[en] Non-backtracking centrality was introduced as a way to correct what may be understood as a deficiency in the eigenvector centrality, since the eigenvector centrality in a network can be artificially increased in high-degree nodes (hubs) because a hub is central because its neighbors are central, but these, in turn, are central just because they are hub neighbors. We define the non-backtracking PageRank as a new measure modifying the well-known classic PageRank in order to avoid the possibility of the random walker returning to the node immediately visited (non-backtracking walk). But, as we show, this measure presents a gap and a remarkable difference between the limit of “no penalty for return trips” and the direct calculation of the non-backtracking PageRank. Also, as it is shown in the applications presented, in certain cases this new measure produces notable variations with respect to the classifications obtained by the classic PageRank.

[en] Let M_n be an n×n real (resp. complex) Wigner matrix and U_nΛ_nU_n^* be its spectral decomposition. Set (y₁,y₂⋯,y_n)^T=U_n^*x, where x = (x₁, x₂, ⋅⋅⋅, x_n)^T is a real (resp. complex) unit vector. Under the assumption that the elements of M_n have 4 matching moments with those of GOE (resp. GUE), we show that the process X_n(t)=√((βn)/2 )∑_i=1^⌊nt⌋(|y_i|²−1/n ) converges weakly to the Brownian bridge for any x satisfying ‖x‖_∞ → 0 as n → ∞, where β = 1 for the real case and β = 2 for the complex case. Such a result indicates that the orthogonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthogonal (resp. unitary) group from a certain perspective

[en] The higher-derivative theories with degenerate frequencies exhibit BRST symmetry [V.O. Rivelles, Phys. Lett. B 577 (2003) 147]. In the present Letter meaning of BRST-invariance condition is analyzed. The BRST symmetry is related to nondiagonalizability of the Hamiltonian and it is shown that BRST condition singles out the subspace spanned by proper eigenvectors of the Hamiltonian.

[en] Some computational methods for the evaluation of eigenvalues and eigenvectors of (square) real matrices are briefly described. The methods of Jacobi, Given and Householder are used for real-symmetric matrices while Lanczos's method, supertriangularization and deflation methods are used for real-non symmetric matrices. (author)

[en] An early, but at the time illuminating, piece of work on how to deal with a general, linearly coupled accelerator lattice is revisited. This work is based on the SLIM formalism developed in 1979-1981

[en] We report an unusual buildup of the quantum coherence in a qubit subjected to non-Hermitian evolution generated by a parity-time ($\mathcal{P}\mathcal{T}$) symmetric Hamiltonian, which is reinterpreted as a Hermitian system in a higher dimensional space using Naimark dilation. The coherence is found to be maximum about the exceptional points (EPs), i.e. the points of coalescence of the eigenvalues as well as the eigenvectors. The nontrivial physics about EPs has been observed in various systems, particularly in photonic systems. As a consequence of enhancement in coherence, the various formulations of Leggett–Garg inequality tests show maximal violation about the EPs. (paper)

[en] Using large $N$ arguments, we propose a scheme for calculating the two-point eigenvector correlation function for non-normal random matrices in the large $N$ limit. The setting generalizes the quaternionic extension of free probability to two-point functions. In the particular case of biunitarily invariant random matrices, we obtain a simple, general expression for the two-point eigenvector correlation function, which can be viewed as a further generalization of the single ring theorem. This construction has some striking similarities to the freeness of the second kind known for the Hermitian ensembles in large $N$. On the basis of several solved examples, we conjecture two kinds of microscopic universality of the eigenvectors — one in the bulk, and one at the rim. The form of the conjectured bulk universality agrees with the scaling limit found by Chalker and Mehlig [JT Chalker, B Mehlig, Phys. Rev. Lett. 81 (1998) 3367] in the case of the complex Ginibre ensemble.

[en] A new algorithm for simultaneous coordinate relaxation is described. For the determination of several extreme eigenvalues and eigenvectors of large, sparse matrices the simultaneous algorithm affords significant advantages in comparison with a coordinate relaxation algorithm applied to determine individual eigenvalues and eigenvectors in turn. Results of application of the algorithm to test matrices are discussed

[en] We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on ℤ^ν-lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.

[en] The spectral problem for the quantum dispersionless Korteweg–de Vries (KdV) hierarchy, aka the quantum Hopf hierarchy, is solved by Dubrovin. In this article, following Dubrovin, we study Buryak–Rossi’s quantum KdV hierarchy. In particular, we prove a symmetry property and a non-degeneracy property for the quantum KdV Hamiltonians. On the basis of this we construct a complete set of common eigenvectors. The analysis underlying this spectral problem implies certain vanishing identities for combinations of characters of the symmetric group. We also comment on the geometry of the spectral curves of the quantum KdV hierarchy and we give a representation of the quantum dispersionless KdV Hamiltonians in terms of multiplication operators in the class algebra of the symmetric group. (paper)