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We propose a natural strategy to deal with compatible and incompatible binary questions, and with their time evolution. The strategy is based on the simplest, non-commutative, Hilbert space $\mathcal{H}=\mathbb{C}^2$, and on the (commuting or not) operators on it. As in ordinary Quantum Mechanics, the dynamics is driven by a suitable operator, the Hamiltonian of the system. We discuss a rather general situation, and analyse the resulting dynamics if the Hamiltonian is a simple Hermitian matrix., In press in Physica A

We calculate the one loop effective potential for the class ( - ( i ϕ ) α ) of PT -symmetric and non-Hermitian field theories in 0 + 1 space-time dimensions. To test the method, we showed that for the massless Hermitian ϕ 4 theory, the method gives the exact power law behavior known from the literature. We show that this order of calculations goes beyond the truncation of the Schwinger-Dyson equations at the two-point green functions applied to the PT -symmetric ( - ϕ 4 ) theory in the literature. We found that the effective potential calculation represents good approximations of the vacuum energies of the class ( - ( i ϕ ) α ) compared to the numerical results. For the vacuum condensate, the method gives also accurate results for the absolute values but gives both positive as well as negative imaginary condensates for even α which again agrees with the prediction of the Schwinger-Dyson equations. Unlike other methods, the effective potential can be directly extended to higher dimensions as it offers a way to implement the PT -symmetric boundary conditions as well as there exist well known methods to regularize the theory at higher dimensions.

We present a polynomial Hybrid Monte Carlo (PHMC) algorithm as an exact simulation algorithm with dynamical Kogut-Susskind fermions. The algorithm uses a Hermitian polynomial approximation for the fractional power of the KS fermion matrix. The systematic error from the polynomial approximation is removed by the Kennedy-Kuti noisy Metropolis test so that the algorithm becomes exact at a finite molecular dynamics step size. We performed numerical tests with $N_f$$=$2 case on several lattice sizes. We found that the PHMC algorithm works on a moderately large lattice of $16^4$ at $\beta$$=$5.7, $m$$=$0.02 ($m_{\mathrm{PS}}/m_{\mathrm{V}}$$\sim$0.69) with a reasonable computational time., Comment: 3 pages, 2 figures, Lattice2002(algor)

In this note we discuss interlacing inequalities relating the eigenvalues of a partitioned Hermitian matrix and the eigenvalues of its blocks.We apply such inequalities to estimate the eigenvalues of the adjacency matrix and the Laplacian of a graph. In particular, we prove that for every r⩾3,c>0, there exists β=β(c,r) such that for every Kr-free graph G=G(n,m) with m>cn2, the smallest eigenvalue μn of G satisfies μn⩽-βn.Similarly, for every r⩾3,cγnfor sufficiently large n.

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson–Kahler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kahler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kahler reduction. An interpretation in terms of constrained mechanical systems is included.

We show that non-Hermitian and nearest-neighbor-interacting perturbations to the Fritzsch textures of lepton and quark mass matrices can make both of them fit current experimental data very well. In particular, we obtain \theta_{23} \simeq 45^\circ for the atmospheric neutrino mixing angle and predict \theta_{13} \simeq 3^\circ to 6^\circ for the smallest neutrino mixing angle when the perturbations in the lepton sector are at the 20% level. The same level of perturbations is required in the quark sector, where the Jarlskog invariant of CP violation is about 3.7 \times 10^{-5}. In comparison, the strength of leptonic CP violation is possible to reach about 1.5 \times 10^{-2} in neutrino oscillations., Comment: 14 pages, 4 figures. More discussions added. Accepted for publication in PLB

In this paper, using a minimum principle for Schur complements of positive semidefinite Hermitian matrices and some estimates of the eigenvalues and the singular values, we obtain some inequalities for the eigenvalues of the Schur complement of the matrix product BAB* in terms of the eigenvalues of the Schur complements of BB* and A.

In this paper, we first present a local Hermitian and skew-Hermitian splitting (LHSS) iteration method for solving a class of generalized saddle point problems. The new method converges to the solution under suitable restrictions on the preconditioning matrix. Then we give a modified LHSS (MLHSS) iteration method, and further extend it to the generalized saddle point problems, obtaining the so-called generalized MLHSS (GMLHSS) iteration method. Numerical experiments for a model Navier–Stokes problem are given, and the results show that the new methods outperform the classical Uzawa method and the inexact parameterized Uzawa method.

We study the eigenvalue bounds for the nonsingular saddle point matrices of Hermitian and indefinite (1,1) and (2,2) blocks without imposing the restrictions that the (1,1) blocks are positive definite on the kernels of the (2,1) blocks.

In a recent paper, Overton and Van Dooren have considered structured indefinite perturbations to a given Hermitian matrix. We extend their results to skew-Hermitian, Hamiltonian and skew-Hamiltonian matrices. As an application, we give a formula for computation of the smallest perturbation with a special structure, which makes a given Hamiltonian matrix own a purely imaginary eigenvalue.

