For a linear and bounded operator T from a Banach space X into a Banach space Y, let ϱ(T[mid ][Iscr ]n, [Rscr ]n) and ϱ(T[mid ][Iscr ]n, [Gscr ]n) denote the Rademacher and Gaussian cotype 2 norm of T computed with n vectors, respectively. It is shown that the sequence ϱ(T[mid ][Iscr ]n, [Rscr ]n) has submaximal behaviour if and only if ϱ(T[mid ][Iscr ]n, [Gscr ]n) has. This means that

Moreover, the class of these operators coincides with the class of operators preserving copies of ln∞ uniformly. The tool connecting these concepts is the equal norm Rademacher cotype of operators.

For a linear and bounded operator T from a Banach space X into a Banach space Y, let ϱ(T[mid ][Iscr ]n, [Rscr ]n) and ϱ(T[mid ][Iscr ]n, [Gscr ]n) denote the Rademacher and Gaussian cotype 2 norm of T computed with n vectors, respectively. It is shown that the sequence ϱ(T[mid ][Iscr ]n, [Rscr ]n) has submaximal behaviour if and only if ϱ(T[mid ][Iscr ]n, [Gscr ]n) has. This means that

[formula here]

Moreover, the class of these operators coincides with the class of operators preserving copies of ln∞ uniformly. The tool connecting these concepts is the equal norm Rademacher cotype of operators.

An n-affine manifold is a differentiable manifold endowed with an atlas whose transition functions are locally affine transformations of ℝn . The paper studies affine dynamic, that is, affine endomorphisms of affine manifolds. The main goal is to relate the dynamical viewpoint to the completeness problem. In particular, it is shown that the Markus conjecture is true when dim Aff(M, [dtri ]) > dim M − 2.

An n-affine manifold is a differentiable manifold endowed with an atlas whose transition functions are locally affine transformations of ℝn . The paper studies affine dynamic, that is, affine endomorphisms of affine manifolds. The main goal is to relate the dynamical viewpoint to the completeness problem. In particular, it is shown that the Markus conjecture is true when dim Aff(M, [dtri ]) > dim M − 2.

The paper gives examples of knots with equal knot polynomials, but distinct signatures, 4-genera, double branched cover homology groups and unknotting numbers. This generalizes some examples given by Lickorish and Millett. It is also shown that there are knots with minimal (crossing number) diagrams of different unknotting number (thus answering a question of Bleiler), and an alternative proof is given of Rudolph's result that there are knots of [les ] 15n crossings with unit Alexander polynomial and 4-genus or unknotting number n.

The paper gives examples of knots with equal knot polynomials, but distinct signatures, 4-genera, double branched cover homology groups and unknotting numbers. This generalizes some examples given by Lickorish and Millett. It is also shown that there are knots with minimal (crossing number) diagrams of different unknotting number (thus answering a question of Bleiler), and an alternative proof is given of Rudolph's result that there are knots of [les ] 15n crossings with unit Alexander polynomial and 4-genus or unknotting number n.

It is proved that there is, up to isotopy, a unique irreducible Heegaard splitting in an orientable, closed, connected Seifert 3-manifold with an orientable elliptical or euclidean orbifold basis. Using Hamilton's and Lawson's results, the topological uniqueness is obtained of closed orientable minimal surfaces of a given genus g [ges ] 2, embedded in a closed orientable Riemannian 3-manifold with strictly positive Ricci curvature.

It is proved that there is, up to isotopy, a unique irreducible Heegaard splitting in an orientable, closed, connected Seifert 3-manifold with an orientable elliptical or euclidean orbifold basis. Using Hamilton's and Lawson's results, the topological uniqueness is obtained of closed orientable minimal surfaces of a given genus g [ges ] 2, embedded in a closed orientable Riemannian 3-manifold with strictly positive Ricci curvature.