Surface registration is widely used in machine vision and medical imaging, where 1-1 correspondences between surfaces are computed to study their variations. Surface maps are usually stored as the 3D coordinates each vertex is mapped to, which often requires lots of storage memory. This causes inconvenience in data transmission and data storage, especially when a large set of surfaces are analyzed. To tackle this problem, we propose a novel representation of surface diffeomorphisms using Beltrami coefficients, which are complex-valued functions defined on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of Beltrami coefficients on the source domain. Hence, every bijective surface map can be represented by a unique Beltrami coefficient. Conversely, given a Beltrami coefficient, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method introduced in this paper. Using this representation, 1/3 of the storage space is saved. We can further reduce the storage requirement by 90% by compressing the Beltrami coefficients using Fourier approximations. We test our algorithm on synthetic data, real human brain and hippocampal surfaces. Our results show high accuracy in the reconstructed data, while the amount of storage is greatly reduced. Our approach is compared with the Fourier compression of the coordinate functions using the same amount of data. The latter approach often shows jaggy results and cannot guarantee to preserve diffeomorphisms.

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We use techniques from both real and complex algebraic geometry to study$K$-theoretic and related invariants of the algebra$C$($X$) of continuous complex-valued functions on a compact Hausdorff topological space$X$. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if$M$is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and$X$is contractible, then every finitely generated projective module over$C$($X$)[$M$] is free. The particular case $$ M = \mathbb{N}_0^n $$ gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic$K$-theory of$C$($X$) follows from the particular case $$ M = {\mathbb{Z}^n} $$ . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on$C$^{*}-algebras, and for a homology theory of commutative algebras to vanish on$C$^{*}-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil$K$-theory implies that commutative$C$^{*}-algebras are$K$-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of$C$($X$). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.

We present a uniformly accurate finite difference method and establish rigorously its uniform error bounds for the Zakharov system (ZS) with a dimensionless parameter 0 < ε ≤ 1, which is inversely proportional to the speed of sound. In the subsonic limit regime, i.e., 0 < ε << 1, the solution propagates highly oscillatory waves and/or rapid outgoing initial layers due to the perturbation of the wave operator in the ZS and/or the incompatibility of the initial data, which is characterized by two parameters α ≥ 0 and β ≥ −1. Specifically, the solution propagates waves with wavelength of O(ε) and O(1) in time and space, respectively, and amplitude at O(ε^{min{2,α,1+β}}). This high oscillation of the solution in time brings significant difficulties in designing numerical methods and establishing their error bounds, especially in the subsonic limit regime. A uniformly accurate finite difference method is proposed by reformulating the ZS into an asymptotic consistent formulation and adopting an integral approximation of the oscillatory term. By applying the energy method and using the limiting equation via a nonlinear Schr ̈odinger equation with an oscillatory potential, we rigorously establish two independent error bounds at O(h^2 + τ^2/ε) and O(h^2 + τ^2 + τε^{α^∗} + ε^{1+α^∗}), respectively, with h the mesh size, τ the time step and α^∗ = min{1, α, 1 + β}. Thus we obtain error bounds at O(h^2 + τ^{4/3}) and O(h^2 + τ^{1+α^∗/(2+α^∗)}) for well-prepared (α^∗ = 1) and ill-prepared (0 ≤ α^∗ < 1) initial data, respectively, which are uniform in both space and time for 0 < ε ≤ 1 and optimal at the second order in space. Other techniques in the analysis include the cut-off technique for treating the nonlinearity and inverse estimates to bound the numerical solution. Numerical results are reported to demonstrate our error bounds.

We show an example of a non-archimedean version of the existence part of the Calabi-Yau theorem in complex geometry. Precisely, we study totally degenerate abelian varieties and certain probability measures on their associated analytic spaces in the sense of Berkovich.