Let $q\in(0,\infty]$ and $\varphi$ be a Musielak-Orlicz function with uniformly lower type $p_{\varphi}^-\in(0,\infty)$ and uniformly upper type $p_{\varphi}^+\in(0,\infty)$. In this article, the authors establish various real-variable characterizations of the Musielak-Orlicz-Lorentz Hardy space $H^{\varphi,q}(\mathbb{R}^n)$, respectively, in terms of various maximal functions, finite atoms, and various Littlewood-Paley functions. As applications, the authors obtain the dual space of $H^{\varphi,q}(\mathbb{R}^n)$ and the summability of Fourier transforms from $H^{\varphi,q}(\mathbb{R}^n)$ to the Musielak-Orlicz-Lorentz space $L^{\varphi,q}(\mathbb{R}^n)$ when $q\in(0,\infty)$ or from the Musielak-Orlicz Hardy space $H^{{\varphi}}({\mathbb{R}^n})$ to $L^{\varphi,\infty}(\mathbb{R}^n)$ in the critical case. These results are new when $q\in(0,\infty)$ and also essentially improve the existing corresponding results (if any) in the case $q=\infty$ via removing the original assumption that $\varphi$ is concave. To overcome the essential obstacles caused by both that $\varphi$ may not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown, the authors make full use of the obtained atomic characterization of $H^{\varphi,q}(\mathbb{R}^n)$, the corresponding results related to weighted Lebesgue spaces, and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.

In this paper, we study the multiplicity and concentration of positive solutions for a Schrödinger-Poisson system involving sign-changing potential and the nonlinearity $K(x)|u|^{p-2}u$ $(2 < p < 4)$ in $\mathbb{R}^3$. Such a problem cannot be studied by variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold since its (PS) sequence may not be bounded. By some new analytic techniques and the Ljusternik-Schnirelmann category theory, we relate the concentration and the number of positive solutions to the category of the global minima set of a suitable ground energy function. Furthermore, we investigate the asymptotic behavior of the solutions. In particular, we do not use Pohozaev equality in this work.

Tree-based models have been widely applied in both academic and industrial settings due to the natural interpretability, good predictive accuracy, and high scalability. In this paper, we focus on improving the single-tree method and propose the segmented linear regression trees (SLRT) model that replaces the traditional constant leaf model with linear ones. From the parametric view, SLRT can be employed as a recursive change point detect procedure for segmented linear regression (SLR) models, which is much more efficient and flexible than the traditional grid search method. Along this way, we propose to use the conditional Kendall's $\tau$ correlation coefficient to select the underlying change points. From the non-parametric view, we propose an efficient greedy splitting method that selects the splits by analyzing the association between residuals and each candidate split variable. Further, with the SLRT as a single-tree predictor, we propose a linear random forest approach that aggregates the SLRTs by a weighted average. Both simulation and empirical studies showed significant improvements than the CART trees and even the random forest.

In this paper, we establish a weighted maximal $L_2$ estimate of operator-valued Bochner-Riesz means. The proof is based on noncommutative square function estimates and a sharp weighted noncommutative Hardy-Littlewood maximal inequality.

Measuring and testing tail dependence is important in finance, insurance, and risk management. This paper proposes two tail dependence matrices based on classic rank correlation coefficients, which possess the desired population properties and interpretability. Their nonparametric estimators with strong consistency and asymptotic distributions are derived using the limit theory of $U$-processes. The simulation and application studies show that, compared to the tail dependence matrix based on Spearman's $\rho$ with large deviation, the Kendall-based tail dependence measure has stable variances under different tail conditions; thus, it is an effective approach to testing and quantifying tail dependence between random variables.

