This paper studies singular optimal control problems for systems described by nonlinear-controlled stochastic differential equations of mean-field type (MFSDEs in short), in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. The control variable has two components, the first being absolutely continuous and the second singular. We establish necessary as well as sufficient conditions for optimal singular stochastic control where the system evolves according to MFSDEs. These conditions of optimality differs from the classical one in the sense that here the adjoint equation turns out to be a linear mean-field backward stochastic differential equation. The proof of our result is based on convex perturbation method of a given optimal control. The control domain is assumed to be convex. A linear quadratic stochastic optimal control problem of mean-field type is discussed as an illustrated example.

The study of real-life network modeling has become very popular in recent years. An attractive model is the scale-free percolation model on the lattice ${\mathbb Z}^d$, $d\ge 1$, because it fulfills several stylized facts observed in large real-life networks. We adopt this model to continuum space which leads to a heterogeneous random-connection model on ${\mathbb R}^d$: Particles are generated by a homogeneous marked Poisson point process on ${\mathbb R}^d$, and the probability of an edge between two particles is determined by their marks and their distance. In this model we study several properties such as the degree distributions, percolation properties and graph distances.

We propose a method that combines isogeometric analysis (IGA) with the discontinuous Galerkin (DG) method for solving elliptic equations on 3-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted across the patch interfaces to glue the multiple patches, while the traditional IGA, which is very suitable for solving partial differential equations (PDEs) on (3D) surfaces, is employed within each patch. Our method takes advantage of both IGA and the DG method. Firstly, the time-consuming steps in mesh generation process in traditional finite element analysis (FEA) are no longer necessary and refinements, including $h$-refinement and $p$-refinement which both maintain the original geometry, can be easily performed by knot insertion and order-elevation (Farin, in Curves and surfaces for CAGD, 2002). Secondly, our method can easily handle the cases with non-conforming patches and different degrees across the patches. Moreover, due to the geometric flexibility of IGA basis functions, especially the use of multiple patches, we can get more accurate modeling of more complex surfaces. Thus, the geometrical error is significantly reduced and it is, in particular, eliminated for all conic sections. Finally, this method can be easily formulated and implemented. We generally describe the problem and then present our primal formulation. A new ideology, which directly makes use of the approximation property of the NURBS basis functions on the parametric domain rather than that of the IGA functions on the physical domain (the former is easier to get), is adopted when we perform the theoretical analysis including the boundedness and stability of the primal form, and the error analysis under both the DG norm and the ${\mathcal {L}}^{2}$ norm. The result of the error analysis shows that our scheme achieves the optimal convergence rate with respect to both the DG norm and the ${\mathcal {L}}^{2}$ norm. Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.

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.

This paper establishes probabilistic and statistical properties of the extension of time-invariant coefficients asymmetric $\log $ GARCH processes to periodically time-varying coefficients ($P\log $ GARCH) one. In these models, the parameters of $\log -$volatility are allowed to switch periodically between different seasons. The main motivations of this new model are able to capture the asymmetry and hence leverage effect, in addition, the volatility coefficients are not a subject to positivity constraints. So, some probabilistic properties of asymmetric $P\log $ GARCH models have been obtained, especially, sufficient conditions ensuring the existence of stationary, causal, ergodic (in periodic sense) solution and moments properties are given. Furthermore, we establish the strong consistency and the asymptotic normality of the quasi-maximum likelihood estimator (QMLE) under extremely strong assumptions. Finally, we carry out a simulation study of the performance of the QML and the $P\log $ GARCH is applied to model the crude oil prices of Algerian Saharan Blend.

In this article, the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss, and discuss their relations with Besov and Triebel–Lizorkin spaces. As an application, the authors establish the difference characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type. A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space ${{\mathcal {X}}}$ via using the geometrical property of ${{\mathcal {X}}}$ expressed by its dyadic reference points, dyadic cubes, and the (local) lower bound. Moreover, some results when $p\le 1$ but near to 1 are new even when ${{\mathcal {X}}}$ is an RD-space.

For the $p$-harmonic function with strictly convex level sets, we find an auxiliary function which comes from the combination of the norm of gradient of the $p$-harmonic function and the Gaussian curvature of the level sets of $p$-harmonic function. We prove that this curvature function is concave with respect to the height of the $p$-harmonic function. This auxiliary function is an affine function of the height when the $p$-harmonic function is the $p$-Green function on ball.

We introduce a four-parameter lifetime distribution called the odd log-logistic generalized Gompertz model to generalize the exponential, generalized exponential and generalized Gompertz distributions, among others. We obtain explicit expressions for the moments, moment-generating function, asymptotic distribution, quantile function, mean deviations and distribution of order statistics. The method of maximum likelihood estimation of parameters is compared by six different methods of estimations with simulation study. The applicability of the new model is illustrated by means of a real data set.

We characterize bounded and compact positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space.

