In this paper we investigate narrow linear operators in the setting of Köthe--Bochner spaces of vector-valued measurable functions. We prove that for an atomless finite measure space $(A, \Sigma, \mu)$, a Köthe $F$-space $E$ on $(A, \Sigma, \mu)$ with the absolutely continuous norm, a Banach space $X$ and an $F$-space $Y$ every bounded linear operator $T \colon E(X) \to Y$ from a Köthe--Bochner space $E(X)$ to $Y$ is narrow, whenever the density of the range space $Y$ is strictly less than the density of $E(B)$ for every $B \in \Sigma$ with $\mu(B) > 0$. Our main result is based on methods and technique of the set-theoretic topology, the measure theory and the theory of $F$-spaces. In this connection, we mention the work of Popov, Semenov and Vatsek \cite{PSV}, where sufficient conditions on a rearrangement invariant space $E$ were established for the orthogonal projection onto the closed linear span of the Rademacher system to be a hereditarily narrow operator. Moreover, in~\cite{PSV} the authors posed a number of open problems turned out to be important important for the further development of the theory of narrow operators.
В работе исследуются общие операторы усреднения, определенные на пространстве ограниченных вещественных последовательностей. Доказаны достаточные условия того, что эти операторы отображают пространство почти сходящихся последовательностей в себя и в подпространство сходящихся последовательностей. В качестве следствия из этих результатов получены достаточные условия существования банаховых пределов, инвариантных относительно этих операторов усреднения. Исследованы свойства множеств таких банаховых пределов, в частности, вычислен их диаметр. Для нескольких конкретных операторов усреднения установлена функциональная характеристика. Во всех случаях она оказалась непрерывной, но отличной от функциональной характеристики банаховых пределов, инвариантных относительно оператора Чезаро. Таким образом, показано, что множества банаховых пределов, инвариантных относительно этих операторов усреднения, дизъюнктны и между собой, и с множеством банаховых пределов, инвариантных относительно оператора Чезаро. В противоположном направлении получены достаточные условия совпадения множеств банаховых пределов, инвариантных относительно разных операторов усреднения.
Previously, the author considered closed convex polyhedra in $E^3$ with symmetrical rhombic vertices and regular faces, the so-called $RR$-polyhedra, in two cases: 1) regular faces of such polyhedra are of the same type, 2) regular faces are of different types, but the polyhedra are composite. From the author's previously published works, it follows that in both cases the list of $RR$-polyhedra is complete. In~the first case, there were twenty-four of them, and in the second, fifty-four. A vertex $V$ of a polyhedron is called rhombic if the vertex star $\Star(V)$ consists of $n$ equal and identically located rhombi (not squares) having a common vertex $V$. If the vertex $V$ lies on the rotation axis of order $n$ of the star $\Star(V)$, then $V$ is called symmetric. A symmetric rhombic vertex $V$ is called obtuse-angled if the rhombi of the star $\Star(V)$ at the vertex $V$ meet at their obtuse angles. The star $\Star(V)$ is then called symmetric rhombic. The faces of an $RR$-polyhedron can be divided into two non-empty disjoint sets: the set of equal symmetric rhombic stars that do not have common edges, and the set of regular faces. An example of an $RR$-polyhedron is an elongated rhombododecahedron. If~an~$RR$-polyhedron has regular faces of one type, then such an $RR$-polyhedron is classified as the first type. Some regular-faced polyhedra with conditional edges, such as the extended rhombododecahedron and the doubly complemented cuboctahedron, are special cases of $RR$-polyhedrons. In this case, the rhombic faces of regular-faced polyhedra with conditional edges, unlike the rhombi of $RR$-polyhedra, are composed only of regular triangles. In this paper, constructive proofs are given for the existence of four non-composite $RR$-polyhedra. For two of them the author previously found the angles of the rhombi that form symmetrical faceted stars of the vertices. In the case of the 25-sided polyhedron, the rhombic vertex has an acute-angled faceted star, i.\,e. the rhombi in it meet at their acute angles. And in the case of a 31-sided polyhedron the faceted star is obtuse. The existence of two other non-composite $RR$-polyhedra is~proven on the basis of the regular dodecahedron and the cube.
In this paper, we investigate the translation-invariant Gibbs measures (TIGM) for the Hard-Core (HC) lattice model with a countable set $\mathbb{Z}$ of spin values on the Cayley tree of order $k \geq 2$. This statistical mechanics model is uniquely characterized by an infinite set of parameters governed by a~positive activity function $\lambda_{i}>0$, $i \in \mathbb{Z}$. We explicitly derive the functional consistency equation and establish the exact normalizability conditions required for the boundary laws of the system. The analysis specifically focuses on the convergence of the activity series partitions, denoted by $\sum_{j\in\mathbb{Z}_{0}}\lambda_{2j}=\Delta$ and ${\sum_{j\in\mathbb{Z}}\lambda_{2j+1}=\Psi}$. In the symmetric subcase where $\Delta=\Psi\equiv\gamma$, we determine the precise critical value $\gamma_{cr}(k)$, proving that a unique TIGM exists for $\gamma \le \gamma_{cr}(k)$, while there exist exactly three distinct TIGMs when $\gamma>\gamma_{cr}(k)$ on Cayley trees of orders $k=2,3,4,5$. Furthermore, a generalized lower bound for the total number of TIGMs is formulated for all orders $k \geq 2$. Finally, in the asymmetric subcase ($\Delta \neq \Psi$), we provide a comprehensive mathematical description and classification of all translation-invariant Gibbs measures on Cayley trees of orders $k=1,2,3$.
In this paper, we investigate a class of elliptic nonlocal boundary value problems posed in a Banach space and subject to the Samarskii--Ionkin boundary condition, with the principal part of the equa\-ti\-on generated by a positive linear operator. The analysis is carried out within an abstract operator-theoretic framework, which allows us to treat a wide range of concrete problems arising in applications in a unified manner. First, we prove the fundamental well-posedness of the corresponding differential problem by establishing the existence, uniqueness, and continuous dependence of solutions on the given data. Particular attention is paid to the role of the positivity of the operator in ensuring these properties. Furthermore, we obtain coercive stability estimates for the solutions, which provide quantitative bounds in appropriate normed spaces. These estimates are derived for four distinct classes of elliptic nonlocal boundary value problems with Samarskii--Ionkin conditions, thereby demonstrating the robustness and generality of the proposed approach
