Recent zbMATH articles in MSC 15https://www.zbmath.org/atom/cc/152022-05-16T20:40:13.078697ZWerkzeugCombinatorial properties of the enhanced principal rank characteristic sequence over finite fieldshttps://www.zbmath.org/1483.050162022-05-16T20:40:13.078697Z"Dukes, Peter J."https://www.zbmath.org/authors/?q=ai:dukes.peter-james"Martínez-Rivera, Xavier"https://www.zbmath.org/authors/?q=ai:martinez-rivera.xavierSummary: The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix \(B \in \mathbb{F}^{n \times n}\) is defined as \(\ell_1 \ell_2 \cdots \ell_n\), where \(\ell_j \in\) \{\texttt{A,S,N}\} according to whether all, some but not all, or none of the principal minors of order \(j\) of \(B\) are nonzero. Building upon the second author's recent classification of the epr-sequences of symmetric matrices over the field \(\mathbb{F} = \mathbb{F}_2\) [\textit{X. Martínez-Rivera}, Linear Multilinear Algebra 65, No. 8, 1581--1599 (2017; Zbl 1370.15005)], we initiate a study of the case \(\mathbb{F}= \mathbb{F}_3\). Moreover, epr-sequences over finite fields are shown to have connections to Ramsey theory and coding theory.Graphs \(G\) with nullity \(2c(G) + p(G) - 1\)https://www.zbmath.org/1483.050922022-05-16T20:40:13.078697Z"Chang, Sarula"https://www.zbmath.org/authors/?q=ai:chang.sarula"Tam, Bit-Shun"https://www.zbmath.org/authors/?q=ai:tam.bit-shun"Li, Jianxi"https://www.zbmath.org/authors/?q=ai:li.jianxi"Zheng, Yirong"https://www.zbmath.org/authors/?q=ai:zheng.yirongSummary: The nullity \(\eta(G)\) of a (simple) graph \(G\) is the multiplicity of \(0\) as an eigenvalue of the adjacency matrix of \(G\). It was shown by \textit{H. Ma} et al. [Linear Algebra Appl. 438, No. 1, 331--341 (2013; Zbl 1257.05026)] that for a connected graph \(G\) with \(n(G)\) vertices, \(e(G)\) edges and \(p(G)(\geq 1)\) pendant vertices, \(\eta(G) \leq 2c(G) + p(G) - 1\), where \(c(G) = e(G) - n(G) + \theta(G)\) is the cyclomatic number of \(G\), \(\theta(G)\) being the number of connected components of \(G\). In this paper connected graphs \(G\) and trees \(T\) that satisfy respectively the conditions \(\eta(G) = 2c(G) + p(G) - 1\) and \(\eta(T) = p(T) - 1\) are characterized. Besides, we identify the matched and mismatched vertices in a tree \(T\) that satisfies \(\eta(T) = p(T) - 1\) and characterize trees \(T\) that have a pendant vertex matched in \(T\) and satisfy \(\eta(T) = p(T) - 2\). We obtain new sharp lower and upper bounds for a tree \(T\) with \(p(T) \geq 3\), given in terms of \(p(T)\) and the numbers of two special kinds of internal paths in \(T\). It is also proved that for any integer \(c \geq 2\), the set of values attained by \(\eta(G) - n(G) + 2m(G)\), as \(G\) runs through all connected, leaf-free graphs \(G\) with cyclomatic number \(c\), is \(\{-c + 1, -c + 2, \ldots, 2c - 3, 2c - 2\}\).Real symmetric matrices and their negative eigenvalueshttps://www.zbmath.org/1483.050962022-05-16T20:40:13.078697Z"Mohammadian, Ali"https://www.zbmath.org/authors/?q=ai:mohammadian.aliSummary: For two integers \(k \geqq 0\) and \(q \geqq 1\), consider symmetric matrices \(M\) with \(k\) negative eigenvalues counted with multiplicities and \(q\) pairwise distinct values of entries such that the rows of \(M\) are mutually distinct and the largest diagonal entry of \(M\) is less than or equal to the smallest off-diagonal entry of \(M\). It is shown that the number of such matrices is finite when \(k\) and \(q\) are fixed. This generalizes some known results on the adjacency matrices of graphs. It is conjectured that any twin-free graph on \(n\) vertices with no isolated vertices has at least \(- 1 + \log_2(n + 2)\) negative adjacency eigenvalues.The completely delocalized region of the Erdős-Rényi graphhttps://www.zbmath.org/1483.051512022-05-16T20:40:13.078697Z"Alt, Johannes"https://www.zbmath.org/authors/?q=ai:alt.johannes"Ducatez, Raphaël"https://www.zbmath.org/authors/?q=ai:ducatez.raphael"Knowles, Antti"https://www.zbmath.org/authors/?q=ai:knowles.anttiSummary: We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on \(N\) vertices with edge probability \(\frac{d}{N}\). We determine the full region of delocalization by determining the critical values of \(\frac{d}{\log N}\) down to which delocalization persists: for \(\frac{d}{\log N}> \frac{1}{\log 4-1}\) all eigenvectors are completely delocalized, and for \(\frac{d}{\log N}> 1\) all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [\textit{J. Alt} et al., Commun. Math. Phys. 388, No. 1, 507--579 (2021; Zbl 1477.15029); ``Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs'', Preprint, \url{arXiv:2106.12519}] that localized eigenvectors exist in the corresponding spectral regions.Ihara zeta function and cycle fluctuations of random Ramanujan graph generated by \(p\)-adic pseudorandom numbershttps://www.zbmath.org/1483.051632022-05-16T20:40:13.078697Z"Naito, Koichiro"https://www.zbmath.org/authors/?q=ai:naito.koichiroSummary: Using the sequences of the pseudorandom numbers generated by the \(p\)-adic logistic map, we construct some pseudorandom adjacency matrices and their various nonregular graphs, which are characterized as weak Ramanujan graphs. For these various types of random graphs, which have some clustering properties, we statistically calculate the fitting curves to the distributions of these cycle numbers by using their cycle basis and estimating the numbers of each cycle in these graphs. We can estimate their fluctuation exponents and we call this value the cycle-fluctuation exponent (CFE). Furthermore, using the distributions of poles of the Ihara zeta function, we introduce the Ramanujan radius ratio (RRr) and the Ramanujan radius standard deviation (RRd). Comparing the value CFE to the other parameters, the small-worldness coefficient (SWC), RRr and RRd and clarifying the relations among these parameters numerically, we characterize the clustering and random properties of our random Ramanujan graphs.Combinatorics in the exterior algebra and the Bollobás two families theoremhttps://www.zbmath.org/1483.051932022-05-16T20:40:13.078697Z"Scott, Alex"https://www.zbmath.org/authors/?q=ai:scott.alexander-d"Wilmer, Elizabeth"https://www.zbmath.org/authors/?q=ai:wilmer.elizabeth-lSummary: We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. Using initial monomials, projections of the underlying vector space onto subspaces, and the interior product, we find analogs of local and global LYM inequalities, the Erdős-Ko-Rado theorem, and the Ahlswede-Khachatrian bound for \(t\)-intersecting hypergraphs. Using these tools, we prove a new extension of the Two Families Theorem of Bollobás, giving a weighted bound for subspace configurations satisfying a skew cross-intersection condition. We also verify a recent conjecture of \textit{D. Gerbner} et al. [``Set systems related to a house allocation problem'', Preprint, \url{arXiv:1910.04666}] on pairs of set systems satisfying both an intersection and a cross-intersection condition.Matrices over noncommutative rings as sums of \(k\)th powershttps://www.zbmath.org/1483.112152022-05-16T20:40:13.078697Z"Katre, S. A."https://www.zbmath.org/authors/?q=ai:katre.shashikant-a"Wadikar, Kshipra"https://www.zbmath.org/authors/?q=ai:wadikar.kshipraSummary: The first author and \textit{A. S. Garge} [Proc. Am. Math. Soc. 141, No. 1, 103--113 (2013; Zbl 1276.11160)] obtained necessary and sufficient trace conditions for matrices over commutative rings with unity to be sums of \(k\)th powers. In this paper, we show that these conditions hold for matrices over noncommutative rings too. For \(n \geq 2\), we deduce Vaserstein's result for sums of squares of matrices and also obtain nice trace conditions for \(n \times n\) matrices to be sums of cubes. For an integer \(n \geq k \geq 2\), we get a sufficient condition \(\pmod k\) for an \(n \times n\) matrix over a noncommutative ring to be a sum of \(k\)th powers.Cramer's rule over quaternions and split quaternions: a unified algebraic approach in quaternionic and split quaternionic mechanicshttps://www.zbmath.org/1483.112512022-05-16T20:40:13.078697Z"Wang, Gang"https://www.zbmath.org/authors/?q=ai:wang.gang.4|wang.gang.1|wang.gang|wang.gang.5|wang.gang.3|wang.gang.2"Zhang, Dong"https://www.zbmath.org/authors/?q=ai:zhang.dong"Guo, Zhenwei"https://www.zbmath.org/authors/?q=ai:guo.zhenwei"Jiang, Tongsong"https://www.zbmath.org/authors/?q=ai:jiang.tongsongThe paper treats solutions of matrix equations over the so called ``$v$-quaternions'', where ``$v$'' stands for the square of the pseudo-scaler, distinguishing between the classical Hamiltonian $(v = -1)$ and the split-quaternion $(v = 1)$ setting. In other words, both algebras are considered as particular cases of an infinite one-parameter set, rather than real forms of the biquaternions. Using the standard $2\times 2$ complex matrix representation, the determinant and the adjoint (hence, the inverse) are introduced in this more general setting. Finally, the classical Crammer's rules are derived and applied for systems of $v$-quaternionic liner equations.
The style of the exposition is quite approachable, well suited even for students. No particular background in Clifford algebras is assumed, the brief introduction fills that possible gap quite efficiently. There are comprehensive examples and a nice selection of related literature which may interest researchers focused on the applications of algebra in both algorithms and theoretical physics.
Reviewer: Danail Brezov (Sofia)Invariant rings of sums of fundamental representations of \(\operatorname{SL}_n\) and colored hypergraphshttps://www.zbmath.org/1483.130152022-05-16T20:40:13.078697Z"Braun, Lukas"https://www.zbmath.org/authors/?q=ai:braun.lukasIn the paper under review, the author describes a new symbolic method, by using colored hypergraphs and by using this new method, the invariant rings of (sums of) fundamental representations of \(\operatorname{SL}_n\) are classified for \(n=4, 5\) and the relations (so called syzygies) are also obtained for \(\operatorname{SL}_4.\)
The research is somewhat an extension of the previous work of \textit{F. D. Grosshans} et al. [Invariant theory and superalgebras. Providence, RI: American Mathematical Society (AMS) (1987; Zbl 0648.15020)] which involves superalgebras and also the author's recent work about the classification of \(\operatorname{SL}_n\) representations for which their invariant rings form a complete intersection [Transform. Groups 26, No. 1, 115--144 (2021; Zbl 1482.20025)].
The paper includes many simple examples to follow the new method and showing by examples where the classical symbolic method (i.e., bracket monomials) fails but it is more convenient to do this with colored hypergraphs.
The graphsums \(\Upsilon_1,\dotsc,\Upsilon_k\) are called \emph{reducibly independent} if the only reducible linear combination \(\sum a_i\Upsilon_i\) is the trivial one. Based on this definition, the main result of the paper declares that there is a bijection between maximal reducibly independent graphsums and generators of the invariant ring \(\mathbb{C}[W]^{\operatorname{SL}_n}.\)
It is also worth to mention that the colored hypergraphs used are nicely plotted making it easy to follow the arguments. Moreover, in the last part (Section 7) several possible extensions of this technique are described.
Reviewer: Ugur Madran (Eqaila)Spectrally simple zeros of zeon polynomialshttps://www.zbmath.org/1483.130212022-05-16T20:40:13.078697Z"Stacey Staples, G."https://www.zbmath.org/authors/?q=ai:stacey-staples.gWorking over any base field \(K\), the \emph{\(n\)-particle zeon algebra} \(K\mathfrak{Z}_n\) is defined as \(K[\zeta_{\{1\}}, \zeta_{\{2\}}, \ldots, \zeta_{\{n\}}]/(\zeta_{\{1\}}^2, \zeta_{\{2\}}^2, \ldots, \zeta_{\{n\}}^2)\). Zeon algebras generalize the ring of dual numbers \(K[\epsilon]/(\epsilon^2)\), and every element of a zeon algebra (\emph{zeon} for short) is either nilpotent or invertible, depending on whether or not the zeon's constant term (its \emph{complex part}) is \(0\). Zeon algebras using the base field \(K=\mathbb{R}\) have been used in combinatorics and differential equations; this paper concerns an analogue of the Fundamental Theorem of Algebra for the \(n\)-particle complex zeon algebra \(\mathbb{C}\mathfrak{Z}_n\), whose elements are called \emph{complex zeons}.
After the introduction, there are three sections building up smaller results about the complex zeon algebra. Section 2 contains a proof of the special case that every invertible complex zeon has exactly \(k\) distinct \(k\)th roots for each natural number \(k\geq 1\), and constructs those roots explicitly using a recursive formula. Section 3 shows that polynomial division-with-remainder is still well-defined for polynomials whose coefficients are complex zeons, so long as the leading coefficient of the dividing polynomial is invertible. Section 4 classifies the zeon solution sets to polynomial equations with ordinary complex coefficients, showing that each zeon roots of a complex polynomial consists of an ordinary complex root plus a nilpotent zeon, where the nilpotent zeon raised to the power of the complex root's multiplicity must vanish. In particular, a complex polynomial with only simple roots in \(\mathbb{C}\) has no additional roots among the complex zeons.
The main result in section 5 is an analogue of the Fundamental Theorem of Algebra for monic polynomials with complex zeon coefficients. For such a polynomial \(\phi\), let \(f\) be the ordinary complex polynomial obtained by taking the complex part of each coefficient of \(\phi\). Then then main theorem states that for each simple root \(z\) of \(f\), there is a unique complex zeon \(\zeta\) with complex part \(z\) that is a root of \(\phi\); the proof is constructive with a formula and examples. The hypothesis that \(z\) be a simple root of \(f\) is necessary: section 5 contains an example of a nonconstant polynomial \(\phi\) with no complex zeon roots, and section 6 shows that if \(\phi\) has two zeon roots with the same complex part, then it has infinitely many such roots.
Reviewer: Owen Biesel (Northfield)An analogue of a result of Tits for transvection groupshttps://www.zbmath.org/1483.130262022-05-16T20:40:13.078697Z"Chattopadhyay, Pratyusha"https://www.zbmath.org/authors/?q=ai:chattopadhyay.pratyushaLet \(R\) be a commutative ring with \(R=2R\) and let \(I\) be an ideal of \(R\). Let \((P,\langle\,,\rangle)\) be a symplectic \(R\)-module with \(P\) a finitely generated projective \(R\)-module of even rank at least 4. Let \((Q,\langle\,,\rangle)\) be the orthogonal direct sum of \((P,\langle\,,\rangle)\) with a hyperbolic plane. One now has elementary symplectic transvections, generating a group \(\mathrm{ETrans_{Sp}}(Q,\langle\,,\rangle)\), and a subgroup \(\mathrm{ETrans_{Sp}}(IQ,\langle\,,\rangle)\) generated by ``relative elementary symplectic transvections'', congruent to the identity modulo \(I\). The group \(\mathrm{ETrans_{Sp}}(IQ,\langle\,,\rangle)\) depends on the chosen orthogonal direct sum decomposition of \((Q,\langle\,,\rangle)\). Now the result is that \(\mathrm{ETrans_{Sp}}(IQ,\langle\,,\rangle)\) contains a normal subgroup of \(\mathrm{ETrans_{Sp}}(Q,\langle\,,\rangle)\) that contains \(\mathrm{ETrans_{Sp}}(I^4Q,\langle\,,\rangle)\). There are similar results for orthogonal \(R\)-modules and for linear \(R\)-modules.
