The following is a discussion regarding a specific set of operators acting on the space of functions analytic on the unit disk. A diagonal operator is said to admit spectral synthesis if all of its invariant subspaces can be expressed as a closed linear span of a subset of its eigenvectors. This article employs various techniques for verifying the convergence of infinite products and infinite sums as a means of demonstrating that a certain class of operators fail to admit spectral synthesis.

Let $T_{TRP}$ be the tree corresponding to the weighted trapezoid graph $G=(V,E)$. The eccentricity $e(v)$ of the vertex $v$ is defined as the sum of the weights of the vertices from $v$ to the vertex farthest from $v \in T_{TRP}$. A vertex with minimum eccentricity in the tree $T_{TRP}$ is called the 1-center of that tree. In an inverse 1-center location problem, the parameter of the tree $T_{TRP}$ corresponding to the weighted trapezoid graph $G=(V,E)$, like vertex weights, have to be modified at minimum total cost such that a pre-specified vertex $s \in V$ becomes the 1-center of the trapezoid graph $G$. In this paper, we present an optimal algorithm to find an inverse 1-center location on the weighted tree $T_{TRP}$ corresponding to the weighted trapezoid graph $G=(V,E)$, where the vertex weights can be changed within certain bounds. The time complexity of our proposed algorithm is $O(n)$, where $n$ is the number of vertices of the trapezoid graph $G$.

The decomposition of a UP-filter is first discussed, and then the notion of a shift UP-filter is introduced, and several properties are investigated. Relations between a UP-filter, a comparative UP-filter, and a shift UP-filter are considered. Conditions for a UP-filter to be a shift UP-filter, and for a comparative UP-filter to be a shift UP-filter are provided. Characterizations of a shift UP-filter are considered, and an extension property for a shift UP-filter is established.

When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although results for continuous functions such as the Fundamental Theorem of Calculus follow immediately, a much more satisfying approach would be to provide direct proofs not relying on the Riemann integral. This note provides those proofs.

The notion of uni-hesitant fuzzy commutative ideals in BCK-algebras is introduced, and related properties are investigated. Relations between uni-hesitant fuzzy commutative ideals and uni-hesitant fuzzy ideals are discussed. Characterizations of uni-hesitant fuzzy commutative ideals are considered. Conditions for a uni-hesitant fuzzy ideal to be a uni-hesitant fuzzy commutative ideal are provided. Extension property for a uni-hesitant fuzzy commutative ideal is established.

We present different methods to generalize sums of powers of positive integers in terms of recurrence relations using the Taylor series, and in closed form using a finite difference method and an integral method. The result gained through the integral method is similar to Bernoulli's sum formula, but it is expressed in terms of a certain recursive sequence $H_i$.

We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in $\mathbb{Z}[A^2, A^{-2}, h]$. The decomposition of the resulting polynomial into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]h$ yields the decomposition of the Kauffman-Jones polynomial of virtual singular links into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]A^2$, where both components are invariants for virtual singular links.