Annihilating polynomials, étale algebras, trace forms and the Galois number

2000 ◽  
Vol 75 (2) ◽  
pp. 116-120 ◽  
Author(s):  
D. W. Lewis ◽  
S. McGarraghy
Keyword(s):  
Trace Forms ◽  
Galois Number

The Galois number

1997 ◽  
Vol 309 (1) ◽  
pp. 81-96 ◽  
Author(s):  
Pat W. Beaulieu ◽  
Tommy C. Palfrey
Keyword(s):  
Galois Number

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 Corporeal Musical Structure: A Gestural-Kinesthetic Approach to Toru Takemitsu’s Rain Tree Sketch II

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Jocelyn Ho
Keyword(s):  
The Body ◽  
Musical Experience ◽  
Musical Structure ◽  
Trace Forms ◽  
Starting Point

The music of Tōru Takemitsu’s Rain Tree Sketch II (1994) entails a procession of discrete gestures that are delineated by moments of repose. The performer’s grasp of the piece lies in its physicality of movement: each gesture and in-between stillness are both heard and felt as an aggregate of velocities, directions, and intentions of the body. Drawing upon Carrie Noland’s concept of “vitality affects,” I take the performative gesture, encompassing both visually accessible movement and inwardly felt kinesthesia, as a starting point for the analysis of Rain Tree Sketch II. Concepts of effort and shape taken from Rudolf Laban’s dance theory provide a framework for creating a new methodology of enhanced trace-forms to analyze gesture and kinesthesia. The analysis of gestures reveals the coexistence of opposite effort qualities and shapes in an expanded corporeal space, resonating with Takemitsu’s ideal of reconciling contradictory sounds, as noted in his collection of essays Confronting Silence (1995). Husserl’s notions of retention and protention, viewed through the lens of embodiment, and Laban’s concepts of effort states and effort recovery are brought to bear on the still moments, showing the piece to have a throbbing, embodied rhythmic structural arc. This new methodology centering on gestural-kinesthetic details provides the tools to articulate structural sensations that are often overlooked but lie at the center of musical experience.

scholarly journals On PAC extensions and scaled trace forms

2010 ◽  
Vol 175 (1) ◽  
pp. 113-124 ◽  
Author(s):  
Lior Bary-Soroker ◽  
Dubi Kelmer
Keyword(s):  
Trace Forms

scholarly journals Orthogonal groups of some trace forms

1983 ◽  
Vol 29 (3) ◽  
pp. 297-312 ◽  
Author(s):  
Gerald J. Janusz
Keyword(s):  
Orthogonal Groups ◽  
Trace Forms

scholarly journals On trace forms of higher degree

1996 ◽  
Vol 246 ◽  
pp. 313-333 ◽  
Author(s):  
Manuel O'Ryan ◽  
Daniel B. Shapiro
Keyword(s):  
Trace Forms ◽  
Forms Of Higher Degree
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