On λ-linear functionals arising from p-adic integrals on $\mathbb{Z}_{p}$

The aim of this paper is to determine the λ-linear functionals sending any given polynomial $p(x)$ p ( x ) with coefficients in $\mathbb{C}_{p}$ C p to the p-adic invariant integral of $P(x)$ P ( x ) on $\mathbb{Z}_{p}$ Z p and also to that of $P(x_{1}+\cdots +x_{r})$ P ( x 1 + ⋯ + x r ) on $\mathbb{Z}_{p}^{r}$ Z p r . We show that the former is given by the generating function of degenerate Bernoulli polynomials and the latter by that of degenerate Bernoulli polynomials of order r. For this purpose, we use the λ-umbral algebra which has been recently introduced by Kim and Kim (J. Math. Anal. Appl. 493(1):124521 2021).

Some applications of degenerate poly-Bernoulli numbers and polynomials

In this paper, we consider degenerate poly-Bernoulli numbers and polynomials associated with a polylogarithmic function and a p-adic invariant integral on {\mathbb{Z}_{p}} . By using umbral calculus, we derive some identities of those numbers and polynomials.

A Note on Type 2 w-Daehee Polynomials

In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results.

Ways of application of the provisions of mechanics of bodies with cracks to the calculation of asphalt concrete on strength and plasticity

The report discusses the principles of two major theories of fracture mechanics of bodies with cracks, which include the theory of accumulation of damage Kachanov–Rabotnov and theory of brittle fracture Griffith–Irwin, including the invariant integral Cherepanov–Rice, describing the criterion of growth the crack. To assess the application of these theories to the calculation of asphalt concrete, laboratory test data are given and based on their analysis the appropriate conclusions.