus }\{0\}$

The purpose of this paper is to prove strong-type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the boundedness of commutators of these one-sided operators.

This corrigendum corrects a result in [1].

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$. We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell ^{2}_{s}(\hbar \mathbb {Z}^n)$, $s \in \mathbb {R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal {S}'(\hbar \mathbb {Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb {R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb {Z}^n$.

Let M be a closed 4-manifold with a free circle action. If the orbit manifold N3 satisfies an appropriate fibering condition, then we show how to represent a cone in H2(M; ℝ) by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández et al. In the case that M is the product 4-manifold S1 × N, our construction complements our previous results and allows us to determine completely the symplectic cone of such 4-manifolds.

1. At the present moment there is much discussion regarding the means by which properties are hereditarily transmitted from a parent organism to its offspring, and of the extent to which the Mendelian theory is capable of accounting for the facts. In this note it is not proposed to discuss the general question but to investigate the conditions under which the theory of correlation may be applied to Mendelian groupings. Two important papers on this subject have already been published: one by Professor Pearson, entitled “A Generalized Theory of Mendelian Inheritance”; the other, which is largely a criticism of this, by Professor Udny Yule. In Professor Pearson's paper the results produced when two organisms with any number of pairs of different zygotes mate indiscriminately are fully considered. He finds that such a population once established is stable, and he then deduces the parental and fraternal correlation coefficients.

The middle cells are present throughout the whole length of the spinal cord. They are situated in the middle region of the grey matter, between the free anterior and free posterior cornua; but they sometimes also extend into the regions usually occupied by the anterior cornual cells, by the intermedio-lateral tract or by Clarke's column. The small cells in the base of the anterior horn (scattered cells) cannot be sharply separated from the middle cells, nor can most of the small cells about the base of the free posterior horn.

Although some of the middle cells may be found in all this wide area of distribution at practically any level of the cord, there are certain arrangements of cells which may be looked upon as typical of each segment, or at least of each region of the cord. In the upper cervical region they are not on the whole very abundant (this differs from Waldeyer's account), but are best developed in the central and para-central fields. In the cervical enlargement they are much more numerous, particularly within a broad band extending from the formatio reticularis to about the anterior grey commissure. Throughout the dorsal region they are again comparatively few in number, and may be scattered irregularly; but small groups are often found, most commonly in the central area, about on a level with the central canal, and in the post-central area, between Clarke's column and the reticular group of the intermedio-lateral tract. In the lumbar segments the middle cells are abundant, particularly centrally and in the base of the anterior horn. Their field extends anteriorly, until in the lower sacral region they come to be found over the whole area of the anterior cornua in addition to their more usual situations.

At all levels, as studied in serial sections, the distribution of the middle cells is seemingly erratic and casual. No regular plan of arrangement can be made out, and there is most certainly no segmentation (as Argutinski described) like that so well seen in the intermedio-lateral tract. The cells are sometimes just dotted here and there singly; sometimes they are scattered fairly thickly and evenly over the whole or part of the regions they are to be found in; or, again, they may occur more thickly in one part, or be aggregated into a distinct little cell group or nest, but even then these cell groups are seldom so closely packed as are the cell groups of the intermedio-lateral tract. The duration of any one of these types of cell arrangement is inconstant; and although there are levels where cells seem for a bit to be almost persistent in one place, this much can be stated as a general rule, that no middle cell group lasts through more than a very few serial sections. If traced further, the group is found either to shift to some other area or to die out altogether. Occasionally there appears to be a variation in number of cells parallel with the oscillations of the intermedio-lateral tract, but closer study shows that this is by no means absolute, and is probably only a local accidental variation.

