An n×n matrix A consisting of nonnegative integers is a general magic square of ordern if the sum of elements in each row, column, and main diagonal is the same. A general magic square A of order n is called a magic square, denoted by MS(n), if the entries of A are distinct. A magic square A of order n is normal if the entries of A are n^{2} consecutive integers. Let A^{*d} denote the matrix obtained by raising each element of A to the d-th power. The matrix A is a d-multimagic square, denoted by MS(n, d), if A^{*e} is an MS(n) for 1 ≤ e ≤ d. In this paper we investigate the existence of normal bimagic squares of order 2u and prove that there exists a normal bimagic square of order 2u, where u and 6 are coprime and u ≥ 5.
The frequentist model averaging (FMA) and the focus information criterion (FIC) under a local framework have been extensively studied in the likelihood and regression setting since the seminal work of Hjort and Claeskens in 2003. One inconvenience, however, of the existing works is that they usually require the involved criterion function to be twice differentiable which thus prevents a direct application to the case of quantile regression (QR). This as well as some other intrinsic merits of QR motivate us to study the FIC and FMA in a locally misspecified linear QR model. Specifically, we derive in this paper the explicit asymptotic risk expression for a general submodel-based QR estimator of a focus parameter. Then based on this asymptotic result, we develop the FIC and FMA in the current setting. Our theoretical development depends crucially on the convexity of the objective function, which makes possible to establish the asymptotics based on the existing convex stochastic process theory. Simulation studies are presented to illustrate the finite sample performance of the proposed method. The low birth weight data set is analyzed.
An orbit code is a special constant dimension subspace code, which is an orbit of a subgroup of a general linear group acting on the set of all subspaces in the given ambient space. This paper presents some methods of constructing new orbit codes from known orbit codes. Firstly, we introduce the sum operation, intersection operation and union operation of subspace codes, and then we give some methods to obtain new orbit codes from known orbit codes by fully applying the sub-orbits of permutation groups and the direct product operation of the groups. Finally, as a special application, partial spread codes are researched and a condition of orbit codes with constant distance is given.
The paper gives a new method to identify significant effects in two-level factorial experiments, and compares the new method with the exiting methods using Monte Carlo simulation. The existing methods only perform well under the assumption of effect sparsity, but the new method performs well without effect sparsity.
In this paper, we give a Gröbner-Shirshov basis of quantum group of type C_{3} by using the RingelHall algebra approach. For this, first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type C_{3} by using an “inductive” method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew-commutator relations; contrarily, we compute the “easier” skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first, then “inductive” others from these “easier” ones and this in turn gives Hall polynomials as a byproduct. Then we prove that the set of these relations is closed under composition. So they constitutes a minimal Gröbner-Shirshov basis of the positive part of quantum group of type C_{3}. Dually, we get a Gröbner-Shirshov basis of the negative part of quantum group of type C_{3}. And finally we give a Gröbner-Shirshov basis for the whole quantum group of type C_{3}.
In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and time-dependent coefficients utt + △^{2}u + k_{1}(t)|u_{t}|^{m-2}u_{t} = k_{2}(t)|u|^{p-2}u, x ∈ Ω, t > 0, where Ω is a bounded domain in R^{n}. Under certain conditions on k_{1}(t), k_{2}(t) and the initial-boundary data, the upper bound for blow-up time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if p > m. Also, a lower bound of blow-up time are obtained by using a Sobolev-type inequality and a first order differential inequality technique for n = 2, 3 and n > 4.
In the paper, we establish some exponential inequalities for non-identically distributed negatively orthant dependent (NOD, for short) random variables. In addition, we also establish some exponential inequalities for the partial sum and the maximal partial sum of identically distributed NOD random variables. As an application, the Kolmogorov strong law of large numbers for identically distributed NOD random variables is obtained. Our results partially generalize or improve some known results.
We consider a nonlocal boundary value problem for a viscoelastic equation with a Bessel operator and a weighted integral condition and we prove a general decay result. We also give an example to show that our general result gives the optimal decay rate for ceratin polynomially decaying relaxation functions. This result improves some other results in the literature.
