Scaled in Cartesian Coordinates Ab Initio Molecular Force Fields of DNA Bases: Application to Canonical Pairs

The model of Regularized Quantum Mechanical Force Field (RQMFF) was applied to the joint treatment of ab initio and experimental vibrational data of the four primary nucleobases using a new algorithm based on the scaling procedure in Cartesian coordinates. The matrix of scaling factors in Cartesian coordinates for the considered molecules includes diagonal elements for all atoms of the molecule and off-diagonal elements for bonded atoms and for some non-bonded atoms (1–3 and some 1–4 interactions). The choice of the model is based on the results of the second-order perturbation analysis of the Fock matrix for uncoupled interactions using the Natural Bond Orbital (NBO) analysis. The scaling factors obtained within this model as a result of solving the inverse problem (regularized Cartesian scale factors) of adenine, cytosine, guanine, and thymine molecules were used to correct the Hessians of the canonical base pairs: adenine–thymine and cytosine–guanine. The proposed procedure is based on the block structure of the scaling matrix for molecular entities with non-covalent interactions, as in the case of DNA base pairs. It allows avoiding introducing internal coordinates (or coordinates of symmetry, local symmetry, etc.) when scaling the force field of a compound of a complex structure with non-covalent H-bonds.

Combination Synchronization of Fractional Systems Involving the Caputo–Hadamard Derivative

The main aim of this paper is to investigate the combination synchronization phenomena of various fractional-order systems using the scaling matrix. For this purpose, the combination synchronization is performed by considering two drive systems and one response system. We show that the combination synchronization phenomenon is achieved theoretically. Moreover, numerical simulations are carried out to confirm and validate the obtained theoretical results.

Scaling the Potential of Compact City Development: The Case of Lahore, Pakistan

With increasing urban populations, high vehicle miles have made the concept of a compact city imperative. A compact city is characterized by high-density development and mixed land use with no urban sprawl. City managers are trying hard to make their cities compact and livable. The potential conformance to a compact city development requires scaling before any significant intervention. Several studies have been conducted on the different aspects of the compact city in the developed world, but there is limited understanding in the South Asian context. This study aimed to fill this research gap and proposes a theoretical matrix to gauge the potential compactness of Lahore, Pakistan. It comprises some key attributes, such as landscape ecology, measurement of density, density distribution, transportation network, accessibility, dispersion index, and mixed-use land consumption, which were analyzed in this research. The data were analyzed using Geographical Information System (GIS) and ERDAS IMAGINE software to make a scaling matrix. The research findings show that Lahore is a semi-compact city, with high potential to become a true compact city. The paper recommends that the urban extent should not be extended until targeted colonization is achieved, and the spatial growth of the city should be managed by encouraging infilled development, high-density living, and public transport provision. This research will help policymakers, urban planners, and transport planners devising policies for compact city development.

A Method of Improving Time Integration Algorithm Accuracy for Long-Term Dynamic Simulation

This paper aims to improve the accuracy of time integration algorithms (TIAs) for long-term simulation of structural dynamics. To this end, a new method of reducing the period elongation (numerical dispersion) was proposed by utilizing the mass scaling matrix. Firstly, the period elongation of explicit Gui-[Formula: see text] algorithm as a representative was analyzed, and the strategy of enhancing the algorithm accuracy was investigated. Subsequently, the period elongation was reduced by introducing a parameter to change the mass matrix, which is weighted by the original mass matrix and stiffness matrix. The bisection method is utilized to determine the parameter according to the formulation of period elongation. Since just the mass matrix of original algorithm is changed slightly, the convergence rate of original TIA remains unchanged and the proposed mass scaling method imposes little influence on the stability condition of original algorithm. Moreover, this method has few modifications to the computer program of TIA and is easy to implement. Finally, both linear and nonlinear long-term dynamic response analyses for multiple-degree-of-freedom systems indicated that the proposed mass scaling method is effective and convenient to reduce the period elongation for several typical TIAs.

Complex Unitary Recurrent Neural Networks Using Scaled Cayley Transform

Recurrent neural networks (RNNs) have been successfully used on a wide range of sequential data problems. A well known difficulty in using RNNs is the vanishing or exploding gradient problem. Recently, there have been several different RNN architectures that try to mitigate this issue by maintaining an orthogonal or unitary recurrent weight matrix. One such architecture is the scaled Cayley orthogonal recurrent neural network (scoRNN) which parameterizes the orthogonal recurrent weight matrix through a scaled Cayley transform. This parametrization contains a diagonal scaling matrix consisting of positive or negative one entries that can not be optimized by gradient descent. Thus the scaling matrix is fixed before training and a hyperparameter is introduced to tune the matrix for each particular task. In this paper, we develop a unitary RNN architecture based on a complex scaled Cayley transform. Unlike the real orthogonal case, the transformation uses a diagonal scaling matrix consisting of entries on the complex unit circle which can be optimized using gradient descent and no longer requires the tuning of a hyperparameter. We also provide an analysis of a potential issue of the modReLU activiation function which is used in our work and several other unitary RNNs. In the experiments conducted, the scaled Cayley unitary recurrent neural network (scuRNN) achieves comparable or better results than scoRNN and other unitary RNNs without fixing the scaling matrix.

A New Class of Scaling Matrices for Scaled Trust Region Algorithms

A new class of affine scaling matrices for the interior point Newton-type methods is considered to solve the nonlinear systems with simple bounds. We review the essential properties of a scaling matrix and consider several well-known scaling matrices proposed in the literature. We define a new scaling matrix that is the convex combination of these matrices. The proposed scaling matrix inherits those interesting properties of the individual matrices and satisfies all the desired requirements. The numerical experiments demonstrate the superiority of the new scaling matrix in solving several important test problems.

Finite equal norm Parseval wavelet frames over prime fields

In the framework of wave packet analysis, finite wavelet systems are particular classes of finite wave packet systems. In this paper, using a scaling matrix on a permuted version of the discrete Fourier transform (DFT) of system generator, we derive a locally-scaled version of the DFT of system generator and obtain a finite equal-norm Parseval wavelet frame over prime fields. We also give a characterization of all multiplicative subgroups of the cyclic multiplicative group, for which the associated wavelet systems form frames. Finally, we present some concrete examples as applications of our results.