Split Mapping Research Articles

Height coincidences in products of the projective line

We consider hypersurfaces in $${\mathbb {P}}_1^n$$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $$n-1$$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $$n-1$$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.

Planck intermediate results

We present the NPIPE processing pipeline, which produces calibrated frequency maps in temperature and polarization from data from the Planck Low Frequency Instrument (LFI) and High Frequency Instrument (HFI) using high-performance computers. NPIPE represents a natural evolution of previous Planck analysis efforts, and combines some of the most powerful features of the separate LFI and HFI analysis pipelines. For example, following the LFI 2018 processing procedure, NPIPE uses foreground polarization priors during the calibration stage in order to break scanning-induced degeneracies. Similarly, NPIPE employs the HFI 2018 time-domain processing methodology to correct for bandpass mismatch at all frequencies. In addition, NPIPE introduces several improvements, including, but not limited to: inclusion of the 8% of data collected during repointing manoeuvres; smoothing of the LFI reference load data streams; in-flight estimation of detector polarization parameters; and construction of maximally independent detector-set split maps. For component-separation purposes, important improvements include: maps that retain the CMB Solar dipole, allowing for high-precision relative calibration in higher-level analyses; well-defined single-detector maps, allowing for robust CO extraction; and HFI temperature maps between 217 and 857 GHz that are binned into 0′.9 pixels (Nside = 4096), ensuring that the full angular information in the data is represented in the maps even at the highest Planck resolutions. The net effect of these improvements is lower levels of noise and systematics in both frequency and component maps at essentially all angular scales, as well as notably improved internal consistency between the various frequency channels. Based on the NPIPE maps, we present the first estimate of the Solar dipole determined through component separation across all nine Planck frequencies. The amplitude is (3366.6 ± 2.7) μK, consistent with, albeit slightly higher than, earlier estimates. From the large-scale polarization data, we derive an updated estimate of the optical depth of reionization of τ = 0.051 ± 0.006, which appears robust with respect to data and sky cuts. There are 600 complete signal, noise and systematics simulations of the full-frequency and detector-set maps. As a Planck first, these simulations include full time-domain processing of the beam-convolved CMB anisotropies. The release of NPIPE maps and simulations is accompanied with a complete suite of raw and processed time-ordered data and the software, scripts, auxiliary data, and parameter files needed to improve further on the analysis and to run matching simulations.

PASA: Identifying More Credible Structural Variants of Hedou12.

Although plenty of structural variant detecting approaches for human genomes can be looked up in the literatures, little has been acknowledged on the effectiveness of those structural variant softwares for plant genomes. Moreover, it has been demonstrated frequent occurrences for those structural variant detecting softwares to find too many false structural variants. In this paper, we devote to detect deletions, insertions, and inversions, in total of three kinds of structural variants occurring in Hedou12 genome in contrast to Williams82 genome. To find more potential structural variants, we try to develop new principles to detect discordant and split read map sets supporting structural variants. Aiming to enhance the precision of structural variant detections, we propose two new sequencing characteristic based probability models, which use the sequencing parameters of Hedou12 genome as well as the parameters for Hedou12 paired-end reads to be aligned onto Williams82, to evaluate the probability for a potential structural variant to occur in. To remove the false members from those potential structural variants, we propose a set cover problem model to describe formally on which potential structural variants it should accept to achieve as high as possible a probability summation. This will achieve a solution with more credible structural variants, which can be verified by comparing with DELLY version 0.5.8 and LUMPY version 0.2.2.3. Our algorithm has been verified to be able to find deletions, insertions, and inversions in Hedou12 in contrast to Williams82 DELLY as well as LUMPY fails to find.

Approximations and Mittag-Leffler conditions the tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [20], [14], [19]. If R is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [16], and it does not provide for approximations when R has cardinality ≤ ℵ0, [8]. We remove the cardinality restriction on R in the latter result. We also prove an extension of the Countable Telescope Conjecture [23]: a cotorsion pair (A, B) is of countable type whenever the class B is closed under direct limits. In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs’ problem on module approximations for classes of modules associated with tilting [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [26]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [22].

