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Abstract: Abstract This paper studies variable selection using the penalized likelihood method for distributed sparse regression with large sample size n under a limited memory constraint. This is a much needed research problem to be solved in the big data era. A naive divide-and-conquer method solving this problem is to split the whole data into N parts and run each part on one of N machines, aggregate the results from all machines via averaging, and finally obtain the selected variables. However, it tends to select more noise variables, and the false discovery rate may not be well controlled. We improve it by a special designed weighted average in aggregation. Although the alternating direction method of multiplier can be used to deal with massive data in the literature, our proposed method reduces the computational burden a lot and performs better by mean square error in most cases. Theoretically, we establish asymptotic properties of the resulting estimators for the likelihood models with a diverging number of parameters. Under some regularity conditions, we establish oracle properties in the sense that our distributed estimator shares the same asymptotic efficiency as the estimator based on the full sample. Computationally, a distributed penalized likelihood algorithm is proposed to refine the results in the context of general likelihoods. Furthermore, the proposed method is evaluated by simulations and a real example. PubDate: 2023-02-01

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Abstract: Abstract In this paper, we consider relativization of measure-theoretical- restricted sensitivity. For a given topological dynamical system, we define conditional measure-theoretical-restricted asymptotic rate with respect to sensitivity and obtain that it equals to the reciprocal of the Brin–Katok local entropy for almost every point under the conditional measure. PubDate: 2023-01-31

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Abstract: Abstract The Harnack inequality for stochastic differential equation driven by G-Brownian motion with multiplicative noise is derived by means of the coupling by change of measure, which extends the corresponding results derived in Wang (Probab. Theory Related Fields 109:417–424) under the linear expectation. Moreover, we generalize the gradient estimate under nonlinear expectation appeared in Song (Sci. China Math. 64:1093–1108). PubDate: 2023-01-31

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Abstract: Abstract In this article, we prove that a quasi-isometric map between rank one symmetric spaces is within bounded distance from an f-harmonic map. PubDate: 2023-01-30

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Abstract: Abstract With the help of the local isomorphism between the Hurwitz space \(M_{0;k-1,l-k-r+1,r-1}\) and the orbit space \(\mathcal {M}_{k,k+r}(A_l)\) , we will show the existence of a Frobenius manifold structure on the orbit space \(\mathcal {M}_{k,k+r}(A_l)\setminus \Sigma _r\) of the extended affine Weyl group \(\widetilde{W}^{(k,k+r)}(A_l)\) for \(1\le k<k+r\le l\) . PubDate: 2023-01-29

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Abstract: Abstract Let \(T_n\) be the number of triangles in the random intersection graph G(n, m, p). When the mean of \(T_n\) is bounded, we obtain an upper bound on the total variation distance between \(T_n\) and a Poisson distribution. When the mean of \(T_n\) tends to infinity, the Stein–Tikhomirov method is used to bound the error for the normal approximation of \(T_n\) with respect to the Kolmogorov metric. PubDate: 2023-01-29

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Abstract: Abstract In this paper, the functional central limit theorem is established for martingale like random vectors under the framework sub-linear expectations introduced by Shige Peng. As applications, the Lindeberg central limit theorem for independent random vectors is established, the sufficient and necessary conditions of the central limit theorem for independent and identically distributed random vectors are found, and a Lévy’s characterization of a multi-dimensional G-Brownian motion is obtained. PubDate: 2023-01-20

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Abstract: Abstract In this paper, the structure of finite groups in which maximal subgroups of some Sylow subgroups have a \(\sigma \) -soluble or \(\sigma \) -nilpotent supplement, where \(\sigma \) is a partition of the set of all prime numbers, is investigated. Some solubility, \(\sigma \) -solubility and \(\sigma \) -nilpotency criteria leading to some significant improvements of earlier results are given. PubDate: 2023-01-16

