Recent zbMATH articles in MSC 15https://zbmath.org/atom/cc/152023-05-31T16:32:50.898670ZWerkzeugProof of a conjecture involving derangements and roots of unityhttps://zbmath.org/1508.050132023-05-31T16:32:50.898670Z"Wang, Han"https://zbmath.org/authors/?q=ai:wang.han"Sun, Zhi-Wei"https://zbmath.org/authors/?q=ai:sun.zhiwei|sun.zhi-wei.1|sun.zhi-weiSummary: Let \(n>1\) be an odd integer, and let \(\zeta\) be a primitive \(n\) th root of unity in the complex field. Via the eigenvector-eigenvalue identity, we show that
\[
\sum_{\tau\in D(n-1)}\,\,\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1+\zeta^{j-\tau(j)}}{1-\zeta^{j-\tau(j)}}=(-1)^{\frac{n-1}{2}}\frac{((n-2)!!)^2}{n},
\]
where \(D(n-1)\) is the set of all derangements of \(1, \ldots, n-1\). This confirms a previous conjecture of \textit{Z.-W. Sun} [New conjectures in number theory and combinatorics (in Chinese). Harbin: Institute of Technology Press (2021)]. Moreover, for each \(\delta=0,1\) we determine the value of \(\det[x+m_{jk}]_{1\leqslant j,k\leqslant n-1}\) completely, where
\[
m_{jk}=\begin{cases}(1+\zeta^{j-k})/(1-\zeta^{j-k})&\text{ if } j\not=k,\\ \delta&\text{
if }j=k. \end{cases}
\]Locating eigenvalues of unbalanced unicyclic signed graphshttps://zbmath.org/1508.051022023-05-31T16:32:50.898670Z"Belardo, Francesco"https://zbmath.org/authors/?q=ai:belardo.francesco"Brunetti, Maurizio"https://zbmath.org/authors/?q=ai:brunetti.maurizio"Trevisan, Vilmar"https://zbmath.org/authors/?q=ai:trevisan.vilmarSummary: A signed graph is a pair \(\Gamma=(G,\sigma)\), where \(G\) is a graph, and \(\sigma:E(G)\longrightarrow\{+1,-1\}\) is a signature of the edges of \(G\). A signed graph \(\Gamma\) is said to be unbalanced if there exists a cycle in \(\Gamma\) with an odd number of negatively signed edges. In this paper it is presented a linear time algorithm which computes the inertia indices of an unbalanced unicyclic signed graph. Additionally, the algorithm computes the number of eigenvalues in a given real interval by operating directly on the graph, so that the adjacency matrix is not needed explicitly. As an application, the algorithm is employed to check the integrality of some infinite families of unbalanced unicyclic graphs. In particular, the multiplicities of eigenvalues of signed circular caterpillars are studied, getting a geometric characterization of those which are non-singular and sufficient conditions for them to be non-integral. Finally, the algorithm is also used to retrieve the spectrum of bidegreed signed circular caterpillars, none of which turns out to be integral.On the eigenvalue and energy of extended adjacency matrixhttps://zbmath.org/1508.051042023-05-31T16:32:50.898670Z"Ghorbani, Modjtaba"https://zbmath.org/authors/?q=ai:ghorbani.modjtaba"Li, Xueliang"https://zbmath.org/authors/?q=ai:li.xueliang"Zangi, Samaneh"https://zbmath.org/authors/?q=ai:zangi.samaneh"Amraei, Najaf"https://zbmath.org/authors/?q=ai:amraei.najafSummary: The extended adjacency matrix of graph \(G\), \(\mathcal{A}_{ex}\) is a symmetric real matrix that if \(i\neq j\) and \(u_iu_j\in E(G)\), then the \(ij\)th entry is \(d_{u_i}^2+d_{u_j}^2/2d_{u_i} d_{u_j}\), and zero otherwise, where \(d_u\) indicates the degree of vertex \(u\). In the present paper, several investigations of the extended adjacency matrix are undertaken and then some spectral properties of \(\mathcal{A}_{ex}\) are given. Moreover, we present some lower and upper bounds on extended adjacency spectral radii of graphs. Besides, we also study the behavior of the extended adjacency energy of a graph \(G\).On two problems related to anti-adjacency (eccentricity) matrixhttps://zbmath.org/1508.051092023-05-31T16:32:50.898670Z"Sorgun, Sezer"https://zbmath.org/authors/?q=ai:sorgun.sezer"Küçük, Hakan"https://zbmath.org/authors/?q=ai:kucuk.hakanSummary: The anti-adjacency (aka eccentricity) matrix \(\mathcal{A} ( G )\) of a graph \(G\) is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. The motivation for this paper is to find answers to two problems: ``For which connected graphs the eccentricity (anti-adjacency) matrix is either reducible or irreducible?'' and ``Characterize the graphs with a small number of distinct anti-adjacency eigenvalues''. In this context, we get the sufficient and necessary conditions of the graphs which have an irreducible matrix. For the second problem, we characterize the graphs with exactly two distinct eigenvalues different from 0. We also obtain some spectral properties of the anti-adjacency matrix.The first eigenvector of a distance matrix is nearly constanthttps://zbmath.org/1508.051102023-05-31T16:32:50.898670Z"Steinerberger, Stefan"https://zbmath.org/authors/?q=ai:steinerberger.stefanSummary: Let \(x_1, \dots, x_n\) be points in a metric space and define the distance matrix \(D \in \mathbb{R}^{n \times n}\) by \(D_{ij} = d(x_i, x_j)\). The Perron-Frobenius theorem implies that there is an eigenvector \(v \in \mathbb{R}^n\) with non-negative entries associated to the largest eigenvalue. We prove that this eigenvector is nearly constant in the sense that the inner product with the constant vector \(\1\in \mathbb{R}^n\) is large
\[
\langle v, \1\rangle \geq \frac{1}{\sqrt{2}} \cdot \|v\|_{\ell^2} \cdot \|\1\|_{\ell^2}
\]
and that each entry satisfies \(v_i \geq \| v \|_{\ell^2} / \sqrt{4n}\). Both inequalities are sharp.Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomialshttps://zbmath.org/1508.051652023-05-31T16:32:50.898670Z"Xin, Guoce"https://zbmath.org/authors/?q=ai:xin.guoce"Zhong, Yueming"https://zbmath.org/authors/?q=ai:zhong.yuemingSummary: In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as \(A050446\) on OEIS. This number is exactly the two variable array \(T(n, m)\) defined by the recursion \(T(n, m) = T(n, m - 1) + \sum_{k = 0}^{\lfloor \frac{ n - 1}{ 2} \rfloor} T(2 k, m - 1) T(n - 1 - 2 k, m)\), where \(T(n, 0) = T(0, m) = 1\) for all nonnegative integers \(m, n\). Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to both the row and column generating function of \(T(n, m)\). In this paper, we give explicit formula of the column generating function, which is also the generating function \(F(n, x)\) studied by \textit{M. Bóna} et al. [Discrete Appl. Math. 155, No. 11, 1481--1496 (2007; Zbl 1149.90162)]. We also get trig function representations by using Chebyshev polynomials of the second kind. This allows us to prove all these conjectures.A point-plane incidence theorem in matrix ringshttps://zbmath.org/1508.110142023-05-31T16:32:50.898670Z"The, Nguyen"https://zbmath.org/authors/?q=ai:the.nguyen-van"Vinh, Le Anh"https://zbmath.org/authors/?q=ai:vinh.le-anhSummary: In this paper, we study a point-hyper plane incidence theorem in matrix rings, which generalizes all previous works in literature of this direction.On generalized Jacobsthal and Jacobsthal-Lucas numbershttps://zbmath.org/1508.110192023-05-31T16:32:50.898670Z"Bród, Dorota"https://zbmath.org/authors/?q=ai:brod.dorota"Michalski, Adrian"https://zbmath.org/authors/?q=ai:michalski.adrianSummary: Jacobsthal numbers and Jacobsthal-Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal-Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal-Lucas sequence. We give generating functions, Binet's formulas for these numbers. Moreover, we obtain some identities, among others Catalan's, Cassini's identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal-Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal-Lucas numbers. Additionally, we give a matrix representation of the presented numbers.A parametric type of Bernoulli polynomials with higher levelhttps://zbmath.org/1508.110332023-05-31T16:32:50.898670Z"Komatsu, Takao"https://zbmath.org/authors/?q=ai:komatsu.takao|komatsu.takao.1Summary: In this paper, we introduce a parametric type of Bernoulli polynomials with higher level and study their characteristic and combinatorial properties. We also give determinant expressions of a parametric type of Bernoulli polynomials with higher level. The results are generalizations of those with level 2 by Masjed-Jamei, Beyki and Koepf [\textit{M. Masjed-Jamei} et al., Mediterr. J. Math. 15, No. 3, Paper No. 138, 17 p. (2018; Zbl 1422.11051)] and with level 3 by the author [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114, No. 3, Paper No. 151, 19 p. (2020; Zbl 1476.11046)].A matrix viewpoint for various algebraic extensionshttps://zbmath.org/1508.120042023-05-31T16:32:50.898670Z"Abrams, Gene"https://zbmath.org/authors/?q=ai:abrams.gene-d"Ánh, Pham Ngoc"https://zbmath.org/authors/?q=ai:anh.pham-ngocLet \(K\) be a field and \(f(x)\in K[x]\) be an irreducible polynomial of degree \(n\). The existence of algebraic extensions of \(K\) is usually introduced via Kronecker's construction \(F:=K[x]/f(x)K[x]\) as a quotient ring. The authors argue that students may find it more appealing to use a concrete matrix approach in which \(F:=K[A_{f}]\subseteq M_{n}(K)\) where \(M_{n}(K)\) is the ring of \(n\times n\) matrices over \(K\) and \(A_{f}\) is the companion matrix of \(f\). This idea is not new but the paper includes details which would be useful for someone wishing to explore this approach. For example, suppose \( B,C\in M_{n}(K)\) and \(f(B)=f(C)=0\) then: \(B\ \)and \(C\) are similar (in particular \(B\) is similar to its transpose); \(K[B]\) is a maximal commutative subring of \(M_{n}(K)\) and hence a maximal subfield; and \(K[B]\) has the double centralizer property. The paper also shows how to construct the algebraic closure of a field as a subring of \(\mathrm{End}_{K}(V)\) where \(V\) is an infinite dimensional vector space over \(K\), and briefly indicates how companion matrices can be used to parametrise algebraic curves.
Reviewer: John D. Dixon (Ottawa)Introduction to linear algebrahttps://zbmath.org/1508.150012023-05-31T16:32:50.898670Z"Strang, Gilbert"https://zbmath.org/authors/?q=ai:strang.gilbertSee the reviews of the German edition in [Zbl 1042.15001] and the third English edition in [Zbl 1046.15001].Tensor algebra and analysis for engineers. With applications to differential geometry of curves and surfaceshttps://zbmath.org/1508.150022023-05-31T16:32:50.898670Z"Vannucci, Paolo"https://zbmath.org/authors/?q=ai:vannucci.paoloPublisher's description: In modern theoretical and applied mechanics, tensors and differential geometry are two almost essential tools. Unfortunately, in university courses for engineering and mechanics students, these topics are often poorly treated or even completely ignored. At the same time, many existing, very complete texts on tensors or differential geometry are so advanced and written in abstract language that discourage young readers looking for an introduction to these topics specifically oriented to engineering applications.
This textbook, mainly addressed to graduate students and young researchers in mechanics, is an attempt to fill the gap. Its aim is to introduce the reader to the modern mathematical tools and language of tensors, with special applications to the differential geometry of curves and surfaces in the Euclidean space. The exposition of the matter is sober, directly oriented to problems that are ordinarily found in mechanics and engineering. Also, the language and symbols are tailored to those usually employed in modern texts of continuum mechanics.
Though not exhaustive, as any primer textbook, this volume constitutes a coherent, self-contained introduction to the mathematical tools and results necessary in modern continuum mechanics, concerning vectors, 2nd- and 4th-rank tensors, curves, fields, curvilinear coordinates, and surfaces in the Euclidean space. More than 100 exercises are proposed to the reader, many of them complete the theoretical part through additional results and proofs. To accompany the reader in learning, all the exercises are entirely developed and solved at the end of the book.On the relationship between the matrix operators, vech and vecdhttps://zbmath.org/1508.150032023-05-31T16:32:50.898670Z"Nagakura, Daisuke"https://zbmath.org/authors/?q=ai:nagakura.daisukeSummary: We introduce a matrix operator, which we call ``vecd'' operator. This operator stacks up ``diagonals'' of a symmetric matrix. This operator is more convenient for some statistical analyses than the commonly used ``vech'' operator. We show an explicit relationship between the vecd and vech operators. Using this relationship, various properties of the vecd operator are derived. As applications of the vecd operator, we derive concise and explicit expressions of the Wald and score tests for equal variances of a multivariate normal distribution and for the diagonality of variance coefficient matrices in a multivariate generalized autoregressive conditional heteroscedastic (GARCH) model, respectively.The Moore-Penrose inverse of symmetric matrices with nontrivial equitable partitionshttps://zbmath.org/1508.150042023-05-31T16:32:50.898670Z"Alazemi, Abdullah"https://zbmath.org/authors/?q=ai:alazemi.abdullah"Anđelić, Milica"https://zbmath.org/authors/?q=ai:andelic.milica"Cvetković-Ilić, Dragana"https://zbmath.org/authors/?q=ai:cvetkovic-ilic.dragana-sSummary: In this paper we consider symmetric matrices that admit nontrivial equitable partitions. We determine some sufficient conditions for the quotient matrix of the Moore-Penrose inverse of the initial matrix to be equal to the Moore-Penrose inverse of its quotient matrix. We also study several particular cases when the computation of the Moore-Penrose inverse can be reduced significantly by establishing the formula for its computation based on the Moore-Penrose inverse of the quotient matrix. Among others we consider the adjacency matrix of a generalized weighted threshold graph.Representations for the weak group inversehttps://zbmath.org/1508.150052023-05-31T16:32:50.898670Z"Mosić, Dijana"https://zbmath.org/authors/?q=ai:mosic.dijana"Stanimirović, Predrag S."https://zbmath.org/authors/?q=ai:stanimirovic.predrag-sSummary: The aim of this paper is to provide new representations and characterizations of the weak group inverse. We give a relation between the weak group inverse and a corresponding nonsingular border matrix. Continuity of weak group inverse is studied. Several limit representations, integral representations and perturbation formulae for the weak group inverse are proposed. Various expressions for the weak group and core-EP inverses of upper block triangular matrix are established and their sign patterns are considered. The splitting method for finding weak group inverse is proved as well as the successive matrix squaring algorithm for computing weak group inverse. Also, the weak group inverse can be used in solving appropriate systems of linear equations.Matrix computations with the Omega calculushttps://zbmath.org/1508.150062023-05-31T16:32:50.898670Z"Francisco Neto, Antônio"https://zbmath.org/authors/?q=ai:neto.antonio-franciscoThe author considers an extension of the formalism of matrix analysis, referred here as omega matrix calculus. Let \(\textbf{a}=(a,\dots,a)\in\mathbb{C}^N\) with \(a\in\mathbb{N}_0\), \([n]=\{1,\dots,n\}\), \([x^k]f(x)=a_k\). If \(f(x)=\sum_{k\ge 0}a_kx^k\), with \(a_k=0\) for all but a finite number of coefficients, then \(f(x)\) is a polynomial and \(\deg f(x)\) is the highest power of \(x\).