Fixed points of n-valued maps on surfaces and the Wecken property—a configuration space approach

In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focusing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp. the 2-sphere S2) has the Wecken property for n-valued maps for all n ∈ ℕ (resp. all n ≥ 3). In the case n = 2 and S2, we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q: X → X with a subset of the coordinate maps of a lift of the n-valued split map .

On the non-existence of right almost split maps

We show that, over any ring, a module C is the codomain of a right almost split map, if and only if C is finitely presented with a local endomorphism ring. Thus we answer a 40-year-old question by Maurice Auslander.

Subpixel Multiplexing Method for 3D Lenticular Display

Three-dimensional display technologies based on lenticular sheet overlaid onto spatial light modulator screen have been studied for decades. However, the quality of these displays still suffers from insufficient number of views and zone-jumping between views. We present herein a subpixel multiplexing method in this paper. We propose to split mapping and alignment into two separate tasks, processed in parallel threads. Alignment thread deals with the task of computing the geometrical relationship between lenticular sheet and Liquid Crystal Display (LCD) panel for multiplexing. Afterwards, we conduct the multiplexing procedure through a box-constrained integer least squares algorithm. After multiplexing, each subpixel aggregated on the lenticular sheet is a multiplexing one that mixes up a number of subpixels in local region on the LCD plane. As a result, we multiplex subpixels on the synthetic image up to 27 views with a resolution of $1080 \times 1920$ and the rendering speed is 73.34 frames per second (fps).

Monocular SLAM with Conditionally Independent Split Mapping

The recovery of structure from motion in real time over extended areas demands methods that mitigate the effects of computational complexity and arithmetical inconsistency. In this paper, we develop SCISM, an algorithm based on relative frame bundle adjustment, which splits the recovered map of 3D landmarks and keyframes poses so that the camera can continue to grow and explore a local map in real time while, at the same time, a bulk map is optimized in the background. By temporarily excluding certain measurements, it ensures that both maps are consistent, and by using the relative frame representation, new results from the bulk process can update the local process without disturbance. The paper first shows how to apply this representation to the parallel tracking and mapping (PTAM) method, a real-time bundle adjuster, and compares results obtained using global and relative frames. It then explains the relative representation's use in SCISM and describes an implementation using PTAM. The paper provides evidence of the algorithm's real-time operation in outdoor scenes, and includes comparison with a more conventional submapping approach.

Genome-Wide Detection of Spontaneous Chromosomal Rearrangements in Bacteria

Genome rearrangements have important effects on bacterial phenotypes and influence the evolution of bacterial genomes. Conventional strategies for characterizing rearrangements in bacterial genomes rely on comparisons of sequenced genomes from related species. However, the spectra of spontaneous rearrangements in supposedly homogenous and clonal bacterial populations are still poorly characterized. Here we used 454 pyrosequencing technology and a ‘split mapping’ computational method to identify unique junction sequences caused by spontaneous genome rearrangements in chemostat cultures of Salmonella enterica Var. Typhimurium LT2. We confirmed 22 unique junction sequences with a junction microhomology more than 10 bp and this led to an estimation of 51 true junction sequences, of which 28, 12 and 11 were likely to be formed by deletion, duplication and inversion events, respectively. All experimentally confirmed rearrangements had short inverted (inversions) or direct (deletions and duplications) homologous repeat sequences at the endpoints. This study demonstrates the feasibility of genome wide characterization of spontaneous genome rearrangements in bacteria and the very high steady-state frequency (20–40%) of rearrangements in bacterial populations.

