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.

In this paper, we investigate a class of nonlinear fractional differential equations with integral boundary condition. By means of Krasnosel'skiȋ 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.

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equationDue to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent α(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 α(x).

We apply singularity theory to study bifurcation problems with trivial solutions. The approach is based on a new equivalence relation called t-equivalence which preserves the trivial solutions. We obtain a sufficient condition for recognizing such bifurcation problems to be t-equivalent and discuss the properties of the bifurcation problems with trivial solutions. Under the action of t-equivalent group, we classify all bifurcation problems with trivial solutions of codimension three or less.

In this paper, we give a basis for the derivation module of the cone over the Shi arrangement of the type D ℓ explicitly.

We discuss in this paper a deterministic multi-group MSIR epidemic model with a vaccination rate, the basic reproduction number ℛ0, a key parameter in epidemiology, is a threshold which determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show if ℛ0 is greater than 1 and the deterministic model obeys some conditions, then the disease will prevail, the infective persists and the endemic state is asymptotically stable in a feasible region. If ℛ0 is less than or equal to 1, then the infective disappear so the disease dies out. In addition, stochastic noises around the endemic equilibrium will be added to the deterministic MSIR model in order that the deterministic model is extended to a system of stochastic ordinary differential equations. In the stochastic version, we carry out a detailed analysis on the asymptotic behavior of the stochastic model. In addition, regarding the value of ℛ0, when the stochastic system obeys some conditions and ℛ0 is greater than 1, we deduce the stochastic system is stochastically asymptotically stable. Finally, the deterministic and stochastic model dynamics are illustrated through computer simulations.

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.

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:where, a i ∈ R, a n = 0, and f : [0, T] × R → R is a continuous operation. We get the Green's function in terms of the Laplace transform. We obtain the existence and uniqueness of solution for the class of multi-order fractional differential equations. We investigate the blowing-up solutions to the special case f (t, u(t)) = |u(t)| p , a i ≥ 0, and give an upper bound on the blow-up time T max .