Abstract Let $P_1,P_2,\dots , P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or Chebyshev polynomials, and $S$ the semigroup under composition generated by P_k$. We show all elements share a measure maximal entropy if only intersection principal left ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.

For the Fr'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. Th...

The problem of finding the minimum number colors to color a graph properly without containing any bicolored copy fixed family subgraphs has been widely studied. Most well-known examples are star coloring and acyclic graphs (Grunbaum, 1973) where copies \(P_4\) cycles not allowed, respectively. We introduce variation these problems study proper path length provide general bounds for all graphs. ...

Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of stable manifold zero solution. In particular, for $n$ dimensional case, solution globally asymptotically if and only this $n.$ Fixed $n,$ let $X$ be random variable that assigns to each system index, $p_k$ with $k=0,1,\ldots,n,$ denote probabilities $P(X=k)$. paper we obtain eith...

Given a graph $G = (V,E)$ and two its distinct vertices $u$ $v$. The $(u,v)$-$P_k$-{\em addition graph} of $G$ is the $G_{u,v,k-2}$ obtained from disjoint union path $P_k: x_0,x_1,..,x_{k-1}$, $k \geq 2$, by identifying $x_0$, $v$ $x_{k-1}$. We prove that (a) $ \gamma(G)-1 \leq \gamma(G_{u,v,k})$ for all 1$, (b) $\gamma(G_{u,v,k}) > \gamma(G)$ when 5$. also provide necessary sufficient conditio...

We establish $L^{p_1}\times\cdots\times L^{p_k}\to L^r$ and $\ell^{p_1}\times\cdots\times \ell^{p_k}\to \ell^r$ type bounds for multilinear maximal operators associated to averages over isometric copies of a given non-degenerate $k$-simplex in both the continuous discrete settings. These provide natural extensions $L^p\to L^p$ $\ell^p\to \ell^p$ Stein's spherical operator operator, with each th...

The class of graphs that do not contain an induced path on k vertices, Pk-free graphs, plays a prominent role in algorithmic graph theory. This motivates the search for special structural properties of Pkfree graphs, including alternative characterizations. Let G be a connected Pk-free graph, k ≥ 4. We show that G admits a connected dominating set that induces either a Pk−2-free graph or a grap...

We introduce a natural class of multicomponent local Poisson structures $$\mathcal P_k + \mathcal P_1$$ , where is bracket order one and P_k$$ homogeneous odd $$k$$ under the assumption that has Darboux coordinates (Darboux–Poisson bracket) nondegenerate. For such brackets, we obtain general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples), provide description...