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Record W2046511471 · doi:10.1142/s0219199705001945

RATIONAL JORDAN DECOMPOSITION OF BILINEAR FORMS

2005· article· en· W2046511471 on OpenAlex
Dragomir Ž. Ðoković, Kaiming Zhao

Why this work is in the frame

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affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.
fundA Canadian funder is recorded on the work.

Bibliographic record

VenueCommunications in Contemporary Mathematics · 2005
Typearticle
Languageen
FieldMathematics
TopicAlgebraic structures and combinatorial models
Canadian institutionsWilfrid Laurier UniversityUniversity of Waterloo
FundersNatural Sciences and Engineering Research Council of Canada
KeywordsMathematicsAlgebraically closed fieldAlgebraic closurePure mathematicsField (mathematics)Unimodular matrixBilinear interpolationClosure (psychology)Jordan matrixBilinear formDecompositionField extensionSymmetric bilinear formAlgebra over a fieldDiscrete mathematicsMathematical analysis

Abstract

fetched live from OpenAlex

This is a continuation of our previous work on Jordan decomposition of bilinear forms over algebraically closed fields of characteristic 0. In this note, we study Jordan decomposition of bilinear forms over any field K 0 of characteristic 0. Let V 0 be an n-dimensional vector space over K 0 . Denote by [Formula: see text] the space of bilinear forms f : V 0 × V 0 → K 0 . We say that f = g + h, where f, g, [Formula: see text], is a rational Jordan decomposition of f if, after extending the field K 0 to an algebraic closure K, we obtain a Jordan decomposition over K. By using the Galois group of K/K 0 , we prove the existence of rational Jordan decompositions and describe a method for constructing all such decompositions. Several illustrative examples of rational Jordan decompositions of bilinear forms are included. We also show how to classify the unimodular congruence classes of bilinear forms over an algebraically closed field of characteristic different from 2 and over the real field.

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Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.139
Threshold uncertainty score0.562

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0010.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.131
GPT teacher head0.389
Teacher spread0.258 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it