Two-dimensional Representation Research Articles

Formulation of Special Pythagorean Triangles through Integer Solutions of the Hyperbola 𝑦𝟐 = (𝑘𝟐 + 𝟐𝑘)𝑥𝟐 + 𝟏

Objectives: The objective of this research paper is to formulate Pythagorean triangles, each of which satisfies the relation: Hypotenuse = (𝑘 + 1) times the leg with even values added with unity, through employing the integer solutions to the Hyperbola 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1. Methods: The sides of the Pythagorean triangle satisfying the requirement are obtained by suitably choosing its generators consisting of the integer solutions to the considered hyperbola. Findings : There are plenty of Pythagorean triangles satisfying the given characterization for each value of k > 0 in the binary quadratic equation given by 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1.The characterizations of special pattern of the Pythagorean triangle for 𝑘 = 2is illustrated. Novelty: From the integer solutions of a binary quadratic equation, namely, a hyperbola (Two-dimensional geometrical representation), it is possible to obtain integer solutions to a ternary quadratic equation, namely, Pythagorean triangle (Threedimensional geometrical representation). Keywords: Binary quadratic equation; Ternary quadratic equation; Hyperbola; Pythagorean triangle; Integer solutions

A bi-consolidating model for joint relational triple extraction

Current methods to extract relational triples directly make a prediction based on a possible entity pair in a raw sentence without depending on entity recognition. The task suffers from a serious semantic overlapping problem, in which several relation triples may share one or two entities in a sentence. In this paper, based on a two-dimensional sentence representation, a bi-consolidating model is proposed to address this problem by simultaneously reinforcing the local and global semantic features relevant to a relation triple. This model consists of a local consolidation component and a global consolidation component. The first component uses a pixel difference convolution to enhance semantic information of a possible triple representation from adjacent regions and mitigate noise in neighbouring neighbours. The second component strengthens the triple representation based a channel attention and a spatial attention, which has the advantage to learn remote semantic dependencies in a sentence. They are helpful to improve the performance of both entity identification and relation type classification in relation triple extraction. After evaluated on several publish datasets, the bi-consolidating model achieves competitive performance. Analytical experiments demonstrate the effectiveness of our model for relational triple extraction and give motivation for other natural language processing tasks.

Optoelectronic and photoacoustic waves in rotating excited semiconductor subjected to thermal shock

In this study, the impact of rotation on the transmission of optical acoustic waves, arising from the movement of the optical carrier within an elastic thermal environment, is examined through theoretical analysis. The theoretical framework incorporates the interaction between acoustic waves and thermomechanics to formulate governing equations specific to a semiconductor medium. The fundamental equations of the physical model, influenced by photothermal and thermoelastic principles, are deduced mathematically while considering the presence of rotation. The model studied the medium under a thermal shock wave from thermal stress resulting from light-induced temperature elevation that is not fixed but decaying over time. The mathematical model is solvable using the normal mode method. The model’s numerical solutions provide access to all physical fields within the physical domain, encompassing displacement, temperature, acoustic pressure, mechanical stress, and carrier density diffusion. Utilizing the harmonic wave method, a two-dimensional graphical representation of the rotation parameter with and without the influence of the decay parameter is obtained. The theoretical analysis involves comparison, analysis, and discussion of the effects of these parameters investigated.

Intelligent Fault Diagnosis of Planetary Gearbox Across Conditions Based on Subdomain Distribution Adversarial Adaptation.

Sensory data are the basis for the intelligent health state awareness of planetary gearboxes, which are the critical components of electromechanical systems. Despite the advantages of intelligent diagnostic techniques for detecting intricate fault patterns and improving diagnostic speed, challenges still persist, which include the limited availability of fault data, the lack of labeling information and the discrepancies in features across different signals. Targeting this issue, a subdomain distribution adversarial adaptation diagnosis method (SDAA) is proposed for faults diagnosis of planetary gearboxes across different conditions. Firstly, nonstationary vibration signals are converted into a two-dimensional time-frequency representation to extract intrinsic information and avoid frequency overlapping. Secondly, an adversarial training mechanism is designed to evaluate subclass feature distribution differences between the source and target domain. A conditional distribution adaptation is employed to account for correlations among data from different subclasses. Finally, the proposed method is validated through experiments on planetary gearboxes, and the results demonstrate that SDAA can effectively diagnose faults under crossing conditions with an accuracy of 96.7% in diagnosing gear faults and 95.2% in diagnosing planet bearing faults. It outperforms other methods in both accuracy and model robustness. This confirms that this approach can refine domain-invariant information for transfer learning with less information loss from the sub-class level of fault data instead of the overall class level.

