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.

Seven species belonging to the Echinoderm order CRINOIDEA, were procured during the “Porcupine” dredging expeditions of 1869 and 70. Four of these belong to the free section of the order, and are referred to the genus Antedon.

The subject of the production within Diatoms of spores, en docysts, &c., has been worked at by several observers, principally Castracane in recent years, and before him O'Meara and Rabenhorst. Some of these observers have noted ciliated, spore-like bodies, probably parasitic or casual intrusive organisms; others, notably Castracane, have recorded oval and other cysts winch have not been figured and the characters of which are difficult to realise. Miquel has made an experimental examination of the subject without being able to confirm Castracane's results. Castracane has also recorded the presence, as he thinks, of “gonids or embryonal forms” within a fossil Diatom (Coscinodiscus punctatus) in a marine deposit of miocene age. During March of this year I tow-netted near the Bell Rock a large quantity of Coscinodiscus concinnus — with many dead valves separated at the girdle. These frequently contained not only other and smaller individuals of the same species, but other species of diatoms and casual objects as well, and the observation makes one cautious about accepting evidence of the character brought forward by Castracane as to his fossil form.

Lauder, in describing forms of Bacteriastrum and Chœtoceros, observed the formation of endocysts. “At certain times, or under certain circumstances, the endochrome does not divide after the lengthening of the frustule, but secretes a siliceous envelope and becomes a gonidium or sporangium, consisting of a cell with two rounded ends, and a connecting hoop, one end being smaller than the other ….

We consider a class of generalized nonlocal $p$-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.

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.

Let λ > 0 and let

be the Bessel operator on ℝ+ := (0,∞). We show that the oscillation operator 𝒪(RΔλ,∗) and variation operator 𝒱ρ(RΔλ,∗) of the Riesz transform RΔλ associated with Δλ are both bounded on Lp(ℝ+, dmλ) for p ∈ (1,∞), from L1(ℝ+, dmλ) to L1,∞(ℝ+, dmλ), and from L∞(ℝ+, dmλ) to BMO(ℝ+, dmλ), where ρ ∈ (2,∞) and dmλ(x) := x2λ dx. As an application, we give the corresponding Lp-estimates for β-jump operators and the number of up-crossings.

The compact weak topology (kw) on a Banach space is defined as the finest topology that agrees with the weak topology on weakly compact subsets. It appears in a natural manner in the study of certain classes of continuous and holomorphic maps between Banach spaces. In this paper we treat the kw topology and the finest locally convex topology contained in kw, which we call the ckw topology. We prove that kw = ckw if and only if the space is reflexive or Schur, and we derive characterisations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. We also show that ckw is the topology of uniform convergence on (L)-subsets of the dual space. As a consequence, Banach spaces with the reciprocal Dunford–Pettis property are characterised.

The paper is devoted to the development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. We mainly focus on the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results involve boundary value techniques and reproducing kernel space methods, and they include an explicit functional model for the class of stable C-symmetries. Some of the results are specialized further by studying the case where S has defect numbers 〈2,2〉 in detail.

We explore some necessary conditions for quasiconvexity in an attempt to show that rank-one convexity does not imply quasiconvexity when the target space for deformations is two- dimensional. An interesting construction is presented, showing how rank-one directions may fit with each other, making the task harder than in higher dimensions.

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.

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.

Given the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.