Exponential Operators Research Articles

Convergence of Special Sequences of Semi-Exponential Operators

Several papers, mainly written by J. de la Call and co-authors, contain modifications of classical sequences of positive linear operators to obtain new sequences converging to limits which are not necessarily the identity operator. Such results were obtained using probabilistic methods. Recently, results of this type have been obtained with analytic methods. Semi-exponential operators have also been introduced, extending the theory of exponential operators. We combine these two approaches, applying the semi-exponential operators in a new context and enlarging the list of operators representable as limits of other operators.

Exact time-evolution of a generalized two-dimensional quantum parametric oscillator in the presence of time-variable magnetic and electric fields

The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei–Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin–Man’ko–Trifonov and the Lewis–Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy–Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.

Telegraph model with fractional differential operators: Nonsingular kernels

In this paper, a system of partial differential equations that was suggested to depict the behavior of the telegraph is considered where the time differential operators used to evaluate the rate of change is converted to nonlocal operators with the aim to include into mathematical formulation nonlocal behaviors. Three differential operators are employed in this conversion, including the Caputo-power fractional derivative, which helps to include power-law behaviors into a model, the Caputo-Fabrizio exponential decay kernel differential operator, which helps to include fading memory behaviors into the model, and finally, the Atangana-Baleanu fractional derivative, that helps to introduce crossover from stretched exponential to a power law. We presented a numerical solution using the technique based on Newton’s polynomial interpolation. Using Laplace transform, we attempted to derive exact solutions to these equations.

Tensor-Train Split-Operator KSL (TT-SOKSL) Method for Quantum Dynamics Simulations.

Numerically exact simulations of quantum reaction dynamics, including nonadiabatic effects in excited electronic states, are essential to gain fundamental insights into ultrafast chemical reactivity and rigorous interpretations of molecular spectroscopy. Here, we introduce the tensor-train split-operator KSL (TT-SOKSL) method for quantum simulations in tensor-train (TT)/matrix product state (MPS) representations. TT-SOKSL propagates the quantum state as a tensor train using the Trotter expansion of the time-evolution operator, as in the tensor-train split-operator Fourier transform (TT-SOFT) method. However, the exponential operators of the Trotter expansion are applied using a rank-adaptive TT-KSL scheme instead of using the scaling and squaring approach as in TT-SOFT. We demonstrate the accuracy and efficiency of TT-SOKSL as applied to simulations of the photoisomerization of the retinal chromophore in rhodopsin, including nonadiabatic dynamics at a conical intersection of potential energy surfaces. The quantum evolution is described in full dimensionality by a time-dependent wavepacket evolving according to a two-state 25-dimensional model Hamiltonian. We find that TT-SOKSL converges faster than TT-SOFT with respect to the maximally allowed memory requirement of the tensor-train representation and better preserves the norm of the time-evolving state. When compared to the corresponding simulations based on the TT-KSL method, TT-SOKSL has the advantage of avoiding the need to construct the matrix product state Laplacian by exploiting the linear scaling of multidimensional tensor-train Fourier transforms.

Global strong solutions and large‐time behavior of 2D tropical climate model with zero thermal diffusion

In this article, we study the global well‐posedness and large‐time behaviors of solutions to the two‐dimensional tropical climate system with zero thermal diffusion for small initial data in the whole space. The main approaches include high‐ and low‐frequency decomposition method and exploiting the structure of system (1) to obtain the estimates of thermal dissipation. We utilize the time decay properties of the kernels to a linear differential equation to obtain the decay rates of solutions of the low‐frequency part and the decay property of exponential operator for the high‐frequency part. The key ingredient here is the explicit large‐time decay rate of solutions.

