Variable selection has been a hot topic, with various popular methods including lasso, SCAD, and elastic net. These penalized regression algorithms remain sensitive to noisy data. Furthermore, “concept drift” fundamentally distinguishes streaming data learning from batch learning. This article presents a method for noise-resistant regularization and variable selection in noisy data streams with multicollinearity, dubbed canal-adaptive elastic net, which is similar to elastic net and encourages grouping effects. In comparison to lasso, the canal adaptive elastic net is especially advantageous when the number of predictions (p) is significantly larger than the number of observations (n), and the data are multi-collinear. Numerous simulation experiments have confirmed that canal-adaptive elastic net has higher prediction accuracy than lasso, ridge regression, and elastic net in data with multicollinearity and noise.

In this paper, we develop an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems. As an application of the theorem, we study the quintic nonlinear Schrödinger equation on the two-dimensional torus $\begin{equation*} {\mathrm {i}}u_{t}-\Delta u + {|u|}^4u = 0,\quad x\in \mathbb {T}^2\coloneq \mathbb {R}^2/(2\pi \mathbb {Z})^2,\quad t\in \mathbb {R}. \end{equation*}$ We obtain a Whitney smooth family of small-amplitude quasi-periodic solutions for the equation. The overall strategy in the proof of the KAM theorem is a normal form techniques sparsing angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. The idea in our proof comes from Geng, Xu, and You [Adv. Math. 226 (2011), pp. 5361–5402], which however has to be substantially developed to deal with the equation above.

In this paper, we develop an infinite dimensional Kolmogorov-Arnold-Moser theorem. As an application, we prove the existence of almost-periodic solutions for a nonlinear Schrödinger equation $ {\rm i}u_{t}-u_{xx}+\mu u+\psi(\omega t)f(|u|^2)u = 0, \ \mu >0, \ t\in\mathbb{R}, \ x\in [0, 2\pi] $ subject to periodic boundary conditions, where $ \psi(\omega t) $ is analytic and almost-periodic, $ f $ is real analytic in some neighborhood of the origin. For some technical difficulties in some known results, the almost-periodic potential requires a special structure. The overall strategy to overcome such difficulties is the set of Diophantine frequencies inspired by Montalto and Procesi [SIAM J. Math. Anal. 53(1)(2021), pp.386-434].

This article is devoted to the study of a two-dimensional $(2D)$ quasi-periodically forced beam equation $ u_{tt}+\Delta^2 u+ \varepsilon\phi(t)(u+{u}^3) = 0, \quad x\in\mathbb{T}^2, \quad t\in\mathbb{R} $ under periodic boundary conditions, where $\varepsilon$ is a small positive parameter, $\phi(t)$ is a real analytic quasi-periodic function in $t$ with frequency vector $\omega = (\omega_1, \omega_2 \ldots, \omega_m)$. We prove that the equation possesses a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. By solving the measure estimation of infinitely many small divisors, we construct a symplectic coordinate transformation which can reduce the linear part of Hamiltonian system to constant coefficients. And we construct some conversion of coordinates which can change the Hamiltonian of the equation into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we prove that there are many quasi-periodic solutions for the above equation via an abstract KAM theorem.

This work deals with the Feigenbaum’s functional equation in the broad sense where ϕ² is the 2-fold iteration of ϕ f(x) is a strictly increasing continuous function on r0; 1s and satisfies f(0)=0; f(x) <x; x Ɛ (0, 1]). Using constructive method, we discuss the existence of single-valley-extended continuous solutions of the above equation.

This work deals with Feigenbaum’s functional equationf(φ(x))=φp(f(x)),φ(0)=1,0≤φ(x)≤1,x∈ [0, 1], wherep≥2is an integer,φpis thep-fold iteration ofφ, andf(x)is a strictly increasing continuous function on[0, 1]that satisfiesf(0)=0,f(x)<x,(x∈(0, 1]). Using a constructive method, we discuss the existence of non-single-valley continuous solutions of the above equation.