Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system

Graded commutative algebras: examples, classification, open problems

We consider \G-graded commutative algebras, where \G is an abelian group. Starting from a remarkable example of the classical algebra of quaternions and, more generally, an arbitrary Clifford algebra, we develop a general viewpoint on the subject. We then give a recent classification result and formulate an open problem

Global bifurcation for asymptotically linear Schr\"odinger equations

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schr\"odinger equations \begin{equation}\label{1} \{{array}{lr} \D u + f(x,u)u = \lam u \quad \text{in} \ {\mathbb R}^N, u \in H^1({\mathbb R}^N)\setmimus\{0\}, \quad N \ge 1. {array}. \end{equation} The method is topological, based on recent developments of degree theory. We use the inversion $u\to v:= u/\Vert u\Vert_X^2$ in an appropriate Sobolev space $X=W^{2,p}({\mathbb R}^N)$, and we first obtain bifurcation from the line of trivial solutions for an auxiliary problem in the variables (\lambda,v) \in {\mathbb R} \x X. This problem has a lack of compactness and of regularity, requiring a truncation procedure. Going back to the original problem, we obtain global branches of positive/negative solutions 'bifurcating from infinity'. We believe that, for the values of $\lambda$ covered by our bifurcation approach, the existence result we obtain for positive solutions of \eqref{1} is the most general so fa

Nuclear phytochrome a signaling promotes phototropism in Arabidopsis.

Phototropin photoreceptors (phot1 and phot2 in Arabidopsis thaliana) enable responses to directional light cues (e.g., positive phototropism in the hypocotyl). In Arabidopsis, phot1 is essential for phototropism in response to low light, a response that is also modulated by phytochrome A (phyA), representing a classical example of photoreceptor coaction. The molecular mechanisms underlying promotion of phototropism by phyA remain unclear. Most phyA responses require nuclear accumulation of the photoreceptor, but interestingly, it has been proposed that cytosolic phyA promotes phototropism. By comparing the kinetics of phototropism in seedlings with different subcellular localizations of phyA, we show that nuclear phyA accelerates the phototropic response, whereas in the fhy1 fhl mutant, in which phyA remains in the cytosol, phototropic bending is slower than in the wild type. Consistent with this data, we find that transcription factors needed for full phyA responses are needed for normal phototropism. Moreover, we show that phyA is the primary photoreceptor promoting the expression of phototropism regulators in low light (e.g., PHYTOCHROME KINASE SUBSTRATE1 [PKS1] and ROOT PHOTO TROPISM2 [RPT2]). Although phyA remains cytosolic in fhy1 fhl, induction of PKS1 and RPT2 expression still occurs in fhy1 fhl, indicating that a low level of nuclear phyA signaling is still present in fhy1 fhl

Cancellation of probe effects in measurements of spin polarized momentum density by electron positron annihilation

Measurements of the two dimensional angular correlation of the electron-positron annihilation radiation have been done in the past to detect the momentum spin density and the Fermi surface. We point out that the momentum spin density and the Fermi Surface of ferromagnetic metals can be revealed within great detail owing to the large cancellation of the electron-positron matrix elements which in paramagnetic multiatomic systems plague the interpretation of the experiments. We prove our conjecture by calculating the momentum spin density and the Fermi surface of the half metal CrO2, who has received large attention due to its possible applications as spintronics material

A series of algebras generalizing the octonions and Hurwitz-Radon identity

International audienceWe study non-associative twisted group algebras over (ℤ2)n with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: (a21+⋯+a2N)(b21+⋯+b2ρ(N))=c21+⋯+c2N , where c k are bilinear functions of the a i and b j and where ρ(N) is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop

Cantilever-Enhanced Photoacoustic Spectroscopy of Radioactive Methane

We report the first high-resolution spectroscopy study of radiocarbon methane, 14CH4. Several absorption lines of the fundamental vibrational band v3 were measured using a continuous-wave mid-infrared optical parametric oscillator with cantilever-enhanced photoacoustic spectroscopy. © 2020 OSA.Peer reviewe