On certain constructions of p-adic families of Siegel modular forms of even genus

Suppose that p > 5 is a rational prime. Starting from a well-known p-adic analytic family of ordinary elliptic cusp forms of level p due to Hida, we construct a certain p-adic analytic family of holomorphic Siegel cusp forms of arbitrary even genus and of level p associated with Hida's p-adic analytic family via the Duke-Imamoglu lifting provided by Ikeda. Moreover, we also give a similar results for the Siegel Eisenstein series of even genus with trivial Nebentypus

Higher congruences between newforms and Eisenstein series of squarefree level

Let $p\geq 5$ be prime. For elliptic modular forms of weight 2 and level $\Gamma_0(N)$ where $N>6$ is squarefree, we bound the depth of Eisenstein congruences modulo $p$ (from below) by a generalized Bernoulli number with correction factors and show how this depth detects the local non-principality of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to give an infinite class of examples where the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert--Samuel multiplicities.Comment: 19 pages. Minor revisions. Accepted for publication in The Journal de Th\'eorie des Nombres de Bordeau

Special function methods for bursty models of transcription

We explore a Markov model used in the analysis of gene expression, involving the bursty production of pre-mRNA, its conversion to mature mRNA, and its consequent degradation. We demonstrate that the integration used to compute the solution of the stochastic system can be approximated by the evaluation of special functions. Furthermore, the form of the special function solution generalizes to a broader class of burst distributions. In light of the broader goal of biophysical parameter inference from transcriptomics data, we apply the method to simulated data, demonstrating effective control of precision and runtime. Finally, we propose and validate a non-Bayesian approach for parameter estimation based on the characteristic function of the target joint distribution of pre-mRNA and mRNA

Special Function Methods for Bursty Models of Transcription

We explore a Markov model used in the analysis of gene expression, involving the bursty production of pre-mRNA, its conversion to mature mRNA, and its consequent degradation. We demonstrate that the integration used to compute the solution of the stochastic system can be approximated by the evaluation of special functions. Furthermore, the form of the special function solution generalizes to a broader class of burst distributions. In light of the broader goal of biophysical parameter inference from transcriptomics data, we apply the method to simulated data, demonstrating effective control of precision and runtime. Finally, we suggest a non-Bayesian approach to reducing the computational complexity of parameter inference to linear order in state space size and number of candidate parameters.Comment: Body: 15 pages, 2 figures, 2 tables. Supplement: 10 pages, 1 figur

Hexagon OPE Resummation and Multi-Regge Kinematics

We analyse the OPE contribution of gluon bound states in the double scaling limit of the hexagonal Wilson loop in planar N=4 super Yang-Mills theory. We provide a systematic procedure for perturbatively resumming the contributions from single-particle bound states of gluons and expressing the result order by order in terms of two-variable polylogarithms. We also analyse certain contributions from two-particle gluon bound states and find that, after analytic continuation to the $2\to 4$ Mandelstam region and passing to multi-Regge kinematics (MRK), only the single-particle gluon bound states contribute. From this double-scaled version of MRK we are able to reconstruct the full hexagon remainder function in MRK up to five loops by invoking single-valuedness of the results.Comment: 29 pages, 3 figures, 4 ancillary file

Optimal disclosure risk assessment

Protection against disclosure is a legal and ethical obligation for agencies releasing microdata les for public use. Consider a microdata sample of size n from a nite population of size n = n + n, with > 0, such that each sample record contains two disjoint types of information: identifying categorical information and sensitive information. Any decision about releasing data is supported by the estimation of measures of disclosure risk, which are dened as discrete functionals of the number of sample records with a unique combination of values of identifying variables. The most common measure is arguably the number 1 of sample unique records that are population uniques. In this paper, we rst study nonparametric estimation of 1 under the Poisson abundance model for sample records. We introduce a class of linear estimators of 1 that are simple, computationally ecient and scalable to massive datasets, and we give uniform theoretical guarantees for them. In particular, we show that they provably estimate 1 all of the way up to the sampling fraction ( + 1)1 / (log n)1, with vanishing normalized mean-square error (NMSE) for large n. We then establish a lower bound for the minimax NMSE for the estimation of 1, which allows us to show that: i) (+1)1 / (log n)1 is the smallest possible sampling fraction for consistently estimating 1; ii) estimators' NMSE is near optimal, in the sense of matching the minimax lower bound, for large n. This is the main result of our paper, and it provides a rigorous answer to an open question about the feasibility of nonparametric estimation of 1 under the Poisson abundance model and for a sampling fraction ( + 1)1 < 1=2