We prove the non-degeneracy of the extremals of the Sobolev inequality

\[ \int_{\mathbb R^N}|\nabla u|^p\,\rd x\ge \mathcal S_p\int_{\open R^N}|u|^\frac{Np}{N-p}\,\rd x,\quad u\in \mathcal D^{1,p}(\open R^N) \]

We show that silting modules are closely related with localizations of rings. More precisely, every partial silting module gives rise to a localization at a set of maps between countably generated projective modules and, conversely, every universal localization, in the sense of Cohn and Schofield, arises in this way. To establish these results, we further explore the finite-type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects. In particular, we prove that silting modules are of finite type.

Let R be a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes of R-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasi-isomorphism at a certain fixed stage of the associated spectral sequence. For filtered complexes, we relate the different model structures obtained, when we vary the stage of the spectral sequence, using the functors shift and décalage.

We provide an estimate of the energy of the solutions to the Poisson equation with constant data and Dirichlet boundary conditions in a convex domain Ω ⊂ Rn. This estimate is obtained by restricting the variational formulation of the problem to the space of functions depending only on the distance from the boundary of Ω. The main tool in the proof is an isoperimetric inequality for convex domains, which is a consequence of the Brunn-Minkowski theorem.

A method for counting the solutions for Penrose–Fife-type phase field equations is derived. The method used is similar to that developed recently for obtaining a precise count for the number of solutions for the Cahn–Hilliard equation [9], and is based on the derivation of an extended system of Picard–Fuchs equations as well as on estimates obtained in [11]. The Penrose–Fife-type phase field equations represent a thermodynamically consistent model for phase separation of a conserved order parameter (typically concentration) in binary systems in which latent heat effects are important in the phase separation process.

We characterize the weighted Hardy inequalities for monotone functions in In dimension n = 1, this recovers the standard theory of Bp weights. For n > 1, the result was previously only known for the case p = 1. In fact, our main theorem is proved in the more general setting of partly ordered measure spaces.

Given a sequence of polynomials $\{x_k(q)\}_{k \ges 0}$, define the transformation

$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$

$n\ges 0$

$[x_{i+j}(q)]_{i,j \ges 0}$

$\{y_n(q)\}_{n\ges 0}$

In this paper we show that conforming Galerkin approximations for p-harmonic functions tend to ∞-harmonic functions in the limit p → ∞ and h → 0, where h denotes the Galerkin discretization parameter.

In this paper, we study the following Ginzburg–Landau functional:

where (G, C), and p is a smooth bounded and non-negative map, having minima on the boundary of Ḡ. We give the location of the singularities in the case where the degree around each singularity is equal to 1.

In this paper we analyse possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a non-local eigenvalue problem of fractional type that approximates, when taking a suitable limit, the classical Steklov eigenvalue problem.

In this paper, we analyse the semilinear fourth-order problem ( − Δ)2u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and

$$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$

Consider the two-dimensional autonomous systems of differential equations

$$ \dot{x}=-y+\lambda x+P(x,y),\qquad\dot{y}=x+\lambda y+Q(x,y), $$

where $\lambda$ is a real constant and $P$ and $Q$ are $\mathcal{C}^{\infty}$-functions of order greater than or equal to two. These systems, so-called centre-focus-type systems, have either a centre or a focus at the origin. In this work, we give necessary and sufficient conditions of isochronicity using normal forms. We characterize the systems which have either an isochronous centre or an isochronous focus at the origin by means of the existence of a commutator of the field. Moreover, we prove that the maximum order of a weak isochronous focus for quadratic systems is two, and that for systems with cubic nonlinearities is three.

The kindness of Professor Cunningham has enabled me to carry out the dissection of a gibbon in his possession. The dissection was performed in 1905, in the Anatomical Department of the University of Edinburgh. The following paper is a description of the short muscles of the hand of the animal, to which is added a consideration of the primitive position and function of the short muscles of the human hand.

Hyperasymptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric U-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.

In May 1908 an adult female Sowerby's whale, Mesoplodon bidens, was stranded in St Andrews Bay, about a mile from the clubhouse. Its capture and external characters were recorded by Professor W. C. M‘Intosh in the Annals and Magazine of Natural History, December 1908. The skeleton was obtained by him for the Gatty Marine Laboratory, and at his request I have examined and prepared this report on its characters.

It is well known from the work of Noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law. In this paper, we prove that each such conservation law arises directly as the Euler-Lagrange equation for the functional on taking suitable variations around a minimizer.

1. Endodermal cells frequently show considerable increase in tangential dimension at a late stage in their existence, as a result of which the continuity of the endodermis may be maintained, despite the rapid expansion of the intra-endodermal cylinder following secondary growth.

2. This tangential increase in size of the endodermal cells is usually accompanied by their division. The divisions may actually cause the increase in size of the cells, that is, there is active growth of the cells. Or they may follow a passive stretching. In either case the new walls fulfil a mechanical function, lending added strength to the stretched walls of the original cell.

3. Two types of division are distinguishable:

(a) Those occurring in primary endodermal cells. The new walls soon develop Caspary strips, linking up with those of the parent cells.

(b) Divisions in tertiary endodermal cells. The new walls here show neither Caspary strip nor suberin lamella.

4. It is pointed out that in neither case does the development of new walls interfere with the impermeability of the endodermis.

5. The occurrence of the divisions is considered in relation to the supposed impermeability of the suberin lamella of the secondary endodermal cell, and to the conditions resulting in the deposition of the Caspary strip.

A classical theorem of Hartogs gives conditions on the singularity set of an analytic function of several complex variables in order for such a set to be an analytic variety. A result of E. Bishop from 1963 gives an analogous condition of the maximal ideal space of a uniform algebra in order for this space to have analytic structure. We show that algebras of functions satisfying a maximum principle serve to explain both of these results.

We consider an inverse parabolic problem of reconstruction of the source function, together with the traditional solution. In contrast with older literature, we consider non-homogeneous and time-dependent boundary conditions. We are able to prove a general result of convergence to a stationary state, and of asymptotic expansion as t → ∞.

In commutative Banach algebras with factorisation, the existence of an identity (bounded approximate identity) modulo a topologically nilpotent radical implies the existence of a global identity (bounded approximate identity), respectively.