Unit N-ball Research Articles

On proper holomorphic mappings among irreducible bounded symmetric domains of rank at least $2$

Proper holomorphic mappings among domains on Euclidean spaces is a classical topic in Several Complex Variables. The literature can date back to the earliest results like the theorem of H. Alexander [1] which says that any proper holomorphic self-map of the complex unit n-ball is a biholomorphism if n ≥ 2. Since then, the study of the proper holomorphic mappings between complex unit balls of different dimensions has become a very popular topic in the field. Many important inputs from various perspectives have been made, like Algebraic Geometry, Chern-Moser Theory, Segre variety and Bergman kernel, etc. It is apparent by now that the complexity of the problem grows with the codimension and one in general must impose certain regularity assumptions on the proper maps in order to give any satisfactory classification.

The Volume of the Unit n-Ball

SummaryTwo simple proofs are given for the fact that the volume of the unit ball in n-dimensional Euclidean space approaches 0 as n approaches ∞. (Some authors use the term “unit sphere” for what is here called the unit ball.) One argument involves covering the unit ball by simplices. The other argument involves covering the unit ball by rectangular solids.

On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ≥ n, given by F1(z) = (0, In;m(z)) with the isometric constant equal to (m + 1)/(n + 1) and F2(z) = (In;m(z), In;m(z)) with the isometric constant equal to 2(m + 1)/(n + 1). Here \({I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}\) is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.

Minimax Problems Related to Cup Powers and Steenrod Squares

If F is a family of mod 2 k-cycles in the unit n-ball, we lower bound the maximal volume of any cycle in F in terms of the homology class of F in the space of all cycles. We give examples to show that these lower bounds are fairly sharp.

The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space

The purpose of this paper is to study some uniqueness, existence and regularity properties of the Dirichlet problem at infinity for proper harmonic maps from the hyperbolic m-space to the open unit n-ball with a specific incomplete metric. When m=n=2, harmonic solutions of this Dirichlet problem yield complete constant mean curvature surfaces in the hyperbolic 3-space.

Singular Values and the Spectral Theorem

The action of an arbitrary real or complex m x n-matrix is not at all arbitrary: it takes a suitable orthonormal basis of n-space into some orthogonal set in m-space; consequently it maps the unit n-ball onto a (possibly lower-dimensional) ellipsoid. This is the geometric view of the Singular Value Theorem, which is not only one of the nicest matrix theorems to state and to visualize but also one of the easiest to prove and to apply. In any introductory course on matrices it deserves a place near the center. Theorem 1 below is a pure matrix version of this result. One of its main consequences, Theorem 2 below, leads straight to the orthogonal diagonalization of certain matrices, i.e., the Spectral Theorem. It is customary to prove Theorem 2 independently, applying either the Fundamental Theorem of Algebra to a characteristic polynomial or Lagrange Multipliers to a quadratic form, and even to deduce Theorem 1 from it. The analytic equipment needed for the following proof is more modest: it suffices to know that a continuous real-valued function on any compact set has a maximum. For the sake of simplicity the argument will first be given for real matrices and then (trivially) extended to complex ones. To prevent any suspicion that extraneous subtleties are tacitly used, the prefix eigen will be avoided.

A Sharp Inequality of J. Moser for Higher Order Derivatives

In fact go = nwl/(1), where wnis the area of the surface of the unit n-ball. Also, he notes that if /3 exceeds go then (1) can no longer hold with some co independent of the u, satisfying (2). The fact that this result holds for some /3 > 0 independent of u has been observed now by several authors with a variety of proofs; see [24], [20], [23], [22], [10], [4], [1], [2], [13], [14], for example. The earliest of these arguments expands the exponential function in a power series. Then observing that the norm in the imbedding

A representation in beta series for the exact density of some random volume contents

Consider r independent and identically distributed random points in a unit n-ball of which p are in the interior and rp are on the surface. These r points, via their convex hull, generate an r-simplex. This article deals with the exact density of the r-content when the points are uniformly distributed. The exact density of the r-content is obtained for the general values of the parameters r, n and p. A representation of the density is given as a mixture of beta type-1 densities so that one can evaluate various types of probabilities by using incom-plete beta tables.

On uniformly distributed random points in an n-Ball

Consider a set of r+1 independently and identically and uniformly distributed random points X0, X1,…,Xr in RnThese points determine almost surely via their convex hull a unique r-simplex in Re This article deals with the exact density of the r-content of this random r-simplex when the points are such that p of them are in the interior and r+l−p of them are on the surface of a unit n-ball. This problem is transformed into a distribution problem connected with multivariate test statistics. Various possible representations of the exact density in the general case, are also pointed out.

Acute triangles in the n-ball

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝ n , with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

Acute triangles in the n-ball

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝn, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

The volume of a random simplex in an n-ball is asymptotically normal

A proof is given of a conjecture in the literature of geometrical probability that the r-content of the r-simplex whose r + 1 vertices are independent random points of which p are uniform in the interior and q uniform on the boundary of a unit n-ball (1 ≦ r ≦ n; 0 ≦ p, q ≦ r + 1, p + q = r + 1) is asymptotically normal (n →∞) with asymptotic mean and variance and , respectively.

The volume of a random simplex in an n-ball is asymptotically normal

A proof is given of a conjecture in the literature of geometrical probability that the r-content of the r-simplex whose r + 1 vertices are independent random points of which p are uniform in the interior and q uniform on the boundary of a unit n-ball (1 ≦ r ≦ n; 0 ≦ p, q ≦ r + 1, p + q = r + 1) is asymptotically normal (n →∞) with asymptotic mean and variance and , respectively.

Integral inequalities and theorems of Liouville type

Throughout this paper x = (x1 ,... , x,) denotes a point of real Euclidean space En, r = / x [ is the distance to the origin, and F and P > 0 are continuous functions of r, 0 < r < co. We use dr and da for the volume and surface elements of integration respectively, while a, is the area of the surface of the unit n-ball in E”. Our immediate objective is to sharpen and generaiize some theorems of Putnam [2], which are as follows: