The paper considers stationary critical points of the heat flow in sphere SN and in hyperbolic space ℍN, and proves several results corresponding to those in Euclidean space ℝN which have been proved by Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy problems for the heat equation, it is shown that the solution u has a stationary critical point if and only if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say x0, is fixed and initial-boundary value problems are considered for the heat equation on bounded domains containing x0. It is shown that for any initial data satisfying the balance law with respect to x0 (or being centrosymmetric with respect to x0) the corresponding solution always has x0 as a stationary critical point, if and only if the domain is a geodesic ball centred at x0 (or is centrosymmetric with respect to x0, respectively).

The paper considers stationary critical points of the heat flow in sphere SN and in hyperbolic space ℍN, and proves several results corresponding to those in Euclidean space SN which have been proved by Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy problems for the heat equation, it is shown that the solution u has a stationary critical point if and only if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say x0, is fixed and initial-boundary value problems are considered for the heat equation on bounded domains containing x0. It is shown that for any initial data satisfying the balance law with respect to x0 (or being centrosymmetric with respect to x0) the corresponding solution always has x0 as a stationary critical point, if and only if the domain is a geodesic ball centred at x0 (or is centrosymmetric with respect to x0, respectively).

The notion of an attractor for a random dynamical system with respect to a general collection of deterministic sets is introduced. This comprises, in particular, global point attractors and global set attractors. After deriving a necessary and sufficient condition for existence of the corresponding attractors it is proved that a global set attractor always contains all unstable sets of all of its subsets. Then it is shown that in general random point attractors, in contrast to deterministic point attractors, do not support all invariant measures of the system. However, for white noise systems it holds that the minimal point attractor supports all invariant Markov measures of the system.

The notion of an attractor for a random dynamical system with respect to a general collection of deterministic sets is introduced. This comprises, in particular, global point attractors and global set attractors. After deriving a necessary and sufficient condition for existence of the corresponding attractors it is proved that a global set attractor always contains all unstable sets of all of its subsets. Then it is shown that in general random point attractors, in contrast to deterministic point attractors, do not support all invariant measures of the system. However, for white noise systems it holds that the minimal point attractor supports all invariant Markov measures of the system.

Let E(ℤ) = {einx}n∈ℤ denote the trigonometrical exponential system. It is well known that E(ℤ) forms an orthogonal basis in the space L2(0, 2π). In 1964, H. Landau discovered that the trigonometrical system has the following property: certain small perturbations of E(ℤ) yield exponential systems which are complete in L2 on any finite union of 2π-periodic translations of any interval (ε, 2π−ε), 0 < ε < π.

Let E(ℤ) = {einx}n∈ℤ: denote the trigonometrical exponential system. It is well known that E(ℤ) forms an orthogonal basis in the space L2(0, 2π). In 1964, H. Landau discovered that the trigonometrical system has the following property: certain small perturbations of E(ℤ) yield exponential systems which are complete in L2 on any finite union of 2π-periodic translations of any interval (ε, 2π−ε), 0 < ε < π.

The derivation problem for a locally compact group G is to decide whether for each derivation D from L1(G) into L1(G) there is a bounded measure μ∈M(G) with D(a) = aμ−μa (a ∈ L1(G)). In this paper we obtain an affirmative answer for the case of connected groups. To explain the contents of this paper we give an equivalent formulation of the problem. Suppose that the group G acts as a group of homeomorphisms of the locally compact space X. Related to this there is an action of G on M(X). A bounded crossed homomorphism from G to M(X) is a map Φ with bounded range and satisfying Φ(gh) = gΦ(h)+Φ(g) (g, h ∈ G). The problem for bounded crossed homomorphisms is to decide if for each such Φ there is an element μ of M(X) with Φ(g) = gμ− μ (g ∈ G). The derivation problem is equivalent to this bounded crossed homomorphism problem for the special case X = G where G acts on X by conjugation (together with some mild continuity hypotheses about the map Φ:G→M(X) which are often automatically satisfied). The bounded crossed homomorphism problem always has a positive solution if G is amenable and a closely related calculation shows that in solving the bounded crossed homomorphism problem we need only solve it for functions Φ which are zero on H where H is a given amenable subgroup of G. It can happen that this condition of being zero on H forces Φ to be zero even when H is a comparatively small subgroup of G. If h is an element of G such that ‘hnx → ∞’ as n → ∞ for all x ∈ X then for any two measures μ and ν, for large values of n, μ and hnν have little overlap so ∥μ + hNν∥ → ∥μ∥ + ∥ν∥. Thus if H is the subgroup generated by h, for any g ∈ G

The derivation problem for a locally compact group G is to decide whether for each derivation D from L1(G) into L1(G) there is a bounded measure μ∈M(G) with D(a) = aμ−μa (a ∈ L1(G)). In this paper we obtain an affirmative answer for the case of connected groups. To explain the contents of this paper we give an equivalent formulation of the problem. Suppose that the group G acts as a group of homeomorphisms of the locally compact space X. Related to this there is an action of G on M(X). A bounded crossed homomorphism from G to M(X) is a map Φ with bounded range and satisfying Φ(gh) = gΦ(h)+Φ(g) (g, h ∈ G). The problem for bounded crossed homomorphisms is to decide if for each such Φ there is an element μ of M(X) with Φ(g) = gμ− μ (g ∈ G). The derivation problem is equivalent to this bounded crossed homomorphism problem for the special case X = G where G acts on X by conjugation (together with some mild continuity hypotheses about the map Φ:G→M(X) which are often automatically satisfied). The bounded crossed homomorphism problem always has a positive solution if G is amenable and a closely related calculation shows that in solving the bounded crossed homomorphism problem we need only solve it for functions Φ which are zero on H where H is a given amenable subgroup of G. It can happen that this condition of being zero on H forces Φ to be zero even when H is a comparatively small subgroup of G. If h is an element of G such that ‘hnx → ∞’ as n → ∞ for all x ∈ X then for any two measures μ and ν, for large values of n, μ and hnν have little overlap so ∥μ + hnν∥ → ∥μ∥ + ∥ν∥. Thus if H is the subgroup generated by h, for any g ∈ G

[formula here]

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.