Assume that \(X_{\mathbf{a}}\in\mathbb{C}^{N\times N}\) for each \(\mathbf{a}\in\mathbb{Z}^n\) and \(\lambda^{\mathbf{a}}=\lambda_1^{a_1}\cdots \lambda_n^{a_n}\). Linear operators act on absolutely convergent matrix valued expansions as follows:
\[
\mathop{\Omega}_{\ge}^\lambda\sum_{a_1=-\infty}^{\infty}\cdots\sum_{a_n=-\infty}^{\infty}X_{\mathbf{a}}\lambda^{\mathbf{a}}=\sum_{a_1=0}^{\infty}\cdots\sum_{a_n=0}^{\infty}X_{\mathbf{a}} ,
\]
and
\[
\mathop{\Omega}_{=}^\lambda\sum_{a_1=-\infty}^{\infty}\cdots\sum_{a_n=-\infty}^{\infty}X_{\mathbf{a}}\lambda^{\mathbf{a}}=X_{\mathbf{0}_n} ,
\]
in an open neighborhood of the complex circle \(|\lambda_i|=1\).
The author gives the basic definitions in the context of omega matrix calculus. In particular, he introduces the omega representations along with closed form expressions for the powers of the companion tridiagonal and triangular matrices.
Reviewer: Ali Morassaei (Zanjan)Simple proof of existence of a complex eigenvalue of a complex square matrix \(\ldots\)and yet another proof of the fundamental theorem of algebra with linear algebrahttps://zbmath.org/1508.150072023-05-31T16:32:50.898670Z"Magalhães, Luis T."https://zbmath.org/authors/?q=ai:magalhaes.luis-tThe author provides a new proof of the fundamental theorem of algebra (every polynomial of degree \(n\) with complex number coefficients has \(n\) complex roots) in the context of linear algebra using the Cauchy-Schwarz inequality and the Weierstrass extreme value theorem for continuous functions defined on nonempty compact subsets of \({\mathbb C}^n\).
Reviewer: Tin Yau Tam (Reno)Doubly stochastic and permutation solutions to \(AXA = XAX\) when \(A\) is a permutation matrixhttps://zbmath.org/1508.150082023-05-31T16:32:50.898670Z"Djordjević, Bogdan D."https://zbmath.org/authors/?q=ai:djordjevic.bogdan-dAssume that \(A\in\mathbb{C}^{n\times n}\) (or \(A \in \mathbb{R}^{n\times n}\)) is a nonzero square matrix. A Yang-Baxter-like matrix equation is
\[
AXA=XAX ,
\]
which has two obvious solutions, namely \(X=0\) and \(X=A\).
The author finds a sufficient and necessary condition for the solvability of the above matrix equation in the set of doubly stochastic and permutation matrices. In particular, all permutation matrix solutions are found.
Reviewer: Ali Morassaei (Zanjan)Upper bounds and lower bounds for the Frobenius norm of the solution to certain structured Sylvester equationhttps://zbmath.org/1508.150092023-05-31T16:32:50.898670Z"Fu, Chunhong"https://zbmath.org/authors/?q=ai:fu.chunhong"Chen, Jiajia"https://zbmath.org/authors/?q=ai:chen.jiajia"Xu, Qingxiang"https://zbmath.org/authors/?q=ai:xu.qingxiangSummary: This paper studies the Frobenius norm upper bounds and lower bounds of the unique solution to \(AX+XB=AC+DB\), where \(A\in\mathbb{C}^{m\times m}\) and \(B\in\mathbb{C}^{n\times n}\) are Hermitian positive definite, and \(C,D\in\mathbb{C}^{m\times n}\) are arbitrary. Some theoretical improvements of the known results are made. Numerical tests to illustrate the sharpness of the newly obtained upper bounds are dealt with, and numerical examples associated with the positivity of lower bounds are also provided.Notes on the quadraticity of linear combinations of a cubic matrix and a quadratic matrix that commutehttps://zbmath.org/1508.150102023-05-31T16:32:50.898670Z"Petik, Tuğba"https://zbmath.org/authors/?q=ai:petik.tugba"Özdemir, Halim"https://zbmath.org/authors/?q=ai:ozdemir.halim"Gökmen, Burak Tufan"https://zbmath.org/authors/?q=ai:gokmen.burak-tufanSummary: Let \(A_1\) and \(A_2\) be an \(\{\alpha_1, \beta_1, \gamma_1\}\)-cubic matrix and an \(\{\alpha_2, \beta_2\}\)-quadratic matrix, respectively, with \(\alpha_1 \neq \beta_1\), \(\beta_1 \neq \gamma_1\), \(\alpha_1 \neq \gamma_1\) and \(\alpha_2\neq \beta_2\). In this work, we characterize all situations in which the linear combination \(A_3 = a_1A_1 + a_2A_2\) with the assumption \(A_1A_2 = A_2A_1\) is an \(\{\alpha_3, \beta_3\}\)-quadratic matrix, where \(a_1\) and \(a_2\) are unknown nonzero complex numbers.Construct a matrix with prescribed Ritz values of the order less than or equal to threehttps://zbmath.org/1508.150112023-05-31T16:32:50.898670Z"Nazari, Alimohammad"https://zbmath.org/authors/?q=ai:nazari.alimohammad"Nezami, Atiyeh"https://zbmath.org/authors/?q=ai:nezami.atiyehSummary: The Ritz values of a matrix are the set of all the eigenvalues of the leading principal submatrices. In this paper, assuming that the set of Ritz values is given from the dimension of maximum three, we find a matrix such that the given set is its Ritz values. The conditions for the existing solution are also studied.Solutions of the matrix inequality \(AXA\overset{?}{\leq}A\) in some partial ordershttps://zbmath.org/1508.150122023-05-31T16:32:50.898670Z"Wang, Hongxing"https://zbmath.org/authors/?q=ai:wang.hongxing"Liu, Xiaoji"https://zbmath.org/authors/?q=ai:liu.xiaojiSummary: In this paper, we consider the matrix inequality \(AXA\overset{?}{\leq}A\) in the star, sharp and core partial orders, respectively. We get general solutions of those matrix inequalities and prove \(\mathcal{D}_*\subseteq\mathcal{S}_*\) and \(\mathcal{D}_{\#} \subseteq\mathcal{S}_{\#}\), although \(\mathcal{D}_{O\#}\nsubseteq \mathcal{S}_{O\#}\).On majorizations and singular valueshttps://zbmath.org/1508.150132023-05-31T16:32:50.898670Z"Huang, Hong"https://zbmath.org/authors/?q=ai:huang.hong.1|huang.hongSummary: The purpose of this paper is to establish some new weak majorizations concerning the singular values of the powers of a matrix, which generalize previous related results of \textit{R. B. Bapat} [Linear Multilinear Algebra 21, 211--214 (1987; Zbl 0651.15014)].On norms of iterations of \(\{0,1\}\)-matriceshttps://zbmath.org/1508.150142023-05-31T16:32:50.898670Z"Wei, Chun"https://zbmath.org/authors/?q=ai:wei.chun|wei.chun.1"Wen, Fan"https://zbmath.org/authors/?q=ai:wen.fanSummary: Let \(M\) be a \(b \times b\) nonzero \(\{0, 1\}\)-matrix. Let \(\rho(M)\) be its spectral radius and let \(\| M^n \|\) be the norm of its \(n\)-th iteration. In the case \(\rho(M) > 1\), we see from the spectral radius formula that \(\{\| M^n \|\}_{n = 1}^\infty\) tends to \(\infty\) exponentially as \(n \to \infty\). In the case \(\rho(M) = 1\), \(\{\| M^n \|\}_{n = 1}^\infty\) can be bounded or tend to \(\infty\) depending on \(M\). The fine behavior of this sequence is completely characterized in the present paper.Tensor slice rank and Cayley's first hyperdeterminanthttps://zbmath.org/1508.150152023-05-31T16:32:50.898670Z"Amanov, Alimzhan"https://zbmath.org/authors/?q=ai:amanov.alimzhan"Yeliussizov, Damir"https://zbmath.org/authors/?q=ai:yeliussizov.damirThe authors study \(d\)-dimensional tensors \(T \in V^{\otimes d}\), i.e., multilinear elements of a vector space \(V \otimes \ldots \otimes V\). For a base \((e_i)\) each tensor has the coordinate representation
\[
T = \sum_{i_1, \dots , i_d} T(i_1, \dots , i_d) \, \cdot \, e_{i_1} \otimes \ldots \otimes e_{i_d}.
\]
A \textit{simple tensor} is a non-zero tensor that is completely factorizable:
\[
T (i_1, \dots , i_d) = v_1(i_1) \cdot \ldots \cdot v_d(i_d), \, \, \forall i_1, \dots i_d \in \{1 \dots n\}.
\]
Every tensor \(T\) can be expressed as a sum of simple tensors, and the minimum number of simple tensors that sum to \(T\) is known as the rank of \(T\). The main rank-properties of \(T\) are considered as follows:
\begin{itemize}
\item The \textit{slice rank} \(\operatorname{srank}(T)\) is defined if the simple tensors are decomposable along some coordinates:
\[
T(i_1, \dots , i_d) = v(i_k) \, \cdot \, T(i_1, \dots , i_{k-1}, \, \, i_{k+1}, \dots, i_d), \forall i_1, \dots i_d \in \{1 \dots n\}.
\]
\item The \textit{partition rank} \(\operatorname{prank}(T)\) plays a role if simple tensors are decomposable with respect to a partitions of coordinates, i.e.,
\[
T(i_1, \dots , i_d) = T_1(i_{a_1}, \dots , i_{a_d}) \cdot T_2(i_{b_1}, \dots , i_{b_{d-k}}), \, \, \forall i_1, \dots i_d \in \{1 \dots n\}
\]
for some partitions \(\{a_1 < \dots < a_k\} \cup \{b_1 < \dots < b_{d-k}\}\) of the set \(\{ 1, \dots, d \}\).
\item If one partition is of odd size, but does not contain the element \(1\), the authors define finally the \textit{odd partition rank} \(\operatorname{oprank}(T)\) of \(T\).
\end{itemize}
Along with these features the authors study the (first) hyperdeterminant of Cayley:
\[
\det ( T ) = \sum_{\sigma_2, \dots, \sigma_d \in S_n} \operatorname{sgn}(\sigma_2 \dots \sigma_d) \cdot \prod_{i=1}^n T(i, \sigma_2(i), \dots, \sigma_d(i) ).
\]
The hyperdeterminant is linear, skew-symmetric and invariant under \(\operatorname{GL}(n)^d\). Using sum and product formulae for \(\det(T)\) the authors prove some theorems for the different versions of ranks, as for example:
\begin{itemize}
\item \(\operatorname{oprank}(T) < n \Rightarrow \det(T) = 0\), i.e., \(\det(T) \neq 0 \Rightarrow \operatorname{oprank}(T) = n\);
\item \(\operatorname{rank}(T) \geq n\), and if \(d\) is even then \(\operatorname{srank}(T) = n\);
\item A tensor over a field of characteristic \(p > 0\) with \(\operatorname{prank}(T) < \frac{n}{p-1}\) has \(\det(T) = 0\); thus \(\det(T) \neq 0 \Rightarrow \operatorname{prank}(T) \geq \frac{n}{p-1}\).
\end{itemize}
As an application the authors consider \(d\)-colored sum-ordered sets in \(\mathbb{F}_p^n\) with \(p\) prime and \(d > 2\). It is shown that its size \(N\) is \(N= \mathcal{O}(\gamma^n)\) for a constant \(1 \leq \gamma < p\).
Reviewer: Lienhard Wimmer (Isny)New S-type inclusion theorems for the M-eigenvalues of a 4th-order partially symmetric tensor with applicationshttps://zbmath.org/1508.150162023-05-31T16:32:50.898670Z"He, Jun"https://zbmath.org/authors/?q=ai:he.jun"Liu, Yanmin"https://zbmath.org/authors/?q=ai:liu.yanmin"Xu, Guangjun"https://zbmath.org/authors/?q=ai:xu.guangjunSummary: Two new S-type inclusion theorems for the M-eigenvalues of a 4th-order partially symmetric tensor are established. These inclusion theorems provide upper bounds for the M-spectral radius of 4th-order partially symmetric tensors. Finally, the M-positive definiteness for 4th-order partially symmetric tensors are further studied and two sufficient conditions are also presented based on the two new S-type M-eigenvalue inclusion theorems.Matrix completion with sparse measurement errorshttps://zbmath.org/1508.150172023-05-31T16:32:50.898670Z"Petrov, Sergey"https://zbmath.org/authors/?q=ai:petrov.sergey-s|petrov.sergey-v"Zamarashkin, Nikolai"https://zbmath.org/authors/?q=ai:zamarashkin.nikolai-lSummary: The problem of completion of low-rank matrices is considered in a special setting, where each element of the matrix may be erroneous with a limited probability. Although such a perturbation is extremely sparse on a given mask of known elements, it is not incoherent and may cause instabilities in the commonly used projected gradient method. A new iterative method is proposed that is insensitive to rare observation errors and is more stable for ill-conditioned solutions. The method can also be used for finding a matrix approximation in the format of a sum of a low-rank and a sparse matrix.Improving formulas for the eigenvalues of finite block-Toeplitz tridiagonal matriceshttps://zbmath.org/1508.150182023-05-31T16:32:50.898670Z"Abderramán Marrero, J."https://zbmath.org/authors/?q=ai:abderraman-marrero.j|abderraman-marrero.jesus"Aiat Hadj, D. A."https://zbmath.org/authors/?q=ai:aiat-hadj.driss-ahmedSummary: After a short overview, improvements (based on the Kronecker product) are proposed for the eigenvalues of \((N \times N)\) block-Toeplitz tridiagonal (block-TT) matrices with \((K \times K)\) matrix-entries, common in applications. Some extensions of the spectral properties of the Toeplitz-tridiagonal matrices are pointed-out. The eigenvalues of diagonalizable symmetric and skew-symmetric block-TT matrices are studied. Besides, if certain matrix square-root is well-defined, it is proved that every block-TT matrix with commuting matrix-entries is isospectral to a related symmetric block-TT one. Further insight about the eigenvalues of hierarchical Hermitian block-TT matrices, of use in the solution of PDEs, is also achieved.Inverses of \(k\)-Toeplitz matrices with applications to resonator arrays with multiple receivershttps://zbmath.org/1508.150192023-05-31T16:32:50.898670Z"Alberto, José"https://zbmath.org/authors/?q=ai:alberto.jose"Brox, Jose"https://zbmath.org/authors/?q=ai:brox.joseSummary: We find closed-form algebraic formulas for the elements of the inverses of tridiagonal 2- and 3-Toeplitz matrices which are symmetric and have constant upper and lower diagonals. These matrices appear, respectively, as the impedance matrices of resonator arrays in which a receiver is placed over every 2 or 3 resonators. Consequently, our formulas allow to compute the currents of a wireless power transfer system in closed form, allowing for a simple, exact and symbolic analysis thereof. Small numbers are chosen for illustrative purposes, but the elementary linear algebra techniques used can be extended to \(k\)-Toeplitz matrices of this special form with \(k\) arbitrary, hence resonator arrays with a receiver placed over every \(k\) resonators can be analysed in the same way.Upper Hessenberg and Toeplitz Bohemianshttps://zbmath.org/1508.150202023-05-31T16:32:50.898670Z"Chan, Eunice Y. S."https://zbmath.org/authors/?q=ai:chan.eunice-y-s"Corless, Robert M."https://zbmath.org/authors/?q=ai:corless.robert-m"Gonzalez-Vega, Laureano"https://zbmath.org/authors/?q=ai:gonzalez-vega.laureano"Sendra, J. Rafael"https://zbmath.org/authors/?q=ai:sendra.juan-rafael"Sendra, Juana"https://zbmath.org/authors/?q=ai:sendra.juana"Thornton, Steven E."https://zbmath.org/authors/?q=ai:thornton.steven-eSummary: A set of matrices with entries from a fixed finite population \(P\) is called ``Bohemian''. The mnemonic comes from BOunded HEight Matrix of Integers, BOHEMI, and although the population \(P\) need not be solely made up of integers, it frequently is. In this paper we look at Bohemians, specifically those with population \(\{- 1, 0, + 1 \}\) and sometimes other populations, for instance \(\{0, 1, i, - 1, - i \}\). More, we specialize the matrices to be upper Hessenberg Bohemian. We then study the characteristic polynomials of these matrices, and their height, that is the infinity norm of the vector of monomial basis coefficients. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the number of normal matrices and the number of stable matrices in these families.Quantitative results for banded Toeplitz matrices subject to random and deterministic perturbationshttps://zbmath.org/1508.150212023-05-31T16:32:50.898670Z"O'Rourke, Sean"https://zbmath.org/authors/?q=ai:orourke.sean"Wood, Philip Matchett"https://zbmath.org/authors/?q=ai:wood.philip-matchettThe authors study the behavior of the eigenvalues of a non-normal matrix perturbed both by a deterministic matrix and a random matrix. In particular, banded Toeplitz matrices are involved. Several asymptotic and non-asymptotic results are provided.
