On the shape of avalanches

In this paper we study a dimensionless model of granular matter. The model can be rewritten into a system of balance laws. We prove that, every sufficiently small, compactly supported perturbation of a Lipschitz continuous decoupled initial data gives decoupled solution in finite time. Moreover, no gradient catastrophe occurs, i.e., the solution does not develop discontinuities.     

On the shape of decomposable trees

A n-vertex graph is said to be decomposable if, for any partition (λ₁,…,λ_p) of the integer n, there exists a sequence (V₁,…,V_p) of connected vertex-disjoint subgraphs with |V_i|=λ_i. The aim of the paper is to study the homeomorphism classes of decomposable trees. More precisely, we show that homeomorphism classes containing decomposable trees with an arbitrarily large minimal distance between all pairs of distinct vertices of degree different from 2, is exactly the set of combs.     

Aperiodic tiles

A set of tiles (closed topological disks) is calledaperiodic if there exist tilings of the plane by tiles congruent to those in the set, but no such tiling has any translational symmetry. Several aperiodic sets have been discussed in the literature. We consider a number of aperiodic sets which were briefly described in the recent bookTilings and Patterns, but for which no proofs of their aperiodic character were given. These proofs are presented here in detail, using a technique with goes back to R. M. Robinson and Roger Penrose. The research of Branko Grünbaum was supported in part by NSF Grants MCS8301971 and DMS-8620181.     

On the optimal shape of magnetic swimmers

Magnetic actuation of elasto-magnetic devices has long been proposed as a simple way to propel fluid and achieve locomotion in environments dominated by viscous forces. Under the action of an oscillating magnetic field, a permanent magnet, when attached to an elastic tail, is able to generate bending waves and sufficient thrust for propulsion. We study the hydrodynamical effects of the magnetic head geometry using a geometrically exact formulation for the elastic tail elastohydrodynamics.We show that the spherical head geometry fails to take full advantage of the propulsive potential from the flexible tail. Nevertheless, while elongated prolate spheroids demonstrate a superior swimming performance, this is still regulated by the nature of the imposed magnetic field. Interestingly, the highest swimming speed is observed when the magnitude of the magnetic field is weak due to delays between the orientation of the magnetic moment and the oscillating magnetic field. This allows the stored elastic energy from the deformed tail to relax during the phase lag between the imposed magnetic field and the swimmer’s magnetic moment, favouring in this way the net propulsion. In particular, this result suggests the existence of optimal magnetic actuations that are non-smooth, and even discontinuous in time, in order to fully explore the propulsive potential associated with the relaxation dynamics of periodically deformed elastic filaments.     

On the optimal shape of a hydrofoil

On the basis of the results on domain shape optimization for elliptic systems developed in Ref. 1, the problem of the optimal shape of a hydrofoil, moving slowly in a viscous incompressible fluid, is formulated and studied. The otpimization problem consists in finding the shape of the hydrofoil having minimal drag, while satisfying certain constraints on the volume and hydrodynamic lift.     

On shape preserving semigroups

Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce a definition of shape preserving operator semigroups and analyze their fundamental properties. In particular, we prove that the class of shape preserving semigroups is preserved by perturbations and taking limits. These results are applied to partial delay differential equations.     

On shape of product spaces

It is known that if X is a compactum and Y is metrizable Sh₅(X × Y) is not generally determined by Sh₅(X) and Sh₅(Y), where Sh₅(Z) is the strong shape of Z in the sense of Borsuk. In this paper it is proved that Sh(X × Y) is uniquely determined by Sh(X) and Sh(Y), where Sh(Z) is the shape of Z in the sense of Fox. If X is an FANR and Y is an MANR, then X × Y is an MANR.     

Topology of crystallographic tiles

We study self-affine tiles which tile the n-dimensional real vector space with respect to a crystallographic group. First we define classes of graphs that allow to determine the neighbors of a given tile algorithmically. In the case of plane tiles these graphs are used to derive a criterion for such tiles to be homeomorphic to a disk. As particular application, we will solve a problem of Gelbrich, who conjectured that certain examples of tiles which tile with respect to the ornament group p2 are homeomorphic to a disk.      

On the shape of the mean residual lifetime function

Some general properties of the mean residual life (MRL) function are studied. The analysis is based on the shape of the corresponding failure rate. The conditions under which the failure rate and the reciprocal to the MRL function have asymptotically equivalent behaviour as t→∞ are discussed. The simplest non‐monotone shapes of the functions under consideration (bathtub and upside down bathtub) are also considered. The MRL functions for mixtures of distributions are described via the corresponding conditional probability density functions. The direct proportional model of mixing is characterized and some asymptotic results on the shape of the mixture MRL are obtained. Some simple examples are given. Copyright © 2002 John Wiley & Sons, Ltd.     

Each four-celled animal tiles the plane

An animal A is a set of unit squares in the plane, parallel to the axes, and with corners at integer lattice points. We show that any animal A with four cells tiles the plane, in the sense that infinitely many copies of A, translated by integer vectors and possibly rotated through 90°, 180°, or 270°, can be placed so as to fill plane exactly without overlap.     

On some coarse shape invariants

The author and N. Ugleši? have recently introduced a new classification of topological spaces, strictly coarser than the shape type classification. The corresponding coarse shape theory is founded on the coarse shape category. In this paper several topological and shape invariants are considered with respect to the coarse shape. It is shown that the coarse shape domination preserves connectedness, shape dimension, movability, n-movability and strong movability. Further, stability is a coarse shape invariant. Moreover, the coarse shape and shape coincide on the class of stable spaces.     

On permutation polynomials of prescribed shape

We count permutation polynomials of ${\mathbb{F}}_q$ which are sums of $m+1$ $(?2)$ monomials of prescribed degrees. This allows us to prove certain results about existence of permutation polynomials of prescribed shape.     

On the shape of the convex hull of random points

Summary Denote by E _n the convex hull of n points chosen uniformly and independently from the d-dimensional ball. Let Prob(d, n) denote the probability that E _n has exactly n vertices. It is proved here that Prob(d, 2 ^d/2 d ^-)1 and Prob(d, 2 ^d/2 d ^(3/4)+)0 for every fixed >0 when d. The question whether E _n is a k-neighbourly polytope is also investigated.     

On the shape of the moduli of spherical minimal immersions

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of . In this paper we define and study a sequence of metric invariants , , associated to a compact convex body with base point in the interior of . The invariant measures how lopsided is in dimension with respect to . The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate for convex polytopes.