We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post-composition with a Möbius transformation by its critical points. Bull. London Math. Soc. 30 (1998) 62–66 Stony Brook IMS Preprint #1996/7 June 1996 This note will give a brief proof of the following known theorem: Theorem 1. Let c1, · · · , cd be given (not necessarily distinct) points...

We compare the following three notions of uniformity for a finitely additive probability measure on the set of natural numbers: that it extend limiting relative frequency, that it be shift-invariant, and that it map every residue class mod m to 1/m. We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set L of extensions of limiting relative fr...

Definition Let K be a field. An elliptic curve over K is a pair (X, O) where X is a smooth absolutely integral proper curve over K with genus one and O is a K-rational point on X. Terminology (see Liu [Liu] and Stichtenoth [Sti]) A curve over K is an equidimensional variety over K of dimension one. A variety over K is a separated scheme of finite type over Spec(K). Examples A triangle with equa...

Traditionally, image registration of multi-modal and multi-temporal images is performed satisfactorily before land cover mapping. However, since multi-modal and multi-temporal images are likely to be obtained from different satellite platforms and/or acquired at different times, perfect alignment is very difficult to achieve. As a result, a proper land cover mapping algorithm must be able to co...

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, and that the set of critical points of the square of the moment map is a countable discrete union of compact sets. We show that certain integrals of equivariant co...

In the parallel map theory, the hippocampus encodes space with 2 mapping systems. The bearing map is constructed primarily in the dentate gyrus from directional cues such as stimulus gradients. The sketch map is constructed within the hippocampus proper from positional cues. The integrated map emerges when data from the bearing and sketch maps are combined. Because the component maps work in pa...

By base change, i′n is also a closed embedding, hence affine. We have the unit map F → in∗i ′∗ n (F ). Applying Rf∗ gives a map Rf∗(F ) → Rf∗i ′ n∗i ′∗ n (F ). Since i ′ n is affine, Ri′n∗ ≃ i ′ n∗. By Leray’s spectral sequence, Rf∗i ′ n∗ ≃ R(f ◦ i ′ n)∗ ≃ R(in ◦ fn)∗ ≃ in∗Rfn∗. Applying H, we have a map Rf∗(F ) → R f∗i ′ n∗i ′∗ n (F ) ≃ in∗R fn∗(i ′∗ n (F )). Applying in∗i ∗ n to both sides an...

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map....