Let $K=\mathbb{Q}\left(\alpha \right)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f\left(x\right)=x^{18}-m$, $m\ne \mp 1$, is a square free rational integer. We prove that if $m\equiv 2$ or $3(\mathrm{mod}4)$ and $m\neg \equiv \mp 1(\mathrm{mod}9)$, then the number field $K$ is monogenic. If $m\equiv 1(\mathrm{mod}4)$ or $m\equiv 1(\mathrm{mod}9)$, then the number field $K$ is not monogenic.

The aim of this paper is to prove the following Theorem Theorem Let $K$ be an octic subfield of the field $Q({\zeta }_p+{\zeta }_p^{-1})$ and let $p=n^4+16$ be prime. Then $p$ divides $h_K$ if and only if $p$ divides $B_j$ for some $j=\frac{p-1}{8}$, $3\frac{p-1}{8}$, $5\frac{p-1}{8}$, $7\frac{p-1}{8}$.

The information divergence of a probability measure $P$ from an exponential family $ℰ$ over a finite set is defined as infimum of the divergences of $P$ from $Q$ subject to $Q\in ℰ$. All directional derivatives of the divergence from $ℰ$ are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for $P$ to be a maximizer of the divergence from $ℰ$ are presented, including new ones when $P$ is not projectable...

Let $p\in (0,1)$ be a real number and let $n\ge 2$ be an even integer. We determine the largest value $c_n\left(p\right)$ such that the inequality $$\sum _{i=1}^n{\left|a_i\right|}^p\ge c_n\left(p\right)$$ holds for all real numbers $a_1,...,a_n$ which are pairwise distinct and satisfy ${min}_{i\ne j}|a_i-a_j|=1$. Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value $c_n\left(p\right)$ in the case $p>0$ and $n$ odd, and in the case $p\ge 1$ and $n$ even.

Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and $U_{n+1}=PUₙ-Q{U}_{n-1}$ for n ≥ 1, and V₀ = 2, V₁ = P and $V_{n+1}=PVₙ-Q{V}_{n-1}$ for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and $V_r\ne 1$. We show that there is no integer x such that $Vₙ=V_{r}Vₘx²$ when m ≥ 1 and r is an even integer. Also we completely solve the equation $Vₙ=VₘV_{r}x²$ for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...

Let $P$ be a discrete multidimensional probability distribution over a finite set of variables $N$ which is only partially specified by the requirement that it has prescribed given marginals $\{P_A;A\in 𝒮\}$, where $𝒮$ is a class of subsets of $N$ with $\bigcup 𝒮=N$. The paper deals with the problem of approximating $P$ on the basis of those given marginals. The divergence of an approximation $\widehat{P}$ from $P$ is measured by the relative entropy $H\left(P\right|\widehat{P})$. Two methods for approximating $P$ are compared. One of them uses formerly introduced...

Let q > 2 be a prime power and $f=-x+t{x}^q+x^{2q-1}$, where $t\in {*}_q$. We prove that f is a permutation polynomial of ${}_{q²}$ if and only if one of the following occurs: (i) q is even and $T{r}_{q/2}(1/t)=0$; (ii) q ≡ 1 (mod 8) and t² = -2.