We derive a compatible condition between coefficients and negative exponents, which is optimal for H01-solutions of strongly singular problems. This allows us to establish the existence of H01-solutions for p ≥ 3. Our proof is based on a variational argument.

The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limits

are positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

A semigroup is amiable if there is exactly one idempotent in each ℛ*-class and in each ℒ*-class. A semigroup is adequate if it is amiable and if its idempotents commute. We characterize adequate semigroups by showing that they are precisely those amiable semigroups that do not contain isomorphic copies of two particular non-adequate semigroups as subsemigroups.

The question of parallelizability and stable parallelizability of a family of manifolds obtained as a quotient of circle action on the complex Stiefel manifolds is studied and settled.

We consider a selection–mutation equation for the density of individuals with respect to a continuous phenotypic evolutionary trait. We assume that the competition term for an individual with a given trait depends on the traits of all the other individuals, therefore giving an infinite-dimensional nonlinearity. Mutations are modelled by means of an integral operator. We prove existence of steady states and show that, when the mutation rate goes to zero, the asymptotic profile of the population is a Cauchy distribution.

We consider a general class of parabolic equations of the type

with Dirichlet boundary conditions and with a right-hand side belonging to L1 + Lp′ (W−1, p′). Using the framework of renormalized solutions we prove uniqueness results under appropriate growth conditions and Lipschitz-type conditions on a(u, ∇u), K(u) and H(∇u).

A third-order system of ordinary differential equations, modelling two predators competing for a single prey species, is analysed in this paper. A delay term modelling the delayed logistic growth of the prey is included. Fixed points of the system are identified, and a linearized stability analysis is carried out. For some parameter regime, there exists a continuum of equilibria and these equilibria may undergo a zip bifurcation. The main results presented herein are that this zip bifurcation is ‘unsustainable’ for certain ranges of values of the time-delay parameter. Finally, spatial diffusion is incorporated in the delay differential equation model, and it is shown that the zip bifurcation remains unsustainable.

In this paper, we investigate the Dirichlet eigenvalue problems of the poly-Laplacian with any order and the quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first k eigenvalues.

Polynomial vector fields which admit a prescribed Darboux integrating factor are quite well understood when the geometry of the underlying curve is non-degenerate. In the general setting, morphisms of the affine plane may remove degeneracies of the curve, and thus allow more structural insight. In the present paper we establish some properties of integrating factors subjected to morphisms, and we discuss in detail one particular class of morphisms related to finite reflection groups. The results indicate that degeneracies for the underlying curve generally impose additional restrictions on vector fields admitting a given integrating factor.

We study the conflict-free chromatic number χCF of graphs from extremal and probabilistic points of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős–Rényi random graph G(n,p) and give the asymptotics for p = ω(1/n). We also show that for p ≥ 1/2 the conflict-free chromatic number differs from the domination number by at most 3.

It was pointed out by Crisan and Ghazali that the error estimate for the cubature on Wiener space algorithm developed by Lyons and Victoir requires an additional assumption on the drift. In this paper we demonstrate that it is straightforward to adopt the analysis of Kusuoka to obtain a general estimate without an additional assumptions on the drift. In the process we slightly sharpen the bounds derived by Kusuoka.

In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n-bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef, Birk, Jayram and Kol, IEEE Trans. Inform. Theory, 2011).

We show a polynomial-time algorithm that, given an n-vertex graph G with minrank k, finds a linear index code for G of length Õ(nf(k)), where f(k) depends only on k. For example, for k = 3 we obtain f(3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring (J. Assoc. Comput. Mach., 1998) and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank.

At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász ϑ-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a1n × a2n × ⋅⋅⋅ × adn box in $\mathbb{R}^d$ (for constants a1, . . ., ad > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1 − p. We show that if p = λ/n, then there exists λc > 0, which is the positive root of a degree d polynomial whose coefficients depend on a1, . . ., ad, such that for λ < λc the largest component has O(log n) vertices (w.h.p. as n → ∞), and for λ > λc the largest component has $(1-q) \lambda \bigl(\prod_i a_i \bigr) n^{d-1} + o (n^{d-1})$ vertices and the second largest component has O(log n) vertices w.h.p. An implicit formula for q < 1 is also given. The value of λc that we find is distinct from the critical value for the emergence of a giant component in bond percolation on the Hamming torus.

We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group.

Let $\mathfrak{F}$ be a locally compact nonarchimedean field with residue characteristic $p$, and let $\mathrm{G} $ be the group of $\mathfrak{F}$-rational points of a connected split reductive group over $\mathfrak{F}$. For $k$ an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke $k$-algebra ${\mathrm{H} }^{\prime } $ and of the pro-$p$ Iwahori–Hecke $k$-algebra $\mathrm{H} $ of $\mathrm{G} $. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of $\mathrm{G} $. If $\mathrm{G} $ is semisimple, we also show that this upper bound is sharp, that both $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $ are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of $\mathrm{H} $ (respectively ${\mathrm{H} }^{\prime } $). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $.

When $k$ has characteristic $p$, we prove that in ‘most’ cases $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $ have infinite global dimension. In particular, we deduce that the category of smooth $k$-representations of $\mathrm{G} = {\mathrm{PGL} }_{2} ({ \mathbb{Q} }_{p} )$ generated by their invariant vectors under the pro-$p$ Iwahori subgroup has infinite global dimension (at least if $k$ is algebraically closed).

We associate to every JB*-triple system a so-called universal enveloping ternary ring of operators (TRO). We compute the universal enveloping TROs of the finite dimensional Cartan factors.

We prove that if f is an integrable function on the unit sphere S in ℝn, g is its symmetric decreasing rearrangement and u, v are the harmonic extensions of f, g in the unit ball , then v has larger convex integral means over each sphere rS, 0 < r < 1, than u has. We also prove that if u is harmonic in with |u| < 1 and u(0) = 0, then the convex integral mean of u on each sphere rS is dominated by that of U, which is the harmonic function with boundary values 1 on the right hemisphere and −1 on the left one.

When can two fibrewise maps be deformed in a fibrewise fashion until they are coincidence free? In order to get a thorough understanding of this problem (and, more generally, of minimum numbers that are closely related to it) we study the strength of natural geometric obstructions, such as ω-invariants and Nielsen numbers, as well as the related Nielsen theory.

In the setting of sphere bundles, a certain degree map degB turns out to play a decisive role. In many explicit cases it also yields good descriptions of the set ℱ of fibrewise homotopy classes of fibrewise maps. We introduce an addition on ℱ, which is not always single valued but still very helpful. Furthermore, normal bordism Gysin sequences and (iterated) Freudenthal suspensions play a crucial role.

Nevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.