Acta Mathematica Sinica-English Series最新文献

We study the transfer between small special unipotent representations for all equal rank real forms of type E₆ and E₇. As a consequence, one can verify these modules are unitarity using the results of Wallach and Zhu. Moreover, the K-spectra of these modules can be obtained explicitly.

The first passage time has many applications in fields like finance, econometrics, statistics, and biology. However, explicit formulas for the first passage density have only been obtained for a few cases. This paper derives an explicit formula for the first passage density of Brownian motion with two-sided piecewise continuous boundaries which may have some points of discontinuity. Approximations are used to obtain a simplified formula for estimating the first passage density. Moreover, the results are also generalized to the case of two-sided general nonlinear boundaries. Simulations can be easily carried out with Monte Carlo method and it is demonstrated for several typical two-sided boundaries that the proposed approximation method offers a highly accurate approximation of first passage density.

Let W be a real symplectic space and (G, G′) an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let (widetilde {rm{G}}) be the preimage of G in the metaplectic group (widetilde {{rm{Sp}}}({rm{W}})). Given an irreducible unitary representation Π of (widetilde {rm{G}}) that occurs in the restriction of the Weil representation to (widetilde {rm{G}}), let Θ_Π denote its character. We prove that, for a suitable embedding T of (widetilde {{rm{Sp}}}({rm{W}})) in the space of tempered distributions on W, the distribution T(Θ̌_Π) admits an asymptotic limit, and the limit is a nilpotent orbital integral. As an application, we compute the wave front set of Π′, the representation of (widetilde {{G^prime}}) dual to Π, by elementary means.

Consider the polynomial differential system of degree m of the form

where μ and ν are real numbers such that ((mu^{2}+nu^{2})(mu+nu(m-2))(a_{1}^{2}+a_{2}^{2})ne 0,m > 2) and Ω_m−1(x,y) is a homogenous polynomial of degree m − 1. A conjecture, stated in J. Differential Equations 2019, suggests that when ν = 1, this differential system has a weak center at the origin if and only if after a convenient linear change of variable (x,y) → (X,Y) the system is invariant under the transformation (X,Y,t) → (−X,Y, −t). For every degree m we prove the extension of this conjecture to any value of ν except for a finite set of values of μ.

Let (mathfrak{g}) be a classical complex simple Lie algebra and (mathfrak{q}) be a parabolic subalgebra. Let M be a generalized Verma module induced from a one dimensional representation of (mathfrak{q}). Such M is called a scalar generalized Verma module. In this paper, we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand–Kirillov dimension of the corresponding highest weight modules.

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of G = GL_n(ℂ) possessing the minimal Gelfand–Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of G of type (n − 1,1). We give the transition matrix between the two bases for the corresponding coherent families.

Abstract

K-UR, K-LUR and K-R are the generalizations of UR, LUR and R respectively, which are of great significance in Banach space theory. While in Orlicz–Lorentz function space (Lambda_{varphi,omega}^{circ}[0,gamma)) equipped with the Orlicz norm, the research methods of K-UR, K-LUR and K-R are much more complicated than those of UR, LUR and R. In this paper we obtain some criteria of K-UR, K-LUR and K-R of (Lambda_{varphi,omega}^{circ}[0,gamma)) by means of the norm of dual space and H_μ property of (Lambda_{varphi,omega}^{circ}[0,gamma)) .

Abstract

In this paper we consider the theta correspondence over a non-Archimedean local field. Using the homological method and the theory of derivatives, we show that under a mild condition the big theta lift is irreducible.

In this paper, we prove that in a hyperconvex domain Ω in ℍⁿ, if a non-negative Borel measure is dominated by a quaternionic Monge–Ampère measure, then it is a quaternionic Monge–Ampère measure of a function in the class ({cal E}(Omega )).

We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial W, with coefficients in a field of characteristic 2, is a square matrix Q of polynomial entries satisfying Q² = W · Id. We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold ℝP² ⊂ ℂP² and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.