In this paper, we consider the existence and uniqueness of the mild solutions for a class of fractional non-autonomous evolution equations with delay and Caputo’s fractional derivatives. By using the measure of noncompactness, β-resolvent family, fixed point theorems and Banach contraction mapping principle, we improve and generalizes some related results on this topic. At last, we give an example to illustrate the application of the main results of this paper.
A path in an edge-colored graph G is called a rainbowpath if no two edges of the path are colored the same color. The minimum number of colors required to color the edges of G such that every pair of vertices are connected by at least k internally vertex-disjoint rainbow paths is called the rainbow k-connectivity of the graph G, denoted by rc_{k}(G). For the random graph G(n, p), He and Liang got a sharp threshold function for the property rc_{k}(G(n, p)) ≤ d. For the random equi-bipartite graph G(n, n, p), Fujita et. al. got a sharp threshold function for the property rc_{k}(G(n, n, p)) ≤ 3. They also posed the following problem: For d ≥ 2, determine a sharp threshold function for the property rc_{k}(G) ≤ d, where G is another random graph model. This paper is to give a solution to their problem in the general random bipartite graph model G(m, n, p).
A new kind of tangent derivative, M-derivative, for set-valued function is introduced with help of a modified Dubovitskij-Miljutin cone. Several generalized pseudoconvex set-valued functions are introduced. When both the objective function and constraint function are M-derivative, under the assumption of near conesubconvexlikeness, by applying properties of the set of strictly efficient points and a separation theorem for convex sets, Fritz John and Kuhn-Tucker necessary optimality conditions are obtained for a point pair to be a strictly efficient element of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn-Tucker sufficient optimality condition is obtained for a point pair to be a strictly efficient element of set-valued optimization problem.
This paper considers a competing risks model for right-censored and length-biased survival data from prevalent sampling. We propose a nonparametric quantile inference procedure for cause-specific residual life distribution with competing risks data. We also derive the asymptotic properties of the proposed estimators of this quantile function. Simulation studies and the unemployment data demonstrate the practical utility of the methodology.
In recent years, there has been a large amount of literature on missing data. Most of them focus on situations where there is only missingness in response or covariate. In this paper, we consider the adequacy check for the linear regression model with the response and covariates missing simultaneously. We apply model adjustment and inverse probability weighting methods to deal with the missingness of response and covariate, respectively. In order to avoid the curse of dimension, we propose an empirical process test with the linear indicator weighting function. The asymptotic properties of the proposed test under the null, local and global alternative hypothetical models are rigorously investigated. A consistent wild bootstrap method is developed to approximate the critical value. Finally, simulation studies and real data analysis are performed to show that the proposed method performed well.
In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.
In this paper we prove that every compact invariant subset A associated with the semigroup {S_{n,k}(t)}_{t≥0} generated by wave equations with variable damping, either in the interior or on the boundary of the domain Ω, where Ω⊂R^{3} is a smooth bounded domain, in H_{1}^{0} (Ω)×L^{2}(Ω) is in fact bounded in D(B_{0})×H_{1}^{0}(Ω). As an application of our results, we obtain the upper-semicontinuity for global attractor of the weakly damped semilinear wave equation in the norm of H^{1}(Ω)×L^{2}(Ω) when the interior variable damping converges to the boundary damping in the sense of distributions.
In present note, we apply the Leibniz formula with the nabla operator in discrete fractional calculus (DFC) due to obtain the discrete fractional solutions of a class of associated Bessel equations (ABEs) and a class of associated Legendre equations (ALEs), respectively. Thus, we exhibit a new solution method for such second order linear ordinary differential equations with singular points.
In the paper, we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Littlewood maximal function Mf and the centered Hardy-Littlewood maximal function M_{c}f on R^{n}. As two applications, we can easily deduce that M_{c}f and Mf are continuous if f is continuous, and Mf is continuous if f is of local bounded variation on R.