The life and letters of Charles Darwin, including an autobiographical chapter, ed. by his son Francis Darwin ...

1

The Cambridge natural history, ed. by S.F. Harmer ... and A.E. Shipley ...

v.6 (1901)

Studi sui Ragni Malesi e Papuani / per T. Thorell. Parte IV : dell'Indo-Malesia, raccolti da O. Beccari, G. Doria, H. Forbes, J. G. H. Kinberg Ed Altri.

v.2:p.4 (1891-1892)

Ethnology of the Mayas of southern and central British Honduras / by J. Eric Thompson -- Berthold Laufer -- ed.

v.17:no.2(1930)

Studi sui Ragni Malesi e Papuani / per T. Thorell. Parte IV : dell'Indo-Malesia, raccolti da O. Beccari, G. Doria, H. Forbes, J. G. H. Kinberg Ed Altri.

v.1:p.4 (1889-1890)

Infinite time aggregation for the critical Patlak-Keller-Segel model in R2

We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 Π/Χ. Actually, we prove that solutions blow-up as a delta dirac at the center of mass when t→∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t→∞ if initially bounded.

Convergence of the mass-transport steepest descent scheme for the sub-critical Patlak-Keller-Segel model

Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recover the recent result about the global in time existence of weak-solutions to the modified Patlak-Keller-Segel equation for the logarithmic interaction kernel in any dimension in the sub-critical case. Moreover, we show how this method performs numerically in one dimension. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudo-inverse of the cumulative distribution function. We demonstrate its capabilities to reproduce easily without the need of mesh-refinement the blow-up of solutions for super-critical masses.

Gestión de centros deportivos: Virtual Trainer Web Ed.

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Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions

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