Positive Solutions for a Class of Quasilinear Elliptic Equations with a Dirichlet Problem

In this paper, we study the following problem -Δ_p u = h(x)u^q + f(u), u∈W_0^{1,p}(Ω), u > 0 in Ω, where Ω is a bounded smooth domain in R^N (N ≥ 3), 0 < q < 1. By using Mountain Pass Theorem, we prove that there exists at least two positive solutions under suitable assumptions on the nonlinearity. Key Words: Quasilinear elliptic equation; Positive solution; Dirichlet problem; Mountain Pass Theore

POD-DEIM Global-Local Model Reduction for Multi-phase Flows in Heterogeneous Porous Media

Many applications such as production optimization and reservoir management are computationally demanding due to a large number of forward simulations. Typically, each forward simulation involves multiple scales and is computationally expensive. The main objective of this dissertation is to develop and apply both local and global model-order reduction techniques to facilitate subsurface flow modeling. We develop a POD-DEIM global model reduction method for multi-phase flow simulation. The approach entails the use of Proper Orthogonal Decomposition (POD)-Galerkin projection, and Discrete Empirical Interpolation Method (DEIM). POD technique constructs a small POD subspace spanned by a set of global basis that can approximate the solution space. The reduced system is set up by projecting the full-order system onto the POD subspace. Discrete Empirical Interpolation Method (DEIM) is used to reduce the nonlinear terms in the system. DEIM overcomes the shortcomings of POD in the case of nonlinear PDEs by retaining nonlinearities in a lower dimensional space. The POD-DEIM global reduction method enjoys the merit of significant complexity reduction. We also propose an online adaptive global-local POD-DEIM model reduction method. This unique global-local online combination allows (1) developing local indicators that are used for both local and global updates; (2) computing global online modes via local multiscale basis functions. The multiscale basis functions consist of offline and some online local basis functions. The main contribution of the method is that the criteria for adaptivity and the construction of the global online modes are based on local error indicators and local multiscale basis functions which can be cheaply computed. The approach is particularly useful for situations where one needs to solve the reduced system for inputs or controls that result in a solution outside the span of the snapshots generated in the offline stage. Another aspect of my dissertation is the development of a local model reduction method for multiscale problems. We use global coupling in the coarse grid level via the mortar framework to link the sub-grid variations of neighboring coarse regions. The mortar framework offers some advantages, such as the flexibility in the constructions of the coarse grid and sub-grid capturing tools. By following the framework of the Generalized Multiscale Finite Element Method (GMsFEM), we design an enriched multiscale mortar space. Using the proposed multiscale mortar space, we (1) construct a multiscale finite element method to solve the flow problem on a coarse grid; (2) design two-level preconditioners as exact solver for the flow problem

Numerical study on free vibration characteristics of encastre clinched joints

The present paper deals with free vibration analysis of single lap encastre clinched joints using three dimensional finite element methods. The focus of the analysis is to reveal the influence on the natural frequencies, natural frequency ratios and mode shapes of these joints caused by variations in the material properties of the sheet materials. Numerical examples show that natural frequencies of single lap encastre clinched joints increase significantly as the Young’s modulus of the sheets increase, but only slight changes are encountered for variations of Poisson’s ratios. The mode shapes show that there are different deformations in the jointed section of clinched joints. These different deformations may cause different natural frequency values and different stress distributions. In both cases of transverse free vibration and torsional free vibration, odd mode shapes were found to be symmetrical about the mid-length position and even mode shaps were anti-symmetrical. The amplitudes of vibration at the mid-length of the joints are different for the odd and even modes. The geometry of the lap section is therefore very important and has a very significant effect on the dynamic response of the single lap encastre clinched joints. The main goal of this paper is to give an outline of free vibration characteristics of encastre clinched joints by finite element methods and to provide a basis for further experimental research

Power Amplification and Coherent Combination Techniques for Terahertz Quantum Cascade Lasers

Power amplification and coherent combination are important ways to improve the output power and beam quality of single‐mode terahertz quantum cascade lasers (THz QCLs). Up to date, the tapered waveguide is the most convenient way to amplify the power of THz QCLs. The self‐focusing effect in tapered THz QCLs induces non‐monotonic behaviours of the peak power and far‐field beam divergence, which lead to the existence of optimal structural parameters. The surface and lateral grating techniques have also been employed in tapered THz QCLs to further improve the spectral purity. For coherent combinations, the progress of facet‐emitting phase‐locked arrays of THz QCLs is still limited due to both the lack of the understanding of dynamics of coupled QCLs and the difficulties in designing high‐performance coupled waveguides. We will briefly review the developments of coherent arrays of THz QCLs and present a design of monolithic QCL arrays with common coupled cavity to achieve the optical mutual injection, which may provide a new way for coherent combination of THz QCLs

