Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.     

How Fast can a Subharmonic Function Decay Along a Ray?

Much is known about the connection between the growth and decayof subharmonic functions. The results indicate that there isa general principle: asubharmonic function cannot decay ‘toofast’ relative to its growth.Three theorems are provedwhich, together with work previously published elsewhere, givea fairly complete account of how this principle works out fora subharmonic function having extremal decay along a ray. 1991Mathematics Subject Classification: 30D20, 31A05.     

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form Y ²=f ₃ X ⁶+f ₂ X ⁴+f ₁ X ²+f ₀, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ. Received: 27 November 1998 / Revised version: 4 June 1999     

Characterization of the 1st and 2nd EF-hands of NADPH oxidase 5 by fluorescence, isothermal titration calorimetry, and circular dichroism

Background

Superoxide generated by non-phagocytic NADPH oxidases (NOXs) is of growing importance for physiology and pathobiology. The calcium binding domain (CaBD) of NOX5 contains four EF-hands, each binding one calcium ion. To better understand the metal binding properties of the 1^st and 2^nd EF-hands, we characterized the N-terminal half of CaBD (NCaBD) and its calcium-binding knockout mutants.

Results

The isothermal titration calorimetry measurement for NCaBD reveals that the calcium binding of two EF-hands are loosely associated with each other and can be treated as independent binding events. However, the Ca²⁺ binding studies on NCaBD(E31Q) and NCaBD(E63Q) showed their binding constants to be 6.5 × 10⁵ and 5.0 × 10² M^-1 with ??Hs of -14 and -4 kJ/mol, respectively, suggesting that intrinsic calcium binding for the 1^st non-canonical EF-hand is largely enhanced by the binding of Ca²⁺ to the 2^nd canonical EF-hand. The fluorescence quenching and CD spectra support a conformational change upon Ca²⁺ binding, which changes Trp residues toward a more non-polar and exposed environment and also increases its ??-helix secondary structure content. All measurements exclude Mg²⁺-binding in NCaBD.

Conclusions

We demonstrated that the 1^st non-canonical EF-hand of NOX5 has very weak Ca²⁺ binding affinity compared with the 2^nd canonical EF-hand. Both EF-hands interact with each other in a cooperative manner to enhance their Ca²⁺ binding affinity. Our characterization reveals that the two EF-hands in the N-terminal NOX5 are Ca²⁺ specific.

Graphical abstract

     

New measurements of proton-proton total cross section at the CERN intersecting storage rings

Measurements of the proton-proton total cross section have been made with increased precision (±0.6%) over the ISR energy range $\text{s}\text{=23.5?62.7}\text{GeV}$. Two different experimental methods gave consistent results, showing that the total cross section increases 10% over the ISR range and in addition that the absolute value of the ISR luminosity can be measured to ±0.9%.