Medical image segmentation is a critical step in various healthcare applications, aiding in diagnosis, treatment planning, and disease monitoring. In this study, we investigate the efficacy of a segmentation approach based on the 2D wavelet transform. Leveraging a dataset comprising 10 diverse medical images, we evaluate the performance of our segmentation method using three key metrics: accuracy, precision, and recall. Our findings demonstrate that the proposed approach enhances segmentation accuracy, offering promising results compared to existing methods. By harnessing the multi-resolution feature extraction capabilities of the 2D wavelet transform, our method achieves improved delineation of medical image structures, paving the way for more accurate and efficient healthcare interventions.

The paper presents an error model of a measurement chain containing link performing discrete wavelet transform algorithm which is most often the last stage of measurement signal processing. The goal is to determine the uncertainty budget of the input quantities of the wavelet transform algorithm. The error model takes into account parts of analog, analog-to-digital and digital processing, describing the properties of subsequent fragments of the chain using their transmittance and processing functions. The proposed model enables the description of both deterministic and non-deterministic properties of signal errors. The proposed model was validated using an example measurement chain created for this purpose.

Diabetic retinopathy (DR) is the primary factor leading to vision impairment and blindness in diabetic people. Uncontrolled diabetes can damage the retinal blood vessels. Initial detection and prompt medical intervention are vital in preventing progressive vision impairment. Today's growing medical field poses more significant workload and diagnostic demands on medical professionals. In the proposed study, a convolutional neural network (CNN) is employed to detect the stages of DR. This research is crucial for studying DR because of its innovative methodology incorporating two different public datasets. This strategy enhances the model's capacity to generalize unseen DR images, as each dataset encompasses unique demographics and clinical circumstances. The network can learn and capture complicated hierarchical image features with asymmetric weights. Each image undergoes preprocessing using contrast-limited adaptive histogram equalization (CLAHE) and the discrete wavelet transform (DWT). The model is trained and validated using the combined datasets of Dataset for Diabetic Retinopathy (DDR) and the Asia Pacific Tele-Ophthalmology Society (APTOS). The CNN model is tuned with different learning rates and optimizers. The CNN model with Adam optimizer achieved 70% accuracy and an area under curve (AUC) score of 0.82. The recommended study results may reduce diabetes-related vision impairment by early DR severity identification.

The electroencephalogram (EEG) is a record of brain activity. Brain Computer Interface (BCI) technology formed by the EEG signal has become one of the hotspots at present. How to extract the feature signal of EEG is the most basic research of BCI technology. In this paper, A new method of recognizing fatigue, conscious, concentrated state of human brain is proposed by the combination of discrete wavelet transform and the neural network based on EEG signal. First of all, the law signal is preprocessed by the wavelet denoising method because the law EEG signal contains a large number of high frequency noise, which is decomposed into multi-layer high frequency signal and low frequency signal. thus, δ wave, θ wave, α wave, β wave are obtained by the wavelet transform. And then, frequency band energy of the different wave is regards as the feature signal of EEG. In the experiment, the feature signal is classified by radial basic function (RBF) and annealed chaotic competitive learning network (ACCLN). RBF and ACCLN networks are trained with 500 sets of sample data and are tested by 100 sets of samples in different mental states. The experimental results show that the average accuracy of RBF network under three conditions are 88.75%, 88.25%, 88.5%, respectively, and the correct rate of ACCLN network is 97%, 98%, 98%, respectively.

This paper presents the methodology to detect and identify the type of fault that occurs in shunt connected static synchronous compensator (STATCOM) transmission line using a combination of Discrete Wavelet Transform (DWT) and Naive Bayes classifier. To study this, the network model is designed using Mat-lab/Simulink. The different faults such as Line to Ground (LG), Line to Line (LL), Double Line to Ground (LLG) and three-phase (LLLG) fault are applied at different zones of system with and without STATCOM considering the effect of varying fault resistance. The three-phase fault current waveforms obtained are decomposed into several levels using daubechies mother wavelet of db4 to extract the features such as standard deviation and Energy values. The extracted features are used to train the classifiers such as Multi-Layer Perceptron Neural Network (MLP), Bayes and Naive Bayes (NB) classifier to classify the type of fault that occurs in the system. The results reveal that the proposed NB classifier outperforms in terms of accuracy rate, misclassification rate, kappa statistics, mean absolute error (MAE), root mean square error (RMSE), relative absolute error (RAE) and root-relative square error (RRSE) than MLP and Bayes classifier.

Lamb wave-based structural health monitoring is widely acknowledged as a reliable 11 method for damage identification, classification, localization and quantification. However, due to 12 the complexity of Lamb wave signals, especially after interacting with structural components and 13 defects, interpreting these waves and extracting useful information about the structure's health is 14 still a major challenge. However, deep learning-based strategy offers a great opportunity to address 15 such challenges as the algorithm can operate directly on raw discrete time-domain signals. Unlike 16 traditional methods, which often require careful feature engineering and preprocessing, deep learn-17 ing can automatically extract relevant features from the raw data. This paper proposes an autoen-18 coder based on a bidirectional long short-term memory network with maximal overlap discrete 19 wavelet transform layer to detect the signal anomaly and determine the location of the damage in 20 the composite structure. This approach has the potential to greatly enhance our ability to detect and 21 locate structural damage in composite structures, thereby increasing safety and efficiency.

This paper presents a method to detect and classify the high impedance fault that occur in the medium voltage distribution network using discrete wavelet transform (DWT) and adaptive neuro-fuzzy inference system (ANFIS). The network is designed using Matlab software and various faults such as high impedance, symmetrical and unsymmetrical fault have been applied to study the effectiveness of the proposed ANFIS classifier method. This is achieved by training the ANFIS classifier using the features (standard deviation values) extracted from the three phase fault current signal by DWT technique for various cases of fault with different values of fault resistance in the system. The success and discrimination rate obtained for identifying and classifying the high impedance fault from the proffered method is 100% whereas the values are 66.7% and 85% respectively for conventional fuzzy based approach. The results indicate that the proposed method is more efficient to identify and discriminate the high impedance fault accurately from other power system faults in the system.

The paper delves into the critical issue of damage detection within the guy cable of a truss steel mast, a pivotal component in structural integrity. It introduces a model wherein damage manifests as a localized cross-section reduction in a single element of the cable. Employing Discrete Wavelet Transform (DWT), a pioneering methodology, the study scrutinizes the behavior of the affected elements through static structural analyses. Signal decomposition via the Mallat pyramid algorithm facilitates comprehensive examination. Static displacements at the connection point between the cable and the truss mast serve as the measured variables. Through systematic investigation, the paper evaluates the impact of damage size and location, external loading force, and cable tension force on the efficacy of the proposed approach. Utilizing data derived from Finite Element Method (FEM) computations for wavelet analysis the authors substantiate the findings with numerical examples, thus offering valuable insights into damage detection strategies for structural health monitoring and engineering applications.

This algorithm is designed to perform Discrete Fourier Transforms (DFT) to convert temporal data into spectral data. This algorithm obtains the Fourier Transforms by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate. What is especially beneficial about this DFT algorithm is that it can produce spectral data at any user-defined resolution.

During routine bridge maintenance, edge detection allows the partial condition of the bridge to be viewed. However, many edge detection methods often have unsatisfactory performances when dealing with images with complex backgrounds. Moreover, the processing often involves the manual selection of thresholds, which can result in repeated testing and comparisons. To address these problems in this paper, the wavelet transform modulus maxima method is used to detect the target image, and then the threshold value of the image can be determined automatically according to the OTSU method to remove the pseudo-edges. Thus, the real image edges can be detected. The results show that the information entropy and SSIM of the detection results are the highest when compared with the commonly used Canny and Laplace algorithms, which means that the detection quality is optimal. To more fully illustrate the advantages of the algorithms, images with more complex backgrounds were detected and the processing results of the algorithms in this paper are still optimal. In addition, the automatic selection of thresholds saves the operator’s effort and improves the detection efficiency. Thanks to the combined use of the above two methods, detection quality and efficiency are significantly improved, which has a good application in engineering practice.

Discrete Double Hilbert Exponential sums along polynomials. This is preprint version (not peer reviewed) And we now check the proof of (ii) of Lemma 3.2 whether it is really true.

High Efficiency Video Coding (HEVC) uses an 8-point filter and a 7-point filter, which are based on the discrete cosine transform (DCT), for the 1/2-pixel and 1/4-pixel interpolations, respectively. In this paper, discrete sine transform (DST)-based interpolation filters (IF) are proposed. The first proposed DST-based IFs (DST-IFs) use 8-point and 7-point filters for the 1/2-pixel and 1/4-pixel interpolations, respectively. The final proposed DST-IFs use 12-point and 11-point filters for the 1/2-pixel and 1/4-pixel interpolations, respectively. These DST-IF methods are proposed to improve the motion-compensated prediction in HEVC. The 8-point and 7-point DST-IF methods showed average BD-rate reductions of 0.7% and 0.3% in the random access (RA) and low delay B (LDB) configurations, respectively. The 12-point and 11-point DST-IF methods showed average BD-rate reductions of 1.4% and 1.2% in the RA and LDB configurations for the Luma component, respectively.

Inverse Sumudu transform multiple shifting properties are used to design methodology for solving ordinary differential equations. Then algorithm applied to solve Whittaker and Zettl equations to get their new exact solutions and profiles which shown through Maple complex graphicals. Table of inverse Sumudu transforms for elementary functions given for supporting the differential equations solving using inverse Sumudu transform.

