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The article examines the probability and integral solutions to the fixed Kolmogorov equation, L*μ = 0. It analyzes the circumstances that probability solutions are given unique integral solution up to multiplication by a constant. It notes that the uniqueness in the class of probability solutions and the uniqueness in the class of integrable solutions are varied constructs that are already in dimension d = 1. It considers the validity of the given theorems, asserting that theorems 3 and 5 need additional conditions on the manifold.

The article discusses the problem from the context of the classical problem in extremal hypergraph theory. It explains that the bounds proved for the problem differ for different relations between the parameters as it relies on two parameters. It outlines the formulation of the problem's results and shows its theorem. However, it points out that there is uncertainty on whether or not the theorem can be improved. The equations, theorem, as well as the proof for the bound of the problem are also presented.

The article investigates the optimality conditions in optimal control problems with state constraints through the use of Pontryagin's maximum principle (MP). It presents an MP in the form of Gamkrelidze without considering a priori assumption on the regularity of optimal trajectory. It notes the non-restrictiveness of the assumption on the compatibility of state constraints with the terminal ones. It reveals the theorem which explains that nontriviality condition in the MP can be strengthened in a regular optimal process. The necessary optimality conditions in control problem is also presented.

The article presents a study regarding the vector integrals of the Fubini and Prokhorov Theorems which reduces the lag between locally convex spaces (LCSs) and Banach spaces. It highlights the values in the vector-dual LCS which involves the measure as an abstract measure space and the domain of the function. It states that the Fubini theorem is invalid since the multiplication by the initial transition measure may not commute. It mentions that a criterion in the Prokhorov theorem had no infinite-dimensional analogue which justifies the analogue as a general class of LCSs.

The article discusses the non-existence of distance-regular graphs with intersection arrays {55, 36, 11; 1, 4, 45} and {56, 36, 9; 1, 3, 48}. It proves the theorem that these intersection arrays, which were listed as admissible for distance-regular graphs, do not exist. It provides an overview of some definitions and notations regarding the arrays' non-existence. It explores the significance of the result of a study, by J. H. Koolen and J. Park, which involves several auxiliary assertions in proving the theorems.

The article discusses the integration of functions rd-continuous on time scales. The notes the development of the theory of dynamic equations on time scales which aims to establish a uniform theory of ordinary differential equations with discontinuous argument. It explains that the definition of integral invokes doubts in the rightness of results based on an integration method. It reveals the availability of an approach to differential equations on time scales according to a natural extension of solutions on the corresponding general integral theory adaptation.