In this paper, by combining modular forms and characteristic forms, we obtain general anomaly cancellation formulas of any dimension. For 4 k+ 2 dimensional manifolds, our results include the gravitational anomaly cancellation formulas of Alvarez-Gaum and Witten in dimensions 2, 6 and 10 (\cite {AW}) as special cases. In dimension 4 k+ 2, we derive anomaly cancellation formulas for index gerbes. In dimension 4 k+ 2, we obtain certain results about eta invariants, which are interesting in spectral geometry.

Recent proposals for implementation of kernel-based nonparametric curve estimators are seen to be faster than naive direct implementations by factors up into the hundreds. The main ideas behind the two different approaches are made clear. Careful speed comparisons in a variety of settings and using a variety of machines and software are done. Various issues on computational accuracy and stability are also discussed. Our speed tests show that the fast methods are as fast or somewhat faster than methods traditionally considered very fast, such as LOWESS and smoothing splines.

Let R be a not necessarily commutative ring with 1. In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on R. We proceed by uniformly defining a coarsening relation ≤ on the set Q(R) of all quasi-orderings on R. One of our main results states that (Q(R),≤′) is a rooted tree for some slight modification ≤′ of ≤, i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that (Q(R),≤′) is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with 1. We conclude this paper by studying Q(R) as a topological space.

This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invariance of the big quantum rings under ordinary flops of splitting type. To achieve that, several new ingredients are introduced. One is a quantum Leray--Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D module of Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray--Hirsch theorem will be discussed in subsequent papers. In particular, a quantum splitting principle is developed in Part III which reduces the general ordinary flops to the split case solved here.

LWE/RLWE-based cryptosystems require sampling error term from discrete Gaussian distribution. However, some existing samplers are somehow slow under certain circumstances therefore efficiency of such schemes is restricted. In this paper, we introduce a more efficient discretized Gaussian sampler based on ziggurat sampling algorithm. We also analyze statistical quality of our sampler to prove that it can be adopted in LWE/RLWE-based cryptosystems. Compared with ziggurat-based sampler by Buchmann et al. related samplers by Peikert, Ducas et al. and Knuth-Yao, our sampler achieves more than 2x speedup when standard deviation is large. This can benefit constructions rely on noise flooding (e.g., homomorphic encryption). We also present two applications: First, we use our sampler to optimize the RLWE-based authenticated key exchange (AKE) protocol by Zhang et al. We achieve 1.14x speedup on total runtime of this protocol over major parameter choices. Second, we give practical post-quantum Transport Layer Security (TLS) ciphersuite. Our ciphersuite inherits advantages from TLS and the optimized AKE protocol. Performance of our proof-of-concept implementation is close to TLS v1.2 ciphersuites and one post-quantum TLS construction.