We study germs of holomorphic vector fields at the origin of ℂ^3 that arise from singular second-order ordinary differential equations and are tangent to codimension-one holomorphic foliations. Such a pair is called a 2-flag. Under a natural primitivity condition, we prove that the singular set of the codimension-one foliation is contained in that of the one-dimensional foliation. We also prove rigidity of the meromorphic quotient defining the equation and exhibit several nonlinear families that admit 2-flags. Finally, explicit transversality and trace conditions yield a nonconstant formal first integral, and hence a formal 2-flag, for a class of higher-order equations.
We establish the existence of Carathéodory–Filippov solutions for various types of boundary value problems involving second-order ordinary differential equations with discontinuous right-hand sides. By a Carathéodory–Filippov solution, we mean a function that is simultaneously a Carathéodory solution and a Filippov solution. Our main results rely on novel local transversality conditions on the discontinuities of the nonlinear terms, combined with classical tools such as the method of lower and upper solutions and Nagumo conditions. We also prove the existence of extremal Carathéodory–Filippov solutions between given lower and upper solutions, namely, the least and the greatest solutions.
We introduce a new class of state-dependent delay differential equations for which the delay associated to the term (t, u(t)) is described by an iteration process. We study the existence and uniqueness of solutions and in the last section some examples on partial evolution differential equations with state-dependent delay are presented.
We characterize the convexity of the Berezin range of finite-rank operators acting on the Fock space. First, it is proved that the Berezin range of the rank-one operator T(f)=⟨ f,z^n ⟩ z^n acting on the Fock space ℱ^2(ℂ), is convex in ℂ. The Berezin range of this operator gradually expands with increasing n, and for sufficiently large values of n, it almost becomes the whole of the positive real axis. We have also established the Berezin range of the finite-rank operator T(f)=∑ _i=1^n ⟨ f,a_iz^i ⟩ a_iz^i, a_i ∈ℂ, acting on ℱ^2(ℂ) , and surprisingly we have found a connection between this Berezin range and the only positive real root of the algebraic equation 1+x+2x^2+3x^3+⋯ +(n-1)x^n-1-x^n=0. It has been shown that the Berezin range of the rank-one operator of the form T(f)=⟨ f,z^m ⟩ z^n, m,n ∈ℕ, which is either a disc or an interval depending on the values of m and n, is always convex in ℂ . Furthermore, it has been proved that this Berezin range almost occupies the entire complex plane or the whole positive real axis for very large values of m and n. Several illustrative examples are discussed to support our theoretical findings. In addition, geometrical interpretations have also been employed.
Tensor-product Bernstein operators connect approximation of convex-valued maps on a hypercube with a degenerate diffusion in the space of support functions. We establish pointwise and uniform Voronovskaja formulas and quantitative Hausdorff estimates from exact centered moments, including the optimal fourth-moment constant. We also obtain simultaneous approximation of parameter derivatives. The saturation class consists of maps affine in each coordinate with respect to Minkowski addition; the complete approximation class of order n^-1 follows from the weighted scalar inequalities of Ditzian and Zhou. We characterize first-order Minkowski corrections and express the best correction errors as distances to the support-function cone and its negative. Spherical Hessian inequalities give sufficient conditions for generalized Hukuhara differences at large degrees. The associated strongly continuous convex-valued semigroup admits an Aumann-integral representation, and its Bernstein product formula converges uniformly over all non-negative times for each initial map. Its stationary states coincide with the saturation class. An explicit smooth planar family shows that both generalized Hukuhara relations can fail at every degree and between every pair of distinct times, despite positive curvature and exact asymptotic formulas.
