MathSciDoc: An Archive for Mathematician ∫

In this work we study the following cutting pattern problem. Given a triangulated surface (i.e. a two-dimensional simplicial complex), assign each triangle with a triple of 1, one integer per edge, such that the assignment is both complete (i.e. every triangle has integers of both signs) and consistent (i.e. every edge shared by two triangles has opposite signs in these triangles). We show that this problem is the major challenge in converting a volumetric mesh consisting of prisms into a mesh consisting of tetrahedra, where each prism is cut into three tetrahedra. In this paper we provide a complete solution to this problem for topological disks under various boundary conditions ranging from very restricted one to the most flexible one. For each type of boundary conditions, we provide efficient algorithms to compute valid assignments if there is any, or report the obstructions otherwise. For all the proposed algorithms

We give a simplified and more algebraic proof of the finiteness of the families of Calabi-Yau n-folds with non-vanishing of Yukawa-coupling over a fixed base curve and with fixed degeneration locus. We also give a generalization of this result. Our method is variation of Hodge structure and poly-stability of Higgs bundles.

In this continuation of [L-Y1],[LLSY],[L-Y2], and [L-Y3](arXiv: 0709.1515 [math. AG], arXiv: 0809.2121 [math. AG], arXiv: 0901.0342 [math. AG], arXiv: 0907.0268 [math. AG]), we study D-branes in a target-space with a fixed B -field background B along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of B -field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to B , are given. For holomorphic D-strings, we prove a valuation-criterion property of this moduli space. The setting is then extended to take into account also the deformation-quantization-type noncommutative geometry effect of B -field to both the D-brane world-volume and the superstring target-space (-time) B . This brings the notion of twisted B -modules that are realizable as twisted locally-free coherent modules with a flat connection into the study. We use this to realize the notion of both the classical and the quantum spectral covers as morphisms from Azumaya schemes with a fundamental module (with a flat connection in the latter case) in a very special situation. The 3rd theme (subtitled" Sharp vs. Polchinski-Grothendieck") of Sec. 2.2 is to be read with the work [Sh3](arXiv: hep-th/0102197) of Sharp while Sec. 5.2 (subtitled less appropriately" Dijkgraaf-Holland-Sukowski-Vafa vs. Polchinski-Grothendieck") is to be read with the related sections in [DHSV](arXiv: 0709.4446 [hep-th]) and [DHS](arXiv: 0810.4157 [hep-th]) of Dijkgraaf, Hollands, Sukowski, and Vafa.

In this sequel to [L-Y1],[LLSY], and [L-Y2](respectively arXiv: 0709.1515 [math. AG], arXiv: 0809.2121 [math. AG], and arXiv: 0901.0342 [math. AG]), we study a D-brane probe on a conifold from the viewpoint of the Azumaya structure on D-branes and toric geometry. The details of how deformations and resolutions of the standard toric conifold Y can be obtained via morphisms from Azumaya points are given. This should be compared with the quantum-field-theoretic/D-brany picture of deformations and resolutions of a conifold via a D-brane probe sitting at the conifold singularity in the work of Klebanov and Witten [KW](arXiv: hep-th/9807080) and Klebanov and Strasser [KS](arXiv: hep-th/0007191). A comparison with resolutions via noncommutative desingularizations is given in the end.

Mirror symmetry conjecture identifies the complex geometry of a Calabi $ Yau manifold with the symplectic geometry of its mirror Calabi $ Yau man $ ifold. Using the SYZ mirror transform, we argue that (i) the mirror of an elliptic Calabi $ Yau manifold admits a twin Lagrangian fibration structure and (ii) the mirror of the Fourier $ Mukai transform for dual elliptic fibra $ tions is a symplectic Fourier $ Mukai transform for dual twin Lagrangian fibrations, which is essentially an identity transformation in this case.

Structured light system using a digital video projector is increasingly used for a 3-D shape measurement because of its digital nature. However, the nonlinear gamma of the projector causes the projected fringe patterns to be non-sinusoidal, which results in phase error therefore shape measurement error. Previous work showed that, by using a small look-up-table (LUT), this type of phase error can be reduced significantly for a three-step phase-shifting algorithm. In this research, we prove that this type of phase error compensation method is not limited to a three-step phase-shifting algorithm. It is generic for any phase-shifting algorithm. The phase error compensation algorithm is able to theoretically eliminate the phase error caused by the gamma of the projector completely. It is based on our finding that in phase domain, the phase error due to the projector's gamma is preserved for arbitrary object's surface

In this paper we introduce our preliminary research results for segmentation and labeling of 3-dimensional microscopy neuron image. We segment each of stacked 2-dimensional image slices using a partial differential equation (PDE) based algorithm and project previous slice segmentation result to the next slide as an initialization condition. Then we label neurons using an efficient connected component labeling algorithm. We show sample results obtained from real neuron image data

In this paper, we give a priori estimation of derivatives of solution of Duncan-Mortensen-Zakai (DMZ) equation up to 2nd order. We also prove the existence and uniqueness of weak solution for DMZ equation, which is a fundamental equation in nonlinear filtering.

In this short survey paper, we shall discuss certain recent results in classical gravity. Our main attention will be restricted to two topics in which we have been involved; the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantum-mechanical particles (Part II).

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