Image representation of vibration signals and its application on fault diagnosis of rotating machinery with convolutional neural network: A benchmark study

Intelligent fault diagnosis methods on the basis of two-dimensional (2D) image representation of vibration signals (IRVS) and the convolutional neural network (CNN) have been extensively applied in rotating machinery. However, as a large number of IRVS methods are being used, their performance has not been fairly and comprehensively evaluated, which leads to difficulties for researchers to make choices. To address this issue, this paper conducted a benchmark study aimed at comparing the comprehensive performance of different IRVS methods, including computation time, fault diagnosis accuracy, and noise resistance performance. Firstly, the IRVS methods and the fault diagnosis method combining IRVS and CNN were summarized. Then, the general process of the IRVS-CNN method was proposed. Finally, 17 types of IRVS methods were selected, and the performance of the IRVS method was compared based on three datasets and two classic CNN models. The results indicate that the computing time taken to generate 2D images using unthresholded recurrence plot (UTRP), and frequency-domain unthresholded recurrence plot (FDUTRP) methods is relatively longer. The vast majority of frequency-domain methods and time-frequency domain methods can achieve or approach a high accuracy rate in all experiments, demonstrating their outstanding performance. Frequency-domain color vibration image (FDCVI), FDUTRP, and continuous wavelet transform (CWT) have good noise resistance. These results provide a reference for future researchers in selecting IRVS methods.

A two-phase adaptive large neighborhood search algorithm for the electric location routing problem with range anxiety

In response to environmental pollution and global warming, electric vehicles (EVs) have been widely applied in supply chain, and lots of researches have been focused on the electric location routing problem (ELRP) to optimize the location of battery-swapping stations (BSSs). However, few studies have simultaneously considered the EV’s collaborative behavior and range anxiety in ELRP. Therefore, this study focuses on formulating an ELRP with battery-swapping stations considering both collaborative behavior and range anxiety. The model takes into account the collaboration between the supply and demand sides to share supply chain resources effectively. The detour probability function is employed to handle the uncertainty caused by drivers’ range anxiety during the trip, which can be evaluated through a proposed range anxiety function. In addition, a new two-dimensional matrix-based solution representation is proposed to explore ELRP optimal solutions intuitively and efficiently. To solve the proposed model effectively, a two-phase adaptive large neighborhood search (TALNS) algorithm that integrates the adaptive large neighborhood search (ALNS) algorithm and the extended binary particle swarm optimization (EBPSO) algorithm is proposed. In the EBPSO algorithm, a new position update mechanism and local search strategy are used to strengthen the local search ability. In the ALNS algorithm, multiple destroy and repair operators are developed for the proposed model. Further, to validate the effectiveness and performance of the proposed algorithm, comparison experiments with the Gurobi optimizer and other baseline heuristic algorithms are conducted.

Integrated optical wave analyzer using the discrete fractional Fourier transform

In this work, we detail a proposal for optical signals to be represented and analyzed in phase-space. Our proposal aims to integrate a series of operations in waveguide realization, as a compact all-together platform that takes an initial wavefield and returns a two-dimensional representation of the information. We show, step by step, that the quantum harmonic oscillator can be considered as a propagator of initial fields, and when a discretized version is implemented, the fractional order Fourier transform emerges. This last is crucial, since the Wigner-Radon theorem is used to establish a path between the propagated wavefield and the phase-space representation. We show by example that this integration offers a direct and efficient method for characterizing optical signals by reconstructing their Wigner phase-space in the scope of integrated optics.

