In this paper, we will define a signed Lattice measure on σ-algebras, as well as give the definition of positive and negative Lattice. Herein, we will show that the Hahn Decomposition Theorem decomposes any space X into a positive lattice A and a negative Lattice B such that A∨B =X and the signed Lattice measure of A ∧ B is 0.

elementary classes (AEC) were introduced by Shelah to generate a common framework to treat those classes of models that, in spite of being non-elementary, behave similarly to elementary classes. Part of the motivation was the study of classes of models for theories in infinitary languages. As stated in the introduction, Rami Grossberg and Olivier Lessmann presented an axiomatic framework which ...

Introduction We develop in this paper a theory of complexity for pairs (M,X) where M is a compact 3-manifold such that χ(M) = 0, and X is a collection of trivalent graphs, each graph τ being embedded in one component C of ∂M so that C \ τ is one disc. In the special case where M is closed, so X = ∅, our complexity coincides with Matveev’s [6]. Extending his results we show that complexity of pa...

Compressive Sensing techniques are used in a short proof of Kashin’s decomposition theorem generalized to `p-spaces for p ≤ 1. The proof is based on the observation that the null-space of a properly-sized matrix with restricted isometry property is almost Euclidean when endowed with the `p-quasinorm. Kashin’s decomposition theorem states that, for any integer m ≥ 1, `2m 1 is the orthogonal sum ...

In this article we show that every smooth closed oriented fourmanifold admits a decomposition into two submanifolds along common boundary. Each of these submanifolds is a complex manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-manifold is a Stein domain in the the complement of a certain contractible 2-complex.

The Chemical Organization Theory (COT) is an abstract reaction network model that has a deep connection to autopoiesis as they share the same central topic: Organization. The main characteristic of autopoietic systems is that they preserve their own organization; this constitutes their identity. In terms of COT, organizations are special reaction networks which are closed and self-maintaining. ...

A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. In this series of papers we give a structural description of all claw-free graphs. In this paper, we achieve a major part of that goal; we prove that every claw-free graph either belongs to one of a few basic classes, or admits a decomposition in a useful way.

In this paper we prove that every nite Markov chain can be de composed into a cascade product of a Bernoulli process and several simple permutation reset deterministic automata The original chain is a state homomorphic image of the product By doing so we give a positive answer to an open question stated in Paz concerning the decomposability of probabilistic systems Our result is based on the ob...

We propose an algorithm for solving the maximum weighted stable set problem on claw-free graphsthat runs in O(n)−time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present paper. Despite being weaker than the well-known structure result for claw-free graphs given...

In this paper we introduce and study rational slice monogenic functions. After proving a decomposition theorem for such functions, we are able to prove the Runge approximation theorem for slice monogenic functions. We then show how a similar argument can be used to obtain an analogue of the Runge approximation theorem in the slice regular setting.