Let $\varOmega$ be a domain in $\mathbb{R}^N$ (possibly unbounded), $N\geq2$, $1<p<\infty$, and let $V\in L_{\mathrm{loc}}^\infty(\varOmega)$. Consider the energy functional $\mathcal{Q}_V$ on $C_{\mathrm{c}}^\infty(\varOmega)$ and its Gâteaux derivative $\mathcal{Q}_V^\prime$, respectively, given by

$$ \mathcal{Q}_V(u)\eqdef\frac{1}{p}\int_\varOmega(|\nabla u|^p+V|u|^p)\,\mathrm{d} x,\qquad\mathcal{Q}_V^\prime(u)= \mathrm{div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u, $$

for $u\in C_{\mathrm{c}}^\infty(\varOmega)$. Assume that $\mathcal{Q}_V>0$ on $C_{\mathrm{c}}^{\infty}(\varOmega)\setminus\{0\}$ and that $\mathcal{Q}_V$ does not have a ground state (in the sense of a null sequence for $\mathcal{Q}_V$ that converges in $L_{\mathrm{loc}}^p(\varOmega)$ to a positive function $\varphi\in C_{\mathrm{loc}}^1(\varOmega)$, a ground state). Finally, let $f\in\mathcal{D}^\prime(\varOmega)$ be such that the functional $u\mapsto \mathcal{Q}_V(u)-\langle u,f\rangle:C_{\mathrm{c}}^\infty(\varOmega)\to\mathbb{R}$ is bounded from below. Then the equation

$$ \mathcal{Q}_V^\prime(u)=f $$

has a solution $u_0\in W^{1,p}_{\mathrm{loc}}(\varOmega)$ in the sense of distributions. This solution also minimizes the functional $u\mapsto\mathcal{Q}_V^{**}(u)-\langle u,f\rangle:C_{\mathrm{c}}^\infty(\varOmega)\to \mathbb{R}$, where $\mathcal{Q}_V^{**}$ denotes the bipolar ($\varGamma$-regularization) of $\mathcal{Q}_V$ and $\mathcal{Q}_V^{**}$ is the largest convex, weakly lower semicontinuous functional on $C_{\mathrm{c}}^\infty(\varOmega)$ that satisfies $\mathcal{Q}_V^{**}\leq\mathcal{Q}_V$. (The original energy functional $\mathcal{Q}_V$ is not necessarily convex.)

The structure of positive boundary blow-up solutions to semilinear problems of the form −Δu = λf(u) in Ω, u = ∞ on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied for a class of nonlinearities f ∈ C1 ([0, ∞)\{z2}) satisfying f (0) = f(z1) = f (z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1)∪(z2, ∞), f > 0 in (z1, z2). Two positive boundary-layer solutions and infinitely many positive spike-layer solutions are obtained for λ sufficiently large.

In this paper, a theory of fractional calculus is developed for certain spaces D′p,μ of generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.

In this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann algebra. We show that if M is a von Neumann subalgebra of B(H) which is complemented in B(H) and isomorphic to M⊗M, then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part, we show that any complemented reflexive subspace of a C*-algebra is necessarily linearly isomorphic to a Hilbert space.

A fractionalisation of the Fourier transform was developed formally by Namias. By workingin ℐ, we found that this theory could be made mathematically rigorous. Here we extend our previous results to the space ℐ′ to consider fractional powers of the distributional Fourier transform. The paper concludes with an application to an ordinary differential equation which shows how a distributional approach can sometimes be more useful than a classical approach.

The usual asymptotic expansion for the solutions of an elliptic linear problem with oscillatory periodic coefficients is known to not be accurate near the boundary. In order to obtain a better approximation it is necessary to add to this expansion a boundary-layer term. This term has been obtained by other authors in the case of a plane boundary, such that its normal is proportional to some period. We consider the case where the normal is arbitrary.špace{-8pt}

In this paper, we consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and full, in another word, the ideal quotient becomes abelian. Moreover, a new equivalent characterization of cluster tilting subcategories is given by applying homological methods according to this functor. As an application, we show that in a connected 2-Calabi-Yau triangulated category ℬ, a functorially finite, extension closed subcategory 𝒯 of ℬ is cluster tilting if and only if ℬ /𝒯 is an abelian category.

Let H be a Hilbert space in which a symmetric operator S with a dense domain Ds is given and let S have a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov type

and a method for calculating the best possible constants Cn,m(S).

Moreover, let φ be a symmetric bilinear functional with a dense domain Dφ such that Ds ⊂ Dφ and φ(f, g) = (Sf, g) for all f ∈ Ds, g ∈ Dφ. A necessary and sufficient condition for validity of the inequality

as well as a method for calculating the best possible constant K are obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the form

The paper is concluded by establishing the best possible constants in the inequalities

where T is an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

The Mullins–Sekerka sharp-interface model for phase transitions interpolates between attachment-limited and diffusion-limited kinetics if kinetic drag is included in the Gibbs–Thomson interface condition. Heuristics suggest that the typical length-scale of patterns may exhibit a crossover in coarsening rate from l(t) ˜ t1/2 at short times to l(t) ˜ t1/3 at long times. We establish rigorous, universal one-sided bounds on energy decay that partially justify this understanding in the monopole approximation and in the associated Lifshitz–Slyozov–Wagner mean-field model. Numerical simulations for the Lifshitz–Slyozov–Wagner model illustrate the crossover behaviour. The proofs are based on a method for estimating coarsening rates introduced by Kohn and Otto, and make use of a gradient-flow structure that the monopole approximation inherits from the Mullins–Sekerka model by restricting particle geometry to spheres.