In the process of image feature extraction, noise in the image will greatly affect the accuracy of edge detection. In this paper, the image is filtered to remove noise before edge detection by using the algorithm of wavelet transformation. Different wavelet functions are used to decompose image. Based on the experimental results, the best denoising wavelet function is selected. Canny algorithm is used to detect the edge of the denoised image, and the result of edge detection is evaluated according to the Pratt quality factor. It is proved that wavelet transform can improve the edge detection results.

The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform (DHT) and inverse DFT sequence can be exploited for efficient code. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.

In this paper we define a continuous wavelet transform of a Schwartz tempered distribution $f \in S^{'}(\mathbb R^n)$ with wavelet kernel $\psi \in S(\mathbb R^n)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{'}(\mathbb R^n)$. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

Estimating the remaining useful life (RUL) of components is a crucial task to enhance the reliability, safety, productivity, and to reduce maintenance cost. In general, predicting the RUL of a component includes constructing a health indicator ( ) to infer the current condition of the component, and modelling the degradation process, to estimate the future behavior. Although many signal processing and data-driven based methods were proposed to construct the , most of the existing methods are based on manual feature extraction techniques, and need the prior knowledge of experts, or rely on a large amount of failure data. Therefore, in this study, a new data-driven method based on the convolutional autoencoder (CAE) is presented to construct the . For this purpose, the continuous wavelet transform (CWT) technique is used to convert the raw acquired vibrational signals into a two-dimensional image; then, the CAE model is trained by the healthy operation dataset. Finally, the Mahalanobis Distance (MD) between the healthy and failure stages is measured as the . The proposed method is tested on a benchmark bearing dataset and compared with several other traditional construction models. Experimental results indicate that the constructed exhibits a monotonically increasing degradation trend and has a good performance to detect incipient faults.

This addendum contains clarifications and a sharpening of some of the results on the VIT transform framework developed in [1]. The focus is on the right-coefficient and left- coefficient forms of the transform, the extraction of a first-order term from a left polynomial fraction, and the application to linear time-varying systems.

The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier Transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel Transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.

The dynamic evolution and variation of electrical loads is now a priority for optimum management and, above all, forecasting. Indeed, these dynamic load variations require computer tools able to implement optimal load forecasting models. Scientific research into automated models for forecasting electrical loads therefore represents a challenge for scientific researchers. Several research studies have therefore been carried out. These include machine learning approaches such as: LSTM (Long Short-Term Memory), Support Vector Machine (SVM) and others. This article proposes a comparison study between a hybrid model based on the wavelet transform coupled with the adaboost method, and LSTM, for forecasting electricity consumption loads. Python 3.10 was used for the studies. The first results obtained consisted in the study of electricity load forecasts over four (04) data sets of the year (January, April, August and December). These results are very satisfactory, showing a good correlation between the real results of the electrical load consumption data, and the results of the implemented model. The electrical load consumption data are collected from a renewable energy production source: photovoltaic solar energy. The model test results (R2) obtained range from 0.919 to 0.958. These results are representative of real data and reflect the model's performance. The second set of results concerns a comparative study between the methods used by Y. Xie et al (CNN, MLP, LSTM-AM and LSTM-AM-MLP), and our approach. The prediction results obtained between the approaches show a prediction gap ranging from 0.15 to 0.78 for RMSE and 0.83 for MAPE. These results show significant differences between the methods. The minimization of errors by our model, therefore, reflects the model's performance in terms of accuracy, which is necessary for the optimal management of consumer electricity load forecasts in order to ensure balance between supply and demand.

Visual object tracking is an important technology in camera based sensor networks, which has a wide range of practicability in auto drive system. A transformer is a deep learning model that adopts the mechanism of self-attention, and it differentially weights the significance of each part of the input data. It has been widely applied in the field of visual tracking. Unfortunately, the security of the transformer model is unclear. It makes such transformer-based applications be exposed to security threats. In this work, the security of the transformer model is investigated with the important component of autonomous driving, visual tracking. Such deep-learning-based visual tracking is vulnerable to adversarial attacks, so adversarial attacks are implemented as the security threats to conduct the investigation. First, adversarial examples are generated on top of video sequences to degrade tracking performance, and the frame-by-frame temporal motion is taken into consideration when generating perturbations over the predicted tracking results. Then, the influence of perturbations on performance is sequentially investigated and analyzed. Finally, numerous experiments on OTB100, VOT2018, and GOT-10k data sets demonstrate that the executed adversarial examples are effective on the performance drops of the transformer-based visual tracking.

The diagnosis of Cardio-Vascular diseases (CVD) is highly dependent on analysis of ECG signals. ECG analysis can be helpful in estimating the underlying cause and condition of heart in cardiac abnormality. The effectiveness of ECG signal analysis in detection of CVDs is widely accepted by professional healthcare service provider. Many algorithms have been proposed but almost all of them have some kind of limitations, and these limitations largely influence the effectiveness of ECG analysis. The performed research work is dedicated for design of unique self-organizing maps (SOMs) based neural network for classification of arrhythmia according to a particular ECG signal, the generation of SOMs is based on the certain unique signatures of ECG signals and have potential to classify different cardiac conditions. For extraction of unique features from ECG signals, we have proposed to use Fast Discrete Stockwell Transform (FDST). Since the proposed technique is a result of combining two different techniques hence called as hybrid technology. The purpose of using FDST is to identify unique signatures of ECG signals in a more improved manner than existing one, the term improved is used because it has several advantages over existing techniques such as wavelet and Fourier Transform based methods. Results obtained from the implementation of the technique are capable in visualizing the ECG sinus rhythm and arrhythmia conditions in form of unique SOM for each associated arrhythmia condition. This unique SOM based classification makes them ideal for being used as a diagnostic tool. This ability of arrhythmia classification using FDST and SOMs makes the technique unique and useful providing valuable information about patient condition. Using proposed technology a portable diagnosis tool for monitoring of patient at their site may be facilitated later, that will improve the quality of life of the patient by diagnosing cardiac condition.

Background: The electroretinogram is a clinical test used to assess the function of the photoreceptors and retinal circuits of various cells in the eye, with the recorded waveform being the result of the summated response of neural generators across the retina. Methods: The present investigation involved an analysis of the electroretinogram waveform in both the time and time-frequency domain through the utilization of the discrete wavelet transform and continuous wavelet transform techniques. The primary aim of this study was to monitor and evaluate the effects of treatment in a New Zealand rabbit model of endophthalmitis via electroretinogram waveform analysis and to compare these with normal human electroretinograms Results: The wavelet scalograms were analyzed using various mother wavelets, including the Daubechies, Ricker, Wavelet Biorthogonal 3.1 (bior3.1), Morlet, Haar, and Gaussian wavelets. Distinctive variances were identified in the wavelet scalograms between rabbit and human electroretinograms. The wavelet scalograms in the rabbit model of endophthalmitis showed recovery with treatment in parallel with the time -domain features. Conclusions: The study compared Adult, Child, and Rabbit electroretinogram responses using DWT and CWT, finding that Adult signals had higher power than Child signals, and Rabbit signals showed differences in a-wave and b-wave depending on the type of response tested, while Haar Wavelet was found to be superior in visualizing frequency components in electrophysiological signals in following the treatment of endophthalmitis and may give additional outcome measures for the management of retinal disease.

Bearings are widely used in various types of electrical machinery and equipment. As their core components, failures will often cause serious consequences . At present, most methods of parameter adjustment are still manual adjustment of parameters. This adjustment method is susceptible to prior knowledge and easy to fall into the local optimal solution, failing to obtain the global optimal solution and requires a lot of resources.Therefore, this paper proposes a new method of bearing fault diagnosis based on wavelet packet transform and convolutional neural network optimized by simulated annealing algorithm.The experimental results show that the method proposed in this paper has a more accurate effect in feature extraction and fault classification compared with traditional bearing fault diagnosis methods. At the same time, compared with the traditional artificial neural network parameter adjustment, this paper introduces the simulated annealing algorithm to automatically adjust the parameters of the neural network, thereby obtaining an adaptive bearing fault diagnosis method. To verify the effectiveness of the method, the Case Western Reserve University bearing database was used for testing, and the traditional intelligent bearing fault diagnosis method was compared. The results show that the method proposed in this paper has good results in bearing fault diagnosis. Provides a new way of thinking in the field of bearing fault diagnosis in parameter adjustment and fault classification algorithms

Recently, significant progress has been made in developing computer-aided diagnosis (CAD) systems for identifying glaucoma abnormalities using fundus images. Despite their drawbacks, methods for extracting features such as wavelets and their variations, along with classifier like support vector machines (SVM), are frequently employed in such systems. This paper introduces a practical and enhanced system for detecting glaucoma in fundus images. This system adresses the chanallages encountered by other existing models in recent litrature. Initially, we have employed contrast limited adaputive histogram equalization (CLAHE) to enhanced the visualization of input fundus inmages. Then, the discrete ripplet-II transform (DR2T) employing a degree of 2 for feature extraction. Subsequently, a golden jackal optimization algorithm (GJO) employed to select the optimal features to reduce the dimension of the extracted feature vector. During the classification stage the least square support vector machine (LS-SVM) with three kernels called as linear, polynomial and radial basis function(RBF), for classifying of fundus images as glaucoma or healthy. The proposed method is validated with the current state-of-the-art models on two standard datasets, namely, G1020 and ORIGA. The results obtained from our experimental result demonstrate that our best suggested approach DR2T+GJO+LS-SVM-RBF obtains better classification accuracy 93.38% and 97.31% for G1020 and ORIGA dataset with less number of features. It establishes a more concise network structure when contrasted with traditional classifiers.

The Doppler band compensated by the keystone transform (KT) is limited. Therefore, it needs to be used in conjunction with the Doppler ambiguity compensation function to correct the range migration (RM) caused by maneuvering targets with Doppler ambiguity. However, the KT implemented by sinc interpolation suffers from significant performance loss at boundaries of compensation Doppler bands. Additionally, in a multi-target scenario, KT implementation methods occupy high complexity when the Doppler range of targets spans over two compensation Doppler bands. To address aforementioned issues, this study presents a variable Doppler starting point keystone transform (VDSPKT) method, where a new form of ambiguity compensation function is constructed, turning the Doppler starting point of the compensation band in KT variable. Firstly, the position of the compensation Doppler band is changed from fixed to adjustable as needed, enhancing the flexibility of KT. Crucially, the connection points of the compensation Doppler bands in sinc interpolation are reset, avoiding the performance loss at their boundaries. Also, the compensation band is adjusted to cover the narrow Doppler frequency range caused by targets, significantly improving the computational efficiency. Finally, the simulation results demonstrate that the proposed approach effectively addresses the performance degradation and high computational complexity of KT in the aforementioned scenarios, resulting in a computational load reduced by approximately 50% compared to traditional methods.

Analysis of non-stationary and nonlinear sound signals obtained from dynamical processes is one of the greatest challenges in signal processing. Turning machine operation is a highly dynamic process persuaded by many events such as dynamical responses, chip formations and operational conditions of machining. Traditional and widely used fast Fourier transformation and spectrogram is not suitable for processing sound signals acquired from dynamical systems as its result have significant deficiencies because of stationary assumptions and having priori basis. The relatively new technique, Discrete Wavelet Transform, which uses Wavelet decomposition and the recently developed technique, Hilbert Huang Transform (HHT) which uses Empirical Mode Decomposition have notably better properties in the analysis of nonlinear and non-stationary sound signals. This paper presents a comparative study on the decomposition of multicomponent sound signals using Empirical Mode Decomposition and Wavelet Decomposition to highlight the suitability found in HHT to analyze tool-emitted sound signals received from turning processes. Apart from the short mathematical and theoretical foundations of the transformations, this paper demonstrates their decomposition strength using an experimental case study of tool flank wear monitoring in turning. It also concludes HHT is more suitable than DWT to analyze tool-emitted sound signals received from turning processes.

Recognition, decoding and tracking of vortex patterns is of increasing importance in many fields reaching from astronomical observation of distant galaxies to turbulence phenomena in liquids or gases. Currently, coherent light beams with orbital angular momentum (OAM) are of particular interest for optical communication, metrology, micro-machining or particle manipulation. One common task is to identify characteristic spiral patterns in pixelated intensity maps at real-world signal-to-noise ratios. Recently, Fast Fourier Transform (FFT) was applied to the quantitative analysis of spiral interference patterns of OAM beams. Our study considers this approach in more detail. It is demonstrated that specific information can be extracted by combining conformal mapping by polar transform with determining one- or two-dimensional angular frequency spectra either in spatially fixed or scanning mode. Results of proof-of-principle experiments are presented.

Real time condition monitoring and precision health assessment system is a mandatory need for effective maintenance program in industrial sector. Rapid advancement in the information technology and other engineering technologies have invited more proactive attention from research and development in industrial sectors and particularly in condition monitoring of machines and related Industrial processes. In this work, the drill bit condition monitoring techniques have been developed based on the wavelet analysis and artificial neural network (ANN) as automatic drill bit fault detection and classification. An experimental work has been conducted to capture the vibration signals for analysis. In this experiment, the CNC drill machine is used with high carbon steel drill bit and mild steel material as work piece. The cutting condition parameters are kept constant and the wear level is changed. The Data Acquisition system (DAQ) with Lab VIEW software is used to capture the vibration signals for drill bit with different wear conditions. The captured vibrations data are analyzed using continuous wavelet transform (CWT) with Morlet wavelet and Daubechies wavelet as a prime function. In general, the CWT coefficient is used to generate the inputs features to ANN for automatic tool condition classification, with two outputs (0, 1) for healthy and (1, 0) for faulty. The results show the effectiveness of the combed WT and ANN for automatic classification of tool wear conditions with high success rate.

To assure the opacity, an optimal problem is to retain the occurring events sequences as many as possible. Contrary to such problem, the other optimal problem is to preserve the minimal occurring events sequences. In the paper, based on the choosing cost, an optimal opacity-enforcing model is presented to minimize the sum of discount choosing cost, which is not only to preserve the opacity, but also to retain the secret to the maximum. To solve the model, two scenarios on opacity are considered. For the two scenarios, some algorithms are presented to achieve the optimal solution for the model by using the method of dynamic programming. And then, the solutions produced by the algorithms are proved correctly by theoretical proof. Finally, some illustrations for the algorithms are given.

The accompanying mechanical behavior during the flow of ore-rock dispersion is an important factor leading to the instability and failure of the orepass in the ore storage section. It is of great significance to accurately understand the instability characteristics of the well wall in the ore section, during the ore drawing process by understanding the flow characteristics and internal mechanical transfer mechanism of ore-rock dispersion. Using the discrete element method to analyze the flow characteristics, contact compactness, stress distribution characteristics and contact force probability distribution of the ore-rock dispersion, it realizes the quantitative characterization of the damage degree of ore-rock flow, and reveals the damage mechanism of mine storage wall. The results show that: (1) The friction between ore-rock particles, between ore-rock and well wall is an important reason for the change of the macroscopic flow form of ore-rock from “one” to “V” type. (2) The compactness and stress concentration of ores are increasing with the increase of storage depth, which is shown that with the increase of storage depth, the greater the probability of well wall instability. (3) The probability distribution of contact force strength between particles decreases exponentially in the whole process of ore drawing, in which the strong chain plays a major role in the stability of the dispersion system. (4) The number of overpressure and the overpressure coefficient can be used to quantitatively characterize the wall damage degree under the action of ore-rock flow. The dynamic side pressure decreases periodically in the form of exponential fluctuations, and the dynamic load formed by the ore-rock flow mainly played a role in the lower area of the ore storage section.

This paper proposed an optimal tracking control scheme through adaptive dynamic programming(ADP) for a class of partially unknown discrete-time nonlinear systems based on radial basis function neural network(RBF-NN). In order to acquire the unknown system dynamics, we use two RBF-NNs, the one is used to construct the identifier, and the another is used to directly approximate the steady-state control input, where a novel adaptive law is proposed to update neural network weights. While the optimal feedback control and the cost function are derived via feedforward neural networks approximating, it is proposed to regulate the tracking error, the critic network and the actor network are then trained online to obtain the solution of the associated Hamilton–Jacobi–Bellman (HJB) equation being built under the ADP framework. Simulations verify the effectiveness of the optimal tracking control technique using the neural networks.

The paper presents the problem of damage detection in steel girders. Static displacements at the selected point of the structure play the role of measured variables. Structural response signal decomposition is performed according to the Mallat pyramid algorithm, which is used to perform the discrete wavelet transform (DWT). This procedure allows us to quite well determine the location of structural damage. The geometry and the placement of any defective part of the structure may have a random character. It can be assumed that the random processes occurring in the broadly understood structure mechanics are Gaussian in nature. The first four probabilistic moments are estimated using three approaches independent: semi-analytical (SAM), perturbative (SPT), and Monte-Carlo simulations (MCS). The semi-analytical random approach seems to be the most optimal due to the necessary computation time. The incorporation of the mathematical stochastic apparatus into the classical (deterministic) analysis of the statics of the structure makes it possible to estimate the reliability measures of the analyzed girder.

This article presents a concise survey of basic discrete and semi-discrete nonlinear models which produce two- and three-dimensional (2D and 3D) solitons, and a summary of main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ⁽²⁾) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of basic species of multidimensional discrete modes, including fundamental and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

We present an idea how to use long-wavelength multi-wavelength radio continuum observations of spiral galaxies to isolate magnetic structures which were previously accessible from short-wavelength observations only. The approach is based on the RM-synthesis and 2D continuous wavelet transform. Wavelet analysis helps to recognize a configuration of small-scale structures which are produced by Faraday dispersion. We find that these structures can trace galactic magnetic arms for the case of the galaxy NGC 6946 observed at $\lambda = 17 \div 22$~cm. We support this interpretation by an analysis of a synthetic observation obtained using a realistic model of a galactic magnetic field.

This paper is devoted to the study of multi-dimensional integral transform with Fox H-function in the kernel in weighted spaces integrable functions in the domain R+n with positive coordinates. Mapping properties such as the boundedness, the range, the representations of the considered transformation are established.

In Indonesia, types of meat are still identified manually due to increasing prices for beef and buf-falo ahead of Idul Fitri. Traders mix beef with pork to thwart the beef scam. Haar wavelet and GLCM, GLCM with angles of 0°, 45°, 90°, and 135°, as well as matrices using contrast, correlation, energy, homogeneity, and entropy, are used in feature extraction. The following are the results of meat image classification testing using k-NN, Haar wavelet, and GLCM: The k-NN algorithm shows superiority in identifying fresh (99%), frozen (99%), and rotten (96%) meat texture images with the most fantastic accuracy results in every situation. Meanwhile, GLCM routinely provides good results, especially regarding the texture image of fresh meat (183.21) and rotten meat (115.79). However, despite delivering less throughput than k-NN and GLCM, Haar wavelets are still helpful, especially when dealing with fresh meat texture images (89,96).