We address the well-posedness of the 2D (Euler)–Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich’s type data, which gives a positive answer to part of the questions raised in Lai (Arch Ration Mech Anal 199(3):739–760, 2011). Our analysis on the the polygonal-like domains essentially relies on the recent elliptic regularity results for such domains proved in Bardos et al. (J Math Anal Appl 407(1):69–89, 2013) and Di Plinio (SIAM J Math Anal 47(1):159–178, 2015).

In this paper, we consider Marshall and Olkin’s family of distributions. The parent (baseline) distribution is taken to be a scaled family of distributions. Two models: (i) modified proportional hazard rate scale and (ii) modified proportional reversed hazard rate scale, are considered. Some stochastic comparison results in terms of the usual stochastic, hazard rate and reversed hazard rate orders are studied to compare order statistics formed from two sets of independent observations following these models. Most of the sufficient conditions are obtained depending on various majorization-type partial orderings. Further, the setup with multiple-outlier model is taken. Various stochastic orders between the smallest and largest order statistics are developed. Several numerical examples are provided to illustrate the effectiveness of the established theoretical results.

In the present paper, we study the structure of cyclic DNA codes of even length over the ring ${F}_2+u{F}_2+u^2{F}_2$ where $u^3=0$. We investigate two presentations of cyclic codes of even length over ${F}_2+u{F}_2+u^2{F}_2$ satisfying the reverse constraint and the reverse-complement constraint.

Let $\sigma =\{\sigma _i |i\in I\}$ be some partition of all primes ${\mathbb {P}}$ and G a finite group. A subgroup H of G is said to be $\sigma $-subnormal in G if there exists a subgroup chain $H=H_0\le H_1\le \cdots \le H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $\sigma _j$-group for some $j \in I$ for $i = 1, \ldots , n$. We call a finite group G a $T_{\sigma }$-group if every $\sigma $-subnormal subgroup is normal in G. In this paper, we analyse the structure of the $T_{\sigma }$-groups and give some characterisations of the $T_{\sigma }$-groups.

We consider the tail behavior of the product of two dependent random variables X and $\Theta $. Motivated by Denisov and Zwart (J Appl Probab 44:1031–1046, 2007), we relax the condition of the existing $\alpha \,+\,\epsilon $ th moment of $\Theta $ in Breiman’s theorem to the existing $\alpha $th moment and obtain the similar result as Breiman’s theorem of the dependent product $X \Theta $, while X and $\Theta $ follow a copula function. As applications, we consider a discrete-time insurance risk model with dependent insurance and financial risks and derive the asymptotic tail behaviors for the (in)finite-time ruin probabilities.

Let $M$ be a complete Kähler surface and $\Sigma $ be a symplectic surface which is smoothly immersed in $M$. Let $\alpha $ be the Kähler angle of $\Sigma $ in $M$. In the previous paper Han and Li (JEMS 12:505–527, 2010) 2010, we study the symplectic critical surfaces, which are critical points of the functional $L=\int _{\Sigma }\frac{1}{\cos \alpha }d\mu $ in the class of symplectic surfaces. In this paper, we calculate the second variation of the functional $L$ and derive some consequences. In particular, we show that, if the scalar curvature of $M$ is positive, $\Sigma $ is a stable symplectic critical surface with $\cos \alpha \ge \delta >0$, whose normal bundle admits a holomorphic section $X\in L^2(\Sigma )$, then $\Sigma $ is holomorphic. We construct symplectic critical surfaces in $\mathbf{C}^2$. We also prove a Liouville theorem for symplectic critical surfaces in $\mathbf{C}^2$.

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.

Truncated elliptical distributions occur naturally in theoretical and applied statistics and are essential for the study of other classes of multivariate distributions. Two members of this class are the multivariate truncated normal and multivariate truncated t distributions. We derive statistical properties of the truncated elliptical distributions. Applications of our results establish new properties of the multivariate truncated slash and multivariate truncated power exponential distributions.

In this article, a novel susceptible–infected–recovered epidemic model with nonmonotonic incidence and treatment rates is proposed and analyzed mathematically. The Monod–Haldane functional response is considered for nonmonotonic behavior of both incidence rate and treatment rate. The model analysis shows that the model has two equilibria which are named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The stability analysis has been performed for the local and global behavior of the DFE and EE. With the help of the basic reproduction number

, we investigate that DFE is locally asymptotically stable when

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{0} < 1 $$\end{document}

and unstable when

R_0>1

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{0} > 1 $$\end{document}

. The local stability of DFE at

R_0=1

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{0} = 1 $$\end{document}

has been analyzed, and it is obtained that DFE exhibits a forward transcritical bifurcation. Further, we identify conditions for the existence of EE and show the local stability of EE under certain conditions. Moreover, the global stability behavior of DFE and EE has been investigated. Lastly, numerical simulations have been done in the support of our theoretical findings.

The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincaré–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Köthe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.

A subgroup $E$ of a finite group $G$ is called hypercyclically embedded in $G$ if every chief factor of $G$ below $E$ is cyclic. Let $A$ be a subgroup of a group $G$. Then we call any chief factor $H/A_{G}$ of $G$ a $G$-boundary factor of $A$. For any $G$-boundary factor $H/A_{G}$ of $A$, we call the subgroup $(A\cap H)/A_{G}$ of $G/A_{G}$ a $G$-trace of $A$. On the basis of these notions, we give some new characterizations of hypercyclically embedded subgroups.