Reviewer: Yaogan Mensah (Lomé)On Hadamard powers of positive semi-definite matriceshttps://zbmath.org/1508.150222023-05-31T16:32:50.898670Z"Baslingker, Jnaneshwar"https://zbmath.org/authors/?q=ai:baslingker.jnaneshwar"Dan, Biltu"https://zbmath.org/authors/?q=ai:dan.biltuLet denote by \(\mathcal{P}^{+}_{n}\) the set of positive semi-definite matrices of size \(n\times n\) with non-negative entries. For any \(A\in \mathcal{P}^{+}_{n}\) and \(\alpha>0\) the \(\alpha\)-Hadamard power of \(A=[a_{i,j}]_{i,j=1}^{n}\) is the matrix \(A^{\alpha}= [a^{\alpha}_{i,j}]_{i,j=1}^{n}\). Let us consider the set \(S_{A}= \{\alpha \geq 0: A^{\alpha}\in \mathcal{P}^{+}_{n}\}\). According to the Schur product theorem (see [\textit{R. A. Horn} and \textit{C. R. Johnson}, Topics in matrix analysis. Cambridge etc.: Cambridge University Press (1991; Zbl 0729.15001)]), every non-negative number belongs to \(S_{A}\) for every \(A\in \mathcal{P}^{+}_{n}\).
In [\textit{C. H. FitzGerald} and \textit{R. A. Horn}, J. Math. Anal. Appl. 61, 633--642 (1977; Zbl 0406.15006)], it is proved that \(\bigcap_{A \in \mathcal{P}^{+}_{n}} S_{A}= \{0,1, \dots, n-3\} \cup [n-2, \infty)\), i.e., \(n - 2\) is the least number for which \( A^{\alpha}\in \mathcal{P}^{+}_{n}\) for every \(A\in \mathcal{P}^{+}_{n}\) and every \(\alpha\geq n-2\). The description of the set \(S_{A}\) for a fixed matrix \(A\in \mathcal{P}^{+}_{n}\) is an interesting problem. In all examples considered in the literature such a set turns out to be the union of a finite set and a semi-infinite interval.
In this paper, assuming that \(n\geq4\), the authors provide an example where \(S_{A}\) has at least two interval components. Indeed, for \(n\) distinct positive real numbers \(x_{k}, k=1, 2, \dots, n\), let \(A_{n}= [1 + x_{i} x_{j}]_{i,j= 1}^{n} \in \mathcal{P}^{+}_{n}\). \textit{T. Jain} [Linear Algebra Appl. 528, 147--158 (2017; Zbl 1398.15038)] proves that \(S_{A_ {n}}= \{0,1, \dots, n-3\} \cup [n-2, \infty)\). For \(n\geq4\) and \(k\in \{2, 3, \dots, n-2\}\) let us introduce the matrix \(A_{n, k,\epsilon} = A_{n} + \epsilon R_{n,k}\), where \(R_{n,k}\) is a matrix of size \(n\times n\) with \(1\) in the last \(k\) diagonal entries and \(0\) otherwise. Thus, for a small enough non-negative real number \(\epsilon\), depending on \(A_{n}\), the authors prove that \(S_{A_{n, k,\epsilon} }\) has at least \(k\) interval components, one around each of the integers from \(n-k-1\) to \(n-3\) and the last one is the semi-infinite interval containing \(n-2\). Notice that \(S_{A_{n,k,\epsilon} }\) can have more than \(k\) interval components but all of such extra intervals should be contained in \([n-k-2, n-2]\).
Notice that for any \(A\in \mathcal{P}^{+}_{n}\), the number of interval components of \(S_{A}\) is at most \(n!\).
Reviewer: Francisco Marcellán (Leganes)Generalized row substochastic matrices and majorizationhttps://zbmath.org/1508.150232023-05-31T16:32:50.898670Z"Mohammadhasani, Ahmad"https://zbmath.org/authors/?q=ai:mohammadhasani.ahmad"Sayyari, Yamin"https://zbmath.org/authors/?q=ai:sayyari.yaminSummary: The square and real matrix \(A\) is called a generalized row substochastic matrix, if the sum of the absolute values of the entries in each row is less than or equal to one. Let \(x ,y\in\mathbb{R}^n\). We say that \(x\) is right \(B\)-majorized (resp. left \(B\)-majorized) by \(y\), denoted by \(x\prec_{rB} y\) (\(x\prec_{lB} y\)), if there exists a substochastic matrix \(D\), such that \(x = yD\) (resp. \(x = Dy\)). In this article, we have found all the vectors such as \(x\) that \(x\) is right \(B\)-majorized (resp. left \(B\)-majorized) by \(y\), for all row vector \(y\) (resp. column vector). Also, we show \(x\sim_{lB}y\) if and only if \(\|x\|_\infty = \|y\|_\infty\) and prove \(x \sim_{rB} y\) if and only if \(\|x\|_1 =\|y\|_1\). We have also created conditions in which the left \(B\)-majorization is equivalent to the left majorization, and created conditions in which the right \(B\)-majorization is equivalent to the right majorization.Nonuniversality of fluctuations of outliers for Hermitian polynomials in a complex Wigner matrix and a spiked diagonal matrixhttps://zbmath.org/1508.150242023-05-31T16:32:50.898670Z"Capitaine, Mireille"https://zbmath.org/authors/?q=ai:capitaine.mireilleSummary: We study the fluctuations associated to the a.s. convergence of the outliers established by Belinschi-Bercovici-Capitaine of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the nonuniversality phenomenon established by
\textit{M. Capitaine} et al. [Ann. Probab. 37, No. 1, 1--47 (2009; Zbl 1163.15026); Ann. Inst. Henri Poincaré, Probab. Stat. 48, No. 1, 107--133 (2012; Zbl 1237.60007)]
for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The result is described using the operator-valued subordination functions of free probability theory.Explicit formulas concerning eigenvectors of weakly non-unitary matriceshttps://zbmath.org/1508.150252023-05-31T16:32:50.898670Z"Dubach, Guillaume"https://zbmath.org/authors/?q=ai:dubach.guillaumeSummary: We investigate eigenvector statistics of the Truncated Unitary ensemble \(\mathrm{TUE}(N,M)\) in the weakly non-unitary case \(M=1\), that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when \(M\geq N\), suggesting possible extensions to \(\mathrm{TUE}(N,M)\) for all values of \(M\).On the Cartan decomposition for classical random matrix ensembleshttps://zbmath.org/1508.150262023-05-31T16:32:50.898670Z"Edelman, Alan"https://zbmath.org/authors/?q=ai:edelman.alan-s"Jeong, Sungwoo"https://zbmath.org/authors/?q=ai:jeong.sungwooSummary: We complete Dyson's dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan's symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely one coordinate system on the symmetric space, albeit a beautiful one. However, other matrix factorizations, especially the generalized singular value decomposition from numerical linear algebra, reveal themselves to be perfectly valid coordinate systems that one symmetric space can lead to many classical random matrix theories. We establish the connection between this numerical linear algebra viewpoint and the theory of generalized Cartan decompositions. This, in turn, allows us to produce yet more random matrix theories from a single symmetric space. Yet, again, these random matrix theories arise from matrix factorizations, though ones that we are not aware have appeared in the literature.
{\copyright 2022 American Institute of Physics}Spectral statistics of Dirac ensembleshttps://zbmath.org/1508.150272023-05-31T16:32:50.898670Z"Khalkhali, Masoud"https://zbmath.org/authors/?q=ai:khalkhali.masoud"Pagliaroli, Nathan"https://zbmath.org/authors/?q=ai:pagliaroli.nathanSummary: In this paper, we find spectral properties in the large \(N\) limit of Dirac operators that come from random finite noncommutative geometries. In particular, for a Gaussian potential, the limiting eigenvalue spectrum is shown to be universal, regardless of the geometry, and is given by the convolution of the semicircle law with itself. For simple non-Gaussian models, this convolution property is also evident. In order to prove these results, we show that a wide class of multi-trace multimatrix models have a genus expansion.
{\copyright 2022 American Institute of Physics}Commutators of random matrices from the unitary and orthogonal groupshttps://zbmath.org/1508.150282023-05-31T16:32:50.898670Z"Palheta, Pedro H. S."https://zbmath.org/authors/?q=ai:palheta.pedro-h-s"Barbosa, Marcelo R."https://zbmath.org/authors/?q=ai:barbosa.marcelo-r"Novaes, Marcel"https://zbmath.org/authors/?q=ai:novaes.marcelSummary: We investigate the statistical properties of \(C = uvu^{-1}v^{-1} \), when \(u\) and \(v\) are independent random matrices, uniformly distributed with respect to the Haar measure of the groups \(U(N)\) and \(O(N)\). An exact formula is derived for the average value of power sum symmetric functions of \(C\), and also for products of the matrix elements of \(C\), similar to Weingarten functions. The density of eigenvalues of \(C\) is shown to become constant in the large-\(N\) limit, and the first \(N^{-1}\) correction is found.
{\copyright 2022 American Institute of Physics}Eigenvalue distribution of large random matrices arising in deep neural networks: orthogonal casehttps://zbmath.org/1508.150292023-05-31T16:32:50.898670Z"Pastur, L."https://zbmath.org/authors/?q=ai:pastur.leonid-andreevich|pastur.luc-rSummary: This paper deals with the distribution of singular values of the input-output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem has been considered in the several recent studies of the field for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Based on a free probability argument, it was claimed in those papers that, in the limit of infinite width (matrix size), the singular value distribution of the Jacobian coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. In this paper, we justify the claim for random Haar distributed weight matrices and Gaussian biases. This, in particular, justifies the validity of the mean field approximation in the infinite width limit for the deep untrained neural networks and extends the macroscopic universality of random matrix theory to this new class of random matrices.
{\copyright 2022 American Institute of Physics}Power spectrum of the circular unitary ensemblehttps://zbmath.org/1508.150302023-05-31T16:32:50.898670Z"Riser, Roman"https://zbmath.org/authors/?q=ai:riser.roman"Kanzieper, Eugene"https://zbmath.org/authors/?q=ai:kanzieper.eugeneThe sequence of \(N\) ordered energy levels of a given quantum system can naturally be interpreted as a discrete time random process, the times being the indices of the ordered energy levels.
A usual key assumption is then the independence of the mean values on their index (``stationarity''). Without making such a restrictive assumption, the authors first develop a nonperturbative theory to compute the Fourier transform of the covariance matrix (``power spectrum''). Applying it to the circular unitary ensemble (CUE) of random matrices, they express this power spectrum with the tau function of the most general Painlevé function, the sixth one, thus defining a universal probability law.
The additional interest of this paper is the representation of the above result by three different, equivalent methods: a differential equation (the sixth Painlevé), a Fredholm determinant, a Toeplitz determinant.
The paper, which avoids unnecessary esoteric terminology, is clearly written and very pleasant to read.
Reviewer: Robert Conte (Gif-sur-Yvette)Real tensor eigenvalue/vector distributions of the Gaussian tensor model via a four-Fermi theoryhttps://zbmath.org/1508.150312023-05-31T16:32:50.898670Z"Sasakura, Naoki"https://zbmath.org/authors/?q=ai:sasakura.naokiSummary: Eigenvalue distributions are important dynamic quantities in matrix models, and it is an interesting challenge to study corresponding quantities in tensor models. We study real tensor eigenvalue/vector distributions for real symmetric order-three random tensors with a Gaussian distribution as the simplest case. We first rewrite this problem as the computation of a partition function of a four-fermi theory with \(R\) replicated fermions. The partition function is exactly computed for some small-\(N\), \(R\) cases, and is shown to precisely agree with Monte Carlo simulations. For large-\(N\), it seems difficult to compute it exactly, and we apply an approximation using a self-consistency equation for two-point functions and obtain an analytic expression. It turns out that the real tensor eigenvalue distribution obtained by taking \(R = 1/2\) is simply the Gaussian within this approximation. We compare the approximate expression with Monte Carlo simulations, and find that, if an extra overall factor depending on \(N\) is multiplied to the the expression, it agrees well with the Monte Carlo results. It is left for future study to improve the approximation for large-\(N\) to correctly derive the overall factor.Toeplitz band matrices with small random perturbationshttps://zbmath.org/1508.150322023-05-31T16:32:50.898670Z"Sjöstrand, Johannes"https://zbmath.org/authors/?q=ai:sjostrand.johannes"Vogel, Martin"https://zbmath.org/authors/?q=ai:vogel.martinSummary: We study the spectra of \(N \times N\) Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime \(N \gg 1\). We prove a probabilistic Weyl law, which provides a precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on \(N\), with probability sub-exponentially (in \(N)\) close to 1. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most \(\mathcal{O} ( N^{-1 + \varepsilon})\), for all \(\varepsilon > 0\), to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.Generalized tournament matrices with the same principal minorshttps://zbmath.org/1508.150332023-05-31T16:32:50.898670Z"Boussaïri, A."https://zbmath.org/authors/?q=ai:boussairi.abderrahim"Chaïchaâ, A."https://zbmath.org/authors/?q=ai:chaichaa.abdelhak"Chergui, B."https://zbmath.org/authors/?q=ai:chergui.brahim"Lakhlifi, S."https://zbmath.org/authors/?q=ai:lakhlifi.soufianeA tournament matrix \(M=(m_{ij})\) is the adjacency matrix of a tournament, that is, for all \(i,j\), \(m_{ij}\) is zero or one and \(M+M^t=J-I\), where \(t\) is the transpose, \(J\) is the all ones matrix and \(I\) is the identity matrix. A generalized tournament matrix \(M=(m_{ij})\) is a matrix such that for all \(i,j\), \(m_{ij}\in [0,1]\) and \(M+M^t=J-I\).
Motivated by several papers, the present authors characterize generalized tournament matrices with the same principal minors of orders 2, 3 and 4.
For this purpose, they define the operation \textit{clan reversal} (see Section 2) and prove the main result (Theorem 2.4).
This paper provides an important generalization of previous results.
