A class of identities satisfied by both Bernoulli and Euler polynomials is established. Recurrence relations for Bernoulli and Euler numbers are derived.

The Bernoulli polynomials Bk restricted to 0, 1 and extended by periodicity have nth sine and cosine Fourier coefficients of the formCk/n . In general, the Fourier coefficients of any polynomial restricted to 0, 1 are linear combinations of terms of the form 1/n . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a...

A sequence of random variables, each taking values 0 or 1, is called a Bernoulli sequence. We say that a string of length d occurs in a Bernoulli sequence if a success is followed by exactly (d − 1) failures before the next success. The counts of such d-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic d-cycle counts in random permuta...

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulli στ -shift rules and eight hyper Bernoulli στ -shift rules, the latter including such famous rules 30 and 110 . All exhibit a bizarre composite wave dynamics with arbitrar...

In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. Importantly the model can estimate not only the main effects and pairwise interactions among the nodes b...

This paper proposes a design for a precision positioning miniature walking robot using piezoelectric unimorph actuators. The theoretical working equations of a uniform piezoelectric unimorph beam are derived based on the linear piezoelectric relations, Hamilton’s principle, and Euler-Bernoulli beam equation. The modal superposition method is used to determine the response of the forced transver...

Arithmetic properties of integer sequences counting periodic points are studied, and applied to the case of linear recurrence sequences, Bernoulli numerators, and Bernoulli denominators.

in this paper, the modified euler-bernoulli beam model is presented to examine the influence of surface elasticity and residual surface tension on the critical force of axial buckling of nanotubes in the presence of rotary inertia. an explicit solution is derived for the buckling loads of microscaled euler beams considering surface effects. the size-dependent buckling behavior of the nanotube d...

We present a unified statistical framework for quantifying a collection of stroke lesion images that have been segmented in diffusion weighted images. Although Bernoulli models are often used in modeling a collection of binary images, the Bernoulli models actually break down for testing statistical significance of common overlap. To remedy the limitation of the Bernoulli models, we propose to a...