This paper deals with analytical and numerical methods for constructing a minimax (generalized) solution to the Dirichlet problem for the Hamilton–Jacobi equation. The case of a closed planar nonconvex boundary set is considered, where the boundary points have a smoothness defect in the coordinate functions with respect to third-order derivatives. These points belong to the pseudo-vertices of the boundary set. Pseudo-vertices generate branches of a singular set, which are one-dimensional manifolds where the smoothness of the minimax solution breaks down. To construct a branch of a singular set, it is necessary to find markers, i.e., numerical characteristics of the corresponding pseudo-vertex. The markers (left and right ones) establish a link between the characteristics of the Hamilton–Jacobi equation and the geometry of the boundary set. For the markers, a relation with the structure of the equation at a fixed point is obtained. An iterative procedure for calculating a solution based on Newton's method is proposed. The convergence of the procedure to the pseudo-vertex marker is proved. An example of constructing a minimax solution is given, demonstrating the effectiveness of the developed approaches for solving nonsmooth boundary value problems.
This paper introduces a generalization of asymptotic notation using filters, a topological structure. We present key properties of this generalization, including reflexivity, symmetry, and transitivity, supported by illustrative examples. Our research demonstrates that classical asymptotic notations imply their filter-generalized counterparts, but we provide examples showing the converse is not universally true. We also propose a characterization of traditional asymptotic notations using filters, offering a new perspective on these fundamental concepts. Furthermore, we establish relationships between bounded or vanishing sequences and filter-based asymptotic notations, enabling the determination of properties central to this study. This generalization provides a more nuanced framework for analyzing algorithmic complexity, potentially capturing behaviors overlooked by classical notations and opening new avenues for theoretical computer science research.
This paper discusses the exact controllability of linear and nonlinear impulsive Caputo fractional systems. The exact controllability of a linear impulsive system is studied using the concept of generators and functional analysis. In contrast, the controllability of a nonlinear system is discussed using nonlinear functional analysis. An example is provided in the paper to support the results.
We give a construction of a new family of Deza graphs of girth at least 5 that possess an arc-transitive group of automorphisms isomorphic to a Suzuki simple group \(Sz(q)\). To study their combinatorical properties, we elaborate some group-theoretic arguments involving classical results on the groups of given type.
The Efficient 2-Terminal Reliability Problem is a nonlinear optimization problem aimed at designing a minimal-cost network within the reliability guaranties. Recent research has provided Branch-and-Price (BnP) solution based on probability relaxation, the Dantzig–Wolfe decomposition, followed by the column generation technique, and Branch-and-Bound scheme. Unfortunately, the performance of this algorithm deteriorates in cases of high-density graphs and stringent unreliability thresholds. By extending our recent approach, we introduce an improved BnP algorithm supplemented with novel valid inequalities, more efficient nonlinear integer pricing problem solver, primal heuristics, and branching strategies. Evaluation results on benchmarking instances demonstrate significant performance advantage of the proposed method.
