Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients

In this paper, we study the following degenerate critical elliptic equations with anisotropic coefficients $$-div(|x_{N}|^{2\alpha}\nabla u)=K(x)|x_{N}|^{\alpha\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N}$$ where $x=(x_{1},...,x_{N})\in\mathbb{R}^{N},$ $N\geq 3,$ $\alpha>1/2,$ $0\leq s\leq 2$ and $2^{*}(s)=2(N-s)/(N-2).$ Some basic properties of the degenerate elliptic operator $-div(|x_{N}|^{2\alpha}\nabla u)$ are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation. As a consequence, we obtain some nonexistence results for solutions of this equation.Comment: 29 page

Numerical algorithm and complexity analysis for diagonalization of multivariate homogeneous polynomials

We study the computational complexity of a criterion and an algorithm for diagonalization of multivariate homogeneous polynomials, that is, expressing them as sums of powers of independent linear forms. They are based on Harrison's center theory and only require solving linear and quadratic systems of equations. Detailed descriptions and computational complexity of each step of the algorithm are provided. The complexity analysis focuses on the impacts of problem sizes, including the number of variables and the degree of given polynomials. We show that this algorithm runs in polynomial time and validate it through numerical experiments. Other diagonalization algorithms are reviewed and compared in terms of complexity

INSOLVENCY PREDICTION OF AUSTRALIAN CONSTRUCTION COMPANIES USING DEEP LEARNING WITH BIDIRECTIONAL LSTM AUTOENCODER

Business insolvency in the building and construction industry is a major concern on a worldwide scale, and it is particularly pervasive in the Australian construction industry. Many Australian construction companies frequently uses high levels of borrowing and poor profit margins, which increases the likelihood of insolvency. This paper develops a novel, intelligent insolvency prediction model for the Australian construction companies. The proposed framework with bidirectional long short-term memory (BiLSTM) models and autoencoder techniques contains not only the financial variables but also other important indicators that are linked to the features of the sector that have previously been disregarded. Finally, numerical experiments show that the proposed neural network model outperforms several existing models for predicting the insolvency of construction companies

Nontrivial Solutions for Periodic Schrödinger Equations with Sign-Changing Nonlinearities

We prove a new infinite-dimensional linking theorem. Using this theorem, we obtain nontrivial solutions for strongly indefinite periodic Schrödinger equations with sign-changing nonlinearities

Characterizing Urban Household Waste Generation and Metabolism Considering Community Stratification in a Rapid Urbanizing Area of China

The relationship between social stratification and municipal solid waste generation remains uncertain under current rapid urbanization. Based on a multi-object spatial sampling technique, we selected 191 households in a rapidly urbanizing area of Xiamen, China. The selected communities were classified into three types: work-unit, transitional, and commercial communities in the context of housing policy reform in China. Field survey data were used to characterize household waste generation patterns considering community stratification. Our results revealed a disparity in waste generation profiles among different households. The three community types differed with respect to family income, living area, religious affiliation, and homeowner occupation. Income, family structure, and lifestyle caused significant differences in waste generation among work-unit, transitional, and commercial communities, respectively. Urban waste generation patterns are expected to evolve due to accelerating urbanization and associated community transition. A multi-scale integrated analysis of societal and ecosystem metabolism approach was applied to waste metabolism linking it to particular socioeconomic conditions that influence material flows and their evolution. Waste metabolism, both pace and density, was highest for family structure driven patterns, followed by lifestyle and income driven. The results will guide community-specific management policies in rapidly urbanizing areas

Nontrivial Solutions for Periodic Schrödinger Equations with Sign-Changing Nonlinearities

We prove a new infinite-dimensional linking theorem. Using this theorem, we obtain nontrivial solutions for strongly indefinite periodic Schrödinger equations with sign-changing nonlinearities