Limit Cycles on Piecewise Smooth Vector Fields with Coupled Rigid Centers

We provide an upper bound for the maximum number of limit cycles of the class of discontinuous piecewise differential systems formed by two differential systems separated by a straight line presenting rigid centers. These two rigid centers are polynomial differential systems with a linear part and a nonlinear homogeneous part. We study the maximum number of limit cycles that such a class of piecewise differential systems can exhibit.

The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

Planar vector fields with central symmetry: roughness and first degree of non-roughness

Рассматривается пространство гладких векторных полей, заданных в замкнутой области D на плоскости, инвариантных относительно центральной симметрии и трансверсальных границе D. Описано множество векторных полей, грубых относительно этого пространства; показано, что оно открыто и всюду плотно. Во множестве всех негрубых векторных полей выделено открытое всюду плотное подмножество, состоящее из векторных полей первой степени негрубости. We consider the space of smooth vector fields defined in a closed domain D on the plane, invariant under the central symmetry and transversal to the boundary D. The set of vector fields that are rough with respect to this space is described; it is shown that it is open and everywhere dense. In the set of all non-rough vector fields, an open everywhere dense subset consisting of vector fields of the first degree of non-roughness is distinguished.

Transversal local rigidity of discrete abelian actions on Heisenberg nilmanifolds

In this paper we prove a perturbative result for a class of ${\mathbb Z}^2$ actions on Heisenberg nilmanifolds that have Diophantine properties. Along the way we prove cohomological rigidity and obtain a tame splitting for the cohomology with coefficients in smooth vector fields for such actions.

Optimality conditions and Mond–Weir duality for a class of differentiable semi-infinite multiobjective programming problems with vanishing constraints

In this paper, the class of differentiable semi-infinite multiobjective programming problems with vanishing constraints is considered. Both Karush–Kuhn–Tucker necessary optimality conditions and, under appropriate invexity hypotheses, sufficient optimality conditions are proved for such nonconvex smooth vector optimization problems. Further, vector duals in the sense of Mond–Weir are defined for the considered differentiable semi-infinite multiobjective programming problems with vanishing constraints and several duality results are established also under invexity hypotheses.

G-α-preinvex functions and non-smooth vector optimization problems

In this paper, we proposed the non-smooth G-?-preinvexity by generalizing ?-invexity and G-preinvexity, and discussed some solution properties about non-smooth vector optimization problems and vector variational-like inequality problems under the condition of non-smooth G-?-preinvexity. Moreover, we also proved that the vector critical points, the weakly efficient points and the solutions of the non-smooth weak vector variational-like inequality problem are equivalent under non-smooth pseudo-G-?-preinvexity assumptions.

A two-weight Sobolev inequality for Carnot-Carathéodory spaces

Let $$X = \{X_1,X_2, \ldots ,X_m\}$$ X = { X 1 , X 2 , … , X m } be a system of smooth vector fields in $${{\mathbb R}^n}$$ R n satisfying the Hörmander’s finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Carathéodory space $$\mathbb G$$ G associated to system X$$\begin{aligned} \left( \frac{1}{\int _{B_R} K(x)\; dx} \int _{B_R} |u|^{t} K(x) \; dx \right) ^{1/t} \le C\, R \left( \frac{1}{\int _{B_R}\frac{1}{K(x)} \; dx} \int _{B_R} \frac{|X u|^2}{K(x)} \; dx \right) ^{1/2}, \end{aligned}$$ 1 ∫ B R K ( x ) d x ∫ B R | u | t K ( x ) d x 1 / t ≤ C R 1 ∫ B R 1 K ( x ) d x ∫ B R | X u | 2 K ( x ) d x 1 / 2 , where Xu denotes the horizontal gradient of u with respect to X. We assume that the weight K belongs to Muckenhoupt’s class $$A_2$$ A 2 and Gehring’s class $$G_{\tau }$$ G τ , where $$\tau $$ τ is a suitable exponent related to the homogeneous dimension.