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In this paper, we introduce the limit, unique solution of the nonlinear equations, geodesic property, tolerance relations and pinch on the spectral geometric mean for two positive definite operators. We show that the spectral geometric mean is a geodesic with respect to some semi-metric. We also prove that the tolerance relation on determinant one matrices can be characterized by the spectral geometric mean. Moreover, two positive tuples can be pinched by the spectral geometric mean. [ABSTRACT FROM AUTHOR]

We establish some new properties of spectral geometric mean. In particular, we prove a log majorization relation between $ (B^{ts/2}A^{(1-t)s} B^{ts/2})^{1/s} $ (B ts / 2 A (1 − t) s B ts / 2 ) 1 / s and the t-spectral mean $ A\natural_t B :=(A^{-1}\sharp B)^{t}A(A^{-1}\sharp B)^{t} $ A ♮ t B := (A − 1 ♯B) t A (A − 1 ♯B) t of two positive semidefinite matrices A and B, where $ A\sharp B $ A ♯B is the geometric mean, and the t-spectral mean is the dominant one. The limit involving t-spectral mean is also studied. We then extend all the results in the context of symmetric spaces of negative curvature. [ABSTRACT FROM AUTHOR]

For k>b≥0 $k\gt b\ge 0$, let Yk,b ${Y}_{k,b}$ be the k $k$‐uniform hypergraph with two edges intersecting in b $b$ vertices. Our main result is that any n $n$‐vertex 3‐uniform hypergraph with at least n3−n−m+13+o(n3) $\left(\genfrac{}{}{0.0pt}{}{n}{3}\right)-\left(\genfrac{}{}{0.0pt}{}{n-m+1}{3}\right)+o({n}^{3})$ edges contains a collection of m $m$ vertex‐disjoint copies of Y3,2 ${Y}_{3,2}$, for m≤n∕7 $m\le n\unicode{x02215}7$. The bound on the number of edges is asymptotically best possible. This problem generalizes the Matching Conjecture of Erdős. We then use this result combined with the absorbing method to determine the asymptotically best possible minimum (k−3) $(k-3)$‐degree threshold for ℓ $\ell $‐Hamiltonicity in k $k$‐graphs, where k≥7 $k\ge 7$ is odd and ℓ=(k−1)∕2 $\ell =(k-1)\unicode{x02215}2$. Moreover, we give related results on Yk,b ${Y}_{k,b}$‐tilings and Hamilton ℓ $\ell $‐cycles with d $d$‐degree for some other values of k,ℓ,d $k,\ell ,d$. [ABSTRACT FROM AUTHOR]

In this paper, cluster synchronization in community network with nonidentical nodes is investigated. Through introducing proper adaptive strategy into impulsive control scheme, adaptive impulsive controllers are designed for achieving the cluster synchronization. In this adaptive impulsive control scheme, for any given networks, the impulsive gains can adjust themselves to proper values according to the proposed adaptive strategy when the impulsive intervals are fixed. The impulsive instants can be estimated by solving a sequence of maximum value problems when the impulsive gains are fixed. Both community networks without and with coupling delay are considered. Based on the Lyapunov function method and mathematical analysis technique, two synchronization criteria are derived. Several numerical examples are performed to verify the effectiveness of the derived theoretical results. [ABSTRACT FROM AUTHOR]