Reviewer: Rosário Fernandes (Lisboa)Jordan \(\mathcal{G}_n\)-derivations on path algebrashttps://zbmath.org/1508.160522023-05-31T16:32:50.898670Z"Adrabi, Abderrahim"https://zbmath.org/authors/?q=ai:adrabi.abderrahim"Bennis, Driss"https://zbmath.org/authors/?q=ai:bennis.driss"Fahid, Brahim"https://zbmath.org/authors/?q=ai:fahid.brahimSummary: Recently, Brešar's Jordan \(\{g,h\}\)-derivations have been investigated on triangular algebras [\textit{M. Brešar}, Linear Multilinear Algebra 64, No. 11, 2199--2207 (2016; Zbl 1372.16046)]. As a first aim of this paper, we extend this study to an interesting general context. Namely, we introduce the notion of Jordan \(\mathcal{G}_n\)-derivations, with \(n \geq 2\), which is a natural generalization of Jordan \(\{g,h\}\)-derivations. Then, we study this notion on path algebras. We prove that, when \(n > 2\), every Jordan \(\mathcal{G}_n\)-derivation on a path algebra is a \(\{g,h\}\)-derivation. However, when \(n = 2\), we give an example showing that this implication does not hold true in general. So, we characterize when it holds. As a second aim, we give a positive answer to a variant of Lvov-Kaplansky conjecture on path algebras. Namely, we show that the set of values of a multi-linear polynomial on a path algebra \(KE\) is either \(\{0\},\, KE\) or the space spanned by paths of a length greater than or equal to \(1\).Derivations of upper triangular matrix semiringshttps://zbmath.org/1508.160582023-05-31T16:32:50.898670Z"Vladeva, D. I."https://zbmath.org/authors/?q=ai:vladeva.dimitrinka-iSummary: In this paper, the author gives a description of the derivatives in \(\mathrm{UTM}_n(S)\), the semiring of upper triangular matrices over an additively idempotent semiring \(S\). The main result states that an arbitrary derivation in the semiring \(\mathrm{UTM}_n(S)\) is a linear combination of finite number of well-known derivations.Derivations of octonion matrix algebrashttps://zbmath.org/1508.170052023-05-31T16:32:50.898670Z"Petyt, Harry"https://zbmath.org/authors/?q=ai:petyt.harrySummary: It is well-known that the exceptional Lie algebras \(\mathfrak{f_4}\) and \(\mathfrak{g_2}\) arise from the octonions as the derivation algebras of the \(3\times 3\) hermitian and \(1\times 1\) antihermitian matrices, respectively. Inspired by this, we compute the derivation algebras of the spaces of hermitian and antihermitian matrices over an octonion algebra in all dimensions.On nilpotent generators of the symplectic Lie algebrahttps://zbmath.org/1508.170082023-05-31T16:32:50.898670Z"Chistopolskaya, Alisa"https://zbmath.org/authors/?q=ai:chistopolskaya.alisaThe paper deals with the case of the symplectic Lie algebra over an algebraically closed field \(\mathbb{K}\) of characteristic zero. Motivated by the works of \textit{D. H. Collingwood} and \textit{W. M. McGovern} [Nilpotent orbits in semisimple Lie algebras. New York, NY: Van Nostrand Reinhold Company (1993; Zbl 0972.17008)], the idea of the proof here is to reduce the problem to the case when \(X\) is the lowest weight vector in the adjoint representation there. Specific examples of nilpotent matrices generating \(\mathfrak{sp}_{2n}(\mathbb{K})\) are given. It is first conjectured that for any nonzero nilpotent element \(X\) in an arbitrary simple Lie algebra \(\mathfrak{g}\), there exists another nonzero nilpotent element \(Y\), such that \(X\) and \(Y\) generate \(\mathfrak{g}\). It is finally conjectured that for all \(n\geq2\) one can find three one-parameter subgroups \(U_{1}, U_{2}, U_{3}\) of symplectomorphisms of \(\mathbb{A}^{2n}\) such that the group \(H=\{U_{1}, U_{2}, U_{3}\}\) acts infinitely transitively on \(\mathbb{A}^{2n}\). This paper would be of great interest to advanced researchers in Lie algebra looking to solve any existing open problem.
Reviewer: Jervin Zen Lobo (Mapusa)Tridiagonal pairs of \(q\)-Serre type and their linear perturbationshttps://zbmath.org/1508.170212023-05-31T16:32:50.898670Z"Karan, Aayush"https://zbmath.org/authors/?q=ai:karan.aayushSummary: A tridiagonal pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on the eigenspaces of the other in a block-tridiagonal fashion. We consider a tridiagonal pair \((A, A^\ast)\) of \(q\)-Serre type; for such a pair the maps \(A\) and \(A^\ast\) satisfy the \(q\)-Serre relations. There is a linear map \(K\) in the literature that is used to describe how \(A\) and \(A^\ast\) are related. We investigate a pair of linear maps \(B = A\) and \(B^\ast = t A^\ast +(1 - t) K\), where \(t\) is any scalar. Our goal is to find a necessary and sufficient condition on \(t\) for the pair \((B, B^\ast)\) to be a tridiagonal pair. We show that \((B, B^\ast)\) is a tridiagonal pair if and only if \(t \neq 0\) and \(P(t (q - q^{- 1})^{- 2}) \neq 0\), where \(P\) is a certain polynomial attached to \((A, A^\ast)\) called the Drinfel'd polynomial.The group generated by Riordan involutionshttps://zbmath.org/1508.200602023-05-31T16:32:50.898670Z"Luzón, Ana"https://zbmath.org/authors/?q=ai:luzon.ana"Morón, Manuel A."https://zbmath.org/authors/?q=ai:moron.manuel-alonso"Prieto-Martínez, L. Felipe"https://zbmath.org/authors/?q=ai:prieto-martinez.luis-felipeSummary: We prove that any element in the group generated by the Riordan involutions is the product of at most four of them. We also give a description of this subgroup as a semidirect product of a special subgroup of the commutator subgroup and the Klein four-group.On Hedenmalm-Shimorin type inequalitieshttps://zbmath.org/1508.300032023-05-31T16:32:50.898670Z"Han, Yong"https://zbmath.org/authors/?q=ai:han.yong"Qiu, Yanqi"https://zbmath.org/authors/?q=ai:qiu.yanqi"Wang, Zipeng"https://zbmath.org/authors/?q=ai:wang.zipengThe authors directly prove an Hedenmalm-Shimorin inequality for short antidiagonals. The main results are given in the following theorems.
Theorem 1.1. For any infinite complex-valued matrix \(M=\{m_{j,k}\}_{j,k=1}^{\infty}\), we have
\[
\sum_{l=2}^{\infty}s^l\left|\sum_{j+k=l}\frac{m_{j,k}}{\sqrt{jk}}\right|^2\leq(\|M\|_{1\to2}^2+\|M\|_{2\to\infty}^2)s\log\frac{1}{1-s},\;\; 0\leq s\leq1,
\]
provided that the two quantities defined as follows
\[
\|M\|_{1\to2}^2=\sup_{k\geq1}\sum_{j=1}^{\infty}|m_{j,k}|^2\;\;\text{and}\;\;\|M\|_{2\to\infty}^2= \sup_{j\geq1}\sum_{k=1}^{\infty}|m_{j,k}|^2
\]
are both finite.
Theorem 1.3. Let \(\{m_{i,j,k}\}_{i,j,k=1}^{\infty}\) be a sequence of complex numbers such that
\[
\sup_{j,k\geq1}\sum_{i=1}^{\infty}|m_{i,j,k}|^2+\sup_{i,k\geq1}\sum_{j=1}^{\infty}|m_{i,j,k}|^2+ \sup_{i,j\geq1}\sum_{k=1}^{\infty}|m_{i,j,k}|^2\leq1.
\]
Then
\[
\sum_{l=3}^{\infty}\frac{s^l}{l+1}\left|\sum_{i+j+k=l}\frac{m_{i,j,k}}{\sqrt{ijk}}\right|^2\leq\frac{s}{2}\left(\log\frac{1}{1-s}\right)^2, \;\;0\leq s\leq1.
\]
Reviewer: Dmitri V. Prokhorov (Saratov)Adaptive Fourier decomposition of slice regular functionshttps://zbmath.org/1508.300912023-05-31T16:32:50.898670Z"Jin, Ming"https://zbmath.org/authors/?q=ai:jin.ming"Leong, Ieng Tak"https://zbmath.org/authors/?q=ai:leong.iengtak|leong.ieng-tak"Qian, Tao"https://zbmath.org/authors/?q=ai:qian.tao"Ren, Guangbin"https://zbmath.org/authors/?q=ai:ren.guangbinSummary: In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator \(\mathcal{S}_a\) with the decomposition process
\[
f = e_a\langle f, e_a\rangle + B_a\ast\mathcal{S}_a f,
\]
where \(e_a\) denotes the slice normalized Szegö kernel and \(B_a\) the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka-Malmquist orthonormal system.Biquaternion extensions of analytic functionshttps://zbmath.org/1508.300942023-05-31T16:32:50.898670Z"Oba, Roger M."https://zbmath.org/authors/?q=ai:oba.roger-mSummary: This paper presents a three-step program for extension of functions of complex analysis to the biquaternions by means of Cauchy's integral formula: I. \textit{Investigate biquaternion bases consisting of roots of} \(-1\). A complex valued \textit{standard function} (standardization factor) determines roots of \(-1\). A root of \(-1\) with a non-zero imaginary part, can uniquely determine a biquaternion ortho-standard basis. II. \textit{A single reference basis element determines two subspaces, one the span of scalars and the reference element, the other pure vector biquaternions orthogonal to the reference.} The subspaces represent the distinct parts of the generalized Cayley-Dickson form. The Peirce decomposition projects into two subspaces: one is the span of the related idempotents and the other of the nilpotents. III. \textit{Using invertible elements in each of these subspaces, biquaternion functional extensions of holomorphic functions follow by Cauchy's integral formula.} Extensions retain analyticity in each biquaternion component. Cauchy integral formula uses separate idempotent and nilpotent representations of biquaternion reciprocals to define holomorphic function extensions. The Peirce projections allow extension to all viable biquaternions.Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalueshttps://zbmath.org/1508.341212023-05-31T16:32:50.898670Z"Yu, Jianduo"https://zbmath.org/authors/?q=ai:yu.jianduo"Li, Chuanzhong"https://zbmath.org/authors/?q=ai:li.chuanzhong|li.chuanzhong.1"Zhu, Mengkun"https://zbmath.org/authors/?q=ai:zhu.mengkun"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1Summary: We discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with a singularly perturbed Gaussian weight \(w(z) = |z|^\alpha\exp\left(-z^2 - t/z^2\right)\), \(z\in\mathbb{R}\), \(t > 0\), \(\alpha > 1\). Based on the ladder operator approach, two auxiliary quantities are defined. We show that the auxiliary quantities and the recurrence coefficients satisfy some equations with the aid of three compatibility conditions, which will be used to derive the Riccati equations and Painlevé III. We show that the Hankel determinant has an integral representation involving a particular \(\sigma\)-form of Painlevé III and to calculate the asymptotics of the Hankel determinant under a suitable double scaling, i.e., \(n\rightarrow\infty\) and \(t\rightarrow0\) such that \(s = (2n + 1 + \lambda)t\) is fixed, where \(\lambda\) is a parameter with \(\lambda := (\alpha\mp 1)/2\). The asymptotic behaviors of the Hankel determinant for large \(s\) and small \(s\) are obtained, and Dyson's constant is recovered here. They have generalized the results in the literature [\textit{C. Min} et al., Nucl. Phys., B 936, 169--188 (2018; Zbl 1400.33020)] where \(\alpha = 0\). By combining the Coulomb fluid method with the orthogonality principle, we obtain the asymptotic expansions of the recurrence coefficients, which are applied to derive the relationship between second order differential equations satisfied by our monic orthogonal polynomials and the double-confluent Heun equations as well as to calculate the smallest eigenvalue of the large Hankel matrices generated by the above weight. In particular, when \(\alpha = t = 0\), the asymptotic behavior of the smallest eigenvalue for the classical Gaussian weight \(\exp(-z^2)\) [\textit{G. Szegö}, Trans. Am. Math. Soc. 40, 450--461 (1936; Zbl 0015.34603; JFM 62.0405.01)] is recovered.
{\copyright 2022 American Institute of Physics}Glimpses of soliton theory. The algebra and geometry of nonlinear PDEshttps://zbmath.org/1508.350012023-05-31T16:32:50.898670Z"Kasman, Alex"https://zbmath.org/authors/?q=ai:kasman.alexPublisher's description: Solitons are nonlinear waves which behave like interacting particles. When first proposed in the 19th century, leading mathematical physicists denied that such a thing could exist. Now they are regularly observed in nature, shedding light on phenomena like rogue waves and DNA transcription. Solitons of light are even used by engineers for data transmission and optical switches. Furthermore, unlike most nonlinear partial differential equations, soliton equations have the remarkable property of being exactly solvable. Explicit solutions to those equations provide a rare window into what is possible in the realm of nonlinearity.
Glimpses of Soliton Theory reveals the hidden connections discovered over the last half-century that explain the existence of these mysterious mathematical objects. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant explanation of something seemingly miraculous.
Assuming only multivariable calculus and linear algebra, the book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass \(\wp\)-functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Hierarchy, and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians.
Notable features of the book include: careful selection of topics and detailed explanations to make the subject accessible to undergraduates, numerous worked examples and thought-provoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of Mathematica to facilitate computation and animate solutions.
The second edition refines the exposition in every chapter, adds more homework exercises and projects, updates references, and includes new examples involving non-commutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition.
See the review of the first edition in [Zbl 1216.35106].Benedicks-Amrein-Berthier type theorem and local uncertainty principles in Clifford algebrashttps://zbmath.org/1508.420122023-05-31T16:32:50.898670Z"Tyr, Othman"https://zbmath.org/authors/?q=ai:tyr.othman"Daher, Radouan"https://zbmath.org/authors/?q=ai:daher.radouanSummary: This paper uses some basic notions and results in real Clifford algebra analysis. A new uncertainty principle for the Clifford-Fourier transform CFT is obtained. This result is an extension of a result of\textit{W. O. Amrein} and \textit{A. M. Berthier} [J. Funct. Anal. 24, 258--267 (1977; Zbl 0355.42015)] and \textit{M. Benedicks} [J. Math. Anal. Appl. 106, 180--183 (1985; Zbl 0576.42016)], it states that a non-zero function \(f\) and its Fourier transform \(\widehat{f}\) cannot both have the support of finite measure. A Donoho-Stark's local uncertainty principle to the CFT is also obtained.The first cohomology group and weak amenability of a Morita context Banach algebrahttps://zbmath.org/1508.460352023-05-31T16:32:50.898670Z"Lakzian, H."https://zbmath.org/authors/?q=ai:lakzian.hosein"Vishki, H. R. Ebrahimi"https://zbmath.org/authors/?q=ai:vishki.hamid-reza-ebrahimi"Barootkoob, S."https://zbmath.org/authors/?q=ai:barootkoob.sedighehSummary: Motivated by the elaborate works of
\textit{B.~E. Forrest} and \textit{L.~W. Marcoux} [Trans. Am. Math. Soc. 354, No. 4, 1435--1452 (2002; Zbl 1014.46017)] and
\textit{Y.~Zhang} [Trans. Am. Math. Soc. 354, No.~10, 4131--4151 (2002; Zbl 1008.46019)]
on determining the first cohomology group and studying \(n\)-weak amenability of triangular and module extension Banach algebras, we investigate the same notions for a Morita context Banach algebra \[\mathbb{G}=\left[ \begin{matrix}\mathbb{A} &\mathbb{M}\\ \mathbb{N}& \mathbb{B}\end{matrix}\right],\] where \(\mathbb{A}\) and \(\mathbb{B}\) are Banach algebras, \( \mathbb{M}\) and \(\mathbb{N}\) are Banach \((\mathbb{A},\mathbb{B})\) and \((\mathbb{B},\mathbb{A})\)-bimodules, respectively. We describe the \(n^{\text{th}} \)-dual \(\mathbb{G}^{(n)}\) of \(\mathbb{G}\) and characterize the structure of derivations from \(\mathbb{G}\) to \(\mathbb{G}^{(n)}\) for studying the first cohomology group \(\mathbf{H}^1\left( \mathbb{G}, \mathbb{G}^{(n)}\right)\) and characterizing the \(n\)-weak amenability of \(\mathbb{G} \). Our study provides some improvements of certain known results on the triangular Banach algebras. The results are then applied to the full matrix Banach algebras \(\mathbb{M}_k(\mathbb{A})\). Some examples illustrating the results are also included, and several questions are also left undecided.The royal road to automatic noncommutative real analyticity, monotonicity, and convexityhttps://zbmath.org/1508.460462023-05-31T16:32:50.898670Z"Pascoe, J. E."https://zbmath.org/authors/?q=ai:pascoe.james-eldred"Tully-Doyle, Ryan"https://zbmath.org/authors/?q=ai:tully-doyle.ryanThe primary goal of this article is to simplify the proofs of the multivariable generalizations of Löwner's theorem on matrix monotone functions and Kraus' theorem on matrix convex functions. A function \(f:(a,b)\to\mathbb{R}\) is matrix monotone if, for any self-adjoint matrices \(A,B\) of the same size with spectrum in \((a,b)\), we have \(A\leq B\ \Rightarrow \ f(A) \leq f(B)\). Löwner's theorem gives a complete characterization of matrix monotone functions in one variable in terms of analytic continuation to the upper half-plane.