Reviewer: Wilberd van der Kallen (Utrecht)Derandomization and absolute reconstruction for sums of powers of linear formshttps://www.zbmath.org/1483.130442022-05-16T20:40:13.078697Z"Koiran, Pascal"https://www.zbmath.org/authors/?q=ai:koiran.pascal"Skomra, Mateusz"https://www.zbmath.org/authors/?q=ai:skomra.mateuszIn this paper, given a homogeneous polynomial \(f (x_1,\dots, x_n)\) of degree 3, the problem to decide whether it can be written as a sum of cubes of linearly independent linear forms is studied. In other words, an algorithm is given to check if, modulo a linear change of coordinates, we can re-write \(f (A(x_1, \dots, x_n)) = f (y_1,\dots, y_n) = a_1y_1^3 + a_2y_2^3 +\dots+ a_ny_n^3\), where \(\mathbf{\underline{y}} = A\mathbf{\underline{x}}\) is a linear change of coordinates and each \(a_i\in \{0, 1\}\).
The given algorithm works over \(\mathbb{C}\) and is algebraic, i.e., it performs only arithmetic operations and equality tests on the coefficients of the input polynomial \(f\) (also inequalities if working over \(\mathbb{R}\)); moreover the algorithm runs in polynomial time if \(f \in \mathbb{Q}[x_1, \dots, x_n]\) and it does not make any kind of polynomial factorization.
The algorithm can run in polynomial time because it just answers the question whether \(f\) can or cannot be decomposed in a sum of \(r \leq n\) sum of powers of linearly independent linear forms; it may not give the linear forms in question, whose coefficients may not be hard computable with arithmetic operations from those of \(f\) (examples are given).
The algorithms is obtained by viewing the input polynomial \(f\) as symmetric tensor \(T\) of size \(n\) and order 3 and considering the 3 ``slices'' of \(T\) (i.e. symmetric \(n\times n\) matrices) and to find a simultaneous diagonalization of them; this procedure uses the factorization properties of the Hessian determinant of \(f\).
Reviewer: Alessandro Gimigliano (Bologna)Relative EP matriceshttps://www.zbmath.org/1483.150012022-05-16T20:40:13.078697Z"Ferreyra, D. E."https://www.zbmath.org/authors/?q=ai:ferreyra.david-e"Malik, Saroj B."https://www.zbmath.org/authors/?q=ai:malik.saroj-bThis paper introduces a new concept that generalizes EP matrices to the rectangular case. Some interesting characterizations for this kind of matrices are given using both the singular value decomposition and the Hartwig-Spindelbock decomposition. Relationships to other known classes of matrices are presented.
Reviewer: Néstor Thome (Valencia)The composition of polynomials is a determinanthttps://www.zbmath.org/1483.150022022-05-16T20:40:13.078697Z"Carrillo, Sergio A."https://www.zbmath.org/authors/?q=ai:carrillo.sergio-aSummary: We show that the composition of two univariate polynomials can be written as a characteristic polynomial using a block-companion matrix. This matrix is useful to determine whether a polynomial is the composition of two other polynomials. Moreover, it provides bounds for the moduli of roots of such polynomials in the complex case.A combinatorial algorithm for computing the degree of the determinant of a generic partitioned polynomial matrix with \(2\times 2\) submatriceshttps://www.zbmath.org/1483.150032022-05-16T20:40:13.078697Z"Iwamasa, Yuni"https://www.zbmath.org/authors/?q=ai:iwamasa.yuniSummary: In this paper, we consider the problem of computing the degree of the determinant of a block-structured symbolic matrix (a generic partitioned polynomial matrix) \(A = (A_{\alpha \beta } x_{\alpha \beta } t^{d_{\alpha \beta }})\), where \(A_{\alpha \beta }\) is a \(2 \times 2\) matrix over a field \(\mathbf{F}\), \(x_{\alpha \beta }\) is an indeterminate, and \(d_{\alpha \beta }\) is an integer for \(\alpha , \beta = 1,2,\dots , n\), and \(t\) is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum perfect bipartite matching problem.
The main result of this paper is a combinatorial \(O(n^5)\)-time algorithm for the deg-det computation of a \((2 \times 2)\)-type generic partitioned polynomial matrix of size \(2n \times 2n\). We also present a min-max theorem between the degree of the determinant and a potential defined on vector spaces. Our results generalize the classical primal-dual algorithm (Hungarian method) and min-max formula (Egerváry's theorem) for maximum weight perfect bipartite matching.
For the entire collection see [Zbl 1476.90004].Eigenvalue problems for a class of infinite complex symmetric tridiagonal matrices with related three-term recurrence relationhttps://www.zbmath.org/1483.150042022-05-16T20:40:13.078697Z"Asai, Nobuyoshi"https://www.zbmath.org/authors/?q=ai:asai.nobuyoshi"Miyazaki, Yoshinori"https://www.zbmath.org/authors/?q=ai:miyazaki.yoshinoriSummary: We consider the eigenvalue problem of a class of infinite complex symmetric tridiagonal matrices whose diagonal and off-diagonal elements diverge in modulus, and which have a compact inverse. We regard the matrix as a linear operator mapping a maximal domain in Hilbert space \(\ell^2\) into \(\ell^2\). This paper aims to extend the work of Ikebe et al. on a class of eigenvalue problems and for which asymptotic error estimates have been obtained. In this paper we focus on the following points: (1) considering what class of zero-finding problems of three-term recurrence relations can be reformulated as eigenvalue problems of the class of infinite tridiagonal matrices stated above; (2) determining a class of matrices for which obtaining good approximate eigenvalues is guaranteed by using those of truncated principal sub-matrices; and (3) determining a class of matrices that permits us the asymptotic error estimates computed as in (2).Location of Ritz values in the numerical range of normal matriceshttps://www.zbmath.org/1483.150052022-05-16T20:40:13.078697Z"Dela Rosa, Kennett L."https://www.zbmath.org/authors/?q=ai:dela-rosa.kennett-l"Woerdeman, Hugo J."https://www.zbmath.org/authors/?q=ai:woerdeman.hugo-jSummary: Let \(\mu_1\) be a complex number in the numerical range \(W(A)\) of a normal matrix \(A\). In the case when no eigenvalues of \(A\) lie in the interior of \(W(A)\), we identify the smallest convex region containing all possible complex numbers \(\mu_2\) for which \(\begin{bmatrix} \mu_1 & \ast \\ 0 & \mu_2 \end{bmatrix}\) is a 2-by-2 compression of \(A\).Correction to: ``Equivalence classes of e-matrices and associated eigenvalue localization regions''https://www.zbmath.org/1483.150062022-05-16T20:40:13.078697Z"Marsli, Rachid"https://www.zbmath.org/authors/?q=ai:marsli.rachid"Hall, Frank J."https://www.zbmath.org/authors/?q=ai:hall.frank-jTwo typographical errors in Theorem 3.10 in the authors' paper [ibid. 68, No. 5, 915--930 (2020; Zbl 1456.15009)] are corrected.Algorithmic characterization of pentadiagonal ASSR matriceshttps://www.zbmath.org/1483.150072022-05-16T20:40:13.078697Z"Alonso, P."https://www.zbmath.org/authors/?q=ai:alonso.pedro"Peña, J. M."https://www.zbmath.org/authors/?q=ai:pena.juan-manuel"Serrano, M. L."https://www.zbmath.org/authors/?q=ai:serrano.maria-luisaSummary: A matrix is sign regular if all its minors of the same order are of the same sign. A very important subclass of sign regular matrices is formed by those matrices whose non-trivial minors are non-zero. These matrices are called almost strictly sign regular matrices. In this work, an algorithmic characterization of pentadiagonal almost strictly sign regular matrices is presented.The inverse eigenvalue problem for linear treeshttps://www.zbmath.org/1483.150082022-05-16T20:40:13.078697Z"Johnson, Charles R."https://www.zbmath.org/authors/?q=ai:johnson.charles-richard-jun|johnson.charles-royal"Wakhare, Tanay"https://www.zbmath.org/authors/?q=ai:wakhare.tanay-vSummary: We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in [\textit{C. R. Johnson} et al., Discrete Math. 333, 39--55 (2014; Zbl 1298.05068)]. This is the most general class of trees for which the inverse eigenvalue problem has been solved. We explore many consequences, including the Degree Conjecture for possible spectra, upper bounds for the minimum number of eigenvalues of multiplicity \(1\), and the equality of the diameter of a linear tree and its minimum number of distinct eigenvalues.A unified proof of interlacing properties of eigenvalues of totally positive matriceshttps://www.zbmath.org/1483.150092022-05-16T20:40:13.078697Z"Chen, Xi"https://www.zbmath.org/authors/?q=ai:chen.xi|chen.xi.1|chen.xi.2|chen.xi.4|chen.xi.5"Zheng, Sai-Nan"https://www.zbmath.org/authors/?q=ai:zheng.sainanThe manuscript presents a complete simple proof of interlacing of eigenvalues for the class of totally positive matrices, i.e., matrices with nonnegative minors. The proof is based on the characterizations of both interlacing and compatibility properties of polynomials with real zeros.
Reviewer: Iveta Hnetynkova (Praha)Sharp nonzero lower bounds for the Schur product theoremhttps://www.zbmath.org/1483.150102022-05-16T20:40:13.078697Z"Khare, Apoorva"https://www.zbmath.org/authors/?q=ai:khare.apoorvaTo sketch the background of this interesting paper, let us recall a few basic facts from matrix analysis and linear algebra. Fix \(m, n \in \mathbb{N}\). Let \(M_{m,n}(\mathbb{F}) \equiv \mathbb{F}^{m \times n}\) denote the set of all \(m \times n\) matrices with entries in \(\mathbb{F}\), where \(\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}\). A matrix \(A \in M_n(\mathbb{F}) \equiv M_{n,n}(\mathbb{F})\) is positive semidefinite (resp., positive definite) in \(\mathbb{M}_n(\mathbb{F})\) if and only if the following two conditions are satisfied:
\begin{itemize}
\item[(i)] \(A = A^\ast\) (i.e., \(A\) is Hermitian);
\item[(ii)] \(x^\ast A x \geq 0\) (respectively \(x^\ast A x > 0\)) for all \(x \in {\mathbb{F}^n}\setminus\{0\}\).
\end{itemize}
If \(A \in \mathbb{M}_n(\mathbb{R})\), then \(A^\ast = A^\top\). In the complex case, condition (i) is unnecessary. However, the inclusion of (i) guarantees that \(A \in M_n(\mathbb{R})\) is positive semidefinite in \(M_n(\mathbb{R})\) if and only if \(A \in M_n(\mathbb{R}) \subseteq M_n(\mathbb{C})\) is positive semidefinite in \(M_n(\mathbb{C})\). To recognise this fact, we just have to examine the example of the non-symmetric real \(2 \times 2\)-matrix \(\left(\begin{smallmatrix} 0 & 1\\ -1 & 0 \end{smallmatrix}\right)\).
Let \(\mathbb{P}_n(\mathbb{F}) \subseteq \mathbb{M}_n(\mathbb{F})\) denote the set of all positive semidefinite \(n \times n\) matrices with entries in \(\mathbb{F}\) (in the following simply abbreviated as PSD).
The considerably rich structure of the convex cone \(\mathbb{P}_n(\mathbb{F})\) is not only essential in linear algebra and matrix analysis itself. \(\mathbb{P}_n(\mathbb{F})\) also plays a key role in conic optimisation (in particular, semidefinite programming), quantum information theory, computational complexity and spectral graph theory.
If \(A, B \in \mathbb{P}_n(\mathbb{F})\), it is a well-known fact that in general the standard matrix product \(AB\) is not positive semidefinite. In fact, \(AB \in \mathbb{P}_n(\mathbb{F})\) if and only if \(AB\) is Hermitian, which is equivalent to \(AB = BA\), i.e., \(A\) and \(B\) commute (see [\textit{A. R. Meenakshi} and \textit{C. Rajian}, Linear Algebra Appl. 295, No. 1--3, 3--6 (1999; Zbl 0940.15022)]).
The situation changes completely when the Hadamard product of matrices is considered instead. If \(C = (c_{ij}) \in \mathbb{M}_{m,n}(\mathbb{F})\) and \(D = (d_{ij}) \in \mathbb{M}_{m,n}(\mathbb{F})\), the \textit{Hadamard product of \(C\) and \(D\)} is defined as
\[
C \circ D : = (c_{ij}\,d_{ij}) \quad ((i,j) \in [m] \times [n])\,.
\]
The Hadamard product is sometimes called the \textit{entrywise product} for obvious reasons, or the \textit{Schur product}, because of some early and basic results about the product obtained by Schur (see [\textit{R. A. Horn} and \textit{C. R. Johnson}, Topics in matrix analysis. Cambridge etc.: Cambridge University Press (1991; Zbl 0729.15001)]).
(NB: In my opinion, the symbolic notation \(\circ\) perhaps could lead to a minor ambiguity, since quite regularly, \(\circ\) denotes composition of mappings. In [\textit{V. Paulsen}, Completely bounded maps and operator algebras. Cambridge: Cambridge University Press (2002; Zbl 1029.47003)], the symbol \(\ast\) is used instead. Also \(\odot\) could be a useful substitution for \(\circ\).)
Like the usual matrix product, the distributive law also holds for the Hadamard product: \(A\circ(B + C) = A \circ B + A \circ C\). Unlike the usual matrix product, the Hadamard product is commutative: \(A \circ B = B \circ A\).
In the real subspace of Hermitian matrices, the most common order relation is the \textit{Loewner partial order}. It is induced by the cone \(\mathbb{P}_n(\mathbb{F})\). By definition, \(A \geq B\) if and only if both \(A, B\) are Hermitian and \(A-B \in \mathbb{P}_n(\mathbb{F})\). A seminal result by \textit{I. Schur} [J. Reine Angew. Math. 140, 1--28 (1911; JFM 42.0367.01)], nowadays known as ``Schur product theorem'', asserts that if \(A \geq 0\) and \(B \geq 0\), then also \(A \circ B \geq 0\) (see [Horn and Johnson, loc. cit.], Chapter 5.2). Moreover, if we implement the (unique) positive semidefinite root of a PSD matrix (\(A = A^{1/2}\,A^{1/2}\) for all \(A \in \mathbb{P}_n(\mathbb{F})\)) into the trace, we reobtain the well-known fact that \(\mathbb{P}_n(\mathbb{F})\) is a self-dual cone (for both fields), meaning that
\[
A \geq 0 \text{ if and only if } \langle A, B\rangle_F : = \operatorname{tr}(A B^\ast) = \operatorname{tr}(A B) \geq 0 \ \text{ for all } B \in \mathbb{P}_n(\mathbb{F}).
