Sentence examples for basis set of functions from inspiring English sources

Given a complete orthonormal basis set of functions {φn} in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as: :f = \sum_{n=1}^\infty \alpha_n \varphi_n.

Introduced is a general framework for performing group-level analyses of fMRI data using any basis set of two functions (i.e., the canonical hemodynamic response function and its first derivative) to model the hemodynamic response to neural activity.

Amplitude and phase modulation protocols are crafted leveraging insights from functional synthesis and the basis set of Walsh functions.

We are able to effectively bound the synthesis space while still achieving highly effective gates by synthesizing relevant time-domain control fields in the basis set of Walsh functions - square wave analogues of the sines and cosines [51, 61] - using the concept of functional analysis.

In particular, since we were interested in the residual semantic processing occurring after the subjects responded to the stimulus, we examined the ROIs' activation profile in more detail by modeling the hemodynamic time course with a more general basis set of spline functions.

The first stage constructs a basis set of nonlinear functions (such as a set of orthogonal polynomials or B-splines) of the exposure conditional on the IV, and the second stage fits the outcome as a linear combination of these nonlinear functions.

The electronic equations of motion for a leads unit cell, independent of N, may be expressed in a square matrix form, with an orthonormal minimal basis set of local wave functions as follows: ( E I − M d ) × c ( k, E ) = 0. (6).

In this model, the hemodynamic response during MES and CHOICE was modeled with a basis set of seven cubic spline functions spaced one TR (2.5 s) apart and spanning the interval from 0 to 15 seconds post trial onset.

Except for states of the same symmetry of diatomic molecules, basis sets of strictly diabatic electronic wave functions do not exist, apart from the "trivial" basis set made of functions ϕ n that are independent of the nuclear coordinates Q.

Calculation of the electronic structure of mQCA gates composed of these molecules is challenging, since the number of basis set functions grows exponentially as the number of molecules and atoms are increased.

Figure 3 illustrates the convergence performance of the proposed method for the electron electron correlation energy of a HF molecule with the 6-31G** baset set as a function of the number of employed SDs.