Optical Coherence Tomography in Myocardial Infarction Management: Enhancing Precision in Percutaneous Coronary Intervention

In acute myocardial infarction (AMI), the urgency of coronary revascularization through percutaneous coronary intervention (PCI) is paramount, offering notable advantages over pharmacologic treatment. However, the persistent risk of adverse events, including recurrent AMI and heart failure post-revascularization, underscores the necessity for enhanced strategies in managing coronary artery disease. Traditional angiography, while widely employed, presents significant limitations by providing only two-dimensional representations of complex three-dimensional vascular structures, hampering the accurate assessment of plaque characteristics and stenosis severity. Intravascular imaging, specifically optical coherence tomography (OCT), significantly addresses these limitations with superior spatial resolution compared to intravascular ultrasound (IVUS). Within the context of AMI, OCT serves dual purposes: as a diagnostic tool to accurately identify culprit lesions in ambiguous cases and as a guide for optimizing PCI procedures. Its capacity to differentiate between various mechanisms of acute coronary syndrome, such as plaque rupture and spontaneous coronary dissection, enhances its diagnostic potential. Furthermore, OCT facilitates precise lesion preparation, optimal stent sizing, and confirms stent deployment efficacy. Recent meta-analyses indicate that OCT-guided PCI markedly improves safety and efficacy in revascularization, subsequently decreasing the risks of mortality and complications. This review emphasizes the critical role of OCT in refining patient-specific therapeutic approaches, aligning with the principles of precision medicine to enhance clinical outcomes for individuals experiencing AMI.

Distance-based loss function for deep feature space learning of convolutional neural networks

Convolutional Neural Networks (CNNs) have been on the forefront of neural network research in recent years. Their breakthrough performance in fields such as image classification has gathered efforts in the development of new CNN-based architectures, but recently more attention has been directed to the study of new loss functions. Softmax loss remains the most popular loss function due mainly to its efficiency in class separation, but the function is unsatisfactory in terms of intra-class compactness. While some studies have addressed this problem, most solutions attempt to refine softmax loss or combine it with other approaches. We present a novel loss function based on distance matrices (LDMAT), softmax independent, that maximizes interclass distance and minimizes intraclass distance. The loss function operates directly on deep features, allowing their use on arbitrary classifiers. LDMAT minimizes the distance between two distance matrices, one constructed with the model’s deep features and the other calculated from the labels. The use of a distance matrix in the loss function allows a two-dimensional representation of features and imposes a fixed distance between classes, while improving intra-class compactness. A regularization method applied to the distance matrix of labels is also presented, that allows a degree of relaxation of the solution and leads to a better spreading of features in the separation space. Efficient feature extraction was observed on datasets such as MNIST, CIFAR10 and CIFAR100.

Soliton solutions of cubic quintic septimal nonlinear Schrödinger wave equation with conformable derivative by two distinct algorithms

This paper investigates the cubic-quintic-septimal nonlinear Schrödinger wave equation with a conformable derivative, which governs the evolution of light beams in a weak nonlocal medium. The analysis utilizes the Kudryashov method and the enhanced modified tanh expansion method. By utilizing these analytical integration schemes, various optical wave solutions are derived within the present conformable model. The paper demonstrates the significance of these optical soliton solutions by illustrating different soliton solutions, including kink-type, bell-shaped, singular, dark, and wave soliton solutions, depicted via contour, three-dimensional, and two-dimensional representations. Moreover, it is crucial to emphasize the importance of analyzing the cubic-quintic-septimal nonlinear Schrödinger wave equation, which finds utility across a spectrum of fields including optics, quantum mechanics, and the study of nonlinear wave propagation. Moving forward, these approaches hold promise for investigating diverse sets of differential equations within multiple domains of applied sciences. This governing equation also has numerous applications in nonlinear optics, such as describing the propagation of laser beams through materials with nonlinear optical properties. The inclusion of these nonlinearities illustrates the interaction and behavior of light beams in weakly non-local media.

Breaking the CO2 Gridlock: Can Renewables Lead the Way for the OECD?

