Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

We prove the operator norm inequality, for a positive matrix partitioned into four blocks in [Formula: see text], [Formula: see text] where [Formula: see text] is the diameter of the largest possible disc in the numerical range of [Formula: see text]. This shows that the inradius [Formula: see text] satisfies [Formula: see text] Several eigenvalue inequalities are derived. In particular, if [Formula: see text] is a normal matrix whose spectrum lies in a disc of radius [Formula: see text], the third eigenvalue of the full matrix is bounded by the second eigenvalue of the sum of the diagonal block, [Formula: see text] We think that [Formula: see text] is optimal and we propose a conjecture related to a norm inequality of Hayashi.

Unimodular matrix and bernoulli map on text encryption algorithm using python

One of the encryption algorithms is the Hill Cipher. The square key matrix in the Hill Cipher method must have an inverse modulo. The unimodular matrix is one of the few matrices that must have an inverse. A unimodular matrix can be utilized as a key in the encryption process. This research aims to demonstrate that there is another approach to protect text message data. Symmetric cryptography is the sort of encryption utilized. A Bernoulli Map is used to create a unimodular matrix. To begin, the researchers use an identity matrix to generate a unimodular matrix. The Bernoulli Map series of real values in (0,1) is translated to integers between 0 and 255. The numbers are then inserted into the unimodular matrix's top triangular entries. To acquire the full matrix as the key, the researchers utilize Elementary Row Operations. The data is then encrypted using modulo matrix multiplication.

A Modified Wavenumber Algorithm of Multi-Layered Structures with Oblique Incidence Based on Full-Matrix Capture

Full-matrix capture (FMC)-based ultrasonic imaging provides good sensitivity to small defects in non-destructive testing and has gradually become a mainstream research topic. Many corresponding algorithms have been developed, e.g., the total focusing method (TFM). However, the efficiency of the TFM is limited, especially in multi-layered structures. Although the appearance of wavenumber algorithms, such as extended phase-shift migration (EPSM) methods, has improved imaging efficiency, these methods cannot be applied to cases with oblique incidence. Therefore, a modified wavenumber method for full-matrix imaging of multi-layered structures with oblique array incidence is proposed. This method performs a coordinate rotation in the frequency domain to adapt it to the oblique incidence. It then utilizes wave-field extrapolation to migrate the transmitting and receiving wave field to each imaging line, and a correlation imaging condition is used to reconstruct a total focused image. The proposed method can deal with any incident angle without precision loss. Moreover, it inherits the computational efficiency advantages of the wavenumber algorithms. The simulation and experimental results show that the proposed method performs better in terms of accuracy and efficiency than the TFM. Specifically, it is nearly 60 times faster than the TFM when processing an FMC dataset with a size of 4096 × 64 × 64.

Multi-defect detection based on ultrasonic Lamb wave sign phase coherence factor imaging method

The ultrasonic Lamb wave total focusing method (TFM) only uses the amplitude of the defective scattered signal for virtual focused imaging, while ignoring the phase information of the scattered signal and the dispersion characteristics of the Lamb wave, resulting in low imaging resolution and easily produced artefacts in imaging. To solve this problem, an ultrasonic Lamb wave imaging method based on phase coherence is proposed in this paper and the sign coherence factor (SCF) in the full matrix scattering signal is extracted. Moreover, the method uses the SCF to weight the amplitude of the full matrix scattering signal, suppresses the side lobes of the defect echo signal and the Lamb wave dispersion effect, improves the ultrasonic Lamb wave imaging resolution and weakens the artefacts. Finally, single- and multiplehole defects in aluminium plates are detected for experimental validation using an ultrasonic phased array. The array performance indicator and signal-to-noise ratio are used as indicators for quantitative assessment of imaging performance. The results show that compared with the TFM imaging, the SCF imaging can effectively suppress the noise and scattered signal side lobes, improve the array performance indicator (API) by 69.1% and improve the signal-to-noise ratio (SNR) by 73.9%. In addition, the SCF imaging can effectively weaken the interference of scattered signals between multiple through-hole defects, resulting in fewer artefacts in imaging.

Ultrasonic Total Focusing Imaging Method of Multilayer Composite Structures Using the Root-Mean-Square (RMS) Velocity

Multilayer composite structures have been widely used in industrial manufacturing, and nondestructive testing of these multilayer structures is to ensure their reliable quality and performance. Currently, ultrasonic total focusing method (TFM) imaging using full-matrix capture (FMC) technology has been proven to sense small defects in a single homogeneous medium and improve the imaging signal-to-noise ratio. However, these algorithms cannot be accurately applied to imaging of multilayer composite structures, due to the acoustic impedance variation and because reflection and refraction occur at the interface between the layers, which makes it very difficult to calculate the ultrasonic propagation path and time. To solve this problem, a root-mean-square (RMS) velocity algorithm for total focusing imaging of multilayer structures is proposed in the article. Based on the theory of RMS velocity for processing of seismic data, the approximated delays can be easily and quickly calculated by a hyperbolic time-distance relationship under circumstances of short lateral distance and horizontal layers. The performance of the proposed algorithm is evaluated by total focusing imaging of a two-layer medium with drilled holes and conducted by the finite element simulation. To further improve imaging efficiency, the partial data in the full-matrix data were used for imaging which is the simplified matrix focusing method (SFM). The results verify that the proposed methods are capable of total focusing imaging of two-layered structures. However, the imaging performance and efficiency of these algorithms are different.

Modeling and Imaging of Ultrasonic Array Inspection of Side Drilled Holes in Layered Anisotropic Media

There has been an increase in the use of ultrasonic arrays for the detection of defects in composite structures used in the aerospace industry. The response of a defect embedded in such a medium is influenced by the inherent anisotropy of the bounding medium and the layering of the bounding medium and hence poses challenges for the interpretation of the full matrix capture (FMC) results. Modeling techniques can be used to understand and simulate the effect of the structure and the defect on the received signals. Existing modeling techniques, such as finite element methods (FEM), finite difference time domain (FDTD), and analytical solutions, are computationally inefficient or are singularly used for structures with complex geometries. In this paper, we develop a novel model based on the Gaussian-based recursive stiffness matrix approach to model the scattering from a side-drilled hole embedded in an anisotropic layered medium. The paper provides a novel method to calculate the transmission and reflection coefficients of plane waves traveling from a layered anisotropic medium into a semi-infinite anisotropic medium by combining the transfer matrix and stiffness matrix methods. The novelty of the paper is the developed model using Gaussian beams to simulate the scattering from a Side Drilled Hole (SDH) embedded in a multilayered composite laminate, which can be used in both immersion and contact setups. We describe a method to combine the scattering from defects with the model to simulate the response of a layered structure and to simulate the full matrix capture (FMC) signals that are received from an SDH embedded in a layered medium. The model-assisted correction total focusing method (MAC-TFM) imaging is used to image both the simulated and experimental results. The proposed method has been validated for both isotropic and anisotropic media by a qualitative and quantitative comparison with experimentally determined signals. The method proposed in this paper is modular, computationally inexpensive, and is in good agreement with experimentally determined signals, and it enables us to understand the effects of various parameters on the scattering of a defect embedded in a layered anisotropic medium.