论埃尔特《30首随想曲》(Op.107)长笛演奏技巧

卡格·埃尔特创作的《30首随想曲》(Op.107)曲目是长笛专业教材体系里重要的作品之一,在高等音乐学院长笛专业教学中,是学生们必学的曲目,对提高长笛的演奏技巧以及提高音乐情感的处理有很大帮助。目前国内发表的论文中,没有看到关于埃尔特《30首随想曲》(Op.107)的研究论文,所以本文通过对其中的几首随想曲,进行谱面、演奏技巧和情感表达等方面的分析与理解,并结合自身的学习心得,提出一些见解,希望对想练习这首作品的学生们提供一些帮助。

Wavepacket Dynamics in a Non-Hermitian PT Symmetric Tight-Binding Chain

We examine the time development of a wavepacket in a Non-Hermitian PT symmetric tight-bindingchain based on the Bethe ansatz solutions.The complete orthogonal sets in both unbroken and broken PT symmetryregions are established with respect to an inner product in terms of CPT conjugation.The dynamical development ofa stationary wavepacket is exhibited,showing the impacts from additional imaginary potentials within unbroken andbroken symmetric regimes.Our result indicates that there are distinct dynamic characteristics for both two regions.Theunbroken symmetric imaginary end potentials can be regarded as a special kind of boundary condition in the sense ofHermitian quantum mechanics.

Extension and Application of the Wielandt-Hoffman Theorem

The Wielandt-Hoffman theorem of the real symmetric matrix is extended into a plural matrix. On the basis of it, a similar theory about the trace of a matrix for the arithmetic mean, geometric mean inequality, Holder inequality and Minkowski inequality is proved.

Hermite interpolation of scattered data based on the assistant surface

An assistant surface was constructed on the base of boundary that being auto-matically extracted from the scattered data.The parameters of every data point corre-sponding to the assistant surface and their applied fields were calculated respectively.Inevery applied region,a surface patch was constructed by a special Hermite interpolation.The final surface can be obtained by a piecewise bicubic Hermite interpolation in the ag-gregate of applied regions of metrical data.This method avoids the triangulation problem.Numerical results indicate that it is efficient and accurate.

Entanglement Involved in Pair Coherent State Studied via Wigner Function Formalism

We calculate Wigner function, tomogram of the pair coherent state by using its Sehmidt decomposition in the coherent state representation. It turns out that the Wigner function can be seen as the quantum entanglement （QE） between two two-variable Hermite polynomials （TVHP） and the tomogram is further simplified as QE of two single-variable Hermite polynomials. The Husimi function of pair coherent state is also calculated.

A Verifiable Multi-Secret Sharing Scheme Based on Hermite Interpolation

A threshold scheme, which is introduced by Shamir in 1979, is very famous as a secret sharing scheme. We can consider that this scheme is based on Lagrange＇s interpolation formula. A secret sharing scheme has one key. On the other hand, a multi-secret sharing scheme has more than one key, that is, a multi-secret sharing scheme has p （〉_ 2） keys. Dealer distribute shares of keys among n participants. Gathering t （〈 n） participants, keys can be reconstructed. Yang et al. （2004） gave a scheme of a （t, n） multi-secret sharing based on Lagrange＇s interpolation. Zhao et al. （2007） gave a scheme of a （t, n） verifiable multi-secret sharing based on Lagrange＇s interpolation. Recently, Adachi and Okazaki give a scheme of a （t, n） multi-secret sharing based on Hermite interpolation, in the case ofp 〈 t. In this paper, we give a scheme ofa （t, n） verifiable multi-secret sharing based on Hermite interpolation.

On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems

For the Hermitian inexact Rayleigh quotient iteration （RQI）, we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for the residual, whose norm is ξk, of the linear system obtained by the Lanczos method at outer iteration k ＋ 1. Based on them, we make a refined analysis and establish new local convergence results. It is proved that （i） the inexact RQI with Lanezos converges quadratically provided that ξk ≤ξ with a constant ξ≥） 1 and （ii） the method converges linearly provided that ξk is bounded by some multiple of 1/‖τk‖ with ‖τk‖ the residual norm of the approximate eigenpair at outer iteration k. The results are fundamentally different from the existing ones that always require ξk 〈 1, and they have implications on effective implementations of the method. Based on the new theory, we can design practical criteria to control ξk to achieve quadratic convergence and implement the method more effectively than ever before. Numerical experiments confirm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.

A system of matrix equations and its applications

We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu's recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854-1891).

The Teodorescu Operator in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation in- variant differential operator called the Dirac operator. More recently, Hermitian Clifford analysis has emerged as a new branch, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions, called Hermitian monogenic functions, to two Hermitian Dirac operators and which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Teodorescu operator is the right inverse of the Dirac operator __0. In this paper, Teodorescu operators for the Hermitian Dirac operators c9~_ and 0_~, are constructed. Moreover, the structure of the Euclidean and Hermitian Teodor- escu operators is revealed by analyzing the more subtle behaviour of their components. Finally, the obtained inversion relations are still refined for the differential operators is- suing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. Their relationship with several complex variables theory is discussed.

CPT-Frames for Non-Hermitian Hamiltonians

Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on Cd, it is proved that H has unbroken PT- symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed, These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.

