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In this paper, we investigate a class of nonlinear fractional differential equations with integral boundary condition. By means of Krasnosel’skiĭ fixed point theorem and contraction mapping principle we prove the existence and uniqueness of solutions for a nonlinear system. By means of Bielecki-type metric and the Banach fixed point theorem we investigate the Ulam–Hyers and Ulam–Hyers–Rassias stability of nonlinear fractional differential equations. Besides, we discuss an example for illustration of the main work.

Let V be a smooth variety. A hypersurface arrangement 𝓜 in V is a union of smooth hypersurfaces, which locally looks like a union of hyperplanes. We say 𝓜 is free if all these local models can be chosen to be free hyperplane arrangements. In this paper, we use Saito’s criterion to study the freeness of hypersurface arrangements consisting of hyperplanes and spheres, and construct the bases for the derivation modules explicitly.

In this paper, we define the supersolvable order of hyperplanes in a supersolvable arrangement, and obtain a class of inductively free arrangements according to this order. Our main results improve the conclusion that every supersolvable arrangement is inductively free. In addition, we assert that the inductively free arrangement with the required induction table is supersolvable.

In this paper, we generalize the algorithm described by Rump and Graillat to compute verified and narrow error bounds such that a slightly perturbed matrix is guaranteed to have an eigenvalue with geometric multiplicity q within computed error bounds. The corresponding invariant subspace can be directly obtained by our algorithm. Our verification method is based on border matrix technique. We demonstrate the performance of our algorithm for matrices of dimension up to hundreds with non-defective and defective eigenvalues.

The position vectors of regular rectifying curves always lie in their rectifying planes. These curves were well investigated by B.Y.Chen. In this paper, the concept of framed rectifying curves is introduced, which may have singular points. We investigate the properties of framed rectifying curves and give a method for constructing framed rectifying curves. In addition, we reveal the relationships between framed rectifying curves and some special curves.

We consider one-parameter null hypersurfaces associated with spacelike curves. The spacelike curves are in anti-de Sitter 3-space while one-parameter null hypersurfaces lie in 4-dimensional semi-Euclidean space with index 2. We classify the generic singularities of the hypersurfaces, which are cuspidal edges and swallowtails. And we reveal the geometric meanings of the singularities of such hypersurfaces by the singularity theory.

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation $-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u ^{\alpha(x)}}$ with $f\in L^{m}(\Omega)$ ( $m\geqslant1$ ) and $\alpha(x)>0$ . Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent $\alpha(x)$ , some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of $\alpha(x)$ .

In this paper, we analyze the boundary value problem of a class of multi-order fractional differential equations involving the standard Caputo fractional derivative with the general periodic boundary conditions: $$ \textstyle\begin{cases} L(D)u(t) = f(t,u(t)),\quad t\in[0,T], T>0, \\ u(0) = u(T)>0,\qquad u'(0)=u'(T)>0, \end{cases} $$ where $L(D)=\sum^{n}_{i=0}a_{i}D^{S_{i}}$ , $1\leq S_{0}

This paper considers an optimal control problem for a class of controlled hybrid dynamical systems (HDSs) with prescribed switchings. By using Ekeland’s variational principle and a matrix cost functional, a minimum principle for HDSs is derived, which provides a necessary condition of the aforementioned problem. The results given in this paper include both pure continuous systems and pure discrete-time systems as special cases.

This thesis takes researches on Network Intrusion Detection Based On Cluster Analysis And Multiple Core Set. The distributed immune intrusion detection system and the packet marking theory are used to find out the network data features of the real-time analyses; and the immune intrusion detection system is used to guide the dynamically processing of path signs technology; what's more, the path signs technology is adopted to dynamically adaptive different methods of characteristics of network data. After that, the attack paths can be quickly identified to provide path information for feature detector on attack path in the immune intrusion detection system. Experiment results show that this scheme can quickly reconstruct the attack path information, and the performance on the aspects of the convergence is with efficiency rate and false positive rate, which is superior to the current probabilistic packet marking algorithm and can provide characteristic path information for immune intrusion detection system.

In this paper, we give a basis for the derivation module of the cone over the Shi arrangement of the type $D_{\ell}$ explicitly.