Regular Approximation Research Articles

Strong convergence of two regularized relaxed extragradient schemes for solving the split feasibility and fixed point problem with multiple output sets

We study the split feasibility and fixed point problem with multiple output sets in real Hilbert spaces. Using Tikhonov's regularization and viscosity approximation techniques, we propose two relaxed extragradient algorithms for finding approximate solutions to this problem and prove that the sequences they generate converge strongly. Finally, we give some consequences and an application of our results, and also present numerical examples to validate the efficiency of our algorithms.

Instability analysis and regularization approximation to the forward/backward problems for fractional damped wave equations with random noise

Considered herein is the forward/backward problems for fractional damped wave equations utt+α(−Δ)s1ut+β(−Δ)s2u=F(u,x,t) subject to the random Gaussian white noise initial and final data. First of all, we construct the mild solutions to forward/backward problems under all cases for parameters α, β, s1 and s2, and then investigate their stability and instability properties in the sense of Hadamard. Secondly, we propose the regularized solutions by using the Fourier truncation method and establish well-posedness for regularized solutions in above unstable cases. Thirdly, we derive some error estimates between the exact solutions and their regularized solutions in E‖⋅‖22-norm, and theoretically characterize the Fourier truncation approximation effect from regularized solutions to exact solutions. Finally, we give a series of numerical examples used to illuminate the regularization approximation effect of above method.

Analysis and Simulation of Hydromagnetic Nano-Fluid Flow Passing an Inclined Heated Sheet

The examination of nano-fluid in recent time has been encouraging just as its research interest cuts across some disciplines such as mathematics, mechanical, chemical, civil engineering, physics, earth and natural sciences. Its applicability in the industrial and technological advancement helps in controlling the rate at which heat is conducted and diffuses in a given medium, as well as improving thermal transportation through convection, radiation and conduction. In this work, the analysis of hydromagnetic nano-fluid flow past an inclined heated surface with temperature dependent non-uniform heat source/sink and thermal radiation under heat and mass transmission perspective is considered. Adequate similarity variables are employed in recovering the nonlinear coupled ordinary differential equations (ODEs) from the partial differential equations (PDEs) which describe the equation of the boundary layer. The transformed equations (ODEs) are addressed both analytically and numerically with the aid of regular series approximation technique and Wolfram Mathematica package. The numerically simulated obtained results are graphically presented with legends. The computations opined that velocity declines while energy increases as the magnetic field number enhances. Similarly, improvement on angle of inclination, breeds velocity deceleration. Furthermore, the rate of energy transfer increases as the temperature dependent non-uniform parameters due to thermal generation enhances. In addition, the nanoparticle concentration decreases with increasing values of Schmidt and thermal radiation factors.

Regular approximation of singular third-order differential operators

In this paper, regular approximation of third-order differential operators with two singular endpoints is investigated. Firstly the induced restriction operator is constructed and proved to be a self-adjoint operator in the Hilbert space. Moreover the convergence of operators and spectrum is obtained by strong resolvent convergence. In particular, norm resolvent convergence is given when the two endpoints are both limit circle.

Isotherm Theoretical Study of the AlxGa1-xAsySb1-y Quaternary Alloy Using the Regular Solution Approximation for Its Possible Growth via Liquid-Phase Epitaxy at Low Temperatures.

This work presents the theoretical calculation of isotherm diagrams for quaternary alloys of III-V semiconductor compounds with the form IIIxIII1-xVyV1-y. In particular, the isotherm diagrams for the AlxGa1-xAsySb1-y quaternary alloy at low temperatures were calculated (500 °C, 450 °C, 400 °C, and 350 °C). The AlxGa1-xAsySb1-y quaternary alloy was formed from four binary compounds such as GaAs, AlAs, AlSb, and GaSb, all with direct bandgaps. The regular solution approximation was used to find the quaternary isotherm diagrams, represented in four linearly independent equations, which were solved using Parametric Technology Corporation Mathcad 14.0 software for different arsenic and antimony atomic fractions. The results support the possible growth of layers via liquid-phase epitaxy in a range of temperatures from 500 °C to 350 °C, where the crystalline quality could be improved at low temperatures. These semiconductor layers could have applications for optoelectronic devices in photonic communications, thermophotovoltaic systems, and microwave devices with good crystalline quality.

ZORA Gaussian basis sets for Fr, Ra, and Ac.