PRISM: Pair-read informed split-read mapping for base-pair level detection of insertion, deletion and structural variants

The development of high-throughput sequencing technologies has enabled novel methods for detecting structural variants (SVs). Current methods are typically based on depth of coverage or pair-end mapping clusters. However, most of these only report an approximate location for each SV, rather than exact breakpoints. We have developed pair-read informed split mapping (PRISM), a method that identifies SVs and their precise breakpoints from whole-genome resequencing data. PRISM uses a split-alignment approach informed by the mapping of paired-end reads, hence enabling breakpoint identification of multiple SV types, including arbitrary-sized inversions, deletions and tandem duplications. Comparisons to previous datasets and simulation experiments illustrate PRISM's high sensitivity, while PCR validations of PRISM results, including previously uncharacterized variants, indicate an overall precision of ~90%. PRISM is freely available at http://compbio.cs.toronto.edu/prism.

YAHA: fast and flexible long-read alignment with optimal breakpoint detection

Motivation: With improved short-read assembly algorithms and the recent development of long-read sequencers, split mapping will soon be the preferred method for structural variant (SV) detection. Yet, current alignment tools are not well suited for this.Results: We present YAHA, a fast and flexible hash-based aligner. YAHA is as fast and accurate as BWA-SW at finding the single best alignment per query and is dramatically faster and more sensitive than both SSAHA2 and MegaBLAST at finding all possible alignments. Unlike other aligners that report all, or one, alignment per query, or that use simple heuristics to select alignments, YAHA uses a directed acyclic graph to find the optimal set of alignments that cover a query using a biologically relevant breakpoint penalty. YAHA can also report multiple mappings per defined segment of the query. We show that YAHA detects more breakpoints in less time than BWA-SW across all SV classes, and especially excels at complex SVs comprising multiple breakpoints.Availability: YAHA is currently supported on 64-bit Linux systems. Binaries and sample data are freely available for download from http://faculty.virginia.edu/irahall/YAHA.Contact: imh4y@virginia.edu

SVM 2 : an improved paired-end-based tool for the detection of small genomic structural variations using high-throughput single-genome resequencing data

Several bioinformatics methods have been proposed for the detection and characterization of genomic structural variation (SV) from ultra high-throughput genome resequencing data. Recent surveys show that comprehensive detection of SV events of different types between an individual resequenced genome and a reference sequence is best achieved through the combination of methods based on different principles (split mapping, reassembly, read depth, insert size, etc.). The improvement of individual predictors is thus an important objective. In this study, we propose a new method that combines deviations from expected library insert sizes and additional information from local patterns of read mapping and uses supervised learning to predict the position and nature of structural variants. We show that our approach provides greatly increased sensitivity with respect to other tools based on paired end read mapping at no cost in specificity, and it makes reliable predictions of very short insertions and deletions in repetitive and low-complexity genomic contexts that can confound tools based on split mapping of reads.

Detecting genomic indel variants with exact breakpoints in single- and paired-end sequencing data using SplazerS

The reliable detection of genomic variation in resequencing data is still a major challenge, especially for variants larger than a few base pairs. Sequencing reads crossing boundaries of structural variation carry the potential for their identification, but are difficult to map. Here we present a method for 'split' read mapping, where prefix and suffix match of a read may be interrupted by a longer gap in the read-to-reference alignment. We use this method to accurately detect medium-sized insertions and long deletions with precise breakpoints in genomic resequencing data. Compared with alternative split mapping methods, SplazerS significantly improves sensitivity for detecting large indel events, especially in variant-rich regions. Our method is robust in the presence of sequencing errors as well as alignment errors due to genomic mutations/divergence, and can be used on reads of variable lengths. Our analysis shows that SplazerS is a versatile tool applicable to unanchored or single-end as well as anchored paired-end reads. In addition, application of SplazerS to targeted resequencing data led to the interesting discovery of a complete, possibly functional gene retrocopy variant. SplazerS is available from http://www.seqan.de/projects/ splazers. Supplementary data are available at Bioinformatics online.

SPM: split piecewise mapping for high quality wireless video delivery

Multiple prioritised access category (AC) queues and virtual collision handling among them are the main mechanism of the IEEE 802.11e EDCA for multimedia service differentiation. However, the AC for video (AC_VI) is characteristic of FIFO, and thus its performance is limited. For further performance enhancement, differentiation among video packets can help. This study presents a novel cross-layer design combining the error propagation length (EPL) scheme in the application layer allowing each video frame in a group of pictures to have its own importance level, and our proposed split piecewise mapping (SPM) algorithm in the MAC layer offering a corresponding mapping probability function for each importance level of video packets among the AC queues. The results show that the proposed EPL+SPM design outperforms the existing ones, with largest achieved gains as high as 2.8 dB over I/P+PM, 4.1 dB over I/P+AM and 7.4 dB over EDCA.

A key module over pure-semisimple hereditary rings

Let R be a hereditary, indecomposable, left pure semisimple ring. Inspired by [I. Reiten, C.M. Ringel, Infinite dimensional representations of canonical algebras, Canad. J. Math. 58 (2006) 180–224], we investigate the perfect cotorsion pair ( C , D ) in R - Mod generated by the preinjective component q. We show that there is a finitely generated product-complete tilting and cotilting left R-module W such that C = Cogen W and D = Gen W . The finite subcategory w of R - mod given by the indecomposable summands of W stores important information on R. For example, if we assume R of infinite representation type, then by [B. Zimmermann-Huisgen, Strong preinjective partitions and representation type of artinian rings, Proc. Amer. Math. Soc. 109 (1990) 309–322] there are non-preinjective indecomposable modules occurring as direct summands of products of preinjective modules, and it turns out that w is precisely the class of such modules. Moreover, we prove that R has finite representation type if and only if every module in w is source of a left almost split map in R - mod . Finally, we address the question when W is endofinite.

Parity Splits by Triple Point Distances in X-Trees

At the conference Dress defined parity split maps by triple point distance and asked for a characterisation of such maps coming from binary phylogenetic X-trees. This article gives an answer to that question. The characterisation for X-trees can be easily described as follows: If all restrictions of a split map to sets of five or fewer elements is a parity split map for an X-tree, then so is the entire map. To ensure that the parity split map comes from an X-tree which is binary and phylogenetic, we add two more technical conditions also based on studying at most five points at a time.

Copure semisimple categories and almost split maps

We study copure semisimple Grothendieck categories, i.e. Krull–Schmidt categories satisfying the artinian condition on morphisms between finitely presented indecomposable objects. It is shown that these categories have many attractive properties, e.g. the endofiniteness of finitely presented objects and the existence of left almost split maps, among others. Applications are given to pure semisimple Grothendieck categories, and categories of locally finite representation type. In particular, we prove the existence of almost split sequences over Grothendieck categories of locally finite representation type with enough projectives. Our methods are based on functor categories and a module theory over Krull–Schmidt rings with enough idempotents.

Top exterior powers over commutative rings

We recall that the class of FGC rings arc those rings for which every finitely generated R-module may be written as a direct sum of cyclic R-modules and demonstrate that it is coincident with a new class of rings. When the top exterior power of an R module M is cyclic, we define the non-negative integer p = crank(M), and show that under the hypothesis that the annihilator of the top exterior power is contained in the Jacobson radical, it counts the maximum number of times that the top exterior power can appear as a direct summand of M. If P is finitely generated projective of constant rank n, then an isomorphism is set up. Finally, a unimodular condition is shown to be a consequence whenever right multiplication is a split map onto the top exterior power of P.

Regular semigroups with a split map

Regular semigroups with a split map

Strong preinjective partitions and representation type of Artinian rings

It is shown that for every ring of left pure global dimension zero (i.e., for every ring all of whose left modules are direct sums of finitely generated modules), the finitely generated left modules can be grouped to a unique "strong preinjective partition" while the finitely presented right modules possess a "strong preprojective partition"; these strong partitions are upgraded versions of the partitions introduced by Auslander and Smaløfor Artin algebras. One direct consequence is that a ring of left pure global dimension zero has finite representation type if and only if there exist sufficiently many almost split maps among its finitely generated left modules. This provides a very elementary proof for Auslander’s theorem saying that for Artin algebras vanishing of the left pure global dimension is equivalent to finiteness of the representation type.