\]
Thereby,
\[
\mathbb{M}_{m,n}(\mathbb{F}) \times \mathbb{M}_{m,n}(\mathbb{F}) \ni (C, D) \mapsto \langle C, D\rangle_F : = \operatorname{tr}(C D^\ast) = \operatorname{tr}(D^\ast\,C)
\]
denotes the Frobenius inner product. Because of Hölder's inequality, the matrix \(C \in \mathbb{M}_{m,n}(\mathbb{F})\) can be identified as bounded linear operator from \(l_p^n\) to \(l_q^m\) (of finite rank) which satisfies \(\Vert A \Vert \leq \big(\sum_{i = 1}^m \sum_{j = 1}^n \vert a_{ij}\vert^q\big)^{1/q}\), for \(1 \leq p, q \leq \infty\) with \(\frac{1}{p} + \frac{1}{q} = 1\). Thus, if we view \(C\) as Hilbert-Schmidt operator from the Hilbert space \(l_2^n\) to the Hilbert space \(l_2^m\), the Frobenius inner product coincides with the Hilbert-Schmidt inner product of \(C \in \mathcal{S}_2(l_2^n, l_2^m)\).
The paper consists of three parts. In the first part, the main result (Theorem A) enhances a result of \textit{J. Vybíral}, developed in [Adv. Math. 368, Article ID 107140, 8 p. (2020; Zbl 1441.15024), Theorem 1]. The author improves Vybíral's \textit{positive} lower bound (with respect to the Loewner partial order, induced by PSD matrices). That lower bound appears in a stronger version of the Schur product theorem, also introduced by Vybíral. Moreover, Theorem A unveils that the emerging positive constant as part of the lower bound is the maximum possible one. It cannot be further increased.
Besides an application of the Cauchy-Schwarz inequality with respect to the Frobenius inner product, the trace equality (1.12) plays a significant role in the proof of Theorem A.
We would like to emphasize that (1.12) itself can be deduced from the following simple fact, which however reveals a further direct link between the Hadamard product and the standard matrix product:
\[
M \circ uv^\top = D_u\,M\,D_v \text{ for all } M \in \mathbb{M}_{m,n}(\mathbb{F}), (u,v) \in \mathbb{F}^m \times \mathbb{F}^n\,,
\]
where \(D_x\) denotes the \(p \times p\)-diagonal matrix, whose \((i,j)\)'th entry is given by \(\delta_{ij}\,x_j\), for any \(x = (x_1, \ldots, x_p)^\top \in \mathbb{F}^p\). Here \(\delta_{ij}\) is the Kronecker delta.
The first part concludes with a few (fairly technical) refinements of Theorem A including the determination of an upper bound and an in-depth discussion of special cases of Theorem A which were published in the past, primarily by Vybíral.
In the second part, the author develops a ``suitably modified'' version of Theorem A which produces a non-trivial positive lower bound in the case of Hilbert-Schmidt operators between arbitrary, not necessarily finite-dimensional Hilbert spaces (see Theorem 2.4).
In the third and last part a few applications of Theorem A, including the important entrywise calculus on classes of positive matrices are touched briefly. The latter also plays a crucial role regarding an approximation of the upper bound of the real and complex Grothendieck constant in the famous Grothendieck inequality; a fact, based on my own research activities, not covered here. Complex kernels with lower bounds are introduced, and related non-trivial PSD matrices are listed. Even indications for future research problems are sketched. In this respect, I would like to add a further research question: could Theorem A and its applications to the entrywise calculus on classes of positive matrices even become useful to improve the already existing \textit{lower} bounds of the complex and real Grothendieck constant?
Finally, as a minor and weak ``criticism'', let me point out that it would be very helpful for the reader to see explicitly on which field \(\mathbb{F} \in \{\mathbb{R},\mathbb{C}\}\) a statement about \(\mathbb{P}_n(\mathbb{F})\) is referred to, mainly to work out what results hold for both fields simultaneously. In my view, the primarily considered field is \(\mathbb{C}\).
Reviewer: Frank Oertel (London)Partial determinant inequalities for positive semidefinite block matriceshttps://www.zbmath.org/1483.150112022-05-16T20:40:13.078697Z"Li, Yongtao"https://www.zbmath.org/authors/?q=ai:li.yongtao"Lin, Xiqin"https://www.zbmath.org/authors/?q=ai:lin.xiqin"Feng, Lihua"https://www.zbmath.org/authors/?q=ai:feng.lihuaThere are many reasons to study block matrices \(A=[A_{ij}]\) with each block \(A_{ij}\) square. Let \(f\) be a functional on the space of square matrices. Applying \(f\) to each block of \(A\) results in a smaller (sized) matrix \([f(A_{ij})]\). Research around \([f(A_{ij})]\) was initiated by several mathematicians in the 60s of the last century. For specific \(f\), people now call \([\text{tr}(A_{ij})]\) and \([\det(A_{ij})]\) the partial trace and the partial determinant of \(A\), respectively. The authors consider a general situation when \(f\) is any generalized matrix function, named by them partial matrix function. Some known inequalities on determinants and partial determinants are extended. In the final part of the paper, an interesting conjecture is proposed.
Reviewer: Minghua Lin (Xi'an)A note on the boundary of the Birkhoff-James \(\epsilon\)-orthogonality setshttps://www.zbmath.org/1483.150122022-05-16T20:40:13.078697Z"Katsouleas, Georgios"https://www.zbmath.org/authors/?q=ai:katsouleas.georgios"Panagakou, Vasiliki"https://www.zbmath.org/authors/?q=ai:panagakou.vasiliki"Psarrakos, Panayiotis"https://www.zbmath.org/authors/?q=ai:psarrakos.panayiotis-jThe numerical range of a matrix polynomial \(P\) is a generalization of the numerical range (field of values) of a matrix \(A\). The classical case is obtained for \(P(z) = zI - A\). Recently, the notions of Birkhoff-James \(\varepsilon\)-orthogonality sets of rectangular matrices and elements of a complex normed linear space have been introduced as natural generalizations of the standard numerical range. In this paper, a characterization for the corners of the Birkhoff-James-\(\varepsilon\)-sets of vectors and vector-valued polynomials is obtained. A randomized algorithm for approximating their boundaries is also proposed.
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Matricial radius: a relation of numerical radius with matricial rangehttps://www.zbmath.org/1483.150132022-05-16T20:40:13.078697Z"Kian, Mohsen"https://www.zbmath.org/authors/?q=ai:kian.mohsen"Dehghani, Mahdi"https://www.zbmath.org/authors/?q=ai:dehghani.mahdi"Sattari, Mostafa"https://www.zbmath.org/authors/?q=ai:sattari.mostafaThe matricial range of an operator (on a Hilbert space) is a matrix valued extension of the numerical range. It would be natural to define the matricial radius of an operator to be the maximum norm of the elements of its matricial range. This definition can be not very interesting, as the matricial radius defined in this way would be equal to the norm of the operator. This paper proposes two other candidates for a more meaningful notion of matricial radius. The authors find that neither of them is a ``proper'' extension of the numerical radius based on the matricial range and wonder whether such an extension exists at all.
Reviewer: Minghua Lin (Xi'an)Asymptotically optimal multi-pavinghttps://www.zbmath.org/1483.150142022-05-16T20:40:13.078697Z"Ravichandran, Mohan"https://www.zbmath.org/authors/?q=ai:ravichandran.mohan"Srivastava, Nikhil"https://www.zbmath.org/authors/?q=ai:srivastava.nikhilAs a consequence of the solution of the Kadison-Singer problem in [\textit{A. W. Marcus} et al., Ann. Math. (2) 182, No. 1, 327--350 (2015; Zbl 1332.46056)], it is known that every zero-diagonal matrix admits a nontrivial paving with dimension independent bounds.
In this nice paper, it is shown that given \(k\) zero-diagonal \(n\times n\) Hermitian matrices \(A_1, \dots, A_k\) and \(\varepsilon > 0\), there is a simultaneous paving, i.e., there are diagonal projections \(P_1,\dots , P_r\) with \(r \le 18k/{\varepsilon^2}\) and \(\sum_{j\le r} P_j = I\) such that \(\|P_jA_iP_j\| \le \varepsilon \|A_i\|\). Furthermore, it is proved that every square zero-diagonal complex matrix can be \(\varepsilon\)-paved using \(O(\varepsilon^{-2})\) blocks. This is also used to strengthen a result by \textit{W. B. Johnson} et al. [Proc. Natl. Acad. Sci. USA 110, No. 48, 19251--19255 (2013; Zbl 1297.15027)]
on commutator representations of zero trace matrices.
Reviewer: Cătălin Badea (Villeneuve d'Ascq)On the structure of ternary Clifford algebras and their irreducible representationshttps://www.zbmath.org/1483.150152022-05-16T20:40:13.078697Z"Abłamowicz, Rafał"https://www.zbmath.org/authors/?q=ai:ablamowicz.rafalThe author studies the structure of the algebra generated over \(\mathbb{C}\) by generators \(e_1,\dots,e_n\) whose third powers are equal to 1 and \(e_k e_\ell=\omega e_\ell e_k\) for each \(k<\ell\) where \(\omega=e^{\frac{2\pi i}{3}}\). He concludes that when \(n\) is even, this algebra is isomorphic to \(M_{\frac{n}{2}}(\mathbb{C})\), and when it is odd, it is isomorphic the direct sum of three complex matrix algebras of degree \(\frac{n-1}{2}\).
Reviewer's note: The results reported in the paper appeared earlier in a much greater generality, see for instance [\textit{M. Vela}, Commun. Algebra 30, No. 4, 1995--2021 (2002; Zbl 1011.16030)].
Reviewer: Adam Chapman (Tel Hai)Exponentials of general multivector in 3D Clifford algebrashttps://www.zbmath.org/1483.150162022-05-16T20:40:13.078697Z"Dargys, Adolfas"https://www.zbmath.org/authors/?q=ai:dargys.adolfas"Acus, Artūras"https://www.zbmath.org/authors/?q=ai:acus.arturasThe authors present closed-form expressions to compute the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) \(Cl_{p,q}\) for \(n = p+q = 3\). The exponential function is expanded into a sum of basis elements (grades); for this purpose some computer algebra (Mathematica package) is used.
The finite formulas obtained are rather complicated, however their analysis makes it possible to identify obstacles in constructing the GA coordinate-free formulas. The general multivector (MV) in GA spaces is expanded in the orthonormal basis. The authors start from the geometric algebra \(Cl_{0,3}\), where the expanded exponential in the coordinate form has the simplest MV coefficients. The formulas are very complicated, but the authors verify everything with the help of the Mathematica package.
They present the final exponential formulas as well as the specific cases that follow from these formulas and study the exponential function of multivectors in \(Cl_{0,3}\) algebra, \(Cl_{3,0}\), \(Cl_{1,2}\), and in \(Cl_{2,1}\) algebra. The exponential of a multivector is another multivector. For each case, a theorem is given and proved, thus providing for the multivector \(A\) the formulas for the real coefficients of the multivector \(\exp(A)\). In the next part of the paper, the authors derive the exponentials of blades for \(Cl_{3,0}\) and \(Cl_{0,3}\) algebras. For mixed signature algebras, to the best of authors' knowledge, such formulas are presented here for the first time. In particular, they study exponential of bivectors, exponential of pure vectors, and exponential of scalars and pseudoscalars.
Further, trigonometric and hyperbolic functions of a multivector \(A\) can be expressed through the exponentials, in particular functions \(\sin(A), \cos(A), \sinh(A)\), and \(\cosh(A)\). Various relations between GA trigonometric and hyperbolic functions are analogues of well-known scalar relations, as for example
\(\cos^2(A)+\sin^2(A)=1\) or \(\cosh^2(A)-\sinh^2(A)=1\). Also, it should be noted that GA sine and cosine functions, as well as hyperbolic GA sine and cosine functions, commute: \[ \sin(A)\, \cos(A) = \cos(A)\, \sin(A),\]
\[ \sinh(A)\,\cosh(A) = \cosh(A)\,\sinh(A). \] As an example, the authors consider the spinor evolution under the action of magnetic field.
In the final section, numerical comparison between exact formulas and their series expansion is given.
In conclusions, the author say that they are able to expand the GA exponential function of a general argument into MV in the coordinate form for all four three-dimensional Clifford geometric algebras. They think that such an expansion of the exponential will be useful in solving GA differential equations, in signal and image processing, in automatic control and robotics.
Reviewer: Drahoslava Janovská (Praha)Hermitian and skew-Hermitian splitting methods for solving a tensor equationhttps://www.zbmath.org/1483.150172022-05-16T20:40:13.078697Z"Li, Tao"https://www.zbmath.org/authors/?q=ai:li.tao.3"Wang, Qing-Wen"https://www.zbmath.org/authors/?q=ai:wang.qingwen"Zhang, Xin-Fang"https://www.zbmath.org/authors/?q=ai:zhang.xinfangSummary: The present paper deals with the numerical solution of non-Hermitian positive definite tensor equation \(\mathcal{A} \ast_N \mathcal{X} = \mathcal{B}\) under the Einstein product. Firstly, we extend the Hermitian and skew-Hermitian splitting (HSS) method to solve the tensor equation. Then we propose a new Hermitian splitting (NHS) method under some certain conditions, which is expected to converge faster than the HSS iteration. We also present the optimal parameters of both the HSS and NHS methods. Moreover, we apply the Smith technique to give two modified methods which can greatly accelerate the convergence rate. The performed numerical examples illustrate that the proposed methods are feasible and efficient.Fourth-order partially symmetric tensors: theory and algorithmhttps://www.zbmath.org/1483.150182022-05-16T20:40:13.078697Z"Li, Mengzhen"https://www.zbmath.org/authors/?q=ai:li.mengzhen"Chen, Haibin"https://www.zbmath.org/authors/?q=ai:chen.haibin"Zhou, Guanglu"https://www.zbmath.org/authors/?q=ai:zhou.guangluSummary: With the coming of the big data era, high-order tensors have received much attention of researches in recent years, and this makes it to be a useful tool in data analysis. As a special kind of structured tensor, the fourth-order partially symmetric tensor receives a special concern due to its wide applications in nonlinear elasticity materials. In this review, we will give a survey on recent advances of fourth-order partially symmetric tensors such as M-eigenvalue inclusion intervals, M-positive definiteness, strong ellipticity condition and algorithms for computing the largest M-eigenvalue. Some potential research directions in the future are also listed in the paper.A min-plus analogue of the Jordan canonical form associated with the basis of the generalized eigenspacehttps://www.zbmath.org/1483.150192022-05-16T20:40:13.078697Z"Nishida, Yuki"https://www.zbmath.org/authors/?q=ai:nishida.yuki"Sato, Kohei"https://www.zbmath.org/authors/?q=ai:sato.kohei"Watanabe, Sennosuke"https://www.zbmath.org/authors/?q=ai:watanabe.sennosukeThe min-plus algebra is a kind of idempotent semiring. The arithmetic operations of a min-plus algebra are addition \(a\oplus b:=\mathrm{min}\{a, b\}\) and multiplication \(a\otimes b:=a+b\) for any \(a,b\in\mathbb{R}\cup\{+\infty\}\). Extending these operations to matrices and vectors, one obtains the min-plus matrices and min-plus vectors, which are closely related with the shortest path problem on weighted directed graphs. In this paper, the authors generalize the notion of Jordan canonical forms to min-plus matrices and deduce a sufficient and necessary condition for a min-plus matrix to admit a Jordan canonical form in terms of the weighted directed graph associated with the matrix.
