In this paper we study a problem in multilinear algebra which consists of finding small values of a certain quotientμ/α. Here μ is the minimal eigenvalue of a positive definite operator determinant Δ of the type introduced by F. V. Atkinson, and α is the minimum of the quadratic form corresponding to Δ with respect to all decomposable tensors of unit norm. Our results are connected with earlier results of P. Binding.

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

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.

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

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

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

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

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.

The harmonic mapping problem asks when there exists a harmonic homeomorphism between two given domains. It arises in the theory of minimal surfaces and in the calculus of variations, specifically in hyperelasticity theory. We investigate this problem for doubly connected domains in the plane, where it already presents a considerable challenge and leads to several interesting open questions.

The convergence and blow-up results are established for the evolution of non-simple closed curves in an area-preserving curvature flow. It is shown that the global solution starting from a locally convex curve converges to an m-fold circle if the enclosed algebraic area A0 is positive, and evolves into a point if A0 = 0.

By applying a general large-deviation theorem of Kifer and Ruelle's Smale space technique, some large-deviation estimates are proved for Axiom A endomorphisms.

We consider a time-varying inclusion in a thermal conductor specimen. In particular, the thermal conductivity is a variable function depending on space and time with a jump of discontinuity along the interface of the unknown anomalous region. Provided with some a priori information on the conductivity and its support, we study the continuous dependence of the inclusion from infinitely many thermal measurements taken on an open portion of the boundary of our specimen. We prove a rate of continuity of logarithmic type showing, in addition, its optimality.

We consider a class of second-order systems , with q(t) ∊ℝn, for which the potential energy V: ℝn\S→ℝ admits a (possibly unbounded) singular set S ⊂ℝn and has a unique absolute maximum at 0 ∈ℝn. Under some conditions on S and V, we prove the existence of several solutions homoclinic to 0.

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.

Up to the present time no sympathetic nervous system has been described in Lepidosiren paradoxa, and the following note is an attempt to fill this gap in our knowledge.

Within comparatively recent times a large amount of work has been done on the cathode fall of potential. As a result it is now regarded by many investigators that no sudden fall of potential occurs close to the cathode, at least in the case of low voltage discharges. A comprehensive treatment of recent work on this subject is to be found in a paper by Stücklen. Further, F. W. Aston, in his investigations on the cathode fall, came to the conclusion that practically the whole fall occurred in the dark space between the cathode and the negative glow, and that if there were any force in the negative glow it was negligible compared with that in the dark space. The potentials used by Aston across his discharge tubes were comparatively small, however, never exceeding 1000 volts. With voltages of this order he showed further that the electric force in the dark space was very nearly in linear relation with the distance from the negative glow.

Our knowledge of the anatomy of Dicksonia is mainly based on the description of D. Barometz given by Gwynne-Vaughan. The stem of this species possesses a radial dictyostele, which, on account of the shortness of the leaf gaps, is not far removed from the solenostelic condition. The leaf trace arises as a broad ribbon of tissue with inwardly folding margins, which soon, however, breaks up to form a horse-shoe of isolated strands.

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.

The article is devoted to Hardy type inequalities on closed manifolds. By means of various weighted Ricci curvatures, we establish several sharp Hardy type inequalities on closed weighted Riemannian manifolds. Our results complement in several aspects those obtained recently in the non-compact Riemannian setting.

The four transverse galvanomagnetic and thermomagnetic effects were determined in the case of iron, nickel, cobalt, silver, copper, zinc, cadmium, aluminium.

The results were considered in the light of the electron theory of conduction in metals as formulated by G. H. Livens, and were found to be in fair accord with the theory as regards the ratios of the effects, but they could not be reconciled with the expressions given for the effects themselves.

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.

Transition probabilities are calculated which make the construction of diffusions on finitely ramified fractals straightforward. In contrast to former approaches using Brouwer's Fixed Point Theorem, we consider an approximation procedure based on the iteration of a nonlinear map L. Physically, this is done by ‘coarse-graining-renormalisation of finite electric resistor networks’. Mathematically, it is a convergence problem for quotients of Dirichlet forms on finite graphs. These graphs approximate finitely ramified fractals. The basic tool is a contraction theorem for the renormalisation map L which allows the use of known results about nested fractals for non-nested (p.c.f. self-similar) ones. In general, the above contraction is not strict because several linear independent fixed points occur.