Rational Subgroups Research Articles

The rational torsion subgroup of $$J_0(\mathfrak {p}^r)$$

The rational torsion subgroup of $$J_0(\mathfrak {p}^r)$$

Generalized Rational Evaluation Subgroups of the Inclusion between Complex Projective Spaces

We use a model of mapping spaces to compute the generalized rational Gottlieb groups of the inclusion $i_{n,k}: \mathbb{C}P^n \hookrightarrow \mathbb{C}P^{n+k}$ between complex projective spaces.

Quantitative estimates for fractional Sobolev mappings in rational homotopy groups

Let N⊂RM be a smooth simply connected compact manifold without boundary. A rational homotopy subgroup of πN(N) is represented by a homomorphism deg:πN(N)→R.For maps f:SN→N we give a quantitative estimate of its rational homotopy group element deg([f])∈R in terms of its fractional Sobolev-norm. That is, we show that for all β∈(β0(deg),1], |deg([f])|≤C(deg)[f]Wβ,Nβ(SN)N+L(deg)β.Here C(deg)>0, L(deg)∈N, β0(deg)∈(0,1) are computable from the rational homotopy group represented by deg. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov’s integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivière.

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

AbstractFor a positive integer N, let be the modular curve over and its Jacobian variety. We prove that the rational cuspidal subgroup of is equal to the rational cuspidal divisor class group of when for any prime p and any squarefree integer M. To achieve this, we show that all modular units on can be written as products of certain functions , which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on under a mild assumption.

The rational torsion subgroup of J0(N)

Let N be a positive integer and let J0(N) be the Jacobian variety of the modular curve X0(N). For any prime p≥5 whose square does not divide N, we prove that the p-primary subgroup of the rational torsion subgroup of J0(N) is equal to that of the rational cuspidal divisor class group of X0(N), which is explicitly computed in [33]. Also, we prove the same assertion holds for p=3 under the extra assumption that either N is not divisible by 3 or there is a prime divisor of N congruent to −1 modulo 3.

Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3

We introduce an algorithm to compute the structure of the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points over finite fields using the Kummer variety and to check whether they are rational using heights. Both have been made explicit for Jacobians of hyperelliptic curves of genus 3 by Stoll. This article is partially based on the second-named author’s Master thesis.

An efficient adaptive r-MD multivariate single control chart for nonlinear multiple quality characteristics

An efficient multivariate single control chart to simultaneously monitor ‘location or mean’ and ‘scale or variabilities’, based on individual observations of multiple quality characteristics (MQC), is a continuous research endeavour. Compared to rational subgroups, time-ordered individual sample observations of MQC are common in manufacturing. However, few researchers proposed parametric and nonparametric multivariate control charts to monitor individual observations in this context. The proposed parametric approach has limited applicability due to the restrictive distribution assumption of MQC. In addition, most of the suggested parametric or nonparametric approaches recommend multiple charts to monitor process ‘location’ and ‘scale’ parameters simultaneously. Multiple charts can increase the control complexities and overall false-alarm-rate . MQC can also have significant nonlinear correlation structures and follow unknown or nonnormal distributions in real-life scenarios. Single parametric or nonparametric multivariate charts for individual observations can exhibit poor performance in such situations. In this study, an adaptive Mahalanobis depth and k-nearest neighbour (k-NN) rule-based r-AMD chart is proposed and verified to address the lacuna mentioned above. Monte-Carlo simulation studies are used to evaluate the performance of the r-AMD chart. Four real-life manufacturing scenarios are also considered to validate the suitability and superiority of r-AMD over the existing ‘r-MD’ and other competing charts.

Consistências dos índices de capacidade baseados na distribuição normal de probabilidade

Abstract: Capability analysis seeks to estimate the probability that a process will produce compliant products. The capability indices are dimensionless parameters that measure how well the process can meet specifications. In the literature, eight capability indices are listed, among others, considering a stable process under statistical control and based on the normal probability distribution, defined by: Cp, Pp, Cpk, Ppk, Cpm, Ppm, Cpmk, and Ppmk. Basically, the index formulas differ in the calculations of the variability within and total, and of the shifts of the mean in relation to the nominal value and the nearest specification limit. The objective of this article was to compare these capacity indexes, and for that, it was chosen the most consistent estimator, that is, the one that improved the accuracy and efficiency as the number of observations increased. Thus, a simulation of 30,000 values of a normal random variable with a mean equal to zero and a standard deviation equal to one was performed. This made it possible to sample this process 1,000 times using 5, 10, 15, 20, 25, and 30 rational subgroups with individual observations or sample elements. Subsequently, 20 mean shifts were provoked, with values ranging from 0.1 to 2 and varying by 0.1 unit. According to the results, it was concluded that the indexes Cpk and Ppk were the most consistent in presenting higher accuracy and efficiency for at least 15 rational subgroups or sample elements, regardless of the magnitude of the mean displacement in relation to the nominal value.

