For Fr$acute{mathbf{text{e}}}$chet algebras $(A, (p_n))$ and $(B, (q_n))$, a linear map $T:Arightarrow B$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(Tab - Ta Tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{N}$, $a, b in A$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$...

The asymptotic behaviour of the sequence with general term $P_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $P_n-6/pi^2$ is found.

Abstract We establish the mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related recurrence results. A consequence our main result is that every set integers positive upper density contains patterns form $\{m,m+[p_n^a], m+[p_n^b]\}$ , where $a,b$ are nonintegers $p_n$ denotes n th prime, a property fails if or b natural number. Our...

for fr$acute{mathbf{text{e}}}$chet algebras $(a, (p_n))$ and $(b, (q_n))$, a linear map $t:arightarrow b$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(tab - ta tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{n}$, $a, b in a$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$,...

We study linear transformations $$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]$$ of the form $$T[x^n]=P_n(x)$$ where $$\{P_n(x)\}$$ is a real orthogonal polynomial system. With $$T=\sum \tfrac{Q_k(x)}{k!}D^k$$ , we seek to understand behavior transformation T by studying roots $$Q_k(x)$$ . prove four main things. First, show that only case are constant and an system when $$P_n(x)$$ shifted set ...

This paper concerns the study of asymptotic behavior solutions to a family fractional type problems on bounded domain, satisfying homogeneous Dirichlet boundary conditions. The differential operators includes $$p_n$$ -Laplacian when $$p_n\rightarrow \infty $$ as particular case, tough it could be extended function Hölder quotient order s, whose primitive is an Orlicz appropriated growth limit e...

Billey, Konvalinka, Petersen, Solfstra, and Tenner recently presented a method for counting parabolic double cosets in Coxeter groups, used it to compute $p_n$, the number of $S_n$, $n\leq13$. In this paper, we derive new formula $p_n$ an efficient polynomial time algorithm evaluating formula. We use these results $n\leq5000$ prove asymptotic that was conjectured by Billey et al.

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the polynomials $$ P_N(\theta) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(n\theta), and show that $2k$-th moments on unit circle \int_0^1 \big| \big|^{2k}\, d\theta tend to Gaussian in sense of mean-square convergence, uniformly for $k \ll (\log N / \log N)^{1/3}$, but contrast case i.i.d. coefficients, this behavior...

Let \(G=(V(G),E(G))\) be a simple graph. A set \(S \subseteq V(G)\) is strong edge geodetic if there exists an assignment of exactly one shortest path between each pair vertices from S, such that these paths cover all the edges E(G). The cardinality smallest number \(\mathrm{sg_e}(G)\) G. In this paper, problem studied on Cartesian product two paths. exact value computed for \(P_n \,\square \,P...