The Bj\"orling problem for non-minimal constant mean curvature surfaces

The classical Bj\"orling problem is to find the minimal surface containing a given real analytic curve with tangent planes prescribed along the curve. We consider the generalization of this problem to non-minimal constant mean curvature (CMC) surfaces, and show that it can be solved via the loop group formulation for such surfaces. The main result gives a way to compute the holomorphic potential for the solution directly from the Bj\"orling data, using only elementary differentiation, integration and holomorphic extensions of real analytic functions. Combined with an Iwasawa decomposition of the loop group, this gives the solution, in analogue to Schwarz's formula for the minimal case. Some preliminary examples of applications to the construction of CMC surfaces with special properties are given.Comment: 18 Pages, 16 figures. Typographical corrections in version

Reducing “Structure from Motion”: a general framework for dynamic vision. 1. Modeling

The literature on recursive estimation of structure and motion from monocular image sequences comprises a large number of apparently unrelated models and estimation techniques. We propose a framework that allows us to derive and compare all models by following the idea of dynamical system reduction. The “natural” dynamic model, derived from the rigidity constraint and the projection model, is first reduced by explicitly decoupling structure (depth) from motion. Then, implicit decoupling techniques are explored, which consist of imposing that some function of the unknown parameters is held constant. By appropriately choosing such a function, not only can we account for models seen so far in the literature, but we can also derive novel ones

On the omega-limit sets of tent maps

For a continuous map f on a compact metric space (X,d), a subset D of X is internally chain transitive if for every x and y in D and every delta > 0 there is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) < delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed subset D of X is internally chain transitive if and only if D is an omega-limit set for some point in X, and that the same is also true for the tent map with slope equal to 2. In this paper, we prove that for tent maps whose critical point c=1/2 is periodic, every closed, internally chain transitive set is necessarily an omega-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an omega-limit set. Together, these results lead us to conjecture that for those tent maps with shadowing (or pseudo-orbit tracing), the omega-limit sets are precisely those sets having internal chain transitivity.Comment: 17 page

On monodromy representation of period integrals associated to an algebraic curve with bi-degree (2,2)

We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve $\cal Y$ with bi-degree (2,2) in a product of projective lines ${\Bbb P}^{1} \times {\Bbb P}^{1}$. We calculate two differenent monodromy representations of period integrals for the affine variety ${\cal X}^{(2,2)}$ obtained by the dual polyhedron mirror variety construction from $\cal Y$. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analytic continuation of the Mellin-Barnes integrals that gives us a proper subgroup of the monodromy group. It turns out both representations admit a Hermitian quadratic invariant form that is given by a Gram matrix of a split generator of the derived category of coherent sheaves on on $\cal Y$ with respect to the Euler form