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In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L : ℂ q × q → ℂ n × n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L ( V ∗ ) = L ( V ) ∗ , via representations of the form L ( V ) = ∑ k , l = 1 m H k l A l V A k ∗ , V ∈ ℂ q × q , for matrices A 1 , … , A m ∈ ℂ n × q and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A 1 , … , A m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H = H k l explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗ -linearity on the linear map L .

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the simple Lie group $E_{6(-14)}$, which is of Hermitian symmetric type. Each FS-scattered Dirac series of $E_{6(-14)}$ is realized as a composition factor of certain $A_{\mathfrak{q}}(\lambda)$ module. Along the way, we have also obtained all the fully supported irreducible unitary representations of $E_{6(-14)}$ with integral infinitesimal characters., Comment: 32 pages, some strings are folded

Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its many-body topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems., 15 pages, 5 figures

Definite Determinantal polynomials play a crucial role in semidefinite programming problems. Helton and Vinnikov proved that real zero (RZ) bivariate polynomials are definite determinantals. Indeed, in general, it is a difficult problem to decide whether a given polynomial is definite determinantal, and if it is, it is of paramount interest to determine a definite determinantal representation of that polynomial. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree d. In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points.

Let $G$ be an arbitrary group. We define a gain-line graph for a gain graph $(\Gamma,\psi)$ through the choice of an incidence $G$-phase matrix inducing $\psi$. We prove that the switching equivalence class of the gain function on the line graph $L(\Gamma)$ does not change if one chooses a different $G$-phase inducing $\psi$ or a different representative of the switching equivalence class of $\psi$. In this way, we generalize to any group some results proven by N. Reff in the abelian case. The investigation of the orbits of some natural actions of $G$ on the set $\mathcal H_\Gamma$ of $G$-phases of $\Gamma$ allows us to characterize gain functions on $\Gamma$, gain functions on $L(\Gamma)$, their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph., Comment: 28 pages, 6 figures, 1 table

We extend the continuity equation of the Kahler metrics introduced by La Nave & Tian and the Hermitian metrics introduced by Sherman & Weinkove to the almost Hermitian metrics, and establish its interval of maximal existence. As an example, we study the continuity equation on the (locally) homogeneous manifolds in more detail.

Let $A$ be an $n \times n$ positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of $A$ as the integral of some explicit log-concave function on ${\Bbb R}^{2n}$. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for the permanent of $A$., 5 pages

We analyze both the facial structure of the Gram spectrahedron $\mathrm{Gram}(f)$ and of the Hermitian Gram spectrahedron $\mathcal{H}^{\scriptscriptstyle+}(f)$ of a nonnegative binary form $f \in \mathbb{R}[x, y]_{2d}$. We show that if $F \subseteq \mathcal{H}^{\scriptscriptstyle+}(f)$ is a polyhedral face of dimension $k$ then $\binom{k+1}{2} \leq d$. Conversely, for all $k \in \mathbb{N}$ and $d \geq \binom{k+1}{2}$ we show that the Hermitian Gram spectrahedron of a general positive binary form $f \in \mathbb{R}[x, y]_{2d}$ with distinct roots contains a face $F$ which is a $k$-simplex and whose extreme points are rank-one tensors. For all $k \in \mathbb{N}$ and $d \geq (k+1)^2$ the (symmetric) Gram spectrahedron of a general positive binary form $f \in \mathbb{R}[x, y]_{2d}$ contains a polyhedral face $F$ with $(\mathrm{rk}(F), \dim(F)) = (2(k+1), k)$., 16 pages

We generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case where it is non-Hermitian. New admissibility conditions are introduced to replace the Hermiticity condition. We prove new variational estimates of the Rayleigh quotient for non-Hermitian matrices. As an application, a new Delsarte-Hoffman-type bound on the size of the largest independent pair in a directed graph is developed. Our techniques consist in quantifying the impact of breaking the Hermitian symmetry of a matrix and are broadly applicable.

By introducing a regularization matrix and an additional iteration parameter, a new class of regularized deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioners are proposed for generalized saddle point linear systems. Compared with the well-known Hermitian and skew-Hermitian splitting (HSS) preconditioner and the regularized HSS preconditioner (Bai, 2019) studied recently, the new RDPSS preconditioners have much better computing efficiency especially when the (1,1) block matrix is non-Hermitian. It is proved that the corresponding RDPSS stationary iteration method is unconditionally convergent. In addition, clustering property of the eigenvalues of the RDPSS preconditioned matrix is studied in detail. Two numerical experiments arising from the meshfree discretization of a static piezoelectric equation and the finite element discretization of the Navier–Stokes equation show the effectiveness of the new proposed preconditioners.

