The existence of solitons in non-commutative scalar field theories is proved for large values of the non-commutativity parameter using functional analysis. In the case of even phase-space dimensions greater than or equal to four, the solitons include some that are not rotationally symmetric.

Consider the Cauchy problem for a one-dimensional compressible flow through porous media,Hsiao and Liu showed that the solution (υ, u) behaves as the diffusion wave (ῡ, ū), i.e. the solution of the porous-media equation due to the Darcy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When υ0(x) has the same constant state at x = ±∞, the convergence rate ‖(υ − ῡ)(·,t)‖L∞ = O(t−1) obtained is ‘optimal’, since ‖ῡ(·,t)‖∞ = O(t−1/2). However, this ‘optimal’ convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the ‘truly optimal’ convergence rate by choosing suitably located diffusion waves.

Let Ω be a bounded C2,α domain in R2. We prove that the boundary-value problem Δυ = 0 in Ω, ∂υ/∂n = λsinh(υ) on ∂Ω, has infinitely many (classical) solutions for any given λ > 0. These solutions are constructed by means of a variational principle. We also investigate the limiting behaviour as λ → 0+; indeed, we prove that each of our solutions, as λ → 0+, after passing to a subsequence, develops a finite number of singularities located on ∂Ω.

We classify the Morse indices for rank-convex quadratic forms defined on the space of linear elastic strains in two- and three-dimensional linear elasticity. For the higher-dimensional case n > 3, we give a universal lower bound of the largest possible Morse index and various upper bound of this index. We show in the three-dimensional case that the Morse index is at most 1, and in this case the nullity cannot exceed 2. Examples are given that show that the estimates can be reached. We apply the results to study the critical points for smooth rank-one convex functions defined on the space of linear strains. We also examine an example and construct a quasiconvex function that vanishes in a finite set in the direct sum of the null subspace and the negative subspace of the rank-one quadratic form.

Special operator-ideal approximation properties (APs) of Banach spaces are employed to solve the problem of whether the distance functions S ↦ dist(S*, I(F*, E*)) and S ↦ dist(S, I*(E, F)) are uniformly comparable in each space L(E, F) of bounded linear operators. Here, I*(E, F) = {S ∈ L(E, F) : S* ∈ I(F*, E*)} stands for the adjoint ideal of the closed operator ideal I for Banach spaces E and F. Counterexamples are obtained for many classical surjective or injective Banach operator ideals I by solving two resulting ‘asymmetry’ problems for these operator-ideal APs.

We investigate equilibrium configurations for a polymer-stabilized liquid-crystal material subject to an applied magnetic field. The configurations are determined by energy minimization, where the energies of the system include those of bulk, surface and external field. The Euler–Lagrange equation is a nonlinear partial differential equation with nonlinear boundary conditions defined on a perforated domain modelling the cross-section of the liquid-crystal–polymer-fibre composite. We analyse the critical values for the external magnetic field representing Fredericks transitions and describe the equilibrium configurations under any magnitude of the external field. We also discuss the limit of the critical values and configurations as the number of polymer fibres approaches infinity. In the case where, away from the boundary of the composite, the fibres are part of a periodic array, we prove that non-constant configurations develop order-one oscillations on the scale of the array's period. Furthermore, we determine the small-scale structure of the configurations as the period tends to zero.

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.

We define a functional version of the Fučik spectrum for the Laplacian and we prove that this functional spectrum has empty interior.

We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone K1 (taking values in another closed cone K2) to the whole of K1.

We give conditions, in considerable generality (for cones in both finite- and infinite-dimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semi-continuous.

Maps f defined on the interior of the non-negative cone K in RN, which are both homogeneous of degree 1 and order preserving, are non-expanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that such an extension must have at least one eigenvector in K – {0}. In the case where the cycle time χ(f) of the original map does not exist, such eigenvectors must lie in ∂K – {0}.