Partial energies monotonicity and holomorphicity of Hermitian pluriharmonic maps

In this paper, we introduce the notion of Hermitian pluriharmonic maps from Hermitian manifold into Kiihler manifold. Assuming the domain manifolds possess some special exhaustion functions and the vecotor field V = JMδJM satisfies some decay conditions, we use stress-energy tensors to establish some monotonicity formulas of partial energies of Hermitian pluriharmonic maps. These monotonicity inequalities enable us to derive some holomorphicity for these Hermitian pluriharmonic maps.

THE RECONSTRUCTION OF AN HERMITIAN TOEPLITZ MATRIX WITH PRESCRIBED EIGENPAIRS

This paper concerns the reconstruction of an hermitian Toeplitz matrix with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, the authors first consider the eigenstructure of hermitian Toeplitz matrices and then discuss a related reconstruction problem. The authors show that the dimension of the subspace of hermitian Toeplitz matrices with two given eigenvectors is at least two and independent of the size of the matrix, and the solution of the reconstruction problem of an hermitian Toeplitz matrix with two given eigenpairs is unique.

Anti-(conjugate) linearity

This is an introduction to antilinear operators. In following Wigner the terminus antilinear is used as it is standard in Physics.Mathematicians prefer to say conjugate linear. By restricting to finite-dimensional complex-linear spaces, the exposition becomes elementary in the functional analytic sense. Nevertheless it shows the amazing differences to the linear case. Basics of antilinearity is explained in sects. 2, 3, 4, 7 and in sect. 1.2: Spectrum, canonical Hermitian form, antilinear rank one and two operators,the Hermitian adjoint, classification of antilinear normal operators,(skew) conjugations, involutions, and acq-lines, the antilinear counterparts of 1-parameter operator groups. Applications include the representation of the Lagrangian Grassmannian by conjugations, its covering by acq-lines. As well as results on equivalence relations. After remembering elementary Tomita-Takesaki theory, antilinear maps, associated to a vector of a two-partite quantum system, are defined. By allowing to write modular objects as twisted products of pairs of them, they open some new ways to express EPR and teleportation tasks. The appendix presents a look onto the rich structure of antilinear operator spaces.

Investigation of P T-symmetric Hamiltonian Systems from an Alternative Point of View

Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones.Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart,this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged.In order to give the positive definite inner product for the PT-symmetric systems,a new operator V,instead of C,can be introduced.The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics,however,it can be constructed,as an advantage,directly in terms of Hamiltonians.The spectra of the two non-Hermitian PT-symmetric systems are obtained,which coincide with that given in literature,and in particular,the Hilbert spaces associated with positive definite inner products are worked out.

Eventual Positivity of Hermitian Polynomials and Integral Operators

Quillen proved that if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.

Time reversal of a discrete system coupled to a continuum based on non-Hermitian flip

Time reversal in quantum or classical systems described by an Hermitian Hamiltonian is a physically allowed process, which requires in principle inverting the sign of the Hamiltonian. Here we consider the problem of time reversal of a subsystem of discrete states coupled to an external environment characterized by a continuum of states, into which they generally decay. It is shown that, by flipping the discrete-continuum coupling from an Hermitian to a non-Hermitian interaction, thus resulting in a non unitary dynamics, time reversal of the subsystem of discrete states can be achieved, while the continuum of states is not reversed. Exact time reversal requires frequency degeneracy of the discrete states,or large frequency mismatch among the discrete states as compared to the strength of indirect coupling mediated by the continuum. Interestingly, periodic and frequent switch of the discrete-continuum coupling results in a frozen dynamics of the subsystem of discrete states.

Adapted Metrics and Webster Curvature in Finslerian 2-Dimensional Geometry

The Webster scalar curvature is computed for the sphere bundle T_1S of a Finsler surface(S, F) subject to the Chern-Hamilton notion of adapted metrics. As an application,it is derived that in this setting(T_1S, g Sasaki) is a Sasakian manifold homothetic with a generalized Berger sphere, and that a natural Cartan structure is arising from the horizontal 1-forms and the author associates a non-Einstein pseudo-Hermitian structure. Also, one studies when the Sasaki type metric of T_1S is generally adapted to the natural co-frame provided by the Finsler structure.

Comparison of Different Approaches of Finding the Positive Definite Metric in Pseudo-Hermitian Theories

To develop a unitary quantum theory with probabilistic description for pseudo-Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator. There are different approaches to find such metric operators. We compare the different approaches of calculating positive definite metric operators in pseudo-Hermitian theories with the help of several explicit examples in non-relativistic as well as in relativistic situations. Exceptional points and spontaneous symmetry breaking are also discussed in these models.

Convergence of Yang-Mills-Higgs flow for twist Higgs pairs on Riemann surfaces

We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle(E,H)over Riemann surface X.It is already known the gradient flow with initial data(A0,φ0)converges to a critical point(A∞,φ∞).Using a modified Chern-Weil type inequality,we prove that the limiting twist Higgs bundle(E,d′′A∞,φ∞)coincides with the graded twist Higgs bundle defined by the HarderNarasimhan-Seshadri filtration of the initial twist Higgs bundle(E,d′′A0,φ0),generalizing Wilkin’s results for untwist Higgs bundle.

The Fitzpatrick-Neville-Type Algorithm for Multivariate Vector-Valued Osculatory Rational Interpolation

In this paper,the authors first apply the Fitzpatrick algorithm to multivariate vectorvalued osculatory rational interpolation.Then based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis,the authors present a Fitzpatrick-Neville-type algorithm for multivariate vector-valued osculatory rational interpolation.It may be used to compute the values of multivariate vector-valued osculatory rational interpolants at some points directly without computing the interpolation function explicitly.