In this paper, we consider the complex Hessian equation involving additional gradient terms on both left and right hand sides. We derive the second order estimates for χ ˜ -plurisubharmonic solutions to this equation on compact Hermitian manifolds. Such an estimate may be helpful to study the existence and regularity for the solution of the equation.

In the setting of various complex Banach spaces we consider the questions of when a square of a Hermitian operator is also Hermitian, and when a Hermitian operator has a Hermitian square root. We also describe the structure of Hermitian projections and Hermitian square roots of the identity operator.

We prove that the determinant of a matrix polynomial with Hermitian positive definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients.

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.

Let F q 2 be the finite field with q 2 elements. Most of the known F q 2 -maximal curves arise as quotient curves of the F q 2 -maximal Hermitian curve H q . After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of H q , but their complete determination is a challenging open problem. In this paper we determine completely the spectrum of genera of quotients of H q for any q ≡ 1 ( mod 4 ) .

The singular values of a matrix A are defined as the square roots of the eigenvalues of A ⁎ A , and the energy of A denoted by E ( A ) is the sum of its singular values. The energy of a graph G, E ( G ) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = ( B D D ⁎ C ) , then E ( A ) ≥ 2 E ( D ) . Also, we show that if G is a graph and H is a spanning subgraph of G such that E ( H ) is an edge cut of G, then E ( H ) ≤ E ( G ) , i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E ( G ) ≥ 4 m + n ( n − 2 ) | det ⁡ ( A ) | 2 n . Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E ( G ) given in Oboudi (2019) [14] .

We extend the positive-definite and skew-Hermitian splitting (PSS) iterative method to EPSS (for extended PSS) for solving non-Hermitian positive-definite systems of linear equations. In this kind of extension, the shift matrix is replaced by a Hermitian positive-definite matrix Σ . It is proved that the method is unconditionally convergent. Then, the EPSS method with a 2 × 2-block diagonal matrix Σ as the shift matrix is applied to generalized saddle point problems and the convergence of the method is verified. Naturally, the induced preconditioner can be applied to generalized saddle point problems. In some special cases, this preconditioner coincides with some existing preconditioners. A special case of the EPSS preconditioner, say SEPSS, is used to accelerate the convergence of GMRES. Effectiveness of this preconditioner is investigated by some numerical experiments.

An extended regularized Hermitian and skew-Hermitian splitting iteration method, for solving the double saddle point linear system arising from PDE optimal control problem, is proposed. The unconditional convergence of the method is proved. Furthermore, some spectral properties of the preconditioned matrix are investigated. Numerical experiments show that the iteration method is feasible and efficient.

A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set { v 1 , … , v n } , is the matrix H = [ h i j ] n × n , where h i j = − h j i = i if there is a directed edge from v i to v j , h i j = 1 if there exists an undirected edge between v i and v j , and h i j = 0 otherwise. The Hermitian spectrum of a mixed graph is defined to be the spectrum of its Hermitian adjacency matrix. In this paper we study mixed graphs which are determined by their Hermitian spectrum (DHS). First, we show that each mixed cycle is switching equivalent to either a mixed cycle with no directed edges ( C n ), a mixed cycle with exactly one directed edge ( C n 1 ), or a mixed cycle with exactly two consecutive directed edges with the same direction ( C n 2 ) and we determine the spectrum of these three types of cycles. Next, we characterize all DHS mixed paths and mixed cycles. We show that all mixed paths of even order, except P 8 and P 14 , are DHS. It is also shown that mixed paths of odd order, except P 3 , are not DHS. Also, all cospectral mates of P 8 , P 14 and P 4 k + 1 and two families of cospectral mates of P 4 k + 3 , where k ≥ 1 , are introduced. Finally, we show that the mixed cycles C 2 k and C 2 k 2 , where k ≥ 3 , are not DHS, but the mixed cycles C 4 , C 4 2 , C 2 k + 1 , C 2 k + 1 2 , C 2 k + 1 1 and C 2 j 1 except C 7 1 , C 9 1 , C 12 1 and C 15 1 , are DHS, where k ≥ 1 and j ≥ 2 .