Subsequent work has provided multivariable versions of Löwner's theorem (see the introduction of the article for references). The first such results were proven for functions of commuting families of matrices, but the proofs are difficult. As explained by the present authors, ``the difficulty is a symptom of the fact that the variety of commuting tuples of matrices is full of holes -- that is, it is not convex and, thus, unnatural for understanding monotonicity.'' Subsequent results were obtained for functions of non-commuting matrices, with proofs relying on the commutative case. In this article, the authors give a direct proof of the non-commutative case, without passing by the commutative case. The key philosophy of the article is that this direct proof is in fact more natural than the case of commuting variables, thus providing the ``royal road'' of the title.
Analogous results are also proven for matrix convex functions of non-commutating variables.
Reviewer: Robert Yuncken (Metz)On supercyclicity for abelian semigroups of matrices on \(\mathbb{R}^n\)https://zbmath.org/1508.470122023-05-31T16:32:50.898670Z"Herzi, Salah"https://zbmath.org/authors/?q=ai:herzi.salah"Marzougui, Habib"https://zbmath.org/authors/?q=ai:marzougui.habibSummary: We give a complete characterization of supercylicity for abelian semigroups of matrices on \(\mathbb{R}^n\), \(n \geqslant 1\). We solve the problem of determining the minimal number of matrices over \(\mathbb{R}\) which form a supercyclic abelian semigroup on \(\mathbb{R}^n\). In particular, we show that no abelian semigroup generated by \([\frac{n-1}{2}]\) matrices on \(\mathbb{R}\) can be supercyclic. (\([\cdot]\) denotes the integer part.) This answers a question raised by the second author in [\textit{H. Marzougui}, Monatsh. Math. 175, No. 3, 401--410 (2014; Zbl 1318.47015)]. Furthermore, we show that supercyclicity and \(\mathbb{R}_+\)-supercyclicity are equivalent.Similarity preserving linear maps on \(\mathscr{J}\)-subspace lattice algebrashttps://zbmath.org/1508.470842023-05-31T16:32:50.898670Z"Qin, Zijie"https://zbmath.org/authors/?q=ai:qin.zijie"Lu, Fangyan"https://zbmath.org/authors/?q=ai:lu.fangyanSummary: We describe the structure of similarity preserving linear maps of reflexive algebras with \(\mathscr{J}\)-subspace lattices. This result can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.Morse theoretic aspects of Plücker embeddingshttps://zbmath.org/1508.510122023-05-31T16:32:50.898670Z"Ozawa, Tetsuya"https://zbmath.org/authors/?q=ai:ozawa.tetsuyaThe author gives necessary and sufficient conditions for the Plücker embeddings of real and complex Grassmannians to be taut. A convexity property of such Plücker embeddings is also examined.
Reviewer: Georgi Hristov Georgiev (Shumen)Fluctuations in the spectrum of non-Hermitian i.i.d. matriceshttps://zbmath.org/1508.600042023-05-31T16:32:50.898670Z"Cipolloni, Giorgio"https://zbmath.org/authors/?q=ai:cipolloni.giorgioSummary: We consider large non-Hermitian random matrices \(X\) with independent identically distributed real or complex entries. In this paper, we review recent results about the eigenvalues of \(X\): (i) universality of local eigenvalue statistics close to the edge of the spectrum of \(X\) [\textit{G. Cipolloni} et al., Probab. Theory Relat. Fields 179, No. 1--2, 1--28 (2021; Zbl 1461.60008)], which is the non-Hermitian analog of the celebrated Tracy-Widom universality; (ii) central limit theorem for linear eigenvalue statistics of macroscopic test functions having \(2 + \varepsilon\) derivatives [\textit{G. Cipolloni}, \textit{L. Erdős} and \textit{D. Schröder}, ``Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices'', Commun. Pure Appl. Math., 89 p. (2021; \url{doi:10.1002/cpa.22028}); Electron. J. Probab. 26, Paper No. 24, 61 p. (2021; Zbl 1477.60016)]. The main novel ingredients in the proof of these results are local laws for products of two resolvents of the Hermitization of \(X\) at two different spectral parameters, coupling of weakly dependent Dyson Brownian motions, and the lower tail estimate for the smallest singular value of \(X - z\) in the transitional regime \(|z| \approx 1\) [\textit{G. Cipolloni} et al., Probab. Math. Phys. 1, No. 1, 101--146 (2020; Zbl 1485.15041)].
{\copyright 2022 American Institute of Physics}Sparse matrices: convergence of the characteristic polynomial seen from infinityhttps://zbmath.org/1508.600052023-05-31T16:32:50.898670Z"Coste, Simon"https://zbmath.org/authors/?q=ai:coste.simonSummary: We prove that the reverse characteristic polynomial \(\det ({I_n}-z{A_n})\) of a random \(n\times n\) matrix \({A_n}\) with iid Bernoulli\((d/ n)\) entries converges in distribution towards the random infinite product
\[\prod \limits_{\ell =1}^\infty(1-{z^\ell})^{Y_\ell}\] where \({Y_{\ell }}\) are independent Poisson\(({d^{\ell }}/ \ell )\) random variables. We show that this random function is a Poisson analog of more classical Gaussian objects such as the Gaussian holomorphic chaos. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erdős-Rényi digraphs: for every \(d > 1\), the greatest eigenvalue of \({A_n}\) is close to \(d\) and the second greatest is smaller than \(\sqrt{d} \), a Ramanujan-like property for irregular digraphs. For \(d > 1\), the only non-zero eigenvalues of \({A_n}\) converge to a Poisson multipoint process on the unit circle.
Our results also extend to the semi-sparse regime where \(d\) is allowed to grow to \(\infty\) with \(n\), slower than \({n^{o(1)}}\). We show that the reverse characteristic polynomial converges towards a more classical object written in terms of the exponential of a log-correlated real Gaussian field. In the semi-sparse regime, the empirical spectral distribution of \({A_n}/ \sqrt{{d_n}}\) converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.Column randomization and almost-isometric embeddingshttps://zbmath.org/1508.600072023-05-31T16:32:50.898670Z"Mendelson, Shahar"https://zbmath.org/authors/?q=ai:mendelson.shaharSummary: The matrix \(A: \mathbb{R}^n\to\mathbb{R}^m\) is \((\delta, k)\)-regular if for any \(k\)-sparse vector \(x\),
\[
\left|\|Ax\|_2^2 - \|x\|_2^2\right|\leq\delta\sqrt{k}\|x\|_2^2.
\]
We show that if \(A\) is \((\delta, k)\)-regular for \(1 \leq k \leq 1/\delta^2\), then by multiplying the columns of \(A\) by independent random signs, the resulting random ensemble \(A_\varepsilon\) acts on an arbitrary subset \(T\subset\mathbb{R}^n\) (almost) as if it were Gaussian, and with the optimal probability estimate: if \(\ell_\ast(T)\) is the Gaussian mean-width of \(T\) and \(d_T = \sup_{t \in T}\|t\|_2\), then with probability at least \(1-2\exp(-c(\ell_\ast(T)/d_T)^2)\),
\[
\sup_{t\in T}\left|\|A_\varepsilon t\|_2^2 - \|t\|_2^2\right| \leq C\left(\varLambda d_T\delta\ell_\ast(T) + (\delta\ell_\ast(T))^2\right),
\]
where \(\varLambda = \max\{1, \delta^2\log(n\delta^2)\}\). This estimate is optimal for \(0 < \delta \leq 1/\sqrt{\log n}\).The embedding problem for Markov matriceshttps://zbmath.org/1508.600742023-05-31T16:32:50.898670Z"Casanellas, Marta"https://zbmath.org/authors/?q=ai:casanellas.marta"Fernández-Sánchez, Jesús"https://zbmath.org/authors/?q=ai:fernandez-sanchez.jesus"Roca-Lacostena, Jordi"https://zbmath.org/authors/?q=ai:roca-lacostena.jordiSummary: Characterizing whether a Markov process of discrete random variables has a homogeneous continuous-time realization is a hard problem. In practice, this problem reduces to deciding when a given Markov matrix can be written as the exponential of some rate matrix (a Markov generator). This is an old question known in the literature as the \textit{embedding problem} [\textit{G. Elfving}, Acta Soc. Sci. Fennicae, N. Ser. A 2, No. 8, 1--17 (1937; Zbl 0017.31603)], which has been solved only for matrices of size \(2\times 2\) or \(3\times 3\). In this paper, we address this problem and related questions and obtain results along two different lines. First, for matrices of any size, we give a bound on the number of Markov generators in terms of the spectrum of the Markov matrix. Based on this, we establish a criterion for deciding whether a generic (distinct eigenvalues) Markov matrix is embeddable and propose an algorithm that lists all its Markov generators. Then, motivated and inspired by recent results on substitution models of DNA, we focus on the \(4\times 4\) case and completely solve the embedding problem for any Markov matrix. The solution in this case is more concise as the embeddability is given in terms of a single condition.Fast algorithms for finding the solution of CUPL-Toeplitz linear system from Markov chainhttps://zbmath.org/1508.650182023-05-31T16:32:50.898670Z"Fu, Yaru"https://zbmath.org/authors/?q=ai:fu.yaru"Jiang, Xiaoyu"https://zbmath.org/authors/?q=ai:jiang.xiaoyu"Jiang, Zhaolin"https://zbmath.org/authors/?q=ai:jiang.zhaolin"Jhang, Seongtae"https://zbmath.org/authors/?q=ai:jhang.seongtaeSummary: In this paper, the nonsingular CUPL-Toeplitz linear system from Markov chain is solved. We introduce two fast approaches whose complexity could be considered to be \(O(n\log n)\) based on the splitting method of the CUPL-Toeplitz matrix which equals to a Toeplitz matrix minus a rank-one matrix. Finally, we confirm the performance of the new algorithms by three numerical experiments.A generalization of the ABS algorithms and its application to some special real and integer matrix factorizationshttps://zbmath.org/1508.650192023-05-31T16:32:50.898670Z"Golpar-Raboky, E."https://zbmath.org/authors/?q=ai:golpar-raboky.effat"Mahdavi-Amiri, N."https://zbmath.org/authors/?q=ai:mahdavi-amiri.nezamSummary: In 1984, \textit{J. Abaffy} et al. [Numer. Math. 45, 361--376 (1984; Zbl 0535.65009)] introduced a class of the so-called ABS algorithms to solve systems of real linear equations. Later, the scaled ABS, the extended ABS, the block ABS, and the integer ABS algorithms were introduced leading to various well-known matrix factorizations. Here, we present a generalization of ABS algorithms containing all matrix factorizations such as triangular, \(W Z\), and \(ZW\). We discuss the octant interlocking factorization and make use of the generalized ABS algorithm as a more general approach for producing the octant interlocking factorization.A matrix splitting preconditioning method for solving the discretized tempered fractional diffusion equationshttps://zbmath.org/1508.650212023-05-31T16:32:50.898670Z"Tang, Shi-Ping"https://zbmath.org/authors/?q=ai:tang.shi-ping"Huang, Yu-Mei"https://zbmath.org/authors/?q=ai:huang.yumei|huang.yu-meiSummary: The initial boundary value problem of the tempered fractional diffusion equations is a kind of important equations arising in many application fields. In this paper, the Crank-Nicolson scheme is applied in the discretization of the tempered fractional diffusion equations. We then get the discretized system of linear equations with the coefficient matrix having the structure of the sum of an identity matrix and the product of a diagonal and a symmetric positive-definite Toeplitz matrix. A scaled diagonal and Toeplitz-approximate splitting (SDTAS) preconditioner is developed, and the GMRES method combined with this preconditioner is applied to solve the linear system. The spectral distribution of the preconditioned matrix is analyzed and some theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method.A filter in constructing the preconditioner for solving linear equation systems of radiation diffusion problemshttps://zbmath.org/1508.650232023-05-31T16:32:50.898670Z"Ye, Shuai"https://zbmath.org/authors/?q=ai:ye.shuai"An, Hengbin"https://zbmath.org/authors/?q=ai:an.hengbin"Xu, Xiaowen"https://zbmath.org/authors/?q=ai:xu.xiaowen"Xu, Xinhai"https://zbmath.org/authors/?q=ai:xu.xinhai"Yang, Xuejun"https://zbmath.org/authors/?q=ai:yang.xuejunSummary: The coefficient matrices of the linear equation systems arising from the radiation diffusion problems usually have orders of magnitude difference between their off-diagonal entries. While solving these linear equations with a preconditioned iterative method, the entries with small magnitudes may be insignificant to the preconditioner efficiency. In this paper, we use a filter to remove such small entries in the coefficient matrix while constructing the preconditioner. The proposed filter eliminates the small entries first according to a so-called weak dependence matrix, which relies on the conception of the strength of connections in algebraic multigrid. The preconditioner is then built based on the filtered matrix instead of the original one. Four strategies of filtering out entries are designed and investigated. Numerical results for various model-type problems and two real application problems, i.e., the multi-group radiation diffusion equations and the three temperature energy equations, are provided to show the effectiveness of the proposed method. In particular, this paper provides a practical approach to choose a proper parameter in the proposed method, which should help solve linear equation systems of radiation diffusion problems.A symbol-based analysis for multigrid methods for block-circulant and block-Toeplitz systemshttps://zbmath.org/1508.650252023-05-31T16:32:50.898670Z"Bolten, Matthias"https://zbmath.org/authors/?q=ai:bolten.matthias"Donatelli, Marco"https://zbmath.org/authors/?q=ai:donatelli.marco"Ferrari, Paola"https://zbmath.org/authors/?q=ai:ferrari.paola"Furci, Isabella"https://zbmath.org/authors/?q=ai:furci.isabellaSummary: In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an elegant form in the case of Toeplitz matrices generated by a scalar-valued function. In the block-Toeplitz setting, that is, in the case where the matrix entries are small generic matrices instead of scalars, some algorithms have already been proposed regarding specific applications, and a first rigorous convergence analysis has been performed in [\textit{M. Donatelli} et al., Numer. Linear Algebra Appl. 28, No. 4, e2356, 20 p. (2021; Zbl 07396240)]. However, with the existent symbol-based theoretical tools, it is still not possible to prove the convergence of many multigrid methods known in the literature. This paper aims to generalize the previous results, giving more general sufficient conditions on the symbol of the grid transfer operators. In particular, we treat matrix-valued trigonometric polynomials which can be nondiagonalizable and singular at all points, and we express the new conditions in terms of the eigenvectors associated with the ill-conditioned subspace. Moreover, we extend the analysis to the V-cycle method, proving a linear convergence rate under stronger conditions, which resemble those given in the scalar case. In order to validate our theoretical findings, we present a classical block structured problem stemming from an FEM approximation of a second order differential problem. We focus on two multigrid strategies that use the geometric and the standard bisection grid transfer operators and prove that both fall into the category of projectors satisfying the proposed conditions. In addition, using a tensor product argument, we provide a strategy to construct efficient V-cycle procedures in the block multilevel setting.Perron communicability and sensitivity of multilayer networkshttps://zbmath.org/1508.650312023-05-31T16:32:50.898670Z"El-Halouy, Smahane"https://zbmath.org/authors/?q=ai:el-halouy.smahane"Noschese, Silvia"https://zbmath.org/authors/?q=ai:noschese.silvia"Reichel, Lothar"https://zbmath.org/authors/?q=ai:reichel.lotharSummary: Modeling complex systems that consist of different types of objects leads to multilayer networks, where nodes in the different layers represent different kinds of objects. Nodes are connected by edges, which have positive weights. A multilayer network is associated with a supra-adjacency matrix. This paper investigates the sensitivity of the communicability in a multilayer network to perturbations of the network by studying the sensitivity of the Perron root of the supra-adjacency matrix. Our analysis sheds light on which edge weights to make larger to increase the communicability of the network, and which edge weights can be made smaller or set to zero without affecting the communicability significantly.A fast iterative algorithm for near-diagonal eigenvalue problemshttps://zbmath.org/1508.650322023-05-31T16:32:50.898670Z"Kenmoe, Maseim"https://zbmath.org/authors/?q=ai:kenmoe.maseim"Kriemann, Ronald"https://zbmath.org/authors/?q=ai:kriemann.ronald"Smerlak, Matteo"https://zbmath.org/authors/?q=ai:smerlak.matteo"Zadorin, Anton S."https://zbmath.org/authors/?q=ai:zadorin.anton-sSummary: We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schrödinger perturbation theory and termed \textit{iterative perturbative theory} (IPT). Contrary to standard eigenvalue algorithms, which are either ``direct'' (to compute all eigenpairs) or ``iterative'' (to compute just a few), IPT computes any number of eigenpairs with the same basic iterative procedure. Thanks to this perfect parallelism, IPT proves