In this paper, convergence rates of solutions towards stationary solutions for the outflow problem of planar magnetohydrodynamics (MHD) are investigated. Inspired by the relationship between MHD and Navier-Stokes, we prove that the global solutions of the planar MHD converge to the corresponding stationary solutions of Navier-Stokes equations. We obtain the corresponding convergence rates based on the weighted energy method when the initial perturbation belongs to some weighted Sobolev space.

One considers the linear differential systems where is a (not necessarily diagonal) matrix and one relates the computation of a general multiplicity defined from this system to the corresponding multiplicity of some eigenvalues of . Then applying these conclusions, one gives simple conditions ensuring the existence of odd or even periodic solutions for systems having the form .

We present a new result on the existence of periodic solutions for the equation:

for all positive parameters λ sufficiently large. Our fundamental assumption is the following monotonicity property: if ø ≧ ψ (ø and ψ are data) then x(ø) ≧ x(ψ). The proof consists in applying the global Hopf bifurcation theorem. The main steps are: (i) a classical estimation of the periods; (ii) an a priori estimate for the solutions along a connected branch; (iii) a transformation acting on periodic solutions.

This paper is concerned with the fractional system

\begin{cases} (-\Delta)^{\frac{\alpha}{2}} u(x) = \vert x \vert ^a v^p(x), &x\in\mathbb{R}^n_+,\\ (-\Delta)^{\frac{\beta}{2}} v(x) = \vert x \vert ^b u^q(x), &x\in\mathbb{R}^n_+,\\ u(x)=v(x)=0, &x\in\mathbb{R}^n{\setminus}\mathbb{R}^n_+, \end{cases}

$p \leqslant \frac {n+\alpha +2a}{n-\beta }$

$q \leqslant \frac {n+\beta +2b}{n-\alpha }$

$p+q<\frac {n+\alpha +2a}{n-\beta }+\frac {n+\beta +2b}{n-\alpha }$

In regular state stabilization of a class of linear parabolic systems, the number of the sensors wk and the actuators hk are required to be greater than or equal to the maximum of the multiplicities of the unstable eigenvalues. In this paper, we ask what control theoretic results we can establish with smaller numbers of wk and hk. The enhancement of stability of output or the output stabilization, a concept weaker than regular state stabilization, gives an answer to the question. Conditions of a fairly different nature than those for regular state stabilization appear; these are the rank conditions of the products of the observability and controllability matrices. The result is applied to a parabolic system with boundary observation/control via two different approaches.

In this paper we obtain, for a semilinear elliptic problem in RN, families of solutions bifurcating from the bottom of the spectrum of −Δ. The problem is variational in nature and we apply a nonlinear reduction method that allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

We consider the generalisation from central series to marginal series in groups and set up firstly various basic results. The main section of the paper is concerned with the study of which group theoretical properties may be transferred from a marginal factor in a group to the corresponding verbal subgroup and which properties may be transferred from one factor of a lower marginal series to successive factors of the series.

We consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In [3], it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schrödinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.

We describe a new general method for the computation of the group Aut(X) of self-homotopy equivalences of a space. It is based on the decomposition of Aut(X) induced by a factorization of X into a product of simpler spaces. Normally, such decompositions require assumptions (‘induced equivalence property’, ‘diagonalizability’), which are strongly restrictive and difficult to check. We derive computable homological criteria for an analogous assumption, called reducibility, and then show that these criteria are satisfied when the so-called atomic decomposition of the space is used. This essentially reduces the computation of Aut(X) to the computation of the group of self-equivalences of its atomic factors, and the computation of certain homotopy sets between those factors.

We consider a purely quasi-incompressible elasticity model. We rigorously establish asymptotic expansions of near- and far-field measurements of the transient elastic wave induced by a small elastic anomaly. Our proof uses layer potential techniques for the modified Stokes system. Based on these formulae, we design asymptotic imaging methods leading to a quantitative estimation of elastic and geometrical parameters of the anomaly.

This paper concerns with the existence of multiple solutions for a class of elliptic problems with discontinuous nonlinearity. By using dual variational methods, properties of the Nehari manifolds and Ekeland's variational principle, we show how the ‘shape’ of the graph of the function A affects the number of nontrivial solutions.

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ ≪ ε and slow time scales ε ≪ τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio ε/τ > 0 is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.

Let be the Hurwitz zeta function. Furthermore, let p > 1 and α ≠ 0 be real numbers and n ≥ 2 be an integer. We determine the best possible constants a(p, α, n), A(p, α, n), b(p, n) and B(p, n) such that the inequalities and hold for all positive real numbers x1,…,xn.

