In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely generated. (2) Even if a submodule of a finitely generated module is finitely generated, the minim...

In this paper, by introducing some new concepts in minimal spaces, we prove a generalized form of the Fan-KKM theorem vector spaces. A class F-implicit variational inequality problems and, as an application is investigated. Moreover, existence for kind under suitable assumptions spaces given.

The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A, and is isomorphic to the mod two cohomology of BO , the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., minimal generators and minimal relations. From this we produce minimal presentations for vario...

A jump process on configuration space whose transition rates are governed by the state vector of a quantum field theory was first considered by J.S. Bell. This process and its variants involving “minimal” jump rates are reviewed, and a recent proof of global existence of Bell’s process is sketched. As an outlook, it is suggested how methods of this proof could be applied to similar global exist...

This article considers the n-number Ducci game as a linear map over the n dimensional vector space ZZn 2 . Period lengths are characterized as orders of factors of the minimal polynomial of that transformation. The minimal polynomial can be explicitly obtained for any n. We provide algebraic proofs of divisibility conditions, initially found by Ehrlich. In addition, all periodic cycle lengths f...

a few papers have been interested by the fixed switching frequency direct torque control fed by direct matrix converters, where we can find just the use of direct torque controlled space vector modulated method. In this present paper, we present an improved method used for a fixed switching frequency direct torque control (DTC) using a direct matrix converter (DMC). This method is characterized...

We present vector space semantic parsing (VSSP), a framework for learning compositional models of vector space semantics. Our framework uses Combinatory Categorial Grammar (CCG) to define a correspondence between syntactic categories and semantic representations, which are vectors and functions on vectors. The complete correspondence is a direct consequence of minimal assumptions about the sema...

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...