Search Results (10)

In this paper, we present a mathematical approach for studying the changes in pressure and temperature variables in flames. This conception extends beyond the traditional second-order Laplacian diffusion model by considering the p-Laplacian operator and a bi-stable reaction term, thereby providing a more generalized framework for flame diffusion analysis. Given the structure of our equations, we provide the boundedness and uniqueness of the solutions in a weak sense from both analytical and numerical approaches. We further reformulate the governing equations in the context of traveling wave solutions, applying singular geometric perturbation theory to derive the analytical expressions of these profiles. This theoretical development is complemented by numerical assessments, which not only validate our theoretical predictions, but also optimize the traveling wave speed to minimize the error between numerical and analytical solutions. Additionally, we explore self-similar structured solutions. The paper then concludes with a perspective on future research, with emphasis being placed on the need for experimental validation in laboratory settings. Such empirical studies could test the robustness of our model and allow for refinement based on actual measurements, thereby broadening the applicability and accuracy of our findings in practical scenarios. Full article

To address the problem that traditional spectral clustering algorithms cannot obtain the complete structural information of networks, this paper proposes a spectral clustering community detection algorithm, PMIK-SC, based on the point-wise mutual information (PMI) graph kernel. The kernel is constructed according to the point-wise mutual information between nodes, which is then used as a proximity matrix to reconstruct the network and obtain the symmetric normalized Laplacian matrix. Finally, the network is partitioned by the eigendecomposition and eigenvector clustering of the Laplacian matrix. In addition, to determine the number of clusters during spectral clustering, this paper proposes a fast algorithm, BI-CNE, for estimating the number of communities. For a specific network, the algorithm first reconstructs the original network and then runs Monte Carlo sampling to estimate the number of communities by Bayesian inference. Experimental results show that the detection speed and accuracy of the algorithm are superior to other existing algorithms for estimating the number of communities. On this basis, the spectral clustering community detection algorithm PMIK-SC also has high accuracy and stability compared with other community detection algorithms and spectral clustering algorithms. Full article

An important challenge in the study of complex systems is to identify appropriate effective variables at different times. In this paper, we explain why structures that are persistent with respect to changes in length and time scales are proper effective variables, and illustrate how persistent structures can be identified from the spectra and Fiedler vector of the graph Laplacian at different stages of the topological data analysis (TDA) filtration process for twelve toy models. We then investigated four market crashes, three of which were related to the COVID-19 pandemic. In all four crashes, a persistent gap opens up in the Laplacian spectra when we go from a normal phase to a crash phase. In the crash phase, the persistent structure associated with the gap remains distinguishable up to a characteristic length scale ${ϵ}^{*}$ where the first non-zero Laplacian eigenvalue changes most rapidly. Before ${ϵ}^{*}$, the distribution of components in the Fiedler vector is predominantly bi-modal, and this distribution becomes uni-modal after ${ϵ}^{*}.$ Our findings hint at the possibility of understanding market crashs in terms of both continuous and discontinuous changes. Beyond the graph Laplacian, we can also employ Hodge Laplacians of higher order for future research. Full article

The external electric field (E-field), which is an important stimulus, can change the decomposition mechanism and sensitivity of energetic materials. As a result, understanding the response of energetic materials to external E-fields is critical for their safe use. Motivated by recent experiments and theories, the two-dimensional infrared (2D IR) spectra of 3,4-bis (3-nitrofurazan-4-yl) furoxan (DNTF), which has a high energy, a low melting point, and comprehensive properties, were theoretically investigated. Cross-peaks were observed in 2D IR spectra under different E-fields, which demonstrated an intermolecular vibration energy transfer; the furazan ring vibration was found to play an important role in the analysis of vibration energy distribution and was extended over several DNTF molecules. Measurements of the non-covalent interactions, with the support of the 2D IR spectra, indicated that there were obvious non-covalent interactions among different DNTF molecules, which resulted from the conjugation of the furoxan ring and the furazan ring; the direction of the E-field also had a significant influence on the strength of the weak interactions. Furthermore, the calculation of the Laplacian bond order, which characterized the C-NO₂ bonds as trigger bonds, predicted that the E-fields could change the thermal decomposition process of DNTF while the positive E-field facilitates the breakdown of the C-NO₂ in DNTFⅣ molecules. Our work provides new insights into the relationship between the E-field and the intermolecular vibration energy transfer and decomposition mechanism of the DNTF system. Full article

