The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for which the recognition problem is solved.

Let A be a finite-dimensional local algebra over an algebraically closed field, let J be the radical of A. The modules we are interested in are the finitely generated left A-modules. Projective modules are always reflexive, and an algebra is self-injective iff all modules are reflexive. We discuss the existence of non-projective reflexive modules in case A is not self-injective. We assume that A is short (this means that $ J^3 = 0$). In a joint paper with Zhang Pu, it has been shown that 6 is the smallest possible dimension of A that can occur and that in this case the following conditions have to be satisfied: $ J^2$ is both the left socle and the right socle of A and there is no uniform ideal of length 3. The present paper is devoted to showing the converse.

In this paper, we will first give the numerical simulation of the sub-fractional Brownian motion through the relation of fractional Brownian motion instead of its representation of random walk. In order to verify the rationality of this simulation, we propose a practical estimator associated with the LSE of the drift parameter of mixed sub-fractional Ornstein–Uhlenbeck process, and illustrate the asymptotical properties according to our method of simulation when the Hurst parameter $H>1/2$.

In this article, we carry out stochastic comparisons on the maximum order statistics arising from two batches of multiple-outlier gamma random variables with different shape and scale parameters. It is proved that, under certain conditions, the majorization order between the vectors of shape parameters together with the weak majorization order [p-larger order] between the vectors of scale parameters implies the likelihood ratio order [hazard rate order] between the largest order statistics. The results established here strengthen and generalize some known ones in the literature.

The quantile-based entropy measures possess some unique properties than their distribution function approach. The present communication deals with the study of the quantile-based Shannon entropy for record statistics. In this regard a generalized model is considered for which cumulative distribution function or probability density function does not exist and various examples are provided for illustration purpose. Further we consider the dynamic versions of the proposed entropy measure for record statistics and also give a characterization result for that. At the end, we study $F^{\alpha }$-family of distributions for the proposed entropy measure.

Process regression models, such as Gaussian process regression model (GPR), have been widely applied to analyze kinds of functional data. This paper introduces a composite of two T-process (CT), where the first one captures the smooth global trend and the second one models local details. The CT has an advantage in the local variability compared to general T-process. Furthermore, a composite T-process regression (CTP) model is developed, based on the composite T-process. It inherits many nice properties as GPR, while it is more robust against outliers than GPR. Numerical studies including simulation and real data application show that CTP performs well in prediction.

This article deals with some new chain imputation methods by using two auxiliary variables under missing completely at random (MCAR) approach. The proposed generalized classes of chain imputation methods are tested from the viewpoint of optimality in terms of MSE. The proposed imputation methods can be considered as an efficient extension to the work of Singh and Horn (Metrika 51:267–276, 2000), Singh and Deo (Stat Pap 44:555–579, 2003), Singh (Stat A J Theor Appl Stat 43(5):499–511, 2009), Kadilar and Cingi (Commun Stat Theory Methods 37:2226–2236, 2008) and Diana and Perri (Commun Stat Theory Methods 39:3245–3251, 2010). The performance of the proposed chain imputation methods is investigated relative to the conventional chain-type imputation methods. The theoretical results are derived and comparative study is conducted and the results are found to be quite encouraging providing the improvement over the discussed work.

We develop a fractional-degree expectation dependence which is the generalization of the first-degree and second-degree expectation dependence. The motivation for introducing such a dependence notion is to conform with the preferences of decision makers who are mostly risk averse but would be risk seeking at some wealth levels. We investigate some tractable equivalent properties for this new dependence notion, and explore its properties, including the invariance under increasing and concave transformations, and the invariance under convolution. We also extend our results to a combined fractional-degree expectation dependence notion including $\varepsilon $-almost first-degree expectation dependence. Two applications on portfolio diversification problem and optimal investment in the presence of a background risk illustrate the usefulness of the approaches proposed in the present paper.

Achieving higher true positive rate when decreasing false positive rate is always a great challenge to the imbalance learning community. This work combines penalized empirical likelihood method, lower bound algorithm and Nyström method and applies these techniques along with kernel method to density ratio model. The resulting classifier, density ratio classifier (DRC), is a combination of kernelization, regularization, efficient implementation and threshold moving, all of which are critical to enable DRC to be an effective and powerful method for solving difficult imbalance problems. Compared with other methods, DRC is competitive in that it is widely applicable and it is simple and easy to use without additional imbalance handling skills. In addition, the convergence rate of the estimate of log density ratio is discussed as well. And the results of numerical analysis also show that DRC outperforms other methods in AUC and G-mean score.

In this paper, we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph (CVR graph). We propose a novel structure, by which we offer an effective and easy operative method for constructing the basis functions of the biquadratic spline space. The mapping we construct is an isomorphism. The basis functions of the biquadratic spline space hold the properties such as linear independence, completeness and the property of partition of unity, which are the same as the properties for the basis functions of piecewise constant space over the CVR graph. To demonstrate that the new basis functions are efficient, we apply the basis functions to fit some open surfaces.

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

This paper presents an extension of the factor analysis model based on the normal mean–variance mixture of the Birnbaum–Saunders in the presence of nonresponses and missing data. This model can be used as a powerful tool to model non-normal features observed from data such as strongly skewed and heavy-tailed noises. Missing data may occur due to operator error or incomplete data capturing therefore cannot be ignored in factor analysis modeling. We implement an EM-type algorithm for maximum likelihood estimation and propose single imputation of possible missing values under a missing at random mechanism. The potential and applicability of our proposed method are illustrated through analyzing both simulated and real datasets.