At present, there are few systematic researches on macro-scale heterogeneous modeling and numerical simulation of dynamic mechanical properties of 3-D braided composites. In this paper, the parametric virtual simulation model of 3D five-directional braided composites is realized in the way of “point-line-solid” based on the integrated design idea of process-structure-performance. And the impact compression numerical simulation of the material is carried out by using multi-scale analysis method. The effects of strain rate and braiding angle on transverse impact compression properties and fracture characteristics of composites is studies and verified by comparing the test results with the numerical simulation results systematically. The dynamic failure mechanism of the matrix and fiber bundles during the impact compression process is revealed. The results show that the macro-scale heterogeneous simulation model of 3D five-directional braided composites established is effective, and the numerical simulation results agree well with the test results. The matrix fracture and shear failure of fiber bundles are presented simultaneously under transverse impact compression. The failure of fiber bundles and matrix mainly concentrates on two main fracture shear planes. And the included angle between the fracture shear planes and the vertical direction is consistent with the corresponding internal braiding angle of specimens.

The paper extensively examined the intricate components underpinning innovation ability, culminating in the construction of a linear spatial model delineating innovation and entrepreneurship prowess. This paper analyzed the components of the connotation of innovation ability, then constructs a linear spatial model of innovation and entrepreneurship ability, proposes a multi-objective function model of the utilization efficiency and allocation efficiency of education resources, and uses the grey correlation algorithm The experimental simulation and model solution are carried out. The simulation results show that, through the optimization, the utilization efficiency and allocation efficiency of the educational resources for innovation and entrepreneurship for all are increased by 18.72% and 20.98% respectively, and tend to be in equilibrium, which can achieve the optimization of educational resources allocation. Among all people, the correlation value with ideal entrepreneurship is 0.3177, achieving the most excellent innovation and entrepreneurship education.

For a graph \(G\), two vertices \(x,y\in G\) are said to be resolved by a vertex \(s\in G\), if \(d(x|s)\neq d(y|s)\). The minimum cardinality of such a resolving set \(\textit{R}\) in \(G\) is called its metric dimension. A resolving set \(\textit{R}\) is said to be fault-tolerant, if for every \(p\in R\), \(R-p\) preserves the property of being a resolving set. A fault-tolerant metric dimension of \(G\) is a minimal possible order fault-tolerant resolving set. A wide variety of situations, in which connection, distance, and connectivity are important aspects, call for the utilisation of metric dimension. The structure and dynamics of complex networks, as well as difficulties connected to robotics network design, navigation, optimisation, and facility positioning, are easier to comprehend as a result of this. As a result of the relevant concept of metric dimension, robots are able to optimise their methods of localization and navigation by making use of a limited number of reference locations. As a consequence of this, numerous applications of robotics, including collaborative robotics, autonomous navigation, and environment mapping, have become more precise, efficient, and resilient. The arithmetic graph \(A_l\) is defined as the graph with its vertex set as the set of all divisors of \(l\), where \(l\) is a composite number and \(l = p^{\gamma_1}_{1} p^{\eta_2}_{2}, \dots, p^{n}_{n}\), where \(p_n \geq 2\) and the \(p_i\)’s are distinct primes. Two distinct divisors \(x, y\) of \(l\) are said to have the same parity if they have the same prime factors (i.e., \(x = p_{1}p_{2}\) and \(y = p^{2}_{1}p^{3}_{2}\) have the same parity). Further, two distinct vertices \(x, y \in A_l\) are adjacent if and only if they have different parity and \(\gcd(x, y) = p_i\) (greatest common divisor) for some \(i \in \{1, 2, \dots, t\}\). This article is dedicated to the investigation of the arithmetic graph of a composite number \(l\), which will be referred to throughout the text as \(A_{l}\). In this study, we compute the fault-tolerant resolving set and the fault-tolerant metric dimension of the arithmetic graph \(A_{l}\), where \(l\) is a composite number.

In this paper, we identify LWO graphs, f\-ind the minimum \(\lambda\) for decomposition of \(\lambda K_n\) into these graphs, and show that for all viable values of \(\lambda\), the necessary conditions are suf\-f\-icient for LWO–decompositions using cyclic decompositions from base graphs.

For a graph \( G \) and a subgraph \( H \) of a graph \( G \), an \( H \)-decomposition of the graph \( G \) is a partition of the edge set of \( G \) into subsets \( E_i \), \( 1 \leq i \leq k \), such that each \( E_i \) induces a graph isomorphic to \( H \). In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the \( \lambda \)-fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

We classify the geometric hyperplanes of the Segre geometries, that is, direct products of two projective spaces. In order to do so, we use the concept of a generalised duality. We apply the classification to Segre varieties and determine precisely which geometric hyperplanes are induced by hyperplanes of the ambient projective space. As a consequence we find that all geometric hyperplanes are induced by hyperplanes of the ambient projective space if, and only if, the underlying field has order \(2\) or \(3\).

A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

The hub cover pebbling number, \(h^{*}(G)\), of a graph $G$, is the least non-negative integer such that from all distributions of \(h^{*}(G)\) pebbles over the vertices of \(G\), it is possible to place at least one pebble each on every vertex of a set of vertices of a hub set for \(G\) using a sequence of pebbling move operations, each pebbling move operation removes two pebbles from a vertex and places one pebble on an adjacent vertex. Here we compute the hub cover pebbling number for wheel related graphs.

An outer independent double Roman dominating function (OIDRDF) on a graph \( G \) is a function \( f : V(G) \to \{0, 1, 2, 3\} \) having the property that (i) if \( f(v) = 0 \), then the vertex \( v \) must have at least two neighbors assigned 2 under \( f \) or one neighbor \( w \) with \( f(w) = 3 \), and if \( f(v) = 1 \), then the vertex \( v \) must have at least one neighbor \( w \) with \( f(w) \ge 2 \) and (ii) the subgraph induced by the vertices assigned 0 under \( f \) is edgeless. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number \( \gamma_{oidR}(G) \) is the minimum weight of an OIDRDF on \( G \). The \( \gamma_{oidR} \)-stability (\( \gamma^-_{oidR} \)-stability, \( \gamma^+_{oidR} \)-stability) of \( G \), denoted by \( {\rm st}_{\gamma_{oidR}}(G) \) (\( {\rm st}^-_{\gamma_{oidR}}(G) \), \( {\rm st}^+_{\gamma_{oidR}}(G) \)), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the outer independent double Roman domination number. In this paper, we determine the exact values on the \( \gamma_{oidR} \)-stability of some special classes of graphs, and present some bounds on \( {\rm st}_{\gamma_{oidR}}(G) \). In addition, for a tree \( T \) with maximum degree \( \Delta \), we show that \( {\rm st}_{\gamma_{oidR}}(T) = 1 \) and \( {\rm st}^-_{\gamma_{oidR}}(T) \le \Delta \), and characterize the trees that achieve the upper bound.

We introduce a two-player game where the goal is to illuminate all edges of a graph. At each step the first player, called Illuminator, taps a vertex. The second player, called Adversary, reveals the edges incident with that vertex (consistent with the edges incident with the already tapped vertices). Illuminator tries to minimize the taps needed, and the value of the game is the number of taps needed with optimal play. We provide bounds on the value in trees and general graphs. In particular, we show that the value for the path on \( n \) vertices is \( \frac{2}{3} n + O(1) \), and there is a constant \( \varepsilon > 0 \) such that for every caterpillar on \( n \) vertices, the value is at most \( (1 – \varepsilon) n + 1 \).