A new spectral invariant for quantum graphs

The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic $$\chi _G:= |V|-|V_D|-|E|$$ χ G : = | V | - | V D | - | E | , with $$|V_D|$$ | V D | denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic $$\chi _G$$ χ G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic $$\chi _G$$ χ G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic $$\chi _G$$ χ G a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.

Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

Spectral Invariant Provides a Practical Modeling Approach for Future Biophysical Variable Estimations

This paper presents a simple radiative transfer model based on spectral invariant properties (SIP). The canopy structure parameters, including the leaf angle distribution and multi-angular clumping index, are explicitly described in the SIP model. The SIP model has been evaluated on its bidirectional reflectance factor (BRF) in the angular space at the radiation transfer model intercomparison platform, and in the spectrum space by the PROSPECT+SAIL (PROSAIL) model. The simulations of BRF by SIP agreed well with the reference values in both the angular space and spectrum space, with a root-mean-square-error (RMSE) of 0.006. When compared with the widely-used Soil-Canopy Observation of Photochemistry and Energy fluxes (SCOPE) model on fPAR, the RMSE was 0.006 and the R2 was 0.99, which shows a high accuracy. This study also suggests the newly proposed vegetation index, the near-infrared (NIR) reflectance of vegetation (NIRv), was a good linear approximation of the canopy structure parameter, the directional area scattering factor (DASF), with an R2 of 0.99. NIRv was not influenced much by the soil background contribution, but was sensitive to the leaf inclination angle. The sensitivity of NIRv to canopy structure and the robustness of NIRv to the soil background suggest NIRv is a promising index in future biophysical variable estimations with the support of the SIP model, especially for the Deep Space Climate Observatory (DSCOVR) Earth Polychromatic Imaging Camera (EPIC) observations near the hot spot directions.

An improved spectral classification of Boolean functions based on an extended set of invariant operations

Boolean functions expressing some particular properties often appear in engineering practice. Therefore, a lot of research efforts are put into exploring different approaches towards classification of Boolean functions with respect to various criteria that are typically selected to serve some specific needs of the intended applications. A classification is considered to be strong if there is a reasonably small number of different classes for a given number of variables n and it it desir able that classification rules are simple. A classification with respect to Walsh spectral coefficients, introduced formerly for digital system design purposes, appears to be useful in the context of Boolean functions used in cryptography, since it is in a way compatible with characterization of cryptographically interesting functions through Walsh spectral coefficients. This classification is performed in terms of certain spectral invariant operations. We show by introducing a new spectral invariant operation in the Walsh domain, that by starting from n?5, some classes of Boolean functions can be merged which makes the classification stronger, and from the theoretical point of view resolves a problem raised already in seventies of the last century. Further, this new spectral invariant operation can be used in constructing bent functions from bent functions represented by quadratic forms.