Kim-Kim (Russ. J. Math. Phys. 2017, 24, 241-248) defined the degenerate Laplace transform and investigated some of their certain properties. Motivated by this study, in this paper, we introduce the degenerate Sumudu transform and establish some properties and relations. We derive degenerate Sumudu transforms of power functions, degenerate sine, degenerate cosine, degenerate hyperbolic sine, degenerate hyperbolic cosine, degenerate exponential function, and function derivatives. We also acquire a relationship between degenerate Sumudu transform and degenerate gamma function. Moreover, we investigate a scale preserving theorem for the degenerate Sumudu transform. Furthermore, we show that the degenerate Sumudu transform is the theoretical dual transform to the degenerate Laplace transform.

Saif et al. (J. Math. Comput. Sci. 21 (2020) 127-135) considered modified Laplace transform and developed some of their certain properties and relations. Motivated by this work, in this paper, we define modified Sumudu transform and investigate many properties and relations including modified Sumudu transforms of the power function, sine, cosine, hyperbolic sine, hyperbolic cosine, exponential function, and function derivatives. Moreover, we attain two shifting properties and a scale preserving theorem for the modified Sumudu transform. We give modified inverse Sumudu transform and investigate some relations and examples. Furthermore, we show that the modified Sumudu transform is the theoretical dual transform to the modified Laplace transform.

The twisted Wang transform distribution family, defined as the composition of parameter shifted inverse CDF function with an original CDF function, is found to be most suitable for matching low shape factor distributions, characterizing hard to fit or to simulate reinsurance portfolio losses for some perils from our previous study. Among them, the best form for matching a hard-to-fit empirical loss distribution for a specific peril, is the Exponential Fractional Extra Power 0 Distribution in (0,1) with CDF:.The simplest yet still a good form of this family is the Transformed Hyperbolic Tangent Distribution with CDF:,which has analytical formulas for the moments. The twisted Wang transform distribution family is compared and confirmed to be superior to all other well-known distribution families through extensive numerical optimization practice, distribution forms guesses, and computer-aided exploration experiments.

Sentiment analysis to characterize properties of Bitcoin prices and their forecasting is here developed thanks to the capability of the Fuzzy transform to capture stylized facts and mutual connections between time series having different nature. Six years of daily Bitcoin Prices and Google Trends are analyzed in order to establish new perspectives in the management of their dynamics.

In this work, a mixture of Elzaki transform and projected differential transform method is applied to solve a nonlinear wave-like equations with variable coefficients. Nonlinear terms can be easily manipulated by using the projected differential transformation method. The method gives the results show that the proposed method is very efficient, simple and can be applied to other applications.

Marketing is the manner of how to make our sales the best in the market, our prices the most accessible, our clients satisfactory, and thus our brand is the largest distributed. This needs sophisticated and advanced understanding of the total network related. Indeed, marketing data may be seen in different forms such as qualitative and quantitative. However, in the literature, it is easily noticed that large bibliography may be collected about qualitative studies, against few studies on the quantitative point of view. This is a major drawback that makes the marketing science still focusing on the design, although the market is strongly depending on quantities such as money and time. Indeed, marketing data may be a time series such as brand sales per specified periods, brand related-prices over specified periods, market shares, ..., etc. The purpose of the present work is to investigate some marketing models based on time series due to brands. We will precisely study the effect of the time scale on the persistence of brands sales in the market and on the forecasting of such a persistence according to the characteristics of the brand and the related market competition or competitors. Our study is acted on a sample of Saudi brands during the period November 22, 2017 to December 30, 2021.

This paper first introduces the main classification of brain diseases and the main causes of these diseases. It is followed by wavelet analysis, which is a mathematical function or wave-like pattern used to transform data, which can decompose the signal into different wavelet functions, each of which is related to a specific scale or frequency. Wavelet analysis has been applied in many fields such as image processing, data analysis and engineering. In addition, wavelet is also used in the analysis of biological signals. Wavelet analysis breaks down brain signals, such as EEG or fMRI data, into various frequency components at different scales. Wavelet analysis provides time-frequency localization. Pattern recognition can be enhanced by isolating salient features in the data. Wavelet denoising can effectively separate noise from underlying brain signals. Wavelet denoising usually adopts threshold method. Wavelet analysis can enhance pattern recognition by isolating salient features in the data. Researchers can train machine learning models based on these wavelet-derived features to recognize specific patterns associated with different neurological disorders. Wavelet analysis can track these changes by continuously evaluating the frequency content of the neuroimage data over time. Through continuous efforts, wavelet theory and technology have become valuable tools in the field of neuroscience and brain disease research.

By utilizing the Fourier transform, we present a practical method for evaluating a function of a derivative applied to any other arbitrary function of a single complex variable. As an illustration of this approach, we compute the actions of the displacement and squeeze operators on an arbitrary function, as well as the propagation of paraxial fields, Airy and number states, without the need for algebraic quantum operator techniques or the Fresnel integral.

Aiming at the APF algorithm in the collision-free path planning and LQR algorithm tracking process, there are problems such as unreachable target points, local optimization, low safety coefficient of autonomous lane change collision avoidance tracking control and poor stability, this paper proposes an optimized APF method, which solves the defects of the traditional artificial potential field method by adding the distance adjustment factor, the safety distance protection model and the sub-target virtual potential field, and at the same time generates the autonomous driving car collision-free path. The path tracking control system is designed based on feed-forward control and DLQR controller, in which the vehicle dynamics state space model is used as the basis, the model is linearized and discretized, and the corresponding constraint variables are set to minimize the control command objective cost function under the constraints, and the objective of the DLQR controller.The objective is to accurately determine the steering angle in order to achieve path tracking and eliminate the lateral error between the reference path and the actual path caused by external interference.Finally, the simulink model and CarSim parameter configuration joint simulation platform are built, and different obstacle scenarios and different speeds are selected for collision-free path planning and tracking motion simulation research and analysis, which verifies the accuracy and practicability of the algorithm, and shows more obvious advantages in obstacle avoidance ability, path smoothing degree, and collision avoidance safety stability. The ideas proposed in this paper are expected to be further extended and popularized to actual path planning problems, which has important practical significance and engineering value.

Recently Candida auris has emerged as a multi-resistant fungal pathogen, with a significant clinical impact, and has become a critical issue in WHO fungal priority list, due to its unique ability to persist for long time on human skin and hospital environments. It is therefore fundamental to reinforce hospital surveillance protocols to limit nosocomial outbreaks. The study purpose was the Fourier-Transform Infrared Spectroscopy (FT-IR) application to investigate the phylogenetic relationships between isolated strains within the C. auris outbreak at the university Intensive Care Unit of a Tertiary University hospital in Turin (Italy). In order to calculate a clustering cut-off, intra- and inter-isolates distance values were analysed. Data showed the presence of a major cluster Alfa and a minor cluster Beta with a defined C. auris clustering cut-off. The results were validated by an external C. auris strain and Principal Component and Linear Discriminant Analysis. The application of FT-IR technology had allowed to obtain important information about the phylogenetic relationships between the analysed strains, defining for the first time a “not WGS-based” clustering cut-off with a statistical-mathematical approach. FT-IR could represent a valid alternative to molecular methods for rapid and cost saving typing of C. auris strains with important clinical implications.

This paper reviews how dynamically adaptive wavelet methods can be designed to simulate atmosphere and ocean dynamics in both flat and spherical geometries. We highlight the special features that these models must have in order to be valid for climate modelling applications. These include exact mass conservation and various mimetic properties that ensure the solutions remain physically realistic, even in the under-resolved conditions typical of climate models. Particular attention is paid to the implementation of complex topography in adaptive models. Using \textsc{wavetrisk} as an example, we explain in detail how to build a semi-realistic global atmosphere or ocean model of interest to the geophysical community. We end with a discussion of the challenges that remain to developing a realistic dynamically adaptive atmosphere or ocean climate models. These include scale-aware subgrid scale parameterizations of physical processes, such as clouds. Although we focus on adaptive wavelet methods, many of the topics we discuss are relevant for adaptive mesh refinement (AMR).

Multiple sclerosis is a chronic, autoimmune disease that mainly affects the central nervous system, including the brain, spinal cord, and optic nerve. This disease can cause clinical symptoms such as cognitive decline, muscle weakness, spasms, and fatigue in patients, and the onset tends to be younger. Current medication can only prevent or alleviate symptoms, so early diagnosis of this disease can increase patients' chances of treatment. Although the use of nuclear magnetic resonance detection can improve the efficiency of early auxiliary diagnosis, it still requires experienced doctors to spend too much time and energy on comprehensive judgment. To reduce the time cost and improve the efficiency of multiple sclerosis diagnosis, this article proposes a detection and recognition algorithm for multiple sclerosis based on wavelet entropy and self-adaptive particle swarm optimization. Firstly, a triple discrete wavelet transform is performed on the brain image of multiple sclerosis, and then 10 wavelet entropies are extracted from the decomposed wavelet subbands, which can reduce the feature dimensions of the image; Then, the algorithm of self-adaptive particle swarm optimization is used to optimize the feedforward neural network, in order to obtain the optimal connection weights and thresholds during the training process. The result of the model is with an average sensitivity of 92.29±1.89, specificity of 92.54±0.67, precision of 92.48±0.59, accuracy of 92.42±0.88, and F1 score of 84.85±1.74, Matthews correlation coefficient of 92.37±0.96, and Fowlkes223 Mallows Index of 92.38±0.96. The experimental results indicate that this algorithm has a very important data support role in detecting multiple sclerosis.

