Examples of a topological semidirect product

Let G be a compact topological group, and Aut(G) the group of autohomeomorphisms of G. I have proved some (topological) results about the holomorph G⋋, and now looking for nice examples. I can, of course, take any compact group G, but I want to know if there are “natural” examples to be found. That is, … Read more

What are some interesting examples of cooperative games that can be naturally generalised to a stochastic version of it?

In classical, deterministic cooperative game theory, there are $N$ players that can form $2^{N}$ coalitions. Each of these coalitions is assigned a value by means of the characteristic function $v ( \cdot )$ associated with the particular game, such that $v: 2^{N} \to \mathbb{R}$. The game can be “solved” (i.e. the payoffs can be distributed … Read more

Ricci flow on locally symmetric noncompact manifold

As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the evolution equation $$\frac{\partial}{\partial t}(\nabla \text{Rm})=\Delta(\nabla \text{Rm})+\text{Rm} \ast \nabla \text{Rm}$$ and the uniqueness of the solution due to the compactness of … Read more

Separable metrizable spaces far from being completely metrizable

I came across a kind of separable metrizable space that is “far” from being completely metrizable. Before specifying what I mean with “far”, I recall that a space is said to be Polish if it’s separable and completely metrizable and that $2^\mathbb{N}$, known as the Cantor space, is the space of infinite binary sequences endowed … Read more

Kind of multiplicative total boundedness in Hausdorff compact rings

Let $(R,\cal T)$ be a unital Hausdorff compact topological ring and let $A$ be an open subset of $R$ containing $1$. Is there a finite set $B$ with $AB=R$? Answer Not in general. Following your title let us say that a monoid $(M,\cdot,1)$ is totally bounded if for every identity nbd $A$ there exists a … Read more

Very canonical constructions

You have two categories C1 and C2. We call a map of the classes Ob(C1)→Ob(C2) a construction. Sometimes you can find a functor C1→C2 inducing this map, then you call your construction functorical or canonical. Let us all a construction very canonical if there is a functor inducing it and between any two such functors … Read more

When does Ext1C(M,Ni)=0\operatorname{Ext}_C^1(M,N_i)=0 imply Ext1C(M,lim\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0?

Let C be an abelian category. Suppose that (N_i)_{i\in I} is an inverse system of objects in C. Under which conditions does the hypothesis that \operatorname{Ext}_C^1(M,N_i)=0\quad\forall i\in I\tag{1} imply \operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0?\tag{2} The functor \operatorname{Ext}^1_C is the Yoneda \operatorname{Ext}^1-functor. So we don’t have to worry whether C has enough injectives/projectives. However, I would be satisfied if … Read more

Push-out in the category of coherent sheaves over the complex projective plane

I’m trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal slope) but it does not admit subbundles with greater slope. This is the simplest example I have in mind in order … Read more

examples of Chow rings of surfaces

Can somone provide me (articles/literature) with examples of Chow rings of surfaces? (e.g. here: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf Chapter 9) What I want is a list of (smooth projective surface X/Fq, CH∗(X)=CH0(X)⊕CH1(X)⊕CH2(X), the product structure on this, deg:CH2(X)→Z and the ample cone). Answer For rational surfaces, there is the paper by Colliot-Thélène Hilbert’s Theorem 90 for K2, with … Read more

Counterexample for closedness under union of $\prec_{\infty,\kappa}$ chains

Assume $\kappa$ is uncountable and $\phi$ is an $L_{\infty,\kappa}$ sentence. Let $K$ be the collection of models of $\phi$ partially ordered by $\prec_{\infty,\kappa}$. It is well-known that $K$ is closed under unions of increasing chains of length $\kappa^+$, but the argument fails for increasing chains of length $<\kappa$. Do you have an easy counterexample, either … Read more