Reviewer: Xiangqian Guo (Waterloo)Determinant of a matrix over min-plus algebrahttps://www.zbmath.org/1483.150202022-05-16T20:40:13.078697Z"Siswanto"https://www.zbmath.org/authors/?q=ai:siswanto.nurhadi"Gusmizain, A."https://www.zbmath.org/authors/?q=ai:gusmizain.a"Wibowo, S."https://www.zbmath.org/authors/?q=ai:wibowo.santoso|wibowo.sasmito-hSummary: Max-plus algebra is a set \(\mathbb{R}_{max} = \mathbb{R} \cup \{\varepsilon \}\), with \(\mathbb{R}\) is the set of all real numbers and \(e = -\infty\) equipped with \(\oplus \) (maximum) and \(\otimes \) (plus) operation. Another algebra that can be learned is the min-plus algebra. Min-plus algebra is a set \(\mathbb{R}_{min} = \mathbb{R} \cup\{\varepsilon' \}\), with \(\mathbb{R}\) is the set of all real number and \(\varepsilon = +\infty\) equipped with \(\oplus' \) (minimum) and \(\otimes \) (plus) operation. Any square matrices over max-plus algebra can be related to the determinant. The determinant term of max-plus algebra is represented by the permanent and dominant. This study will discuss the determinant of a matrix over min-plus algebra.Closure of the simple image set of linear mapping interval max-plushttps://www.zbmath.org/1483.150212022-05-16T20:40:13.078697Z"Siswanto"https://www.zbmath.org/authors/?q=ai:siswanto.nurhadi"Kurniawan, Vika Yugi"https://www.zbmath.org/authors/?q=ai:kurniawan.vika-yugi"Pangadi"https://www.zbmath.org/authors/?q=ai:pangadi."Santoso, B. W."https://www.zbmath.org/authors/?q=ai:santoso.b-wSummary: Max-plus algebra is a set \(\mathbb{R}_\varepsilon=\mathbb{R}\cup\{\varepsilon\}\), where \(\mathbb{R}\) is a set of all real numbers that is equipped with operations \(\oplus\) and \(\otimes\). For each \(a,b\in\mathbb{R}\) operations \(\oplus\) and \(\otimes\) are defined as \(a\oplus b =\max(a,b)\) and \(a\otimes b=a+b\). The set matrices of size \(m \times n\) whose elements are element of \(\mathbb{R}_\varepsilon\) is called the matrices over \(\mathbb{R}_\varepsilon\) and denoted as \(\mathbb{R}_\varepsilon^{m\times n}\). Given the system of linear equations \(A\otimes x=b\) with \(A\otimes x=b\), with \(A\in\mathbb{R}_\varepsilon^{m\times n}\) and \(b\in\mathbb{R}_\varepsilon^n\). The concept of the simple image set of strongly regular matrices is related to the concept of linear equations systems. It has been discussed about closure of the simple image set of strongly regular matrices over max-plus algebra. Interval max-plus algebra is an extension of max-plus algebra. The purpose of this research is to determine closure of the simple image set of strongly regular matrices which is image of \(k\)-iteration the matrix after the matrix is normalized. Matrices that is discussed is matrices over interval max-plus algebra.Cayley-Hamilton theorem in the min-plus algebrahttps://www.zbmath.org/1483.150222022-05-16T20:40:13.078697Z"Siswanto"https://www.zbmath.org/authors/?q=ai:siswanto.nurhadi"Nurhayati, N."https://www.zbmath.org/authors/?q=ai:nurhayati.nunung"Pangadi"https://www.zbmath.org/authors/?q=ai:pangadi.Summary: Max-plus algebra is the set of \(\mathbb{R}_\varepsilon = \mathbb{R} \cup\{\varepsilon \}\) where \(\mathbb{R}\) is a set of all real numbers and \(\varepsilon = -\infty\) which is equipped with max \((\oplus)\) and plus \((\otimes)\) operations. The structure of max-plus algebra is semi-field. Max-plus algebra has structural similarities to conventional algebra. Because of these similarities, concept in conventional algebra such as the Cayley-Hamilton theorem has max-algebraic equivalence. Another semi-field is min-plus algebra, so that the Cayley-Hamilton theorem also has a min-plus algebraic equivalence. In this paper, we will show how the Cayley-Hamilton theorem can be proved in the min-plus algebra.Recognizing Cartesian products of matrices and polytopeshttps://www.zbmath.org/1483.150232022-05-16T20:40:13.078697Z"Aprile, Manuel"https://www.zbmath.org/authors/?q=ai:aprile.manuel"Conforti, Michele"https://www.zbmath.org/authors/?q=ai:conforti.michele"Faenza, Yuri"https://www.zbmath.org/authors/?q=ai:faenza.yuri"Fiorini, Samuel"https://www.zbmath.org/authors/?q=ai:fiorini.samuel"Huynh, Tony"https://www.zbmath.org/authors/?q=ai:huynh.tony"Macchia, Marco"https://www.zbmath.org/authors/?q=ai:macchia.marcoSummary: The 1-\textit{product} of matrices \(S_1 \in \mathbb{R}^{m_1 \times n_1}\) and \(S_2 \in \mathbb{R}^{m_2 \times n_2}\) is the matrix in \(\mathbb{R}^{(m_1+m_2) \times (n_1n_2)}\) whose columns are the concatenation of each column of \(S_1\) with each column of \(S_2\). Our main result is a polynomial time algorithm for the following problem: given a matrix \(S\), is \(S\) a 1-product, up to permutation of rows and columns? Our main motivation is a close link between the 1-product of matrices and the Cartesian product of polytopes, which relies on the concept of slack matrix. Determining whether a given matrix is a slack matrix is an intriguing problem whose complexity is unknown, and our algorithm reduces the problem to irreducible instances. Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information in information theory. We also give a polynomial time algorithm to recognize a more complicated matrix product, called the 2-\textit{product}. Finally, as a corollary of our 1-product and 2-product recognition algorithms, we obtain a polynomial time algorithm to recognize slack matrices of 2-level matroid base polytopes.
For the entire collection see [Zbl 1465.05002].Enumeration of some matrices and free linear codes over commutative finite local ringshttps://www.zbmath.org/1483.150242022-05-16T20:40:13.078697Z"Sirisuk, Siripong"https://www.zbmath.org/authors/?q=ai:sirisuk.siripongThis work generalizes results of \textit{K. Abdel-Ghaffar} [Linear Algebra Appl. 436, No. 7, 2665--2669 (2012; Zbl 1236.15043)], who enumerated \(m\times n\) matrices of rank \(r\) over finite fields having a given number of rows of single unit. A vector of single unit is a vector with exactly one entry equal to a unit and all other entires equal to \(0\).
In this article, the setting is generalized to finite commutative local rings \(R\). Such rings have the advantage that one can work in the residue field and lift the enumeration results back to \(R\) with some modifications. The author enumerates the number of \(m\times n\) matrices of McCoy rank \(r\) (see [\textit{N. H. McCoy} [Rings and ideals. La Salle, IL: Open Court Publishing (1948; Zbl 0041.36406)]) over \(R\) with exactly \(k\) rows of unit weight and, furthermore, counts the number of free linear codes of \(R^n\) of rank \(r\) containing exactly \(k\) standard basis vectors. Free linear codes of \(R^n\) are free submodules of \(R^n\). The formulae for these two enumeration problems closely resemble their finite field counterparts.
Reviewer: Joshua Maglione (Bielefeld)(0,1)-matrices and discrepancyhttps://www.zbmath.org/1483.150252022-05-16T20:40:13.078697Z"Beasley, Leroy B."https://www.zbmath.org/authors/?q=ai:beasley.leroy-bLet \(R=(r_1,r_2,\ldots,r_m)\) and \(S=(s_1,s_2,\ldots,s_n)\) be sequences of length \(m\) and \(n\) consisting of integers from the sets \(\left\{0,1,2,\ldots,n\right\}\), \(\left\{0,1,2,\ldots,m\right\}\), respectively, and let \(\mathcal A(R,S)\) denote the set of \(m\times n\) \((0,1)\)-matrices with \(r_i\) \(1\)'s in row \(i\) and \(s_j\) \(1\)'s in column \(j\). The most part of the paper deals with the case when \(R\) and \(S\) are decreasing. In this case, for any pair of sequences \(R\), \(S\) satisfying condition \begin{displaymath} r_1+r_2+\cdots+r_m=s_1+s_2+\cdots+s_n, \end{displaymath} there exists in \(\mathcal A(R,S)\) a unique matrix whose \(i\)-th row is of the form \begin{displaymath} (\underbrace{1,1,\ldots,1}_{r_i},\underbrace{0,0,\ldots,0}_{n-r_i}). \end{displaymath} It is called a Ferrers matrix and it is denoted by \(F_{R,n}\). For any \(A\in\mathcal A(R,S)\), the minimum number of \(1\)'s that exchanged with \(0\)'s in the same row result in a Ferrers matrix is called the discrepancy of \(A\), and is denoted by \(\mathrm{disc}(A)\). As the author points out, two matrices may have equal discrepancies, although the discrepancies of their transposes are different. The paper is devoted to the relation between \(\mathrm{disc}(A)\) and \(\mathrm{disc}(A^t)\). It is proved that if \(R\), \(S\) are decreasing and \(\mathrm{disc}(A)=l\), then \(\lceil\frac l2\rceil\leqslant\mathrm{disc}(A^t)\leqslant 2l\). In the last chapter, the author considers the case when \(R\), \(S\) are not necessarily decreasing. In such case, the isomorphic discrepancy of \(A\), denoted \(d_I(A)\) is defined as the \(\min\left\{\mathrm{disc}(AQ)\colon Q\in\mathcal P_n(S)\right\}\), where \(\mathcal P_n(S)\) is the set of all permutation matrices satisfying condition \(A\in\mathcal A(R,S)\) iff \(AQ\in\mathcal A(R,S)\). The author shows that if \(R\), \(S\) are non-increasing and \(\mathrm{disc}(A)=k\), then \(d_I(A)\geqslant \frac 14k\).
The paper ends with the following open problem. Let \(R\), \(S\) be non-increasing. Find a matrix in \(\mathcal A(R,S)\) whose discrepancy is greater that twice the isomorphic discrepancy or show that for any \(A\in\mathcal A(R,S)\) \(d_I(A)\geqslant\frac 12\mathrm{disc}(A)\).
Reviewer: Roksana Słowik (Gliwice)Powers of rational matriceshttps://www.zbmath.org/1483.150262022-05-16T20:40:13.078697Z"Garcia, Stephan Ramon"https://www.zbmath.org/authors/?q=ai:garcia.stephan-ramon"Horn, Roger A."https://www.zbmath.org/authors/?q=ai:horn.roger-a"Luca, Florian"https://www.zbmath.org/authors/?q=ai:luca.florian"Sarma, Kuldeep"https://www.zbmath.org/authors/?q=ai:sarma.kuldeep(no abstract)3 order Fibonacci-Lucas matriceshttps://www.zbmath.org/1483.150272022-05-16T20:40:13.078697Z"Wu, Weicai"https://www.zbmath.org/authors/?q=ai:wu.weicaiSummary: We obtain 3 order Fibonacci-Lucas matrices.Eigenvalue distribution of some nonlinear models of random matriceshttps://www.zbmath.org/1483.150282022-05-16T20:40:13.078697Z"Benigni, Lucas"https://www.zbmath.org/authors/?q=ai:benigni.lucas"Péché, Sandrine"https://www.zbmath.org/authors/?q=ai:peche.sandrineSummary: This paper is concerned with the asymptotic empirical eigenvalue distribution of some non linear random matrix ensemble. More precisely we consider \(M=\frac{1}{m}YY^{\ast}\) with \(Y=f(WX)\) where \(W\) and \(X\) are random rectangular matrices with i.i.d. centered entries. The function \(f\) is applied pointwise and can be seen as an activation function in (random) neural networks. We compute the asymptotic empirical distribution of this ensemble in the case where \(W\) and \(X\) have sub-Gaussian tails and \(f\) is real analytic. This extends a result of [\textit{J. Pennington} and \textit{P. Worah}, J. Stat. Mech. Theory Exp. 2019, No. 12, Article ID 124005, 14 p. (2019; Zbl 1459.60012)] where the case of Gaussian matrices \(W\) and \(X\) is considered. We also investigate the same questions in the multi-layer case, regarding neural network and machine learning applications.Multiplicative semigroup automorphisms of the strictly upper triangular matrices over a commutative ringhttps://www.zbmath.org/1483.201012022-05-16T20:40:13.078697Z"Ou, Shikun"https://www.zbmath.org/authors/?q=ai:ou.shikun"Fan, Lijun"https://www.zbmath.org/authors/?q=ai:fan.lijun"Wang, Dengyin"https://www.zbmath.org/authors/?q=ai:wang.dengyin(no abstract)Regular matrix semigroupshttps://www.zbmath.org/1483.201062022-05-16T20:40:13.078697Z"Zhu, Yong Wen"https://www.zbmath.org/authors/?q=ai:zhu.yongwen(no abstract)Certain properties of the Laguerre-Sheffer polynomialshttps://www.zbmath.org/1483.330062022-05-16T20:40:13.078697Z"Khan, Subuhi"https://www.zbmath.org/authors/?q=ai:khan.subuhi"Nahid, Tabinda"https://www.zbmath.org/authors/?q=ai:nahid.tabindaSummary: The present paper intends to study certain properties of the Laguerre-Sheffer polynomials utilizing matrix algebra. Some indispensable properties such as the recursive formulas, differential equations and identities for the Laguerre-Sheffer, Laguerre-associated Sheffer and Laguerre-Appell polynomials are established. This approach is mainly based upon the properties and relationships between the Pascal functional and Wronskian matrices. Examples providing the corresponding results for some hybrid special polynomials are derived which serve as a focal theme of the study.Eigenvalues of a class of fourth-order boundary value problems with transmission conditions using matrix theoryhttps://www.zbmath.org/1483.340372022-05-16T20:40:13.078697Z"Ao, Ji-jun"https://www.zbmath.org/authors/?q=ai:ao.jijun"Sun, Jiong"https://www.zbmath.org/authors/?q=ai:sun.jiongThe authors study the differential equation \[ (p u'')'' + q u = \lambda w u \] on \(J = (a, c) \cup (c, b)\) for finite \(a < c < b\), together with boundary conditions \[ A U (a) + B U (b) = 0, \quad U = \begin{pmatrix} u \\
u' \\
p u'' \\
(p u'')' \end{pmatrix}, \quad A, B \in M_4 (\mathbb{R}), \] and transmission conditions \[ C U (c-) + D U (c+) = 0. \] Here \(C, D\) are real-valued \(4 \times 4\)-matrices with positive determinants and the coefficient functions \(1/p, q, w\) are integrable on \(J\). Moreover, conditions on the matrices \(A, B\) are imposed which make the problem self-adjoint. As the main result, sufficient conditions are provided under which this eigenvalue problem is equivalent to a matrix eigenvalue problem of the form \((\mathbb P + \mathbb Q) \mathbb Y = \lambda \mathbb W \mathbb Y\), where \(\mathbb {P, Q, W}\) are constructed explicitly.
