We show that the universal covering space of a connected component of a regular level set of a smooth complex valued function on \({\mathbb{C}}^2\), which is a smooth affine Riemann surface, is \({\mathbb{R} }^2\). This implies that the orbit space of the action of the covering group on \({\mathbb{R} }^2\) is the original affine Riemann surface.
This paper investigates the stationary probability distribution of the well-known stochastic logistic equation under regime switching. Sufficient conditions for the asymptotic stability of both the zero solution and the positive equilibrium are derived. The stationary distribution of the logistic equation under Markovian switching is obtained by computing the weighted mean of the stationary distributions of its subsystems. The weights correspond to the limiting distribution of the underlying Markov chain.
In medical and biomedical research, real data sets often exhibit characteristics such as bimodality, unimodality, or asymmetry. Among the generalized regression models commonly employed for analyzing such data are the Kumaraswamy and gamma-normal models. This study introduces two new generalized regression models based on the Harmonic Mixture Weibull-Normal distribution: one with varying dispersion and the other with constant dispersion. Additionally, a novel experimental design model was developed using the same distribution framework. The proposed models demonstrated the capability to effectively capture symmetric, asymmetric, and bimodal response variables. Model parameters were estimated using the maximum likelihood method, and simulation experiments were conducted to assess the behavior of the model coefficients. Empirical results revealed that the newly developed models outperformed several established alternatives, making them more practical for biomedical applications. Residual analysis further confirmed the adequacy of the proposed models, supporting their suitability for analyzing complex data in biomedical research.
The paper aims to investigate the existence and uniqueness of weak solution, using the Browder Theorem method, for the nonlocal \((p,q)\)-Kirchhoff system:
\[\begin{cases}
-K_{1}\big(\int_{\Omega}|\nabla \phi|^{p}\big)\Delta_{p}\phi+\lambda a(x)|\phi|^{p-2}\phi=f_1(x,\phi,\psi), & x\in \Omega \\
-K_{2}\big(\int_{\Omega}|\nabla \psi|^{q}\big)\Delta_{q}\psi+\lambda b(x)|\psi|^{q-2}v=f_2(x,\phi,\psi), & x\in \Omega \\
\phi=\psi=0, & x \in \partial\Omega
\end{cases}\]
where \(\Omega\) is a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary \(\partial\Omega\), with \(K_{1},K_{2}\) be continuous functions and \(f_1,f_2\) be Carathéodory functions.