Let Ω be a domain of $(\mathbb{R}^n)$ with n ≥ 2 and p(·) be a local Lipschitz funcion in Ω with 1 < p(x) < ∞ in Ω. We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation -Δ_p(·)^Nu = 0 in Ω as well as the corresponding inhomogeneous equation -Δ_p(·)^Nu=f in Ω with f ∈ C⁰(Ω). In particular, given any viscosity solution u to -Δ_p(·)^Nu= 0 in Ω, we prove the following:
(i) in dimension $n=2$, for any subdomain $U \Subset \Omega$ and any $\beta \geq 0$, one has $|D u|^\beta D u \in L_{\text {loc }}^{2+\delta}(U)$ with a quantitative upper bound, and moreover, the map $\left(x_1, x_2\right) \rightarrow|D u|^\beta\left(u_{x_1},-u_{x_2}\right)$ is quasiregular in $U$ in the sense that
$\left|D\left[|D u|^\beta D u\right]\right|^2 \leq-C \operatorname{det} D\left[|D u|^\beta D u\right] \quad$ a.e. in $U$.
(ii) in dimension $n \geq 3$, for any subdomain $U \Subset \Omega$ with $\inf _U p(x)>1$ and $\sup _U p(x)<3+\frac{2}{n-2}$, one has $D^2 u \in L_{\text {loc }}^{2+\delta}(U)$ with a quantitative upper bound, and also with a pointwise upper bound
$\left|D^2 u\right|^2 \leq-C$ $\sum\limits_{1 \le i < j \le n} {} $ $\left[u_{x_i x_j} u_{x_j x_i}-u_{x_i x_i} u_{x_j x_j}\right]$ a.e. in $U$.
Here constants $\delta>0$ and $C \geq 1$ are independent of $u$. These extend the related results obtaind by Adamowicz-Hästö [Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, 10, 1940-1965 (2010)] when $n=2$ and $\beta=0$.

We establish the Strassen's law of the iterated logarithm (LIL for short) for independent and identically distributed random variables with $\hat{\mathbb{E}}\left[X_1\right]=\hat{\mathcal{E}}\left[X_1\right]=0$ and $C_{\mathrm{V}}\left[X_1^2\right]<\infty$ under a sub-linear expectation space with a countably sub-additive capacity V. We also show the LIL for upper capacity with $\sigma$=$\bar{\sigma}$ under some certain conditions.

Quantile regression is widely used in variable relationship research for statistical learning. Traditional quantile regression model is based on vector-valued covariates and can be efficiently estimated via traditional estimation methods. However, many modern applications involve tensor data with the intrinsic tensor structure. Traditional quantile regression can not deal with tensor regression issues well. To this end, we consider a tensor quantile regression with tensor-valued covariates and develop a novel variational Bayesian estimation approach to make estimation and prediction based on the asymmetric Laplace model and the CANDECOMP/PARAFAC decomposition of tensor coefficients. To incorporate the sparsity of tensor coefficients, we consider the multiway shrinkage priors for marginal factor vectors of tensor coefficients. The key idea of the proposed method is to efficiently combine the prior structural information of tensor and utilize the matricization of tensor decomposition to simplify the complexity of tensor coefficient estimation. The coordinate ascent algorithm is employed to optimize variational lower bound. Simulation studies and a real example show the numerical performances of the proposed method.

The classical Mumford stability condition of vector bundles on a complex elliptic curve $X$, can be viewed as a Bridgeland stability condition on $D^b({\rm Coh}\,X)$, the bounded derived category of coherent sheaves on $X$. This point of view gives us infinitely many $t$-structures and hearts on $D^b({\rm Coh}\, X)$. In this paper, we answer the question which of these hearts are Noetherian or Artinian.

Ridge regression is an effective tool to handle multicollinearity in regressions. It is also an essential type of shrinkage and regularization methods and is widely used in big data and distributed data applications. The divide and conquer trick, which combines the estimator in each subset with equal weight, is commonly applied in distributed data. To overcome multicollinearity and improve estimation accuracy in the presence of distributed data, we propose a Mallows-type model averaging method for ridge regressions, which combines estimators from all subsets. Our method is proved to be asymptotically optimal allowing the number of subsets and the dimension of variables to be divergent. The consistency of the resultant weight estimators tending to the theoretically optimal weights is also derived. Furthermore, the asymptotic normality of the model averaging estimator is demonstrated. Our simulation study and real data analysis show that the proposed model averaging method often performs better than commonly used model selection and model averaging methods in distributed data cases.

This paper will deal with a nonlocal geometric flow in centro-equiaffine geometry, which keeps the enclosed area of the evolving curve and converges smoothly to an ellipse. This model can be viewed as the affine version of Gage's area-preserving flow in Euclidean geometry.