Segmented all-electron basis sets of double and triple zeta valence qualities plus polarization functions (DZP and TZP) for the elements Fr, Ra, and Ac to be used with the zeroth-order regular approximation (ZORA) were presented. These sets were constructed from the reoptimization of the contraction coefficients of the corresponding non-relativistic basis sets. In order to adequately describe electrons distant from the atomic nuclei, these sets were augmented with diffuse functions and were, respectively, designated as ADZP-ZORA and ATZP-ZORA. At the ZORA-B3LYP theory level, the relativistic sets were employed to calculate ionization energies of Fr, Ra, and Ac as well as bond lengths, dissociation energies, harmonic vibrational frequencies, and static mean dipole polarizabilities of some diatomics. Comparing with benchmark theoretical results and with experimental data available in the literature, it can be verified that our basis sets are able to produce reliable and accurate results. Evaluation of the performances of ZORA and second-order Douglas-Kroll-Hess Hamiltonians was performed.

Regular Approximations of the Fastest Motion of Mobile Robot under Bounded State Variables

Regular Approximations of the Fastest Motion of Mobile Robot under Bounded State Variables

Relativistic and QED corrections to one-bond indirect nuclear spin-spin couplings in X2 2+ and X3 2+ ions (X = Zn, Cd, Hg).

The indirect spin-spin coupling tensor, J, between mercury nuclei in systems containing this element can be of the order of a few kHz and one of the largest measured. We analyzed the physics behind the electronic mechanisms that contribute to the one- and two-bond couplings nJHg-Hg (n = 1, 2). For doing so, we performed calculations for J-couplings in the ionized X2 2+ and X3 2+ linear molecules (X = Zn, Cd, Hg) within polarization propagator theory using the random phase approximation and the pure zeroth-order approximation with Dirac-Hartree-Fock and Dirac-Kohn-Sham orbitals, both at four-component and zeroth-order regular approximation levels. We show that the "paramagnetic-like" mechanism contributes more than 99.98% to the total isotropic value of the coupling tensor. By analyzing the molecular and atomic orbitals involved in the total value of the response function, we find that the s-type valence atomic orbitals have a predominant role in the description of the coupling. This fact allows us to develop an effective model from which quantum electrodynamics (QED) effects on J-couplings in the aforementioned ions can be estimated. Those effects were found to be within the interval (0.7;1.7)% of the total relativistic effect on isotropic one-bond 1J coupling, though ranging those corrections between the interval (-0.4; -0.2)% in Zn-containing ions, to (-1.2; -0.8)% in Hg-containing ions, of the total isotropic coupling constant in the studied systems. The estimated QED corrections show a visible dependence on the nuclear charge Z of each atom X in the form of a power-law proportional to ZX 5.

Evaluation of mixed micellar interactions of CnBCl and SDBS mixtures using dissociated Margules model and influence of different salts

AbstractThe conductivity measurement has been used to determine the first and second CMC's (CMC1 and CMC2) of alkyldimethylbenzylammonium chloride (CnBCl; n = 12, 14 and 16) as well as gemini surfactant trimethylene‐1,2‐bis‐(dodecyldimethylammonium bromide) (12‐2‐12) with sodium dodecylbenzenesulphonate (SDBS) in aqueous solutions using Rubingh's the regular theory approximation as well as the dissociated Margules Model. The use of Margules model allows us to calculate activity coefficients of the constituents and hence provides better values of micellar parameters for asymmetric mixed systems as compared to the regular solution theory. The CMC1 values for CnBCl/SDBS mixtures were seen to be lower than those predicted from their ideal mixtures suggesting synergistic interactions although the synergism increases with chain length (n) of the cationic surfactant. Both CMC1 and CMC2 experience a decrease with rising values of n. The addition of salts including NaCl, KCl, NaBr, Na3PO4, and Na2SO4 in mixtures of C16BCl and SDBS suggests that salt counterions have a considerable impact on CMC1 when either surfactant is in excess. Zeta potential (ζ) measurements provide more evidence in favor of these observations. The thermodynamic features of micellization have also been scrutinized using isothermal titration calorimetry (ITC).

Non-self-adjoint relativistic point interaction in one dimension

The one-dimensional Dirac operator with a singular interaction term which is formally given by A⊗|δ0〉〈δ0|, where A is an arbitrary 2×2 matrix and δ0 stands for the Dirac distribution, is introduced as a closed not necessarily self-adjoint operator. We study its spectral properties, find its non-relativistic limit and also address the question of regular approximations. In particular, we show that, contrary to the case of local approximations, for non-local approximating potentials, coupling constants are not renormalized in the limit.

