查看更多>>摘要：? 2022 Elsevier Inc.In this article we present new numerical radius inequalities for sectorial matrices, and their analogues. For example, we show that if A is a sectorial matrix with sectorial index [Formula presented], then [Formula presented] where w(A) is the numerical radius of A. The significance of this result is the geometric meaning of the obtained inequality. In particular, we will show how this interpolates some well known inequalities in the literature, in a way that depends on the angle of the sector that contains the numerical range of the sectorial matrix A. Many other interpolating inequalities will be presented to show how the ratio [Formula presented] depends mainly on the angle of the sector that contains the numerical range of the matrix A. Our discussion will lead to a new vision about those matrices which contain the origin in the interior of their numerical ranges.
查看更多>>摘要：? 2022 Elsevier Inc.The L-index (resp. Q-index) of a graph G is the largest eigenvalue of the Laplacian matrix (resp. signless Laplacian matrix) of G. Very recently, Lou, Guo and Wang [6] determined the graph with fixed size and diameter having the maximum Q-index (resp. L-index). As a continuance of their result, in this paper we order all the graphs with given size and diameter from the second to the [Formula presented]th via their Q-indices. Consequently, we identify all the graphs of given size and diameter from the second to the [Formula presented]th via their L-indices. Furthermore, the graph of given size and diameter with at least one cycle having the largest Q-index is also characterized.
查看更多>>摘要：? 2022 Elsevier Inc.We prove new results about the robustness of well-known convex noise-blind optimization formulations for the reconstruction of low-rank matrices from an underdetermined system of random linear measurements. Specifically, our results address random Hermitian rank-one measurements as used in a version of the phase retrieval problem; that is, each measurement can be represented as the inner product of the unknown matrix and the outer product of a given realization of the standard complex Gaussian random vector. We obtain our results by establishing that with high probability the measurement operator consisting of independent realizations of such a random rank-one matrix exhibits the so-called Schatten-1 quotient property, which corresponds to a lower bound for the inradius of their image of the nuclear norm (Schatten-1) unit ball. We complement our analysis by numerical experiments comparing the solutions of noise-blind and noise-aware formulations. These experiments confirm that noise-blind optimization methods exhibit comparable robustness to noise-aware formulations.
查看更多>>摘要：? 2022 Elsevier Inc.Consideration of the observabilities of linear hybrid descriptor systems implies the distinguishability of these systems to be imperative. We have obtained some results related to the distinguishability of the descriptor systems. Also, we have attained equivalent criteria for input distinguishability of descriptor systems with a regular pencil.
查看更多>>摘要：? 2022 Elsevier Inc.Let T be a tree of order n with μ as an eigenvalue of multiplicity mT(μ). Wong et al. showed that [Formula presented] when n>6 and μ2 is an integer at least 2. In this paper, we generalize this result by first showing that if T is a tree of order n and t is a positve integer, then (i) if n≤2t+2, then mT(t)≤1; (ii) If n≥2t+3, then [Formula presented], with equality if and only if T has a vertex v such that T?v=kK1,t for some integer k≥2.
查看更多>>摘要：? 2022 Elsevier Inc.Let T be a tree on n(≥7) vertices with λ as a positive eigenvalue of multiplicity k. If λ2≥2 is an integer, then we prove that [Formula presented] and all extremal graphs attaining the upper bound are characterized. This result revises and improves the main conclusion of Wong, Zhou and Tian (2020). Moreover, applying this result we investigate the eigenvalue multiplicity of unicyclic graphs. Let G be a unicyclic graph of order n(≥11), which contains λ (λ2≥2 is an integer) as a positive eigenvalue of multiplicity m. Then it is proved that [Formula presented], and all extremal graphs attaining the upper bound are determined. These two upper bounds improve the conclusions of Rowlinson (2010, 2011), respectively.
查看更多>>摘要：? 2022 Elsevier Inc.Suppose that Σ is a signed graph with n vertices and m edges. Let λ1≥λ2≥?≥λn be the eigenvalues of Σ.A signed graph is called balanced if each of its cycles contains an even number of negative edges, and unbalanced otherwise. Let ωb be the balanced clique number of Σ, which is the maximum order of a balanced complete subgraph of Σ. In this paper, we prove that [Formula presented] This inequality extends a conjecture of ordinary graphs, which was confirmed by Nikiforov (2002) [8], to the signed case. In addition, we completely characterize the signed graphs with ?1≤λ2≤0.
查看更多>>摘要：? 2022 Elsevier Inc.The Extended Randomized Kaczmarz method is a well known iterative scheme which can find the Moore-Penrose inverse solution of a possibly inconsistent linear system and requires only one additional column of the system matrix in each iteration in comparison with the standard randomized Kaczmarz method. Also, the Sparse Randomized Kaczmarz method has been shown to converge linearly to a sparse solution of a consistent linear system. Here, we combine both ideas and propose an Extended Sparse Randomized Kaczmarz method. We show linear expected convergence to a sparse least squares solution in the sense that an extended variant of the regularized basis pursuit problem is solved. Moreover, we generalize the additional step in the method and prove convergence to a more abstract optimization problem. We demonstrate numerically that our method can find sparse least squares solutions of real and complex systems if the noise is concentrated in the complement of the range of the system matrix and that our generalization can handle impulsive noise.
查看更多>>摘要：? 2022 Elsevier Inc.We consider graphs with two cut vertices joined by a path with one or two edges, and prove that there can be no quantum perfect state transfer between these vertices, unless the graph has no other vertex. We achieve this result by applying the 1-sum lemma for the characteristic polynomial of graphs, the neutrino identities that relate entries of eigenprojectors and eigenvalues, and variational principles for eigenvalues (Cauchy interlacing theorem, Weyl inequalities and Wielandt minimax principle). We see our result as an intermediate step to broaden the understanding of how connectivity plays a key role in quantum walks, and as further evidence of the conjecture that no tree on four or more vertices admits state transfer. We conclude with some open problems.
查看更多>>摘要：? 2022 Elsevier Inc.Let F be a field of characteristic zero and M2,1(F) the algebra of 3×3 matrices over F endowed with non-trivial Z2-grading. The transpose involution t on M2,1(F) preserves the homogeneous components of the grading and so, we consider (M2,1(F),t) as a superalgebra with graded involution. We study the (Z2,?)-identities of this algebra and make explicit the decomposition of the space of multilinear (Z2,?)-identities into the sum of irreducibles under the action of the group (Z2×Z2)?Sn in order to determine all the irreducible characters appearing with non-zero multiplicity in the decomposition of the ?-graded cocharacter of (M2,1(F),t). Along the way, using the representation theory of the general linear group, we determine all the (Z2,?)-identities of (M2,1(F),t) up to degree 3.
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