A combinatorial batch code has strong practical motivation in the distributed storage and retrieval of data in a database. In this survey, we give a brief introduction to the combinatorial batch codes and some progress.

Determining the search direction and the search step are the two main steps of the nonlinear optimization algorithm, in which the derivatives of the objective and constraint functions are used to determine the search direction, the one-dimensional search and the trust domain methods are used to determine the step length along the search direction. One dimensional line search has been widely discussed in various textbooks and references. However, there is a less-known technique—arc-search method, which is relatively new and may generate more efficient algorithms in some cases. In this paper, we will survey this technique, discuss its applications in different optimization problems, and explain its potential improvements over traditional line search method.

It is not completely clear which elements constitute the frame sets of the B-splines currently, but some considerable results have been obtained. In this paper, firstly, the background of frame set is introduced. Secondly, the main progress of the frame sets of the B-splines in the past more than twenty years are reviewed, and particularly the progress for the frame set of the 2 order B-spline and the frame set of the 3 order B-spline are explained, respectively.

In this paper,we give all primitive solutions of a parameterized family of quartic Thue equations:

　　　　　 x^4-4c{x}^{3}y+(6c+2)x^2{y}^2+4cx{y}^3+y^4=96c+169,c>0.

By utilizing the improvement function, we change the nonsmooth convex constrained optimization into an unconstrained optimization, and construct an infeasible quasi-Newton bundle method with proximal form. It should be noted that the objective function being minimized in unconstrained optimization subproblem may vary along the iterations (it does not change if the null step is made, otherwise it is updated to a new function). It is necessary to make some adjustment in order to obtain the convergence result. We employ the main idea of infeasible bundle method of Sagastizàbal and Solodov, and under the circumstances that each iteration point may be infeasible for primal problem, we prove that each cluster point of the sequence generated by the proposed algorithm is the optimal solution to the original problem. Furthermore, for BFGS quasi-Newton algorithm with strong convex objective function, we obtain the condition which guarantees the boundedness of quasi-Newton matrices and the R-linear convergence of the iteration points.

In this paper, a three-term derivative-free projection method is proposed for solving nonlinear monotone equations. Under some appropriate conditions, the global convergence and R-linear convergence rate of the proposed method are analyzed and proved. With no need of any derivative information, the proposed method is able to solve large-scale nonlinear monotone equations. Numerical comparisons show that the proposed method is effective.

In this note, we consider a class of Fourier integral operators with rough amplitudes and rough phases. When the index of symbols in some range, we prove that they are bounded on L^1 and construct an example to show that this result is sharp in some cases. This result is a generalization of the corresponding theorems of Kenig-Staubach and Dos Santos Ferreira-Staubach.

As the extension of classical Hardy operator and Cesàro operator, Hausdorff operator plays an important role in the harmonic analysis, so it is significant to discuss the boundedness of this kind of operator on various function spaces. The article explores the boundedness of a kind of Hausdorff operators on Lebesgue spaces and calculates the optimal constants for the operators to be bounded on such spaces. In addition, the paper also obtains the necessary and sufficient for a kind of multilinear Hausdorff operators to be bounded on Lebesgue spaces and their optimal constants.

In this work, we use the variant fountain theorem to study the existence of nontrivial solutions for the superquadratic fractional difference boundary value problem:

　　　　　　　　 \begin{array}{l}\{\begin{array}{l}{}_T{\mathrm{\Delta }}_{t-1}^{\nu }\left(t{\mathrm{\Delta }}_{\nu -1}^{\nu }x\left(t\right)\right)=f\left(x\right(t+\nu -1\left)\right),t\in [0,T{]}_{{\mathbb{N}}_0},\hfill \\ x(\nu -2)={\left[{}_t{\mathrm{\Delta }}_{\nu -1}^{\nu }x\left(t\right)\right]}_{t=T}=0.\hfill \end{array}\hfill \end{array}

The existence of nontrivial solutions is obtained in the case of super quadratic growth of the nonlinear term f by change of fountain theorem.

Let p and q be two distinct odd primes and let d=(p-1,q-1). In this paper, we construct d-ary generalized two-prime Sidelnikov sequences and study the autocorrelation values and linear complexity.