In the paper we generalize some classic results on limit cycles of Liénard system \[\dot x=\phi(y)-F(x), \quad \dot y=-g(x)\] having a unique equilibrium to that of the system with several equilibria. As applications, we strictly prove the number of limit cycles and obtain the distribution of limit cycles for three classes of Liénard systems, in which we correct a mistake in the literature.

A graph G is edge-$k$-choosable if, for any assignment of lists $L(e)$ of at least $k$ colors to all edges $e\in E(G)$, there exists a proper edge coloring such that the color of $e$ belongs to $L(e)$ for all $e\in E(G)$. One of Vizing's classic conjectures asserts that every graph is edge-$(\Delta+1)$-choosable. It is known since 1999 that this conjecture is true for general graphs with $\Delta\leq4$. More recently, in 2015, Bonamy confirmed the conjecture for planar graph with $\Delta\geq8$, but the conjecture is still open for planar graphs with $5\leq\Delta\leq7$. We confirm the conjecture for planar graphs with $\Delta\ge 6$ in which every 7-cycle (if any) induces a $C_7$ (so, without chords), thereby extending a result due to Dong, Liu and Li.

Despite of the wide use of the factor models, the issue of determining the number of factors has not been resolved in the statistics literature. An ad hoc approach is to set the number of factors to be the number of eigenvalues of the data correlation matrix that are larger than one, and subsequent statistical analysis proceeds assuming the resulting factor number is correct. In this work, we study the relation between the number of such eigenvalues and the number of factors, and provide the if and only if conditions under which the two numbers are equal. We show that the equality only relies on the properties of the loading matrix of the factor model. Guided by the newly discovered condition, we further reveal how the model error affects the estimation of the number of factors.

Let $G$ be a finite group. We denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. In this paper, we characterize the structure of finite groups $G$ with lower bounds $\frac{1}{p}$, $\frac{p^2+8}{9p^2}$ and $\frac{p+3}{4p}$ on $\nu(G)$, where $p$ is a prime divisor of $|G|$.

In this paper, we study a class of Finsler metrics of cohomogeneity two on $\mathbb{R} \times \mathbb{R}^n$. They are called weakly orthogonally invariant Finsler metrics. These metrics not only contain spherically symmetric Finsler metrics and Marcal-Shen's warped product metrics but also partly contain another "warping" introduced by Chen-Shen-Zhao. We obtain differential equations that characterize weakly orthogonally invariant Finsler metrics with vanishing Douglas curvature, and therefore we provide a unifying frame work for Douglas equations due to Liu-Mo, Mo-Solórzano-Tenenblat and Solórzano. As an application, we obtain a lot of new examples of weakly orthogonally invariant Douglas metrics.

Single-index model offers the greater flexibility of modelling than generalized linear models and also retains the interpretability of the model to some extent. Although many standard approaches such as kernels or penalized/smooothing splines were proposed to estimate smooth link function, they cannot approximate complicated unknown link functions together with the corresponding derivatives effectively due to their poor approximation ability for a finite sample size. To alleviate this problem, this paper proposes a semiparametric least squares estimation approach for a single-index model using the rectifier quadratic unit (ReQU) activated deep neural networks, called deep semiparametric least squares (DSLS) estimation method. Under some regularity conditions, we show non-asymptotic properties of the proposed DSLS estimator, and evidence that the index coefficient estimator can achieve the semiparametric efficiency. In particular, we obtain the consistency and the convergence rate of the proposed DSLS estimator when response variable is conditionally sub-exponential. This is an attempt to incorporate deep learning technique into semiparametrically efficient estimation in a single index model. Several simulation studies and a real example data analysis are conducted to illustrate the proposed DSLS estimator.

This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and cost functional depend on the joint distribution of the state and the control. In our game, both the (lower and upper) value functions and the (lower and upper) second-order Bellman--Isaacs equations are defined on the Wasserstein space $\mathcal{P}_2(\mathbb{R}^n)$ which is an infinite dimensional space. The dynamic programming principle for the value functions is proved. If the (upper and lower) value functions are smooth enough, we show that they are the classical solutions to the second-order Bellman--Isaacs equations. On the other hand, the classical solutions to the (upper and lower) Bellman--Isaacs equations are unique and coincide with the (upper and lower) value functions. As an illustrative application, the linear quadratic case is considered. Under the Isaacs condition, the explicit expressions of optimal closed-loop controls for both players are given. Finally, we introduce the intrinsic notion of viscosity solution of our second-order Bellman--Isaacs equations, and characterize the (upper and lower) value functions as their viscosity solutions.

