1 Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 2 Heegner points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 3 The regulator term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 4 The conjecture . . . . . . . . . . . . . . ....

The paper presents a novel fast calculation method for broadband Electromagnetic Scattering analysis. In this work, the isogeometric boundary element is used to solve Helmholtz equations electromagnetic scattering problems. non-uniform rational B-splines are employed construct structural geometries and discretize electric magnetic field integral [1,2]. To avoid timeconsuming multi-frequency cal...

This study focuses on the topology optimization of structures using a hybrid of level set method (LSM) incorporating sensitivity analysis and isogeometric analysis (IGA). First, the topology optimization problem is formulated using the LSM based on the shape gradient. The shape gradient easily handles boundary propagation with topological changes. In the LSM, the topological gradient method as ...

Abstract Coherent lines are generally more complex spatial curves, and the study of coherent is basis for studying cutting trajectories machines. In this paper, geometry problem transformed into an algebraic solving by establishing a mathematical model line to find its parametric equations. Interpolation refers process data densification. speed planning method that adapts curvature change Non-U...

This paper presents a new kind of uniform splines, called hyperbolic polynomial B-splines, generated over the space Ω = span{sinh t, cosh t, tk−3, tk−4, . . . , t,1} in which k is an arbitrary integer larger than or equal to 3. Hyperbolic polynomial B-splines share most of the properties as those of the B-splines in the polynomial space. We give the subdivision formulae for this new kind of cur...

A method for reconstructing 3D rational B-spline surfaces from multiple views is proposed. The method takes advantage of the projective invariance properties of rational Bsplines. Given feature correspondences in multiple views, the 3D surface is reconstructed via a four step framework. First, corresponding features in each view are given an initial surface parameter value (s; t), and a 2D B-sp...

We introduce expo-rational B-splines for curves and surfaces and explore for the first time some properties of these new splines which depend not only on their knot vector but also on their intrinsic parameters and the geometry and parametrization of the local curves/surfaces. We consider several examples, discuss some computational aspects, and address potential applications in shape design. §

This paper presents a short, simple, and general proof showing that the control polygons generated by subdivision and degree elevation converge to the underlying splines, box-splines, or multivariate B~zier polynomials, respectively. The proof is based only on a Taylor expansion. Then the results are carried over to rational curves and surfaces. Finally, an even shorter but as simple proof is p...