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Matroid theory
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Crapo and Rota [350] proved the following remarkable theorem. In view of the last corollary, this theorem is stated for simple matroids. Recall that a linear functional on a vector space is a linear map from the vector space into its underlying field.
More About Dual Spaces
Published in James K. Peterson, Basic Analysis III, 2020
Comment 11.1.1 Thus J: X → X″ is what is called a congruence between X and J(X) ⊂ X″. A congruence is thus a linear map which is a norm preserving bijection. The symbol we use for congruence is ≅; so we would say X ≅ J(X).
Hilbert spaces
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
We close this section with a brief discussion of conditioning. See [St], for example, for additional information. Suppose first that L is an invertible linear map on a finite-dimensional real or complex vector space. When solving the linear equation Lu = v numerically, one needs to bound the relative error in the solution in terms of the relative error in the right-hand side. Assume that u is an exact solution to Lu = v. Let δu and δυ denote the errors in measuring u and υ. These errors are vectors. Then v+δv=L(u+δu)=Lu+L(δu).
On the asymptotically cubic fractional Schrödinger–Poisson system
Published in Applicable Analysis, 2021
Wenbo Wang, Yuanyang Yu, Yongkun Li
By the definition, we get Now we analyse . We first prove that Taking , we obtain Thus, by (14), it holds that Therefore, It remains to prove the uniqueness of weak solution. Define bilinear map and linear map So by Hlder inequality and Sobolev embedding, it holds that By the way, is used here again to ensure that 4s+2t−3>0. By Lax–Milgram theorem, there exists a unique such that By (13), is the unique weak solution of .
Solving SDP completely with an interior point oracle
Published in Optimization Methods and Software, 2021
Bruno F. Lourenço, Masakazu Muramatsu, Takashi Tsuchiya
Although our focus is on semidefinite programming, most of the results will be proved for the following primal and dual pair of general conic linear programs: where is a linear map, , . Semidefinite programming corresponds to the specific case where and .
ADMM for multiaffine constrained optimization
Published in Optimization Methods and Software, 2020
Wenbo Gao, Donald Goldfarb, Frank E. Curtis
is a multilinear map of . For every X, is a linear map of Y, and thus is a linear map of Y.