We prove some stability results for certain classes of C*-algebras. We prove that, whenever A is a finite-dimensional C*-algebra, B is a C*-algebra and ϕ: A → B is approximately a *-homomorphism, there is an actual *-homomorphism close to ϕ by a factor depending only on how far ϕ is from being a *-homomorphism and not on A or B.

1. Following upon a previous study of spirit specimens of Glyptonotus, observations have been made on live examples of Chiridotea cœca and C. tuftsii with the object of elucidating points relating to the natural history and structural peculiarities of the Glyptonoteinæ.

2. Though wholly carnivorous, Chiridotea is not a predaceous animal. Subsisting on portions of animals, already dead, which it discovers by persevering search aided by sense of smell, it is a scavenger rather than a hunter of living prey.

3. The uncommon division of the thoracic limbs in Chiridotea is associated with an exceptional range of locomotor accomplishments. The animal not only walks and swims like other members of the Idoteidæ, but also tunnels its way for long distances under the surface of sand.

4. The three pairs of gnathopods subserve a number of functions. They are used in ordinary reptant progression. They are used to seize and hold on to food, but not to tear or rend it. Working in conjunction with the mandibles, they play an essential part in the process of manducation. They also come into action during sand-tunnelling.

5. On closer observation the characteristic type of limb-taxis of Chiridotea proves to be specially related to the tunnelling habit. During the process of tunnelling the mesially situated gnathopods and pleopods, working together, deal with one body of sand, the laterally projecting peræopods with another.

6. Certain common features in the bodily conformation of Serolis, of Apus, of Limulus, and of Trilobites are interpreted as an adaptation, by convergence, to a habitat on the surface of mud.

7. Owing to the transparency of its body, the movements of the alimentary canal of Chiridotea can be observed during ingestion and the movements of the hepatic cæca during digestion of food.

8. The midgut exhibits, according to circumstances, two different types of peristaltic movement. At the very commencement of a meal, and also during long continued vain effort to masticate a tough morsel of food, a succession of forward-running waves of contraction is observed. When food begins to be swallowed these are replaced by backward-coursing waves of contraction.

9. There are two pairs of hepatic cæca. As the midgut fills the more mesial pair passes out of sight. The laterally situated cæca increase greatly in volume and exhibit a series of dilatations or pockets, which enter into vigorous and long-sustained rhythmic contraction.

10. The mesial split in the thoracic sternites is a device for allowing distension of the body after a meal.

11. Its eyes being wholly dorsal, Chiridotea shows no conspicuous colour change in response to its background. It has both brown and black chromatophores, the latter contractile.

12. A new point of specific difference between C. cœca and C. tuftsii is recorded.

13. The direction of rotation by which the uropods of Valvifera have assumed their present orientation is discussed anew, the conclusion again being that the anterior border was originally mesial.

14. Objections are raised against the elevation by Racovitza and Sevastos of the northern representatives of Miers' Glyptonoteinæ to independent sub-family rank.

The simplest and most common form of reproduction amongst the Ciliata is binary fission. In this ordinary, possibly vegetative, method of reproduction the plane of division is generally perpendicular to the long axis of the creature, To this generalisation the Vorticellidse form an apparent exception, but on the view that their evident long axis really corresponds to the dorso-ventral axis of other ciliates their case falls into line with that of the rest of the sub-class. Formerly many instances of fission in the direction of the long axis were described; they may safely be considered to have been mere instances of conjugation.

We deal with the large-time behaviour of scalar hyperbolic conservation laws with source terms

which are often called hyperbolic balance laws. Fan and Hale have proved existence of a global attractor for this equation with x ∈ S1. consists of spatially homogeneous equilibria, a large number of rotating waves and of heteroclinic orbits between these objects. In this paper, we solve the connection problem and show which equilibria and rotating waves are connected by a heteroclinic orbit. Apart from existence results, our approach via generalized characteristics also gives geometric information about the heteroclinic solutions, e.g. about the shock curves and their strength.

We give an example of an indefinite weight Sturm-Liouville problem whose eigenfunctions form a Riesz basis under Dirichlet boundary conditions but not under anti-periodic boundary conditions.

For a system of semilinear elliptic partial differential equations with a small parameter, denned on a bounded multi-dimensional smooth domain, we show the existence of solutions with internal layers. The high-dimensionality of the domain gives rise to quite interesting an outlook in the analysis, dramatically different from that in one-dimensional settings. Our analysis indicates, in a certain situation, an occurrence of an infinite series of bifurcation phenomena accumulating as the small parameter goes to zero. We also present a related free boundary problem with a possible approach to its resolution.

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial-value problem, and these solutions need to be supplemented with further admissibility conditions. This paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns.

In a certain factor analysis model sampling formulae for maximum likelihood estimators of the parameters are found by inverting an augmented information matrix. The formulae were derived previously by other methods; but the techniques here employed may be of some general interest and applicable to other problems.

The homogenisation of a linearly viscoelastic composite material is performed without any periodicity assumptions and with no restrictions on the initial data. A term of fading memory is evidenced in the expression of the homogenised stress tensor.

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$, to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$, it is given by a condition on the homology group $H_2(\mathcal {R}_K)$, whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$.

We make some remarks about rank-one convex and polyconvex functions on the set of all real n × n matrices that vanish on the subset Kn consisting of all conformal matrices and grow like a power function at infinity. We prove that every non-negative rank-one convex function that vanishes on Kn and grows below a power of degree n/2 must vanish identically. In odd dimensions n ≧ 3, we prove that every non-negative polyconvex function that vanishes on Kn must vanish identically if it grows below a power of degree n; while in even dimensions, such polyconvex functions can exist that also grow like a power of half-dimension degree.