more efficient than classical methods (LAPACK or CUSOLVER for the full-spectrum problem, preconditioned Davidson solvers for extremal eigenvalues). We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices, including one from quantum chemistry. The code is available at \url{http://github.com/msmerlak/IterativePerturbationTheory.jl}.Singular quadratic eigenvalue problems: linearization and weak condition numbershttps://zbmath.org/1508.650332023-05-31T16:32:50.898670Z"Kressner, Daniel"https://zbmath.org/authors/?q=ai:kressner.daniel"Šain Glibić, Ivana"https://zbmath.org/authors/?q=ai:glibic.ivana-sainSummary: The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of \(\delta \)-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases \(\delta \)-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.Verified error bounds for all eigenvalues and eigenvectors of a matrixhttps://zbmath.org/1508.650342023-05-31T16:32:50.898670Z"Rump, Siegfried M."https://zbmath.org/authors/?q=ai:rump.siegfried-michaelSummary: A verification method is presented to compute error bounds for all eigenvectors and eigenvalues, including clustered and/or multiple ones of a general, real, or complex matrix. In case of a narrow cluster, error bounds for an invariant subspace are computed because computation of a single eigenvector may be ill-posed. Computer algebra and verification methods have in common that the computed results are correct with mathematical certainty. Unlike a computer algebra method, a verification method may fail in the sense that only partial or no inclusions at all are computed. That may happen for very ill conditioned problems being too sensitive for the arithmetical precision in use. That cannot happen for computer algebra methods which are ``never-failing'' because potentially infinite precision is used. In turn, however, that may slow down computer algebra methods significantly and may impose limitations on the problem size. In contrast, verification methods solely use floating-point operations so that their computing time and treatable problem size is of the order of that of purely numerical algorithms. For our problem it is proved that the union of the eigenvalue bounds contains the whole spectrum of the matrix, and bounds for corresponding invariant subspaces are computed. The computational complexity to compute inclusions of all eigenpairs of an \(n\times n\)-matrix is \(\mathcal{O}(n^3)\).Subspaces analysis for random projection UTV frameworkhttps://zbmath.org/1508.650352023-05-31T16:32:50.898670Z"Jian, Yuan"https://zbmath.org/authors/?q=ai:jian.yuanSummary: The UTV decompositions are promising and computationally efficient alternatives to the singular value decomposition (SVD), which can provide high-quality information about rank, range, and nullspace. However, for large-scale matrices, we want more computationally efficient algorithms. Recently, randomized algorithms with their surprising reliability and computational efficiency have become increasingly popular in many application areas. In this paper, we analyze the subspace properties of a random projection UTV framework and give the bounds of subspace distances between the exact and the approximate singular subspaces, the approximate methods including random projection ULV, random projection URV, and random projection SVD. Numerical experiments demonstrate the effectiveness of the proposed bounds.Orthogonal Cauchy-like matriceshttps://zbmath.org/1508.650372023-05-31T16:32:50.898670Z"Fasino, Dario"https://zbmath.org/authors/?q=ai:fasino.darioSummary: Cauchy-like matrices arise often as building blocks in decomposition formulas and fast algorithms for various displacement-structured matrices. A complete characterization for orthogonal Cauchy-like matrices is given here. In particular, we show that orthogonal Cauchy-like matrices correspond to eigenvector matrices of certain symmetric matrices related to the solution of secular equations. Moreover, the construction of orthogonal Cauchy-like matrices is related to that of orthogonal rational functions with variable poles.An efficient numerical method for condition number constrained covariance matrix approximationhttps://zbmath.org/1508.650382023-05-31T16:32:50.898670Z"Wang, Shaoxin"https://zbmath.org/authors/?q=ai:wang.shaoxinSummary: In the high-dimensional data setting, the sample covariance matrix is singular. In order to get a numerically stable and positive definite modification of the sample covariance matrix in the high-dimensional data setting, in this paper we consider the condition number constrained covariance matrix approximation problem and present its explicit solution with respect to the Frobenius norm. The condition number constraint guarantees the numerical stability and positive definiteness of the approximation form simultaneously. By exploiting the special structure of the data matrix in the high-dimensional data setting, we also propose some new algorithms based on efficient matrix decomposition techniques. Numerical experiments are also given to show the computational efficiency of the proposed algorithms.An iterative method based on ADMM for solving generalized Sylvester matrix equationshttps://zbmath.org/1508.650392023-05-31T16:32:50.898670Z"Shafiei, Soheila Ghoroghi"https://zbmath.org/authors/?q=ai:shafiei.soheila-ghoroghi"Hajarian, Masoud"https://zbmath.org/authors/?q=ai:hajarian.masoudThe generalized Sylvester matrix equation is \(AXB+CXD=E,\) with \(A\) and \(C,\) \( B\) and \(D\), and \(E\) being matrices of sizes \(m\times n\), \(r\times p\) and \( m\times p\), respectively;\ accordingly, the unknown matrix \(X\) is of size \( n\times r\). The authors assume that \(A\) and \(C\) (\(B\) and \(D\)) are full column (row, respectively) rank. In the solution method they propose, submatrix equations of the type \(AXB=E_{1}\) and \(CXD=E_{2}\) are considered. An equation of the type \(AXB=F\) is reformulated as the optimization problem consisting in minimizing \(\frac{1}{2}\left\Vert ZB-X\right\Vert ^{2}\) subject to \(AX-Z=0\). Then, an alternating direction augmented Lagrangian method, using the decomposition \(f(X)+g(Z)\) of the objective function given by \(f(X):=0\) and \(g(Z):=\frac{1}{2}\left\Vert ZB-X\right\Vert ^{2}\), is proposed. A convergence theorem is proved and numerical examples are presented.
Reviewer: Juan-Enrique Martínez-Legaz (Barcelona)Structured matrix approximations via tensor decompositionshttps://zbmath.org/1508.650422023-05-31T16:32:50.898670Z"Kilmer, Misha E."https://zbmath.org/authors/?q=ai:kilmer.misha-e"Saibaba, Arvind K."https://zbmath.org/authors/?q=ai:saibaba.arvind-krishnaSummary: We provide a computational framework for approximating a class of structured matrices; here, the term structure is very general, and may refer to a regular sparsity pattern (e.g., block banded), or be more highly structured (e.g., symmetric block Toeplitz). The goal is to uncover \textit{additional latent structure} that will in turn lead to computationally efficient algorithms when the new structured matrix approximations are employed in place of the original operator. Our approach has three steps: map the structured matrix to tensors, use tensor compression algorithms, and map the compressed tensors back to obtain two different matrix representations -- sum of Kronecker products and block low-rank format. The use of tensor decompositions enables us to uncover latent structure in the problem and leads to compressed representations of the original matrix that can be used efficiently in applications. The resulting matrix approximations are memory efficient, easy to compute with, and preserve the error that is due to the tensor compression in the Frobenius norm. Our framework is quite general. We illustrate the ability of our method to uncover block-low-rank format on structured matrices from two applications: system identification and space-time covariance matrices. In addition, we demonstrate that our approach can uncover the sum of structured Kronecker products structure on several matrices from the SuiteSparse collection. Finally, we show that our framework is broad enough to encompass and improve on other related results from the literature, as we illustrate with the approximation of a three-dimensional blurring operator.Estimating the trace of matrix functions with application to complex networkshttps://zbmath.org/1508.650452023-05-31T16:32:50.898670Z"Fuentes, Rafael Díaz"https://zbmath.org/authors/?q=ai:diaz-fuentes.rafael"Donatelli, Marco"https://zbmath.org/authors/?q=ai:donatelli.marco"Fenu, Caterina"https://zbmath.org/authors/?q=ai:fenu.caterina"Mantica, Giorgio"https://zbmath.org/authors/?q=ai:mantica.giorgioSummary: The approximation of \(\mathrm{trace}(f(\Omega ))\), where \(f\) is a function of a symmetric matrix \(\Omega \), can be challenging when \(\Omega\) is exceedingly large. In such a case even the partial Lanczos decomposition of \(\Omega\) is computationally demanding and the stochastic method investigated by \textit{Z. Bai} et al. [J. Comput. Appl. Math. 74, No. 1--2, 71--89 (1996; Zbl 0870.65035)] is preferred. Moreover, in the last years, a partial global Lanczos method has been shown to reduce CPU time with respect to partial Lanczos decomposition. In this paper we review these techniques, treating them under the unifying theory of measure theory and Gaussian integration. This allows generalizing the stochastic approach, proposing a block version that collects a set of random vectors in a rectangular matrix, in a similar fashion to the partial global Lanczos method. We show that the results of this technique converge quickly to the same approximation provided by Bai et al. [loc. cit.], while the block approach can leverage the same computational advantages as the partial global Lanczos. Numerical results for the computation of the Von Neumann entropy of complex networks prove the robustness and efficiency of the proposed block stochastic method.Practical leverage-based sampling for low-rank tensor decompositionhttps://zbmath.org/1508.650462023-05-31T16:32:50.898670Z"Larsen, Brett W."https://zbmath.org/authors/?q=ai:larsen.brett-w"Kolda, Tamara G."https://zbmath.org/authors/?q=ai:kolda.tamara-gSummary: The low-rank canonical polyadic tensor decomposition is useful in data analysis and can be computed by solving a sequence of overdetermined least squares subproblems. Motivated by consideration of sparse tensors, we propose sketching each subproblem using leverage scores to select a subset of the rows, with probabilistic guarantees on the solution accuracy. We randomly sample rows proportional to leverage score upper bounds that can be efficiently computed using the special Khatri-Rao subproblem structure inherent in tensor decomposition. Crucially, for a \((d+1)\)-way tensor, the number of rows in the sketched system is \(O(r^d/\epsilon)\) for a decomposition of rank \(r\) and \(\epsilon\)-accuracy in the least squares solve, independent of both the size and the number of nonzeros in the tensor. Along the way, we provide a practical solution to the generic matrix sketching problem of sampling overabundance for high-leverage-score rows, proposing to include such rows deterministically and combine repeated samples in the sketched system; we conjecture that this can lead to improved theoretical bounds. Numerical results on real-world large-scale tensors show the method is significantly faster than deterministic methods at nearly the same level of accuracy.A sampling algorithm to compute the set of feasible solutions for nonnegative matrix factorization with an arbitrary rankhttps://zbmath.org/1508.650472023-05-31T16:32:50.898670Z"Laursen, Ragnhild"https://zbmath.org/authors/?q=ai:laursen.ragnhild"Hobolth, Asger"https://zbmath.org/authors/?q=ai:hobolth.asgerSummary: Nonnegative matrix factorization (NMF) is a useful method to extract features from multivariate data, but an important and sometimes neglected concern is that NMF can result in nonunique solutions. Often, there exist a set of feasible solutions (SFS), which makes it more difficult to interpret the factorization. This problem is especially ignored in cancer genomics, where NMF is used to infer information about the mutational processes present in the evolution of cancer. In this paper the extent of nonuniqueness is investigated for two mutational counts data, and a new sampling algorithm that can find the SFS is introduced. Our sampling algorithm is easy to implement and applies to an arbitrary rank of NMF. This is in contrast to state of the art, where the NMF rank must be smaller than or equal to four. For lower ranks we show that our algorithm performs similar to the polygon inflation algorithm that is developed in relation to chemometrics. Furthermore, we show how the size of the SFS can have a high influence on the appearing variability of a solution. Our sampling algorithm is implemented in the R package SFS (\url{https://github.com/ragnhildlaursen/SFS}).A columnwise update algorithm for sparse stochastic matrix factorizationhttps://zbmath.org/1508.650482023-05-31T16:32:50.898670Z"Xiao, Guiyun"https://zbmath.org/authors/?q=ai:xiao.guiyun"Bai, Zheng-Jian"https://zbmath.org/authors/?q=ai:bai.zhengjian"Ching, Wai-Ki"https://zbmath.org/authors/?q=ai:ching.wai-kiSummary: Nonnegative matrix factorization arises widely in machine learning and data analysis. In this paper, for a given factorization of rank \(r\), we consider the sparse stochastic matrix factorization (SSMF) of decomposing a prescribed \(m\)-by-\(n\) stochastic matrix \(V\) into a product of an \(m\)-by-\(r\) stochastic matrix \(W\) and an \(r\)-by-\(n\) stochastic matrix \(H\), where both \(W\) and \(H\) are required to be sparse. With the prescribed sparsity level, we reformulate the SSMF as an unconstrained nonconvex-nonsmooth minimization problem and introduce a columnwise update algorithm for solving the minimization problem. We show that our algorithm converges globally. The main advantage of our algorithm is that the generated sequence converges to a special critical point of the cost function, which is nearly a global minimizer over each column vector of the \(W\)-factor and is a global minimizer over the \(H\)-factor as a whole if there is no sparsity requirement on \(H\). Numerical experiments on both synthetic and real data sets are given to demonstrate the effectiveness of our proposed algorithm.Optimal error estimation of two fast structure-preserving algorithms for the Riesz fractional sine-Gordon equationhttps://zbmath.org/1508.651082023-05-31T16:32:50.898670Z"Ma, Tingting"https://zbmath.org/authors/?q=ai:ma.tingting"Zheng, Qianqian"https://zbmath.org/authors/?q=ai:zheng.qianqian"Fu, Yayun"https://zbmath.org/authors/?q=ai:fu.yayunSummary: The primary purpose of this paper is to develop and analyze two fast and conservative algorithms for the fractional sine-Gordon equation. The numerical schemes are derived by using the second-and fourth-order difference method in space and the implicit midpoint rule in time to approximate an equivalent system obtained via the energy quadratic method. In addition, the conservation, existence and uniqueness, and convergence of the two schemes are investigated. Based on the properties of the Toeplitz matrix, a fast algorithm is given in the calculation for the proposed schemes. Numerical experiments verify the theoretical analysis of the schemes, showing their efficiency and excellent behavior.Mathematical physical chemistry. Practical and intuitive methodologyhttps://zbmath.org/1508.810022023-05-31T16:32:50.898670Z"Hotta, Shu"https://zbmath.org/authors/?q=ai:hotta.shuPublisher's description: The second edition of this book has been extensively revised so that readers can gain ready access to advanced topics of mathematical physics including the theory of analytic functions and continuous groups. This easy accessibility helps to create a deeper and clearer insight into mathematical physics, with emphasis on quantum mechanics and electromagnetism along with the theory of linear vector spaces and group theory. The basic nature of the book remains unchanged. The contents are targeted at graduate and undergraduate students majoring in chemistry to supply them with the practical and intuitive methodology of mathematical physics. In parallel, advanced mathematical topics are dealt with in the last chapters of each of the four individual parts so that a close connection among those topics is highlighted. Several important revisions are found in this second edition, however, and they include: (a) a description of set theory and topology that helps to comprehend the essence of the theory of analytic functions and continuous groups; (b) a deep connection between a and continuous groups; (c) development of the theory of exponential functions of matrices, which is useful to solve differential equations; and (d) updated content on lasers and their applications. This new edition thus provides a balanced selection of new and basic material for chemists and physicists.
