Clarke critical values of subanalytic Lipschitz continuous functions

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is locally finite. The proof relies on Pawlucki's extension of the Puiseuxlemma. In the last section we give an example of a continuous subanalytic function which is not constant on a segment of "broadly critical" points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points.

Limit cycles for cubic systems with a symmetry of order 4 and without in nite critical points

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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.

On the critical periods of perturbed isochronous centers

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The Ideas Debate in International and European Studies: Towards a Cartography and Critical Assessment

The appeal to ideas as causal variables and/or constitutive features of political processes increasingly characterises political analysis. Yet, perhaps because of the pace of this ideational intrusion, too often ideas have simply been grafted onto pre-existing explanatory theories at precisely the point at which they seem to get into difficulties, with little or no consideration either of the status of such ideational variables or of the character or consistency of the resulting theoretical hybrid. This is particularly problematic for ideas are far from innocent variables – and can rarely, if ever, be incorporated seamlessly within existing explanatory and/or constitutive theories without ontological and epistemological consequence. We contend that this tendency along with the limitations of the prevailing Humean conception of causality, and associated epistemological polemic between causal and constitutive logics, continue to plague almost all of the literature that strives to accord an explanatory role to ideas. In trying to move beyond the current vogue for epistemological polemic, we argue that the incommensurability thesis between causal and constitutive logics is only credible in the context of a narrow, Humean, conception of causation. If we reject this in favour of a more inclusive (and ontologically realist) understanding then it is perfectly possible to chart the causal significance of constitutive processes and reconstrue the explanatory role of ideas as causally constitutive.

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

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