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Abstract: Abstract The generalized additive partial linear models (GAPLM) have been widely used for flexible modeling of various types of response. In practice, missing data usually occurs in studies of economics, medicine, and public health. We address the problem of identifying and estimating GAPLM when the response variable is nonignorably missing. Three types of monotone missing data mechanism are assumed, including logistic model, probit model and complementary log-log model. In this situation, likelihood based on observed data may not be identifiable. In this article, we show that the parameters of interest are identifiable under very mild conditions, and then construct the estimators of the unknown parameters and unknown functions based on a likelihood-based approach by expanding the unknown functions as a linear combination of polynomial spline functions. We establish asymptotic normality for the estimators of the parametric components. Simulation studies demonstrate that the proposed inference procedure performs well in many settings. We apply the proposed method to the household income dataset from the Chinese Household Income Project Survey 2013. PubDate: 2023-01-13

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Abstract: Abstract Consider the following McKean–Vlasov SDE: $$\begin{aligned} \textrm{d} X_t=\sqrt{2}\textrm{d} W_t+\int _{{\mathbb {R}}^d}K(t,X_t-y)\mu _{X_t}(\textrm{d} y)\textrm{d} t,\ \ X_0=x, \end{aligned}$$ where \(\mu _{X_t}\) stands for the distribution of \(X_t\) and \(K(t,x): {{\mathbb {R}}}_+\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d\) is a time-dependent divergence free vector field. Under the assumption \(K\in L^q_t({\widetilde{L}}_x^p)\) with \(\frac{d}{p}+\frac{2}{q}<2\) , where \({\widetilde{L}}^p_x\) stands for the localized \(L^p\) -space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D Navier–Stokes equations with measure as initial vorticity. PubDate: 2023-01-12

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Abstract: Abstract We construct a Floer type boundary operator for generalised Morse–Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the \(\mathbb {Z}_2\) homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field. PubDate: 2022-12-27

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Abstract: Abstract Katok’s entropy formula is an important formula in entropy theory. It plays significant roles in large deviation theories, multifractal analysis, quantitative recurrence and so on. This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems. We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms. PubDate: 2022-12-26

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Abstract: Abstract To investigate the fractional Hermite–Hadamard-type inequalities, a class of the multiplicative fractional integrals having exponential kernels is introduced. Some estimations of upper bounds for the newly introduced class of integral operators are obtained in terms of the established \(^*\) differentiable identity. And our results presented in this study are substantial generalizations of previous findings given by Ali et al. (Asian Res J Math 12:1–11, 2019). Three examples are also provided to identify the correctness of the results that occur with the change of the parameter \(\alpha \) . PubDate: 2022-12-24

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Abstract: Abstract The power generalized Weibull distribution has been proposed recently by [24] as an alternative to the gamma, Weibull and the exponentiated Weibull distributions. The power generalized Weibull family is suitable for modeling data that indicate nonmonotone hazard rates and can be used in survival analysis and reliability studies. Usefulness and flexibility of the family are illustrated by reanalyzing Efron’s data pertaining to a head-and-neck cancer clinical trial. These data involve censoring and indicate unimodal hazard rate. For this distribution, some recurrence relations are established for the single and product moments of upper record values. Further, using these relations, we have obtained means, variances and covariances of upper record values from samples of sizes up to 10 for various values of the parameters and present them in figures. Real data set is analyzed to illustrate the flexibility and importance of the model. PubDate: 2022-12-23

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Abstract: Abstract In this paper, we prove a version of Livšic theorem for a class of matrix cocycles over a \(C^2\) Axiom A flow. As a by-product, an approximative theorem on Lyapunov exponents is also obtained which assets that Lyapunov exponents of a given ergodic measure can be approximated by those of periodic measures. PubDate: 2022-12-01 DOI: 10.1007/s40304-021-00250-x

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Abstract: Abstract In this paper, the problem of high-dimensional multivariate analysis of variance is investigated under a low-dimensional factor structure which violates some vital assumptions on covariance matrix in some existing literature. We propose a new test and derive that the asymptotic distribution of the test statistic is a weighted distribution of chi-squares of 1 degree of freedom under the null hypothesis and mild conditions. We provide numerical studies on both sizes and powers to illustrate performance of the proposed test. PubDate: 2022-12-01 DOI: 10.1007/s40304-020-00236-1