See the review of the first edition in [Zbl 1411.81002].On complementarity. A universal organizing principlehttps://zbmath.org/1508.810212023-05-31T16:32:50.898670Z"Avrin, Jack"https://zbmath.org/authors/?q=ai:avrin.jack-sPublisher's description: It is not uncommon for the Principle of Complementarity to be invoked in either Science or Philosophy, viz. the ancient oriental philosophy of Yin and Yang whose symbolic representation is portrayed on the cover of the book. Or Niels Bohr's use of it as the basis for the so-called Copenhagen interpretation of Quantum Mechanics. This book arose as an outgrowth of the author's previous book entitled ``Knots, Braids and Moebius Strips,'' published by World Scientific in 2015, wherein the Principle itself was discovered to be expressible as a simple $2\times 2$ matrix that summarizes the algebraic essence of both the well-known Microbiology of DNA and the author's version of the elementary particles of physics. At that point, the possibility of an even wider utilization of that expression of Complementarity arose.
The current book, features Complementarity, in which the matrix algebra is extended to characterize not only DNA itself but the well-known process of its replication, a most gratifying outcome. The book then goes on to explore Complementarity, with and without its matrix expression, as it occurs, not only in much of physics but in its extension to cosmology as well.
Contents:
Preliminaries: Introduction: Complementarity as a Principle, Deoxyribonucleic Acid, The Molecular Ladder to Life on Earth, Alternative Model Taxonomy, The AM/DNA Comparison, The Signature of Complementarity; Replication Unleashed. A (Particular) Connection to Chemistry.
Some Basic Physics: Dynamics, Thermodynamics, Energy-wise Comparison of Two Exercises, Maxwell's Equations and the Electromagnetic Field, Spacetime and The Ethereal Road to Relativity, Noether's Theorem and Gauge Theory.
General Relativity and the Geometry of Spacetime: General Relativity, Differential Geometry and The ``Action Enigma''; an Application, An Authoritative Perspective, And an Inference Thereof, An ``Indigenous Parallel'' to the Higgs, Undulation (or Undulatority). Another look at Our ``Indigenous Higgs Model''.
Quantum Mechanics and the Significance of Scale: Introducing Max Planck and the Quantum, Quantum Mechanics, Phase 1, Phase 2; Through the Looking Glass!, Quantum Mechanics, Radar and the Significance of Scale, The Fourier Transform and the Convolution Theorem, Nonlocality, Entanglement, and complementarity, Spin, Spinors, and the Pauli Connection, PAM Dirac, Mixmaster Extraordinaire.
Statistical Mechanics. Some Additional Alternative Model Topics: ``Deuteronomy'' and Isospin Invariance, Cosmological Complementarity, Gurule Loops
Summary and Conclusions: Recapitulation. Linking Complementarity and The Meaning of ``Is''Minimum distance of the boundary of the set of PPT states from the maximally mixed state using the geometry of the positive semidefinite conehttps://zbmath.org/1508.810292023-05-31T16:32:50.898670Z"Banerjee, Shreya"https://zbmath.org/authors/?q=ai:banerjee.shreya"Patel, Aryaman A."https://zbmath.org/authors/?q=ai:patel.aryaman-a"Panigrahi, Prasanta K."https://zbmath.org/authors/?q=ai:panigrahi.prasanta-kSummary: Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices [\textit{A. A, Patel} and \textit{P. K. Panigrahi}, ``Geometric measure of entanglement based on local measurement'', Preprint, \url{arXiv:1608.06145}], we obtain the minimum distance between the set of bipartite \(n\)-qudit density matrices with a positive partial transpose and the maximally mixed state. This minimum distance is obtained as \(\frac{1}{\sqrt{d^n(d^n-1)}}\), which is also the minimum distance within which all quantum states are separable. An idea of the interior of the set of all positive semidefinite matrices has also been provided. A particular class of Werner states has been identified for which the PPT criterion is necessary and sufficient for separability in dimensions greater than six.Joint measurability of quantum effects and the matrix diamondhttps://zbmath.org/1508.810312023-05-31T16:32:50.898670Z"Bluhm, Andreas"https://zbmath.org/authors/?q=ai:bluhm.andreas"Nechita, Ion"https://zbmath.org/authors/?q=ai:nechita.ionSummary: In this work, we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra. Free spectrahedra typically arise as matricial relaxations of linear matrix inequalities. An example of a free spectrahedron is the matrix diamond, which is a matricial relaxation of the \(\ell_{1}\)-ball. We find that joint measurability of binary positive operator valued measures is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study. This connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements. In particular, we completely characterize the case in which the dimension is exponential in the number of measurements. Conversely, we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond.{
\copyright 2018 American Institute of Physics}Partial separability/entanglement violates distributive ruleshttps://zbmath.org/1508.810402023-05-31T16:32:50.898670Z"Han, Kyung Hoon"https://zbmath.org/authors/?q=ai:han.kyung-hoon"Kye, Seung-Hyeok"https://zbmath.org/authors/?q=ai:kye.seunghyeok"Szalay, Szilárd"https://zbmath.org/authors/?q=ai:szalay.szilardSummary: We found three qubit Greenberger-Horne-Zeilinger diagonal states which tell us that the partial separability of three qubit states violates the distributive rules with respect to the two operations of convex sum and intersection. The gaps between the convex sets involving the distributive rules are of nonzero volume.\(J\)-states and quantum channels between indefinite metric spaceshttps://zbmath.org/1508.810712023-05-31T16:32:50.898670Z"Felipe-Sosa, Raúl"https://zbmath.org/authors/?q=ai:felipe-sosa.raul"Felipe, Raúl"https://zbmath.org/authors/?q=ai:felipe.raulSummary: In the present work, we introduce and study the concepts of state and quantum channel on spaces equipped with an indefinite metric. Exclusively, we will limit our analysis to the matricial framework. As it will be confirmed below, from our research it is noticed that, when passing to the spaces with indefinite metric, the use of the adjoint of a matrix with respect to the indefinite metric is required in the construction of states and quantum channels; which prevents us to consider the space of matrices of certain order \(M_n(\mathbb{C})\) as a \(C^\ast\)-algebra. In our case, this adjoint is defined through a \(J\)-metric, where the matrix \(J\) is a fundamental symmetry of \(M_n(\mathbb{C})\). In our paper, for quantum operators, we include the general setting in the which, these operators map \(J_1\)-states into \(J_2\)-states, where \(J_2\neq\pm J_1\) are two arbitrary fundamental symmetries. In the middle of this program, we carry out a study of the completely positive maps between two different positive matrices spaces by considering two different indefinite metrics on \(\mathbb{C}^n\).Entanglement classification via integer partitionshttps://zbmath.org/1508.811542023-05-31T16:32:50.898670Z"Li, Dafa"https://zbmath.org/authors/?q=ai:li.dafaSummary: In [\textit{M. Walter} et al., Science 340, No. 6137, 1205--1208 (2013; Zbl 1355.81041)], they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification via polytopes and the eigenvalues of the single-particle states. In this paper, for \(4n\) qubits, we show the invariance of algebraic multiplicities (AMs) and geometric multiplicities (GMs) of eigenvalues and the invariance of sizes of Jordan blocks (JBs) of the coefficient matrices under SLOCC. We explore properties of spectra, eigenvectors, generalized eigenvectors, standard Jordan normal forms (SJNFs), and Jordan chains of the coefficient matrices. The properties and invariance permit a reduction in SLOCC classification of \(4n\) qubits to integer partitions (in number theory) of the number \(2^{2n}-k\) and the AMs.Constructions of unextendible entangled baseshttps://zbmath.org/1508.811912023-05-31T16:32:50.898670Z"Shi, Fei"https://zbmath.org/authors/?q=ai:shi.fei"Zhang, Xiande"https://zbmath.org/authors/?q=ai:zhang.xiande"Guo, Yu"https://zbmath.org/authors/?q=ai:guo.yuSummary: We provide several constructions of special unextendible entangled bases with fixed Schmidt number \(k\) (SUEB\(k\)) in \(\mathbb{C}^d\otimes\mathbb{C}^{d'}\) for \(2 \leq k \leq d \leq d'\). We generalize the space decomposition method in [\textit{Y. Guo}, ``Constructing the unextendible maximally entangled basis from the maximally entangled basis'', Phys. Rev. A (3) 94, No. 5, Article ID 052302, 5 p. (2016; \url{doi:10.1103/PhysRevA.94.052302})], by proposing a systematic way of constructing new SUEB\(k\)s in \(\mathbb{C}^d\otimes\mathbb{C}^{d'}\) for \(2 \leq k < d \leq d'\) or \(2\leq k = d < d'\). In addition, we give a construction of a (\(pqdd'-p(dd'-N)\))-number SUEB\(pk\) in \(\mathbb{C}^{pd}\otimes\mathbb{C}^{qd'}\) from an \(N\)-number SUEB\(k\) in \(\mathbb{C}^d\otimes\mathbb{C}^{d'}\) for \(p \leq q\) by using permutation matrices. We also connect a (\(d(d'-1)+m\))-number UMEB in \(\mathbb{C}^d\otimes\mathbb{C}^{d'}\) with an unextendible partial Hadamard matrix \(H_{m\times d}\) with \(m < d\), which extends the result in [\textit{Y.-L. Wang} et al., Quantum Inf. Process. 16, No. 3, Paper No. 84, 11 p. (2017; Zbl 1373.81090)].The \(H_2\)-reducible matrix in four six-dimensional mutually unbiased baseshttps://zbmath.org/1508.813072023-05-31T16:32:50.898670Z"Liang, Mengfan"https://zbmath.org/authors/?q=ai:liang.mengfan"Hu, Mengyao"https://zbmath.org/authors/?q=ai:hu.mengyao"Chen, Lin"https://zbmath.org/authors/?q=ai:chen.lin.5"Chen, Xiaoyu"https://zbmath.org/authors/?q=ai:chen.xiaoyuSummary: Finding four six-dimensional mutually unbiased bases (MUBs) is a long-standing open problem in quantum information. By assuming that they exist and contain the identity matrix, we investigate whether the remaining three MUBs have an \(H_2\)-reducible matrix, namely a \(6\times 6\) complex Hadamard matrix (CHM) containing a \(2\times 2\) subunitary matrix. We show that every \(6\times 6\) CHM containing at least 23 real entries is an \(H_2\)-reducible matrix. It relies on the fact that the CHM is complex equivalent to one of the two constant \(H_2\)-reducible matrices. They, respectively, have exactly 24 and 30 real entries, and both have more than eighteen \(2\times 2\) subunitary matrices. It turns out that such \(H_2\)- reducible matrices do not belong to the remaining three MUBs. This is the corollary of a stronger claim; namely, any \(H_2\)-reducible matrix belonging to the remaining three MUBs has exactly nine or eighteen \(2\times 2\) subunitary matrices.Several new constructions of mutually unbiased bases derived from functions over finite fieldshttps://zbmath.org/1508.813962023-05-31T16:32:50.898670Z"Qian, Liqin"https://zbmath.org/authors/?q=ai:qian.liqin"Cao, Xiwang"https://zbmath.org/authors/?q=ai:cao.xiwangSummary: A collection \(\mathfrak{B} = \{B_1, B_2, \cdots, B_N\}\) of orthonormal bases of \(\mathbb{C}^K\) is called mutually unbiased bases if \(|\langle v_i|v_j\rangle| = \frac{1}{\sqrt{K}}\) for all \(v_i\in B_i\), \(v_j\in B_j\) and \(1 \leq i < j \leq N\). In this paper, we present several new series of mutually unbiased bases constructed by utilizing \(p\)-ary weakly regular bent functions, permutation polynomials and PN functions over finite fields. Specifically, we are the first to use weakly regular bent functions to construct MUBs. In addition, we obtain a complete set of MUBs by employing linearized permutation polynomials over finite fields.The construction of 7-qubit unextendible product bases of size tenhttps://zbmath.org/1508.813972023-05-31T16:32:50.898670Z"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.5"Chen, Lin"https://zbmath.org/authors/?q=ai:chen.lin.5Summary: The construction of multiqubit unextendible product bases (UPBs) is an important problem in quantum information. We construct a 7-qubit UPB of size 10 by studying the unextendible orthogonal matrices. We apply our result to construct an 8-qubit UPB of size 18. Our results solve an open problem proposed in [\textit{L. Chen} and \textit{D. Ž Đoković}, J. Phys. A, Math. Theor. 51, No. 26, Article ID 265302, 24 p. (2018; Zbl 1396.81029)]. We also investigate the properties of general 7-qubit UPBs of size 10.Solving systems of linear algebraic equations via unitary transformations on quantum processor of IBM quantum experiencehttps://zbmath.org/1508.814432023-05-31T16:32:50.898670Z"Doronin, S. I."https://zbmath.org/authors/?q=ai:doronin.sergey-i"Fel'Dman, E. B."https://zbmath.org/authors/?q=ai:feldman.eduard-b"Zenchuk, A. I."https://zbmath.org/authors/?q=ai:zenchuk.alexandre-iSummary: We propose a protocol for solving systems of linear algebraic equations via quantum mechanical methods using the minimal number of qubits. We show that \((M+1)\)-qubit system is enough to solve a system of \(M\) equations for one of the variables leaving other variables unknown, provided that the matrix of a linear system satisfies certain conditions. In this case, the vector of input data (the rhs of a linear system) is encoded into the initial state of the quantum system. This protocol is realized on the 5-qubit superconducting quantum processor of IBM Quantum Experience for particular linear systems of three equations. We also show that the solution of a linear algebraic system can be obtained as the result of a natural evolution of an inhomogeneous spin-1/2 chain in an inhomogeneous external magnetic field with the input data encoded into the initial state of this chain. For instance, using such evolution in a 4-spin chain we solve a system of three equations.Quantum computing based on complex Clifford algebrashttps://zbmath.org/1508.814682023-05-31T16:32:50.898670Z"Hrdina, Jaroslav"https://zbmath.org/authors/?q=ai:hrdina.jaroslav"Návrat, Aleš"https://zbmath.org/authors/?q=ai:navrat.ales"Vašík, Petr"https://zbmath.org/authors/?q=ai:vasik.petrSummary: We propose to represent both \(n\)-qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of dimension \(2n\). In this framework, the Dirac formalism can be realized in straightforward way. We demonstrate its functionality by performing quantum computations with several well known examples of quantum gates. We also compare our approach with representations that use real geometric algebras.Low-rank approximation to entangled multipartite quantum systemshttps://zbmath.org/1508.814892023-05-31T16:32:50.898670Z"Lin, Matthew M."https://zbmath.org/authors/?q=ai:lin.matthew-m"Chu, Moody T."https://zbmath.org/authors/?q=ai:chu.moody-tSummary: Qualifying the entanglement of a