For a measure μ on ℝn (or on a doubling metric measure space) and a Young function Φ, we define two versions of Orlicz–Poincaré inequalities as generalizations of the usual p-Poincaré inequality. It is shown that, on ℝ, one of them is equivalent to the boundedness of the Hardy–Littlewood maximal operator from LΦ(ℝ,μ) to LΦ(ℝ,μ), while the other is equivalent to a generalization of the Muckenhoupt Ap-condition. While one direction in these equivalences is valid only on ℝ, the other holds in the general setting of doubling metric measure spaces. We also characterize both Orlicz–Poincaré inequalities on metric measure spaces by means of pointwise inequalities involving maximal functions of the gradient.

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integrodifferential equations that can be of independent interest.

In 1890 I published a theory of ferromagnetic induction in which it was suggested that the equilibrium of Weber's elementary magnetic particles was due only to magnetic forces. It was shown that when the elementary magnets are made to turn by applying a magnetising force which is progressively increased, the conditions of equilibrium must be such that there is first a small amount of stable (reversible) deflection, then a break away with irreversible deflection into a new position of stability, and finally a reversible approach to the position of complete parallelism which corresponds to saturation. A model was constructed showing that these conditions could be satisfied by a purely magnetic control. It was made up of little magnets, pivoted on fixed centres which were uniformly spaced. The magnets were all free to turn, but controlled one another by their mutual magnetic forces. They tended to form rows, and when an external field was applied the rows broke up and fresh rows were formed more nearly in the direction of the field. With this simple model the known characteristics of the magnetising process were, qualitatively, well reproduced. A recent study of the stability of such rows of magnets has, however, led me to abandon this model, and to design instead a model in which each atom forms a magnetic system comprising a Weber element capable of turning, but controlled by the magnetic forces exerted on it by other parts of the atom which are taken as fixed. As in the old model, the control is wholly magnetic. Various forms of the new model will be described in a later part of this paper, but in the first place it may be useful to give some account of the investigations of stability which have convinced me that the old model fails, quantitatively, to represent the process.

In my former paper, read before the Society in June 1878, I gave an account of observations which I had made on the fungus disease affecting salmon, and described the character of the fungus, which I referred to Saprolegnia ferax.

In the present communication I propose to relate additional observations, and to discuss the theories which have been advanced by different writers in explanation of the cause of the disease. Four theories have been advocated, namely—pollution of rivers, overcrowding, absence of frost, diseased kelts and addled ova.

The anatomy of the primates is responsible for a very considerable volume of contributions to morphological literature. But, quite naturally, those forms which have the closest affinity to man have been more extensively and more carefully observed than those occupying a lower position in the zoological scale. And, further, the most painstaking and accurate work has been done in connection with organs which are very specially modified in man, such, for example, as those of the nervous system. This is perfectly consistent with the desire to obtain any light, and from any source, which will facilitate the unravelling of the many problems of human structure. Moreover, the soft and flaccid viscera do not lend themselves to accurate observation so readily as do the more solid organs which retain their shape after their removal from the body. So it comes to pass that, up to quite recent times, the abdominal viscera, and more especially the hollow organs of the abdomen, such as the stomach and the intestines, have only received scant attention; or, if they have been more carefully examined, the descriptions as given prior to the introduction of modern methods cannot be held as absolutely satisfactory, because of the change of shape and disposition resulting from the removal of their natural support, to wit, the abdominal walls.

In man, in the rabbit, and some other animals the trachea is built up of a series of cartilaginous rings incomplete behind; the rings being completed in this position by the trachealis muscle.

The mucous membrane forms a smooth cylindrical lining for the whole tube, and is covered by a ciliated epithelium.

In the cat and dog the cartilage rings completely encircle the trachea, and overlap posteriorly, and the trachealis muscle, which is well developed, is placed outside the cartilage, and has a powerful action in varying the diameter of the tube.

When this muscle contracts the overlapping ends of the cartilage ride one upon another, and the projecting ends form a vertical ridge down the inside of the trachea, which can be readily seen on slitting the organ open. This ridge is separated by a deep groove from the other end of the plate.

The introduction of standardised methods for describing that characteristic of social behaviour denoted by the word intelligent is of comparatively recent date. Therefore it is not feasible to undertake a study of the intellectual resemblance of relatives along the ancestral line. In two classes of collateral relatives, however, such resemblance can be made the subject of inquiry. These are (a) members of the same fraternity— twins or ordinary sibs—and (b) first cousins. This investigation, which was prompted by a previous inquiry into the intellectual resemblance of twins carried out in the Department of Social Biology, is confined to a study of ordinary sibs and first cousins. Though resemblance between both sibs and first cousins has already been investigated, there are in each case special reasons which justify further inquiry.