scholarly journals Forms over fields and Witt's lemma

2020 ◽  
Vol 126 (3) ◽  
pp. 401-423
Author(s):  
David Sprehn ◽  
Nathalie Wahl
Keyword(s):  
Quadratic Forms ◽  
General Framework ◽  
Vector Spaces ◽  
Group Of Isometries

We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, and describe its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt's lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.

scholarly journals A General Framework for Implicit and Explicit Debiasing of Distributional Word Vector Spaces

2020 ◽  
Vol 34 (05) ◽  
pp. 8131-8138
Author(s):  
Anne Lauscher ◽  
Goran Glavaš ◽  
Simone Paolo Ponzetto ◽  
Ivan Vulić
Keyword(s):  
General Framework ◽  
Semantic Information ◽  
Evaluation Framework ◽  
Vector Spaces ◽  
Racial Biases ◽  
Definition Of ◽  
Cross Lingual ◽  
Experimental Findings ◽  
Implicit And Explicit ◽  
Embedding Methods

Distributional word vectors have recently been shown to encode many of the human biases, most notably gender and racial biases, and models for attenuating such biases have consequently been proposed. However, existing models and studies (1) operate on under-specified and mutually differing bias definitions, (2) are tailored for a particular bias (e.g., gender bias) and (3) have been evaluated inconsistently and non-rigorously. In this work, we introduce a general framework for debiasing word embeddings. We operationalize the definition of a bias by discerning two types of bias specification: explicit and implicit. We then propose three debiasing models that operate on explicit or implicit bias specifications and that can be composed towards more robust debiasing. Finally, we devise a full-fledged evaluation framework in which we couple existing bias metrics with newly proposed ones. Experimental findings across three embedding methods suggest that the proposed debiasing models are robust and widely applicable: they often completely remove the bias both implicitly and explicitly without degradation of semantic information encoded in any of the input distributional spaces. Moreover, we successfully transfer debiasing models, by means of cross-lingual embedding spaces, and remove or attenuate biases in distributional word vector spaces of languages that lack readily available bias specifications.

On the space of generalized theta-series for certain quadratic forms in any number of variables

2019 ◽  
Vol 69 (1) ◽  
pp. 87-98
Author(s):  
Ketevan Shavgulidze
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Theta Series ◽  
Vector Spaces ◽  
Generalized Theta Series

Abstract An upper bound of the dimension of vector spaces of generalized theta-series corresponding to some nondiagonal quadratic forms in any number of variables is established. In a number of cases, an upper bound of the dimension of the space of theta-series with respect to the quadratic forms of five variables is improved and the basis of this space is constructed.

Witts Theorem for Quadratic Forms Over Non-Dyadic Discrete Valuation Rings

1977 ◽  
Vol 29 (5) ◽  
pp. 928-936
Author(s):  
David Mordecai Cohen
Keyword(s):  
Bilinear Form ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Symmetric Bilinear Form ◽  
Finitely Generated ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Group Of Isometries

Let R be a discrete valuation ring, with maximal ideal pR, such that ½ ϵ R. Let L be a finitely generated R-module and B : L × L → R a non-degenerate symmetric bilinear form. The module L is called a quadratic module. For notational convenience we shall write xy = B(x, y). Let O(L) be the group of isometries, i.e. all R-linear isomorphisms φ : L → L such that B((φ(x), (φ(y)) = B(x, y).

scholarly journals Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

2004 ◽  
Vol 175 ◽  
pp. 1-37 ◽  
Author(s):  
Takahiko Ueno
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Zeta Functions ◽  
Vector Spaces ◽  
Parabolic Subgroups ◽  
Converse Theorem ◽  
Half Integral Weight ◽  
Prehomogeneous Vector Spaces ◽  
Integral Weight ◽  
Weil Type

AbstractIn this paper, we prove the functional equations for the zeta functions in two variables associated with prehomogeneous vector spaces acted on by maximal parabolic subgroups of orthogonal groups. Moreover, applying the converse theorem of Weil type, we show that elliptic modular forms of integral or half integral weight can be obtained from the zeta functions.

Collineations of Polar Spaces

1985 ◽  
Vol 37 (2) ◽  
pp. 296-309 ◽  
Author(s):  
Donald G. James
Keyword(s):  
Projective Space ◽  
Quadratic Forms ◽  
Fundamental Theorem ◽  
Valuation Ring ◽  
Finite Dimension ◽  
Vector Spaces ◽  
Polar Spaces ◽  
Hermitian Forms ◽  
Bijective Mapping ◽  
Integral Structure

The fundamental theorem of projective geometry describes the bijective collineations between two projective spaces PV and PV′ of finite dimension (greater than one) over division rings k and k′ in terms of an isomorphism φ:k → k′ and a φ-semilinear bijective mapping between the underlying vector spaces V and V′. Tits [9, Theorem 8.611] has given an extensive generalization of this theorem to embeddable polar spaces induced by polarities coming from either (σ, )-hermitian forms or from (σ, )-quadratic forms with Witt indices at least two. In another direction, Klingenberg [7] and later André [1] and Rado [8], have generalized the fundamental theorem by considering non-injective collineations. Now the isomorphism φ must be replaced by a place φ:k → k′ ∪ ∞ and an integral structure over the valuation ring A = φminus1(k′) is induced into the projective space PV.

scholarly journals On Some Approximation Theorems for Powerq-Bounded Operators on Locally Convex Vector Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ludovic Dan Lemle
Keyword(s):  
General Framework ◽  
Vector Spaces ◽  
Bounded Operators ◽  
Operator Inequalities ◽  
Approximation Theorems ◽  
Locally Convex

This paper deals with the study of some operator inequalities involving the powerq-bounded operators along with the most known properties and results, in the more general framework of locally convex vector spaces.

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