The use of low-carbon energy in power grids is essential for minimizing negative effects on the environment. Energy consumption causes environmental damage to the OECD’s economy. This study aims to investigate the effect of energy consumption, population, and GDP on CO2 emissions using panel data from 17 OECD countries over the period 2000–2023. We use regression approaches, such as partial least squares and principal components, to study the effects of GDP, urban and total population, oil and nuclear use, renewable energy, and industrialization on CO2 emissions. The regression process in this study reduces the data to a two-dimensional representation using a stochastic model and estimation techniques. The findings of this empirical investigation indicate that the United States, Canada, France, Germany, Italy, Korea, Mexico, and the United Kingdom exhibit higher levels of primary energy consumption in comparison to value-added sectors, renewable–geothermal energy, and nuclear energy. We determined the effects of CO2 emissions, GDP, and energy consumption by considering these as the most significant elements. This has made it possible to reduce CO2 emissions by focusing one’s attention and energy on the development of novel technologies, the use of renewable energy sources, and the execution of strategic plans. Attracting increasing attention are technological shifts that deliver enormous quantities of clean energy to combat climate change. Findings from this study can help environmentalists and policymakers better understand the role of structural change and energy consumption processes in the globalization process.

Teaching protein structure and function through molecular visualization.

The function of proteins is governed by their three-dimensional structure. This structure is determined by the chemical characteristics and atomic interactions of amino acids. Students of biochemistry, with a particular focus on protein chemistry, benefit from looking at protein structures and understanding how proteins are built and fold. Due to their three-dimensional nature, static two-dimensional representations in textbooks can be limiting to student learning. Here, we developed a series of tutorials that introduce students to molecular graphics software. The students are challenged to apply the software to look at proteins and to get a deeper understanding of how amino acid properties are linked to structure. We also familiarize students with some of the latest tools in computational structural biology. Students performed the tutorials with visual enthusiasm and reported general satisfaction in being able to visualize theoretical concepts learned during lectures. We further stimulated student engagement by allowing space for self-exploration. We share the tutorial instructions for other teachers to build on them, and we also offer suggestions for further improvement based on student feedback. In summary, we present a series of tutorials aimed at students of an advanced course in protein biochemistry to enable them to explore the universe of protein structures and how those relate to function.

Three-dimensional grid-free sound source localization method based on deep learning

Sound source localization (SSL) technology is a popular method for identifying the locations of noise sources, which serves as a prerequisite for noise control. Deep learning, as a data-driven tool, shows broad perspectives in the field of SSL with its powerful nonlinear fitting ability. The existing deep learning-based SSL methods only provide a two-dimensional (2D) representation of the sound source location and cannot obtain the specific coordinates of the sound source in three-dimensional (3D) space. Although traditional beamforming methods can be directly generalized to 3D scenes in principle, they suffer from the limitations of insufficient vertical resolution and high computational cost. Therefore, a 3D grid-free SSL method (3DGF) informed by deep learning is suggested in this study to enhance the accuracy and computational efficiency of 3D localization. First, the number of data channels is compressed to respect limited memory resources during the training process. Subsequently, a dense convolutional neural network (DenseNet) model is utilized to obtain the 3D spatial coordinates of the sound source using the processed 3D beamforming map as input. Since the coordinates are continuous and are not constrained by the grid of the beamforming map, the grid-free strategy presents more accurate localization results. Then, the effects of the volume of training data and the compression ratio are analyzed, respectively, in simulation, and the localization performance with different signal-to-noise ratios (SNRs) is also tested. Finally, by comparing 3DGF with DAMAS, both simulation and experimental results demonstrate that 3DGF improves the accuracy and efficacy of 3D localization. Meanwhile, its satisfactory generalization ability and robustness against noise highlight its potential for practical applications.

Translation functors for locally analytic representations

Let [Formula: see text] be a [Formula: see text]-adic Lie group with reductive Lie algebra [Formula: see text]. In analogy to the translation functors introduced by Bernstein and Gelfand on the categories of [Formula: see text]-modules, we consider similarly defined functors on the category of modules over the locally analytic distribution algebra [Formula: see text] on which the center of [Formula: see text] acts locally finitely. These functors induce equivalences between certain subcategories of the latter category. Furthermore, these translation functors are naturally related to those on category [Formula: see text] via the functors from category [Formula: see text] to the category of coadmissible modules. We also investigate the effect of the translation functors on locally analytic representations [Formula: see text]la associated by the [Formula: see text]-adic Langlands correspondence for [Formula: see text] to two-dimensional Galois representations [Formula: see text].

