An explicit method for numerical simulation of wave equations

In this paper, a method to develop a hierarchy of explicit recursion formulas for numerical simulation in an irregular grid for scalar wave equations is presented and its accuracy is illustrated via 2-D and 1-D models. Approaches to develop the stable formulas which are of 2M-order accuracy in both time and space with Mbeing a positive integer for regular grids are discussed and illustrated by constructing the second order （M= 1） and the fourth order （M = 2） recursion formulas.

An explicit method for numerical simulation of wave equations: 3D wave motion

In this paper, an explicit method is generalized from 1D and 2D models to a 3D model for numerical simulation of wave motion, and the corresponding recursion formulas are developed for 3D irregular grids. For uniform cubic grids, the approach used to establish stable formulas with 2M-order accuracy is discussed in detail, with M being a positive integer, and is illustrated by establishing second order （M=1） recursion formulas. The theoretical results presented in this paper are demonstrated through numerical testing.

Trend Autoregressive Model Exact Run Length Evaluation on a Two-Sided Extended EWMA Chart

The Extended Exponentially Weighted Moving Average(extended EWMA)control chart is one of the control charts and can be used to quickly detect a small shift.The performance of control charts can be evaluated with the average run length(ARL).Due to the deriving explicit formulas for the ARL on a two-sided extended EWMA control chart for trend autoregressive or trend AR(p)model has not been reported previously.The aim of this study is to derive the explicit formulas for the ARL on a two-sided extended EWMA con-trol chart for the trend AR(p)model as well as the trend AR(1)and trend AR(2)models with exponential white noise.The analytical solution accuracy was obtained with the extended EWMA control chart and was compared to the numer-ical integral equation(NIE)method.The results show that the ARL obtained by the explicit formula and the NIE method is hardly different,but the explicit for-mula can help decrease the computational(CPU)time.Furthermore,this is also expanded to comparative performance with the Exponentially Weighted Moving Average(EWMA)control chart.The performance of the extended EWMA control chart is better than the EWMA control chart for all situations,both the trend AR(1)and trend AR(2)models.Finally,the analytical solution of ARL is applied to real-world data in the healthfield,such as COVID-19 data in the United Kingdom and Sweden,to demonstrate the efficacy of the proposed method.

Exact Run Length Evaluation on a Two-Sided Modified ExponentiallyWeighted Moving Average Chart for Monitoring

A modified exponentially weighted moving average (EWMA) scheme is one of the quality control charts suchthat this control chart can quickly detect a small shift. The average run length (ARL) is frequently used for theperformance evaluation on control charts. This paper proposes the explicit formula for evaluating the average runlength on a two-sided modified exponentially weighted moving average chart under the observations of a first-orderautoregressive process, referred to as AR(1) process, with an exponential white noise. The performance comparisonof the explicit formula and the numerical integral technique is carried out using the absolute relative change forchecking the correct formula and the CPU time for testing speed of calculation. The results show that the ARL ofthe explicit formula and the numerical integral equation method are hardly different, but this explicit formula ismuch faster for calculating the ARL and offered accurate values. Furthermore, the cumulative sum, the classicalEWMA and the modified EWMA control charts are compared and the results show that the latter is better for smalland intermediate shift sizes. In addition, the explicit formula is successfully applied to real-world data in the healthfield as COVID-19 data in Thailand and Singapore.

ON RITZ METHOD OF ANINTEGRO-DIFFERENTIAL EQUAITON

This paper deals with the Ritz method of an integro-differential equation related with Riemann zeta-function.

On Degenerate Array Type Polynomials

In the paper,with the help of the Fa′a di Bruno formula and an identity of the Bell polynomials of the second kind,the authors define degenerateλ-array type polynomials,establish two explicit formulas,and present several recurrence relations of degenerateλ-array type polynomials and numbers.

A Formula of Solution for a Class of Homogeneous Recurrence of Variable Coefficients with Two Indices

In this paper, we obtain an explicit formula of general solution for a class of the homogeneous recurrence of variable coefficients with two indices.

Laplacian Energies of Regular Graph Transformations

Let LE(G) denote the Laplacian energy of a graph G. In this paper the xyz-transformations G^(xyz) of an r-regular graph G for x,y,z∈{0,1, +,-} are considered. The explicit formulas of LE(G^(xyz)) are presented in terms of r,the number of vertices of G for any positive integer r and x,y,z∈{ 0,1},and also for r = 2 and all x,y,z∈{0,1,+,-}. Some Laplacian equienergetic pairs of G^(xyz) for r = 2 and x,y,z∈{0,1, +,-} are obtained. This also provides several ways to construct infinitely many pairs of Laplacian equienergetic graphs.

Explicit Formula for the Hyperbolic Scattering Determinant

We prove the explicit formula for the hyperbolic scattering determinant in the case of a general subgroup F of PSL （2, R）. The class of test functions involved （not necessarily odd nor continuous） is much broader than that previously known. As an application of the technique, a new representation of the Millson-Shintani zeta function is obtained.

A Hilbert-Schmidt integral operator and the Weil distribution

In this paper,a positive operator is given.It is shown that the product of this positive operator and the convolution operator is a trace class Hilbert-Schmidt integral operator and has nonnegative eigenvalues.A formula is given for the trace of this product operator.It seems that this product operator is the closest trace class integral operator which has nonnegative eigenvalues and is related to the Weil distribution in the context of Connes’program for the Riemann hypothesis.A relation is given between the trace of the product operator and the Weil distribution.

A BOUNDARY INTEGRAL METHOD FOR COMPUTING ELASTIC MOMENT TENSORS FOR ELLIPSES AND ELLIPSOIDS

The concept of elastic moment tensor occurs in several interesting contexts, in particular in imaging small elastic inclusions and in asymptotic models of dilute elastic composites. In this paper, we compute the elastic moment tensors for ellipses and ellipsoids by using a systematic method based on layer potentials. Our computations reveal an underlying elegant relation between the elastic moment tensors and the single layer potential.

Mixed eigenvalues of discrete p-Laplacian

This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation： the difference form, the single summation form, and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity （which was known years ago）, as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at presented at the end of Section 2 to infinity is also studied. Two examples are illustrate the value of the investigation.

Formulations of Some Bit Switching Functions in DES

In the data encryption standard （DES） algorithm, there exist several bit-switching functions, including permutations, expansion, and permuted choices. They are generally presented in the form of matrixes and realized by using table look-up technique in the implementation of the cryptosystem. This paper presents explicit formulas for the initial permutation IP, its inverse IP-1 , the expansion function E, and the permuted choice PC_1. It also gives the program realizations of these functions in C＋＋ applying these formulas. With the advantage of the omission of the storage space for these matrixes and the tedious inputs of tables in the implementations of DES, our experimental results shows that the explicit formulas are useful in some situations, such as wireless sensor networks where the memory capacity is limited, especially when the size of file for encrypting is not too large, preferably smaller than 256KB.

Estimates of generalized Chebyshev function on GLm

In this paper, we study the generalized Chebyshev function related to automorphic L-functions of GLm（AQ）, and estimate its asymptotic behavior with respect to the parameters of the original automorphic objects.