This paper studies the duration patterns of unemployment spells for immigrants and the determinants of unemployment’s completion into one of a number of studied labour market states in Finland. We estimate a duration model for unemployment with competing risks of its terminating into employment, labour market training or economic inactivity. Taking into account the wide period of observation and opportunities to analyse processes of labour market integration during various periods of economic development in Finland, in combination with the individualistic character of the labour careers of immigrants, this research is beneficial owing to the many various findings concerning labour market integration of immigrants. The approach undertaken in this research has a dualistic “descriptive-dynamic” character under which integration is understood as a never-ending process, which is conditioned by a time period of long-term existence and a context of solitary action. We find that transitions out of unemployment spells have a cyclical character; after every new “cycle” in unemployment, the probability of terminating unemployment decreases further. We also find that ascriptive factors make sense in the process of job-placement of immigrants from unemployment. Therefore, the gender, education and age of immigrants, as well as the effect of the period in which first unemployment occurred, potentially predict transitions out of unemployment and further labour market integration of immigrants.

In many application problems, F-transform algorithms are very efficient. In F-transform techniques, we replace the original signal or image with a finite number of weighted averages. The use of weighted average can be naturally explained, e.g., by the fact that this is what we get anyway when we measure the signal. However, most successful applications of F-transform have an additional not-so-easy-to-explain feature: the partition requirement, that the sum of all the related weighting functions is a constant. In this paper, we show that this seemingly difficult-to-explain requirement can also be naturally explained in signal-measuring terms: namely, this requirement can be derived from the natural desire to have all the signal values at different moments of time estimated with the same accuracy.

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

Different variants of thinning for discrete random variables are studied. The thinning procedure allows to introduce an analog of scale parameter for positive integer-valued random variables. Sufficient and necessary conditions for the existence of such a scale are given.

In view of the problem of cracking easily and partial distortion of images after encryption or decryption, a novel image encryption and decryption algorithm based on Arnod Transform and fractional chaotic is proposed. To begin with, the Arnold transform is used to encrypt. So that the spatial confidence of the original image has been comprehensively disturbed. Secondly, the XOR involving the fractional order chaotic sequence is used to encrypt. The key sequence is dynamically generated to ensure the randomness and difference of key generation. When decryption is required, the first decryption is performed using the key and XOR. Then the second decryption is carried out by using the inverse Arnold transform, and finally the decrypted image is obtained. Experimental results show that the improved algorithm has achieved better performance in encryption and decryption.

This paper presents a novel approach for the recognition of multiple sclerosis (MS) using wavelet entropy and a Particle Swarm Optimization (PSO)-based neural network. MS is a complex neurological disorder with diverse clinical manifestations, often challenging to diagnose accurately. In this study, we leverage wavelet entropy (WE) as a feature extraction method to capture intricate patterns within brain imaging data. These extracted features are then employed to train a neural network model optimized through PSO to enhance classification accuracy. Experimental results on a dataset comprising MS and healthy control subjects demonstrate the effectiveness of our proposed approach, viz., WE-PSONN, which achieves a sensitivity of 91.95±1.15, a specificity of 92.36±0.88, a precision of 92.28±0.88, an accuracy of 92.16±0.90. The combination of advanced signal processing techniques and machine learning optimization holds promise for improving the early diagnosis and management of MS, offering potential benefits to both patients and healthcare providers.

This article examines the wavelet transforms and the machine learning methods in accelerating large eddy simulations of turbulent flows. An overarching challenge of large eddy simulations is to accurately represent the cascade of kinetic energy across the cut-off length scale. Taylor hypothesis suggests that the energy cascade occurs through the process of vortex stretching. However, Richardson hypothesis assumes the self-amplification of the strain field and a hierarchical break down of large eddies lead the energy cascade. The large eddy simulations typically employs the self-amplification of the strain in formulating subgrid models. However, several studies also proposed subgrid models based on vortex stretching. The wavelet-based coherent vortex simulation of turbulence directly accounts for vortex stretching in overall forward scatter of energy, while allowing local backscatter. The wavelet-based large eddy simulation adapts the grid to capture the creation of small-scale eddies, while adopting subgrid models based on the self-amplification of strain. The advancement of artificial intelligence in turbulence modelling is currently evolving around accelerating the numerical simulations of turbulent flow. However, there is a clear connection between the application of wavelet transfroms and neural networks for directly solving the Navier-Stokes equation, indicating some potential benefits of wavelet methods over the neural networks.

The paper investigates new robust code constructions based on bent functions and spline–wavelet transformation. Implementation of bent functions in codes construction increases the probability of error detection in the data channel and cryptographic devices. Meanwhile, use of spline–wavelets theory for constructing the codes gives the possibility to increase system security from the actions of an attacker. Presented constructions combines spline–wavelets functions and bent functions. Developed robust codes, compared to existing ones, have a higher parameter of maximum error masking probability. Illustrated codes are ensuring the security of transmitted information. Some of the granted constructions were implemented on FPGA.

With the rapid development of intelligent logistics and network environment, it has become an urgent problem to efficiently and accurately handle and analyse the huge amount of uncertain decision information. Traditional decision making methods often fail to make the best use of complex and incomplete information, especially in the field of cyber defence. To address these problems, this paper introduces a new mathematical tool, discrete multidimensional Z-numbers (MZs), for expressing and dealing with uncertainty and reliability in network defence decisions. In this paper, we first introduce the synthesis method of discrete multidimensional Z-numbers, which allows us to consider the uncertainty and reliability of multi-source information in network defence strategy. Then, the hidden probability of MZs is calculated by using the hidden probability model, and the multidimensional Z+−number is proposed, which improves the expressiveness of the model in dealing with information uncertainty. Based on MZs, we define a variety of utility functions and build a multi-attribute group decision framework for network defence around these functions. The MZs provides a novel perspective for analysing and designing response strategies, especially against highly adaptive and covert attacks. The method in this paper is verified by a case of network security assessment of an intelligent logistics company. The results show that this method can significantly improve the accuracy and efficiency of decision making, demonstrating its unique advantages and wide application potential in network defence. This integrated discrete multidimensional Z-number group decision method is not only suitable for intelligent logistics decision making, but also provides an efficient decision support tool for network defence, which helps to improve the intelligence and adaptability of network security defence.

We prove that the Legendre coefficients associated with a function f(x) can be represented as the Fourier coefficients of a suitable Abel-type transform of the function itself. Thus, the computation of N Legendre coefficients can be performed efficiently in O(NlogN) operations by means of a single Fast Fourier Transform of the Abel-type transform of f(x). We also show how the symmetries associated with the Abel-type transform can be exploited to further reduce the computational complexity. The dual problem of calculating the sum of Legendre expansions is also considered. We prove that a Legendre series can be written as the Abel transform of a suitable Fourier series. This fact allows us to state an efficient algorithm for the evaluation of Legendre expansions. Finally, numerical tests are presented to exemplify and confirm the theoretical results.

In this work we present a novel steganography algorithm, based on Fractal Transform domain, combined with the principles of Cryptography. The proposed method uses a private key, which will indicate which segment of the image will be taken and encoded with the Fractal Transform, to then insert the secret bits in the transformed coefficients, thus generating the stego image.

The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its extensive use in optics, engineering, and signal processing. In the present work, we aimed to expand the fractional Hilbert transform to a space of generalized functions known as Boehmians. We introduce a new fractional convolution operator for the fractional Hilbert transform to prove a convolution theorem similar to the classical Hilbert transform and also to extend the fractional Hilbert transform to Boehmians. We also construct a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform.

In this paper we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, we analyze the analytical solution for a class of fractional models associated with Logistic model, Von Foerster model and Bertalanffy model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case.

This article presents a new methodology for modeling and analyzing discrete dynamic systems through the construction of inverse algebraic models, giving rise to the Theory of Inverse Discrete Dynamical Systems. Key concepts such as inverse modeling, structural analysis in inverse algebraic trees, and the establishment of topological equivalences for the analytical transfer of properties between the canonical system and its inverted counterpart are introduced. Central theorems on homeomorphic invariance and topological transport are demonstrated, consolidating the validity of transferring cardinal attributes between both equivalent dynamic representations. As a concrete application, an alternative proof of the historic Collatz Conjecture is presented through the rigorous construction of the associated inverse model and the analytical transfer of properties exhibited in the inverted tree structure. Furthermore, evidence has been presented on Conway's Conjecture in the Game of Life through meticulous inverse modeling in bounded cases, overcoming initial limitations of the theory in combinatorial explosions. Although a complete proof of the conjecture is not achieved, this positively expands the modeling and analysis capacity of the methodology. The impact of properly applying this novel theory to expand understanding and solve open problems in discrete dynamical systems seems vast and profound.

