Source Enumeration Method using Eigenvalue Gap Ratio and Performance Comparison in Rayleigh Fading

In electronic warfare, source enumeration and direction-of-arrival estimation are important. The source enumeration method based on eigenvalues of covariance matrix from received is one of the most used methods. However, there are some drawbacks such as accuracy less than 100 % at high SNR, poor performance at low SNR and reduction of maximum number of estimating sources. We suggested new method based on eigenvalues gaps, which is named AREG(Accumulated Ratio of Eigenvalues Gaps). Meanwhile, FGML(Fast Gridless Maximum Likelihood) which reconstructs the covariance matrix was suggested by Wu et al., and it improves performance of the existing source enumeration methods without modification of algorithms. In this paper, first, we combine AREG with FGML to improve the performance. Second, we compare the performance of source enumeration and direction-of-arrival estimation methods in Rayleigh fading. Third, we suggest new method named REG(Ratio of Eigenvalues Gaps) to reduce performance degradation in Rayleigh Fading environment of AREG.

The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

In this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has $n-1$ n − 1 zero in $( 0,\pi ) $ ( 0 , π ) for $n\in \mathbb{N}$ n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

On the Neumann eigenvalues for second-order Sturm–Liouville difference equations

The paper is concerned with the Neumann eigenvalues for second-order Sturm–Liouville difference equations. By analyzing the new discriminant function, we show the interlacing properties between the periodic, antiperiodic, and Neumann eigenvalues. Moreover, when the potential sequence is symmetric and symmetric monotonic, we show the order relation between the first Dirichlet eigenvalue and the second Neumann eigenvalue, and prove that the minimum of the first Neumann eigenvalue gap is attained at the constant potential sequence.

Canonical Quantum Observables for Molecular Systems Approximated by Ab Initio Molecular Dynamics

It is known that ab initio molecular dynamics based on the electron ground-state eigenvalue can be used to approximate quantum observables in the canonical ensemble when the temperature is low compared to the first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature. The proof uses the semiclassical Weyl law to show that canonical quantum observables of nuclei–electron systems, based on matrix-valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron–nuclei mass ratio. The result covers observables that depend on time correlations. A combination of the Hilbert–Schmidt inner product for quantum operators and Weyl’s law shows that the error estimate holds for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei position and the observables are in the diagonal form with respect to the electron eigenstates.

Adiabatic optimization without local minima

Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wavefunction consists of two ``lobes'' separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.