We study certain pairs of subspaces V and W of C^n we call supports that consist of eigenspaces of the eigenvalues ±‖M‖ of a minimal hermitian matrix M(‖M‖ ≤‖M+D‖ for all real diagonals D). For any pair of orthogonal subspaces we define a non negative invariant δ called the adequacy to measure how close they are to form a support and to detect one. This function δ is the minimum of another map F defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of F in order to approximate δ. These results allow us to prove that the set of supports has interior points in the space of flag manifolds. Fil: Mendoza, Alberto. Universidad Simón Bolívar; Venezuela Fil: Recht, Lázaro. Universidad Simón Bolívar; Venezuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina Fil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

In an earlier work, the author together with Guo [Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory 85 (2017) 217-248] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory., Comment: Linear Algebra Appl. (2019), in press. arXiv admin note: text overlap with arXiv:1505.01321

In this paper, we discuss the following minimum rank matrix approximation problem in the spectral norm: min X r ( X ) subject to ‖ A − B X B ∗ ‖ 2 1 , where A and X are Hermitian matrices. For its application, we also consider the minimum rank matrix approximation problem: min X r ( X ) subject to ‖ A B B ∗ X ‖ 2 1 .

This paper is devoted to the asymptotic behavior of all eigenvalues of the increasing finite principal sections of an infinite symmetric (in general non-Hermitian) Toeplitz matrix. The symbol of the infinite matrix is supposed to be moderately smooth and to trace out a simple loop in the complex plane. The main result describes the asymptotic structure of all eigenvalues. The asymptotic expansions constructed are uniform with respect to the location of the eigenvalues in the bulk of the spectrum.

We examine some connections among phase-retrievable (not necessarily self-adjoint) operator-valued frames, projective group representation frames and representations of quantum channels. We first present some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces, which generalize the known results for vector-valued frames, fusion frames and frames of Hermitian matrices. For an irreducible projective unitary representation of a finite group, the image system is automatically phase-retrievable and, moreover, it is a point-wisely tight operator-valued frames. We generalize this notion to more general operator-valued frames, and prove that point-wise tight operator-valued frames are exactly the ones that are right equivalent to operator-valued tight frames. For an operator system that represent a quantum channel, we show that phase-retrievability of the system is independent of the choices of the representations of the quantum channel.

The purpose of this paper is twofold. We first solve the Dirichlet problem for α -Hermitian–Einstein equations on I ± -holomorphic bundles over bi-Hermitian manifolds. Second, by using Uhlenbeck–Yau’s continuity method, we prove that the existence of approximate α -Hermitian–Einstein structure and the α -semi-stability on I ± -holomorphic bundles over compact bi-Hermitian manifolds are equivalent.

The FEAST algorithm, a contour-integral based eigensolver, was developed for computing the eigenvalues inside a given interval, along with their eigenvectors, of a Hermitian generalized eigenproblem. The FEAST algorithm is a fast and stable technique, and is easily parallelizable. However, it was shown that this algorithm may fail to find the desired eigenpairs when applied to non-Hermitian problems. Efforts have been made to adapt FEAST to non-Hermitian problems. In this work, we aim at formulating a new non-Hermitian scheme for the FEAST algorithm. Our new scheme is based on the partial generalized Schur decomposition. Unlike the existing non-Hermitian FEAST methods, our method seeks approximation to the generalized Schur vectors instead of the potentially ill-conditioned eigenvectors. Our new method can compute not only the eigenvectors but also the right and left generalized Schur vectors corresponding to target eigenvalues. We present standard benchmark numerical experiments in which our method achieves better stability and efficiency comparing with the existing non-Hermitian FEAST algorithms.

We give a classification of Levi-umbilical real hypersurfaces, whose Levi form is proportional to the induced metric, in the complex two plane Grassmannian G 2 ( C m + 2 ) , the complex quadric Q n , or their noncompact dual spaces.

We consider real versions of the classical numerical range and higher rank numerical ranges of a matrix. Our motivations come from descriptions of geometric structures in Euclidean space such as ellipsoids. We establish a number of basic results on these numerical ranges, comparing and delineating from their complex counterparts. We prove a two-dimensional elliptical range theorem for the joint real numerical range. We show that the joint numerical range and the real joint numerical range respectively of a set of m Hermitian matrices of order n is contained in an affine subspace of dimension n 2 − 1 and 1 2 ( n 2 + n − 2 ) respectively.

