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.

Generalized exponential distribution is a class of important distribution in lifedata analysis, especially in some skewed lifedata. The Parameter estimation problem for generalized exponential distribution model with grouped and right-censored data is considered. The maximum likelihood estimators are obtained using the EM algorithm. Some simulations are carried out to illustrate that the proposed algorithm is effective for the model. Finally, a set of medicine data is analyzed by generalized exponential distribution.

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.

We consider the existence of cluster-tilting objects in a d-cluster category such that its endomorphism algebra is self-injective, and also the properties for cluster-tilting objects in d-cluster categories. We get the following results: (1) When d>1, any almost complete cluster-tilting object in d-cluster category has only one complement. (2) Cluster-tilting objects in d-cluster categories are induced by tilting modules over some hereditary algebras. We also give a condition for a tilting module to induce a cluster-tilting object in a d-cluster category. (3) A 3-cluster category of finite type admits a cluster-tilting object if and only if its type is A_1,A_3,D_{2n-1}\left(n>2\right). (4) The (2m+1)-cluster category of type D_{2n-1} admits a cluster-tilting object such that its endomorphism algebra is self-injective, and its stable category is equivalent to the (4m+2)-cluster category of type A_{4mn-4m+2n-1}.

This paper gives the truncated version of the generalized minimum backward error algorithm (GMBACK)—the incomplete generalized minimum backward perturbation algorithm (IGMBACK) for large nonsymmetric linear systems. It is based on an incomplete orthogonalization of the Krylov vectors in question, and gives an approximate or quasi-minimum backward perturbation solution over the Krylov subspace. Theoretical properties of IGMBACK including finite termination, existence and uniqueness are discussed in details, and practical implementation issues associated with the IGMBACK algorithm are considered. Numerical experiments show that, the IGMBACK method is usually more efficient than GMBACK and GMRES, and IMBACK, GMBACK often have better convergence performance than GMRES. Specially, for sensitive matrices and right-hand sides being parallel to the left singular vectors corresponding to the smallest singular values of the coefficient matrices, GMRES does not necessarily converge, and IGMBACK, GMBACK usually converge and outperform GMRES.