We investigate the boundary value problem for biharmonic operators on the Heisenberg group. The inherent features of ${\mathbb{H}}^n$ make it an appropriate environment for studying symmetry rules and the interaction of analysis and geometry with manifolds. The goal of this paper is to prove that a weak solution for a biharmonic operator on the Heisenberg group exists. Our key tools are a version of the Mountain Pass Theorem and the classical variational theory. This paper will be of interest to researchers who are working on biharmonic operators on ${\mathbb{H}}^n.$ Full article

Breast cancer is widespread around the world and can be cured if diagnosed at an early stage. Digital mammograms are used as the most effective imaging modalities for the diagnosis of breast cancer. However, mammography images suffer from low contrast, background noise as well as contrast as non-coherency among the regions, and these factors makes breast cancer diagnosis challenging. These problems can be overcome by using a new image enhancement technique. The objective of this research work is to enhance mammography images to improve the overall process of segmentation and classification of breast cancer diagnosis. We proposed the image enhancement for mammogram images, as well as the ablation of the pectoral muscle. The image enhancement technique involves several steps. In the first step, we process the mammography images in three channels (red, green and blue), the second step is based on the uniformity of the background on morphological operations, and the third step is to obtain a well-contrasted image using principal component analysis (PCA). The fourth step is based on the removal of the pectoral muscle using a seed-based region growth technique, and the last step contains the coherence of the different regions of the image using a second order Gaussian Laplacian (LoG) and an oriented diffusion filter to obtain a much-improved contrast image. The proposed image enhancement technique is tested with our data collected from different hospitals in Qassim health cluster Qassim province Saudi Arabia, and it contains the five Breast Imaging and Reporting System (BI-RADS) categories and this database contained 11,194 images (the images contain carnio-caudal (CC) view and mediolateral oblique(MLO) view of mammography images), and we used approximately 700 images to validate our database. We have achieved improved performance in terms of peak signal-to-noise ratio, contrast, and effective measurement of enhancement (EME) as well as our proposed image enhancement technique outperforms existing image enhancement methods. This performance of our proposed method demonstrates the ability to improve the diagnostic performance of the computerized breast cancer detection method. Full article

In fluid mechanics, the bi-Laplacian operator with Neumann homogeneous boundary conditions emerges when transforming the Navier–Stokes equations to the vorticity–velocity formulation. In the case of problems with a periodic direction, the problem can be transformed into multiple, independent, two-dimensional fourth-order elliptic problems. An efficient method to solve these two-dimensional bi-Laplacian operators with Neumann homogeneus boundary conditions was designed and validated using 2D compact finite difference schemes. The solution is formulated as a linear combination of auxiliary solutions, as many as the number of points on the boundary, a method that was prohibitive some years ago due to the large memory requirements to store all these auxiliary functions. The validation has been made for different field configurations, grid sizes, and stencils of the numerical scheme, showing its potential to tackle high gradient fields as those that can be found in turbulent flows. Full article

The perceptual quality of digital images is often deteriorated during storage, compression, and transmission. The most reliable way of assessing image quality is to ask people to provide their opinions on a number of test images. However, this is an expensive and time-consuming process which cannot be applied in real-time systems. In this study, a novel no-reference image quality assessment method is proposed. The introduced method uses a set of novel quality-aware features which globally characterizes the statistics of a given test image, such as extended local fractal dimension distribution feature, extended first digit distribution features using different domains, Bilaplacian features, image moments, and a wide variety of perceptual features. Experimental results are demonstrated on five publicly available benchmark image quality assessment databases: CSIQ, MDID, KADID-10k, LIVE In the Wild, and KonIQ-10k. Full article

In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian of the well-defined warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given. Full article