Bardi and R. Jensen , A geometric characterization of viable sets for controlled degenerate diffusions. Set-Valued Anal. Barles and C. Imbert , Second-order elliptic integro-differential equations : Viscosity solutions theory revisited.

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## Viability, Invariance and Applications on Apple Books

Rainer , Existence of stochastic control under state constraints. I Math. Buckdahn , D. Goreac and M. Quincampoix , Stochastic optimal control and linear programming approach. Cook , A. Gerber and S. Tapscott , Modelling stochastic gene expression : Implications for haploinsufficiency.

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Practical information : There is no registration fee. The deadline for registration was on 1 December and, therefore, the registration is now closed. The synergy of the Standard Model of particle physics and General Relativity led to a consistent framework that has been confirmed by numerous experiments and observations. In spite of their undeniable success, these theories cannot be considered as complete theories of Nature.

On the one hand, they fail to explain basic observational facts such as the existence of neutrino masses, the presence of a sizable dark matter component or the matter-antimatter asymmetry of the Universe. On the other hand, there is no satisfactory account of tiny dimensionless ratios, such as the Fermi scale over the Planck scale hierarchy problem , or the dark energy density or cosmological constant in units of the Planck mass. The discovery of a relatively light Higgs boson at the LHC together with the absence of new physics beyond the Standard Model rejuvenated scale symmetry as an appealing scenario where to address the hierarchy problem.

This symmetry consists of a common multiplicative scaling of all fields according to their dimension.

No dimensional parameters are allowed to appear in the action. In particular, dilatation symmetry ensures the absence of a Higgs mass term. In order to describe the appearance of physical scales, a viable scale-invariant theory should exhibit dilatation symmetry breaking in one way or another.

This symmetry breaking could be explicit as a consequence of dimensional transmutation, as happens for instance in QCD.

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## Internet Invariants: What Really Matters

In this case, the dilatation symmetry is anomalous and appears only in the vicinity of non-trivial fixed points. In addition, spontaneous scale symmetry breaking driven, for instance, by the vacuum expectation value of a scalar field, could provide particle masses even in the absence of explicit mass terms. Popular Features. New Releases. Description The book is an almost self-contained presentation of the most important concepts and results in viability and invariance. The viability of a set K with respect to a given function or multi-function F, defined on it, describes the property that, for each initial data in K, the differential equation or inclusion driven by that function or multi-function to have at least one solution.

The invariance of a set K with respect to a function or multi-function F, defined on a larger set D, is that property which says that each solution of the differential equation or inclusion driven by F and issuing in K remains in K, at least for a short time. The book includes the most important necessary and sufficient conditions for viability starting with Nagumo's Viability Theorem for ordinary differential equations with continuous right-hand sides and continuing with the corresponding extensions either to differential inclusions or to semilinear or even fully nonlinear evolution equations, systems and inclusions.

In the latter i. Product details Format Hardback pages Dimensions x x Other books in this series. Add to basket.

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Graphs of Groups on Surfaces: Volume White. Truth, Possibility and Probability: Volume R. Topological Algebras with Involution: Volume M. History of Functional Analysis: Volume 49 J. Developmental Psychology C-. Topics in Soliton Theory: Volume R. Table of contents 1.