Let C be a geometrically irreducible curve over a field k, let k be an algebraic closure of k, and let m be any positive integer not divisible by the characteristic of k. The Jacobian variety J of C comes equipped with a principal p~ar ization A, which is in particular an isomorphism from J to its dual variety J . The polarization A gives us an isomorphism between the m-torsion J m of J and its...

Exercise 10.3.2. Let R be a commutative ring with identity. For all positive integers n and m, R ∼= R if and only if n = m. Proof. Let φ : R → R be an isomorphism of R-modules and let I E R be a maximal ideal. Then the map φ̄ : R → R/IR given by φ̄(α) = φ(α) is a morphism of R-modules. Moreover ker φ̄ = {α ∈ R | φ̄(α) = 0} = {α ∈ R | φ(α) ∈ IR} = φ−1(IRm) = IR. Therefore by the first isomorphism th...

Let T(X) be the full transformation semigroup on a nonempty set X. For ? Z Y X, let T(X,Y,Z) = {? T(X):Y? Z}. It is not difficult to see that it generalized form of three well-known semigroups. This paper obtains an isomorphism theorem T(X,Y,Z). In addition, when X finite and rank calculated.

In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q differentiability, i.e. the main theorem isomorphism between solution space and data using Besov Triebel--Lizorkin spaces. The key compatibility conditions initial data. We are able to get unique smooth if satisfying smooth.

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this ex...

Most treatments of elementary abstract algebra include a discussion of finite groups and some work on their classification. However, very little is done with finite rings. For example most beginning texts state and prove the theorem that for p a prime the cyclic group of order p is the only group of order p up to isomorphism. Yet the equally striking and easily proved result that for a prime p ...

A classical theorem of Veldkamp describes the center an enveloping algebra a Lie semi-simple algebraic group in characteristic p . We generalize this result to class algebras with property that they arise as reduction modulo ≫ 0 from g , such has no nontrivial semi-invariants Sym ( ) and is polynomial algebra. As application, we solve derived isomorphism problem for above algebras.