For fixed positive integers $k,q,r$ with $q$ a prime power and large $m$, we investigate matrices with $m$ rows and a maximum number $N_q (m,k,r)$ of columns, such that each column contains at most $r$ nonzero entries from the finite field $GF(q)$ and any $k$ columns are linearly independent over $GF(q)$. For even integers $k \geq 2$ we obtain the lower bounds $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))})$, and $N_q(m,k,r) = \Omega (m^{((k-1)r)/(2(k-2))})$ for odd $k \geq 3$. For $k=2^i$ we show that $N_q(m,k,r) = \Theta ( m^{kr/(2(k-1))})$ if $\gcd(k-1,r) = k-1$, while for arbitrary even $k \geq 4$ with $\gcd(k-1,r) =1$ we have $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))} \cdot (\log m)^{1/(k-1)})$. Matrices which fulfil these lower bounds can be found in polynomial time. Moreover, for $\Char (GF(q)) > 2 $ we obtain $N_q(m,4,r) = \Theta (m^{\lceil 4r/3\rceil/2})$, while for $\Char (GF(q)) = 2$ we can only show that $N_q(m,4,r) = O (m^{\lceil 4r/3\rceil/2})$. Our results extend and complement earlier results from [7, 18], where the case $q=2$ was considered.

Some results are presented relating to questions raised in a recent paper by Anderson, Hayman and Pommerenke regarding the size of the set of boundary points of the unit disc at which a univalent function has a prescribed radial growth.

AMS 2000 Mathematics subject classification: Primary 30C55

A zigzag in a plane graph is a circuit of edges, such that any two, but not three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbours on opposite edges. A graph without a railroad is called tight. We consider the zigzag and railroad structures of general 3-valent plane graph and, especially, of simple two-faced polyhedra, i.e., 3-valent 3-polytopes with only $a$-gonal and $b$-gonal faces, where $3 \leq a < b \leq 6$; the main cases are $(a,b)=(3,6), (4,6)$ and $(5,6)$ (the fullerenes).

We completely describe the zigzag structure for the case $(a,b)\,{=}\,(3,6)$. For the case $(a,b)\,{=}\,(4,6)$ we describe symmetry groups, classify all tight graphs with simple zigzags and give the upper bound 9 for the number of zigzags in general tight graphs. For the remaining case $(a,b)\,{=}\,(5,6)$ we give a construction realizing a prescribed zigzag structure.

This paper deals with local and global existence for the solutions of the heat equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied iswhere Ω ⊂ Rn (n ≥ 1) is a regular and bounded domain, ∂Ω = Γ0 ∪ Γ1, m > 1, 2 ≤ p < r, where r = 2(n − 1)/(n − 2) when n ≥ 3, r = ∞ when n = 1, 2 and u0 ∈ H1(Ω), u0 = 0 on Γ0. We prove local existence of the solutions in H1(Ω) when m > r/(r + 1−p) or n = 1, 2 and global existence when p ≤ m or the initial datum is inside the potential well associated to the stationary problem.

In this paper we study Mumford–Shah-type functionals associated with doubling metric measures or strong A∞ weights in the setting of the perimeter theory in the sense of Ambrosio and Miranda in metric spaces. We prove an existence theorem in a suitably defined class of special BV functions.

We establish some bifurcation results for the boundary-value problem −Δu = g (u) + λ|∇u|p + μf (x, u) in Ω, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth bounded domain in RN, λ, μ ≥ 0, 0 < p ≤ 2, f is non-decreasing with respect to the second variable and g is unbounded around the origin. The asymptotic behaviour of the solution around the bifurcation point is also established, provided g(u) behaves like u−α around the origin, for some 0 < α < 1. Our approach relies on finding explicit sub- and supersolutions combined with various techniques related to the maximum principle for elliptic equations. The analysis we develop in this paper shows the key role played by the convection term |∇u|p.

In this paper we will show that the optimal bounds for certain static and dynamic bifurcation values of periodic solutions of some superlinear differential equations can be expressed explicitly using Sobolev constants.

