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Sufficient conditions are found for the correct and unique solvability of a class of third-order parabolic operator differential equations, whose principal parts have multiple characteristics, in a Sobolev-type space with exponential weight. The estimates for the norms of intermediate derivative operators are obtained and the relationship between these estimates and solvability conditions is established. Besides, the connection is found between the order of exponential weight and the lower bound for the spectrum of abstract operator appearing in the principal part of the equation.

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In the paper we find sufficient conditions on the coefficients of a class of third order quasi-elliptic operator pencils providing the completeness of the part of eigen and associated vectors in a separable Hilbert space. The completeness of the system of decreasing elementary solutions of the corresponding homogeneous equation in the space of solutions of this equation is proved. All these conditions are expressed by the coefficients of the operator pencil. [ABSTRACT FROM AUTHOR]

Introduction: There are several clinical manifestations of celiac disease, an autoimmune enteropathy caused by gluten intake, including extra-intestinal and gastrointestinal symptoms. Although it is linked to dilated cardiomyopathy and exocrine pancreatic insufficiency, it rare for children with celiac disease to have these disorders concurrently. Case presentation: We describe the case of a five-year-old boy with biopsy-confirmed celiac disease who had chronic diarrhea, vomiting, and failure to grow properly despite strictly following a gluten-free diet. Investigations revealed severe deficiency of pancreatic enzymes and dilated cardiomyopathy. Treatment with pancreatic enzyme supplement resulted in significant improvement on follow up. Clinical discussion: Celiac disease is characterized by various clinical features which may involve cardiovascular problems. However, the occurrence of both exocrine pancreatic insufficiency and dilated cardiomyopathy in children with celiac disease is extremely uncommon. There is limited evidence regarding the use of pancreatic enzyme supplements in the treatment of such patients; however, our patient recovered with this treatment unexpectedly. Conclusion: This case highlights the significance of considering uncommon coexisting conditions in children with celiac disease who display atypical symptoms, this is crucial because it can result in life-threatening situations. Consequently, it is essential to promptly identify and treat conditions such as exocrine pancreatic insufficiency and dilated cardiomyopathy and screen these patients for celiac disease in order to enhance patient outcomes.

Abstract Underwater wireless sensor networks (UWSNs) are an emerging research area that is rapidly gaining popularity. However, it has several challenges, including security, node mobility, limited bandwidth, and high error rates. Traditional trust models fail to adapt to the dynamic underwater environment. Thus, to address these issues, we propose a dynamic trust evaluation and update model using a modified decision tree algorithm. Unlike baseline methods, which often rely on static and generalized trust evaluation approaches, our model introduces several innovations tailored specifically for UWSNs. These include energy-aware decision-making, real-time adaptation to environmental changes, and the integration of multiple underwater-specific factors such as water currents and acoustic signal properties. Our model enhances trust accuracy, reduces energy consumption, and lowers data overhead, achieving a 96% accuracy rate with a 2% false positive rate. Additionally, it outperforms baseline models by improving energy efficiency by 50 mW and reducing response time to 20 ms per packet. These innovations demonstrate the proposed model’s effectiveness in addressing the unique challenges of UWSNs, ensuring both security and operational efficiency goals. The proposed model effectively enhances the trust evaluation process in UWSNs, providing both security and operational benefits. These key findings validate the potential of integrating modified decision tree algorithms to improve the performance and sustainability of UWSNs.