Maltsiniotis's first conjecture for K_1

We show that K_1 of an exact category agrees with K_1 of the associated triangulated derivator. More generally we show that K_1 of a Waldhausen category with cylinders and a saturated class of weak equivalences coincides with K_1 of the associated right pointed derivator.Comment: 23 pages, the main results have been generalize

Robust Stability Under Mixed Time Varying, Time Invariant and Parametric Uncertainty

Robustness analysis is considered for systems with structured uncertainty involving a combination of linear time-invariant and linear time-varying perturbations, and parametric uncertainty. A necessary and sufficient condition for robust stability in terms of the structured singular value μ is obtained, based on a finite augmentation of the original problem. The augmentation corresponds to considering the system at a fixed number of frequencies. Sufficient conditions based on scaled small-gain are also considered and characterized

Slowly Oscillating Solution of the Cubic Heat Equation

In this paper, we are considering the Cauchy problem of the nonlinear heat equation $u\_t -\Delta u= u^{3 },\ u(0,x)=u\_0$. After extending Y. Meyer's result establishing the existence of global solutions, under a smallness condition of the initial data in the homogeneous Besov spaces $\dot{B}\_{p}^{-\sigma, \infty}(\mathbb{R}^{3})$, where 3 \textless{} p \textless{} 9 and $\sigma=1-3/p$, we prove that initial data $u\_0\in \mathcal{S}(\mathbb{R}^{3})$, arbitrarily small in ${\dot B^{-2/3,\infty}\_{9}}(\mathbb{R}^{3})$, can produce solutions that explode in finite time. In addition, the blowup may occur after an arbitrarily short time

Wave dispersion derived from the square-root Klein-Gordon-Poisson system

Recently there has been great interest around quantum relativistic models for plasmas. In particular striking advances have been obtained by means of the Klein-Gordon-Maxwell system, which provides a first order approach to the relativistic regimes of quantum plasmas. It is a reliable method as long as the plasma spin dynamics is not a fundamental aspect, to be addressed using more refined (and heavier) models involving the Pauli-Schr\"odinger or Dirac equations. In this work a further simplification is considered, tracing back to the early days of relativistic quantum theory. Namely, we revisit the square-root Klein-Gordon-Poisson system, where the positive branch of the relativistic energy-momentum relation is mapped to a quantum wave equation. The associated linear wave propagation is analyzed and compared to the results in the literature. We determine physical parameters where the simultaneous quantum and relativistic effects can be noticeable in weakly coupled electrostatic plasmas