scholarly journals Matroid connectivity and singularities of configuration hypersurfaces

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
Graham Denham ◽  
Mathias Schulze ◽  
Uli Walther
Keyword(s):  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Smooth Locus

AbstractConsider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.

scholarly journals Supervised learning using a symmetric bilinear form for record linkage

2015 ◽  
Vol 26 ◽  
pp. 144-153 ◽  
Author(s):  
Daniel Abril ◽  
Vicenç Torra ◽  
Guillermo Navarro-Arribas
Keyword(s):  
Supervised Learning ◽  
Bilinear Form ◽  
Record Linkage ◽  
Symmetric Bilinear Form

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

scholarly journals On the realizable classes of the square root of the inverse different in the unitary class group

2017 ◽  
Vol 13 (04) ◽  
pp. 913-932 ◽  
Author(s):  
Sin Yi Cindy Tsang
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Bilinear Form ◽  
Class Group ◽  
Finite Abelian Group ◽  
Symmetric Bilinear Form ◽  
Square Root ◽  
Ring Of Integers ◽  
Trace Map ◽  
Odd Order

Let [Formula: see text] be a number field with ring of integers [Formula: see text] and let [Formula: see text] be a finite abelian group of odd order. Given a [Formula: see text]-Galois [Formula: see text]-algebra [Formula: see text], write [Formula: see text] for its trace map and [Formula: see text] for its square root of the inverse different, where [Formula: see text] exists by Hilbert’s formula since [Formula: see text] has odd order. The pair [Formula: see text] is locally [Formula: see text]-isometric to [Formula: see text] whenever [Formula: see text] is weakly ramified, in which case it defines a class in the unitary class group [Formula: see text] of [Formula: see text]. Here [Formula: see text] denotes the canonical symmetric bilinear form on [Formula: see text] defined by [Formula: see text] for all [Formula: see text]. We will study the set of all such classes and show that a subset of them forms a subgroup of [Formula: see text].

scholarly journals The norming set of a symmetric bilinear form on the plane with the supremum norm

2021 ◽  
Vol 55 (2) ◽  
pp. 171-180
Author(s):  
S. G. Kim
Keyword(s):  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Supremum Norm ◽  
Linear Forms

An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}_s(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and$|T(x_1, \ldots, x_n)|=\|T\|,$ where ${\mathcal L}_s(^n E)$ denotes the space of all symmetric continuous $n$-linear forms on $E.$For $T\in {\mathcal L}_s(^n E),$ we define $$\mathop{\rm Norm}(T)=\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\}.$$$\mathop{\rm Norm}(T)$ is called the {\em norming set} of $T$. We classify $\mathop{\rm Norm}(T)$ for every $T\in {\mathcal L}_s(^2l_{\infty}^2)$.

scholarly journals On Effective Witt Decomposition and the Cartan–Dieudonné Theorem

2007 ◽  
Vol 59 (6) ◽  
pp. 1284-1300 ◽  
Author(s):  
Lenny Fukshansky
Keyword(s):  
Number Field ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Orthogonal Decomposition ◽  
Classical Theorem ◽  
One Dimensional ◽  
Effective Version ◽  
Special Version ◽  
Small Height ◽  
Hyperbolic Planes

AbstractLetKbe a number field, and letFbe a symmetric bilinear form in 2Nvariables overK. LetZbe a subspace ofKN. A classical theorem of Witt states that the bilinear space (Z,F) can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights ofFandZ. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan–Dieudonné theorem. Namely, we show that every isometry σ of a regular bilinear space (Z,F) can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights ofF,Z, and σ.

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).

RIGIDITY OF AFFINE HYPERSURFACES WITH RANK 1 SHAPE OPERATOR

2003 ◽  
Vol 14 (03) ◽  
pp. 211-234
Author(s):  
LUC VRANCKEN
Keyword(s):  
Curvature Tensor ◽  
Bilinear Form ◽  
Affine Connection ◽  
Shape Operator ◽  
Symmetric Bilinear Form ◽  
Positive Definite ◽  
Affine Hypersurface ◽  
3 Dimensional ◽  
Monge Ampere ◽  
Affine Metric

On a non-degenerate hypersurface it is well known how to induce an affine connection ∇ and a symmetric bilinear form, called the affine metric. Conversely, given a manifold M and an affine connection ∇ one can ask whether this connection is locally realizable as the induced affine connection on a nondegenerate affine hypersurface and to what extend this immersion is unique. In case that the image of the curvature tensor R of ∇ is 2-dimensional and M is at least 3-dimensional a rigidity theorem was obtained in [4]. In this paper, we discuss positive definite n-dimensional affine hypersurfaces with rank 1 shape operator (which is equivalent with 1-dimensional image of the curvature tensor) which are non-rigid. We show how to construct such affine hypersurfaces using solutions of (n - 1)-dimensional differential equations of Monge–Ampère type.

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