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.

In this paper, nonparametric estimation for a stationary strongly mixing and manifold-valued process $(X_j)$ is considered. In this non-Euclidean and not necessarily i.i.d setting, we propose kernel density estimators of the joint probability density function, of the conditional probability density functions and of the conditional expectations of functionals of $X_j$ given the past behavior of the process. We prove the strong consistency of these estimators under sufficient conditions, and we illustrate their performance through simulation studies and real data analysis.

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.

The Gruenberg–Kegel graph (or the prime graph) $\varGamma (G)$ of a finite group G is a graph, in which the vertex set is the set of all prime divisors of the order of G and two different vertices p and q are adjacent if and only if there exists an element of order pq in G. The paw is a graph on four vertices whose degrees are 1, 2, 2, 3. We consider the problem of describing finite groups whose Gruenberg–Kegel graphs are isomorphic as abstract graphs to the paw. For example, the Gruenberg–Kegel graph of the alternating group $A_{10}$ of degree 10 is isomorphic as abstract graph to the paw. In this paper, we describe finite non-solvable groups G whose Gruenberg–Kegel graphs are isomorphic as abstract graphs to the paw in the case when G has no elements of order 6 or the vertex of degree 1 of $\varGamma (G)$ divides the order of the solvable radical of G.

This paper investigates the optimal design problem for the prediction of the individual parameters in hierarchical linear models with heteroscedastic errors. An equivalence theorem is established to characterize D-optimality of designs for the prediction based on the mean squared error matrix. The admissibility of designs is also considered and a sufficient condition to simplify the design problem is obtained. The results obtained are illustrated in terms of a simple linear model with random slope and heteroscedastic errors.

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.

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.

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.

This paper shows that the multiplicity of the base point locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant. As a consequence, we get a new proof of the degree formula relating the degree of the surface, the degree of the parametrization, the base point multiplicity and the degree of the rational map induced by the parametrization. In addition, we extend both formulas to the case of dominant rational maps of the projective plane and describe how the base point loci of a parametrization and its reparametrizations are related. As an application of these results, we explore how the degree of a surface reparametrization is affected by the presence of base points.

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.