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Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)We develop and apply the theory of lower and upper compensated convex transforms introduced in [K. Zhang, Compensated convexity and its applications, Anal. Non-Lin. H. Poincaré Inst. 25 (2008) 743-771] to define multiscale, parameterized, geometric singularity extraction transforms of ridges, valleys and edges of function graphs and sets in R n . These transforms can be interpreted as 'tight' opening and closing operators, respectively, with quadratic structuring functions. We show that these geometric morphological operators are invariant with respect to translation, and stable under curvature perturbations, and establish precise locality and tight approximation properties for compensated convex transforms applied to bounded functions and continuous functions. We moreover establish multiscale and Hausdorff stable versions of such transforms. Specifically, the stable ridge transforms can be used to extract exterior corners of domains defined by their characteristic functions. Examples of explicitly calculated prototype mathematical models are given, as well as some numerical experiments illustrating the application of these transforms to 2d and 3d objects.

We model the growth, dispersal and mutation of two phenotypes of a species using reaction-diffusion equations, focusing on the biologically realistic case of small mutation rates. After verifying that the addition of a small linear mutation rate to a Lotka-Volterra system limits it to only two steady states in the case of weak competition, an unstable extinction state and a stable coexistence state, we prove that under some biologically reasonable condition on parameters the spreading speed of the system is linearly determinate. Using this result we show that the spreading speed is a non-increasing function of the mutation rate and hence that greater mixing between phenotypes leads to slower propagation. Finally, we determine the ratio at which the phenotypes occur at the leading edge in the limit of vanishing mutation.

Highlights d There is extreme variation in the probability of food finding in vertebrate species d Decreasing probability increases inter-individual variation in foraging duration d This variability can result in individual or breeding ruin d Apex predators are most likely to incur ruin under changing environmental conditions

In this paper we introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale λ, and provide a sharp regularity result for the squared-distance function to any closed non-empty subset K of R n . Our results exploit properties of the function C l λ (dist 2 (·; K)) obtained by applying the quadratic lower compensated convex transform of parameter λ [59] to dist 2 (·; K), the Euclidean squared-distance function to K. Using a quantitative estimate for the tight approximation of dist 2 (·; K) by C l λ (dist 2 (·; K)), we prove the C 1,1 -regularity of dist 2 (·; K) outside a neighbourhood of the closure of the medial axis M K of K, which can be viewed as a weak Lusintype theorem for dist 2 (·; K), and give an asymptotic expansion formula for C l λ (dist 2 (·; K)) in terms of the scaled squared distance transform to the set and to the convex hull of the set of points that realize the minimum distance to K. The multiscale medial axis map, denoted by M λ (·; K), is a family of non-negative functions, parametrized by λ > 0, whose limit as λ → ∞ exists and is called the multiscale medial axis landscape map, M ∞ (·; K). We show that M ∞ (·; K) is strictly positive on the medial axis M K and zero elsewhere. We give conditions that ensure M λ (·; K) keeps a constant height along the parts of M K generated by two-point subsets with the value of the height dependent on the scale of the distance between the generating points, thus providing a hierarchy of heights (hence, the word 'multiscale') between different parts of M K that enables subsets of M K to be selected by simple thresholding. Asymptotically, further understanding of the multiscale effect is provided by our exact representation of M ∞ (·; K). Moreover, given a compact subset K of R n , while it is well known that M K is not Hausdorff stable, we prove that in contrast, M λ (·; K) is stable under the Hausdorff distance, and deduce implications for the localization of the stable parts of M K . Explicitly calculated prototype examples of medial axis maps are also presented and used to illustrate the theoretical findings. a grass fire lit on the boundary and allowed to propagate uniformly inside the object, closely related definitions of skeleton [17] and cut-locus [53] have since been proposed, and have served for the study of its topological properties [3,22,41,44,51], its stability [23,21] and for the development of fast and efficient algorithms for its computation [1,12,11,39,46]. Applications of the medial axis are ample in scope and nature, ranging from computer vision to image analysis, from mesh generation to computer aided design. We refer to [50] and the references therein for applications and accounts of some recent theoretical developments.An inherent drawback of the medial axis is, however, its sensitivity to boundary details, in the sense that small perturbations of the object (with respect to the Hausdorff distance) can produce huge variations of the corr...

Interfacial energy is often incorporated into variational solid-solid phase transition models via a perturbation of the elastic energy functional involving second gradients of the deformation. We study consequences of such higher-gradient terms for local minimizers and for interfaces. First it is shown that at slightly sub-critical temperatures, a phase which globally minimizes the elastic energy density at super-critical temperatures is an L 1 -local minimizer of the functional including interfacial energy, whereas it is typically only a W 1,∞ -local minimizer of the purely elastic functional. The second part deals with the existence and uniqueness of smooth interfaces between different wells of the multi-well elastic energy density. Attention is focussed on so-called planar interfaces, for which the deformation depends on a single direction x · N and the deformation gradient then satisfies a rank-one ansatz of the form Dy(x) = A + u(x · N ) ⊗ N , where A and B = A + a ⊗ N are the gradients connected by the interface. Mathematics Subject Classification (2000)74A50 · 49J45 · 74B20 · 74G25

We introduce Lipschitz continuous and C 1,1 geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in R n by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of C 1,1 functions. We also prove that our approximation methods, which are differentiation and integration free and not sensitive to sample type, are stable with respect to the Hausdorff distance between samples.

there is an underlying drift (chromatography, convection in chemical reactions or wind effects in biology [9, pp. 292, 322 and 420]). Some work has been done on scalar-valued gradient-dependent problems [14, p. 111]. Here we prove existence theorems which extend that of [14] to gradient-dependent cases. Corresponding stability results are under development and will be presented elsewhere. In Sections 2-4 of this paper, we prove the existence of a monotone travelling-wave solution for the reaction-diffusion-convection systemunder the following hypotheses. (Standard notation is recalled at the end of the Introduction.)(a) A is a positive-definite diagonal matrix.As in [14], f : R N → R N is a continuously differentiable function satisfying (f1) f is locally monotone;(f2) S and T are stable equilibria of f and S < T ; (f3) there is a (necessarily non-zero) finite number of equilibria E of f with S < E < T and each such E is unstable. (g1) G is a continuously differentiable, diagonal-matrix-valued function on R N × R N and there exist continuous functions β, γ :where β is increasing andThus for some c ∈ R, we obtain a solution w of the system (4)Aw + cw + G(w, w )w + f (w) = 0.Note that (g2) can be assumed without loss of generality in a problem where (f3) holds. This is obvious because, by ( f3), dI + G(E, 0) is positive-definite at each of the finite number of equilibria E in [S, T ] provided d is sufficiently large. Now replace the unknown parameter c with a new parameter c = c − d and replace G with G = G + dI. The new system, (4) with c instead of c and G instead of G satisfies (a), (f1)-(f3) and (g1), (g2).The main result of this paper is in Section 5. There the existence of travelling waves is established without assuming (g1) about the growth of G, provided that G is known to be monotone in a certain weak sense. These results are deduced using the theory of the preceding sections, and contain the previous existence theorems as special cases.