Decompositions of some types of quaternionic matrices

The structures of some important types of matrices over the quaternion skew field are discussed, and corresponding decompositions are obtained by virtue of real orthogonal matrices and real anti-symmetric matrices. In particular, the normal forms are found for several classes of quaternionic matrices.     

Dynamical Invariant and Exact Mechanical Analyses for the Caldirola–Kanai Model of Dissipative Three Coupled Oscillators

We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators.     

On the nature of the and from their behavior in chiral dynamics

We study the behavior with the number of colors (N_c) of the Λ(1405) and Λ(1670) resonances obtained dynamically within the chiral unitary approach. The leading order meson–baryon interaction, used as the kernel of the unitarization procedure, manifests a nontrivial N_c dependence of the flavor SU(3) representation for baryons. As a consequence, the SU(3) singlet (or ) component of the Λ(1405) states remains bound in the large N_c limit, while the other components dissolve into the continuum. Introducing explicit SU(3) breaking, we obtain the N_c dependence of the excitation energy, masses and widths of the physical Λ(1405) and Λ(1670) resonances. The N_c behavior of the decay widths is found to be different from the general counting rule for a qqq state, indicating the dynamical origin of these resonances.     

利用单光子的量子对话

提出了一个利用一束单光子对话的方案.在方案中,利用两个不同的幺正操作对光子态进行编码,并且从一束光子中选择较大的子集进行窃听检查,该方案能够有效地抵御截取再发送袭击.此外,由于利用单光子没有利用EPR对,因此该方案是很实际的.该方案是绝对安全的.     

Gaussian unitary ensembles with two jump discontinuities,PDEs, and the coupled Painlevé II and IV systems

We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of Min and Chen [Math. Methods Appl Sci. 2019;42:301‐321] where a second‐order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in Wu and Xu [arXiv: 2002.11240v2] by a study of the Riemann‐Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as , the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second‐order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system.     

Convexity and Haar null sets

It is shown that for every closed, convex and nowhere dense subset of a superreflexive Banach space there exists a Radon probability measure on so that for all . In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a closed convex nowhere dense set which is not Haar null.

     

On numerical methods for discrete least-squares approximation by trigonometric polynomials

Fast, efficient and reliable algorithms for discrete least-squares approximation of a real-valued function given at arbitrary distinct nodes in by trigonometric polynomials are presented. The algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only arithmetic operations as compared to operations needed for algorithms that ignore the structure of the problem. An algorithm which solves this problem with real-valued data and real-valued solution using only real arithmetic is given. Numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general QR decomposition methods for the least-squares problem.

     

An orthogonal family of polynomials on the generalized unit disk and ladder representations of

Inner product structures are given for realizations of the positive spin ladder representations over the generalized unit disk . This is accomplished by combining previous results of the author with the construction of a family of holomorphic polynomials on . These polynomials, which play a crucial role in the present work, are shown to be orthogonal with respect to Lebesgue measure, and their norms are computed. The orthogonal family is then used to invert a certain integral transform, giving the desired inner product structures.