We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami and Rivin. It is proved by A. Kitaev and the author that any closed 3-manifold has a triangulation supporting an angle structure. The moduli space of all angle structures on a triangulated 3-manifold is a bounded open convex polytope in a Euclidean space. The volume of an angle structure is defined. Both the angle structure and the volume are natural generalizations of tetrahedra in the constant sectional curvature spaces and their volume. It is shown that the volume functional can be extended continuously to the compact closure of the moduli space. In particular, the maximum point of the

In this paper, we produce an elementary approach to Thurston's theory of measured laminations on compact surfaces with non-empty boundary. We show that the theory can be derived from a simple inequality for geometric intersection numbers between arcs inside an octagon.

We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein–Gordon equation (NKGE) with weak power-type nonlinearity and O(1) initial data, while the nonlinearity strength is characterized by εp with a constant p ∈ N+ and a dimensionless parameter ε ∈ (0, 1], for the long-time dynamics up to the time at O(ε^{−β}) with 0 ≤ β ≤ p. In fact, when 0 < ε ≪ 1, the problem is equivalent to the long-time dynamics of NKGE with small initial data and O(1) nonlinearity strength, while the amplitude of the initial data (and the solution) is at O(ε). By reformulating the NKGE into a relativistic nonlinear Schr ̈odinger equation, we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at O(h^m + ε^{p−β} τ^2) of the TSFP method with h mesh size, τ time step and m ≥ 2 depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at O(ε^{−β}) and uniformly valid for ε ∈ (0, 1]. Especially, the error bounds are uniformly at the second order rate for the large time step τ = O(ε^{−(p−β)/2}) in the parameter regime 0 ≤ β < p. Numerical results are reported to confirm our error bounds in the long-time regime. Finally, the TSFP method and its error bounds are extended to a highly oscillatory complex NKGE which propagates waves with wavelength at O(1) in space and O(ε^β) in time and wave velocity at O(ε^{−β}).

We present two uniformly accurate numerical methods for discretizing the Zakharovsystem (ZS) with a dimensionless parameter 0< ε ≤ 1, which is inversely proportional to theacoustic speed. In the subsonic limit regime, i.e., 0< ε << 1, the solution of ZS propagates waves with O(ε)- andO(1)-wavelengths in time and space, respectively, and/or rapid outgoing initial layerswith speed O(1/ε) in space due to the singular perturbation of the wave operator in ZS and/or theincompatibility of the initial data. By adopting an asymptotic consistent formulation of ZS, wepresent a time-splitting exponential wave integrator (TS-EWI) method via applying a time-splittingtechnique and an exponential wave integrator for temporal derivatives in the nonlinear Schr ̈odingerequation and wave-type equation, respectively. By introducing a multiscale decomposition of ZS, wepropose a time-splitting multiscale time integrator (TS-MTI) method. Both methods are explicitand convergent exponentially in space for all kinds of initial data, which is uniformly for ε ∈ (0,1].The TS-EWI method is simpler to be implemented and it is only uniformly and optimally accuratein time for well-prepared initial data, while the TS-MTI method is uniformly and optimally accuratein time for any kind of initial data. Extensive numerical results are reported to show their efficiencyand accuracy, especially in the subsonic limit regime. Finally, the TS-MTI method is applied tostudy numerically convergence rates of ZS to its limiting models when ε→0+.

Multivariate public key cryptosystems are being focused on as candidates for post-quantum cryptography. Rainbow is one of the most efficient signature schemes in multivariate public key cryptosystems. The main drawback of Rainbow is that their key size is much larger than that of RSA and ECC. In this paper, we propose an efficient variant of Rainbow that has a shorter secret key (and thus generates signatures faster) than the corresponding original Rainbow. In our scheme, we divide each layer of Rainbow into smaller blocks by using diagonal matrix representations. The size of the smaller blocks can be flexibly selected, and this enables us to carefully choose secure parameters so that our proposed scheme is secure against known attacks such as rank attacks, direct attacks, and UOV attack. We estimate that the secret key size of our proposed scheme with 100-bit security is smaller by about 40% than that of the original Rainbow. In addition, an implementation of our scheme in the C language is seen to generate signature faster by 40%.