In this paper, we classify the finite non-abelian $p$-groups all of whose non-abelian proper subgroups have centers of the same order.

Combining p-values is a well-known issue in statistical inference. When faced with a study involving $m$ p-values, determining how to effectively combine them to arrive at a comprehensive and reliable conclusion becomes a significant concern in various fields, including genetics, genomics, and economics, among others. The literature offers a range of combination strategies tailored to different research objectives and data characteristics. In this work, we aim to provide users with a systematic exploration of the p-value combination problem. We present theoretical results for combining p-values using a logarithmic transformation, which highlights the benefits of this approach. Additionally, we propose a combination strategy together with its statistical properties utilizing the gold section method, showcasing its performance through extensive computer simulations. To further illustrate its effectiveness, we apply this approach to a real-world scenario.

To learn the subgroup structure generated by multidimensional interaction, we propose a novel multiview subgroup integration technique based on tensor decomposition. Compared to the traditional subgroup analysis that can only handle single-view heterogeneity, our proposed method achieves a greater level of homogeneity within the subgroups, leading to enhanced interpretability and predictive power. For computational readiness of the proposed method, we build an algorithm that incorporates pairwise shrinkage-encouraging penalties and ADMM techniques. Theoretically, we establish the asymptotic consistency and normality of the proposed estimators. Extensive simulation studies and real data analysis demonstrate that our proposal outperforms other methods in terms of prediction accuracy and grouping consistency. In addition, the analysis based on the proposed method indicates that intergenerational care significantly increases the risk of chronic diseases associated with diet and fatigue in all provinces while only reducing the risk of emotion-related chronic diseases in the eastern coastal and central regions of China.

With the advent of modern devices, such as smartphones and wearable devices, high-dimensional data are collected on many participants for a period of time or even in perpetuity. For this type of data, dependencies between and within data batches exist because data are collected from the same individual over time. Under the framework of streamed data, individual historical data are not available due to the storage and computation burden. It is urgent to develop computationally efficient methods with statistical guarantees to analyze high-dimensional streamed data and make reliable inferences in practice. In addition, the homogeneity assumption on the model parameters may not be valid in practice over time. To address the above issues, in this paper, we develop a new renewable debiased-lasso inference method for high-dimensional streamed data allowing dependences between and within data batches to exist and model parameters to gradually change. We establish the large sample properties of the proposed estimators, including consistency and asymptotic normality. The numerical results, including simulations and real data analysis, show the superior performance of the proposed method.

Gaussian graphical models (GGMs) are widely used as intuitive and efficient tools for data analysis in several application domains. To address the reproducibility issue of structure learning of a GGM, it is essential to control the false discovery rate (FDR) of the estimated edge set of the graph in terms of the graphical model. Hence, in recent years, the problem of GGM estimation with FDR control is receiving more and more attention. In this paper, we propose a new GGM estimation method by implementing multiple data splitting. Instead of using the node-by-node regressions to estimate each row of the precision matrix, we suggest directly estimating the entire precision matrix using the graphical Lasso in the multiple data splitting, and our calculation speed is $p$ times faster than the previous. We show that the proposed method can asymptotically control FDR, and the proposed method has significant advantages in computational efficiency. Finally, we demonstrate the usefulness of the proposed method through a real data analysis.

We consider the problem of multi-task regression with time-varying low-rank patterns, where the collected data may be contaminated by heavy-tailed distributions and/or outliers. Our approach is based on a piecewise robust multi-task learning formulation, in which a robust loss function—not necessarily to be convex, but with a bounded derivative—is used, and each piecewise low-rank pattern is induced by a nuclear norm regularization term. We propose using the composite gradient descent algorithm to obtain stationary points within a data segment and employing the dynamic programming algorithm to determine the optimal segmentation. The theoretical properties of the detected number and time points of pattern shifts are studied under mild conditions. Numerical results confirm the effectiveness of our method.

