Sentence examples for kind and second from inspiring English sources

He considers the elliptic variational inequalities (EVI) as the first kind and second kind EVI and defines those in a functional context as follows.

It is well known that the Chebyshev polynomials of the first kind and second kind are closely related to Vieta-Fibonacci and Vieta-Lucas polynomials.

Recall that the n th Chebyshev polynomials of the first kind and second kind are denoted by T n ( x ) and U n ( x ), respectively.

In order to do that, we first present the following formulae for the ( p, q ) -Fibonacci and Lucas polynomials which can be also regarded as the generalization of the results of He and Zhang [12] on the Lucas sequences of the first kind and second kind.

One way is the q-analog of degenerate Bernoulli polynomials by using the p-adic q-integral on (mathbb{Z}_{p}) by Kim et al. Another way is considered and studied the partially degenerate Bernoulli numbers and polynomials of the first kind and second kind by Kim and Seo in [21].

It becomes obvious that Theorem 4.1, Remark 4.2, Theorems 4.3 and 4.4 give the results of He and Zhang [12] on the Lucas sequences of the first kind and second kind when p ( x ) and q ( x ) in Definitions 2.1 and 2.2 satisfy p ( x ) = 1, q ( x ) = − 1 and p ( x ) = 2, q ( x ) = − 1, respectively.

Employing the von Mises/Tikhonov distribution, an efficient approximation is proposed via the modified Bessel functions of the first kind and zeroth order.

Since the Laplace distribution recently justified to represent jitter in the CNA breakpoints is not sufficiently accurate to estimate small changes, we propose a more accurate approximation based on the modified Bessel function of the second kind and zeroth order.

And is the modified Bessel function of the first kind and zeroth order.

where J0 is the Bessel function of the first kind and zeroth order.

The first algorithm is proposed for the numerical solution of nonlinear Fredholm integral equations of the second kind, and the second for the numerical solution of nonlinear Volterra integral equations of the second kind.