We calculate the dimension of the locus of Jacobian elliptic surfaces over P1 with a given Picard number, in the corresponding moduli space.

We obtain improved fractional Poincaré inequalities in John domains of a metric space (X,d) endowed with a doubling measure μ under some mild regularity conditions on the measure μ. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.

Blocking is commonly used in randomized experiments to increase efficiency of estimation. A generalization of blocking removes allocations with imbalance in covariate distributions between treated and control units, and then randomizes within the remaining set of allocations with balance. This idea of rerandomization was formalized by Morgan and Rubin (Annals of Statistics, 2012, 40, 1263–1282), who suggested using Mahalanobis distance between treated and control covariate means as the criterion for removing unbalanced allocations. Kallus (Journal of the Royal Statistical Society, Series B: Statistical Methodology, 2018, 80, 85–112) proposed reducing the set of balanced allocations to the minimum. Here we discuss the implication of such an ‘optimal’ rerandomization design for inferences to the units in the sample and to the population from which the units in the sample were randomly drawn. We argue that, in general, it is a bad idea to seek the optimal design for an inference because that inference typically only reflects uncertainty from the random sampling of units, which is usually hypothetical, and not the randomization of units to treatment versus control.

In this paper, we present a new algorithm, such that, for the learning with errors (LWE) problems, if the errors are bounded -- the errors do not span the whole prime finite field F_q but a fixed known subset of size D (D<q), which we call the learning with bounded errors (LWBE) problems, we can solve it with complexity O(n^D).

This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.