Three-phase-lag analysis of transversely isotropic double porous thermoelastic waves with liquid medium

This manuscript presents a mathematical framework for a double porous, thermoelastic solid half-space with transverse isotropy in contact with an inviscid liquid half-space. The analysis takes into account the three-phase-lag model within thermoelasticity theory. To gain deeper insights, we formulate the governing equations in a two-dimensional representation and subsequently employ normal mode analysis for their rigorous solution. The governing equations are influenced by both types of voids, anisotropy, thermal effect, and inviscid liquid. It has been discovered that thermoelastic half-space exhibits five coupled longitudinal waves, and in the liquid half-space, there is one mechanical wave. Secular equations are obtained by using thermal and mechanical conditions, resulting in a compact representation of phase velocity, specific loss, penetrating depth, and attenuation coefficient. Analytical presentations have been made to showcase the effects of anisotropy, voids, and liquid on various wave parameters. These effects are compared by illustrating the results graphically using the MATLAB software, allowing for a visual understanding of the comparative outcomes. This study has practical applications in engineering, materials science, geophysics, and environmental studies, contributing to advancements in various industries and scientific fields.

Abundant solitary wave solutions to the new extended (3+1)-dimensional nonlinear evolution equation arising in fluid dynamics

Nonlinear evolution equations appear in various branches of physics, including fluid dynamics, solid mechanics, quantum mechanics, and other fields. They are essential for describing systems where interactions and nonlinearity play a significant role, providing a more accurate representation of real-world phenomena than linear equations in certain cases. The study of nonlinear evolution equations involves exploring their solutions, stability properties, and the behavior of the systems they describe over time. In this study, we provide plenty of analytical solutions to a very recently proposed water wave model using the modified generalized exponential rational function, modified extended tanh-function, and the [Formula: see text]-expansion methods. To examine the dynamic characteristics of these results, we displayed three-dimensional (3D), contour, and two-dimensional (2D) visual representations of specific solutions. The outcomes reveal a multitude of novel solutions, emphasizing the robustness and effectiveness of the applied methodologies. It is important to note that all the solutions derived in this study are original and have not been previously documented in the existing literature. These novel solutions hold significant value for researchers in the realms of fluids and plasma physics, providing insights into the dynamics of nonlinear waves within various physical systems. Furthermore, this research contributes to a deeper understanding of the nature of nonlinear waves prevalent in seas and oceans, offering valuable knowledge for the broader study of such phenomena.

A deep neural network approach to heart murmur detection using spectrogram and peak interval features

Congenital heart disease affects about 1% of newborns, posing risks like heart failure and mortality. Developing countries often lack resources for diagnosis and treatment. The George B. Moody PhysioNet Challenge 2022 aims to develop systems for detecting murmurs and clinical outcome events using phonocardiogram (PCG) data, offering a cost-effective method for diagnosing cardiac diseases without invasive procedures. We proposed a deep learning model for this task and achieved the 5th rank out of 40 teams in both Track 1 (murmur detection) and Track 2 (clinical outcome prediction). This paper describes our methods and additional experiments. To extract features from the PCG, we employed techniques including Constant Q Transform (CQT) and Mel-scaled spectrogram (Mel-spectrogram) to generate two-dimensional representations in the frequency–time domain. Additionally, we extracted the Peak Interval (PI) feature, which measures the distance between peaks in the PCG data. This feature is useful because the peak intervals should be shorter in PCG recordings with murmurs. We also considered the sequence and mean of PI as additional features. Our proposed system, titled ‘Phonocardiogram-based Heart murmur Detection using Spectrogram and PI features (SpectroHeart),’ employs the Mel-spectrogram and PI features to detect heart murmurs and assess clinical outcomes. We believe that our deep learning-based system has great potential for automatically detecting heart signals from PCG.

