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Hypergeometric decomposition of symmetric K3 quartic pencils.

ABSTRACT: We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard-Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite-field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives.

SUBMITTER: Doran CF

PROVIDER: S-EPMC7194283 | biostudies-literature | 2020

REPOSITORIES: biostudies-literature

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Hypergeometric decomposition of symmetric K3 quartic pencils.

Doran Charles F CF Kelly Tyler L TL Salerno Adriana A Sperber Steven S Voight John J Whitcher Ursula U

Research in the mathematical sciences 20200316 2

We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard-Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite-field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global <i>L</i>-functions. This computation gives a complete, explicit description of the mot ...[more]

PMID: 32382704

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Project description:BackgroundTiling-arrays are applicable to multiple types of biological research questions. Due to its advantages (high sensitivity, resolution, unbiased), the technology is often employed in genome-wide investigations. A major challenge in the analysis of tiling-array data is to define regions-of-interest, i.e., contiguous probes with increased signal intensity (as a result of hybridization of labeled DNA) in a region. Currently, no standard criteria are available to define these regions-of-interest as there is no single probe intensity cut-off level, different regions-of-interest can contain various numbers of probes, and can vary in genomic width. Furthermore, the chromosomal distance between neighboring probes can vary across the genome among different arrays.ResultsWe have developed Hypergeometric Analysis of Tiling-arrays (HAT), and first evaluated its performance for tiling-array datasets from a Chromatin Immunoprecipitation study on chip (ChIP-on-chip) for the identification of genome-wide DNA binding profiles of transcription factor Cebpa (used for method comparison). Using this assay, we can refine the detection of regions-of-interest by illustrating that regions detected by HAT are more highly enriched for expected motifs in comparison with an alternative detection method (MAT). Subsequently, data from a retroviral insertional mutagenesis screen were used to examine the performance of HAT among different applications of tiling-array datasets. In both studies, detected regions-of-interest have been validated with (q)PCR.ConclusionsWe demonstrate that HAT has increased specificity for analysis of tiling-array data in comparison with the alternative method, and that it accurately detects regions-of-interest in two different applications of tiling-arrays. HAT has several advantages over previous methods: i) as there is no single cut-off level for probe-intensity, HAT can detect regions-of-interest at various thresholds, ii) it can detect regions-of-interest of any size, iii) it is independent of probe-resolution across the genome, and across tiling-array platforms and iv) it employs a single user defined parameter: the significance level. Regions-of-interest are detected by computing the hypergeometric-probability, while controlling the Family Wise Error. Furthermore, the method does not require experimental replicates, common regions-of-interest are indicated, a sequence-of-interest can be examined for every detected region-of-interest, and flanking genes can be reported.

Dataset Information

Hypergeometric decomposition of symmetric K3 quartic pencils.

Publications

Hypergeometric decomposition of symmetric K3 quartic pencils.

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