Canonical coordinates for vector spaces and affine spaces

2018 ◽  
pp. 71-98
Author(s):  
Shayne F. D. Waldron
Keyword(s):  
Vector Spaces ◽  
Canonical Coordinates ◽  
Affine Spaces

Vector Spaces, Affine Spaces, and Metric Spaces

2012 ◽  
pp. 13-43
Author(s):  
Jakob Andreas Bærentzen ◽  
Jens Gravesen ◽  
François Anton ◽  
Henrik Aanæs
Keyword(s):  
Metric Spaces ◽  
Vector Spaces ◽  
Affine Spaces

A Normal Form and the Term-linearity of Affine Spaces over GF(3)

2006 ◽  
Vol 13 (04) ◽  
pp. 675-684
Author(s):  
Jung R. Cho ◽  
Woo Hyun Kim
Keyword(s):  
Normal Form ◽  
Word Problem ◽  
Affine Space ◽  
Linear Term ◽  
Binary Trees ◽  
Vector Spaces ◽  
Affine Spaces ◽  
Term Linear

An algebra is called term-linear if every term is equal to a linear term or a constant and every n-ary linear term is essential. The term-linearity is a generalization of the linearity of polynomials over ordinary linear algebras or vector spaces. In this paper, as a sequel to a previous paper, we present an explicit normal form for terms over an affine space over GF(3) by binary trees, which shows that the word problem for affine spaces over GF(3) is solvable. Using this normal form, we count the number of all essentially n-ary linear terms and apply the Csákány formula to show that an affine space over GF(3) is term-linear.

scholarly journals PERVERSE MOTIVES AND GRADED DERIVED CATEGORY

2016 ◽  
Vol 17 (2) ◽  
pp. 347-395 ◽  
Author(s):  
Wolfgang Soergel ◽  
Matthias Wendt
Keyword(s):  
Additional Assumption ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Flag Varieties ◽  
Koszul Duality ◽  
Affine Spaces ◽  
Finite Dimensional ◽  
De Rham ◽  
Weight Structure ◽  
Tate Motives

For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category ${\mathcal{O}}$ and Koszul duality.

Model Based Defect Detection in Multi-Dimensional Vector Spaces

2011 ◽  
Vol 131 (9) ◽  
pp. 1633-1641
Author(s):  
Toshifumi Honda ◽  
Kenji Obara ◽  
Minoru Harada ◽  
Hajime Igarashi
Keyword(s):  
Defect Detection ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Model Based

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

Stable and Supported Semantics in Continuous Vector Spaces

2020 ◽  
Author(s):  
Yaniv Aspis ◽  
Krysia Broda ◽  
Alessandra Russo ◽  
Jorge Lobo
Keyword(s):  
Logic Program ◽  
Matrix Multiplication ◽  
Vector Spaces ◽  
Continuous Space ◽  
Continuous Vector ◽  
Novel Approach ◽  
Gradient Based ◽  
Normal Logic ◽  
Parameter Values ◽  
Normal Logic Program

We introduce a novel approach for the computation of stable and supported models of normal logic programs in continuous vector spaces by a gradient-based search method. Specifically, the application of the immediate consequence operator of a program reduct can be computed in a vector space. To do this, Herbrand interpretations of a propositional program are embedded as 0-1 vectors in $\mathbb{R}^N$ and program reducts are represented as matrices in $\mathbb{R}^{N \times N}$. Using these representations we prove that the underlying semantics of a normal logic program is captured through matrix multiplication and a differentiable operation. As supported and stable models of a normal logic program can now be seen as fixed points in a continuous space, non-monotonic deduction can be performed using an optimisation process such as Newton's method. We report the results of several experiments using synthetically generated programs that demonstrate the feasibility of the approach and highlight how different parameter values can affect the behaviour of the system.

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