f(T1, . . . , Tn) = f(Tσ(1), . . . , Tσ(n)) for all σ ∈ Sn. Example 1. The sum T1 + · · ·+ Tn and product T1 · · ·Tn are symmetric, as are the power sums T r 1 + · · ·+ T r n for any r ≥ 1. As a measure of how symmetric a polynomial is, we introduce an action of Sn on F [T1, . . . , Tn]: (σf)(T1, . . . , Tn) = f(Tσ−1(1), . . . , Tσ−1(n)). We need σ−1 rather than σ on the right side so this is a...

The Direction of a Reaction Suppose we have a reaction such as A + B ↔ C , which has come to equilibrium at some temperature τ . Now we raise the temperature. Does the equilibrium shift to the left (more A and B) or to the right (more C)? The heat of reaction at constant pressure, Qp, is the heat that must be supplied to the system if the reaction goes from left to right. If Qp > 0, heat is abs...

The most widely-used density functionals for the exchange-correlation energy are inexact for one-electron systems. Their self-interaction errors can be severe in some applications. The problem is not only to correct the self-interaction error, but to do so in a way that will not violate size-consistency and will not go outside the standard Kohn-Sham density functional theory. The solution via t...

Noether’s problem then asks whether K(G) is rational (= purely transcendental) over K. Noether’s problem is closely related to the inverse Galois problem. The main results about Noether’s problem for abelian groups can be found in the survey article [Sw]. More recently, Noether’s problem for nonabelian p-groups was investigated in [CHK, CHPK, HK2, Ka1, Ka2]. Let n ≥ 2 be an arbitrary natural nu...

Let X be a real projective algebraic manifold, s numerates connected components of X(R) and 2Br(X) the subgroup of elements of order 2 of cohomological Brauer group Br(X). We study the natural homomorphism ξ :2 Br(X) → (Z/2)s and prove that ξ is epimorphic if H(X(C)/G;Z/2) → H(X(R);Z/2) is injective. Here G = Gal(C/R). For an algebraic surface X with H(X(C)/G;Z/2) = 0 and X(R) 6= ∅, we give a f...

We study the 2D rotational compressible Euler equations with two singular parameters: the Rossby number for rotational forcing and the Froude/Mach number for pressure forcing. The competition of these two forces leads to a newly found parameter δ = τσ−2 that serves as a characteristic scale separating two dynamic regimes: δ 1 for the strong rotation regime [CT08] and δ 1 for the mild/weak rotat...

Jaynes’ information theory formalism of statistical mechanics is applied to the stationary states of open, non-equilibrium systems. First, it is shown that the probability distribution p of the underlying microscopic phase space trajectories over a time interval of length τ satisfies p ∝ exp(τσ /2kB) where σ is the time-averaged rate of entropy production of . Three consequences of this result ...

We illustrate the use of the R-package rstiefel for matrix-variate data analysis in the context of two examples. The first example considers estimation of a reduced-rank mean matrix in the presence of normally distributed noise. The second example considers the modeling of a social network of friendships among teenagers. Bayesian estimation for these models requires the ability to simulate from...

We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model structure, and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads induces a Quillen equivalence between their categories of algebras. We construct an explicit Boardman-Vogt style cofibrant resolution for coloured operads, the...