We introduce a curvature-dimension condition CD ($K$,$N$) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD($K$, ∞). The additional real parameter$N$plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD($K$,$N$) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim($M$) ⩽$N$.

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Let W be an extended affine Weyl group. We prove that the minimal length elements W of any conjugacy class W of W satisfy some nice properties, generalizing results of Geck and Pfeiffer [<i>On the irreducible characters of Hecke algebras</i>, Adv. Math. <b>102</b> (1993), 7994] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some W -adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra W

We give a new, simple algorithm for simultaneous degree elevation and knot insertion for B-spline curves. The method is based on the simple approach of computing derivatives using the control points, resampling the knot vector, and then computing the new control points from the derivatives. We compare our approach with previous algorithms and illustrate it with examples.

We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks$N$_{$u,v$}in the disk that correspond to the choice of a pair ($u, v$) of Coxeter elements in the symmetric group$S$_{$n$}and the corresponding networks $N_{u,v}^\circ$ in the annulus. Boundary measurements for$N$_{$u,v$}represent elements of the Coxeter double Bruhat cell$G$^{$u,v$}⊂GL_{$n$}. The Cartan subgroup$H$acts on$G$^{$u,v$}by conjugation. The standard Poisson structure on the space of weights of$N$_{$u,v$}induces a Poisson structure on$G$^{$u,v$}, and hence on the quotient$G$^{$u,v$}/$H$, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for $N_{u,v}^\circ$ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for$N$_{$u,v$}. The corresponding Poisson bracket on the space of weights of $N_{u,v}^\circ$ induces a Poisson bracket on the certain space $ {\mathcal{R}_n} $ of rational functions, which appeared previously in the context of Toda flows.