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The geometric description of linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
The quadratic form corresponding to a symmetric bilinear form is always described by a homogeneous quadratic polynomial. We have seen that, in odd characteristic, quadratic forms are equivalent to symmetric bilinear forms.
Mesoscale analysis of dry shrinkage fractures in concrete repair systems using FEM and VEM
Published in International Journal of Pavement Engineering, 2022
Hao Jin, Yuliang Zhou, Chen Zhao
In the VEM, the symmetric bilinear form of satisfied the characteristics of consistency and stability. For of element , we had The first term on the right end of Equation (17) was called the consistency term, which ensured that the solution of the original problem was the solution of the discrete problem conforming to the exact solution. Except for the projection operator, the form of this term was consistent with that obtained using the standard finite element process, where the consistency term was expressed as: The element stiffness matrix of the consistency term was: where was an elastic matrix; was a geometric matrix; and was represented by the coefficient matrix in the virtual element.
On the two-dimensional tidal dynamics system: stationary solution and stability
Published in Applicable Analysis, 2020
Let us define the non-symmetric bilinear form: where . If has a smooth second-order derivatives, then We consider and an integration by parts twice yields (using the fact that ) and hence , so that is not symmetric. The bilinear form is continuous and coercive in , i.e. for some positive constant . By means of the Gelfand triple we may consider , given by (4), as a mapping from into its dual .
A long-step feasible predictor–corrector interior-point algorithm for symmetric cone optimization
Published in Optimization Methods and Software, 2019
S. Asadi, H. Mansouri, Zs. Darvay, G. Lesaja, M. Zangiabadi
The Jordan algebra over with the identity element e is Euclidean iff the symmetric bilinear form over is positive definite. The symmetric bilinear form is associative, i.e. . These imply that if is a EJA, then is an inner product. In the sequel, will always denote this inner product, so . The norm induced by this inner product is called the Frobenius norm. The Frobenius norm is defined as follows: For any EJA, the corresponding cone of squares is a SC i.e. the cone that is self-dual and homogeneous (see, e.g. [4]). The following result completely characterizes SCs in terms of EJAs.