In this paper we study Lipschitz solutions of partial diﬀerential relations of the form (1) ∇u(x) ∈ K a.e. in Ω,
where u is a (Lipschitz) mapping of an open set Ω ⊂ Rn into Rm , ∇u(x) is its gradient (i.e. the matrix ∂ui (x)/∂xj , 1 ≤ i ≤ m, 1 ≤ j ≤ n, deﬁned for almost every x ∈ Ω), and K is a subset of the set M m×n of all real m × n matrices. In addition to relation (1), boundary conditions and other conditions on u will also be considered. Relation (1)...
We ﬁnd a sharp combinatorial bound for the metric entropy of sets in Rn and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisﬁes the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in Rn . ...
We resolve these issues as follows. We show that a nonincreasing returns to scale (nrs) model is
usually appropriate when modeling rational choice among investors. We show when multiple risk
and return measures can justiﬁably be combined and identify some suitable measures. We show
we need a nonlinear model to justify the assumption of convexity and to model diversiﬁcation.
We develop a method to approximate a solution to this model as accurately as needed using a
sequence of linear models.
Coherent measures of risk come up again and again in our discussion.
This chapter provides a brief introduction to the theory of morphological signal processing and its
applications toimage analysis andnonlinear filtering. By “morphological signal processing”we mean
a broad and coherent collection of theoretical concepts, mathematical tools for signal analysis, nonlinear
signal operators, design methodologies, and applications systems that are based on or related
to mathematical morphology (MM), a set- and lattice-theoreticmethodology for image analysis. MM
aims at quantitatively describing the geometrical structure of image objects.
báo cáo này, chúng ta xem xét một lớp học của các chức năng starlike thống nhất được định nghĩa bởi nhà điều hành tách rời nhất định. Chúng tôi xác định một điều kiện đủ cho một hàm f được thống nhất starlike chức năng đó cũng là cần thiết khi e có hệ số tiêu cực.