Sorting out the relationship between civic and political education and dual-creation education makes the two fully integrated and can play a synergistic role in educating people. This paper carries out a preliminary analysis on the integration mechanism and path of Civic and Political Education for college students and dual-creation education. Then, we established the civic elements, collected relevant civic cases, revised and updated the civic cases, and completed the construction of the civic case library of “dual-creation” education program. Based on this, a hybrid recommendation algorithm based on collaborative filtering and resource content is proposed. The original collaborative filtering recommendation algorithm and content recommendation algorithm have been improved. The recommendation algorithm’s accuracy is enhanced by the introduction of the popular resource penalty factor coefficient and time decay penalty factor coefficient. The content-based recommendation algorithm is used to assign labels to the Civics case resources, calculate the similarity of different Civics case resources, and combine the similarity obtained through the two recommendation algorithms to complete the prediction of the user’s rating of the resources. Based on this algorithm, a dual-creation course for Civics and Politics has been constructed. The overall architecture and functional modules of the system are designed in detail. The performance of the recommendation algorithm in this paper on the evaluation indexes HR@10, NDCG@10, and MRR@10 is the best among the proposed comparative algorithms. The improvement degree in MRR@10 evaluation indicators reached 11.98%. Students’ satisfaction with the recommendation system and their own development reached a high level of 9.4 and 9.12 points, which shows that students are satisfied with the effect of their personal growth under the recommendation of the dual-creation course, reflecting that the recommendation system designed in this paper possesses the effectiveness and realizes the digital and efficient fit between college students’ civic education and dual-creation education.

Let G be a connected simple graph with the vertex set $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}$, where the distance $d_G(v_i,v_j)$ is the length of a shortest path between $v_i$ and $v_j$, and the distance matrix $D\left(G\right)=\left(d_{ij}\right)n\times n$ is defined by $d_{ij}=d_G(v_i,v_j)$. Being real symmetric and non-negative, $D\left(G\right)$ has real eigenvalues ${\lambda }_1\left(G\right)\ge {\lambda }_2\left(G\right)\ge \dots \ge {\lambda }_n\left(G\right)$, with ${\lambda }_n\left(G\right)$ referred to as the least distance eigenvalue of G. In this paper, we give the maximum value of the least distance eigenvalue, and determine the unique extremal graph with two pendant vertices whose complements attain the value.

We study a parameterized SOR-like (PSOR-like) representation of the parameterized inexact Uzawa (PIU) iteration for generalized saddle point systems. An explicit parameter change establishes equivalence to PIU with an exact first-block solve, and to GSOR when stabilization is zero. A quadratic matrix polynomial characterizes the complete iteration spectrum, including unit relaxation. We give a self-contained necessary and sufficient matrix-inequality convergence criterion for this specialization, considered alongside existing generalized PIU analyses without claiming a new iteration or priority for Uzawa convergence theory. The derivation avoids Rayleigh-quotient denominator assumptions and yields computable bounds for definite preconditioners and an exact relation to the preconditioned spectrum. Sparse experiments on two synthetic families compare the stationary iteration, an explicitly defined CGPIU${}^{*}$ predictor–corrector comparator and GMRES. Archived tests use a common true residual tolerance and report setup and solve times separately. New manufactured-solution tests use a common forward-error threshold. At the common forward-error threshold, PSOR/PIU and CGPIU${}^{*}$ require identical stationary counts; P-GMRES uses fewer inner steps on both synthetic families.