background the hepatitis b virus x (hbx) protein has long been recognized as an important transcriptional transactivator of several genes. human aldo-keto reductase family 1, member c1 (akr1c1), a member of the family of akr1cs, is significantly increased in hbx-expressed cells. objectives this study aimed to investigate the possible mechanism of hbx in regulating akr1c1 expression in hepg2.2.1...

in this paper, we solve the quadratic $alpha$-functional equations $2f(x) + 2f(y) = f(x + y) + alpha^{-2}f(alpha(x-y)); (0.1)$ where $alpha$ is a fixed non-archimedean number with $alpha^{-2}neq 3$. using the fixed point method and the direct method, we prove the hyers-ulam stability of the quadratic $alpha$-functional equation (0.1) in non-archimedean banach spaces.

Defining a scale of $k$-modes the quantum fluctuations during inflation through dynamical horizon crossing condition $k = aH$ we go from physical $t$ variable to $k$ and solve equations cosmological first-order perturbations self consistently, with chaotic $\alpha$-attractor type potentials. This enables us study behaviour $n_{s}$, $r$, $n_{t}$ $N$ in $k$-space. Comparison our results low-$k$ r...