Displaying similar documents to “Solutions of the equation f y u x - f x u y = g ”

Relations among analytic functions. I

Edward Bierstone, P. D. Milman (1987)

Annales de l'institut Fourier

Similarity:

Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let Φ : X → Y be a morphism of real analytic spaces, and let Ψ : 𝒢 → ℱ be a homomorphism of coherent modules over the induced ring homomorphism Φ * : 𝒪 Y → 𝒪 X . We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ℛ a = Ker Ψ ^ a , a ∈ X , are upper semi-continuous in the analytic Zariski topology of X . We prove semicontinuity in many cases (e.g. in the algebraic...

Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module

Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2014)

Formalized Mathematics

Similarity:

In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two Z-modules. In this section, we define homomorphism...

Torsion Z-module and Torsion-free Z-module

Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama (2014)

Formalized Mathematics

Similarity:

In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

Deformations of coherent foliations on a compact normal space

Geneviève Pourcin (1987)

Annales de l'institut Fourier

Similarity:

An universal analytic structure is construted on the set of (singular) holomorphic foliations on a normal compact space. Such a foliation is by definition a coherent subsheaf of the holomorphic tangent sheaf stable by the Lie-bracket