The UK is the most important partner of the EU in terms of economic and other fields due to the geographical proximity. It was one of the largest economies in the EU and its per capita income is higher than the EU average, so it is a net contributor to the EU. With UKs membership of the EU ended on 31 January 2019, there are concerns that the Brexit may have a significant impact on the EU, resulting in social, economic, political, and institutional changes, etc. in EU. While the impact of Brexit on the UK has always been the subject of considerable scholarly interest in recent years, there is relatively little literature on the impact of Brexit on the EU. This paper focuses on the evaluation of the impact of Brexit on the EU economy and other relevant aspects along three dimensions: GDP, PPP, Quarterly GDP growth. Employing powerful quantitative analysis technology that includes vector autoregression model, multivariate time series model with intervention variables, and autoregression integrated moving average, this paper obtains the important and novel evidence about the potential impact of Brexit on the EU economy, pointing out that Brexit is of far-reaching significance to the EU. This analysis uses several statistical models to screen out several key influencing factors, which can be used to predict the total GDP of EU in the next five years. The results show that EU economy will react negatively to "no-deal" Brexit, and its growth rate of economy will slow down significantly in next 5 years. Finally, we put forward relevant policy suggestions on how to deal with the negative impact of Brexit on EU.

In this paper, we consider the initial value problem of a class of fractional differential equations. Firstly, we obtain the existence and uniqueness of the solutions by using Picard’s method of successive approximation. Then we discuss the dependence of the solutions on the initial value.

Frequentist model averaging has received much attention from econometricians and statisticians in recent years. A key problem with frequentist model average estimators is the choice of weights. This paper develops a new approach of choosing weights based on an approximation of generalized cross validation. The resultant least squares model average estimators are proved to be asymptotically optimal in the sense of achieving the lowest possible squared errors. Especially, the optimality is built under both discrete and continuous weigh sets. Compared with the existing approach based on Mallows criterion, the conditions required for the asymptotic optimality of the proposed method are more reasonable. Simulation studies and real data application show good performance of the proposed estimators.

This paper investigates the stabilization and synchronization of two fractional chaotic maps proposed recently, namely the 2D fractional Hénon map and the 3D fractional generalized Hénon map. We show that although these maps have non–identical dimensions, their synchronization is still possible. The proposed controllers are evaluated experimentally in the case of non–identical orders or time–varying orders. Numerical methods are used to illustrate the results.

Anomalous diffusion is a widespread physical phenomenon, and numerical methods of fractional diffusion models are of important scientific significance and engineering application value. For time fractional diffusion-wave equation with damping, a difference (ASC-N, alternating segment Crank-Nicolson) scheme with intrinsic parallelism is proposed. Based on alternating technology, the ASC-N scheme is constructed with four kinds of Saul’yev asymmetric schemes and Crank-Nicolson (C-N) scheme. The unconditional stability and convergence are rigorously analyzed. The theoretical analysis and numerical experiments show that the ASC-N scheme is effective for solving time fractional diffusion-wave equation.

Let G be a simple connected graph with order n. Let L(G) and Q(G) be the normalized Laplacian and normalized signless Laplacian matrices of G, respectively. Let λ_k(G) be the k-th smallest normalized Laplacian eigenvalue of G. Denote by ρ(A) the spectral radius of the matrix A. In this paper, we study the behaviors of λ₂(G) and ρ(L(G)) when the graph is perturbed by three operations. We also study the properties of ρ(L(G)) and X for the connected bipartite graphs, where X is a unit eigenvector of L(G) corresponding to ρ(L(G)). Meanwhile we characterize all the simple connected graphs with ρ(L(G)) = ρ(Q(G)).

Let a, b, k be nonnegative integers with 2 ≤ a < b. A graph G is called a k-Hamiltonian graph if G - U contains a Hamiltonian cycle for any subset U ⊆ V (G) with |U| = k. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. If G - U admits a Hamiltonian [a, b]-factor for any subset U ⊆ V (G) with |U| = k, then we say that G has a k-Hamiltonian [a, b]-factor. Suppose that G is a k-Hamiltonian graph of order n with n ≥ ((a+b-4)(2a+b+k-6))/(b-2) + k and δ(G) ≥ a + k. In this paper, it is proved that G admits a k-Hamiltonian [a, b]-factor if max{d_G(x),d_G(y)} ≥ ((a-2)n+(b-2)k)/(a+b-4) + 2 for each pair of nonadjacent vertices x and y in G.

