Positive definite solutions of some matrix equations

In this paper we investigate some existence questions of positive semi-definite solutions for certain classes of matrix equations known as the generalized Lyapunov equations. We present sufficient and necessary conditions for certain equations and only sufficient for others.     

Recovering differential pencils on compact graphs

We study boundary value problems on compact graphs without circles (i.e. on trees) for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so-called Weyl vector which is a generalization of the Weyl function (m-function) for the classical Sturm-Liouville operator. For this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings.     

Design of 24 GHz Radar with Subspace‐Based Digital Beam Forming for ACC Stop‐and‐Go System

For an adaptive cruise control (ACC) stop‐and‐go system in automotive applications, three radar sensors are needed because two 24 GHz short range radars are used for object detection in an adjacent lane, and one 77 GHz long‐range radar is used for object detection in the center lane. In this letter, we propose a single sensor‐based 24 GHz radar with a detection capability of up to 150 m and ±30° for an ACC stop‐and‐go system. The developed radar is highly integrated with a high gain patch antenna, four channel receivers with GaAs RF ICs, and back‐end processing board with subspace based digital beam forming algorithm.     

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

In this work we derive new comparison results for (finite) eigenvalues of two self‐adjoint linear Hamiltonian eigenvalue problems. The coefficient matrices depend on the spectral parameter nonlinearly and the spectral parameter is present also in the boundary conditions. We do not impose any controllability or strict normality assumptions. Our method is based on a generalization of the Sturmian comparison theorem for such systems. The results are new even for the Dirichlet boundary conditions, for linear Hamiltonian systems depending linearly on the spectral parameter, and for Sturm–Liouville eigenvalue problems with nonlinear dependence on the spectral parameter.     

A symmetric generalization of Sturm–Liouville problems in q-difference spaces

Classical Sturm–Liouville problems of q-difference variables are extended for symmetric discrete functions such that the corresponding solutions preserve the orthogonality property. Some illustrative examples are given in this sense.     

Identities for minors of the Laplacian, resistance and distance matrices

It is shown that if L and D are the Laplacian and the distance matrix of a tree respectively, then any minor of the Laplacian equals the sum of the cofactors of the complementary submatrix of D, up to sign and a power of 2. An analogous, more general result is proved for the Laplacian and the resistance matrix of any graph. A similar identity is proved for graphs in which each block is a complete graph on r vertices, and for q-analogues of such matrices of a tree. Our main tool is an identity for the minors of a matrix and its inverse.     

高清电视电影后期制作问题研究

高清电视电影将凭借良好的解晰度、色彩还原以及丰富的艺术表现力而受到各类播出机构的青睐，影片在不同国家地区播出、发行及交流过程中会遇到要求影片提供不同格式的问题，在后期制作过程中如何具体解决上述问题是本文研究的主要方向。     

A Prüfer angle approach to semidefinite Sturm–Liouville problems with coupling boundary conditions

It is shown how to reduce the periodic/antiperiodic Sturm–Liouville problems to analysis of the Prüfer angle. This provides an alternative to the more usual approaches via operator theory or the Hill discriminant in the definite case, and leads to new results in the semidefinite case. An extension to more general coupling boundary conditions is also given.     

Chebyshev polynomials on symmetric matrices

In this paper we evaluate Chebyshev polynomials of the second kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type.