A Berry–Esseen bound is obtained for self-normalized martingales under the assumption of finite moments. The bound coincides with the classical Berry–Esseen bound for standardized martingales. An example is given to show the optimality of the bound. Applications to Student’s statistic and autoregressive process are also discussed.

In this note we present various extensions of Obata’s rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata’s theorem. Besides analyzing the full rigidity case, we also characterize the geometry and topology of the underlying manifold in more general situations.

Let $pod_3(n)$ denote the number of 3-regular partitions with distinct odd parts (and even parts are unrestricted) of a non-negative integer n. In this paper, we present infinite families of Ramanujan-type congruences modulo 2 and 3 for $pod_3(n)$.

Let $\varGamma $ be a distance-regular graph of diameter 3 with strong regular graph $\varGamma _3$. The determination of the parameters $\varGamma _3$ over the intersection array of the graph $\varGamma $ is a direct problem. Finding an intersection array of the graph $\varGamma $ with respect to the parameters $\varGamma _3$ is an inverse problem. Previously, inverse problems were solved for $\varGamma _3$ by Makhnev and Nirova. In this paper, we study the intersection arrays of distance-regular graph $\varGamma $ of diameter 3, for which the graph ${\bar{\varGamma }}_3$ is a pseudo-geometric graph of the net $PG_{m}(n, m)$. New infinite series of admissible intersection arrays for these graphs are found. We also investigate the automorphisms of distance-regular graph with the intersection array $\{20,16,5; 1,1,16 \}$.

This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta +u{\dot{W}}$ on $[0, \infty )\times {{\mathbb {R}}}^d $ with $d\ge 1$ has a unique random field solution, where W(t, x) is a fractional Brownian sheet on $[0, \infty )\times {{\mathbb {R}}}^d$ and formally $\dot{W} =\frac{\partial ^{d+1}}{\partial t \partial x_1 \cdots \partial x_d} W(t, x)$. When the noise W(t, x) is white in time, our condition is both necessary and sufficient when the initial data u(0, x) is bounded between two positive constants. When the noise is fractional in time with Hurst parameter $H_0>1/2$, our sufficient condition, which improves the known results in the literature, is different from the necessary one.

For discrete spectrum of 1D second-order differential/difference operators (with or without potential (killing), with the maximal/minimal domain), a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators. Interestingly, these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier, simply replacing the ‘finite supremum’ by ‘vanishing at infinity’. Except a dual technique, the main tool used here is a transform in terms of the harmonic function, to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context. All of them are illustrated by examples. The main body of the paper is devoted to the hard part of the story, the easier part but powerful one is delayed to the end of the paper.

In this paper, we will propose a way to accurately model certain naturally occurring collections of data. Through this proposed model, the proportion of d as leading digit, $d\in \llbracket 1,9\rrbracket $, in data is more likely to follow a law whose probability distribution is determined by a specific upper bound, rather than Benford’s Law, as one might have expected. These probability distributions fluctuate nevertheless around Benford’s values. These peculiar fluctuations have often been observed in the literature in such data sets (where the physical, biological or economical quantities considered are upper bounded). Knowing beforehand the value of this upper bound enables to find, through the developed model, a better adjusted law than Benford’s one.

In this paper, a new 4-parameter exponentiated generalized inverse flexible Weibull distribution is proposed. Some of its statistical properties are studied. The aim of this paper is to estimate the model parameters via several approaches, namely, maximum likelihood, maximum product spacing and Bayesian. According to Bayesian approach, several techniques are used to get the Bayesian estimators, namely, standard error function, Linex loss function and entropy loss function. The estimation herein is based on complete and censored samples. Markov Chain Monte Carlo simulation is used to discuss the behavior of the estimators for each approach. Finally, two real data sets are analyzed to obtain the flexibility of the proposed model.

Obtaining correct responses to sensitive questions in social and behavioural researches is an ancient problem in survey research with respondents misreporting on sensitive behaviours or giving false response to protect themselves. This paper develops an alternative unbiased estimator by modifying the dichotomous randomized response technique model to tackle this problem. The proposed estimator was compared numerically with conventional ones by considering different practicable and suitable design choices. Proposed model was also considered when sampling with unequal probabilities with or without replacement. It was observed that the proposed estimator performs efficiently than the conventional ones. As the proposed model captures progressively more people involved in the sensitive attribute, the model outperforms other models considered. Therefore, social and behavioural researchers can now obtain correct and valid responses from sensitive behavioural researches with ease in order to make informed and reliable decisions.

In the present paper, we propose an efficient scrambled estimator of population mean of quantitative sensitive study variable, using general linear transformation of non-sensitive auxiliary variable. Efficiency comparisons with the existing estimators have been carried out both theoretically and numerically. It has been found that our optimal scrambled estimator is always more efficient than most of the existing scrambled estimators and also it is more efficient than few other scrambled estimators under some conditions.