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow inner tolerance $\xi_k\geq 1$ at outer iteration $k$ and can be considerably weaker than the condition $\xi_k\leq\xi, Comment: 27 pages, 4 figures

We compute the number of solutions of the equation Tr F t / F s (f(x)+v. x)=b in F t 2N , where f denote a quadratic hermitian form on F t 2N , v∈ F t 2N and b∈ F s , and we deduce the number of hermitian matrices of order N and rank ρ . This number is well-known since the paper of Carlitz and Hodges (Duke Math. J. 22 (1995) 393), but with a more restrictive definition of hermitian matrices and with a rather different proof. Next, we introduce a linear code Γ ( N , t , s ) on F s constructed with the same method as Reed–Muller one, and compute its weight distribution. Γ ( N , t , s ) is a generalization of the two codes Γ and C studied in Mercier (J. Pure Appl. Algebra 173 (3) (2003) 273) and the method for obtaining its weight distribution is new and more straightforward. Tools are exponential sums and linear algebra on F t .

Heegard, Little and Saints worked out a Gröbner basis algorithm for Hermitian codes. Here we extend such a result for codes on norm-trace curves.

In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation m -sum and we obtain an expression for it that involves the Cholesky factorization of the corresponding Hermitian positive definite matrices associated with the Hessenberg components. This work extends a method to obtain the Hessenberg matrix of the sum of measures from the Hessenberg matrices of the individual measures, introduced recently by the authors for subnormal matrices, to matrices which are not necessarily subnormal. Moreover, we give some examples and we obtain the explicit formula for the m -sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the m -sum of two not subnormal Hessenberg matrices.

Let A =( a ij ) i , j =1 n be a Hermitian matrix of size n ⩾2, and set ρ(A)= 1 n 2 ∑ i,j=1 n a ij , disc (A)= max X,Y⊂[n],X≠∅,Y≠∅ 1 |X||Y| ∑ i∈X ∑ j∈Y (a ij −ρ(A)) . We show that the second singular value σ 2 ( A ) of A satisfies σ 2 (A)⩽C 1 disc (A) log n for some absolute constant C 1 , and this is best possible up to a multiplicative constant. Moreover, we construct infinitely many dense regular graphs G such that σ 2 (A(G))⩾C 2 disc (A(G)) log |G|, where C 2 >0 is an absolute constant and A ( G ) is the adjacency matrix of G . In particular, these graphs disprove two conjectures of Fan Chung.

Let H be a Hermitian matrix, and [email protected]?=D^*HD be its perturbed matrix. In this paper, the multiplicative perturbations for both spectral decompositions and singular value decompositions are studied and some new perturbation bounds for these decompositions are presented. Our results improve some existing bounds.

On a para-quaternionic Kahler manifold ( M ˜ 4 n , Q , g ˜ ) , which is first of all a pseudo-Riemannian manifold, a natural definition of (almost) Kahler and (almost) para-Kahler submanifold ( M 2 m , J , g ) can be given where J = J 1 | T M is a (para-)complex structure on M which is the restriction of a section J 1 of the para-quaternionic bundle Q. In this paper, we extend to such a submanifold M most of the results proved by Alekseevsky and Marchiafava, 2001, where Hermitian and Kahler submanifolds of a quaternionic Kahler manifold have been studied. Conditions for the integrability of an almost (para-)Hermitian structure on M are given. Assuming that the scalar curvature of M ˜ is non-zero, we show that any almost (para-)Kahler submanifold is (para-)Kahler respectively and moreover that M is (para-)Kahler iff it is totally (para-)complex. Considering totally (para-)complex submanifolds of maximal dimension 2n, we identify the second fundamental form h of M with a tensor C = J 2 ∘ h ∈ T M ⊗ S 2 T ⁎ M where J 2 ∈ Q is a compatible para-complex structure anticommuting with J 1 . This tensor, at any point x ∈ M , belongs to the first prolongation S J ( 1 ) of the space S J ⊂ End T x M of symmetric endomorphisms anticommuting with J . When M ˜ 4 n is a symmetric manifold the condition for a (para-)Kahler submanifold M 2 n to be locally symmetric is given. In the case when M ˜ is a para-quaternionic space form, it is shown, by using Gauss and Ricci equations, that a (para-)Kahler submanifold M 2 n is curvature invariant. Moreover it is a locally symmetric Hermitian submanifold iff the u ( n ) -valued 2-form [ C , C ] is parallel. Finally a characterization of parallel Kahler and para-Kahler submanifolds of maximal dimension is given.

