scholarly journals The Gauge-Natural Bilinear Operators Similar to the Dorfman–Courant Bracket

2020 ◽  
Vol 17 (2) ◽  
Author(s):  
Włodzimierz M. Mikulski
Keyword(s):  
Courant Bracket ◽  
Bilinear Operators

scholarly journals On the Courant bracket on couples of vector fields and \(p\)-forms

2018 ◽  
Vol 72 (2) ◽  
pp. 29
Author(s):  
Miroslav Doupovec ◽  
Jan Kurek ◽  
Włodzimierz Mikulski
Keyword(s):  
Vector Fields ◽  
Bilinear Operator ◽  
Courant Bracket ◽  
Bilinear Operators

If \(m\geq p+1\geq 2\) (or \(m=p\geq 3\)), all  natural bilinear  operators \(A\) transforming pairs of couples of vector fields and \(p\)-forms on \(m\)-manifolds \(M\) into couples of vector fields and \(p\)-forms on \(M\) are described. It is observed that  any natural skew-symmetric bilinear operator \(A\) as above coincides with the generalized Courant bracket up to three (two, respectively) real constants.

scholarly journals The natural operators similar to the twisted courant bracket on couples of vector fields and p-forms

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4071-4078
Author(s):  
Włodzimierz Mikulski
Keyword(s):  
Vector Fields ◽  
Jacobi Identity ◽  
Natural Numbers ◽  
Normalization Condition ◽  
Courant Bracket ◽  
Natural Operator ◽  
Bilinear Operators ◽  
Natural Operators

Given natural numbers m and p with m ? p + 2 ? 3, all Mfm-natural operators A sending closed (p+2)-forms H on m-manifolds M into R-bilinear operators AH transforming pairs of couples of vector fields and p-forms on M into couples of vector fields and p-forms on M are found. If m ? p + 2 ? 3, all Mfm-natural operators A (as above) such that AH satisfies the Jacobi identity in Leibniz form are extracted, and that the twisted Courant bracket [-,-]H is the unique Mfm-natural operator AH (as above) satisfying the Jacobi identity in Leibniz form and some normalization condition is deduced.

scholarly journals On the twisted Dorfman-Courant like brackets

2020 ◽  
Vol 40 (6) ◽  
pp. 703-723
Author(s):  
Włodzimierz M. Mikulski
Keyword(s):  
Vector Bundle ◽  
Jacobi Identity ◽  
Courant Bracket ◽  
Bilinear Operators ◽  
Restricted Condition ◽  
Linear Sections ◽  
Natural Operators

There are completely described all \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) which, like to the Dorfman-Courant bracket, send closed linear \(3\)-forms \(H\in\Gamma^{l-\rm{clos}}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^{\infty}\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE\oplus T^*E)\times \Gamma^l_E(TE\oplus T^*E)\to \Gamma^l_E(TE\oplus T^*E)\] transforming pairs of linear sections of \(TE\oplus T^*E\to E\) into linear sections of \(TE\oplus T^*E\to E\). Then all such \(C\) which also, like to the twisted Dorfman-Courant bracket, satisfy both some "restricted" condition and the Jacobi identity in Leibniz form are extracted.

scholarly journals On the gauge-natural operators similar to the twisted Dorfman-Courant bracket

2021 ◽  
Vol 41 (2) ◽  
pp. 205-226
Author(s):  
Włodzimierz M. Mikulski
Keyword(s):  
Vector Bundle ◽  
Jacobi Identity ◽  
Courant Bracket ◽  
Bilinear Operators ◽  
Linear Sections ◽  
Natural Operators

All \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)\] transforming pairs of linear sections of \(TE \oplus T^*E \to E\) into linear sections of \( TE \oplus T^*E \to E\) are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets \(C\) (i.e. \(C\) as above such that \(C_0\) is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear \(3\)-forms \(H\). An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.

scholarly journals Nonperturbative renormalization of bilinear operators with dynamical overlap fermions

2010 ◽  
Vol 81 (3) ◽  
Author(s):  
J. Noaki ◽  
T. W. Chiu ◽  
H. Fukaya ◽  
S. Hashimoto ◽  
H. Matsufuru ◽  
...  
Keyword(s):  
Bilinear Operators ◽  
Nonperturbative Renormalization

scholarly journals Weighted Estimates for Bilinear Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hua Zhu ◽  
Heping Liu
Keyword(s):  
Schrödinger Operator ◽  
Multilinear Operators ◽  
Schrodinger Operator ◽  
Weighted Estimates ◽  
Bilinear Operators ◽  
By Products

We study the boundedness of weighted multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We also investigate weighted estimates for bilinear operators related to Schrödinger operator.

scholarly journals One loop matching factors for staggered bilinear operators with improved glue

2010 ◽  
Author(s):  
Jongjeong Kim ◽  
Weonjong Lee ◽  
Stephen R. Sharpe
Keyword(s):  
Bilinear Operators
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