An Integrated Approach for Assessing Aquatic Ecological Carrying Capacity: A Case Study of Wujin District in the Tai Lake Basin, China

Aquatic ecological carrying capacity is an effective method for analyzing sustainable development in regional water management. In this paper, an integrated approach is employed for assessing the aquatic ecological carrying capacity of Wujin District in the Tai Lake Basin, China. An indicator system is established considering social and economic development as well as ecological resilience perspectives. While calculating the ecological index, the normalized difference vegetation index (NDVI) is extracted from Moderate Resolution Imaging Spectroradiometer (MODIS) time-series images, followed by spatial and temporal analysis of vegetation cover. Finally, multi-index assessment of aquatic ecological carrying capacity is carried out for the period 2000 to 2008, including both static and dynamic variables. The results reveal that aquatic ecological carrying capacity presents a slight upward trend in the past decade and the intensity of human activities still exceeded the aquatic ecological carrying capacity in 2008. In terms of human activities, population has decreased, GDP has quadrupled, and fertilizer application and industrial wastewater discharge have declined greatly in the past decade. The indicators representing aquatic ecosystem conditions have the lowest scores, which are primarily attributed to the water eutrophication problem. Yet the terrestrial ecosystem is assessed to be in better condition since topographic backgrounds and landscape diversity are at higher levels. Based on the work carried out, it is suggested that pollutant emission be controlled to improve water quality and agricultural development around Ge Lake (the largest lake in Wujin District) be reduced

Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimation

We present a supervised learning framework of training generative models for density estimation. Generative models, including generative adversarial networks, normalizing flows, variational auto-encoders, are usually considered as unsupervised learning models, because labeled data are usually unavailable for training. Despite the success of the generative models, there are several issues with the unsupervised training, e.g., requirement of reversible architectures, vanishing gradients, and training instability. To enable supervised learning in generative models, we utilize the score-based diffusion model to generate labeled data. Unlike existing diffusion models that train neural networks to learn the score function, we develop a training-free score estimation method. This approach uses mini-batch-based Monte Carlo estimators to directly approximate the score function at any spatial-temporal location in solving an ordinary differential equation (ODE), corresponding to the reverse-time stochastic differential equation (SDE). This approach can offer both high accuracy and substantial time savings in neural network training. Once the labeled data are generated, we can train a simple fully connected neural network to learn the generative model in the supervised manner. Compared with existing normalizing flow models, our method does not require to use reversible neural networks and avoids the computation of the Jacobian matrix. Compared with existing diffusion models, our method does not need to solve the reverse-time SDE to generate new samples. As a result, the sampling efficiency is significantly improved. We demonstrate the performance of our method by applying it to a set of 2D datasets as well as real data from the UCI repository

Endoplasmic reticulum stress in breast cancer: a predictive model for prognosis and therapy selection

BackgroundBreast cancer (BC) is a leading cause of mortality among women, underscoring the urgent need for improved therapeutic predictio. Developing a precise prognostic model is crucial. The role of Endoplasmic Reticulum Stress (ERS) in cancer suggests its potential as a critical factor in BC development and progression, highlighting the importance of precise prognostic models for tailored treatment strategies.MethodsThrough comprehensive analysis of ERS-related gene expression in BC, utilizing both single-cell and bulk sequencing data from varied BC subtypes, we identified eight key ERS-related genes. LASSO regression and machine learning techniques were employed to construct a prognostic model, validated across multiple datasets and compared with existing models for its predictive accuracy.ResultsThe developed ERS-model categorizes BC patients into distinct risk groups with significant differences in clinical prognosis, confirmed by robust ROC, DCA, and KM analyses. The model forecasts survival rates with high precision, revealing distinct immune infiltration patterns and treatment responsiveness between risk groups. Notably, we discovered six druggable targets and validated Methotrexate and Gemcitabine as effective agents for high-risk BC treatment, based on their sensitivity profiles and potential for addressing the lack of active targets in BC.ConclusionOur study advances BC research by establishing a significant link between ERS and BC prognosis at both the molecular and cellular levels. By stratifying patients into risk-defined groups, we unveil disparities in immune cell infiltration and drug response, guiding personalized treatment. The identification of potential drug targets and therapeutic agents opens new avenues for targeted interventions, promising to enhance outcomes for high-risk BC patients and paving the way for personalized cancer therapy