Gray & Hervella gave a classification of almost Hermitian structures (g,I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g,I,J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we obtain a number of relations that give hyperK��her or locally conformal hyperK��hler structures, thus generalising a result of Hitchin. We also study the types of almost quaternion-Hermitian geometries that arise and tabulate the results., 22 pages, 5 tables

In this paper, on the basis of matrix splitting, two preconditioners are proposed and analyzed, for nonsymmetric saddle point problems. The spectral property of the preconditioned matrix is studied in detail. When the iteration parameter becomes small enough, the eigenvalues of the preconditioned matrices will gather into two clusters—one is near (0,0) and the other is near (2,0)—for the PPSS preconditioner no matter whether A is Hermitian or non-Hermitian and for the PHSS preconditioner when A is a Hermitian or real normal matrix. Numerical experiments are given, to illustrate the performances of the two preconditioners.

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.

When an edge is removed from an undirected graph, there is a limited change that can occur in the multiplicity of an eigenvalue of a Hermitian matrix with that graph. Primarily for trees, we identify the changes that can occur and characterize the circumstances under which they occur. This extends known results for the removal of vertices. A catalog of examples is given to illustrate the possibilities that can occur and to contrast the case of trees with that of general graphs.

The global Arnoldi method can be used to compute exterior eigenpairs of a large non-Hermitian matrix A, but it does not work well for interior eigenvalue problems. Based on the global Arnoldi process that generates an F-orthonormal basis of a matrix Krylov subspace, we propose a global harmonic Arnoldi method for computing certain harmonic F-Ritz pairs that are used to approximate some interior eigenpairs. We propose computing the F-Rayleigh quotients of the large non-Hermitian matrix with respect to harmonic F-Ritz vectors and taking them as new approximate eigenvalues. They are better and more reliable than the harmonic F-Ritz values. The global harmonic Arnoldi method inherits convergence properties of the harmonic Arnoldi method applied to a larger matrix whose distinct eigenvalues are the same as those of the original given matrix. Some properties of the harmonic F-Ritz vectors are presented. As an application, assuming that A is diagonalizable, we show that the global harmonic Arnoldi method is able to solve multiple eigenvalue problems both in theory and in practice. To be practical, we develop an implicitly restarted global harmonic Arnoldi algorithm with certain harmonic F-shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.

This article presented to Combinatorics 2006 is a survey of finite projective planes and the processes used to construct them. All non-translation planes are described, fundamental processes in translation planes are defined and some of these are used to connect semi-field flocks with symplectic spreads. Hermitian ovoids are connected to extensions of derivable nets, and three types of ‘lifting’ methods are discussed. Furthermore, hyperbolic fibrations and ‘regulus-inducing’ central collineation groups are connected to flocks of quadratic cones. Finally, hyper-reguli and multiple hyper-regulus replacement are considered.

For the singular, non-Hermitian, and positive semidefinite linear systems, we propose an alternating-direction iterative method with two parameters based on the Hermitian and skew-Hermitian splitting. The semi-convergence analysis and the quasi-optimal parameters of the proposed method are discussed. Moreover, the corresponding preconditioner based on the splitting is given to improve the semi-convergence rate of the GMRES method. Numerical examples are given to illustrate the theoretical results and the efficiency of the generalized HSS method either as a solver or a preconditioner for GMRES.

In this paper, we consider the Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems with nonzero (2, 2) blocks. The spectral property of the preconditioned matrix is studied in detail. Under certain conditions, all eigenvalues of the preconditioned matrix with the original system being non-Hermitian will form two tight clusters, one is near (0, 0) and the other is near (2, 0) as the iteration parameter approaches to zero from above, so do all eigenvalues of the preconditioned matrix with the original system being Hermitian. Numerical experiments are given to demonstrate the results.

We introduce the notion of Ricci-corrected differentiation in parabolic geometry, which is a modification of covariant differentiation with better transformation properties. This enables us to simplify the explicit formulae for standard invariant operators given in work of Cap, Slovak and Soucek, and at the same time extend these formulae from the context of AHS structures (which include conformal and projective structures) to the more general class of all parabolic structures (including CR structures)., Comment: Substantially revised, shortened and simplified, with new treatment of Weyl structures; 24 pages

S n = 1 n T n 1 / 2 X n X n ∗ T n 1 / 2 , where X n = ( x i j ) is a p × n matrix consisting of independent complex entries with mean zero and variance one, T n is a p × p nonrandom positive definite Hermitian matrix with spectral norm uniformly bounded in p . In this paper, if sup n sup i , j E ∣ x i j 8 ∣ ∞ and y n = p / n 1 uniformly as n → ∞ , we obtain that the rate of the expected empirical spectral distribution of S n converging to its limit spectral distribution is O ( n − 1 / 2 ) . Moreover, under the same assumption, we prove that for any η > 0 , the rates of the convergence of the empirical spectral distribution of S n in probability and the almost sure convergence are O ( n − 2 / 5 ) and O ( n − 2 / 5 + η ) respectively.