In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on GL_2) in new cases with joint ramifications. The construction involves minimal vectors, rather than newforms and their variants. This paper gives a uniform treatment for the matrix algebra and division algebra cases under mild assumptions, and establishes an explicit relation between the size of the local integral and the finite conductor C(π×π_{χ^{−1}}). As an application, we combine the test vector results with the relative trace formula, and prove a hybrid type subconvexity bound which can be as strong as the Weyl bound in proper range.

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The sine-Gordon (SG) equation and perturbed nonlinear Schrdinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called <i>light bullets</i>) in nonlinear dispersive optical media. We begin with the (2+1) SG equation obtained as an asymptotic reduction in the two level dissipationless MaxwellBloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusingdefocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet

We introduce an interactive technique for manipulating simple 3D shapes based on extracting them from a single photograph. Such extraction requires understanding of the components of the shape, their projections, and relations. These simple cognitive tasks for humans are particularly difficult for automatic algorithms. Thus, our approach combines the cognitive abilities of humans with the computational accuracy of the machine to solve this problem. Our technique provides the user the means to quickly create editable 3D parts¡ª human assistance implicitly segments a complex object into its components, and positions them in space. In our interface, three strokes are used to generate a 3D component that snaps to the shape¡¯s outline in the photograph, where each stroke defines one dimension of the component. The computer reshapes the component to fit the image of the object in the photograph as well as to satisfy various inferred geometric constraints imposed by its global 3D structure. We show that with this intelligent interactive modeling tool, the daunting task of object extraction is made simple. Once the 3D object has been extracted, it can be quickly edited and placed back into photos or 3D scenes, permitting object-driven photo editing tasks which are impossible to perform in image-space.

A numerical method for computing the extremal Teichmüller map between multiply-connected domains is presented. Given two multiply-connected domains, there exists a unique Teichmüller map (T-Map) between them minimizing the conformality distortion. The extremal T-Map can be considered as the ‘most conformal’ map between multiply-connected domains. In this paper, we propose an iterative algorithm to compute the extremal T-Map using the Beltrami holomorphic flow (BHF). The BHF procedure iteratively adjusts the initial map based on a sequence of Beltrami coefficients, which are complex-valued functions defined on the source domain. It produces a sequence of quasi-conformal maps, which converges to the T-Map minimizing the conformality distortion. We test our method on synthetic data together with real human face data. Results show that our algorithm computes the extremal T-Map between two multiply-connected domains of the same topology accurately and efficiently.