…will require more conditions than the ones formulated in Theorem 6.1, but it provides more flexibility: Problem PTDR can be solved for the total ring of fractions Q (E) by exploiting the existing summation algorithms for ∗-field extensions (Karr, 1981; Schneider, 2004, 2005a, 2007a, 2008, 2015).
Sample text:```For seeking the optimal value of Wh of the sub-problems which corresponds to the h-th level, we can solve the problem by the smoothing proximal gradient (SPG) method [6] fortunately.```
Sample intent: background
Sample text:```A rook pivoting scheme as in [3] and [11] can be implemented for tridiagonal matrices, but a full rook scheme for general banded matrices would probably lead to a ”chasing” scheme as in [5] or [15] .```
Sample intent: background



Sample text:```The above estimator generalizes the estimator in Sriperumbudur et al., 2017 to conditional densities.```
Sample intent: method



Sample text:```In-between methods can be subdivided into linear approximations [12,24] and quadratic interpolation methods [22,23,25–27] based on the degree of the polynomial approximation employed.```
Sample intent: method



Sample text:```A large part of Shannon’s seminal work [16] is devoted to both theoretical and practical aspects of products, and his invocation of the pastry dough mixing analogy [16, p.```
Sample intent: method



Sample text:```This transition was introduced first by Prince and Elder (2007) and it simplifies formulas for parameter update on Mstep of EM algorithm.```
Sample intent: background
Sample text:```Discrepancies analogous to those between our results and those of simulations have, indeed, recently been observed in the case of the binary perceptron [18], the pspin interaction spin glass [19], and in the case of fluctuating manifolds in random media [20], where dynamically frozen phases were observed in regions of parameter space in which static approaches yielded ergodic phases.```
Sample intent: result



Sample text:```This section contains a generalisation of the basic results on multifractal products of stochastic processes developed in Kahane (1985), Kahane (1987) and Mannersalo et al (2002).```
Sample intent: method



Sample text:```phi is a meta-program interpreted in [112, 24] as a second-order predicate which applies to the predicate p.```
Sample intent: method



Sample text:```This seems to be the main drawback of the current implementation in comparison to the method of Black Jepson (“refined” in Table 1).```
Sample intent: result



Sample text:```For seeking the optimal value of Wh of the sub-problems which corresponds to the h-th level, we can solve the problem by the smoothing proximal gradient (SPG) method [6] fortunately.```
Sample intent: background



Sample text:```A large part of Shannon’s seminal work [16] is devoted to both theoretical and practical aspects of products, and his invocation of the pastry dough mixing analogy [16, p.```
Sample intent: method



Sample text:```The discretization is implemented within the DUNE numerics environment [3,4], centered around dune-gdt [23]: the dune-stuff [24] module provides classes for vectors, matrices and linear solvers (for instance the bicgstab.amg.ilu0 solver used in these experiments), dune-gdt provides the discretization building blocks (such as discrete function spaces, operators, products and functionals), and dune-hdd1 provides parametric discretizations compatible with pyMOR [22].```
Sample intent: background



Sample text:```The items are internally consistent and reproducible and the questionnaire may therefore be considered to have at least as favourable levels of reliability as clinical scores used to assess outcomes (Koran 1975).```
Sample intent: result



Sample text:```In this study, we generalized previous results regarding the quasi-Gaussian shape and the scalar property using the SBF model [13,21,52].```
Sample intent: result



Sample text:```A rook pivoting scheme as in [3] and [11] can be implemented for tridiagonal matrices, but a full rook scheme for general banded matrices would probably lead to a ”chasing” scheme as in [5] or [15] .```
Sample intent: background