In this paper, we study the $p$-Laplacian Choquard equation $$-△_p u+V(x)|u|^{p-2}u=\bigg({\sum_{y\in N^n\atop y\not=x}}\frac{|u(y)|^q}{d(x,\,y)^{n-\alpha}}\bigg)|u|^{q-2}u$$ on a finite lattice graph $N^n$ with $n\in\mathbb{N}_+$, where $p>1,$ $q>1$ and $0\leq\alpha\leq n$ are some constants, $V(x)$ is a positive function on $N^n$. Using the Nehari method, we prove that if 1<p<q<+∞, then the above equation admits a ground state solution. Previously, the $p$-Laplacian Choquard equation on finite lattice graph has not been studied, and our result contains the critical cases $\alpha=0$ and $\alpha=n$, which further improves the study of Choquard equations on lattice graphs.

In this article, we discuss the approach to solving a nonlinear PDE equation, specifically the $p$-Laplacian equation, with a general (nonlinear) boundary condition. We establish the existence and uniqueness of the solution, subject to certain assumptions outlined in this paper. To solve our nonlinear problem using the Finite Element Method (FEM), we derive an appropriate variational formulation. Additionally, we introduce a study of the residual a posteriori-error indicator, establishing both upper and lower bounds to control the error. The upper bound is determined using averaging interpolators in some quasi-norms defined by Barrett and Liu. Furthermore, we prove the equivalence between the residual error and the true error $e=u-u_{h}$. Lastly, we perform a simulation of the $p$-Laplacian problem in the $L$-shape domain using a Matlab program in two-dimensional space.

We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric α-stable Lévy processes (1 2 <α ≤ 1), where the drift coefficient is Hölder continuous in space variable, while the noise coefficient is Lipscitz continuous in space variable, and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance. If the drift coefficient does not depend on distribution variable, our methodology developed in this paper applies to the case α ∈ (0, 1]. The main tool relies on heat kernel estimates for (distribution independent) stable SDEs and Banach’s fixed point theorem.

Let G be a finite group and $\pi(G)$ be the set of prime divisors of $|G|$. The prime graph G of G is the graph with vertex set $\pi(G)$, and different $p,q\in \pi(G)$ are joined by an edge if and only if G has an element of order $pq$. In this paper, we characterize the finite solvable groups whose prime graphs have diameter $3$.

Let C be a triangulated category. We define $m$-term subcategories on C induced by $n$-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on $2$-term subcategories $\mathcal{G}$ and support $\tau$-tilting subcategories on an abelian quotient of $\mathcal{G}$. If an $m$-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.

In this paper, we first prove that the retract of a consonant space (or co-consonant space) is consonant (co-consonant). Simultaneously, we consider the co-consonance of two powerspace constructions and proved that (1) the co-consonance of the Smyth powerspace $P_{S}(X)$ implies the co-consonance of $X$ if $X$ is strongly compact; (2) the co-consonance of $X$ implies the co-consonance of the Smyth powerspace under some conditions; (3) if the lower powerspace $P_{H}(X)$ is co-consonant, then $X$ is co-consonant; (4) for a continuous poset $P$, the lower powerspace $P_{H}(\Sigma P)$ is co-consonant.

Doubly truncated data arise when the survival times of interest are observed only if they fall within certain random intervals. In this paper, we consider a semiparametric additive hazards model with doubly truncated data, and propose a weighted estimating equation approach to estimate the regression coefficients, where the weights are estimated both parametrically and nonparametrically. The asymptotic properties of the resulting estimators are established. Simulation studies demonstrate that the proposed estimators perform well in a finite sample. An application to Parkinson's disease data is provided.

In this paper, we obtain some sufficient and necessary conditions for indecomposable positive definite integral lattices with discriminants 2, 3, 4 and 5 over $\mathbb{Z}$ being additively indecomposable lattices. Using these results, we prove that there exist additively indecomposable positive integral quadratic lattices with discriminants 2, 3, 4 and 5 and rank greater than or equal to 2 but for 35 exceptions. In the exceptions there are no lattices with the desired properties. We also give a lifting theorem of additively indecomposable positive definite integral lattices.