Disentangling cobionts and contamination in long-read genomic data using sequence composition.

The recent acceleration in genome sequencing targeting previously unexplored parts of the tree of life presents computational challenges. Samples collected from the wild often contain sequences from several organisms, including the target, its cobionts, and contaminants. Effective methods are therefore needed to separate sequences. Though advances in sequencing technology make this task easier, it remains difficult to taxonomically assign sequences from eukaryotic taxa that are not well represented in databases. Therefore, reference-based methods alone are insufficient. Here, I examine how we can take advantage of differences in sequence composition between organisms to identify symbionts, parasites, and contaminants in samples, with minimal reliance on reference data. To this end, I explore data from the Darwin Tree of Life project, including hundreds of high-quality HiFi read sets from insects. Visualizing two-dimensional representations of read tetranucleotide composition learned by a variational autoencoder can reveal distinct components of a sample. Annotating the embeddings with additional information, such as coding density, estimated coverage, or taxonomic labels allows rapid assessment of the contents of a dataset. The approach scales to millions of sequences, making it possible to explore unassembled read sets, even for large genomes. Combined with interactive visualization tools, it allows a large fraction of cobionts reported by reference-based screening to be identified. Crucially, it also facilitates retrieving genomes for which suitable reference data are absent.

Physeal Location on the Distal Femoral Cortical Surface Is Consistently Proximal to Apparent Location on the Perfect Lateral Radiograph: A 3-Dimensional Digitally Reconstructed Radiograph Study.

The perfect knee lateral radiograph visualizes anatomic landmarks on the distal femur for clinical and scientific purposes. However, radiographic imaging is a two-dimensional (2D) representation of a three-dimensional (3D) physis. The aim of this study was to characterize the perceived radiographic projection of the femoral physis using perfect lateral digitally reconstructed radiographs (DRRs) and to evaluate discrepancies from this projection to the physis at the lateral and medial cortices. Pediatric patients from a cohort of CT scans were analyzed. Inclusion criteria were an open physis; exclusion criteria were any implant or pathology affecting the physis. CT scans were imported into 3D imaging software and transformed into lateral DRRs and 3D renderings of the femur. The physis was divided into four equal segments, with fiducial markers placed at the "anterior," "midpoint," and "posterior" points. Lines extended from these points in the lateral and medial direction. The vertical distance from these lines, representing the radiographic projection of the physis, was measured relative to the physis at the lateral and medial cortex of the femur on coronal CT slices. Thirty-one patients were included. On the perfect lateral radiograph DRR, the physis on the medial cortex was located proximal to the visualized physis by 6.64 ± 1.74 mm, 11.95 ± 1.67 mm, and 14.30 ± 1.75 mm at the anterior (25%), midpoint (50%), and posterior (75%) locations, respectively. On the lateral side, the physis on the lateral cortex was proximal to the visualized physis by 2.19 ± 1.13 mm, 3.71 ± 1.19 mm, and 6.74 ± 1.25 mm at the anterior, midpoint, and posterior locations, respectively. In this cohort of pediatric patients, the location of the cortical physis was, in all areas measured, proximal to the projection of the visualized physis as seen on the perfect knee lateral DRR. The distance from radiographic physis to cortical physis was greater at the medial cortex compared with the lateral cortex. Descriptive laboratory study. III, observational radiographic anatomic study.

Camouflaged named entity recognition in 2D sentence representation

Organizing all spans of a sentence into a two-dimensional (2D) representation unfolds a semantic plane. It has the advantage to resolve nested semantic structures and to build linguistic dependencies across a whole sentence. The main problem is that neighboring elements in the semantic plane are span representations referred to overlapped phrases. Because these representations share the same contextual features in a sentence, a true entity representation is early faded into the background surroundings. It leads to camouflaged named entities in the 2D sentence representation. In this paper, we propose a finer-scale and coarse-scale sentence representation to support camouflaged named entity recognition. The mixed-scale representation has the ability to encode differential clues between entity representations. It is effective to distinguish entity representations from the background surroundings for recognizing camouflaged named entities. Compared with the state-of-the-art models, the results show that our method achieves competitive performance on five public datasets.