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Abstract: Abstract Experimental design is an effective statistical tool that is extensively applied in modern industry, engineering, and science. It is proved that experimental design is a powerful and efficient means to screen the relationships between input factors and their responses, and to distinguish significant and unimportant factor effects. In many practical situations, experimenters are faced with large experiments having four-level factors. Even though there are several techniques provided to design such experiments, the challenge faced by the experimenters is still daunting. The practice has demonstrated that the existing techniques are highly time-consuming optimization procedures, satisfactory outcomes are not guaranteed, and non-mathematicians face a significant challenge in dealing with them. A new technique that can overcome these defects of the existing techniques is presented in this paper. The results demonstrated that the proposed technique outperformed the current techniques in terms of construction simplicity, computational efficiency and achieving satisfactory results capability. For non-mathematician experimenters, the new technique is much easier and simpler than the current techniques, as it allows them to design optimal large experiments without the recourse to optimization softwares. The optimality is discussed from four basic perspectives: maximizing the dissimilarity among experimental runs, maximizing the number of independent factors, minimizing the confounding among factors, and filling the experimental domain uniformly with as few gaps as possible. PubDate: 2022-12-01 DOI: 10.1007/s40304-021-00241-y

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Abstract: Abstract The p-adic Simpson correspondence due to Faltings (Adv Math 198(2):847–862, 2005) is a p-adic analogue of non-abelian Hodge theory. The following is the main result of this article: The correspondence for line bundles can be enhanced to a rigid analytic morphism of moduli spaces under certain smallness conditions. In the complex setting, Simpson shows that there is a complex analytic morphism from the moduli space for the vector bundles with integrable connection to the moduli space of representations of a finitely generated group as algebraic varieties. We give a p-adic analogue of Simpson’s result. PubDate: 2022-12-01 DOI: 10.1007/s40304-021-00256-5

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Abstract: Abstract Earlier it was proved that some distance-regular graphs of diameter 3 with \(c_2=2\) do not exist. Distance-regular graph \(\varGamma \) with intersection array \(\{17,16,10;1,2,8\}\) has strongly regular graph \(\varGamma _{3}\) (pseudo-geometric graph for the net \(pG_9(17,9)\) ). By symmetrizing the arrays of triple intersection numbers, it is proved that the distance-regular graphs with intersection arrays \(\{17,16,10;1,2,8\}\) and \(\{22,21,4;1,2,14\}\) do not exist. PubDate: 2022-10-02 DOI: 10.1007/s40304-021-00281-4

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Abstract: Abstract When milling part surfaces with a ball-end tool in 5-axis CNC machining, maintaining a constant cutting speed by keeping a fixed inclination angle between the tool axis and surface normal is crucial to ensure safe operation and achieve high quality of the machined surface. Under this constraint, the variation of tool orientation is expected to be “smoothest possible” to reduce the angular speed of the rotary axes for the efficient and robust machining. To address this issue, the spatial tractrix which is the extension of classic tractrix is presented to establish the geometry model of the tool orientation kinematics in the part coordinate system. The proposed model describes the relations between the tilt angle and the variation of ball-end tool orientation. Two spatial tractrix-based methods, synchronizing tractrix-based method and equilibrating tractrix-based method, are developed to minimize the variation of tool orientation by controlling the variation of tilt angle. These methods are used to plan the tool orientation on a part surface modeled by a bicubic spline surface. The performance evaluation carried by intense simulations demonstrates the equilibrating tractrix-based method provide the best results in most cases compared with the existing differential geometry-based methods such as the tractrix-based method and parallel transport method. The synchronizing tractrix-based method works well in some special cases. PubDate: 2022-09-15 DOI: 10.1007/s40304-021-00255-6