Silver Ion-Induced Formation of Unprecedented Thorium Nonamer Clusters via Lacuna-Construction Strategy

Silver Ion-Induced Formation of Unprecedented Thorium Nonamer Clusters via Lacuna-Construction Strategy

Optimization of the r2SCAN-3c Composite Electronic-Structure Method for Use with Slater-Type Orbital Basis Sets.

The “Swiss army knife” composite density functional electronic-structure method r2SCAN-3c (J. Chem. Phys.2021, 154, 064103) is extended and optimized for the use with Slater-type orbital basis sets. The meta generalized-gradient approximation (meta-GGA) functional r2SCAN by Furness et al. is combined with a tailor-made polarized triple-ζ Slater-type atomic orbital (STO) basis set (mTZ2P), the semiclassical London dispersion correction (D4), and a geometrical counterpoise (gCP) correction. Relativistic effects are treated explicitly with the scalar-relativistic zeroth-order regular approximation (SR-ZORA). The performance of the new implementation is assessed on eight geometry and 74 energy benchmark sets, including the extensive GMTKN55 database as well as recent sets such as ROST61 and IONPI19. In geometry optimizations, the STO-based r2SCAN-3c is either on par with or more accurate than the hybrid density functional approximation M06-2X-D3(0)/TZP. In energy calculations, the overall accuracy is similar to the original implementation of r2SCAN-3c with Gaussian-type atomic orbitals (GTO), but basic properties, intermolecular noncovalent interactions, and barrier heights are better described with the STO approach, resulting in a lower weighted mean absolute deviation (WTMAD-2(STO) = 7.15 vs 7.50 kcal mol–1 with the original method) for the GMTKN55 database. The STO-optimized r2SCAN-3c outperforms many conventional hybrid/QZ approaches in most common applications at a fraction of their cost. The reliable, robust, and accurate r2SCAN-3c implementation with STOs is a promising alternative to the original implementation with GTOs and can be generally used for a broad field of quantum chemical problems.

Attribute reduction based on observational consistency in intuitionistic fuzzy multi-covering decision systems

Attribute reduction is an important issue in data mining, machine learning and other applications of big data processing. Covering-based rough set and intuitionistic fuzzy (IF) set models are both the effective theoretical tools of uncertainty or imprecise computation, and thus IF covering rough set model has been acknowledged as a positive approach to attribute reduction. Based on IF covering rough set model, this study explores a kind of parameterized IF observational consistency in IF multi-covering decision system, and proposes an attribute reduction method. This article firstly defines the concepts of regular IF β-covering, parameterized IF observational sets on the regular IF β-covering approximation space. Secondly, the parameterized IF observational consistency is defined to be the principal of attribute reduction in the IF multi-covering decision system, and the related IF discernibility matrix is developed to provide a way of attribute reduction. For multi-observational consistency corresponding to an observational parameters set, an unified multi-observational discernibility matrix is constructed, which avoids the disadvantage of needing to construct multiple corresponding discernibility matrices separately. Furthermore, an attribute reduction algorithm based on iterative dissolving of unified multi-observational discernibility matrix is proposed, and the experiment to demonstrate effectiveness of algorithm is presented. Experiments with UCI datasets shows that, the proposed method is a good way for improving both the rates of attribute-reduced and the classification accuracy of reduced datasets.

Reflexivity for spaces of regular operators on Banach lattices

We prove that if Banach lattices E E and F F are reflexive and each positive linear operator from E E to F F is compact then L r ( E ; F ) {\mathcal L}^r(E;F) , the space of all regular linear operators from E E to F F , is reflexive. Conversely, if E ∗ E^\ast or F F has the bounded regular approximation property then the reflexivity of L r ( E ; F ) {\mathcal L}^r(E;F) implies that each positive linear operator from E E to F F is compact. Analogously we also study the reflexivity for the space of regular multilinear operators on Banach lattices.

Optical Activity of Spin-Forbidden Electronic Transitions in Metal Complexes from Time-Dependent Density Functional Theory with Spin-Orbit Coupling.

The calculation of magnetic transition dipole moments and rotatory strengths was implemented at the zeroth‐order regular approximation (ZORA) two‐component relativistic time‐dependent density functional theory (TDDFT) level. The circular dichroism of the spin‐forbidden ligand‐field transitions of tris(ethylenediamine)cobalt(III) computed in this way agrees very well with available measurements. Phosphorescence dissymmetry factors glum and the corresponding lifetimes are evaluated for three N‐heterocyclic‐carbene‐based iridium complexes, two of which contain helicene moieties, and for two platinahelicenes. The agreement with experimental data is satisfactory. The calculations reproduce the signs and order of magnitude of glum , and the large variations of phosphorescence lifetimes among the systems. The electron spin contribution to the magnetic transition dipole moment is shown to be important in all of the computations.

