In this article, we define a modified Koszul complex, which we call a quantized Koszul complex, on a quantum space ring, and we also prove that it is an acyclic complex.

In this paper, we derive analytical closed results for the first (a, b)-KA index, the Sombor index, the modified Sombor index, the first reduced (a, b)-KA index, the reduced Sombor index, the reduced modified Sombor index, the second reduced (a, b)-KA index and the mean Sombor index mSO_α for the OTIS biswapped networks by considering basis graphs as path, wheel graph, complete bipartite graph and r-regular graphs. Network theory plays a significant role in electronic and electrical engineering, such as signal processing, networking, communication theory, and so on. A topological index (TI) is a real number associated with graph networks that correlates chemical networks with a variety of physical and chemical properties as well as chemical reactivity. The Optical Transpose Interconnection System (OTIS) network has recently received increased interest due to its potential uses in parallel and distributed systems.

Let 𝔽_q be a finite field with odd q elements. In this article, we prove that if E ⊆ 𝔽^d_q, d ≥ 2, and |E| ≥ q, then there exists a set Y ⊆ 𝔽^d_q with |Y| ~ q^d such that for all y ∈ Y, the number of distances between the point y and the set E is ~ q. As a corollary, we obtain that for each set E ⊆ 𝔽^d_q with |E| ≥ q, there exists a set Y ⊆ 𝔽^d_q with |Y| ~ q^d so that any set E ∪ {y} with y ∈ Y determines a positive proportion of all possible distances. The averaging argument and the pigeonhole principle play a crucial role in proving our results.

In this paper, we consider the existence of nonconstant warping functions on a warped product manifold M = B × _f² F, where B is a q(> 2)-dimensional base manifold with a nonconstant scalar curvature S_B(x) and F is a 3- dimensional fiber Einstein manifold and discuss that the resulting warped product manifold is an Einstein manifold, using the existence of the solution of some partial differential equation.

In this paper, we introduce three different concepts of closed sets on the intuitionistic fuzzy topological spaces, i.e., the generalized fuzzy (r, s)-closed, semi-generalized fuzzy (r, s)-closed, and generalized fuzzy (r, s)-semiclosed sets on intuitionistic fuzzy topological spaces in Šostak's sense. Also we investigate their properties and the relationships among these generalized fuzzy closed sets.

Uniform space is a generalization of metric space. The main purpose of this paper is to extend several results contained in [5, 6] which have for an expansive homeomorphism with the pseudo orbit tracing property(POTP in short) on a compact metric space (X, d) for an expansive homeomorphism with the POTP on a compact uniform space (X, 𝒰). we characterize stable and unstable sets, sink and source and saddle, recurrent points for an expansive homeomorphism which has the POTP on a compact uniform space (X, 𝒰).

In this note, we verify that the bounded shadowing property and quasi-hyperbolicity of bounded linear operators on Hilbert spaces are preserved under Aluthge transforms.