Chinese Annals of Mathematics

The chiral ring of classical supersymmetric Yang-Mills theory with gauge group Sp(N) or SO(N) is computed, extending previous work (of Cachazo, Douglas, Seiberg, and the author) for SU(N). The result is that, as has been conjectured, the ring is generated by the usual glueball superfield S ~ TrW_αW^α, with the relation S^h = 0, h being the dual Coxeter number. Though this proposition has important implications for the behavior of the quantum theory, the statement and (for the most part) the proofs amount to assertions about Lie groups with no direct reference to gauge theory.

For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems with zero eigenvalues, the authors establish the local exact controllability with boundary controls acting on one end or on two ends and internal controls acting on a part of equations in the system.

The transition from a deflagration to a detonation (DDT) in gas dynamics is investigated through the process of a deflagration with a finite width flame overtaken by a shock. The problem is formulated as a free boundary value problem in an angular domain with a strong detonation and a reflected shock as boundaries. The main difficulty lies in the fact that the strength of reflected shock is zero at the vertex where the shock speed degenerates to be the same as the characteristic speed. The conclusion is that a strong detonation and a retonation (a reflected shock) form locally. Also the entropy satisfaction of this solution is presented.

The authors obtain function theoretic characterizations of the compactness on the standard weighted Bergman spaces of the two operators formed by multiplying a composition operator with the adjoint of another composition operator.

A random functional central limit theorem is obtained for processes of partial sums and product sums of linear processes generated by non-stationary martingale differences. It develops and improves some corresponding results on processes of partial sums of linear processes generated by strictly stationary martingale differences, which can be found in [5].

This paper considers the pointwise estimate of the solutions to Cauchy problem for quasilinear hyperbolic systems, which bases on the existence of the solutions by using the fundamental solutions. It gives a sharp pointwise estimates of the solutions on domain under consideration. Specially, the estimate is precise near each characteristic direction.

The authors establish some uniform estimates for the distance to halfway points of minimal geodesics in terms of the distantce to end points on some types of Riemannian manifolds, and then prove some theorems about the finite generation of fundamental group of Riemannian manifold with nonnegative Ricci curvature, which support the famous Milnor conjecture.

The authors define the directional hyper Hilbert transform and give its mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the L^p-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein.

The notion of L_p-dichotomy for linear differential equations with possibly unbounded operator is introduced. By help of Banach fixed point theorem sufficient conditions for the existence of bounded solutions of nonlinear differential equations with an L_p-dichotomous linear part are obtained.

In this paper, the authors investigate the joint distribution of end points of excursion away from a closed set straddling on a fixed time and use this result to compute the Lévy system and the Dirichlet form of the boundary process.

The author investigates the hyper order of solutions of the higher order linear equation, and improves the results of M. Ozawa^[15], G. Gundersen^[6] and J. K. Langley^[12].

The Moore–Penrose metric generalized inverse T⁺ of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moore-Penrose metric generalized inverse T⁺ is usually homogeneous and nonlinear in general. By means of the methods of geometry of Banach Space, the necessary and sufficient conditions for existence, continuity, linearity and minimum property of the Moore-Penrose metric generalized inverse T⁺ will be given, and some properties of T⁺ will be investigated in this paper.

This paper derives first order necessary and sufficient conditions for unconstrained cone d.c. programming problems where the underlined space is partially ordered with respect to a cone. These conditions are given in terms of directional derivatives and subdifferentials of the component functions. Moreover, conjugate duality for cone d.c. optimization is discussed and weak duality theorem is proved in a more general partially ordered linear topological vector space (generalizing the results in [11]).

In this paper, the author studies the regularity of solutions to the Dirichlet problem for equation Lu = f, where L is a second order degenerate elliptic operator in divergence form in Ω, a bounded open subset of Rⁿ(n ≥ 3).