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For a hybrid stochastic system, most existing feedback controllers need to observe modes at continuous times, which is feasible when the system's mode is observable and does not incur any cost. However, in most cases, the mode is not readily apparent, and identifying it always incurs a certain expense. Therefore, in order to reduce control costs, when designing a feedback controller, both the state and the mode should be observed at discrete moments. This paper introduces an intermittent feedback controller for stabilizing an unstable hybrid stochastic system through discrete delayed observations of state and mode. By utilizing M-matrix theory, intermittent control approach, and the comparison principle, we propose sufficient conditions for the stabilization theory of hybrid stochastic systems. An illustrative example is taken to validate the proposed theory. [ABSTRACT FROM AUTHOR]

The problem of path detection in graphs has been proposed from the past up to present, and various solutions have been proposed for this purpose, but it is often not an easy task to implement these methods on a computer. In this paper, a technique for detecting paths in a graph will be introduced using matrix algebra, which makes it possible to implement this rule on a computer. This method can be helpful the optimization of tree-spanning trees in networks. At the end of this study, a numerical example is solved using the proposed method.

This article explores the Erdős-Szekeres conjecture, a mathematical problem concerning geometric patterns in sets of points. The conjecture was proposed in 1935 and has been the focus of extensive research. Various mathematicians, including Tóth, Valtr, and Suk, have made significant contributions to proving the conjecture. Suk's proof is described as a moment of brilliance in applying the right tools. The article concludes with a quote from Erdős emphasizing the importance of an open mind in mathematics. [Extracted from the article]