Without expressing any opinion as regards function, I am inclined to divide the cells I have described collectively as middle cells into three groups, basing this division merely on the distribution and arrangement of the cells and on the microscopic appearances of the cell bodies. These divisions are, however, not very sharply defined either as regards the character of the individual cells or in the cell distribution. Still they seem to warrant such a division being made, and to suggest at least that the cells of the three groups may be functionally different. The groups I would make are—

(1) The middle cells proper, or central cells, occurring chiefly in the central area of the grey matter. These cells are of medium size and very similar to those of the apical group of the intermedio-lateral tract, multipolar, polygonal or rounded polygonal in outline, with a comparatively large nucleus and a few chromatic granules round it in the cell substance. Sometimes scattered, but more often present as a small clump of cells. With them may be included the cells in the para-central area, which, although sometimes forming a distinct and separate aggregation of cells, can mostly not be separated off from the central cells. Many of the cells occurring in the reticular area at levels where the reticular group of the intermediolateral tract is not present (especially in the lower cervical region) may probably also be included here.

(2) Anterior central cells in the base of the anterior horn—one of Waldeyer's “scattered” cell groups. As this name implies, these are often simply scattered over the area in question, but sometimes they are gathered into small cell-nests, but rarely very compact ones. Although there is no sharp boundary between them posteriorly and the central cells, they are in the main larger cells, and are further distinguished by their shape. They are not so often definitely multipolar and of approximately equal diameter in different directions, but more frequently appear to be bipolar, with longdrawn-out processes. The axis of elongation may be variously oriented. With these would fall to be included the middle cells in the external central area of the lateral enlargements, particularly in the lumbo-sacral cord. Possibly also some of the para-central cells should be classed with this group, and not with the previous one.

(3) Post-central cells. These lie in the area between Clarke's column and the formatio-reticularis, or in the corresponding region of the grey matter at level where Clarke's column is unrepresented. As with the anterior central cell, there is no sharp boundary between this series and the central middle cells. They are often continuous, or the cells may lie betwixt and between the two areas. But although many of the posterior cells may be as large, they are distinguished on the whole by being smaller in size than the central cells and less definitely polygonal, more rounded in outline. They are generally present either as a small clump or as a band of cells on the outer and posterior aspects of Clarke's column. Posteriorly, it is very difficult to separate this group from the posteriorbasal and posterior-marginal cells of Waldeyer. With this group may be included cells present in the area of Clarke's column, particularly when that column is absent or only represented by occasional cells. Also some of the small cells found in the reticular formation, especially those lying between (vertically) the nuclei of the reticular group of the intermedio-lateral tract.

These subdivisions may require modification from subsequent investigations, particularly by tracing the course and termination of the cell processes. I have attempted to do this by means of Cajal's silver impregnation methods, but, so far, have not succeeded. All I can affirm is that the fibres arising from the middle cells run in a variety of directions to begin with; but as some could be traced far enough to be seen doubling more or less sharply upon themselves, this gives no real clue to their destination. In no case could I follow any one to a termination. In some cells them was observed an endo-cellular fibrillary network, similar to that in the large motor cells of the anterior horn. In this particular cord the following figures give approximate average diameter of the cells of the different groups:—

In conclusion, I have to express my thanks to Dr Alexander Bruce for suggesting this subject for a thesis; to Dr Theodore Shennan, in whose laboratory the preparation of the sections was carried out; to Mr Henry Wade for the use of micro-photographic apparatus; and lastly to the Carnegie Trust, since this thesis is part of work done under the terms of a grant from the Trust for original research.

The paper gives asymptotic estimates of the form

as λ→∞ for the length l(μ)of a gap, centre μ in the essential spectrum associated with second-order singular differential operators. The integer r will be shown to depend on the differentiability properties of the coefficients in the operators and, in fact, r increases with the increasing differentiability of the coefficients. The results extend to all r ≧ – 2 the long-standing ones of Hartman and Putnam [10], who dealt with r = 0, 1, 2.

We completely classify non-degenerate surfaces with parallel second fundamental form in three-dimensional Riemannian and Lorentzian reducible spaces.