Injective spaces via the filter monad
... precisely, the filter space construction is a monad—see [7] for the definition. We show that the monad is of the Kock-Zöberlein type [6] and apply this to obtain a simple proof of Alan Day’s result that its algebras are the continuous lattices [2]. In [3] we proved that, given a category with the s ...
... precisely, the filter space construction is a monad—see [7] for the definition. We show that the monad is of the Kock-Zöberlein type [6] and apply this to obtain a simple proof of Alan Day’s result that its algebras are the continuous lattices [2]. In [3] we proved that, given a category with the s ...
Euler`s Formula Where does Euler`s formula eiθ = cosθ + isinθ come
... Where does Euler’s formula eiθ = cos θ + i sin θ come from? How do we even define, for example, ei ? We can’t multiple e by itself the square root of minus one times. The answer is to use the Taylor series for the exponential function. For any complex number z we define ez by ...
... Where does Euler’s formula eiθ = cos θ + i sin θ come from? How do we even define, for example, ei ? We can’t multiple e by itself the square root of minus one times. The answer is to use the Taylor series for the exponential function. For any complex number z we define ez by ...
Computing Galois groups by specialisation
... The final step merits some explanation. It is clear how to write down a left inverse for φ and hence compute the inverse image of any element. As φ is not a homomorphism, it is not immediately clear how to compute the inverse image of a subgroup. But G is quite close to being Abelian, in that it has ...
... The final step merits some explanation. It is clear how to write down a left inverse for φ and hence compute the inverse image of any element. As φ is not a homomorphism, it is not immediately clear how to compute the inverse image of a subgroup. But G is quite close to being Abelian, in that it has ...
Spaces of measures on completely regular spaces
... for real measures and by S. S. Khurana ([fl Theorem 4) for group valued measures. It will be shown in this paper (Theorem 3) that, if X is completely regular, this result can be improved by assuming only that (p,(Z)),.r converges for those regular open sets T of X for which there exists a continuous ...
... for real measures and by S. S. Khurana ([fl Theorem 4) for group valued measures. It will be shown in this paper (Theorem 3) that, if X is completely regular, this result can be improved by assuming only that (p,(Z)),.r converges for those regular open sets T of X for which there exists a continuous ...
Local isometries on spaces of continuous functions
... there is S ∈ S, possibly depending on x, such that T x = Sx. ‘Pointwise’ should be better than ‘locally’, but we have followed tradition. If each operator that belongs locally to S belongs in fact to S we say that S is algebraically reflexive. When Y = X and S = Iso(X) is the group of isometries of ...
... there is S ∈ S, possibly depending on x, such that T x = Sx. ‘Pointwise’ should be better than ‘locally’, but we have followed tradition. If each operator that belongs locally to S belongs in fact to S we say that S is algebraically reflexive. When Y = X and S = Iso(X) is the group of isometries of ...
Fibre products
... Theorem 4.2.1. Fibre products exist in the category of schemes. Before proving this, let us understand some consequences. First of all, it tells us that products exist. Since Spec Z is the terminal object in the category of schemes. The product is X ⇥ Y = X ⇥Spec Z Y . Secondly, given a point s 2 S ...
... Theorem 4.2.1. Fibre products exist in the category of schemes. Before proving this, let us understand some consequences. First of all, it tells us that products exist. Since Spec Z is the terminal object in the category of schemes. The product is X ⇥ Y = X ⇥Spec Z Y . Secondly, given a point s 2 S ...
Profinite Groups - Universiteit Leiden
... We now begin with the formal definitions. A topological group is a group G which is also a topological space with the property that the multiplication map m:G×G→G (a, b) 7→ ab and the inversion map i:G→G a 7→ a−1 are continuous. Whenever we are given two topological groups, we insist that a homomorp ...
... We now begin with the formal definitions. A topological group is a group G which is also a topological space with the property that the multiplication map m:G×G→G (a, b) 7→ ab and the inversion map i:G→G a 7→ a−1 are continuous. Whenever we are given two topological groups, we insist that a homomorp ...
Homework #3 Solutions (due 9/26/06)
... so applying the case of n > 0 to ab−1 a−1 gives use what we want. ...
... so applying the case of n > 0 to ab−1 a−1 gives use what we want. ...
Group action
In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes of the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.A group action is an extension to the notion of a symmetry group in which every element of the group ""acts"" like a bijective transformation (or ""symmetry"") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.If G is a group and X is a set, then a group action may be defined as a group homomorphism h from G to the symmetric group on X. The action assigns a permutation of X to each element of the group in such a way that the permutation of X assigned to the identity element of G is the identity transformation of X; a product gk of two elements of G is the composition of the permutations assigned to g and k.The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.