Reviewer: Jonathan Rohleder (Stockholm)Complex best \(r\)-term approximations almost always exist in finite dimensionshttps://www.zbmath.org/1483.410102022-05-16T20:40:13.078697Z"Qi, Yang"https://www.zbmath.org/authors/?q=ai:qi.yang"Michałek, Mateusz"https://www.zbmath.org/authors/?q=ai:michalek.mateusz"Lim, Lek-Heng"https://www.zbmath.org/authors/?q=ai:lim.lek-hengSummary: We show that in finite-dimensional nonlinear approximations, the best \(r\)-term approximant of a function \(f\) almost always exists over \(\mathbb{C}\) but that the same is not true over \(\mathbb{R}\), i.e., the infimum \(\inf_{f_1, \dots, f_r \in D} \| f - f_1 - \dots - f_r \|\) is almost always attainable by complex-valued functions \(f_1, \ldots, f_r\) in \(D\), a set (dictionary) of functions (atoms) with some desired structures. Our result extends to functions that possess properties like symmetry or skew-symmetry under permutations of arguments. When \(D\) is the set of separable functions, this is the best rank-\(r\) tensor approximation problem. We show that over \(\mathbb{C}\), any tensor almost always has a unique best rank-\(r\) approximation. This extends to other notions of ranks such as symmetric and alternating ranks, to best \(r\)-block-terms approximations, and to best approximations by tensor networks. Applied to sparse-plus-low-rank approximations, we obtain that for any given \(r\) and \(k\), a general tensor has a unique best approximation by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with a fixed sparsity pattern; a problem arising in covariance estimation of Gaussian model with \(k\) observed variables conditionally independent given \(r\) hidden variables. The existential (but not uniqueness) part of our result also applies to best approximations by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with no fixed sparsity pattern, and to tensor completion problems.Proper improvement of well-known numerical radius inequalities and their applicationshttps://www.zbmath.org/1483.470142022-05-16T20:40:13.078697Z"Bhunia, Pintu"https://www.zbmath.org/authors/?q=ai:bhunia.pintu"Paul, Kallol"https://www.zbmath.org/authors/?q=ai:paul.kallolThe authors establish some new inequalities for the numerical radius of bounded linear operators. The main section starts with a result that gives an inequality involving the operator norm and the Crawford number of bounded linear operators. Here, new inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space \(H\) are mentioned. In particular, it is established that, if \(T\) is a bounded linear operator on a Hilbert space \(H\), then \( w^{2}(T)\leq\min_{ 0\leq\alpha\leq 1} \|\alpha T^*T + (1-\alpha)TT^*\|\), where \(w(T)\) is the numerical radius of \(T\).
The inequalities obtained here are nontrivial improvements of well-known numerical radius inequalities. As an application of the numerical radius inequalities obtained, the authors give a better estimation of the bounds for the zeros of a complex monic polynomial. The paper contains many motivational results.
Reviewer: V. Lokesha (Bangalore)Refinements and reverses of Hölder-McCarthy operator inequality via a Cartwright-field resulthttps://www.zbmath.org/1483.470252022-05-16T20:40:13.078697Z"Dragomir, Silvestru Sever"https://www.zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: By the use of a classical result of \textit{D. I. Cartwright} and \textit{M. J. Field} [Proc. Am. Math. Soc. 71, 36--38 (1978; Zbl 0392.26010)], in this paper, we have obtained new refinements and reverses of Hölder-McCarthy operator inequality in the case of \(p \in (0, 1)\). A~comparison for the two upper bounds obtained showing that neither of them is better in general, has also been performed.Some trace inequalities for operators in Hilbert spaceshttps://www.zbmath.org/1483.470262022-05-16T20:40:13.078697Z"Dragomir, Silvestru Sever"https://www.zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: Some new trace inequalities for operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated and applications for power series of such operators are given. Some trace inequalities for matrices are also derived. Examples for the operator exponential and other similar functions are presented as well.Von Neumann type trace inequalities for Schatten-class operatorshttps://www.zbmath.org/1483.470342022-05-16T20:40:13.078697Z"Dirr, Gunther"https://www.zbmath.org/authors/?q=ai:dirr.gunther"vom Ende, Frederik"https://www.zbmath.org/authors/?q=ai:vom-ende.frederikThe von Neumann inequality states that, if \(A,B\in\mathbb{C}^{n\times n}\) with singular values \(s_1(A)\geq s_2(A)\geq\ldots\geq s_n(A)\) and \(s_1(B)\geq s_2(B)\geq\ldots\geq s_n(B)\), respectively, be given, then \[ \max_{U,V \in \mathcal{U}_n}\vert\operatorname{tr}(AUBV)\vert=\sum\nolimits_{j=1}^ns_j (A)s_j(B), \] where \(\mathcal{U}_n\) denotes the unitary group. A~consequence is the von Neumann inequality for Hermitian matrices that reads as follows: Let \(A,B\in \mathbb{C}^{n\times n}\) Hermitian with respective eigenvalues \((\lambda_j(A))_{j=1}^n\) and \((\lambda_j(B))_{j=1}^n\). Then \[ \sum\nolimits_{j=1}^n \lambda_j^\downarrow(A)\lambda_j^\uparrow(B) \leq \operatorname{tr}(AB)\leq \sum\nolimits_{j=1}^n \lambda_j^\downarrow(A)\lambda_j^\downarrow(B), \] where the superindeces \(\downarrow\) and \(\uparrow\) denote the decreasing and increasing sorting of the eigenvectors, respectively. \par The authors of the present paper employ some recent results on the \(C\)-numerical range of Schatten-class operators to extend the above inequalities to Schatten-class operators between complex Hilbert spaces of infinite dimension.
Reviewer: Mohammad Sal Moslehian (Mashhad)Curvature properties of Riemannian manifolds with circulant structureshttps://www.zbmath.org/1483.530492022-05-16T20:40:13.078697Z"Razpopov, Dimitar"https://www.zbmath.org/authors/?q=ai:razpopov.dimitar"Dzhelepov, Georgi"https://www.zbmath.org/authors/?q=ai:dzhelepov.georgi-dSummary: We study a Riemannian manifold \(M\) equipped with a circulant structure \(Q\), which is an isometry with respect to the metric. We consider two types of such manifolds: a 3-dimensional manifold \(M\) where the third power of \(Q\) is the identity, and a 4-dimensional manifold \(M\) where the fourth power of \(Q\) is the identity. In a single tangent space of \(M\) we have a special tetrahedron constructed by vectors of a \(Q\)-basis. The aim of the present paper is to find relations among the sectional curvatures of the 2-planes associated with the four faces of this tetrahedron and its cross sections passing through the medians and the edges of these faces.On a distinguished family of random variables and Painlevé equationshttps://www.zbmath.org/1483.600062022-05-16T20:40:13.078697Z"Assiotis, Theodoros"https://www.zbmath.org/authors/?q=ai:assiotis.theodoros"Bedert, Benjamin"https://www.zbmath.org/authors/?q=ai:bedert.benjamin"Gunes, Mustafa Alper"https://www.zbmath.org/authors/?q=ai:gunes.mustafa-alper"Soor, Arun"https://www.zbmath.org/authors/?q=ai:soor.arunSummary: A family of random variables \(\mathbf{X}(s)\), depending on a real parameter \(s>-\frac{1}{2} \), appears in the asymptotics of the joint moments of characteristic polynomials of random unitary matrices and their derivatives [\textit{T. Assiotis} et al., ``On the joint moments of the characteristic polynomials of random unitary matrices'', Preprint, \url{arXiv:2005.13961}], in the ergodic decomposition of the Hua-Pickrell measures [\textit{A. Borodin} and \textit{G. Olshanski}, Commun. Math. Phys. 223, No. 1, 87--123 (2001; Zbl 0987.60020); \textit{Y. Qiu}, Adv. Math. 308, 1209--1268 (2017; Zbl 1407.60011], and conjecturally in the asymptotics of the joint moments of Hardy's function and its derivative ([\textit{C. Hughes}, On the characteristic polynomial of a random unitary matrix and the Riemann zeta function. Heslington: University of York (PhD Thesis) (2001)] and [Assiotis et al., loc. cit.]). Our first main result establishes a connection between the characteristic function of \(\mathbf{X}(s)\) and the \(\sigma\)-Painlevé \(\text{III}^\prime\) equation in the full range of parameter values \(s>-\frac{1}{2} \). Our second main result gives the first explicit expression for the density and all the complex moments of the absolute value of \(\mathbf{X}(s)\) for integer values of \(s\). Finally, we establish an analogous connection to another special case of the \(\sigma \)-Painlevé \(\text{III}^\prime\) equation for the Laplace transform of the sum of the inverse points of the Bessel point process.Universality of approximate message passing algorithmshttps://www.zbmath.org/1483.600072022-05-16T20:40:13.078697Z"Chen, Wei-Kuo"https://www.zbmath.org/authors/?q=ai:chen.wei-kuo"Lam, Wai-Kit"https://www.zbmath.org/authors/?q=ai:lam.wai-kitSummary: We consider a broad class of Approximate Message Passing (AMP) algorithms defined as a Lipschitzian functional iteration in terms of an \(n\times n\) random symmetric matrix \(A\). We establish universality in noise for this AMP in the \(n\)-limit and validate this behavior in a number of AMPs popularly adapted in compressed sensing, statistical inferences, and optimizations in spin glasses.Tail bounds for gaps between eigenvalues of sparse random matriceshttps://www.zbmath.org/1483.600082022-05-16T20:40:13.078697Z"Lopatto, Patrick"https://www.zbmath.org/authors/?q=ai:lopatto.patrick"Luh, Kyle"https://www.zbmath.org/authors/?q=ai:luh.kyleThe authors study the eigenvalue gaps of sparse random matrices. The theory of sparse random matrices is of interest in its own right, but it also has innumerable applications in computer science and statistics. In contexts where sparse random matrices have similar spectral guarantees as their dense counterparts, they offer significant advantages as they require less space to store, allow quicker multiplication, and fewer random bits is necessary to generate them. The main contribution of the paper is to go beyond verifying the fact that such matrices have simple spectrum, and prove a tail bound for the minimal eigenvalue gap of these sparse random matrices. So, the first eigenvalue repulsion bound for sparse random matrices is established. As a consequence, it is proved that these matrices have simple spectrum, improving the range of sparsity and error probability from the previous works. It is also proved that for sparse Erdős-Rényi graphs, weak and strong nodal domains are the same.
Reviewer: Yuliya S. Mishura (Kyïv)Matrix analysis for continuous-time Markov chainshttps://www.zbmath.org/1483.601082022-05-16T20:40:13.078697Z"Le, Hung V."https://www.zbmath.org/authors/?q=ai:le.hung-viet"Tsatsomeros, M. J."https://www.zbmath.org/authors/?q=ai:tsatsomeros.michael-jSummary: Continuous-time Markov chains have transition matrices that vary continuously in time. Classical theory of nonnegative matrices, M-matrices and matrix exponentials is used in the literature to study their dynamics, probability distributions and other stochastic properties. For the benefit of Perron-Frobenius cognoscentes, this theory is surveyed and further adapted to study continuous-time Markov chains on finite state spaces.Non-convex projected gradient descent for generalized low-rank tensor regressionhttps://www.zbmath.org/1483.620912022-05-16T20:40:13.078697Z"Chen, Han"https://www.zbmath.org/authors/?q=ai:chen.han"Raskutti, Garvesh"https://www.zbmath.org/authors/?q=ai:raskutti.garvesh"Yuan, Ming"https://www.zbmath.org/authors/?q=ai:yuan.mingSummary: In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polynomial-time algorithms they are often slow in practice. On the other hand, limited theoretical guarantees exist for non-convex methods. In this paper we provide a general framework that provides theoretical guarantees for learning high-dimensional tensor regression models under different low-rank structural assumptions using the projected gradient descent algorithm applied to a potentially non-convex constraint set \(\Theta\) in terms of its \textit{localized Gaussian width} (due to Gaussian design). We juxtapose our theoretical results for non-convex projected gradient descent algorithms with previous results on regularized convex approaches. The two main differences between the
convex and non-convex
approach are: (i) from a computational perspective whether the non-convex projection operator is computable and whether the projection has desirable contraction properties and (ii) from a statistical error bound perspective, the non-convex approach has a superior rate for a number of examples. We provide three concrete examples of low-dimensional structure which address these issues and explain the pros and cons for the non-convex and convex approaches. We supplement our theoretical results with simulations which show that, under several common settings of generalized low rank tensor regression, the projected gradient descent approach is superior both in terms of statistical error and run-time provided the step-sizes of the projected descent algorithm are suitably chosen.Spectrum estimation from a few entrieshttps://www.zbmath.org/1483.620922022-05-16T20:40:13.078697Z"Khetan, Ashish"https://www.zbmath.org/authors/?q=ai:khetan.ashish"Oh, Sewoong"https://www.zbmath.org/authors/?q=ai:oh.sewoongSummary: Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as collaborative filtering and network analysis, we only get a partial observation. Under such scenarios, we consider the fundamental problem of recovering spectral properties of the underlying matrix from a sampling of its entries. In this paper, we address the problem of directly recovering the spectrum, which is the set of singular values, and also in sample-efficient approaches for recovering a spectral sum function, which is an aggregate sum of a fixed function applied to each of the singular values. Our approach is to first estimate the Schatten \(k\)-norms of a matrix for a small set of values of \(k\), and then apply Chebyshev approximation when estimating a spectral sum function or apply moment matching in Wasserstein distance when estimating the
singular values
directly. The main technical challenge is in accurately estimating the Schatten norms from a sampling of a matrix. We introduce a novel unbiased estimator based on counting small structures called network motifs in a graph and provide guarantees that match its empirical performance. Our theoretical analysis shows that Schatten norms can be recovered accurately from strictly smaller number of samples compared to what is needed to recover the underlying low-rank matrix. Numerical experiments suggest that we significantly improve upon a competing approach of using matrix completion methods, below the matrix completion threshold, above which matrix completion algorithms recover the underlying low-rank matrix exactly.On multistage experiments with restrictions in the randomization of treatmentshttps://www.zbmath.org/1483.621402022-05-16T20:40:13.078697Z"Katsaounis, T. I."https://www.zbmath.org/authors/?q=ai:katsaounis.tena-iSummary: This paper considers multistage experiments which involve, in each stage, two-level factors whose levels are hard to change. Because of such factors, in each stage, there are restrictions in the randomization of runs leading to two types of experimental units, called whole plots and split plots. As a result, there are two types of random errors, in each stage, that need to be taken into account when modeling the response variable. It is assumed that a linear mixed effects model is appropriate for analyzing observations, and that the method of generalized least squares estimation is used to obtain estimators for the fixed effects in the model. It is also assumed that the model matrix of the fixed effects is based on a general two-level fractional factorial design. The goal of this paper is to provide an analytic form of the covariance matrix of the generalized least squares estimators of the fixed factorial effects in the model, that is useful for evaluating designs. This form shows how any confounding of (fixed) effects with the whole plots, associated with the different stages, affects the variances of their generalized least squares estimators. Some special cases of this form, which correspond to a model matrix based on either a two-level regular factorial or a two-level full factorial design, are also discussed. Results can be extended to multistage experiments with randomization restrictions of the runs, in each stage, with a model matrix based on a general multilevel fractional factorial design.Tracy-Widom law for the largest eigenvalue of sample covariance matrix generated by VARMAhttps://www.zbmath.org/1483.621552022-05-16T20:40:13.078697Z"Tian, Boping"https://www.zbmath.org/authors/?q=ai:tian.boping"Zhang, Yangchun"https://www.zbmath.org/authors/?q=ai:zhang.yangchun.1"Zhou, Wang"https://www.zbmath.org/authors/?q=ai:zhou.wangA matrix-theoretic spectral analysis of incompressible Navier-Stokes staggered DG approximations and a related spectrally based preconditioning approachhttps://www.zbmath.org/1483.650472022-05-16T20:40:13.078697Z"Mazza, M."https://www.zbmath.org/authors/?q=ai:mazza.mariarosa"Semplice, M."https://www.zbmath.org/authors/?q=ai:semplice.matteo"Serra-Capizzano, S."https://www.zbmath.org/authors/?q=ai:serra-capizzano.stefano"Travaglia, E."https://www.zbmath.org/authors/?q=ai:travaglia.elenaSummary: The incompressible Navier-Stokes equations are solved in a channel, using a Discontinuous Galerkin method over staggered grids. We study the structure and the spectral features of the matrices of the linear systems arising from the discretization. They are of block type, each block showing Toeplitz-like, band, and tensor structure at the same time. After introducing new tools to study Toeplitz-like matrix sequences with rectangular symbols, a quite complete spectral analysis is