Springer Science and Business Media LLC

In this study, properties of spectral characteristic are studied for Sturm–Liouville operators with Coulomb potential which have discontinuity conditions inside a finite interval. Also Weyl function for this problem under consideration has been defined and uniqueness theorems for solution of inverse problem according to this function have been proved.

We prove an almost sure ergodic theorem for abstract quasistatic dynamical systems, as an attempt of taking steps toward an ergodic theory of such systems. The result at issue is meant to serve as a working counterpart of Birkhoff’s ergodic theorem which fails in the quasistatic setup. It is formulated so that the conditions, which essentially require sufficiently good memory-loss properties, could be verified in a straightforward way in physical applications. We also introduce the concept of a physical family of measures for a quasistatic dynamical system. These objects manifest themselves, for instance, in numerical experiments. We then illustrate the use of the theorem by examples.

We develop the method of vector-fields to further study Dispersive Wave Equations. Radial vector fields are used to get a-priori estimates such as the Morawetz estimate on solutions of Dispersive Wave Equations. A key to such estimates is the repulsiveness or nontrapping conditions on the flow corresponding to the wave equation. Thus this method is limited to potential perturbations which are repulsive, that is the radial derivative pointing away from the origin. In this work, we generalize this method to include potentials which are repulsive relative to a line in space (in three or higher dimensions), among other cases. This method is based on constructing multi-centered vector fields as multipliers, cancellation lemmas and energy localization.

We give a new proof that the resolvent of the renormalised Nelson Hamiltonian at fixed total momentum P improves positivity in the (momentum) Fock-representation, for every P. The argument is based on an explicit representation of the renormalised operator and its domain using interior boundary conditions, which allows us to avoid the intermediate steps of regularisation and renormalisation used in other proofs of this result.

We study Lagrangian points on smooth holomorphic curves in ${\rm T}{\mathbb P}^1$ equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of ${\rm T}{\mathbb P}^1$ with the space ${\mathbb L}({\mathbb E}^3)$ of oriented affine lines in Euclidean 3-space ${\mathbb E}^3$ , these Lagrangian curves give rise to ruled surfaces in ${\mathbb E}^3$ , which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in ${\mathbb E}^3$ , called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in ${\mathbb E}^3$ where the number of oriented lines in the complex curve Σ that pass through the point is less than the degree of Σ. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.

Các siêu đường trơn holomorphic được thúc đẩy bởi siêu hình học như một sự tổng quát tự nhiên của các đường trơn holomorphic. Trong việc định nghĩa như một bộ gồm một đường trơn holomorphic và một số phần, chúng tôi xem xét một dạng yếu hơn của các phương trình xác định. Điều này có nghĩa là, khi làm nhiễu chúng để phụ thuộc vào một kết nối, toán tử tuyến tính tương ứng thường là surjective. Nhờ kết quả chuyển giao này, chúng tôi chỉ ra rằng các không gian moduli thu được là những đa tạp mịn, định hướng và có chiều hữu hạn. Cuối cùng, chúng tôi xem xét cách mà chúng phụ thuộc vào sự lựa chọn của dữ liệu tổng quát. Một bài báo đồng hành thiết lập các kết quả về tính chặt chẽ trong ngữ cảnh này.

In this paper, we consider Harnack inequalities (the gradient estimates) of positive solutions for two different heat equations via the use of the maximum principle. In the first part, we obtain the gradient estimate for positive solutions to the following nonlinear heat equation problem $\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$ on the compact Riemannian manifold (M, g) of dimension n and with Ric(M) ≥−K. Here a > 0 and K are some constants and V is a given smooth positive function on M. Similar results are showed to be true in case when the manifold (M, g) has compact convex boundary or (M, g) is a complete non-compact Riemannian manifold. In the second part, we study Harnack inequality (gradient estimate) for positive solution to the following linear heat equation on a compact Riemannian manifold with non-negative Ricci curvature: $ \partial _{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $ where Wi and V only depend on the space variable x ∈ M. The novelties of our paper are the refined global gradient estimates for the corresponding evolution equations, which are not previously considered by other authors such as Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010).