The Turán number, denoted by ${\rm ex}\,(n,H)$, is the maximum number of edges of a graph on $n$ vertices containing no graph $H$ as a subgraph. Denote by $kC_{\ell}$ the union of $k$ vertex-disjoint copies of $C_{\ell}$. In this paper, we present new results for the Turán numbers of vertex-disjoint cycles. Our first results deal with the Turán number of vertex-disjoint triangles ${\rm ex}\,(n, kC_{3})$. We determine the Turán number ${\rm ex}(n, kC_{3})$ for $n\geq\frac{k^{2}+5k}{2}$ when $k\leq4$, and $n\geq k^{2}+2$ when $k\geq4$. Moreover, we give lower and upper bounds for ${\rm ex}\,(n, kC_{3})$ with $3k\leq n\leq\frac{k^{2}+5k}{2}$ when $k\leq4$, and $3k\leq n\leq k^{2}+2$ when $k\geq4$. Next, we give a lower bound for the Turán number of vertex-disjoint pentagons ${\rm ex}\,(n, kC_{5})$. Finally, we determine the Turán number ${\rm ex}\,(n, kC_{5})$ for $n=5k$, and propose two conjectures for ${\rm ex}\,(n, kC_{5})$ for the other values of $n$.

The main purpose of this article is using the elementary techniques and the properties of the character sums to study the computational problem of one kind products of Gauss sums, and give an interesting triplication formula for them.

Finding a highly interpretable nonlinear model has been an important yet challenging problem, and related research is relatively scarce in the current literature. To tackle this issue, we propose a new algorithm called Feat-ABESS based on a framework that utilizes feature transformation and selection for re-interpreting many machine learning algorithms. The core idea behind Feat-ABESS is to parameterize interpretable feature transformation within this framework and construct an objective function based on these parameters. This approach enables us to identify a proper interpretable feature transformation from the optimization perspective. By leveraging a recently advanced optimization technique, Feat-ABESS can obtain a concise and interpretable model. Moreover, Feat-ABESS can perform nonlinear variable selection. Our extensive experiments on 205 benchmark datasets and case studies on two datasets have demonstrated that Feat-ABESS can achieve powerful prediction accuracy while maintaining a high level of interpretability. The comparison with existing nonlinear variable selection methods exhibits Feat-ABESS has a higher true positive rate and a lower false discovery rate.

The well-known Berwald square metric is a positively complete and projectively flat Finsler metric with vanishing flag curvature. In this paper, we study a positively complete square metric on a manifold. We show a rigidity result that if the Ricci curvature is constant, then it must be isometric to the Berwald square metric. This is not true without assumption on the completeness of the metric.

Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories. In this paper, we develop the relative homology aspect of Auslander bijection in extriangulated categories. Namely, we introduce the notion of generalized ARS-duality relative to an additive subfunctor, and prove that there is a bijective triangle which involves the generalized ARS-duality and the restricted Auslander bijection relative to the subfunctor. We give the Auslander's defect formula in terms of the generalized ARS-duality, show the interplay of morphisms being determined by objects and a kind of extriangles, and characterize a class of objects which are somehow controlled by this kind of extriangles. We also give a realization for a functor relating the Auslander bijection and the generalized ARS-duality.

In this short note, we establish a sharp Morrey regularity theory for an even order elliptic system of Rivière type: \begin{equation*} \Delta^{m}u=\sum_{l=0}^{m-1}\Delta^{l} \langle V_{l},du \rangle +\sum_{l=0}^{m-2} \Delta^{l}\delta (w_{l}du)+f\quad \text{ in } B^{2m}\label{eq: Longue-Gastel system} \end{equation*} under minimal regularity assumptions on the coefficients functions $V_l, w_l$ and that $f$ belongs to certain Morrey space. This can be regarded as a further extension of the recent $L^p$-regularity theory obtained by Guo--Xiang--Zheng [J. Math. Pures Appl. (9), 165, 286--324 (2022)], and generalizes [Proc. Amer. Math. Soc., 152(10), 4261--4268 (2024)], [Acta Math. Sci. Ser. B (Engl. Ed.), 44(2), 420--430 (2024)] for second and fourth order elliptic systems.

This paper investigates tangent measures in the sense of Preiss for homogeneous Cantor sets on ${{\mathbb{R}}^d}$ generated by specific iterated function systems that satisfy the strong separation condition. Through the dynamics of “zooming in” on typical point, we derive an explicit and uniform formula for the tangent measures associated with this category of homogeneous Cantor sets on ${{\mathbb{R}}^d}$.