Sentence examples for generalized points from inspiring English sources

The top line is the vector z, the rectangles with dashed line are original group vector (z i ) g, and the rectangles with solid line are the generalized points, which changes the overlapping to non-overlapping.

The generalized ℓ 2,1 norm ∥z∥2,1 can be treated as the generalized ℓ 1 norm of generalized points, whose entry (z i ) g is also a vector, and the absolute value of each entry is treated as the ℓ 2 norm of (z i ) g. See Fig. 1 a intuitively, where the top line is the vector z, the rectangles with dashed line are original (z i ) g, and the rectangles with solid line are the generalized points.

Putting these generalized points (rectangles with solid line in the figure) as the columns of a matrix, we can regard (s|mathbf {z} -mathbf {x}|_{2}^{2}) as the matrix Frobenius norm (|left [ z_{i})_{g}right ]-left [(x_{i})_{g}right ]|_{F}^{2}), where [ z i ) g ] is a matrix as in Fig. 1 a with every line being its row ([(x i ) g ] is similar).

Firstly, an explicit form Green function for a generalized point load acted at an arbitrary point outside the hole is derived.

Here we survey some applications of such generalized point set topologies to chemistry and biology, providing an overview of the underlying mathematical structures.

Here, numerical results for a wide variety of scatterer configurations are presented in order to demonstrate the validity and accuracy of our generalized point-matching technique formulation.

Secondly, the Green function for a generalized point load acted at the rim of the hole, as a special case, is obtained, and then a general solution for the case of arbitrarily distributed mechanical and electric loading on the hole surface is presented based on the superposition principle.

These operators, which provide new examples of generalized point interactions in the sense of Šeba, are defined by the boundary conditions ƒ(0+) = e−zƒ(0−), rƒ(0+) + ƒ′(0+) = ez[rƒ(0−) + ƒ′(0−)], for z ∈ C, r ∈ R. We calculate their spectra, resolvents, and scattering matrices, and show that they can be realized as limits of Schrödinger operators with local short-range potentials.