We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.

In this paper we study the Morse functions on Grassmann manifolds. It is shown that many well-known calibrations can be used to construct degenerate Morse functions on Grassmann manifolds. In the study we use Clifford algebras and Lie groups.

For the family of truncations of the Gaussian Riesz transforms and Poisson integral we study their rate of convergence through the oscillation and variation operators. More precisely, we search for their Lp (dγ)-boundedness properties, where dγ denotes the Gauss measure. We achieve our results by looking at the oscillation and variation operators from a vector-valued point of view.

In this paper we establish the existence of many positive solutions towith Vx a vector-valued nonlinearity.

We prove a sharp Hölder estimate for solutions of linear, two-dimensional, divergence-form, elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has unit determinant. Our result extends some previous work by Piccinini and Spagnolo. The proof relies on a sharp Wirtinger type inequality.

We develop the geometric two-scale convergence on forms in order to describe the homogenization of partial differential equations with random variables on non-flat domain. We prove the compactness theorem and some two-scale behaviours for differential forms. For its applications, we investigate the limiting equations of the n-dimensional Maxwell equations with random coefficients,with given initial and boundary conditions, where are symmetric positive-definite matrices for x ∈ M, and M is an n-dimensional compact oriented Riemannian manifold with smooth boundary. The limiting system of n-dimensional Maxwell equations turns out to be degenerate and it is proven to be well-posed. The homogenized coefficients affected by the geometry of the domain are presented, and compared with the homogenized coefficient of the second order elliptic equation. We present the convergence theorem in order to explain the convergence of the solutions of Maxwell system as a parabolic partial differential equation.

By looking for critical points of functionals defined in some subspaces of , invariant under some subgroups of O (N), we prove the existence of many positive non-radial solutions for the following semilinear elliptic problem involving critical Sobolev exponent on an annulus,where 2* − 1 := (N + 2)/(N − 2) (N ≥ 4), the domain is an annulusand f : R+ × R+ → R is a C1 function, which is a subcritical perturbation.

This paper is concerned with the Holling–Tanner prey–predator model with diffusion subject to the homogeneous Neumann boundary condition. We obtain the existence and non-existence of positive non-constant steady states.

Associated with $T=U|T|$ (polar decomposition) in ${\cal{L}}({\bf H})$ is a related operator $\skew3\tilde{T} = |T|^{\frac{1}{2}}U|T|^{\frac{1}{2}}$, called the Aluthge transform of $T$. In this paper we study some connections between $T$ and $\skew3\tilde{T}$, including the following relations; the single valued extension property, an analogue of the single valued extension property on $W^{m}(D, {\bf H})$, Dunford's property $(C)$ and the property $(\beta)$.

A global existence result for solutions $u(t)$ of the differential equation $x^{\prime \prime }+f(t,x)=p(t)$, $t\geq t_0\geq 1$, that can be written as $u(t)=P(t)+o(1)$ for all large $t$, where $P^{\prime \prime}(t)=p(t)$, is established by means of the Schauder-Tikhonov theorem. It generalizes the recent work of Lipovan [On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations, Glasgow Math. J.45 (2003), 179–87] and allows for a unifying treatment of the existence problems concerning asymptotically linear and oscillatory solutions of second order nonlinear differential equations.

Tensor analogues of right 2-Engel elements in groups were introduced by D. P. Biddle and L.-C. Kappe. We investigate the properties of right 2-Engel tensor elements and introduce the concept of $2_{\otimes}$-Engel margin. With the help of these results we describe the structure of $2_{\otimes}$-Engel groups. In particular, we prove a tensor version of Levi's theorem for 2-Engel groups and determine tensor squares of two-generator $2_{\otimes}$-Engel $p$-groups.

We give characterisations of the linear relations that are injective and open (respectively, almost open with dense range) in terms of the stability of the nullity (respectively, of the deficiency). Results of Mbekhta about bounded operators in Banach spaces (J. Operator Theory35 (1996), 191–201) are covered.

We consider the primes which divide the denominator of the $x$-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.