Applications of Mathematics最新文献

Initial value problem for three dimensional (3D) elastodynamic system in two dimensional (2D) inhomogeneous quasicrystals is considered. An analytical method is studied for the solution of this problem. The system is written in terms of Fourier images of displacements with respect to lateral variables. The resulting problem is reduced to integral equations of the Volterra type. Finally, using Paley Wiener theorem it is shown that the solution of the initial value problem can be found by the inverse Fourier transform. A numerical example is considered for the comparison of the exact solution with the computed solution obtained by using the method.

To have accuracy in the extracted information is the goal of the reliability theory investigation. In information theory, varentropy has recently been proposed to describe and measure the degree of information dispersion around entropy. Theoretical investigation on varentropy of past life has been initiated, however details on its stochastic properties are yet to be discovered. In this paper, we propose a novel stochastic order and introduce new classes of life distributions based on past varentropy. Further, we illustrate some of its applications in reliability modeling and in the diversity measure of Boltzmann distribution.

This paper deals with the problem of estimating a covariance matrix from the data in two classes: (1) good data with the covariance matrix of interest and (2) contamination coming from a Gaussian distribution with a different covariance matrix. The ridge penalty is introduced to address the problem of high-dimensional challenges in estimating the covariance matrix from the two-class data model. A ridge estimator of the covariance matrix has a uniform expression and keeps positive-definite, whether the data size is larger or smaller than the data dimension. Furthermore, the ridge parameter is tuned through a cross-validation procedure. Lastly, the proposed ridge estimator is verified with better performance than the existing estimator from the data in two classes and the traditional ridge estimator only from the good data.

The existence of synchronization is an important issue in complex dynamical networks. In this paper, we study the synchronization of impulsive coupled oscillator networks with the aid of rotating periodic solutions of impulsive system. The type of synchronization is closely related to the rotating matrix, which gives an insight for finding various types of synchronization in a united way. We transform the synchronization of impulsive coupled oscillators into the existence of rotating periodic solutions in a relevant impulsive system. Some existence theorems about rotating periodic solutions for a non-homogeneous linear impulsive system and a nonlinear perturbation system are established by topology degree theory. Finally, we give two examples to show synchronization behaviors in impulsive coupled oscillator networks.

Post-training rounding, also known as quantization, of estimated parameters stands as a widely adopted technique for mitigating energy consumption and latency in machine learning models. This theoretical endeavor delves into the examination of the impact of rounding estimated parameters in key regression methods within the realms of statistics and machine learning. The proposed approach allows for the perturbation of parameters through an additive error with values within a specified interval. This method is elucidated through its application to linear regression and is subsequently extended to encompass radial basis function networks, multilayer perceptrons, regularization networks, and logistic regression, maintaining a consistent approach throughout.

In this work, we consider an inverse backward problem for a nonlinear parabolic equation of the Burgers’ type with a memory term from final data. To this aim, we first establish the well-posedness of the direct problem. On the basis of the optimal control framework, the existence and necessary condition of the minimizer for the cost functional are established. The global uniqueness and stability of the minimizer are deduced from the necessary condition. Numerical experiments demonstrate the effectiveness of this approach.

We analyse a finite-element discretisation of a differential equation describing an axisymmetrically loaded thin shell. The problem is singularly perturbed when the thickness of the shell becomes small. We prove robust convergence of the method in a balanced norm that captures the layers present in the solution. Numerical results confirm our findings.

The evolutions of small and large compressive pulses are studied in a two-phase flow of gas and dust particles with a variable azimuthal velocity. The method of relatively undistorted waves is used to study the mechanical pulses of different types in a rotational, axisymmetric dusty gas. The results obtained are compared with that of nonrotating medium. Asymptotic expansion procedure is used to discuss the nonlinear theory of geometrical acoustics. The influence of the solid particles and the rotational effect of the medium on the distortion are investigated. In a rotational flow it is observed that with the increase in the value of rotational parameter, the steepening of the pulses also increases. The presence of dust in the rotational flow delays the onset of shock formation thereby increasing the distance where the shock is formed first. The rotational and the dust parameters are observed to have the same effect on the shock strength.

In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation.