We develop a systematic method for deriving the phase dynamics of perturbed periodic solutions. The method is to regard periodic solutions as slowly modulated traveling solutions on the circle. There, problems are reduced to the perturbed problems from stationary solutions on the circle. This makes the treatment of periodic solutions far easier and systematic. We also give the rigorous proofs for this method.
The purpose of this article is twofold. The first is to show the Second Main Theorem for degenerate holomorphic curves into Pn(C) with hypersurface targets located in n-subgeneral position. The second is to show the Second Main Theorem with truncated counting functions for nondegenerate holomorphic curves into Pn(C) with hypersurface targets in general position. Finally, by applying the above result, a unicity theorem for algebraically nondegenerate curves into P2(C) with hypersurface targets in general position is also given.
This paper is concerned with partial regularity for weak solutions to nonlinear sub-elliptic systems related to Hörmander' s vector fields. The method of A-harmonic approximation introduced by Simon and developed by Duzaar and Grotowski is adapted to our context, and then a Caccioppoli-type inequality and partial regularity with optimal local Hölder exponent for gradients of weak solutions to the systems under super-quadratic natural growth conditions is established.
Many of the properties of a Weyl algebra An over a base field of non-zero characteristic are explained in terms of connections and curvatures on a vector bundle on an affine space X = A2n. In particular, it is known that an algebra endomorphism φ of An gives rise to a symplectic endomorphism f of X with a gauge transformation g. In this paper we study the converse problem of finding φ from an arbitrary symplectic endomorphism f of X = A2n. It is shown that, given such f , we may construct a projective left An-module (which corresponds to ‘the sheaf of local gauge transformations' ) such that its triviality is equivalent to the existence of the ‘lift' φ. Some properties of such a module will be discussed using the theory of reflexive sheaves.
There are infinitely many generalized hypergeometric differential equations 3E2(a0, a1, a2; b1, b2; x) having the same finite irreducible primitive monodromy group. Their Schwarz images are different plane curves. In this paper, we study the degrees of these curves. We also give the defining equations of Schwarz images of the minimal degrees.
For 1 ≤ p < +∞,every ƒ (≠ 0) ∈Lp(R,dx) defines a sequence space Λp(ƒ) (Honda et al. Proc. Japan Acad. Ser. A 84 (2008), 39-41) which is an additive group but not necessarily a linear space. The main purpose of this paper is to discuss the linearity of Λp(ƒ). First we show that if ƒ is a piecewise monotone function, then Λp(ƒ) is a linear space. Next, specializing the case to p = 2, we characterize Λ2(ƒ) as a set, and discuss the linearity of it. With this aim, we extend the definition of the doubling condition and define the doubling dimension H(φ) of a non-negative function on [0, +∞).Let ƒ be the Fourier transform of ƒ and define a function φƒ associated with ƒ. Then we show that H(|ƒ|)< ∞ implies the linearity of Λ2(ƒ). In addition, we show that if H(φƒ)< 2, then Λ2(ƒ) is linear and give several examples.
In this paper, an asymptotic expansion of the Bergman function at a degenerate point is given for high powers of semipositive holomorphic line bundles on compact Kähler manifolds, whose Hermitian metrics have some kind of quasihomogeneous properties. In the sense of pointwise asymptotics, this expansion is a generalization of the expansion of Tian-Zelditch-Catlin-Lu in the positive line bundle case.
We consider a flat Riemannian manifold M of dimension bigger than two, and a closed and connected subgroup G of the isometries of M, such that the orbit space of the action of G on M is two dimensional. Then we study topological properties of M and the G-orbits of M.
Non-local Dirichlet forms with appropriately chosen jump kernels are used to define Markov pure jump processes on metric measure spaces that do not necessarily possess uniform volume growth. They may be seen as generalizations of stable (or stable-like) processes. Scaling properties and estimates on mean exit and hitting times are established. For some cases they provide enough information to conclude the continuity of related harmonic functions. Typical ultracontractivity arguments entail the existence of transition densities; their joint continuity is then deduced from the preceding results.