Graph burning is a model to describe the spread of social influence. In 2023, Song et al. proposed the Independent Cascade Graph Burning model, where a vertex v can be burned by its burning neighbors u and the influence that u gives to v is larger than a given threshold β . The minimum number of time steps that can be chosen as rounds to burn the whole graph G with the Independent Cascade Graph Burning model called the IC burning number b β ( G ) . In this paper, we determined the IC burning number for some graphs and operation graphs.

In the field of machine learning, support vector machine (SVM) is popular for its powerful performance in classification tasks. However, this method could be adversely affected by data containing noise. One of the main reasons is that many loss functions are too sensitive to sample points far from their classes. In this paper, based on the idea of HTPSVM proposed in (Zhu, Song, et al.), we introduce the SVM classifier with quartic truncated pinball loss (QTPSVM), which can be solved by Bregman modified second accelerated proximal gradient method (BAPGs). Numerical experiments show that QTPSVM is more effective on many real data.

In this article, we demonstrate that embedded vector solitons (EVSs) indeed exist. In other words, we prove that there exist systems of nonlinear partial differential equations (NLPDEs) which have soliton solutions ( u,v ) , where each of the solitons, u( z,t ) and v( z,t ) , is an embedded soliton. We present two systems of NLPDEs with these peculiar EVSs: a system of two cubic-quintic nonlinear Schrödinger (NLS) equations, and a system of two complex modified Korteweg-de Vries (cmKdV) equations. These two systems are the first systems of NLPDEs with EVSs known to date. Moreover, we demonstrate that these two systems also possess dark vector solitons and a highly unusual type of vector soliton: mixed vector solitons, which are composed of a bright soliton and a dark one.

A new Base-X Conjecture was introduced in this paper, and Collatz Conjecture is just one case of Base-X Conjecture - Base-3 (Ternary). Based on Base-X number system property and Collatz Conjecture iteration, it has been proved that for any positive integer D, there are n and m which exist for D n + Y n = 2 m . D n + Y n is just the result built up by collecting divided by 2 of Collatz Conjecture iteration. Divided by 2m will make the Collatz Conjecture get a result of 1 for any positive integer. Also, the Collatz Tree showed that for any odd positive number, there is only one route existing in the Collatz Tree down to 1 on Collatz Conjecture iteration.

Mongolia is a landlocked country with limited infrastructure and high dependence on the Xingang Tianjin port in China for imports. This research examines the potential impacts of establishing a dry port in Zamyn-Uud, Mongolia, utilizing a system dynamics modeling approach via Vensim software. The study evaluates transportation time, costs, inflation, and logistics performance index improvements, revealing that the establishment of the dry port can reduce transportation costs and delays significantly while enhancing economic growth. The findings offer actionable insights for policymakers and stakeholders in addressing logistical inefficiencies and fostering sustainable development in landlocked regions.

In this paper, we derive a second-order Crank-Nicolson finite-difference scheme for the three-dimensional Allen-Cahn equation and present an estimate of the scheme’s truncation error. We prove the existence of the numerical solution and establish stepwise uniqueness, from which the global uniqueness follows given the initial value u 0. We also establish conditional convergence in the L∞ -norm, and demonstrate a discrete maximum principle for the scheme. Numerical Examples 1 and 2 validate the discrete maximum principle, while Numerical Example 3 employs a physics-informed neural network (PINN), using the Crank-Nicolson solution as the reference, to compute the errors between the two approaches in the L2 and L∞ norms. Owing to the high dimensionality of the problem, visualization is performed via isosurface plots on identical time slices; these plots show strong agreement between the Crank-Nicolson and PINN solutions. The numerical results collectively confirm the accuracy and effectiveness of the proposed scheme.

Let G be a complete tripartite graph with n vertices. Its vertex set is U∪V∪W , where | U |=1 , | V |=s , | W |=t , and this graph is denoted as K 1,s,t . Let T 1,s,t ω denote the path-tree corresponding to the vertex ω∈U∪V∪W . In this paper, we mainly present the matching polynomial of the path-tree T 1,s,t ω corresponding to the complete tripartite graph.