We conclude with some discussions and applications to operator-valued means. We also extend our results to an ‘intermediate’ situation, which arises in some important application areas, particularly in the construction of diffusions on certain fractals via maps defined on the interior of cones of Dirichlet forms.

Several comparison results are obtained for solutions to linear elliptic and parabolic equations with a singular potential. Solutions to these equations are singular in many cases, and our results roughly say that they all have comparable singularities, provided that they belong to an appropriate space. We formulate the hypothesis on the potential in terms of an inequality, which in the case of the well-known inverse-square potential, is a consequence of an improvement of Hardy's inequality due to Vázquez and Zuazua.

We provide a simplified proof of the following well-known theorem. The projection to Rn + 1of an n-dimensional rectifiable Legendrian subset of Rn+1 × Snis C2-rectifiable. Such a result generalizes the following simple fact. Let M be a hypersurface whose Gauss map has graph of class C1. Then M is of class C2.

Determining the distribution of the number of empty urns after a number of balls have been thrown randomly into the urns is a classical and well understood problem. We study a generalization: Given a finite alphabet of size σ and a word length q, what is the distribution of the number X of words (of length q) that do not occur in a random text of length n+q−1 over the given alphabet? For q=1, X is the number Y of empty urns with σ urns and n balls. For q[ges ]2, X is related to the number Y of empty urns with σq urns and n balls, but the law of X is more complicated because successive words in the text overlap. We show that, perhaps surprisingly, the laws of X and Y are not as different as one might expect, but some problems remain currently open.

The sizes of the cycles and unicyclic components in the random graph $G(n, n/2\pm s)$, where $n^{2/3}\ll s \ll n$, are studied using the language of point processes. This refines several earlier results by different authors. Asymptotic distributions of various random variables are given: these distributions include the gamma distributions with parameters 1/4, 1/2 and 3/4, as well as the Poisson–Dirichlet and GEM distributions with parameters 1/4 and 1/2.

The bond percolation critical probability of the Kagomé lattice is greater than 0.5209 and less than 0.5291. The proof of these bounds uses the substitution method, comparing the percolative behaviour of the Kagomé lattice bond model with that of the exactly solved hexagonal lattice bond model via stochastic ordering.

A proper vertex coloring of a graph is called equitable if the sizes of colour classes differ by at most 1. In this paper, we find the minimum number l=l(d, Δ) such that every d-degenerate graph with maximum degree at most Δ admits an equitable t-colouring for every t[ges ]l when Δ[ges ]27d.

In this paper we prove that, if a collection of sets [Ascr] is union-closed, then the average set size of $\cal A is at least $\frac{1}{2}\log_2 (\vert{\cal A}\vert)$.

A general spectral bound for the sizes of some vertex subsets, which are mutually at a given minimum distance in a graph, is derived. This unifies and improves some previous results. Some applications to the study of certain metric parameters of the graph are then discussed.

It has been shown [2] that if n is odd and m1,…,mt are integers with mi[ges ]3 and [sum ]i=1tmi=|E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1,…,mt. This result was later generalized [3] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider the corresponding questions for directed graphs. We show that the complete directed graph <?TeX \displaystyle{\mathop{K}^{\raise-2pt\hbox{$\scriptstyle\leftrightarrow$}}}_{n}?> can be decomposed as an edge-disjoint union of directed closed trails of lengths m1,…,mt whenever mi[ges ]2 and <?TeX \sum_{i=1}^t m_{i}=\vert E({\leftrightarrow}_{n})\vert ?>, except for the single case when n=6 and all mi=3. We also show that sufficiently dense Eulerian digraphs can be decomposed in a similar manner, and we prove corresponding results for (undirected) complete multigraphs.

We prove that, for all values of the edge probability $p(n)$, the largest eigenvalue of the random graph $G(n, p)$ satisfies almost surely $\lambda_1(G)=(1+o(1))\max\{\sqrt{\Delta}, np\}$, where Δ is the maximum degree of $G$, and the o(1) term tends to zero as $\max\{\sqrt{\Delta}, np\}$ tends to infinity.