Ring chains with vertex coupling of a preferred orientation

We consider a family of Schrödinger operators supported by a periodic chain of loops connected either tightly or loosely through connecting links of the length [Formula: see text] with the vertex coupling which is non-invariant with respect to the time reversal. The spectral behavior of the model illustrates that the high-energy behavior of such vertices is determined by the vertex parity. The positive spectrum of the tightly connected chain covers the entire halfline while the one of the loose chain is dominated by gaps. In addition, there is a negative spectrum consisting of an infinitely degenerate eigenvalue in the former case, and of one or two absolutely continuous bands in the latter. Furthermore, we discuss the limit [Formula: see text] and show that while the spectrum converges as a set to that of the tight chain, as it should in view of a result by Berkolaiko, Latushkin, and Sukhtaiev, this limit is rather non-uniform.

Temporal stability analysis of jets of lobed geometry

A two-dimensional temporal incompressible stability analysis is performed for lobed jets. The jet base flow is assumed to be parallel and of a vortex-sheet type. The eigenfunctions of this simplified stability problem are expanded using the eigenfunctions of a round jet. The original problem is then formulated as an innovative matrix eigenvalue problem, which can be solved in a very robust and efficient manner. The results show that the lobed geometry changes both the convection velocity and temporal growth rate of the instability waves. However, different modes are affected differently. In particular, mode 0 is not sensitive to the geometry changes, whereas modes of higher orders can be changed significantly. The changes become more pronounced as the number of lobes $N$ and the penetration ratio $\unicode[STIX]{x1D716}$ increase. Moreover, the lobed geometry can cause a previously degenerate eigenvalue ($\unicode[STIX]{x1D706}_{n}=\unicode[STIX]{x1D706}_{-n}$) to become non-degenerate ($\unicode[STIX]{x1D706}_{n}\neq \unicode[STIX]{x1D706}_{-n}$) and lead to opposite changes to the stability characteristics of the corresponding symmetric ($n$) and antisymmetric ($-n$) modes. It is also shown that each eigenmode changes its shape in response to the lobes of the vortex sheet, and the degeneracy of an eigenvalue occurs when the vortex sheet has more symmetric planes than the corresponding mode shape (including both symmetric and antisymmetric planes). The new approach developed in this paper can be used to study the stability characteristics of jets of other arbitrary geometries in a robust and efficient manner.

Interpolating Binary and Multivalued Logical Quantum Gates

A method for synthesizing quantum gates is presented based on interpolation methods applied to operators in Hilbert space. Starting from the diagonal forms of specific generating seed operators with non-degenerate eigenvalue spectrum one obtains for arity-one a complete family of logical operators corresponding to all the one-argument logical connectives. Scaling-up to n-arity gates is obtained by using the Kronecker product and unitary transformations. The quantum version of the Fourier transform of Boolean functions is presented and a Reed-Muller decomposition for quantum logical gates is derived. The common control gates can be easily obtained by considering the logical correspondence between the control logic operator and the binary logic operator. A new polynomial and exponential formulation of the Toffoli gate is presented. The method has parallels to quantum gate-T optimization methods using powers of multilinear operator polynomials. The method is then applied naturally to alphabets greater than two for multi-valued logical gates used for quantum Fourier transform, min-max decision circuits and multivalued adders.

On Sharp Hölder Estimates of the Cauchy-Riemann Equation on Pseudoconvex Domains inCnwith One Degenerate Eigenvalue

LetΩbe a smoothly bounded pseudoconvex domain inCnwith one degenerate eigenvalue and assume that there is a smooth holomorphic curveVwhose order of contact withbΩatz0∈bΩis larger than or equal toη. We show that the maximal gain in Hölder regularity for solutions of the∂¯-equation is at most1/η.

Variational Eigenvalues of Degenerate Eigenvalue Problems for the Weighted p-Laplacian

We prove the existence of nondecreasing sequences of positive eigenvalues of the homogeneous degenerate quasilinear eigenvalue problem − div(a(x)|▽u|

A Modified Algorithm for Generalized Discriminant Analysis

Generalized discriminant analysis (GDA) is an extension of the classical linear discriminant analysis (LDA) from linear domain to a nonlinear domain via the kernel trick. However, in the previous algorithm of GDA, the solutions may suffer from the degenerate eigenvalue problem (i.e., several eigenvectors with the same eigenvalue), which makes them not optimal in terms of the discriminant ability. In this letter, we propose a modified algorithm for GDA (MGDA) to solve this problem. The MGDA method aims to remove the degeneracy of GDA and find the optimal discriminant solutions, which maximize the between-class scatter in the subspace spanned by the degenerate eigenvectors of GDA. Theoretical analysis and experimental results on the ORL face database show that the MGDA method achieves better performance than the GDA method.