Hasselmann and coauthors proposed the discrete interaction approximation (DIA) as the best tool replacing the nonlinear evolution term in a numerical wind-wave model. Much later, Polnikov and Farina radically improved the original DIA by means of location all the interacting four wave vectors, used in the DIA configuration, exactly at the nodes of the numerical frequency-angular grid. This provides nearly two-times enhancing the speed of numerical calculation for the nonlinear evolution term in a wind-wave model. For this reason, the proposed version of the DIA was called as the fast DIA (FDIA). In this paper we demonstrate all details of the FDIA concept for several frequency-angular numerical grids of high resolution, with the aim of active implementation the FDIA in modern versions of world-wide used wind-wave models.

We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer multicomponent mass is broken into fixed number of fragments and calculate the combinatorial multiplicity of all distributions in the set. We define random fragmentation by the condition that the probability of distribution be proportional to its multiplicity and obtain the partition function and the mean distribution in closed form. We then introduce a functional that biases the probability of distribution to produce in a systematic manner fragment distributions that deviate to any arbitrary degree from the random case. We corroborate the results of the theory by Monte Carlo simulation and demonstrate examples in which components in sieve cuts of the fragment distribution undergo preferential mixing or segregation relative to the parent particle.

Breast cancer is the second cause of fatality among all cancers for women. Automatic classification of breast cancer lesions in mammograms is a challenging task due to the irregularity and complexity of the location, size, shape, and texture of these lesions. The intensity dissimilarity has been found between breast cancer tissues and normal tissues, when a multi-spectral anatomical mammographic screening scans have been done. In this work, two approaches have been evaluated to classify the breast tumor lesions. The first one is through Gabor wavelet features and the second one is Statistical features. Subsequently, support vector machine, Multilayer Perceptron and KNN classifiers have been used with computer based method for breast tumor classification.

The electroretinography (ERG) is a diagnostic test that measures the electrical activity of the retina in response to a light stimulus. This test has a long history and has been extensively studied among years. Dr. James Dewar recorded the first electroretinogram signal in 1873. Later he described signal analysis using 4 components, namely amplitude, and latency of a-wave and b-wave. Nowadays, the international electrophysiology community has established the standard for electroretinography in 2008. However, from the point of view of signal analysis, the major change did not happen. ERG analysis is still based on the 4 components evaluation. The article describes ERG database including the classification of signals by using advanced analysis of electroretinograms based on wavelet scalogram processing. To implement an extended analysis of the ERG, the parameters extracted from the wavelet scalogram of the signal were obtained using digital image processing and machine learning methods. The results of the study show that the proposed algorithm implements the classification of adult electroretinogram signals by 19% more accurately and pediatric signals by 20% more accurately than the classical algorithm. The promising use of ERG is differential diagnostics, which may also be used in preclinical toxicology and experimental modeling. The problem of developing methods for electrophysiological signals analysis in ophthalmology is associated with the complex morphological structure of electrophysiological signals components, due to the generation of retina cell electrical responses to light stimulus.

Differences are introduced as outputs of linear systems called differencers, being considered two classes: shift and scale-invariant. Several types are presented, namely: nabla and delta, bilateral, tempered, bilinear, stretching, and shrinking. Both continuous and discrete-time are described. ARMA type systems based on differencers are introduced and exemplified. In passing, the incorrectness of the usual delta difference is shown.

The basis of the discrete-impulse energy supply (DIES) concept is the efficient use of supplied energy. The references describe in detail the general principles of DIES, examine the energy and thermodynamic aspects and the main mechanisms of intensification that can be initiated on the basis of this principle. DIES mechanisms conveniently can be divided into hard and soft ones. The former should be used to stimulate hydromechanical processes, and the latter to accelerate the processes of phase heat and mass transfer, or for the purpose of intensive mixing of multicomponent media. The authors have studied the possibility of using DIES to intensify the hydromechanical processes, in particular emulsification of milk fat (homogenization of milk, preparation of spreads), processing of cream cheese masses. Objects of research were whole non-homogenized milk, fat emulsions, cream cheese mass. In order to evaluate the efficiency of milk homogenization the homogenization coefficient change was studied, which was determined by centrifugation method as the most affordable and accurate one. Emulsions were evaluated according to the degree of destabilization, resistance and dispersion of the fat phase. The rheological characteristics of cheese masses were evaluated by the effective viscosity change.

In this paper we show how to efficiently implement parallel discrete simulations on Multi-Core and GPU architectures through a real example of application: a cellular automata model of laser dynamics. We describe the techniques employed to build and optimize the implementations using OpenMP and CUDA frameworks. We have evaluated the performance on two different hardware platforms that represent different target market segments: high-end platforms for scientific computing, using an Intel Xeon Platinum 8259CL server with48cores and also an NVIDIA Tesla6V100 GPU, both running on Amazon Web Server (AWS) Cloud, and on a consumer-oriented platform, using an Intel Core i9 9900k CPU and an NVIDIA GeForce GTX 1050 TI GPU. Performance results are compared and analysed in detail. We show that excellent performance and scalability can be obtained in both platforms, and we extract some important issues that imply a performance degradation for them. We also found that current Multi-Core CPUs with large core numbers can bring a performance very near to that of GPUs, even similar in some cases.

This paper focuses on establishing boundedness properties and Parseval-Goldstein type relations for the generalized Mehler-Fock transform initially introduced by B. L. J. Braaksma and B. M. Meulenbeld (Compositio Math.,{18}(3):235-287, 1967. Also, we derive an inversion formula for this transform over Lebesgue spaces.

Abstract: The two-dimensional Fuzzy Transform was applied in image compression. The quality of the image compressed with this methodis better than that obtained using methods based on fuzzy relation equations and comparable with that of the JPEG method, with better compression execution times. In this paper we propose a variation of the method based on the two-dimensional Fuzzy Transform of image compression, in which the image is first partitioned into blocks and then each block is compressed with the higher compression rate. The advantage of this method consistsinto a greater compression of the image, guaranteeing a high quality of the reconstructed image.The results show that our method is better than the Fuzzy Transform method improving also the quality of the reconstructed image.

In this paper, we introduce two discrete degenerate random variables, namely the degenerate binomial and degenerate Poisson random variables. We deduce the expectations of the degenerate binomial random variables. We compute the generating function of the moments of the degenerate Poisson random variables, which leads us to define the new type degenerate Bell polynomials, and hence obtain explicit expressions for the moments of those random variables in terms of such polynomials. We also get the variances of the degenerate Poisson random variables. Finally, we illustrate two examples of the degenerate Poisson random variables.

Let be a power function $f_{r,M}(s)$ defined for every $s$ within the finite set $M$ as follows $$f_{r,M}(s)= \begin{cases} s^r, \ &s\in M,\\ 0, \ &\mathrm{otherwise}. \end{cases} $$ Let a discrete convolution of $f_{r,M}(s)$ be denoted as follows $\mathrm{Conv}_{r,M}[n]=(f_{r,M}*f_{r,M})[n]$. Let a real coefficients $A_{m,j}$ be given by the following recurrence $$ A_{m,j} = \begin{cases} 0, & \mathrm{if } \ j<0 \ \mathrm{or } \ j>m, \\ (2j+1)\binom{2j}{j} \sum_{d=2j+1}^{m} A_{m,d} \binom{d}{2j+1} \frac{(-1)^{d-1}}{d-j} B_{2d-2j}, & \mathrm{if } \ 0 \leq j < m, \\ (2j+1)\binom{2j}{j}, & \mathrm{if } \ j=m. \end{cases} $$ In this paper we show that for every $n>0$ the following odd-power identities involving coefficients $A_{m,j}$ and convolution transform $\mathrm{Conv}_{r,M}[n]$ hold $$ \begin{split} n^{2m+1}+1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{N}}[n],\\ n^{2m+1}-1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{Z}_{>0}}[n],\\ n^{2m+1}&=\sum_{r=0}^{m}A_{m,r}\sum_{k=1}^{n} k^r(n-k)^r\\ &=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1} k^r(n-k)^r. \end{split} $$

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals.

The purpose of this work is to study the mechanism of running-in during the friction and to determine the informative parameters characterizing the degree of its completion. During the friction, contact interaction of rough surfaces causes various wave phenomena covering a wide range of frequencies, the subsequent frequency analysis can provide information about the sizes of wave sources and thereby clarify the mechanism of interaction between surface roughness. The using of the wavelet transform for processing of the signals of audible acoustic emission made it possible to determine the beginning and the end of the change in the frequency ranges of the interaction of roughness. The code developed by the authors was used to analyze the acoustic emission signals by using wavelet energy and entropy criteria. The mother wavelet was chosen by carefully evaluating the effectiveness of 54 preliminary candidates for the mother wavelet from 7 wavelet families, ac-cording to three criteria: 1) maximum wavelet energy; 2) Shannon entropy minimum; 3) maximum energy-to-Shannon entropy ratio.

