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The primary aim of this scientific note is first to review the essential background on several special functions in which the Gaussian function in certain complex domains and its integral play fundamental roles, and subsequently to establish (or organize) a number of relevant results together with some of their potential implications.

In this work, we develop a method for rational approximation of the Fourier transform (FT) based on the real and imaginary parts of the complex error function w(z)=e−z2(1−erf(−iz))=K(x,y)+iL(x,y), z=x+iy, where K(x,y) and L(x,y) are known as the Voigt and imaginary Voigt functions, respectively. In contrast to our previous rational approximation of the FT, the expansion coefficients in this method are not dependent on the values of a sampled function. As the values of the Voigt functions remain the same, this approach can be used for rapid computation with help of look-up tables. Mathematica codes with some examples are presented.

In this work we develop a new algorithm for the efficient computation of the Voigt/complex error function. In particular, in this approach we propose a two-domain scheme where the number of the interpolation grid-points is dependent on the input parameter y. The error analysis we performed shows that the MATLAB implementation meets the requirements for radiative transfer applications involving the HITRAN molecular spectroscopic database. The run-time test shows that this MATLAB implementation provides rapid computation, especially at smaller ranges of the parameter x.

Dawson’s integral and related functions in mathematical physics that include the complex error function (Faddeeva’s integral), Fried–Conte (plasma dispersion) function, Jackson function, Fresnel function and Gordeyev’s integral are analytically evaluated in terms of the confluent hypergeometric function. And hence, the asymptotic expansions of these functions on the complex plane C are derived by using the asymptotic expansion of the confluent hypergeometric function.

A rapidly convergent series, based on Taylor expansion of the imaginary part of the complex error function, is presented for highly accurate approximation of the Voigt/complex error function with small imaginary argument y ≤ 0.1. Error analysis and run-time tests in double-precision arithmetic reveals that in the real and imaginary parts, the proposed algorithm provides an average accuracy exceeding 10−15 and 10−16, respectively, and the calculation speed is as fast as that reported in recent publications. An optimized MATLAB code providing rapid computation with high accuracy is presented.

This paper presents a novel approach to deal with the imbalanced data set problem in neural networks by incorporating prior probabilities into a cost-sensitive cross-entropy error function. Several classical benchmarks were tested for performance evaluation using different metrics, namely G-Mean, area under the ROC curve (AUC), adjusted G-Mean, Accuracy, True Positive Rate, True Negative Rate and F1-score. The obtained results were compared to well-known algorithms and showed the effectiveness and robustness of the proposed approach, which results in well-balanced classifiers given different imbalance scenarios.

It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. We study different formulations of the input function space and its effect on the generalization error for 16 different diverse applications. Neural networks are known as universal approximators of continuous functions, but they can also approximate any mathematical operator (mapping a function to another function), which is an important capability for complex systems such as robotics control. A new deep neural network called DeepONet can lean various mathematical operators with small generalization error.

Qubit state measurements are an essential part of any quantum computer, constituting the readout. Accurate measurements are also an integral component of one-way quantum computation and of error correction, which is needed for fault-tolerant quantum computation1. Here, we present a state measurement for neutral-atom qubits based on coherent spatial splitting of the atoms’ wavefunctions. It is reminiscent of the Stern–Gerlach experiment2, but carried out in light traps. For around 160 qubits in a three-dimensional array, we achieve a measurement fidelity of 0.9994, which is roughly 20 times lower error than in previous measurements of neutral-atom arrays3,4. It also greatly exceeds the measurement fidelity of other arrays with more than four qubits, including those with ion and superconducting qubits5,6. Our measurement fidelity is essentially independent of the number of qubits measured, and since the measurement causes no loss, we can reuse the atoms. We also demonstrate that we can replace atoms lost to background gas collisions during the experiment7.A technique based on the coherent splitting of the atoms’ wavefunctions according to their internal states in an optical lattice allows the measurement of neutral-atom qubits in a three-dimensional array with extremely high fidelity, up to 99.94%.

This paper focuses on the trajectory tracking control problem of unmanned underwater vehicles (UUVs) with unknown dead-zone inputs. The primary objective is to design an adaptive trajectory tracking error constraint controller using the fully actuated systems (FAs) approach to enable UUVs to asymptotically track target signals. Firstly, a novel error constraint fully actuated systems (ECFAs) approach is proposed by incorporating the tracking error dependent normalized function and barrier function along with time-varying scaling. Secondly, in order to deal with the model uncertainties of the UUVs, adaptive radial basis function neural networks (RBFNNs) is combined with the ECFAs approach. Then, a positive time-varying integral function is introduced to completely eliminate the effect of the residual effect caused by unknown dead-zone inputs, and it is proved that the trajectory tracking error converges to zero asymptotically based on the Lyapunov functions. Finally, the simulation results demonstrate the effectiveness of the designed adaptive controller.

Our main objective in this work is to investigate complete synchronization (CS) of n -dimensional chaotic complex systems with uncertain parameters. An adaptive control scheme is designed to study the synchronization of chaotic attractors of these systems. We applied this scheme, as an example, to study complete synchronization of chaotic attractors of two identical complex Lorenz systems. The adaptive control functions and the parameters estimation laws are calculated analytically based on the complex Lyapunov function. We show that the error dynamical systems are globally stable. Numerical simulations are computed to check the analytical expressions of adaptive controllers.