A new type of CGO solutions and its applications in corner scattering

We consider corner scattering for the operator ∇ · γ(x)∇ + k2ρ(x) in R2, with γ a positive definite symmetric matrix and ρ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of γ(x) − I or ρ(x) − 1, where I stands for the identity matrix. We assume that γ is a scalar function in a small neighborhood of the corner. We show that any admissible incident field will be scattered by such corners, which are allowed to be concave. Moreover, we provide a brief discussion on the existence of non-scattering waves when γ − I has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics (CGO) solutions.

Combinatorial properties of the enhanced principal rank characteristic sequence over finite fields

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix B ∈ 𝔽 n×n is defined as ℓ1ℓ2· · · ℓ n , where ℓ j ∈ {A, S, N} according to whether all, some but not all, or none of the principal minors of order j of B are nonzero. Building upon the second author’s recent classification of the epr-sequences of symmetric matrices over the field 𝔽 = 𝔽2, we initiate a study of the case 𝔽= 𝔽3. Moreover, epr-sequences over finite fields are shown to have connections to Ramsey theory and coding theory.

Relationship between adjacency and distance matrix of graph of diameter two

The relationship among every pair of vertices in a graph can be represented as a matrix, such as in adjacency matrix and distance matrix. Both adjacency and distance matrices have the same property. Adjacency and distance matrices are both symmetric matrix with diagonals entries equals to 0. In this paper, we discuss relationships between adjacency matrix and distance matrix of a graph of diameter two, which is <em>D=2(J-I)-A</em>. From this relationship, we also determine the value of the determinant matrix <em>A+D</em> and the upper bound of determinant of matrix <em>D</em>.

A Dynamically Adjusted Subspace Gradient Method and Its Application in Image Restoration

In this paper, a new subspace gradient method is proposed in which the search direction is determined by solving an approximate quadratic model in which a simple symmetric matrix is used to estimate the Hessian matrix in a three-dimensional subspace. The obtained algorithm has the ability to automatically adjust the search direction according to the feedback from experiments. Under some mild assumptions, we use the generalized line search with non-monotonicity to obtain remarkable results, which not only establishes the global convergence of the algorithm for general functions, but also R-linear convergence for uniformly convex functions is further proved. The numerical performance for both the traditional test functions and image restoration problems show that the proposed algorithm is efficient.

Can One Hear a Matrix? Recovering a Real Symmetric Matrix from Its Spectral Data

The spectrum of a real and symmetric $$N\times N$$ N × N matrix determines the matrix up to unitary equivalence. More spectral data is needed together with some sign indicators to remove the unitary ambiguities. In the first part of this work, we specify the spectral and sign information required for a unique reconstruction of general matrices. More specifically, the spectral information consists of the spectra of the N nested main minors of the original matrix of the sizes $$1,2,\ldots ,N$$ 1 , 2 , … , N . However, due to the complicated nature of the required sign data, improvements are needed in order to make the reconstruction procedure feasible. With this in mind, the second part is restricted to banded matrices where the amount of spectral data exceeds the number of the unknown matrix entries. It is shown that one can take advantage of this redundancy to guarantee unique reconstruction of generic matrices; in other words, this subset of matrices is open, dense and of full measure in the set of real, symmetric and banded matrices. It is shown that one can optimize the ratio between redundancy and genericity by using the freedom of choice of the spectral information input. We demonstrate our constructions in detail for pentadiagonal matrices.

Prephase-Based Equivalent Amplitude Tailoring for Low Sidelobe Levels of 1-Bit Phase-Only Control Metasurface under Plane Wave Incidence

<p>A prephase synthesis method is proposed for sidelobe level (SLL) suppression of a 1-bit phase-only control metasurface under plane wave incidence. The array factor of the metasurface with N×N unit cells shows that controlling the number of prephases with varying values over the reflective surface realizes equivalent amplitude tailoring. Different from optimizing the prephase distribution, selection of the numbers of 0 and π/2 prephases in specific N regions is used to suppress the SLLs. Therefore, the parameters in the optimization can be dramatically reduced from N² to N. The prephase distribution is then designed based on the optimized number of prephases and a symmetric matrix for SLL suppression in the whole space. The SLLs are further suppressed by optimizing some of the unit cell states based on similar equivalent amplitude tailoring. Simulation and measurement of a set of 1-bit reflective metasurfaces with 20×20 unit cells verify that the phase-only control metasurface realizes SLL suppression to -13 dB for multiple beam directions from -30 to 30 degrees with a 10-degree step under normal plane wave incidence.</p>

Prephase-Based Equivalent Amplitude Tailoring for Low Sidelobe Levels of 1-Bit Phase-Only Control Metasurface under Plane Wave Incidence

<p>A prephase synthesis method is proposed for sidelobe level (SLL) suppression of a 1-bit phase-only control metasurface under plane wave incidence. The array factor of the metasurface with N×N unit cells shows that controlling the number of prephases with varying values over the reflective surface realizes equivalent amplitude tailoring. Different from optimizing the prephase distribution, selection of the numbers of 0 and π/2 prephases in specific N regions is used to suppress the SLLs. Therefore, the parameters in the optimization can be dramatically reduced from N² to N. The prephase distribution is then designed based on the optimized number of prephases and a symmetric matrix for SLL suppression in the whole space. The SLLs are further suppressed by optimizing some of the unit cell states based on similar equivalent amplitude tailoring. Simulation and measurement of a set of 1-bit reflective metasurfaces with 20×20 unit cells verify that the phase-only control metasurface realizes SLL suppression to -13 dB for multiple beam directions from -30 to 30 degrees with a 10-degree step under normal plane wave incidence.</p>