presented, with the target of designing and analyzing fast iterative solvers for the associated large linear systems. Promising numerical results are presented, commented, and critically discussed for elongated two- and three-dimensional geometries.A new randomized vector algorithm for iterative solution of large linear systemshttps://www.zbmath.org/1483.650522022-05-16T20:40:13.078697Z"Sabelfeld, Karl"https://www.zbmath.org/authors/?q=ai:sabelfeld.karl-kSummary: In this letter we suggest a new randomized scalable stochastic-matrix-based algorithms for calculation of large matrix iterations. Special focus is on positive or irreducible nonnegative class of matrices. As an application, a new randomized vector algorithm for iterative solution of large linear systems of algebraic equations governed by M-matrices is constructed. The idea behind these stochastic methods is in a randomized vector representation of matrix iterations. The iterations are performed by sampling random columns only, thus avoiding not only matrix but also matrix vector multiplications. As a result, the algorithm is highly efficient for solving linear equations of high dimension, its computational cost depends linearly on the dimension. Extensions of the suggested randomized iteration method to general classes of matrices are also discussed.The homotopy method for the complete solution of quadratic two-parameter eigenvalue problemshttps://www.zbmath.org/1483.650592022-05-16T20:40:13.078697Z"Dong, Bo"https://www.zbmath.org/authors/?q=ai:dong.boSummary: We propose a homotopy method to solve the quadratic two-parameter eigenvalue problems, which arise frequently in the analysis of the asymptotic stability of the delay differential equation. Our method does not require to form coupled generalized eigenvalue problems with Kronecker product type coefficient matrices and thus can avoid the increasing of the computational cost and memory storage. Numerical results and the applications in the delay differential equations are presented to illustrate the effectiveness and efficiency of our method. It appears that our method tends to be more effective than the existing methods in terms of speed, accuracy and memory storage as the problem size grows.An efficient method for computing the outer inverse \(A_{T,S}^{(2)}\) through Gauss-Jordan eliminationhttps://www.zbmath.org/1483.650612022-05-16T20:40:13.078697Z"Ma, Jie"https://www.zbmath.org/authors/?q=ai:ma.jie"Gao, Feng"https://www.zbmath.org/authors/?q=ai:gao.feng.2|gao.feng.1|gao.feng.3|gao.feng.6|gao.feng"Li, Yongshu"https://www.zbmath.org/authors/?q=ai:li.yongshuSummary: In this paper, we derive a novel expression for the computation of the outer inverse \(A_{T,S}^{(2)}\). Based on this expression, we present a new Gauss-Jordan elimination method for computing \(A_{R(G),N(G)}^{(2)}\). The analysis of computational complexity indicates that our algorithm is more efficient than the existing Gauss-Jordan elimination algorithms for \(A_{R(G),N(G)}^{(2)}\) in the literature for a large class of problems. Especially for the case when \(G\) is a Hermitian matrix, our algorithm has the lowest computational complexity among the existing algorithms. Finally, numerical experiments show that our method for the outer inverse \(A_{R(G),N(G)}^{(2)}\) generally is more efficient than that of the other existing methods in the cases of matrices \(A\) with \(m < n\) or square matrices \(G\) with high rank or Hermitian matrices \(G\) in practice.Sensitivity analysis for the block Cholesky downdating problemhttps://www.zbmath.org/1483.650632022-05-16T20:40:13.078697Z"Farooq, Aamir"https://www.zbmath.org/authors/?q=ai:farooq.aamir"Samar, Mahvish"https://www.zbmath.org/authors/?q=ai:samar.mahvish"Li, Hanyu"https://www.zbmath.org/authors/?q=ai:li.hanyu"Mu, Chunlai"https://www.zbmath.org/authors/?q=ai:mu.chunlaiSummary: Some improved rigorous perturbation bounds with normwise perturbation for the block Cholesky downdating problem are first derived by combining the modified matrix-vector equation approach with the strategy for Lyapunov majorant function and the Banach fixed point theorem. Then, we investigate four distinct kinds of condition numbers, i.e. two normwise ones, and mixed and componentwise ones, for this problem, and present their explicit expressions. Furthermore, using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, we also consider the statistical estimation of these condition numbers and design two algorithms. The obtained results are illustrated by numerical examples.Global Hessenberg and CMRH methods for a class of complex matrix equationshttps://www.zbmath.org/1483.650642022-05-16T20:40:13.078697Z"Gu, Ying"https://www.zbmath.org/authors/?q=ai:gu.ying"Song, Yongzhong"https://www.zbmath.org/authors/?q=ai:song.yongzhongSummary: In this paper, we study how to use Hessenberg-based methods to solve a class of complex matrix equations, which is a general form of many known equations. A generalized global Hessenberg process is presented to construct the basis of complex matrix space. This process requires less work and storage because the basis it uses has more zero components. Then global Hessenberg (Gl-Hess) and global CMRH (Gl-CMRH) methods are proposed, which can be directly realized by the original coefficient matrices. The convergence of the Gl-CMRH method is analyzed. Numerical experiments show the efficiency of the methods and compare them with existing methods.On greedy randomized augmented Kaczmarz method for solving large sparse inconsistent linear systemshttps://www.zbmath.org/1483.650652022-05-16T20:40:13.078697Z"Bai, Zhong-Zhi"https://www.zbmath.org/authors/?q=ai:bai.zhongzhi"Wu, Wen-Ting"https://www.zbmath.org/authors/?q=ai:wu.wentingAn adaptation for iterative structured matrix completionhttps://www.zbmath.org/1483.650662022-05-16T20:40:13.078697Z"Adams, Henry"https://www.zbmath.org/authors/?q=ai:adams.henry"Kassab, Lara"https://www.zbmath.org/authors/?q=ai:kassab.lara"Needell, Deanna"https://www.zbmath.org/authors/?q=ai:needell.deannaSummary: The task of predicting missing entries of a matrix, from a subset of known entries, is known as \textit{matrix completion}. In today's data-driven world, data completion is essential whether it is the main goal or a pre-processing step. Structured matrix completion includes any setting in which data is not missing uniformly at random. In recent work, a modification to the standard nuclear norm minimization (NNM) for matrix completion has been developed to take into account \textit{sparsity-based} structure in the missing entries. This notion of structure is motivated in many settings including recommender systems, where the probability that an entry is observed depends on the value of the entry. We propose adjusting an Iteratively Reweighted Least Squares (IRLS) algorithm for low-rank matrix completion to take into account sparsity-based structure in the missing entries. We also present an iterative gradient-projection-based implementation of the algorithm that can handle large-scale matrices. Finally, we present a robust array of numerical experiments on matrices of varying sizes, ranks, and level of structure. We show that our proposed method is comparable with the adjusted NNM on small-sized matrices, and often outperforms the IRLS algorithm in structured settings on matrices up to size \(1000 \times 1000\).Low-rank tensor approximation of singularly perturbed boundary value problems in one dimensionhttps://www.zbmath.org/1483.650672022-05-16T20:40:13.078697Z"Marcati, Carlo"https://www.zbmath.org/authors/?q=ai:marcati.carlo"Rakhuba, Maxim"https://www.zbmath.org/authors/?q=ai:rakhuba.maxim-v"Ulander, Johan E. M."https://www.zbmath.org/authors/?q=ai:ulander.johan-e-mSummary: We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion boundary value problems in one dimension. Specifically, we show that, independently of the scale of the singular perturbation parameter, a numerical solution with accuracy \(0<\varepsilon <1\) can be represented in the QTT format with a number of parameters that depends only polylogarithmically on \(\varepsilon\). In other words, QTT-compressed solutions converge exponentially fast to the exact solution, with respect to a root of the number of parameters. We also verify the rank bound estimates numerically and overcome known stability issues of the QTT-based solution of partial differential equations (PDEs) by adapting a preconditioning strategy to obtain stable schemes at all scales. We find, therefore, that the QTT-based strategy is a rapidly converging algorithm for the solution of singularly perturbed PDEs, which does not require prior knowledge on the scale of the singular perturbation and on the shape of the boundary layers.Low tubal rank tensor recovery using the Bürer-Monteiro factorisation approach. Application to optical coherence tomographyhttps://www.zbmath.org/1483.650682022-05-16T20:40:13.078697Z"Assoweh, Mohamed Ibrahim"https://www.zbmath.org/authors/?q=ai:assoweh.mohamed-ibrahim"Chrétien, Stéphane"https://www.zbmath.org/authors/?q=ai:chretien.stephane"Tamadazte, Brahim"https://www.zbmath.org/authors/?q=ai:tamadazte.brahimSummary: In this paper, we study the low-tubal-rank tensor completion problem, i.e., the problem of recovering a third-order tensor by observing a subset of its entries, when these entries are selected uniformly at random. We propose a mathematical analysis of an extension of the Bürer-Monteiro factorisation approach to this problem. We then illustrate the use of the Bürer-Monteiro approach on a challenging OCT reconstruction problem on both synthetic and real world data, using an alternating minimisation algorithm.Golub-Kahan bidiagonalization for ill-conditioned tensor equations with applicationshttps://www.zbmath.org/1483.650692022-05-16T20:40:13.078697Z"Beik, Fatemeh P. A."https://www.zbmath.org/authors/?q=ai:beik.fatemeh-panjeh-ali"Jbilou, Khalide"https://www.zbmath.org/authors/?q=ai:jbilou.khalide"Najafi-Kalyani, Mehdi"https://www.zbmath.org/authors/?q=ai:najafi-kalyani.mehdi"Reichel, Lothar"https://www.zbmath.org/authors/?q=ai:reichel.lotharSummary: This paper is concerned with the solution of severely ill-conditioned linear tensor equations. These kinds of equations may arise when discretizing partial differential equations in many space-dimensions by finite difference or spectral methods. The deblurring of color images is another application. We describe the tensor Golub-Kahan bidiagonalization (GKB) algorithm and apply it in conjunction with Tikhonov regularization. The conditioning of the Stein tensor equation is examined. These results suggest how the tensor GKB process can be used to solve general linear tensor equations. Computed examples illustrate the feasibility of the proposed algorithm.A chord-Zhang neural network model for solving absolute value equationshttps://www.zbmath.org/1483.650702022-05-16T20:40:13.078697Z"Cui, Lu-Bin"https://www.zbmath.org/authors/?q=ai:cui.lubin"Hu, Qing"https://www.zbmath.org/authors/?q=ai:hu.qingSummary: We first present a Zhang neural network model for solving the absolute value equation. In order to avoid the changed Jacobian matrix, combined with the chord method, we fix the generalized Jacobian. Then we obtain the chord-Zhang neural network model. The convergence of the proposed models is studied. Moreover, some numerical experiments are presented to illustrate the efficiency of the proposed schemes.A gradient based iterative method and associated preconditioning technique for solving the large multilinear systemshttps://www.zbmath.org/1483.650732022-05-16T20:40:13.078697Z"Khosravi Dehdezi, Eisa"https://www.zbmath.org/authors/?q=ai:khosravi-dehdezi.eisa"Karimi, Saeed"https://www.zbmath.org/authors/?q=ai:karimi.saeedSummary: In this paper, a gradient based iterative method version is proposed for solving the large multilinear systems or tensor equations. A new preconditioner is presented to accelerate the convergence rate of the new iterative methods. Using the linear operator form of the multilinear system, the new methods can be implemented easily for solving several kinds of tensor equations. Finally, some numerical examples are given to illustrate the effectiveness of the proposed method.A novel framework for the NMF methods with experiments to unmixing signals and feature representationhttps://www.zbmath.org/1483.650742022-05-16T20:40:13.078697Z"Teng, Yueyang"https://www.zbmath.org/authors/?q=ai:teng.yueyang"Yao, Yudong"https://www.zbmath.org/authors/?q=ai:yao.yudong"Qi, Shouliang"https://www.zbmath.org/authors/?q=ai:qi.shouliang"Li, Chen"https://www.zbmath.org/authors/?q=ai:li.chen.1"Xu, Lisheng"https://www.zbmath.org/authors/?q=ai:xu.lisheng"Qian, Wei"https://www.zbmath.org/authors/?q=ai:qian.wei"Fan, Fenglei"https://www.zbmath.org/authors/?q=ai:fan.fenglei"Wang, Ge"https://www.zbmath.org/authors/?q=ai:wang.geSummary: Non-negative matrix factorization (NMF) can be used in clustering, feature representation or blind source separation. Many NMF methods have been developed including least squares (LS) error, Kullback-Leibler (KL) divergence, Itakura-Saito (IS) divergence, Bregman-divergence, \(\alpha\)-divergence, \(\beta\)-divergence, \(\gamma\)-divergence, convex, constrained, graph-regularized NMFs. The main contribution of this paper is to develop a framework to generalize the existing NMF methods and also provide new NMF methods. This paper constructs a general optimization model and develops a generic updating rule with a simple structure using a surrogate function, which possesses similar properties as the standard NMF methods. The experimental results, obtained using several standard databases, demonstrate the power of the work in which some new methods provide performance superior to that of the other existing methods.Fast algorithms for multiplication of Foeplitz matrix and vector from interesting inversehttps://www.zbmath.org/1483.650752022-05-16T20:40:13.078697Z"Wei, Yunlan"https://www.zbmath.org/authors/?q=ai:wei.yunlan"Zheng, Yanpeng"https://www.zbmath.org/authors/?q=ai:zheng.yanpeng"Jiang, Zhaolin"https://www.zbmath.org/authors/?q=ai:jiang.zhaolin"Shon, Sugoog"https://www.zbmath.org/authors/?q=ai:shon.sugoogSummary: In this paper, we focus on the multiplication of an \(n\)-by-\(n\) Foeplitz matrix and a vector, whose computational cost is of \(O(n)\). The main idea is to transform the problem into solving the perturbed tridiagonal Toeplitz linear system which is based on the interesting inverse of Foeplitz matrix.Low-rank traffic matrix completion with marginal informationhttps://www.zbmath.org/1483.650762022-05-16T20:40:13.078697Z"Xiong, Zikai"https://www.zbmath.org/authors/?q=ai:xiong.zikai"Wei, Yimin"https://www.zbmath.org/authors/?q=ai:wei.yimin.1|wei.yimin"Xu, Renjie"https://www.zbmath.org/authors/?q=ai:xu.renjie"Xu, Yanwei"https://www.zbmath.org/authors/?q=ai:xu.yanweiSummary: Accurate spatio-temporal traffic data is crucial to intelligent transportation systems. Missing traffic data is an important problem to solve. Low-rank matrix completion provides an effective way to find the missing data. The completion aims to obtain a low-rank matrix that can approximate the known entries as far as possible. Meanwhile, some linear constraint marginal information of the matrix can also be observed in the real application. In this paper, we utilize such marginal information to largely improve the performance of common matrix completion algorithms and propose an alternating direction method of multipliers (ADMM) and conjugate gradient descent method (CGD) based SoftImpute alternative least square (ALS) algorithm. We analyze their convergence rates and prove that the model can always converge to a first-order stationary point. We also utilize ADMM and CGD to largely accelerate the subproblem and make its complexity of each iteration at the same level as the popular SoftImpute-ALS matrix completion algorithm. Furthermore, this algorithm can be used in distributed computation, suitable for large-scale problems. In the numerical experiments, we demonstrate its outstanding matrix completion performance and high speed in several traffic matrix datasets.Finding the nearest passive or nonpassive system via Hamiltonian eigenvalue optimizationhttps://www.zbmath.org/1483.650972022-05-16T20:40:13.078697Z"Fazzi, Antonio"https://www.zbmath.org/authors/?q=ai:fazzi.antonio"Guglielmi, Nicola"https://www.zbmath.org/authors/?q=ai:guglielmi.nicola"Lubich, Christian"https://www.zbmath.org/authors/?q=ai:lubich.christianSummation-by-parts approximations of the second derivative: pseudoinverse and revisitation of a high order accurate operatorhttps://www.zbmath.org/1483.651322022-05-16T20:40:13.078697Z"Eriksson, Sofia"https://www.zbmath.org/authors/?q=ai:eriksson.sofia"Wang, Siyang"https://www.zbmath.org/authors/?q=ai:wang.siyangThe summation-by-parts (SBP)-simultaneous-approximation-term (SAT) finite difference methods lead to energy-stable discretization of PDEs that involve the second derivative including Poisson's equation, the heat equation, and the wave equation. The discretization matrix is singular for equations with Neumann boundary conditions. In this paper, an analytical expression of the Moore-Penrose inverse is derived for that singular discretization matrix. It was proved that the discretization matrix is rank deficient by one for the second and fourth-order schemes. Furthermore, a one-parameter family of the sixth order accurate SBP operator is constructed for the second derivative and it was shown that there can be more than one zero eigenvalue. A detailed analysis is performed in terms of stability, accuracy, and spectral radius with the parameter choices to optimize the corresponding SBP operators for different equations and boundary conditions. Detailed numerical experiments demonstrate the improvements of the SBP operators.
