Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse

We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2＋y2-1＋y（ax＋by＋c）,y=x（ax＋by＋c）,and the ellipse becomes x2＋y2=1.We prove that（i） this quadratic system is analytic integrable if and only if a=0;（ii） if x2＋y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2＋y2=1 is not a limit cycle;and（iii） if x2＋y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.     

A note on the Stefanescu conjecture

We prove the existence of a magnetic field created by a planar configuration of piecewise rectilinear wires having no analytic first integral. This is a counterexample to the Stefanescu conjecture (Rev Roum Phys 31:701–721, 1986) in the analytic setting.     

Weierstrass Integrability of Complex Differential Equations

We characterize the complex differential equations of the form ■ where a_j(x) are meromorphic functions in the variable x for j = 0,..., n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.     

Local density of states and order parameter configurations in layered ferromagnet-superconductor structures

We analyze the local density of states (LDOS) of heterostructures consisting of alternating ferromagnet, F, and superconductor, S, layers. We consider structures of the SFS and SFSFSFS type, with thin nanometer scale F and S layers, within the ballistic regime. The spin-splitting effects of the ferromagnet and the mutual coupling between the S regions, yield several nontrivial stable and metastable pair amplitude configurations, and we find that the details of the spatial behavior of the pair amplitude govern the calculated electronic spectra. These are reflected in discernible signatures of the LDOS. The roles that the magnetic exchange energy, interface scattering strength, and the Fermi wavevector mismatch each have on the LDOS for the different allowed junction configurations, are systematically investigated.     

Biosynthesis and structure of aeruginoside 126A and 126B, cyanobacterial peptide glycosides bearing a 2-carboxy-6-hydroxyoctahydroindole moiety

Aeruginosins represent a group of peptide metabolites isolated from various cyanobacterial genera and from marine sponges that potently inhibit different types of serine proteases. Members of this family are characterized by the presence of a 2-carboxy-6-hydroxyoctahydroindole (Choi) moiety. We have identified and fully sequenced a NRPS gene cluster in the genome of the cyanobacterium Planktothrix agardhii CYA126/8. Insertional mutagenesis of a NRPS component led to the discovery and structural elucidation of two glycopeptides that were designated aeruginoside 126A and aeruginoside 126B. One variant of the aglycone contains a 1-amino-2-(N-amidino-Delta(3)-pyrrolinyl)ethyl moiety at the C terminus, the other bears an agmatine residue. In silico analyses of the aeruginoside biosynthetic genes aerA-aerI as well as additional mutagenesis and feeding studies allowed the prediction of enzymatic steps leading to the formation of aeruginosides and the unusual Choi moiety.     

Robustness of exponential behavior in Banach spaces

We consider a generalization of the notion of exponential dichotomy in which the exponential behaviors on \(\mathbb {R}^+\) and \(\mathbb {R}^-\) need not agree at the origin, although they still satisfy a certain compatibility condition. A nontrivial example is the change from stable to unstable behavior in a given direction, such as in a saddle-node bifurcation. Our main aim is to show that this exponential behavior is robust, in the sense that it persists under sufficiently small linear perturbations. We emphasize that we consider arbitrary evolution families in Banach spaces. This includes any differentiable evolution family obtained from a nonautonomous linear equation \(x^{\prime } =A(t) x\) possibly with A(t) unbounded, although in general we do not require the evolution families to be differentiable.     

Odd triplet pairing in clean superconductor/ferromagnet heterostructures

We study triplet pairing correlations in clean ferromagnet (F)/superconductor (S) nanojunctions, via fully self-consistent solution of the Bogoliubov-de Gennes equations. We consider FSF trilayers, with S being an s-wave superconductor, and an arbitrary angle alpha between the magnetizations of the two F layers. We find that contrary to some previous expectations, triplet correlations, odd in time, are induced in both the S and F layers in the clean limit. We investigate their behavior as a function of time, position, and alpha. The triplet amplitudes are largest at times on the order of the inverse Debye frequency, and at that time scale they are long-ranged in both S and F. The zero temperature condensation energy is found to be lowest when the magnetizations are antiparallel.     

Lyapunov functions versus exponential contractions

For nonautonomous linear equations in a Banach space admitting a nonuniform version of exponential contraction, we give an optimal characterization of the exponential behavior in terms of strict Lyapunov sequences. In particular, we construct explicitly strict Lyapunov sequences for each exponential contraction. We also consider the particular case of quadratic Lyapunov functions, and we use the corresponding characterization of the exponential behavior in terms of these functions to show that the stability of an exponential contraction persists under sufficiently small perturbations.