Advances in machine learning (ML) and artificial intelligence (AI) are transforming the way we treat patients in ways not even imagined a few years ago. Cancer research is at the forefront of this movement. Infertility, though not a life-threatening condition, affects around 15% of couples trying for a pregnancy. Increasing availability of large datasets from various sources creates an opportunity to introduce ML and AI into infertility prevention and treatment. At present in the field of assisted reproduction, very little is done in order to prevent infertility from arising, with the main focus put on treatment when often advanced maternal age and low ovarian reserve make it very difficult to conceive. A shift from this disease-centric model to a health centric model in infertility is already taking place with more emphasis on the patient as an active participator in the process. Poor quality and incomplete data as well as biological variability remain the main limitations in the widespread and reliable implementation of AI in the field of reproductive medicine. That said, one of the areas where this technology managed to find a foothold is identification of developmentally competent embryos. More work is required however to learn about ways to improve natural conception, the detection and diagnosis of infertility, and improve assisted reproduction treatments (ART) and ultimately, develop clinically useful algorithms able to adjust treatment regimens in order to assure a successful outcome of either fertility preservation or infertility treatment. Progress in genomics, digital technologies and advances in integrative biology has had a tremendousimpact on research and clinical medicine. With the rise of ‘big data’, artificial intelligence, and the advances in molecular profiling, there is an enormous potential to transform not only scientific research progress, but also clinical decision making towards predictive, preventive, and personalized medicine. In the field of reproductive health, there is now an exciting opportunity to leverage these technologies and develop more sophisticated approaches to diagnose and treat infertility disorders. In this review, we present a comprehensive analysis and interpretation of different innovation forces that are driving the emergence of a system approach to the infertility sector. Here we discuss recent influential work and explore the limitations of the use of Machine Learning models in this rapidly developing area.

We have presented a controllable and human-readable polynomial neural network (CR-PNN) that is the first human-readable neural network. One can imagine its influence on system identification. Subsequently, we developed a relation spectrum in a medical application, which is likely to stand alongside the Fourier spectrum. However, the system analysis methodology is incomplete in contrast to signal processing methodology. Here, we presented the system filters for the first time. In this paper, we used the simulation system to verify the availability of the system analysis methodology. The system analysis methodology showed great properties in system identification and filter. The contribution of this paper is the system analysis methodology: transform method (CR-PNN), relation spectrum, and system filter design.

This article deals with the conformable double Laplace transforms and their some properties with examples and also the existence Condition for the conformable double Laplace transform is studied. Finally, in order to obtain the solution of nonlinear fractional problems, we present a modified conformable double Laplace that we call conformable double Laplace decomposition methods (CDLDM). Then, we apply it to solve, Regular and singular conformable fractional coupled burgers equation illustrate the effectiveness of our method some examples are given.

Although Shannon mutual information has been widely used, its effective calculation is often difficult for many practical problems, including those in neural population coding. Asymptotic formulas based on Fisher information sometimes provide accurate approximations to the mutual information but this approach is restricted to continuous variables because the calculation of Fisher information requires derivatives with respect to the encoded variables. In this paper, we consider information-theoretic bounds and approximations of the mutual information based on Kullback--Leibler divergence and Rényi divergence. We propose several information metrics to approximate Shannon mutual information in the context of neural population coding. While our asymptotic formulas all work for discrete variables, one of them has consistent performance and high accuracy regardless of whether the encoded variables are discrete or continuous. We performed numerical simulations and confirmed that our approximation formulas were highly accurate for approximating the mutual information between the stimuli and the responses of a large neural population. These approximation formulas may potentially bring convenience to the applications of information theory to many practical and theoretical problems.

In this research article we solve the problem of synchronization and anti-synchronization of chaotic systems described by discrete and time-delayed variable fractional order differential equations. To guarantee the synchronization and anti-synchronization of these systems, we use the well-known PID control theory and the Lyapunov-Krasovskii stability theory for discrete systems of variable fractional order.We illustrate the results obtained through simulation with examples, in which it can be seen that our results are satisfactory, thus achieving synchronization and anti-synchronization of chaotic systems of variable fractional order with discrete time delay.

In this paper, a technique of image watermarking using multiple images as watermarks is presented. The technique is based on transform domain functions including discrete wavelet transform (DWT), discrete cosine transform (DCT) and singular value decomposition (SVD) with an image as the host signal i.e. the watermarks will be used as proofs of the authenticity of the host image. The technique is executed by performing multilevel DWT followed by applying DCT and SVD to both the host and watermark. Multiple watermarks are used for the insurance of better security level. The scheme is immune to common image processing operations & some attacks and exhibits PSNR of 108.3781dB, normalized cross correlation (NCC) over 0.99 and normalized correlation (NC) over 0.99.

Abstract Using the Pauli algebra version of the Dirac equation six symmetric but non-equivalent copies of the QED Lagrangian are constructed. The group of symmetries is the group of outer automorphisms of the finite quaternion group, isomorphic to the group of permutations on three letters. Solutions to corresponding Dirac equations form six sectors of fermion fields. The intersections between different sectors can only contain massless fields, implying that the sectors are distinct. If the symmetry could be extended to the full Standard Model Lagrangian it would predict the ratio of dark matter to ordinary matter of 5 : 1 which is close to the ratio of 5.2 : 1 according to current observations.

Discrete distribution estimation is a fundamental statistical tool, which is widely used to perform data analysis tasks in various applications involving sensitive personal information. Due to privacy concerns, individuals may not always provide their raw information, which leads to unpredictable biases in the final results of estimated distribution. Local Differential Privacy (LDP) is an advanced technique for privacy protection of discrete distribution estimation. Currently, typical LDP mechanisms provide same protection for all items in the domain, which imposes unnecessary perturbation on less sensitive items and thus degrades the utility of final results. Although, several recent works try to alleviate this problem, the utility can be further improved. In this paper, we propose a novel notion called Item-Oriented Personalized LDP (IPLDP), which independently perturbs different items with different privacy budgets to achieve personalized privacy protection. Furthermore, to satisfy IPLDP, we propose the Item-Oriented Personalized Randomized Response (IPRR) based on the observation that the sensitivity of data shows an inverse relationship with the population size of respective individuals. Theoretical analysis and experimental results demonstrate that our method can provide fine-grained privacy protection and improve data utility simultaneously.

A stochastic dominance (SD) relation can be defined by two different perspectives: One from the view of distributions, and the other one from the view of expected utilities. In early days, Fishburn investigated SD from the view of distributions and we refer this perspective as Fishburn’s SD. One of his many results was the development of fractional-order SD for continuous distributions. However, discrete fractional-order SD may not be generalized directly since some properties of fractional calculus do not have a discrete counterpart. In this paper, we develop a discrete analogue of fractional-order SD from the view of distributions. We generalize the order of SD by Lizama’s fractional delta operator, show the preservation of SD hierarchy, and formulate the utility classes that are congruent with our SD relations. This work brings a message that some results of discrete SD cannot be generalized directly from continuous SD. We characterize the difference between discrete and continuous fractional-order SD, as well as the way to handle them.

The author demonstrates a stable Lagrangian solid modeling method, tracking the interactions of solid mass particles rather than using a meshed grid. This numerical method avoids the problem of tensile instability often seen with smooth particle applied mechanics by having the solid particles apply stresses expected with Hooke's law, as opposed to using a smoothing function for neighboring solid particles. This method has been tested successfully with a bar in tension, compression, and shear, as well as a disk compressed into a flat plate, and the numerical model consistently matched the analytical Hooke's law as well as Hertz contact theory for all examples. The solid modeling numerical method was then built into a 2-D model of a pressure vessel, which was tested with liquid water particles under pressure and simulated with smoothed particle hydrodynamics. This simulation was stable, and demonstrated the feasibility of Lagrangian specification modeling for fluid–solid interactions.

3D printing facilitates the straightforward construction of microchannels with complex three-dimensional architectures. Here, we demonstrate 3D-printed modular mixing components that operate on the basis of splitting and recombining fluid streams to decrease interstream diffusion length. These are compared to helical mixers that operate on the principle of chaotic advection.

In this paper we introduce a computational model for the simulation of hypervelocity impact (HVI) phenomena which is based on the Discrete Element Method (DEM). Our paper constitutes the first application of DEM to the modeling and simulating of impact events for velocities beyond 5 kms⁻¹. We present here the results of a systematic numerical study on HVI of solids. For modeling the solids, we use discrete spherical particles that interact with each other via potentials. In our numerical investigations we are particularly interested in the dynamics of material fragmentation upon impact. We model a typical HVI experiment configuration where a sphere strikes a thin plate and investigate the properties of the resulting debris cloud. We provide a quantitative computational analysis of the resulting debris cloud caused by impact and a comprehensive parameter study by varying key parameters of our model. We compare our findings from the simulations with recent HVI experiments performed at our institute. Our findings are that the DEM method leads to very stable, energy–conserving simulations of HVI scenarios that map the experimental setup where a sphere strikes a thin plate at hypervelocity speed. Our chosen interaction model works particularly well in the velocity range where the local stresses caused by impact shock waves markedly exceed the ultimate material strength.

A system of wavelet collocation upwind schemes is constructed for solving hyperbolic conserva-tion laws based on a class of interpolation wavelets. The bias magnitude and symmetry factor are defined to depict the asymmetry of the adopted scaling basis function in wavelet theory. Effects of characteristics of the scaling functions on the schemes are explored based on numerical tests and Fourier analysis. The numerical results reveal that the stability of the constructed scheme is af-fected by the smoothness order, N, and the asymmetry of the scaling function. The dissipation analysis suggests that schemes with Neven have negative dissipation coefficients, leading to unstable behaviors. Only scaling functions with Nodd and bias magnitude of 1 can be used to construct stable upwind schemes due to the non-negative dissipation coefficients. Resolution of the wavelet scheme tends to the spectral resolution as the order of accuracy of the scheme in-creases.