mixed multipartite state by gauging its distance to the nearest separable state of a fixed rank is a challenging but critically important task in quantum technologies. Such a task is computationally demanding partly because of the necessity of optimization over the complex field in order to characterize the underlying quantum properties correctly and partly because of the high nonlinearity due to the multipartite interactions. Representing the quantum states as complex density matrices with respect to some suitably selected bases, this work offers two avenues to tackle this problem numerically. For the rank-1 approximation, an iterative scheme solving a nonlinear singular value problem is investigated. For the general low-rank approximation with probabilistic combination coefficients, a projected gradient dynamics is proposed. Both techniques are shown to converge globally to a local solution. Numerical experiments are carried out to demonstrate the effectiveness and the efficiency of these methods.Large \(n\) limit for the product of two coupled random matriceshttps://zbmath.org/1508.818292023-05-31T16:32:50.898670Z"Silva, Guilherme L. F."https://zbmath.org/authors/?q=ai:silva.guilherme-l-f"Zhang, Lun"https://zbmath.org/authors/?q=ai:zhang.lunSummary: For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem. This vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure. We carry out the steepest descent analysis for a \(4\times 4\) matrix-valued Riemann-Hilbert problem, which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified Bessel functions. A careful study of the vector equilibrium problem, combined with this asymptotic analysis, ultimately leads to the aforementioned convergence result for the limiting mean distribution, an explicit form of the associated spectral curve, as well as local Sine, Meijer-G and Airy universality results for the squared singular values considered.Non-Hermitian extensions of uncertainty relations with generalized metric adjusted skew informationhttps://zbmath.org/1508.818332023-05-31T16:32:50.898670Z"Fan, Yajing"https://zbmath.org/authors/?q=ai:fan.yajing"Cao, Huaixin"https://zbmath.org/authors/?q=ai:cao.huaixin"Wang, Wenhua"https://zbmath.org/authors/?q=ai:wang.wenhua"Meng, Huixian"https://zbmath.org/authors/?q=ai:meng.huixian"Chen, Liang"https://zbmath.org/authors/?q=ai:chen.liang.2Summary: In quantum mechanics, it is well known that the Heisenberg-Schrödinger uncertainty relations hold for two non-commutative observables and density operator. Recently some people start to focus on the uncertainty relations for two non-commutative non-Hermitian operators and density operator. In this paper, we introduce the generalized metric adjusted skew information, generalized metric adjusted correlation measure and the related quantities for non-Hermitian operators. Various properties of them are discussed. Finally, we establish several generalizations of uncertainty relation expressed in terms of the generalized metric adjusted skew information and obtain several results including previous results which can be given as corollaries of our non-Hermitian extensions of Heisenberg-type or Schrödinger-type uncertainty relations.Infrared extensions of the quadratic form of the ground state of scalar field theoryhttps://zbmath.org/1508.819292023-05-31T16:32:50.898670Z"Bolokhov, T. A."https://zbmath.org/authors/?q=ai:bolokhov.timur-aSummary: We extend the quadratic form of the Gaussian functional of the free quantum scalar field theory to the set of functions decreasing in the infinity as \({\left|\overrightarrow{x}\right|}^{-1} \). We use the momentum-space representation (after the Fourier transform and the scalar product is generated by the quadratic form of the Laplace operator (potential term of the quantum Hamiltonian).Investigation of the two-cut phase region in the complex cubic ensemble of random matriceshttps://zbmath.org/1508.820172023-05-31T16:32:50.898670Z"Barhoumi, Ahmad"https://zbmath.org/authors/?q=ai:barhoumi.ahmad"Bleher, Pavel"https://zbmath.org/authors/?q=ai:bleher.pavel-m"Deaño, Alfredo"https://zbmath.org/authors/?q=ai:deano.alfredo"Yattselev, Maxim"https://zbmath.org/authors/?q=ai:yattselev.maxim-lSummary: We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential \(V(M) = - \frac{1}{3}M^3 + tM\), where \(t\) is a complex parameter. As proven in our previous paper [\textit{P. Bleher} et al., J. Stat. Phys. 166, No. 3--4, 784--827 (2017; Zbl 1372.82015)], the whole phase space of the model, \(t\in\mathbb{C}\), is partitioned into two phase regions, \(O_{\mathsf{one}\text{-}\mathsf{cut}}\) and \(O_{\mathsf{two}\text{-}\mathsf{cut}}\), such that in \(O_{\mathsf{one}\text{-}\mathsf{cut}}\) the equilibrium measure is supported by one Jordan arc (cut) and in \(O_{\mathsf{two}\text{-}\mathsf{cut}}\) by two cuts. The regions \(O_{\mathsf{one}\text{-}\mathsf{cut}}\) and \(O_{\mathsf{two}\text{-}\mathsf{cut}}\) are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In [loc. cit.], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter \(t\), but not of the parameter \(t\) itself (so that the Cauchy-Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann-Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of \(S\)-curves and quadratic differentials.
{\copyright 2022 American Institute of Physics}The random normal matrix model: insertion of a point chargehttps://zbmath.org/1508.820462023-05-31T16:32:50.898670Z"Ameur, Yacin"https://zbmath.org/authors/?q=ai:ameur.yacin"Kang, Nam-Gyu"https://zbmath.org/authors/?q=ai:kang.nam-gyu"Seo, Seong-Mi"https://zbmath.org/authors/?q=ai:seo.seong-miSummary: In this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (``Mittag-Leffler fields'') and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for \(\log |p_n(\zeta )|\) where \(p_n\) is the characteristic polynomial of an \(n\):th order random normal matrix.\(\mathcal{UV}\)-theory of a class of semidefinite programming and its applicationshttps://zbmath.org/1508.900552023-05-31T16:32:50.898670Z"Huang, Ming"https://zbmath.org/authors/?q=ai:huang.ming"Yuan, Jin-long"https://zbmath.org/authors/?q=ai:yuan.jinlong"Pang, Li-ping"https://zbmath.org/authors/?q=ai:pang.liping"Xia, Zun-quan"https://zbmath.org/authors/?q=ai:xia.zunquanConvex semidefinite optimization problems are converted into eigenvalue problems by means of exact penalty functions. First-and second-order derivatives of the \(\mathcal{U}\)-Lagrangian of the function of the largest eigenvalues are derived under a transversality condition holds. A conceptual algorithm with superlinear convergence for \(\mathcal{UV}\)-decomposition is proposed, and an application involving an opimization problem containing a bilinear matrix inequality in the constraints is studied.
Reviewer: Sorin-Mihai Grad (Paris)Optimization of realignment criteria and its applications for multipartite quantum stateshttps://zbmath.org/1508.901222023-05-31T16:32:50.898670Z"Shen, Shu-Qian"https://zbmath.org/authors/?q=ai:shen.shuqian"Chen, Lou"https://zbmath.org/authors/?q=ai:chen.lou"Hu, An-Wen"https://zbmath.org/authors/?q=ai:hu.an-wen"Li, Ming"https://zbmath.org/authors/?q=ai:li.ming.3Summary: By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix, and multiple rows and columns from vectorization of reduced density matrices, the authors of [\textit{S-Q. Shen} et al., ``Separability criteria based on the realignment of density matrices and reduced density matrices'', Phys. Rev. A (3) 92, No. 4, Article ID 042332, 5 p. (2015; \url{doi:10.1103/PhysRevA.92.042332})] presented a family of separable criteria to improve the computable cross-norm or realignment criterion [\textit{O. Rudolph}, ``Some properties of the computable cross-norm criterion for separability'', Phys. Rev. A (3) 67, No. 3, Article ID 032312 (2003; \url{doi:10.1103/PhysRevA.67.032312}); \textit{K. Chen} and \textit{L. A. Wu}, Quantum Inf. Comput. 3, No. 3, 193--202 (2003; Zbl 1152.81692)]. In this paper, we first show that these criteria achieve their optimization when the parameterized matrix is chosen to be a constant matrix. It is then proved that the optimized criterion is equivalent to the corresponding criterion with one additional row and one additional column. This reduces the computation cost, since the combined realignment matrix possesses a lower dimension. Finally, the optimized criterion is further used to achieve the separable criterion for multipartite quantum states, which, by using a numerical example, is more efficient than the corresponding previous criteria based on linear contraction methods and sequential realignment methods.Towards a health-aware fault tolerant control of complex systems: a vehicle fleet casehttps://zbmath.org/1508.930792023-05-31T16:32:50.898670Z"Lipiec, Bogdan"https://zbmath.org/authors/?q=ai:lipiec.bogdan"Mrugalski, Marcin"https://zbmath.org/authors/?q=ai:mrugalski.marcin"Witczak, Marcin"https://zbmath.org/authors/?q=ai:witczak.marcin"Stetter, Ralf"https://zbmath.org/authors/?q=ai:stetter.ralfSummary: The paper deals with the problem of health-aware fault-tolerant control of a vehicle fleet. In particular, the development process starts with providing the description of the process along with a suitable Internet-of-Things platform, which enables appropriate communication within the vehicle fleet. It also indicates the transportation tasks to the designated drivers and makes it possible to measure their realization times. The second stage pertains to the description of the analytical model of the transportation system, which is obtained with the max-plus algebra. Since the vehicle fleet is composed of heavy duty machines, it is crucial to monitor and analyze the degradation of their selected mechanical components. In particular, the components considered are ball bearings, which are employed in almost every mechanical transportation system. Thus, a fuzzy logic Takagi-Sugeno approach capable of assessing their time-to-failure is proposed. This information is utilized in the last stage, which boils down to health-aware and fault-tolerant control of the vehicle fleet. In particular, it aims at balancing the exploitation of the vehicles in such a way as to maximize they average time-to-failure. Moreover, the fault-tolerance is attained by balancing the use of particular vehicles in such a way as to minimize the effect of possible transportation delays within the system. Finally, the effectiveness of the proposed approach is validated using selected simulation scenarios involving vehicle-based transportation tasks.Prescribed-time observers of LPV systems: a linear matrix inequality approachhttps://zbmath.org/1508.931312023-05-31T16:32:50.898670Z"Zhang, Jiancheng"https://zbmath.org/authors/?q=ai:zhang.jiancheng"Wang, Zhenhua"https://zbmath.org/authors/?q=ai:wang.zhenhua"Zhao, Xudong"https://zbmath.org/authors/?q=ai:zhao.xudong"Wang, Yan"https://zbmath.org/authors/?q=ai:wang.yan.3"Xu, Ning"https://zbmath.org/authors/?q=ai:xu.ningSummary: This paper considers prescribed-time observer (PTO) designs for a class of linear parameter-varying (LPV) systems. Firstly, a full-order prescribed-time observer with time-varying gains is developed. The existence conditions are given in terms of linear matrix inequalities (LMIs). In addition, the reduced-order PTO is also considered in this paper. Moreover, it is shown that the existence conditions under which the full-order PTO exists can also guarantee the existence of a corresponding reduced-order PTO. The advantages of the full-order and the reduced-order PTOs over the existing asymptotic convergence observers are that (1) they can achieve exact estimations in almost any prescribed convergence time regardless of what the system initial values are. (2) the proposed time-varying gain PTOs can avoid the conservatism of the unknown input decoupling conditions brought about by the traditional polytopic LPV observer design methods. Finally, two examples are given to illustrate the effectiveness of the proposed methods.Blind inverse problems with isolated spikeshttps://zbmath.org/1508.940052023-05-31T16:32:50.898670Z"Debarnot, Valentin"https://zbmath.org/authors/?q=ai:debarnot.valentin"Weiss, Pierre"https://zbmath.org/authors/?q=ai:weiss.pierreSummary: Assume that an unknown integral operator living in some known subspace is observed indirectly, by evaluating its action on a discrete measure containing a few isolated Dirac masses at an unknown location. Is this information enough to recover the impulse response location and the operator with a sub-pixel accuracy? We study this question and bring to light key geometrical quantities for exact and stable recovery. We also propose an in-depth study of the presence of additive white Gaussian noise. We illustrate the well-foundedness of this theory on the challenging optical imaging problem of blind deconvolution and blind deblurring with non-stationary operators.An iterative decoupled method with weighted nuclear norm minimization for image restorationhttps://zbmath.org/1508.940132023-05-31T16:32:50.898670Z"Lv, Xiao-Guang"https://zbmath.org/authors/?q=ai:lv.xiaoguang"Li, Fang"https://zbmath.org/authors/?q=ai:li.fang.4Summary: Recently, iterative methods based on decoupling of deblurring and denoising steps have received remarkable attention in image processing problems. In this paper, we propose an effective iterative decoupled approach for image restoration. The main idea is to split the image restoration problem into two minimization subproblems: deblurring and denoising. An efficient deblurring method using fast transforms is used in the deblurring step and a weighted nuclear norm minimization method is employed in the denoising step. We analyze the convergence of the developed iterative method under some assumptions. Some numerical experiments are given to demonstrate the effectiveness of the proposed method.Additive cyclic complementary dual codes over \(\mathbb{F}_4\)https://zbmath.org/1508.941012023-05-31T16:32:50.898670Z"Shi, Minjia"https://zbmath.org/authors/?q=ai:shi.minjia"Liu, Na"https://zbmath.org/authors/?q=ai:liu.na"Özbudak, Ferruh"https://zbmath.org/authors/?q=ai:ozbudak.ferruh"Solé, Patrick"https://zbmath.org/authors/?q=ai:sole.patrickSummary: An additive cyclic code of length \(n\) over \(\mathbb{F}_4\) can be defined equivalently as an \(\mathbb{F}_2 [x] / \langle x^n + 1 \rangle \)-submodule of \(\mathbb{F}_4 [x] / \langle x^n + 1 \rangle \). In this paper we study additive cyclic and complementary dual codes of odd length over \(\mathbb{F}_4\) with respect to the trace Hermitian inner product and the trace Euclidean inner product. We characterize subfield subcodes and trace codes of these codes by their generators as binary cyclic codes.