RT journal article
T1 Exact solutions to max_||x||=1 sum_i=1^\infty ||T_i(x)||^2 with applications to Physics, Bioengineering and Statistics
A1 García Pacheco, Francisco Javier
A1 Cobos Sánchez, Clemente
A1 Moreno Pulido, Soledad
A1 Sánchez Alzola, Alberto
A2 Matemáticas
K1 Supporting vector
K1 Probability density operator
K1 Truly optimal TMS coil
K1 Statistical optimization
AB The supporting vectors of a matrix A are the solutions of max || x ||_2 =1 {||Ax||_2^2}. The generalized supporting vectors of matrices A_1 , . . . , A_k are the solutions of max || x ||_2 =1 {||A_1x||_2^2 + ||A_2x||_2^2 + ... + ||A_kx||_2^2}. Notice that the previous optimization problem is also a boundary element problem since the maximum is attained on the unit sphere. Many problems in Physics, Statistics and Engineering can be modeled by using generalized supporting vectors. In this manuscript we first raise the generalized supporting vectors to the infinite dimensional case by solving the optimization problem max || x || =1 sum_{i=1}^\infty ||T i (x )||^2 where (T i )_i is a sequence ofbounded linear operators between Hilbert spaces H and K of any dimension. Observe that the previous optimization problem generalizes the first two. Then a unified MATLAB code is presented for computing generalized supporting vectors of a finite number of matrices. Some particular cases are considered and three novel examples are provided to which our technique applies: optimized observable magnitudes by a pure state in a quantum mechanical sys- tem, a TMS optimized coil and an optimal location problem using statistics multivariate analysis. These three examples show the wide applicability of our theoretical and computational model.
PB Elsevier
SN 1007-5704
YR 2020
FD 2020
LK http://hdl.handle.net/10498/30660
UL http://hdl.handle.net/10498/30660
LA eng
DS Repositorio Institucional de la Universidad de Cádiz
RD 07-nov-2026