Computational Prediction of Graphdiyne-Supported Three-Atom Single-Cluster Catalysts

Computational Prediction of Graphdiyne-Supported Three-Atom Single-Cluster Catalysts

Extending NMR Quantum Computation Systems by Employing Compounds with Several Heavy Metals as Qubits

Nuclear magnetic resonance (NMR) is a spectroscopic method that can be applied to several areas. Currently, this technique is also being used as an experimental quantum simulator, where nuclear spins are employed as quantum bits or qubits. The present work is devoted to studying heavy metal complexes as possible candidates to act as qubit molecules. Nuclei such 113Cd, 199Hg, 125Te, and 77Se assembled with the most common employed nuclei in NMR-QIP implementations (1H, 13C, 19F, 29Si, and 31P) could potentially be used in heteronuclear systems for NMR-QIP implementations. Hence, aiming to contribute to the development of future scalable heteronuclear spin systems, we specially designed four complexes, based on the auspicious qubit systems proposed in our previous work, which will be explored by quantum chemical calculations of their NMR parameters and proposed as suitable qubit molecules. Chemical shifts and spin–spin coupling constants in four complexes were examined using the spin–orbit zeroth-order regular approximation (ZORA) at the density functional theory (DFT) level, as well as the relaxation parameters (T1 and T2). Examining the required spectral properties of NMR-QIP, all the designed complexes were found to be promising candidates for qubit molecules.

A version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices

Abstract We give a version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices. As applications, we obtain sufficient conditions for the reflexivity of ${\mathcal P}^r(^nE;F)$, the space of regular n-homogeneous polynomials from a Banach lattice E to a Banach lattice F, and sufficient conditions for the positive Grothendieck property of $\hat{\otimes}_{n,s,|\pi|}E$, the n-fold positive projective symmetric tensor product of a Banach lattice E. Moreover, we also prove that these sufficient conditions are also necessary under the bounded regular approximation property.

Projection in negative norms and the regularization of rough linear functionals

In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as $W_0^{1,q}(\Omega)$, where $1<q<\infty$ and $\Omega$ is a Lipschitz domain, we propose a projection method in negative Sobolev spaces $W^{-1,p}(\Omega)$, $p$ being the conjugate exponent satisfying $p^{-1} + q^{-1} = 1$. Our method is particularly useful when one is dealing with a rough (irregular) functional that is a member of $W^{-1,p}(\Omega)$, though not of $L^1(\Omega)$, but one strives for a regular approximation in $L^1(\Omega)$. We focus on projections onto discrete finite element spaces $G_n$, and consider both discontinuous as well as continuous piecewise-polynomial approximations. While the proposed method aims to compute the best approximation as measured in the negative (dual) norm, for practical reasons, we will employ a computable, discrete dual norm that supremizes over a discrete subspace $V_m$. We show that this idea leads to a fully discrete method given by a mixed problem on $V_m\times G_n$. We propose a discontinuous as well as a continuous lowest-order pair, prove that they are compatible, and therefore obtain quasi-optimally convergent methods. We present numerical experiments that compute finite element approximations to Dirac delta's and line sources. We also present adaptively generate meshes, obtained from an error representation that comes with the method. Finally, we show how the presented projection method can be used to efficiently compute numerical approximations to partial differential equations with rough data.

Hyperspectral Anomaly Detection Based on Improved RPCA with Non-Convex Regularization

The low-rank and sparse decomposition model has been favored by the majority of hyperspectral image anomaly detection personnel, especially the robust principal component analysis(RPCA) model, over recent years. However, in the RPCA model, ℓ0 operator minimization is an NP-hard problem, which is applicable in both low-rank and sparse items. A general approach is to relax the ℓ0 operator to ℓ1-norm in the traditional RPCA model, so as to approximately transform it to the convex optimization field. However, the solution obtained by convex optimization approximation often brings the problem of excessive punishment and inaccuracy. On this basis, we propose a non-convex regularized approximation model based on low-rank and sparse matrix decomposition (LRSNCR), which is closer to the original problem than RPCA. The WNNM and Capped ℓ2,1-norm are used to replace the low-rank item and sparse item of the matrix, respectively. Based on the proposed model, an effective optimization algorithm is then given. Finally, the experimental results on four real hyperspectral image datasets show that the proposed LRSNCR has better detection performance.