Linear and Nonlinear Optical Properties from TDOMP2 Theory

We present a derivation of real-time (RT) time-dependent orbital-optimized Møller–Plesset (TDOMP2) theory and its biorthogonal companion, time-dependent non-orthogonal OMP2 theory, starting from the time-dependent bivariational principle and a parametrization based on the exponential orbital-rotation operator formulation commonly used in the time-independent molecular electronic structure theory. We apply the TDOMP2 method to extract absorption spectra and frequency-dependent polarizabilities and first hyperpolarizabilities from RT simulations, comparing the results with those obtained from conventional time-dependent coupled-cluster singles and doubles (TDCCSD) simulations and from its second-order approximation, TDCC2. We also compare our results with those from CCSD and CC2 linear and quadratic response theories. Our results indicate that while TDOMP2 absorption spectra are of the same quality as TDCC2 spectra, including core excitations where optimized orbitals might be particularly important, frequency-dependent polarizabilities and hyperpolarizabilities from TDOMP2 simulations are significantly closer to TDCCSD results than those from TDCC2 simulations.

Modified artificial bee colony algorithm with differential evolution to enhance precision and convergence performance

Artificial bee colony (ABC) and differential evolution (DE) are the most powerful and operative meta-heuristic algorithms inspired by the nature. Although both algorithms are successful, their successes vary from phase to phase, i.e. while ABC is better in the exploration ability, DE is well in the exploitation capability. Because the diversity of mutation and exponential crossover operators is prominently better than that of onlooker bee; in this study, the exploitation ability of ABC is enhanced by replacing the onlooker bee operator with those of mutation and the crossover phases of DE in order to increase the accuracy and speed up the convergence. We hereby introduce a novel modified algorithm denoted “modified ABC by DE” (mABC). The precision performance of mABC is verified through 20 classical benchmark functions and CEC 2014 test suit by a comprehensive comparison with recent ABC variants and hybrids for 30 and 50 dimensions. The results are interpreted using various statistical evaluations such as Wilcoxon, Friedman, and Nemenyi tests. Moreover, mABC is comparatively examined over convergence plots. In concise, the mean ranks of mABC are 1.4 and 2.3 for classical benchmark functions and CEC 2014, respectively. mABC outperforms the other variants averagely for 14 of 20 classical benchmark functions and 24 of 30 CEC 2014 functions. The results manifest that the proposed mABC is a robust and reliable algorithm as well as better than the existing ABC variants and hybrids with regard to high optimization performance like precision and convergence.

A Novel Energy Flow Analysis and Its Connection With Modal Analysis for Investigating Electromechanical Oscillations in Multi-Machine Power Systems

In this paper, a novel energy flow analysis (EFA) is proposed based on the signal reconstruction and decomposition to investigate the electromechanical oscillations. In contrast to the conventional EFA, the connection between the proposed EFA and modal analysis (MA) can be quantitatively revealed for arbitrary models of synchronous generators in multi-machine power systems. Firstly, the time-domain implementation (TDI) of the proposed EFA is designed. Specifically, the measurements at the terminal of a local generator are reconstructed through an exponential operator and then decomposed with respect to an angular frequency. Then, the mode-screened damping torque coefficient is defined to extract the damping feature with respect to an electromechanical oscillation mode. After that, the frequency-domain implementation (FDI) is derived. Specifically, the Parseval's Theorem is applied to transform the proposed EFA from the time domain to frequency domain. On this basis, the consistency between the proposed EFA and MA is strictly proved, which is applicable for arbitrary models of synchronous generators in multi-machine power systems. Additionally, the application procedure of the proposed EFA in investigating electromechanical oscillations is given. Finally, the proposed EFA are substantially demonstrated in multiple case studies.

Covariance kernels investigation from diffusive wave equations for data assimilation in hydrology

In data assimilation (DA), the estimation of the background error covariance operator is a classical and still open topic. However, this operator is often modeled using empirical information. In order to exploit at best the potential of the knowledge of the physics, the present study proposes a method to derive covariance operators from the underlying equations. In addition, Green’s kernels can be used to model covariance operators and are naturally linked to them. Therefore, Green’s kernels of equations representing physics can provide physically-derived estimates of the background error covariance operator, and also physically-consistent parameters. In this context, the present covariance operators are used in a variational DA (VDA) process of altimetric data to infer bathymetry in the Saint-Venant equations. In order to investigate these new physically-derived covariance operators, the associated VDA results are compared to the VDA results using classical operators with physically-consistent and arbitrary parameters. The physically-derived operators and physically-consistent exponential operator provide better accuracy and faster convergence than empirical operators, especially during the first iterations of the VDA optimization process.

Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants

In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and diffusion terms in the exponential operator. We prove that the midpoint Lawson schemes preserve quadratic invariants and discuss this property as well for the trapezoidal Lawson scheme. Numerical experiments demonstrate that the integration error for highly oscillatory problems is smaller than that of some standard methods.

Optimized Filters for Stabilizing High-Order Large Eddy Simulation

High-order Flux Reconstruction (FR) schemes can be used to simulate unsteady turbulent flows using Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) in the vicinity of complex geometries. However, the application of FR can be limited by non-linear instabilities, which can arise from oscillatory behaviour of the underlying polynomial representation of the solution. In this paper, we explore filtering and its parametrization for stabilizing under-resolved simulations of the Navier–Stokes equations. A new exponential filtering operator is proposed, which is normalized by the time-step size and designed to filter high-frequency modes. Over 14,000 numerical tests are then performed to obtain an optimal set of filtering parameters, with the objective being to stabilize while maintaining high-order accuracy. We then verify that these optimal filters converge to super-accuracy for non-linear problems, and compare filtered and unfiltered simulations of the Taylor–Green vortex using both straight and curved meshes, and turbulent channel flow. These demonstrate that the filtered solutions are generally more accurate than unfiltered ones, while still stabilizing previously unstable simulations. Finally, we demonstrate the utility of these filters for more complex flows, specifically a stalled NACA 0020 airfoil.

Approximation by Durrmeyer Type Exponential Sampling Operators

In the present article, we derive some approximation results concerning the order of convergence for a family of Durrmeyer type exponential sampling operators. Further, we improve the rate of approximation by constructing linear combinations of these operators. At the end, we provide a few examples of the kernel functions to which the presented theory can be applied along with the graphical representation.

Novel Development to the Theory of Dombi Exponential Aggregation Operators in Neutrosophic Cubic Hesitant Fuzzy Sets: Applications to Solid Waste Disposal Site Selection

The neutrosophic cubic hesitant fuzzy set can efficiently handle the complex information in a decision‐making problem because it combines the advantages of the neutrosophic cubic set and the hesitant fuzzy set. The algebraic operations based on Dombi norms and co‐norms are more flexible than the usual algebraic operations as they involve an operational parameter. First, this paper establishes Dombi algebraic operational laws, score functions, and similarity measures in neutrosophic cubic hesitant fuzzy sets. Then, we proposed Dombi exponential operational laws in which the exponents are neutrosophic cubic hesitant fuzzy values and bases are positive real numbers. To use neutrosophic cubic hesitant fuzzy sets in decision‐making, we are developing Dombi exponential aggregation operators in the current study. In the end, we present applications of exponential aggregation operators in multiattribute decision‐making problems.

Mechanical strength evaluation for morsel rubber material via mathematical models

One of the most important concretes for today's environment is green concrete. The global increase in the automobile industry has resulted in a massive quantity of tires waste. Waste tire discarding continues to be a thoughtful scourge to ecosystem shield and human health. The primary goals of this research are to use morsel rubber (MR) as a preferential alternative for sand in M25 concrete mixes at several fractions to create environmentally pleasant concrete. Morsel rubber was mixed in distinct proportions from 0 to 10% with an increment of 2.5%. The mechanical properties of morsel rubber can be prognosticated using mathematical algebraic operators such as the Polynomial operator, Exponential operator, Fourier Gaussian operator, sum of the sine operator, and Power operator. In addition, customs equations are considered in this research to develop an understanding of the behavior of concrete when the waste morsel rubber is added to the concrete at a different percentage.

