A new theorem on quadratic residues modulo primes

Let $p>3$ be a prime, and let $(frac{cdot}p)$ be the Legendre symbol. Let $binmathbb Z$ and $varepsilonin{pm 1}$. We mainly prove that $$left|left{N_p(a,b): 1<a<p text{and} left(frac apright)=varepsilonright}right|=frac{3-(frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with ${x^2+b}_p>{ax^2+b}_p$, and ${m}_p$ with $minmathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.

Improving the Expressive Power of Graph Neural Network with Tinhofer Algorithm

In recent years, Graph Neural Network (GNN) has bloomly progressed for its power in processing graph-based data. Most GNNs follow a message passing scheme, and their expressive power is mathematically limited by the discriminative ability of the Weisfeiler-Lehman (WL) test. Following Tinhofers research on compact graphs, we propose a variation of the message passing scheme, called the Weisfeiler-Lehman-Tinhofer GNN (WLT-GNN), that theoretically breaks through the limitation of the WL test. In addition, we conduct comparative experiments and ablation studies on several well-known datasets. The results show that the proposed methods have comparable performances and better expressive power on these datasets.

Combinatorial identities related to $2times 2$ submatrices of recursive matrices

Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of $2times 2$ minors of certain recursive matrices, the alternating sums of their $2times 2$ minors, and the sums of their $2times 2$ permanents. We obtain some combinatorial identities related to these sums, which generalized the work of Sun and Ma in [{it Electron. J. Combin. 2014}] and [{it European J. Combin. 2014}]. With the help of the computer algebra package {tt HolonomicFunctions}, we further get some new identities involving Narayana polynomials.