Let $\mathcal{H}$ be the Hermitian curve defined over a finite field $\mathbb{F}_{q^2}$. In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over $\mathcal{H}$, started in [1]: if $d$ is the distance of the code, the supports are all the sets of $d$ distinct $\mathbb{F}_{q^2}$-points on $\mathcal{H}$ complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. \texttt{DegRevLex}. For most Hermitian codes, and especially for all those with distance $d\geq q^2-q$ studied in [1], one of the two curves is always the Hermitian curve $\mathcal{H}$ itself, while if $d

In this paper, first we consider the numerical range of C ψ , φ , when φ ( z ) = r z with | r | ≤ 1 . Then we study the numerical range of C ψ , φ , where φ is an elliptic automorphism. Next, the exact value of the norm of some weighted composition operators are obtained in order to investigate their numerical radiuses. Finally, we compute the numerical ranges of all hermitian weighted composition operators.

In this paper, we introduce a new class of linear codes over finite fields, viz. skew multi-twisted codes (or skew generalized quasi-twisted codes), which is a generalization of some well-known classes of linear codes such as cyclic codes and quasi-cyclic codes. We also study algebraic structures of skew multi-twisted codes and their Galois duals (i.e., orthogonal complements with respect to the Galois inner product). We also view skew multi-twisted codes as direct sums of certain concatenated codes, which gives rise to a method to construct these codes. We also determine a lower bound on their minimum Hamming distances using their multilevel concatenated structure. Apart from this, we determine the parity-check polynomial of each skew multi-twisted code, and obtain BCH type bounds on their minimum Hamming distances. We also determine generating sets of Galois duals of some skew multi-twisted codes from generating sets of these codes. We also derive necessary and sufficient conditions under which a skew multi-twisted code is (i) Galois self-dual, (ii) Galois self-orthogonal and (iii) Galois LCD (linear with complementary dual). As an application of this result, we also provide enumeration formulae for all Hermitian self-dual, Hermitian self-orthogonal, Hermitian LCD and Euclidean LCD multi-twisted codes over finite fields. We also obtain many linear codes with best known and optimal parameters from 1-generator skew multi-twisted codes over finite fields F 8 and F 9 .

Recently, by applying the minimum residual technique to the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, a minimum residual HSS (MRHSS) iteration method was proposed for solving non-Hermitian positive definite linear systems. Although the MRHSS iteration method is very efficient, it is conditionally convergent. In this work, we further study the convergence of the MRHSS iteration method, and show that it can unconditionally convergent if its parameters are determined by minimizing a new norm of the residual. Numerical results verify that the MRHSS method discussed in this work is also very efficient.

The FEAST algorithm, due to Polizzi, is a typical contour-integral based eigensolver for computing the eigenvalues, along with their eigenvectors, inside a given region in the complex plane. It was formulated under the circumstance that the considered eigenproblem is Hermitian. The FEAST algorithm is stable and accurate, and has attracted much attention in recent years. However, Yin et al. (2017) observed that the FEAST algorithm may fail to find the target eigenpairs when applied to non-Hermitian problems. This observation motivated them to formulate a non-Hermitian version for the FEAST algorithm. The key to the success of their method is that they use the oblique projection, rather than the orthogonal projection used in the FEAST algorithm, to extract desired eigenpairs. The objective of the present work is two-folded. Firstly, we would like to present a new non-Hermitian scheme for the FEAST algorithm, which also uses the oblique projection technique to extract the desired eigenpairs. Secondly, we would like to study the convergence properties of the non-Hermitian FEAST algorithms based on oblique projection to illustrate the effectiveness. Numerical experiments are reported to demonstrate the numerical performance of the non-Hermitian FEAST algorithms based on oblique projection and to validate the convergence properties studied.

In this paper, we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the Hermitian normalized Laplacian matrix for directed networks. In order to start, we discuss the Courant–Fischer theorem for the eigenvalues of Hermitian normalized Laplacian matrix. Based on the Courant–Fischer theorem, we obtain a similar result towards the normalized Laplacian matrix of undirected networks: for each i ∈ {1, 2,…, n}, any eigenvalue of Hermitian normalized Laplacian matrix λi ∈ [0, 2]. Moreover, we prove some special conditions if 0, or 2 is an eigenvalue of the Hermitian normalized Laplacian matrix L ( X ) . On top of that, we investigate the symmetry of the eigenvalues of L ( X ) and the edge-version for the eigenvalue interlacing result. Finally we present two expressions for the coefficients of the characteristic polynomial of the Hermitian normalized Laplacian matrix. As an outlook, we sketch some novel and intriguing problems to which our apparatus could generally be applied.

