The extended t-process regression model is developed to robustly model functional data with outlier functional curves. This paper applies Bayesian estimation to propose an estimation procedure for the model with independent errors. A Monte Carlo EM method is built to estimate parameters involved in the model. Simulation studies and real examples show the proposed method performs well against outliers.

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.

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.

Let $d^*_k(x)$ be the most likely common differences of arithmetic progressions of length $k+1$ among primes $\le x$. Based on the truth of Hardy–Littlewood Conjecture, we obtain that $\lim \limits _{x\rightarrow +\infty }d^*_k(x)=+\infty $ uniformly in k, and every prime divides all sufficiently large most likely common differences.

This paper provides the explicit and optimal quadrature rules for the cubic $C^1$ spline space, which is the extension of the results in Ait-Haddou et al. (J Comput Appl Math 290:543–552, 2015) for less restricted non-uniform knot values. The rules are optimal in the sense that there exist no other quadrature rules with fewer quadrature points to exactly integrate the functions in the given spline space. The explicit means that the quadrature nodes and weights are derived via an explicit recursive formula. Numerical experiments and the error estimations of the quadrature rules are also presented in the end.

In this paper, we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre’s conjecture will be discussed for a particular class of skew PBW extensions. The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings. Advances for the solution of some of the problems are included.