38 research outputs found
The Steiner tree problem revisited through rectifiable G-currents
The Steiner tree problem can be stated in terms of finding a connected set of
minimal length containing a given set of finitely many points. We show how to
formulate it as a mass-minimization problem for -dimensional currents with
coefficients in a suitable normed group. The representation used for these
currents allows to state a calibration principle for this problem. We also
exhibit calibrations in some examples
Is a nonlocal diffusion strategy convenient for biological populations in competition?
We study the convenience of a nonlocal dispersal strategy in a
reaction-diffusion system with a fractional Laplacian operator. We show that
there are circumstances - namely, a precise condition on the distribution of
the resource - under which a nonlocal dispersal behavior is favored.
In particular, we consider the linearization of a biological system that
models the interaction of two biological species, one with local and one with
nonlocal dispersal, that are competing for the same resource. We give a simple,
concrete example of resources for which the equilibrium with only the local
population becomes linearly unstable. In a sense, this example shows that
nonlocal strategies can become successful even in an environment in which
purely local strategies are dominant at the beginning, provided that the
resource is sufficiently sparse.
Indeed, the example considered presents a high variance of the distribution
of the dispersal, thus suggesting that the shortage of resources and their
unbalanced supply may be some of the basic ingredients that favor nonlocal
strategies
On some geometric properties of currents and Frobenius theorem
In this note we announce some results, due to appear in [2], [3], on the
structure of integral and normal currents, and their relation to Frobenius
theorem. In particular we show that an integral current cannot be tangent to a
distribution of planes which is nowhere involutive (Theorem 3.6), and that a
normal current which is tangent to an involutive distribution of planes can be
locally foliated in terms of integral currents (Theorem 4.3). This statement
gives a partial answer to a question raised by Frank Morgan in [1].Comment: 8 page
Risultati di tipo Nash per le immersioni isometriche
Analizziamo dei risultati di rigiditĂ /flessibilitĂ tipici del principio di omotopia nell'ambito delle immersioni isometriche di varietĂ riemanniane. In particolare, si riprendono il teorema di Nash e le tecniche iterative per dimostrarlo: l'integrazione convessa di Gromov e il teorema di Baire
Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity
In the modeling of dislocations one is lead naturally to energies
concentrated on lines, where the integrand depends on the orientation and on
the Burgers vector of the dislocation, which belongs to a discrete lattice. The
dislocations may be identified with divergence-free matrix-valued measures
supported on curves or with 1-currents with multiplicity in a lattice. In this
paper we develop the theory of relaxation for these energies and provide one
physically motivated example in which the relaxation for some Burgers vectors
is nontrivial and can be determined explicitly. From a technical viewpoint the
key ingredients are an approximation and a structure theorem for 1-currents
with multiplicity in a lattice
An elementary proof of the rank-one theorem for BV functions
We provide a simple proof of a result, due to G. Alberti, concerning a rank-one property for the singular part of the derivative of vector-valued functions of bounded variation
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Is a nonlocal diffusion strategy convenient for biological populations in competition?
We study the convenience of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances - namely, a precise condition on the distribution of the resource - under which a nonlocal dispersal behavior is favored. In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete example of resources for which the equilibrium with only the local population becomes linearly unstable. In a sense, this example shows that nonlocal strategies can become successful even in an environment in which purely local strategies are dominant at the beginning, provided that the resource is sufficiently sparse. Indeed, the example considered presents a high variance of the distribu- tion of the dispersal, thus suggesting that the shortage of resources and their unbalanced supply may be some of the basic ingredients that favor nonlocal strategies