Reviewer: Bülent Karasözen (Ankara)Hadamard matrix guided online hashinghttps://www.zbmath.org/1483.683062022-05-16T20:40:13.078697Z"Lin, Mingbao"https://www.zbmath.org/authors/?q=ai:lin.mingbao"Ji, Rongrong"https://www.zbmath.org/authors/?q=ai:ji.rongrong"Liu, Hong"https://www.zbmath.org/authors/?q=ai:liu.hong.1"Sun, Xiaoshuai"https://www.zbmath.org/authors/?q=ai:sun.xiaoshuai"Chen, Shen"https://www.zbmath.org/authors/?q=ai:chen.shen"Tian, Qi"https://www.zbmath.org/authors/?q=ai:tian.qiSummary: Online image hashing has attracted increasing research attention recently, which receives large-scale data in a streaming manner to update the hash functions on-the-fly. Its key challenge lies in the difficulty of balancing the learning timeliness and model accuracy. To this end, most works follow a supervised setting, i.e., using class labels to boost the hashing performance, which defects in two aspects: first, strong constraints, e.g., orthogonal or similarity preserving, are used, which however are typically relaxed and lead to large accuracy drops. Second, large amounts of training batches are required to learn the up-to-date hash functions, which largely increase the learning complexity. To handle the above challenges, a novel supervised online hashing scheme termed \textbf{H}adamard \textbf{M}atrix Guided \textbf{O}nline \textbf{H}ashing (HMOH) is proposed in this paper. Our key innovation lies in introducing Hadamard matrix, which is an orthogonal binary matrix built via Sylvester method. In particular, to release the need of strong constraints, we regard each column of Hadamard matrix as the target code for each class label, which by nature satisfies several desired properties of hashing codes. To accelerate the online training, LSH is first adopted to align the lengths of target code and to-be-learned binary code. We then treat the learning of hash functions as a set of binary classification problems to fit the assigned target code. Finally, extensive experiments on four widely-used benchmarks demonstrate the superior accuracy and efficiency of HMOH over various state-of-the-art methods. Codes can be available at \url{https://github.com/lmbxmu/mycode}.An alternating nonmonotone projected Barzilai-Borwein algorithm of nonnegative factorization of big matriceshttps://www.zbmath.org/1483.683072022-05-16T20:40:13.078697Z"Li, Ting"https://www.zbmath.org/authors/?q=ai:li.ting"Tang, Jiayi"https://www.zbmath.org/authors/?q=ai:tang.jiayi"Wan, Zhong"https://www.zbmath.org/authors/?q=ai:wan.zhongSummary: In this paper, a new alternating nonmonotone projected Barzilai-Borwein (BB) algorithm is developed for solving large scale problems of nonnegative matrix factorization. Unlike the existing algorithms available in the literature, a nonmonotone line search strategy is proposed to find suitable step lengths, and an adaptive BB spectral parameter is employed to generate search directions such that the constructed subproblems are efficiently solved. Apart from establishment of global convergence for this algorithm, numerical tests on three synthetic datasets, four public face image datasets and a real-world transcriptomic dataset are conducted to show advantages of the developed algorithm in this paper. It is concluded that in terms of numerical efficiency, noise robustness and quality of matrix factorization, our algorithm is promising and applicable to face image reconstruction, and deep mining of transcriptomic profiles of the sub-genomes in hybrid fish lineage, compared with the state-of-the-art algorithms.Robust distribution-based nonnegative matrix factorizations for dimensionality reductionhttps://www.zbmath.org/1483.683132022-05-16T20:40:13.078697Z"Peng, Xinjun"https://www.zbmath.org/authors/?q=ai:peng.xinjun"Xu, Dong"https://www.zbmath.org/authors/?q=ai:xu.dong"Chen, De"https://www.zbmath.org/authors/?q=ai:chen.deSummary: As a popular dimensionality-reduction technique, nonnegative matrix factorization (NMF) has been widely researched since it is consistent with human cognitive processes in the psychology and physiology. This paper presents a novel NMF framework, called robust distribution-based NMF (RDNMF), to learn the robustly discriminative representations for data. In this RDNMF, a Kullback-Leibler divergence to measure the similarity between the data and representations is introduced, which fully preserves the geometrical structure of data. Meanwhile, this RDNMF employs the \(l_{2, 1}\)-norm loss to reduce the influence of noise and outliers. This paper further proposes a semi-supervised RDNMF (SRDNMF) by enforcing the representations of labeled points in the same class to be aligned on the same axis. The proposed RDNMF and SRDNMF are solved by the modified multiplicative update rules. Clustering experiments on seven benchmark datasets demonstrate the effectiveness of our methods in comparison to other state-of-the-art methods.Dual-channel hybrid community detection in attributed networkshttps://www.zbmath.org/1483.683142022-05-16T20:40:13.078697Z"Qin, Meng"https://www.zbmath.org/authors/?q=ai:qin.meng"Lei, Kai"https://www.zbmath.org/authors/?q=ai:lei.kaiSummary: This study considers the problem of hybrid community detection in attributed networks based on the information of network topology and attributes with the aim to address the following two shortcomings of existing hybrid community detection methods. First, many of these methods are based on the assumption that network topology and attributes carry consistent information but ignore the intrinsic mismatch correlation between them. Second, network topology is typically treated as the dominant source of information, with attributes employed as the auxiliary source; the dominant effect of attributes is seldom explored or indeed considered. To address these limitations, this paper presents a novel Dual-channel Hybrid Community Detection (DHCD) method that considers the dominant effects of topology and attributes separately. The concept of transition relation between the topology and attribute clusters is introduced to explore the mismatch correlation between the two sources and learn the behavioral and content diversity of nodes. An extended overlapping community detection algorithm is introduced based on the two types of diversity. By utilizing network attributes, DHCD can simultaneously derive the community partitioning membership and corresponding semantic descriptions. The superiority of DHCD over state-of-the-art community detection methods is demonstrated on a set of synthetic and real-world networks.Providing reliability in recommender systems through Bernoulli matrix factorizationhttps://www.zbmath.org/1483.683982022-05-16T20:40:13.078697Z"Ortega, Fernando"https://www.zbmath.org/authors/?q=ai:ortega.fernando"Lara-Cabrera, Raúl"https://www.zbmath.org/authors/?q=ai:lara-cabrera.raul"González-Prieto, Ángel"https://www.zbmath.org/authors/?q=ai:gonzalez-prieto.angel"Bobadilla, Jesús"https://www.zbmath.org/authors/?q=ai:bobadilla.jesusSummary: Beyond accuracy, quality measures are gaining importance in modern recommender systems, with reliability being one of the most important indicators in the context of collaborative filtering. This paper proposes Bernoulli Matrix Factorization (BeMF), which is a matrix factorization model, to provide both prediction values and reliability values. BeMF is a very innovative approach from several perspectives: a) it acts on model-based collaborative filtering rather than on memory-based filtering, b) it does not use external methods or extended architectures, such as existing solutions, to provide reliability, c) it is based on a classification-based model instead of traditional regression-based models, and d) matrix factorization formalism is supported by the Bernoulli distribution to exploit the binary nature of the designed classification model. The experimental results show that the more reliable a prediction is, the less liable it is to be wrong: recommendation quality improves after the most reliable predictions are selected. State-of-the-art quality measures for reliability have been tested, which shows that BeMF outperforms previous baseline methods and models.Real vector space and related notionshttps://www.zbmath.org/1483.684922022-05-16T20:40:13.078697Z"Nakasho, Kazuhisa"https://www.zbmath.org/authors/?q=ai:nakasho.kazuhisa"Okazaki, Hiroyuki"https://www.zbmath.org/authors/?q=ai:okazaki.hiroyuki"Shidama, Yasunari"https://www.zbmath.org/authors/?q=ai:shidama.yasunariSummary: In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [\textit{G. Bancerek} et al., Lect. Notes Comput. Sci. 9150, 261--279 (2015; Zbl 1417.68201); J. Autom. Reasoning 61, No. 1--4, 9--32 (2018; Zbl 1433.68530)], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among \texttt{TOP-REAL(n)}, \texttt{REAL-NS(n)}, and \texttt{n-VectSp\(\_\)over F\(\_\)Real}. We referred to [\textit{I. Miyadera}, Functional analysis (1972); \textit{K. Yosida}, Functional analysis. 6th ed. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0435.46002); \textit{L. Schwartz}, Analyse I: Théorie des ensembles et topologie. Paris: Hermann, Éditeurs des Sciences et des Arts (1991; Zbl 0872.54001)] in the formalization.Optimal quantum tomography with constrained measurements arising from unitary baseshttps://www.zbmath.org/1483.810232022-05-16T20:40:13.078697Z"Chaturvedi, S."https://www.zbmath.org/authors/?q=ai:chaturvedi.soni|chaturvedi.shashank|chaturvedi.sanchit|chaturvedi.shivam|chaturvedi.s-c|chaturvedi.sidharth|chaturvedi.sanjay-k"Ghosh, S."https://www.zbmath.org/authors/?q=ai:ghosh.sumonta|ghosh.s-n|ghosh.shromona|ghosh.saptarshi.2|ghosh.shinjan|ghosh.sebati|ghosh.sabyasachi|ghosh.shameek|ghosh.shreyoshi|ghosh.sankalpa|ghosh.sutapa|ghosh.saptarshi|ghosh.smriti|ghosh.sasanka|ghosh.sanmitra.1|ghosh.suddhodan|ghosh.subhas-kumar|ghosh.shomit-m|ghosh.sushant-g|ghosh.subhradip|ghosh.samrat|ghosh.shyamal|ghosh.sibasish|ghosh.subhash-chandra|ghosh.santanu|ghosh.subhankar|ghosh.saradindu|ghosh.sudip|ghosh.shounak|ghosh.subail-chandra|ghosh.subal-chandra|ghosh.sri-s-k|ghosh.santosh|ghosh.sikha|ghosh.swarnava|ghosh.satyajit|ghosh.sreeya|ghosh.supratim|ghosh.siddhartha|ghosh.sumit.1|ghosh.sagnik|ghosh.sanmitra|ghosh.shakuntala|ghosh.sayantan.1|ghosh.swapan-kumar.1|ghosh.subhojit|ghosh.somsubhra|ghosh.shamit|ghosh.subhro|ghosh.sharmistha.1|ghosh.sugata|ghosh.shubhashis|ghosh.sukumar|ghosh.sanjoy|ghosh.suman|ghosh.saptarshi.1|ghose.suranjan|ghosh.suranjana|ghosh.sushmita|ghosh.sadhan-kumar|ghosh.samir-kumar|ghosh.shankar|ghosh.sekhar|ghosh.subhendu|ghosh.sunil-kumar|ghosh.souvik|ghosh.sunita|ghosh.subir.1|ghosh.saugata|ghosh.sushil-kumar|ghosh.swarup-narayan|ghosh.samujjwal|ghosh.sugoutam|ghosh.sukhendu|ghosh.satrajit|ghosh.santanu-kumar|ghosh.saumya|ghosh.soumyabrata|ghosh.soumitra|ghosh.sambuddha|ghosh.swapan-kumar|ghosh.surjit-k|ghosh.swarnendu|ghosh.sarbari|ghosh.saibal-ranjan|ghosh.sumanta|ghosh.shaon|ghosh.sumana|ghosh.samiran|ghosh.sucharita|ghosh.santu|ghosh.sourav|ghosh.soumik|ghosh.swaroop|ghosh.sakti-pada|ghosh.shamik|ghosh.shyamali|ghosh.subhajit|ghosh.shamindra-kumar|ghosh.sunayana|ghosh.susmita|ghosh.shuva-j|ghosh.sanchita|ghosh.suvankar|ghosh.soumen|ghosh.saswata|ghosh.sudeshna|ghosh.samiran.1|ghosh.subhas-chandra|ghosh.samik|ghosh.sukesh-k|ghosh.shyamolina|ghosh.sujata|ghosh.sujit-kumar|ghosh.soumendu|ghosh.saptarshi-p|ghosh.shrabonti|ghosh.smita|ghosh.subir-kumar|ghosh.swarnadip|ghosh.supriyo|ghosh.susanta|ghosh.sudipta-kumar|ghosh.shubhrangshu|ghosh.sarthak|ghosh.subrata|ghosh.souparno|ghosh.swarnankur|ghosh.surojit|ghosh.sharmistha|ghosh.subhabrata|ghosh.sadhana|ghosh.saurabh|ghosh.sankar-kumar|ghosh.satyasadhan|ghosh.sanjay-kumar|ghosh.subarna|ghosh.saubhik|ghosh.sandip|ghosh.samit|ghosh.subhasree|ghosh.soumya-k|ghosh.subir|ghosh.surath|ghosh.sujoy|ghosh.subhroshekhar|ghosh.suchismita|ghosh.shubhro|ghosh.sanjoy-kumar|ghosh.saurav|ghosh.soham|ghosh.soumyadip|ghosh.suma|ghosh.sasthi-c|ghosh.s-l|ghosh.sayan|ghosh.somnath"Parthasarathy, K. R."https://www.zbmath.org/authors/?q=ai:parthasarathy.k-r|parthasarathy.kalyanapuram-rangachari"Singh, Ajit Iqbal"https://www.zbmath.org/authors/?q=ai:singh.ajit-iqbalScattering matrix of elementary excitations in the antiperiodic \textit{XXZ} spin chain with \(\eta = \frac{i\pi}{3}\)https://www.zbmath.org/1483.811352022-05-16T20:40:13.078697Z"Sun, Pei"https://www.zbmath.org/authors/?q=ai:sun.pei"Yang, Jintao"https://www.zbmath.org/authors/?q=ai:yang.jintao"Qiao, Yi"https://www.zbmath.org/authors/?q=ai:qiao.yi"Cao, Junpeng"https://www.zbmath.org/authors/?q=ai:cao.junpeng"Yang, Wen-Li"https://www.zbmath.org/authors/?q=ai:yang.wenliSummary: We study the thermodynamic limit of the antiperiodic XXZ spin chain with the anisotropic parameter \(\eta = \frac{\pi i}{3}\). We parameterize eigenvalues of the transfer matrix by their zero points instead of Bethe roots. We obtain patterns of the distribution of zero points. Based on them, we calculate the ground state energy and the elementary excitations in the thermodynamic limit. We also obtain the two-body scattering matrix of elementary excitations. Two types of elementary excitations and three types of scattering processes are discussed in detailed.Gluon gravitational form factors at large momentum transferhttps://www.zbmath.org/1483.811512022-05-16T20:40:13.078697Z"Tong, Xuan-Bo"https://www.zbmath.org/authors/?q=ai:tong.xuan-bo"Ma, Jian-Ping"https://www.zbmath.org/authors/?q=ai:ma.jianping"Yuan, Feng"https://www.zbmath.org/authors/?q=ai:yuan.fengSummary: