The minimal radical and weakly irreducible groups

1973 ◽  
Vol 13 (3) ◽  
pp. 269-273
Author(s):  
K. K. Shchukin
Keyword(s):  
Weakly Irreducible

Separability for weakly irreducible matrices

2014 ◽  
Vol 14 (15&16) ◽  
pp. 1308-1337
Author(s):  
Daniel Cariello
Keyword(s):  
Positive Semidefinite ◽  
Tensor Rank ◽  
Split Decomposition ◽  
Finite Dimensional ◽  
The Matrix ◽  
Separability Problem ◽  
Finite Dimensional Case ◽  
Rank 2 ◽  
Decomposition Theorems ◽  
Weakly Irreducible

This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC $($PPT$)$ matrix can be decomposed into a sum of $s+1$ SPC $($PPT$)$ matrices of which the first $s$ are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC $($PPT$)$ matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC $($PPT$)$ matrices to the case of weakly irreducible SPC $($PPT$)$ matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).

scholarly journals On a new convergence in topological spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1716-1723
Author(s):  
Xiao-jun Ruan ◽  
Xiao-quan Xu
Keyword(s):  
Topological Spaces ◽  
Completely Distributive ◽  
Weakly Irreducible

Abstract In this paper, we introduce a new way-below relation in T0 topological spaces based on cuts and give the concepts of SI2-continuous spaces and weakly irreducible topologies. It is proved that a space is SI2-continuous if and only if its weakly irreducible topology is completely distributive under inclusion order. Finally, we introduce the concept of 𝓓-convergence and show that a space is SI2-continuous if and only if its 𝓓-convergence with respect to the topology τSI2(X) is topological. In general, a space is SI-continuous if and only if its 𝓓-convergence with respect to the topology τSI(X) is topological.

scholarly journals METRICS THAT REALIZE ALL LORENTZIAN HOLONOMY ALGEBRAS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1025-1045 ◽  
Author(s):  
ANTON S. GALAEV
Keyword(s):  
Lorentzian Manifolds ◽  
Weakly Irreducible

All candidates to the weakly-irreducible not irreducible holonomy algebras of Lorentzian manifolds are known. In the present paper metrics that realize all these candidates as holonomy algebras are given. This completes the classification of the Lorentzian holonomy algebras. Also new examples of metrics with the holonomy algebras g2 ⋉ ℝ7 ⊂ 𝔰𝔬(1, 8) and 𝔰𝔭𝔦𝔫(7) ⋉ ℝ8 ⊂ 𝔰𝔬(1, 9) are constructed.

Sharp bounds for spectral radius of nonnegative weakly irreducible tensors

2019 ◽  
Vol 14 (5) ◽  
pp. 989-1015 ◽  
Author(s):  
Lihua You ◽  
Xiaohua Huang ◽  
Xiying Yuan
Keyword(s):  
Spectral Radius ◽  
Sharp Bounds ◽  
Irreducible Tensors ◽  
Weakly Irreducible

scholarly journals The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs

2018 ◽  
Vol 372 (3) ◽  
pp. 2213-2233 ◽  
Author(s):  
Yi-Zheng Fan ◽  
Tao Huang ◽  
Yan-Hong Bao ◽  
Chen-Lu Zhuan-Sun ◽  
Ya-Ping Li
Keyword(s):  
Nonnegative Tensors ◽  
Weakly Irreducible
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