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Bohmian Pathways into Chemistry: A Brief Overview
Published in Xavier Oriols, Jordi Mompart, Applied Bohmian Mechanics, 2019
After using the Born–Oppenheimer approximation, splitting electronic from nuclear degrees of freedom, the essence of electronic structure methodologies is precisely finding the electronic configuration of molecular systems by appealing to the idea of separability, from ab initio methods (wave function approach) to the density functional theory (probability density approach). Of course, once an optimal basis set is found, correlations will appear when appealing to the appropriate symmetrization conditions. For example, in the case of ab initio methodologies, Slater determinants are considered to provide the correct symmetry to the linear combinations of orbitals formed to describe the configuration of the corresponding molecular system (see Section 4.2.2).
Supersymmetric Theory of Stochastics:Demystification of Self-Organized Criticality
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
This symmetrization is the well-known Weyl quantization rule of quantum theory. It guarantees that any real Hamilton function in the pathintegral representation results in a Hermitian Hamiltonian in operator representation of the theory. In our case, the Weyl-Stratonovich bi-graded symmetrization of FP Hamilton function of the pathintegral representation will result (see Reference 30 for details) in the FP operator in Equation 17.45, which is correct as we already established outside the pathintegral formulation of the theory (see Section 17.4.1). In other words, the Stratonovich interpretation of SDEs is correct and it corresponds to the bi-graded Weyl symmetrization on passing from the pathintegral to operator representation of the model.
Results and Conclusions
Published in Krzysztof Wołk, Machine Learning in Translation Corpora Processing, 2019
The baseline system testing was done using the Moses open source SMT toolkit with its Experiment Management System (EMS) [13]. The SRI Language Modeling Toolkit (SRILM) [14] with an interpolated version of the Kneser-Ney discounting (interpolate –unk –kndiscount) was used for 5-gram language model training. The MGIZA++ tool was used for word and phrase alignment. KenLM [15] was used to binarize (transform features of a text entity into vectors of numbers) the language model, with the lexical reordering set to the msd-bidirectional-fe model [16]. The symmetrization method was set to grow-diag-final-and for word alignment processing [13].
On weighted isoperimetric inequalities with non-radial densities
Published in Applicable Analysis, 2019
A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M. R. Posteraro
Assume that the numbers N,k,l and α satisfy the assumptions of Theorem 1.1. Then (1) implies for every smooth set , where is the symmetrization of Ω. We will use (12) to show Theorem 5.1 below. For the sake of simplicity, we will carry out the proof under the additional assumption that is Lipschitz continuous. Note that this assumption is indeed fullfilled when u is sufficiently flat near its maximum, for instance, when the set has positive measure. A detailed proof in the general case will be provided in the forthcoming paper [13].
On the inexact symmetrized globally convergent semi-smooth Newton method for 3D contact problems with Tresca friction: the R-linear convergence rate
Published in Optimization Methods and Software, 2020
R. Kučera, K. Motyčková, A. Markopoulos, J. Haslinger
The projective formulation presented in this paper seems to be natural since it can be directly derived from the weak formulation of contact problems [14]. However, the respective slanting function is given by non-symmetric matrices with a type of non-symmetry that cannot be removed by simple linear algebra tools (linear combinations, eliminations, etc.) unlike the 2D case [27]. Therefore, the inner linear systems in the SSN method are solved by time-consuming algorithms. Fortunately, the slanting function at the minimizer does not contain some terms so that a simple linear algebra based symmetrization is possible. Neglecting these terms a-priori we arrive at an approximation of the slanting function that can be replaced by a symmetric matrix at each step of the SSN method. Then, 3D contact problems can be treated analogously as 2D ones [27]. To implement this idea efficiently, we proceed as follows. First, we propose a dual version of the symmetrized SSN method with inexact solving inner linear systems by a few CG steps. Although numerical experiments indicate a high computational efficiency, the previous superlinear convergence result does not hold in general. Therefore, we propose a monotonous globalization strategy guaranteeing the R-linear convergence rate of the algorithm. This result is similar to the 2D case but its proof differs in many points. To our knowledge, there is no such analysis for 3D frictional contact problems. The globally convergent variant of the SSN method is closely related to the above mentioned active set algorithms [9,12,26]. The main difference consists in different definitions of the active sets.
A new wide neighbourhood primal-dual interior-point algorithm for semidefinite optimization
Published in Optimization, 2019
The paper is organized as follows. In Section 2, we introduce the primal-dual pair of SDO problems and briefly explain how path-following IPMs work. In Section 3, we define our new neighbourhood and prove that it is a wide neighbourhood. In Section 4, we use the square root function for an algebraic equivalent transformation of the centering equation along with the Zhang's symmetrization operator in order to get symmetric search directions. Then, we present the framework of our algorithm. The convergence analysis and theoretical complexity bounds are presented in Section 5. Finally, we present some conclusions and future research aspects in Section 6.