Let T , M ∈ C m × n be such that rank ( M ) = rank ( T ) . Let U T and U M be the partial isometries associated to the generalized polar decompositions T = U T | T | and M = U M | M | . A formula for U M is provided in the Hermitian case, and based on this formula, some necessary and sufficient conditions are derived under which U M = U T .

As the least squares mean for the Riemannian trace metric on the cone of positive definite matrices, the Riemannian mean with its computational and theoretical approaches has been widely studied. Recently the new metric and the least squares mean on the cone of positive definite matrices, which are called the Wasserstein metric and the Wasserstein mean, respectively, have been introduced. In this paper, we explore some properties of Wasserstein mean such as determinantal inequality and find bounds for the Wasserstein mean. Using bounds for the Wasserstein mean, we verify that the Wasserstein mean is the multivariate Lie-Trotter mean.

We present a new method for the study of hemisystems of the Hermitian surface U 3 of PG ( 3 , q 2 ) . The basic idea is to represent generator-sets of U 3 by means of a maximal curve naturally embedded in U 3 so that a sufficient condition for the existence of hemisystems may follow from results about maximal curves and their automorphism groups. In this paper we obtain a hemisystem in PG ( 3 , p 2 ) for each p prime of the form p = 1 + 16 n 2 with an integer n. Since the famous Landau's conjecture dating back to 1904 is still to be proved (or disproved), it is unknown whether there exists an infinite sequence of such primes. What is known so far is that just 18 primes up to 51000 with this property exist, namely 17 , 257 , 401 , 577 , 1297 , 1601 , 3137 , 7057 , 13457 , 14401 , 15377 , 24337 , 25601 , 30977 , 32401 , 33857 , 41617 , 50177 . The scarcity of such primes seems to confirm that hemisystems of U 3 are rare objects.

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding solitons on their nilradicals., 14 pages; section 4 extended; last version, to appear in Differential Geometry and its Applications

Let q be a prime power. For u = ( u 1 , … , u n ) , v = ( v 1 , … , v n ) ∈ F q 2 n let 〈 u , v 〉 : = ∑ i = 1 n u i q v i be the Hermitian form of F q 2 n . For any n × n matrix M over F q 2 set Num k ( M ) : = { 〈 u , M u 〉 | u ∈ F q 2 n , 〈 u , u 〉 = k } . Num 1 ( M ) is the numerical range of q. In this paper we use Num 1 to give a characterization of hermitian matrices, give some examples of 3 × 3 matrices M with Num k ( M ) = F q 2 for all k ∈ F q and study F q and F q 2 linear subspaces V ⊂ M n , n ( F q 2 ) such that either ♯ ( Num 1 ( M ) ) ≤ q for all M ∈ V or ♯ ( Num 1 ( M ) ) > q for all M ∈ V ∖ { 0 } .

Let $X=G/H$ be a symmetric space for a real simple Lie group $G$, equipped with a $G$-invariant complex structure. Then, $X$ is a pseudo-Hermitian manifold, and in this geometric setting, higher Laplacians $L_m$ are defined for each positive integer $m$, which generalize the ordinary Laplace-Beltrami operator. We show that $L_1,L_3,\ldots, L_{2r-1}$ form a set of algebraically independent generators for the algebra $\mathbb{D}_G(X)$ of $G$-invariant differential operators on $X$, where $r$ denotes the rank of $X$. This confirms a conjecture of Engli\v{s} and Peetre, originally stated for the class of Hermitian symmetric spaces., Comment: 26 pages

In this paper, we study the complex symmetric weighted composition operators on the Hilbert space H s of analytic functions over the open unit ball with reproducing kernels K w s ( z ) = ( 1 − 〈 z , w 〉 ) − s , where s ∈ N + . We completely characterize complex symmetric weighted composition operators with respect to two important conjugations J V and J σ , which include Hermitian weighted composition operators and normal weighted composition operators appeared in [17] . Surprisingly, they are essential representatives of general conjugations of the form C ψ , φ J , where C ψ , φ is unitary and J-symmetric. Indeed, these complex symmetric weighted composition operators can only be divided into two classes. The size estimates and some other properties of these complex symmetric weighted composition operators are obtained. Moreover, involutive weighted composition operators are completely characterized too. As an application, we construct concrete conjugations for the involutive composition operators, which solves a problem raised by Garcia and Hammond in a general case.