Analytic gradients in variational quantum algorithms: Algebraic extensions of the parameter-shift rule to general unitary transformations

Optimization of unitary transformations in variational quantum algorithms benefits highly from efficient evaluation of cost function gradients with respect to amplitudes of unitary generators. We propose several extensions of the parameter-shift rule to formulating these gradients as linear combinations of expectation values for generators with general eigenspectra (i.e., with more than two eigenvalues). Our approaches are exact and do not use any auxiliary qubits; instead they rely on a generator eigenspectrum analysis. Two main directions in the parameter-shift-rule extensions are (1) polynomial expansion of the exponential unitary operator based on a limited number of different eigenvalues in the generator and (2) decomposition of the generator as a linear combination of low-eigenvalue operators (e.g., operators with only two or three eigenvalues). These techniques have a range of scalings for the number of needed expectation values with the number of generator eigenvalues from quadratic (for polynomial expansion) to linear and even ${log}_{2}$ (for generator decompositions). This allowed us to propose efficient differentiation schemes for commonly used two-qubit transformations (e.g., match gates, transmon gates, and fSim gates) and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{2}$-conserving fermionic operators for the variational quantum eigensolver.

Some Remarks about Exponential and Semi-Exponential Operators

In this paper, we study the approximation properties of some exponential and semi-exponential operators. We focus on modifications of these operators in King’s sense, examining the rate of convergence for basic and modified operators. The presented line of reasoning emphasizes some symmetry in the modifications of exponential and semi-exponential operators.

Modification of Exponential Type Operators Preserving Exponential Functions Connected with $$x^3$$

Modification of Exponential Type Operators Preserving Exponential Functions Connected with $$x^3$$

Advanced arithmetic optimization algorithm for solving mechanical engineering design problems.

The distributive power of the arithmetic operators: multiplication, division, addition, and subtraction, gives the arithmetic optimization algorithm (AOA) its unique ability to find the global optimum for optimization problems used to test its performance. Several other mathematical operators exist with the same or better distributive properties, which can be exploited to enhance the performance of the newly proposed AOA. In this paper, we propose an improved version of the AOA called nAOA algorithm, which uses the high-density values that the natural logarithm and exponential operators can generate, to enhance the exploratory ability of the AOA. The addition and subtraction operators carry out the exploitation. The candidate solutions are initialized using the beta distribution, and the random variables and adaptations used in the algorithm have beta distribution. We test the performance of the proposed nAOA with 30 benchmark functions (20 classical and 10 composite test functions) and three engineering design benchmarks. The performance of nAOA is compared with the original AOA and nine other state-of-the-art algorithms. The nAOA shows efficient performance for the benchmark functions and was second only to GWO for the welded beam design (WBD), compression spring design (CSD), and pressure vessel design (PVD).

A Road Map to Various Pathways for Calculating the Memory Kernel of the Generalized Quantum Master Equation.

The generalized quantum master equation (GQME) provides a powerful framework for simulating electronic energy, charge, and coherence transfer dynamics in molecular systems. Within this framework, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density matrix is fully captured by a memory kernel superoperator. However, the actual memory kernel depends on the choice of projection operator and is therefore not unique. Furthermore, calculating the memory kernel can be done in multiple ways that use different forms of projection-free inputs. Although the electronic dynamics is invariant to those choices when quantum-mechanically exact projection-free inputs are used, this is not the case when they are obtained via more feasible semiclassical or mixed quantum-classical approximate methods. Furthermore, the accuracy and numerical stability of the resulting electronic dynamics has been observed to be sensitive to the above-mentioned choices when approximate methods are used to calculate the projection-free inputs. In this article, we provide a systematic road map to 30 possible pathways for calculating the memory kernel and highlight how they are related as well as the ways in which they differ. We also compare the performance of different pathways in the context of the spin-boson benchmark model, with the projection-free inputs obtained via a mapping Hamiltonian linearized semiclassical method. In this case, we find that expressing the memory kernel with an exponential operator where the projection operator precedes the Liouvillian yields the most accurate and most numerically stable results.

About the Use of Generalized Forms of Derivatives in the Study of Electromagnetic Problems

The use of non-local operators, defining Riemann–Liouville or Caputo derivatives, is a very useful tool to study problems involving non-conventional diffusion problems. The case of electric circuits, ruled by non-integer derivatives or capacitors with fractional dielectric permittivity, is a fairly natural frame of relevant applications. We use techniques, involving generalized exponential operators, to obtain suitable solutions for this type of problems and eventually discuss specific problems in applications.