We perform a perturbative QCD analysis of the gluonic gravitational form factors (GFFs) of the proton and pion at large momentum transfer. We derive the explicit factorization formula of the GFFs in terms of the distribution amplitudes of hadrons. At the leading power, we find that the gluon GFFs \(A_g\) and \(C_g\) scale as \(A_g^\pi(t) = C_g^\pi(t) \sim 1/(-t)\) for pion, \(A_g^p(t)\sim 1/(-t)^2\) and \(C_g^p(t)\sim \ln^2(-t/\Lambda^2)/(-t)^3\) for proton, respectively, where \(t\) is the momentum transfer and \(\Lambda\) a non-perturbative scale to regulate the endpoint singularity in \(C_g^p\) calculation. Our results provide a unique perspective of the momentum dependence of the GFFs and will help to improve our understanding of the internal pressure distributions of hadrons.Moments of the ground state density for the \(d\)-dimensional Fermi gas in an harmonic traphttps://www.zbmath.org/1483.811722022-05-16T20:40:13.078697Z"Forrester, Peter J."https://www.zbmath.org/authors/?q=ai:forrester.peter-jIntroduction to the algebraic Bethe ansatzhttps://www.zbmath.org/1483.820032022-05-16T20:40:13.078697Z"Slavnov, N. A."https://www.zbmath.org/authors/?q=ai:slavnov.nikita-aThis note is a short introduction to the Algebraic Bethe Ansatz that is one of the essential achievements of the Quantum Inverse Scattering Method. This note is based on the lecture given in the Scientific and Educational Center of Steklov Mathematical Institute in Moscow. The symmetry is limited to \(\mathfrak{sl}_2\) in this note. In Section 2 the \(R\)-matrix \(R(u,v)\) and the monodromy matrix \({T}(u)\) are introduced by the Yang-Baxter relation and the \(RTT\)-relation. The transfer matrix \(\mathcal{T}(u)\) is introduced as the sum of diagonal elements of the monodromy matrix \(T(u)=\left(\begin{smallmatrix} A(u)&B(u)\\ C(u)&D(u) \end{smallmatrix}\right)\). The transfer matrix \(\mathcal{T}(u)\) commutes with the Hamiltonian, hence constructing eigenfunctions of the transfer matrix determines those of the Hamiltonian. In Section 3 the XXX Heisenberg magnet is introduced as an example. In Section 4 the monodromy matrix of the XXZ Heisenberg magnet is introduced. In Section 5 is devoted to construction of the eigenfunctions of the transfer matrix \(\mathcal{T}(u)\). The \(RTT\) relation and the requirements of the spectral parameters called the Bethe equations allow construction of the eigenfunctions of the transfer matrix \(\mathcal{T}(u)\). The spectrums and the Bethe equations of the XXX Heisenberg magnet are given as an example.
For the entire collection see [Zbl 1472.53006].
Reviewer: Takeo Kojima (Yonezawa)The emergence of expanding space-time and intersecting D-branes from classical solutions in the Lorentzian type IIB matrix modelhttps://www.zbmath.org/1483.830762022-05-16T20:40:13.078697Z"Hatakeyama, Kohta"https://www.zbmath.org/authors/?q=ai:hatakeyama.kohta"Matsumoto, Akira"https://www.zbmath.org/authors/?q=ai:matsumoto.akira"Nishimura, Jun"https://www.zbmath.org/authors/?q=ai:nishimura.jun-ichi"Tsuchiya, Asato"https://www.zbmath.org/authors/?q=ai:tsuchiya.asato"Yosprakob, Atis"https://www.zbmath.org/authors/?q=ai:yosprakob.atisSummary: The type IIB matrix model is a promising candidate for a nonperturbative formulation of superstring theory. As such, it is expected to explain the origin of space-time and matter at the same time. This has been partially demonstrated by the previous Monte Carlo studies on the Lorentzian version of the model, which suggested the emergence of (3+1)-dimensional expanding space-time. Here we investigate the same model by solving numerically the classical equation of motion, which is expected to be valid at late times since the action becomes large due to the expansion of space. Many solutions are obtained by the gradient descent method starting from random matrix configurations, assuming a quasi-direct-product structure for the (3+1)-dimensions and the extra 6 dimensions. We find that these solutions generally admit the emergence of expanding space-time and a block-diagonal structure in the extra dimensions, the latter being important for the emergence of intersecting D-branes. For solutions corresponding to D-branes with appropriate dimensionality, the Dirac operator is shown to acquire a zero mode in the limit of infinite matrix size.Copositivity for 3rd-order symmetric tensors and applicationshttps://www.zbmath.org/1483.901072022-05-16T20:40:13.078697Z"Liu, Jiarui"https://www.zbmath.org/authors/?q=ai:liu.jiarui"Song, Yisheng"https://www.zbmath.org/authors/?q=ai:song.yisheng|song.yisheng.1Summary: The strict copositivity of 4th-order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) copositivity of 4th-order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided several analytically sufficient conditions for the copositivity of 3rd-order 2-dimensional (3-dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th-order tensors, the analytically sufficient conditions of copositivity are proved for 4th-order 2-dimensional and 3-dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for \(\mathbb{Z}_3\) scalar dark matter.Two descent Dai-Yuan conjugate gradient methods for systems of monotone nonlinear equationshttps://www.zbmath.org/1483.901612022-05-16T20:40:13.078697Z"Waziri, Mohammed Yusuf"https://www.zbmath.org/authors/?q=ai:waziri.mohammed-yusuf"Ahmed, Kabiru"https://www.zbmath.org/authors/?q=ai:ahmed.kabiruSummary: In this paper, we present two Dai-Yuan type iterative methods for solving large-scale systems of nonlinear monotone equations. The methods can be considered as extensions of the classical Dai-Yuan conjugate gradient method for unconstrained optimization. By employing two different approaches, the Dai-Yuan method is modified to develop two different search directions, which are combined with the hyperplane projection technique of Solodov and Svaiter. The first search direction was obtained by carrying out eigenvalue study of the search direction matrix of an adaptive DY scheme, while the second is obtained by minimizing the distance between two adaptive versions of the DY method. Global convergence of the methods are established under mild conditions and preliminary numerical results show that the proposed methods are promising and more effective compared to some existing methods in the literature.Parametric model order reduction based on parallel tensor compressionhttps://www.zbmath.org/1483.930432022-05-16T20:40:13.078697Z"Li, Zhen"https://www.zbmath.org/authors/?q=ai:li.zhen"Jiang, Yao-Lin"https://www.zbmath.org/authors/?q=ai:jiang.yaolin"Mu, Hong-liang"https://www.zbmath.org/authors/?q=ai:mu.hong-liangSummary: In this paper, we for the first time explore the model order reduction (MOR) of parametric systems based on the tensor techniques and a parallel tensor compression algorithm. For the parametric system characterising multidimensional parameter space and nonlinear parametric dependence, we first approximate the system matrices by tensor functions of the parameters, whose first-order coefficients are third-order tensors. In order to effectively reduce the computational cost and the storage burden, we propose a parallel tensor compression algorithm based on tensor-SVD to deal with the tensors in the tensor functions. Then, we obtain the low-rank approximation in Kruskal form of third-order tensors. After that, by computing the first several expansion coefficients of the state variable with the selected parameter vectors, the projection matrix is constructed to obtain the reduced parametric system. Theoretical analysis shows that the reduced parametric system can match the first several expansion coefficients of the output variable of the original system at the selected parameter vectors. Moreover, the stability of the proposed MOR method is discussed. Finally, the efficiency of the proposed method is illustrated by two numerical examples.An improved predictive control model for stochastic max-plus-linear systemshttps://www.zbmath.org/1483.931532022-05-16T20:40:13.078697Z"Qu, Jingguo"https://www.zbmath.org/authors/?q=ai:qu.jingguo"Zhang, Zilong"https://www.zbmath.org/authors/?q=ai:zhang.zilong"Zhang, Huiqi"https://www.zbmath.org/authors/?q=ai:zhang.huiqiSummary: In order to improve the robustness and stability of the model predictive control system, this paper research the problem by combination of stochastic predictive control and max-plus theory. Based on the analysis of the stochastic predictive control model, the maximum plus stochastic predictive control model is constructed, which is improved by the max-plus algebraic theory. The superiority of the maximum plus stochastic predictive control model is verified by simulation and experiment. The max-plus algebra is an algorithm which is suitable for noise processing of input signal, which can stabilize the input of the control system. The disadvantage of stochastic predictive control model is that the input signal is subjected to random disturbance in the external environment, max-plus algebraic theory can better compensate for the defect. The simulation results show that the stochastic predictive control model has significant advantages in accuracy, stability and robustness.Modelling and feedback control for a class of Petri nets with shared resources subject to strict time constraints using max-plus algebrahttps://www.zbmath.org/1483.931842022-05-16T20:40:13.078697Z"Aberkane, Sofiane"https://www.zbmath.org/authors/?q=ai:aberkane.sofiane"Kara, Redouane"https://www.zbmath.org/authors/?q=ai:kara.redouane"Amari, Saïd"https://www.zbmath.org/authors/?q=ai:amari.saidSummary: We address a modelling framework and a state feedback control problem of discrete event systems with shared resources using max-plus algebra. In this paper, we consider the class of networked conflicting timed event graphs (NCTEGs), which are timed Petri nets with conflicts, subjected to strict temporal constraints. Firstly, an algebraic formalisation in terms of switching max-plus linear systems is proposed to describe the dynamic behaviour of a NCTEG. Secondly, closed-loop control laws are calculated to ensure the compliance of these time constraints imposed on a certain number of places. For this, sufficient conditions for the existence of such control laws have been provided. Finally, these theoretical approaches are illustrated by a realistic example of a time-critical railway network crossing system.An anti-windup approach for nonlinear impulsive system subject to actuator saturationhttps://www.zbmath.org/1483.932452022-05-16T20:40:13.078697Z"Zhu, Chenhong"https://www.zbmath.org/authors/?q=ai:zhu.chenhong"Li, Xiaodi"https://www.zbmath.org/authors/?q=ai:li.xiaodi"Wang, Kening"https://www.zbmath.org/authors/?q=ai:wang.keningSummary: This paper is concerned with the analysis and design of nonlinear impulsive systems subject to actuator saturation, where both impulsive control problem and impulsive disturbance problem are considered. By utilizing a dead-zone function to deal with saturation nonlinearity, some conditions are obtained for local exponential stability and stabilizablity (\textit{LES}) of the considered system. The design of the controller is formulated such that the domain of attraction is as large as possible and moreover, it can be solved as a convex optimization problem with linear matrix inequalites (\textit{LMIs}) constraints. Two numerical examples are used to demonstrate the effectiveness of our proposed results.Formal analysis and control of timed automata with guards using (max, +) and (min, +) algebrashttps://www.zbmath.org/1483.933782022-05-16T20:40:13.078697Z"Ait Oumeziane, F."https://www.zbmath.org/authors/?q=ai:ait-oumeziane.f"Kara, R."https://www.zbmath.org/authors/?q=ai:kara.redouane|kara.rukiye"Amari, S."https://www.zbmath.org/authors/?q=ai:amari.shinji|amari.smain|amari.said|amari.satoshi|amari.shun-ichi|amari.suprasad-vSummary: This paper has introduced new formal approaches to model behaviours and control of a class of uncertain timed discrete event systems represented by timed automata with guards (TAGs). We propose alternative representations for TAGs which approximate their dynamics, since only extremal behaviours are considered. More precisely, recursive equations are proposed in (max, +) and (min, +) algebras to describe the worst and the optimistic behaviours. Thereafter, these developed linear models are used to treat a control problem of TAGs. Finally, the proposed methodologies are illustrated by a realistic study case that corresponds to a job-shop system.Analysis of P-time event graphs in (max,+) and (min,+) semiringshttps://www.zbmath.org/1483.934072022-05-16T20:40:13.078697Z"Špaček, Pavel"https://www.zbmath.org/authors/?q=ai:spacek.pavel"Komenda, Jan"https://www.zbmath.org/authors/?q=ai:komenda.jan"Lahaye, Sébastien"https://www.zbmath.org/authors/?q=ai:lahaye.sebastienSummary: In this paper, we investigate the behaviour of P-time event graphs, a class of time Petri nets with non-deterministic timing of places. Our approach is based on combined linear descriptions in both (max,+) and (min,+) semirings, where lower bounds on the state vector are (max,+)-linear and upper bounds are (min,+)-linear. We present necessary and sufficient conditions for the existence of extremal (fastest and slowest) periodic trajectories that are derived from the new description. The results are illustrated by a realistic example of an electroplating process.