For positive integers n and t, let Γ(n,2,t) be the family of n×n 0–1 matrices A for which every entry of A2 is at most t, and let γ(n,2,t) be the largest possible number of ones in such a matrix. A previous result established the exact value of γ(n,2,t) under a cubic lower bound on n. We reduce this sufficient order condition to a quadratic one. More precisely, for t≥2, the known extremal formula holds when n−t is even and n≥2t2+6t, and when n−t is odd and n≥4t2+9t+2. We also obtain an exchange-rigidity property for extremal matrices. The proof retains the exact deficit associated with the gap between the maximum degree and its extremal value.
In this paper, we study unitary associative G-graded algebras endowed with a graded involution with polynomial codimension growth. For each fixed k≥1, we construct algebras whose codimension sequences grow polynomially of degree k, realizing the maximal and the minimal leading coefficients. We then provide, up to equivalence, a complete classification of finitely generated unitary algebras with quadratic codimension growth. More precisely, we establish a direct correspondence between the nonzero multiplicities appearing in the decomposition of the proper cocharacters and the algebras generating minimal varieties within the given variety. As a consequence, every affine unitary variety with at most quadratic growth is generated by an algebra that decomposes into a direct sum of algebras generating minimal varieties of at most quadratic growth.
The Araki–Lieb–Thirring inequality is a fundamental trace inequality obtained from Araki's log-majorization theorem. In this paper we keep track of the surplus hidden in the log-majorization argument. For positive definite matrices A,B and q>0, setXr=(B1/2AB1/2)r,Yr=Br/2ArBr/2. When r≥1, Araki's theorem gives logλ(Xr)≺logλ(Yr), and we prove the exact representationTrYrq−TrXrq=q2∫−∞∞eqt{Tr(logYr−tI)+−Tr(logXr−tI)+}dt, where the integrand is nonnegative for every real t. Thus the Araki–Lieb–Thirring gap is an integrated upper-tail surplus of logarithmic spectra. We also record the corresponding reverse formula in the range 0<r<1, where the log-majorization and the trace inequality reverse. The proof is based on a general convex-gap identity for majorized vectors and therefore separates the use of Araki's log-majorization from the positive remainder formula. Finally, we identify two special cases: for (r,q)=(2,1), the gap is 12‖[A,B]‖22, while for (r,q)=(2,1/2), it is the trace-norm polar phase defect of AB.
In 2023, Greaves, Syatriadi, and Yatsyna found a set of 57 equiangular lines in R18, breaking the previous record. In 2025, Lin, Munemasa, Taniguchi, and Yoshino constructed a large number of sets of 57 equiangular lines in R18 as affine equiangular sets in an integral overlattice of A92⊕A1. In this paper, we construct further sets of 57 equiangular lines in R18 from Latin squares of order 6 and Pasch configurations, realized as affine equiangular sets in an integral overlattice of A53⊕A14. Unlike the previously known examples, these sets are not strongly maximal. Moreover, some of them have only five distinct Seidel eigenvalues, fewer than any previously known examples.
Let H be a Hilbert space and let B(H) denote the algebra of bounded linear operators on H, equipped with the operator norm ‖⋅‖. We investigate the uniform compressibility of the matched projection, and establish that‖m(Q1)−m(Q2)‖≤‖Q1−Q2‖ for all idempotents Q1,Q2∈B(H), where m(Qi) denotes the matched projection of Qi for i=1,2. Furthermore, in the finite-dimensional setting, we show that this inequality remains valid when the operator norm is replaced by the Frobenius norm.