This paper is concerned with the study of optimality conditions for minimax optimization problems with an infinite number of constraints, denoted by (MMOP). More precisely, we first establish necessary conditions for optimal solutions to the problem (MMOP) by means of employing some advanced tools of variational analysis and generalized differentiation. Then, sufficient conditions for the existence of such solutions to the problem (MMOP) are investigated with the help of generalized convexity functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, some of the obtained results are applied to formulating optimality conditions for weakly efficient solutions to a related multiobjective optimization problem with an infinite number of constraints, and a necessary optimality condition for a quasi ε-solution to problem (MMOP).

In this paper, we develop the quantile regression (QR) estimation for the first-order integer-valued autoregressive (INAR(1)) models by defining the smoothing INAR(1) process. Jittering method is used to derive the QR estimators for the autoregressive coefficient and the quantile of innovations. The consistency and asymptotic normality of the proposed estimators are established. The performances of the proposed estimation procedures are evaluated by Monte Carlo simulations. The results show that the proposed procedures perform well for simulations and a real data application.

In this paper, we explore two conjectures about Rademacher sequences. Let (ε_i) be a Rademacher sequence, i.e., a sequence of independent {-1, 1}-valued symmetric random variables. Set S_n=a₁ε₁+…+a_nε_n for a=(a₁, …, a_n) ∈ Rⁿ. The first conjecture says that P (|S_n|≤||a||) ≥ 1/2 for all a ∈ Rⁿ and n ∈ N. The second conjecture says that P (|S_n|≥||a||) ≥ 7/32 for all a ∈ Rⁿ and n ∈ N. Regarding the first conjecture, we present several new equivalent formulations. These include a topological view, a combinatorial version and a strengthened version of the conjecture. Regarding the second conjecture, we prove that it holds true when n ≤ 7.

In epidemiological and clinical studies, the restricted mean lifetime is often of direct interest quantity. The differences of this quantity can be used as a basis of comparing several treatment groups with respect to their survival times. When the factor of interest is not randomized and lifetimes are subject to both dependent and independent censoring, the imbalances in confounding factors need to be accounted. We use the mixture of additive hazards model and inverse probability of censoring weighting method to estimate the differences of restricted mean lifetime. The average causal effect is then obtained by averaging the differences in fitted values based on the additive hazards models. The asymptotic properties of the proposed method are also derived and simulation studies are conducted to demonstrate their finite-sample performance. An application to the primary biliary cirrhosis (PBC) data is illustrated.

The admissibility of the initial-boundary data, which characterizes the existence of solution for the initial-boundary value problem, is important. Based on the Fokas method and the inverse scattering transformation, an approach is developed to solve the initial-boundary value problem of the nonlinear Schrödinger equation on a finite interval. A necessary and sufficient condition for the admissibility of the initial-boundary data is given, and the reconstruction of the potential is obtained.

Cost effective sampling design is a problem of major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming. In the current paper, a modification of ranked set sampling (RSS) called moving extremes RSS (MERSS) is considered for the estimation of the location parameter for location family. A maximum likelihood estimator (MLE) of the location parameter for this family is studied and its properties are obtained. We prove that the MLE is an equivariant estimator under location transformation. In order to give more insight into the performance of MERSS with respect to (w.r.t.) simple random sampling (SRS), the asymptotic efficiency of the MLE using MERSS w.r.t. that using SRS is computed for some usual location distributions. The relative results show that the MLE using MERSS can be real competitors to the MLE using SRS.

Stochastic gradient descent (SGD) is one of the most common optimization algorithms used in pattern recognition and machine learning. This algorithm and its variants are the preferred algorithm while optimizing parameters of deep neural network for their advantages of low storage space requirement and fast computation speed. Previous studies on convergence of these algorithms were based on some traditional assumptions in optimization problems. However, the deep neural network has its unique properties. Some assumptions are inappropriate in the actual optimization process of this kind of model. In this paper, we modify the assumptions to make them more consistent with the actual optimization process of deep neural network. Based on new assumptions, we studied the convergence and convergence rate of SGD and its two common variant algorithms. In addition, we carried out numerical experiments with LeNet-5, a common network framework, on the data set MNIST to verify the rationality of our assumptions.

In this paper, we concern the Klein-Gordon-Maxwell system with steep potential well
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Without global and local compactness, we can tell the difference of multiple solutions from their norms in L^p(R³).