Explore chapters and articles related to this topic
Algebraic Aspects
Published in Marlos A. G. Viana, Vasudevan Lakshminarayanan, Symmetry in Optics and Vision Studies, 2019
Marlos A. G. Viana, Vasudevan Lakshminarayanan
Linear or vector spaces are therefore specialized modules over a given ring with identity. In analogy with distinct linear spaces of a given field F there are distinct modules of a given ring R: F−linear spaces V,R−modules M.
Operators in the Cowen-Douglas Class and Related Topics
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Let ℳ and N be any two Hilbert modules over the algebra A. Notice that there are two possible module actions on ℳ ⊗ N, i.e., the left action: L ⊗I : (f, h ⊗ k) ↦ f · h ⊗ k and the right action: I ⊗ R: (f, h ⊗ k) ↦ h ⊗ f · k. The module tensor product ℳ⊗AN is defined to be the module obtained by identifying these two actions. Specifically, let I be the closed subspace of ℳ ⊗ N generated by vectors of the form {f⋅h⊗k−h⊗f⋅k:h∈ℳ,k∈Nand f∈A}.
Behaviours, Modules, and Duality
Published in Krzysztof Gałkowski, Jeff David Wood, Multidimensional Signals, Circuits and Systems, 2001
Jeffrey Wood, Eric Rogers, David H. Owens
Annihilators are of great importance in algebraic systems theory. The annihilator oí an arbitrary module M, denoted ann M, is the ideal of all elements r € such that rx — 0 for all X 6 M. Thus the annihilator of a behaviour В is the set of all equations satisfied by each system variable (independently of the values of the others). We have the following useful result:
Toward a unified account of definitions in mathematics education research: a systematic literature review
Published in International Journal of Mathematical Education in Science and Technology, 2023
Hermund André Torkildsen, Tore Alexander Forbregd, Eivind Kaspersen, Trygve Solstad
Definitions may therefore support the development of new definitions, both equivalent and non-equivalent. For example, the definition ‘a number of unifix cubes is an even number if we can pair two and two cubes such that no cube is left over’ does not extend to negative numbers, but the definition ‘an even number is a multiple of two’ easily does (the example space is extended). In abstract algebra a vector space is often defined with a (rather long) list of properties. However, we can define a vector space V to be an -module, where is a ring, by simply requiring that is a field (the example space becomes properly smaller, i.e. all vector spaces are modules, but not all modules are vector spaces).
A dynamical proof of the van der Corput inequality
Published in Dynamical Systems, 2022
Nikolai Edeko, Henrik Kreidler, Rainer Nagel
Let be a right cancellative semigroup. Moreover, let and be right Følner nets for ,E be a pre-Hilbert module over a unital commutative -algebra A, a representation as Markov operators, a representation such that is -dominated for every , and a state for every .Then for every .
A discrete variant of Farkas' lemma and related results
Published in Optimization, 2021
First, we ask whether there are additional assumptions to be added to those already present in Lemma 4.1 so that Farkas condition (5) implies Following [33, Theorem 3.3], we can give a positive answer in the special case when R is the ring of the integer numbers ℤ. Recall that the group or module V is divisible iff, for any positive and for any element , there exists a such that . Then, if the module V is divisible, it is easy to see that formula (6) implies formula (31). Furthermore, it is clear that formula (31) always implies Farkas condition (5). Alternatively, let F be a linearly ordered (commutative or skew) field. Then, we can arrive at the same conclusion if is a commutative subring of the field and the module V is a linearly ordered vector space over the field F. Note that Lemma 4.1 is ‘easy’ in this setting, see [23, Remark 8]. Thus, in these two special cases, statements (5), (6), and (31) are equivalent.