The spherical spectrum of a graded ring

We introduce a functor Sph, the spherical spectrum, which assigns to a graded ringG a space Sph(G) of homogeneous orderings ofG. It combines ideas of concrete geometry in theN-sphere defined by positively homogeneous polynomial equations and inequalities with the abstract notion of the real spectrum of a ring to give a counterpart for real semialgebraic geometry of the functor Proj.     

A Relation between formal and Asymptotic Equivalence for linear Systems Containing A Parameter

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ε^Pdy/dx = A(x,ε)y with A holomorphic at (x₀,0), suppose that x₀ is formally not a turning point in the sense that no singularities appear at x₀ during the standard formal solution procedure with formal fractional power series in ε. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x₀. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.     

Weierstrass preparation of quadratic differentials

Special cases or variants of the following play an important role in the asymptotic analysis of ordinary differential equations with turning points. Theorem.Let a(t, x) be a smooth complex-valued function germ at the origin in C × R ^m which is holomorphic in t. Suppose that a(t, 0) does not vanish identically. Then there is a smooth change of variable t = g(s, x), holomorphic in s, such that a(t, x)dt ² =P(s, x)ds ² where P is a monic polynomial in s.     

NMR chemical shifts of xenon in aqueous solutions of amphiphiles: A new probe of the hydrophobic environment

The chemical shift of elemental xenon in solution is sensitive to the environment. The shift arises from van der Waals interactions in most liquids, but an additional effect is present in aqueous media yielding a larger shift than expected. In water the shift is affected by the presence of low molecular weight amphiphiles, and its variation with composition can reveal the presence of hydrophobic hydration of the amphiphile. The results are similar to the conclusions drawn from other physical studies. Data are presented for aqueous solutions of methanol, ethanol, n-propanol, iso-propanol, tert-butanol, dimethylsulfoxide, p-dioxane, and acetonitrile.     

NMR chemical shifts of xenon in mixed aprotic solvents: A probe of liquid structure

The chemical shift of elemental xenon is extremely sensitive to the environment. In aprotic solvents, the presence of xenon has little effect on the solvent structure, and preferential solvation is not observed in any mixed solvent system. Consequently, xenon shifts can reveal the presence of short range order in certain liquids. Chemical shift data are presented for several model systems, including mixtures of different alkanes, alkanes with benzene, alkanes with acetone, and carbon tetrachloride with dimethylformamide (DMF). In certain cases, the xenon shift is strongly non-linear with composition. This effect arises from a specific interaction between the two solvents in the CCl₄-DMF system, while it reflects short range liquid order in the acetone-alkane systems. This effect is also apparent in the deviation of the densities of the acetone-alkane mixtures from ideality.     

A semialgebraic closure for commutative algebra

Let A⊂R be rings containing the rationals. In R let S be a multiplicatively closed subset such that 1∈S and 0∉S, T a preorder of R (a proper subsemiring containing the squares) such that S⊂T and I an A-submodule of R. Define ρ(I) (or ρ_S,T(I)) to be

ρ(I)={a∈R|sa^2m+t∈I^2m for some m∈N,s∈S and t∈T}.