Sentence examples for by periodicity from inspiring English sources

In addition, in each case the Fourier series of the higher-order Bernoulli functions (B_{m}^{(r)}( langle x rangle)) are expressed in terms of Bernoulli functions which are obtained by extending by periodicity of period 1 the ordinary Bernoulli polynomials (B_{m}(x)) on ([0, 1)).

In this paper, we considered the Fourier series expansion of the higher-order Bernoulli functions (B_{m}^{(r)}( langle x rangle )) which are obtained by extending by periodicity of period 1 the higher-order Bernoulli polynomials (B_{m}^{(r)}(x)) on ([0, 1)). The Fourier series are explicitly determined.

In this paper, we studied the Fourier series expansion of the higher-order Bernoulli functions (B_{m}^{(r)}( langle x rangle )) which are obtained by extending by periodicity of period 1 the higher-order Bernoulli polynomials (B_{m}^{(r)}(x)) on ([0, 1)).

The second one (ii) is the regime characterized by periodicity regions of period increment higher than 1 (we shall see examples with 2 and 3), and by bistability.

One regime (i) is the same existing in the continuous case and it is characterized by periodicity regions following the period increment by 1 rule.

By combining multiple grid modules with different spatial periods one can overcome this fundamental limitation, counteracting the ambiguity caused by periodicity and still preserving the highest resolution at the smallest scale.

However, it is possible this neglects other biologically important details of the nuclear NF-кB response, e.g. classification by time periodicity or by steady-state level.

The time-periodic solution is computed by enforcing periodicity of the element set.

By the periodicity of (3.15), it follows that the periodic solution (v_{1alpha}^(t)) is globally uniformly asymptotically stable.

We observed that cell motion was affected by the presence of the grating and by its periodicity, while cells on flat substrates showed random spatial migration.

In general, at THz frequencies, the spatial dispersion introduced by the periodicity of graphene can be neglected as suggested by [16].