Based on the theory of semi-global C 1 solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems, the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.
This paper is devoted to the study of the subspace ofW m,r of functions that vanish on a part γ 0 of the boundary. The author gives a crucial estimate of the Poincaré constant in balls centered on the boundary of γ 0. Then, the convolution-translation method, a variant of the standard mollifier technique, can be used to prove the density of smooth functions that vanish in a neighborhood of γ 0, in this subspace. The result is first proved for m = 1, then generalized to the case where m ≥ 1, in any dimension, in the framework of Lipschitz-continuous domain. However, as may be expected, it is needed to make additional assumptions on the boundary of γ 0, namely that it is locally the graph of some Lipschitz-continuous function.
The authors investigate the tail probability of the supremum of a random walk with independent increments and obtain some equivalent assertions in the case that the increments are independent and identically distributed random variables with O-subexponential integrated distributions. A uniform upper bound is derived for the distribution of the supremum of a random walk with independent but non-identically distributed increments, whose tail distributions are dominated by a common tail distribution with an O-subexponential integrated distribution.
The authors present an algorithm which is a modification of the Nguyen-Stehle greedy reduction algorithm due to Nguyen and Stehle in 2009. This algorithm can be used to compute the Minkowski reduced lattice bases for arbitrary rank lattices with quadratic bit complexity on the size of the input vectors. The total bit complexity of the algorithm is O(n^2 \cdot (4n!)^n \cdot (\tfrac{{n!}}{{2^n }})^{\tfrac{n}{2}} \cdot (\tfrac{4}{3})^{\tfrac{{n(n - 1)}}{4}} \cdot (\tfrac{3}{2})^{\tfrac{{n^2 (n - 1)}}{2}} \cdot \log ^2 A), where n is the rank of the lattice and A is maximal norm of the input base vectors. This is an O(log2 A) algorithm which can be used to compute Minkowski reduced bases for the fixed rank lattices. A time complexity n! · 3 n(log A) O(1) algorithm which can be used to compute the successive minima with the help of the dual Hermite-Korkin-Zolotarev base was given by Blomer in 2000 and improved to the time complexity n! · (log A) O(1) by Micciancio in 2008. The algorithm in this paper is more suitable for computing the Minkowski reduced bases of low rank lattices with very large base vector sizes.
The authors study the finite decomposition complexity of metric spaces of H, equipped with different metrics, where H is a subgroup of the linear group GL∞(ℤ). It is proved that there is an injective Lipschitz map φ: (F, d S) → (H, d), where F is the Thompson’s group, dS the word-metric of F with respect to the finite generating set S and d a metric of H. But it is not a proper map. Meanwhile, it is proved that φ: (F, d S) → (H, d 1) is not a Lipschitz map, where d 1 is another metric of H.
Let S be a Riemann surface that contains one puncture x. Let ℐ be the collection of simple closed geodesics on S, and let ℱ denote the set of mapping classes on S isotopic to the identity on S ∪ {x}. Denote by t c the positive Dehn twist about a curve c ∈ ℐ. In this paper, the author studies the products of forms (t b −m ∘ t a n) ∘ f k, where a, b ∈ ℐ and f ∈ ℱ. It is easy to see that if a = b or a, b are boundary components of an x-punctured cylinder on S, then one may find an element f ∈ ℱ such that the sequence (t b −m ∘ t n a) ∘ f k contains infinitely many powers of Dehn twists. The author shows that the converse statement remains true, that is, if the sequence (t b −m ∘ t a n) ∘ f k contains infinitely many powers of Dehn twists, then a, b must be the boundary components of an x-punctured cylinder on S and f is a power of the spin map t b −1 ∘ t a.
Based on the Lax operator L and Orlov-Shulman’s M operator, the string equations of the q-KP hierarchy are established from the special additional symmetry flows, and the negative Virasoro constraint generators {L −n, n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.
The bifurcations of orbit flip homoclinic loop with nonhyperbolic equilibria are investigated. By constructing local coordinate systems near the unperturbed homoclinic orbit, Poincaré maps for the new system are established. Then the existence of homoclinic orbit and the periodic orbit is studied for the system accompanied with transcritical bifurcation.
The authors investigate the completeness of the system of eigen or root vectors of the 2 × 2 upper triangular infinite-dimensional Hamiltonian operator H 0. First, the geometrical multiplicity and the algebraic index of the eigenvalue of H 0 are considered. Next, some necessary and sufficient conditions for the completeness of the system of eigen or root vectors of H 0 are obtained. Finally, the obtained results are tested in several examples.
First of all, some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations, nonlinear dissipative dispersive wave equations, nonlinear convection equations, nonlinear reaction diffusion equations and nonlinear hyperbolic equations, respectively.