We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago. Following the results of Sasaki, we try to write and deduce them by different means. Questions of vector fields, mainly those arising from the base, are related as invariants of the classical metric, contact and Hermitian structures. Attention is given to the natural notion of extension or complete lift of a vector field, from the base to the tangent manifold. Few results are original, but finally new equations of the mirror map are considered., To appear in Expositiones Mathematicae; 15 pages

We show that every x -tight set of a Hermitian polar spaces H ( 2 n , q 2 ) , n ≥ 2 , is the union of x disjoint generators of the polar space provided that x ≤ 1 2 ( q + 1 ) . This was known before only when n ∈ { 2 , 3 } . This result is a contribution to the conjecture that the smallest x -tight set of H ( 2 n , q 2 ) that is not a union of disjoint generators occurs for x = q + 1 and is for sufficiently large q an embedded subgeometry.

We present a multistep Rayleigh quotient iteration, as well as its inexact variant, for computing an eigenpair of a large sparse Hermitian matrix. Theoretical analysis shows that both exact and inexact multistep Rayleigh quotient iterations converge much faster than the exact and inexact Rayleigh quotient iterations, respectively. For the inexact multistep Rayleigh quotient iteration, we use the preconditioned conjugate gradient method to solve the inner linear systems, and find that significant saving in the number of inner iteration steps can be achieved when choosing a proper preconditioner. Numerical examples demonstrate effectiveness and superiority of our methods.

In this paper, we develop an arithmetic theory of quadratic and hermitian forms over infinite algebraic extensions of local and global fields. In particular, we prove that the cohomological Hasse principle for H 1 holds for all semisimple simply connected algebraic groups defined over any infinite algebraic extension of any global field and we also show the validity of some local–global principles for (skew-)hermitian forms defined over such infinite extension fields. As applications, we deduce some analogs of well known results such as Landherr's and Kneser's Strong Hasse principle, Hasse–Maass–Schilling Norm Theorem, Albert–Brauer–Hasse–Noether Theorem and Hasse Norm Theorem over such fields.

In this paper, we exploit the numerical solvers for a fractional differential equations (FDE) optimization problem with constraints given by fractional elliptic state equations. The discretized linear system can be rewritten as a two-by-two linear system. By making use of the strategy of modified Hermitian and skew-Hermitian splitting (MHSS) iteration method proposed by Bai, we propose an MHSS-like iteration method. Comparing with the MHSS iteration method, the MHSS-like method only requires one to solve the two-by-two linear systems by a sparse Cholesky factorization and a fast Fourier transform. Hence, the MHSS-like method has less workload than the MHSS method. The convergence properties and the quasi-optimal parameters are analyzed in detail. Numerical examples are used to testify the efficiency of the new method.

In this paper, we study the Fu–Yau equation on compact Hermitian manifolds and prove the existence of solutions of equation on astheno-Kahler manifolds. We also prove the uniqueness of solutions of Fu–Yau equation when the slope parameter α is negative.

We consider a differential expression L V ∇ = ∇ † ∇ + V , where ∇ is a metric covariant derivative on a Hermitian bundle E over a geodesically complete Riemannian manifold ( M , g ) with metric g , and V is a linear self-adjoint bundle map on E . In the language of Everitt and Giertz, the differential expression L V ∇ is said to be separated in L p ( E ) if for all u ∈ L p ( E ) such that L V ∇ u ∈ L p ( E ) , we have V u ∈ L p ( E ) . We give sufficient conditions for L V ∇ to be separated in L 2 ( E ) . We then study the problem of separation of L V ∇ in the more general L p -spaces, and give sufficient conditions for L V ∇ to be separated in L p ( E ) , when 1 p ∞ .

We describe the general form of a Hermitian operator on the injective tensor product of a uniform algebra and a complex Banach space. As an application, we characterize a surjective unital isometry between the injective tensor product of a uniform algebra and a unital factor C ⁎ -algebra. In particular, we give a simple proof that a unital factor C ⁎ -algebra has the Banach–Stone property.

We compute the low lying eigenvalues of the Hermitian Dirac operator in lattice QCD with N f = 2 + 1 + 1 twisted mass fermions . We discuss whether these eigenvalues are in the ϵ -regime or the p -regime of Wilson chiral perturbation theory ( χ PT) for twisted mass fermions. Reaching the deep ϵ -regime is practically unfeasible with presently typical simulation parameters, but still the few lowest eigenvalues of the employed ensemble evince some characteristic ϵ -regime features. With this conclusion in mind, we develop a fitting strategy to extract two low energy constants from analytical ϵ -regime predictions at a fixed index. Thus, we obtain results for the chiral condensate and the low energy constant W 8 . We also discuss how to improve both the theoretical calculation and the lattice computation.