The present paper is concerned with the study of the quality of life index. Such an index has become an important index for measuring the well-being of individuals. However, the quality of life index is always a subjective, intangible, and often hard to quantify with precision due to the lack of quantitative models dealing with. The main goal of the present paper is thus to propose a mathematical, quantitative model for the measurement of a quality of life index. The main novelty is firstly the construction of a wavelet dynamic multiscale model to quantify and investigate the effect of time scale on the quality of life index measuring. The proposed procedure is acted empirically on a sample corresponding to Saudi Arabia as a case of study during the period from 2003 to 2020 as part of the 2030-vision plan. Saudi Arabia has implemented the so-called 2030-vision plan where the quality of life improvement is one of the main goals to be attempted. The findings show that wavelets are capable to localize the time-wise behavior of the index contrarily to classical studies which estimate a global view of the index. Moreover, the study shows the link between the quality of life behavior and many other indices.

Despite the increasing popularity of liquid chromatography-mass spectrometry (LC-MS)-based lipidomics, there is a lack of accepted and validated method for lipid extract quality and quantity assessment prior to LC-MS. Fourier-Transform Infrared Spectroscopy (FTIR) has been reported for quantification of pure lipids, however, the sample complexity and purity complex lipid extract quantification in lipidomics experiments could be impact quantification accuracy. Here, we report comprehensive assessment of the sample matrix on the accuracy of lipid quantification using Attenuated Total Reflectance Fourier-Transform Infrared Spectroscopy (ATR-FTIR). Pure lipids are characterized by CH-and C=O-stretching vibrations on FTIR, with quantitative range of 40–3000 ng and a limit of detection of 12 ng. Sample extraction method and local baseline subtraction during FTIR spectral processing significantly impact lipid quantification by CH-stretching. To facilitate sample quality screening, we developed the Lipid Quality (LiQ) score from a spectral library of common contaminants, using a ratio of peak heights between CH-stretching vibrations maxima and the collective vibrations from amide/amine, CH-stretching minima and sugar moieties. We evaluated LiQ score as a rapid sample quality control method by comparing to total LC-MS intensity of targeted lipidomics of 107 human plasma lipid extracts. Exclusion of poor-quality samples based on LiQ score improved the correlation between FTIR and LC-MS quantification. Finally, the uncertainty of absolute quantification by FTIR was estimated using a 795 ng SPLASH LipidoMix standard to be &lt;10%. In summary, this study identified key parameters for accurate FTIR-based quantification of complex lipid mixture, and developed a rapid workflow requiring only 1 µL of MS-ready sample and &lt; 5 minutes for routine lipidomics sample quality and quantity assessment.

Objective:The Glioma brain tumor detection and segmentation methods are proposed in this paper using machine learning approaches. Methods:The boundary edge pixels are detected using Kirsch’s edge detectors and then contrast adaptive histogram equalization method is applied on the edge detected pixels. Then, Ridgelet transform is applied on this enhanced brain image in order to obtain the Ridgelet multi resolution coefficients. Further, features are derived from the Ridgelet transformed coefficients and the features are optimized using Principal Component Analysis (PCA) method and these optimized features are classified into Glioma or non-Glioma brain images using Co-Active Adaptive Neuro Fuzzy Expert System (CANFES) classifier.Results:The proposed method with PCA and CANFES classification approach obtains 97.6% of se, 98.56% of sp, 98.73% of Acc, 98.85% of Pr, 98.11% of FPR and 98.185 of FNR, then the proposed Glioma brain tumor detection method using CANFES classification approach only.

In this paper we survey some recent results due to the author concerning various inequalities and approximations for the finite Hilbert transform of a function belonging to several classes of functions, such as: Lipschitzian, monotonic, convex or with the derivative of bounded variation or absolutely continuous. More accurate estimates in the case that the higher order derivatives are absolutely continuous, are also provided. Some quadrature rules with error bounds are derived. They can be used in the numerical integration of the finite Hilbert transform and, due to the explicit form of the error bounds, enable the user to predict a priory the accuracy.

The continuous advancements in healthcare technology have empowered the discovery, diagnosis, and prediction of diseases, revolutionizing the field. Artificial intelligence (AI) is expected to play a pivotal role in achieving the goals of precision medicine, particularly in disease prevention, detection, and personalized treatment. This study aims to determine the optimal combination of the mother wavelet and AI model for the analysis of pediatric electroretinogram (ERG) signals. The dataset, consisting of signals and corresponding diagnoses, undergoes Continuous Wavelet Transform (CWT) using commonly used wavelets to obtain a time-frequency representation. Wavelet images were used for the training of five widely used deep learning models: VGG-11, ResNet-50, DensNet-121, ResNext-50, and Vision Transformer, to evaluate their accuracy in classifying healthy and unhealthy patients. The findings demonstrate that the combination of Ricker Wavelet and Vision Transformer consistently yields the highest median accuracy values for ERG analysis, as evidenced by the upper and lower quartile values. The median balanced accuracy of the obtained combination of the three considered types of ERG signals in the article are 0.83, 0.85, and 0.88. However, other wavelet types also achieved high accuracy levels, indicating the importance of carefully selecting the mother wavelet for accurate classification. The study provides valuable insights into the effectiveness of different combinations of wavelets and models in classifying ERG wavelet scalograms.

Microgrids of varying size and applications are regarded as a key feature of modernizing the power system. The protection of those systems, however, has become a major challenge and a popular research topic for the reason that it involves greater complexity than traditional distribution systems. This paper addresses the issue through a novel approach which utilizes detailed analysis of current and voltage waveforms through windowed fast Fourier and wavelet transforms. The fault detection scheme involves bagged decision trees which use input features extracted from the signal processing stage and selected by correlation analysis. The technique was tested on a microgrid model developed using PSCAD/EMTDS, which is inspired from an operational microgrid in Goldwind Sc. Tech. Co. Ltd, in Beijing, China. The results showed great level of effectiveness to accurately identify faults from other non-fault disturbances, precisely locate the fault and trigger opening of the right circuit breaker/s under different operation modes, fault resistances and other system disturbances.

After proposing a natural metric for the space in which particles spin which implements the principle of maximum frequency, E=hf is generalised and its inverse Fourier transform is calculated. As a necessary result, a metric is found for the space in which quantum particles spin, hence the possibility of explanation of correlation of spacelike-separated particles is opened up.

This study advances the field of solar irradiance nowcasting by introducing a discrete-level classification approach, diverging from traditional continuous measurement methods. Grounded in a need for cost-effective and modular solutions, the research employs high-resolution computer vision coupled with a deep learning framework to predict Direct Normal Irradiance (DNI). By harnessing the capabilities of a Logitech C900 1080p HD camera and an NVIDIA Jetson module, the study achieves real-time data processing, pivotal for CSP systems' operational efficiency. The core of the methodology is the ResNet-50 convolutional neural network, refined via transfer learning on a bespoke dataset, culminating in a predictive accuracy of 85.78%. This discrete classification model contrasts with conventional, costly instruments like the MS-57, offering a novel and accessible alternative for DNI estimation. Such innovation not only demonstrates high predictive accuracy but also signifies a shift towards less resource-intensive and more adaptable solar energy forecasting tools, contributing a significant leap towards optimizing renewable energy systems.

By applying a running average (with a window-size= d), we could transform Discrete data to broad-range, Continuous values. When we have more than 2 columns and one of them is containing data about the tags of classification (Class Column), we could compare and sort the features (Non-class Columns) based on the R2 coefficient of the regression for running averages. The parameters tuning could help us to select the best features (the non-class columns which have the best correlation with the Class Column). “Window size” and “Ordering” could be tuned to achieve the goal. this optimization problem is hard and we need an Algorithm (or Heuristics) for simplifying this tuning. We demonstrate a novel heuristics, Called Simulated Distillation (SimulaD), which could help us to gain a somehow good results with this optimization problem.

Pit craters, and pit crater chains, are now recognised as being an important part of the surface morphology and structure of many planetary bodies, and are particularly remarkable on Mars. Pit craters do not possess the elevated rims, ejecta deposits, or other features that are typically associated with impact craters. They are thought to arise from the drainage/collapse of a relatively weak surficial material into an open (or widening) void in a much stronger material below. The creation of such voids has been suggested to be due to extensional fracturing/dilational faulting, shallow dike intrusion, lava tube collapse amongst other hypotheses. These craters have a very distinctive expression, often presenting funnel, cone, or bowl-shaped geometries. Analogue models of pit crater formation provide a map-view picture of their initiation and evolution but give little insight into their internal structure or geometry, but produce pits that typically have steep, nearly conical cross sections. Numerical modelling studies of their formation have been limited and have produced some quite interesting, but nonetheless puzzling, results whereby the simulated pit craters had generally convex (steepening downward) slope profiles with no distinct rim; quite unlike many of those observed on Earth or on Mars. To address these issues, I present here a high-resolution, 2D discrete element model of weak cover (regolith) collapse into either a static or a widening underlying void. I use frictional and frictional-cohesive discrete elements to represent a range of probable cover rheologies. Under Martian gravitational conditions, frictional-cohesive and frictional materials produce cone, bowl and scoop-shaped pit craters. For a given cover thickness, the specific crater shape depends on the amount of underlying void space created for drainage. When void space is small relative to cover thickness, craters have bowl or scoop-shaped geometries. In contrast, when void space is large relative to cover thickness, craters have cone-shaped geometries with essentially planar (nearing angle of repose) slope profiles. Frictional-cohesive materials exhibit more distinct rims than simple frictional materials and thus may reveal some stratigraphic layering on the pit crater walls. In the limit, when drainage from the overlying cover is insufficient to fill the underlying void, ´skylights´ into the deeper structure are created. Implications of these results for the interpretation of pit craters on Earth, Mars, other planets and moons are discussed.

In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.