The Stratified Homotopy Type of the Reductive Borel–Serre Compactification

Abstract We identify the exit path $\infty $-category of the reductive Borel–Serre compactification as the nerve of a $1$-category defined purely in terms of rational parabolic subgroups and their unipotent radicals. As immediate consequences, we identify the fundamental group of the reductive Borel–Serre compactification, recovering a result of Ji–Murty–Saper–Scherk, and we obtain a combinatorial incarnation of constructible complexes of sheaves on the reductive Borel–Serre compactification as elements in a derived functor category.

Another look at rational torsion of modular Jacobians

We study the rational torsion subgroup of the modular Jacobian [Formula: see text] for N a square-free integer. We give a proof of a result of Ohta on a generalization of Ogg's conjecture: For a prime number [Formula: see text], the p-primary part of the rational torsion subgroup equals that of the cuspidal subgroup. Whereas previous proofs of this result used explicit computations of the cardinalities of these groups, we instead use their structure as modules for the Hecke algebra.

A Bivariate Beta from Gamma Ratios for Determining a Potential Variance Change Point: Inspired from a Process Control Scenario

Within statistical process control (SPC), normality is often assumed as the underlying probabilistic generator where the process variance is assumed equal for all rational subgroups. The parameters of the underlying process are usually assumed to be known—if this is not the case, some challenges arise in the estimation of unknown parameters in the SPC environment especially in the case of few observations. This paper proposes a bivariate beta type distribution to guide the user in the detection of a permanent upward or downward step shift in the process’ variance that does not directly rely on parameter estimates, and as such presents itself as an attractive and intuitive approach for not only potentially identifying the magnitude of the shift, but also the position in time where this shift is most likely to occur. Certain statistical properties of this distribution are derived and simulation illustrates the theoretical results. In particular, some insights are gained by comparing the newly proposed model’s performance with an existing approach. A multivariate extension is described, and useful relationships between the derived model and other bivariate beta distributions are also included.

Use of rational subgrouping to identify areas for improving time to ultrasound performance

Background: Safe, high-quality care within emergency departments includes the provision of efficient diagnostic and testing. At the Alberta Children’s Hospital, emergency department (ED) and radiology staff have anecdotally noted delays for some patients in the completion of abdominal and pelvic ultrasounds. Rational subgrouping is an approach that stratifies data into multiple groups for display in control charts to minimize in-group variability and maximize between-group variability. The aim of this project was to identify subgroups based on demographic factors, shift type, and disposition status that experience longer time to completion, in order to identify targeted areas for improvement. 
 Methods: A working group of interprofessional stakeholders was formed, including emergency physicians, an emergency QI nurse, a radiologist and a sonographer. Existing protocols were reviewed. These included recent changes that had been implemented but not yet systemically evaluated, including specifying ultrasound indications that require bladder filling, revising bladder filling protocols, specifying requirements for radiologist approval, and clarifying flow and communication processes.
 Abdominal and pelvic ultrasounds conducted in the Alberta Children’s Hospital ED from May 2019 to April 2021 were included. This range encapsulated data points both before and after the declaration of the COVID-19 pandemic on Mar 11, 2020, which notably reduced ED volumes. Time stamps were obtained from the electronic health record for the time of physician assessment, time of study request and the time of study result. Data was subdivided based on rational subgroups for (1) sex, (2) age (3) shift time availability, and (4) disposition status. Cases with excessive delay were identified and a chart review was conducted to confirm time stamp validity and to identify unique circumstances contributing to delay. An exemption for ethics board review was obtained from the Conjoint Health Research Ethics Board at the University of Calgary due to the primary purpose of the project being quality improvement.
 Evaluation Methods: Statistical process control charts (X-bar and S-chart) were used to establish mean and variation in time from ultrasound order to study result in the total population and each subgroup, measured in minutes. Scrotal ultrasounds were analyzed as a comparator to understand the most efficient time expected from ultrasound order to completion, given that they are typically highly prioritized. Rules to detect special causes were applied, including: (1) Single point outside control limits, defined as 3-sigma (2) Eight or more points above or below the centre line (mean) (3) Six consecutive points that are increasing (trend up) or decreasing (trend down) (4) Two out of three consecutive points near the outer third of a control limit (5) 15 consecutive points in the inner third nearest the centre line (mean).
 Results: The mean time from ordering an abdominal/pelvic ultrasound to having reported findings was 203 minutes, compared to 103 minutes for scrotal ultrasounds. The mean time from physician assessment to having a reported study was 279 minutes, compared to 135 minutes for scrotal ultrasounds. Using subgrouping by sex for abdominal/pelvic ultrasounds, females had an mean order to result time of 226 minutes, versus 178 minutes for males. For the 0 to 3 year age group, the mean order to result time was 167 minutes, which increased for ages 4 to 11 to 193 minutes and for children greater than 12 years old, it was 223 minutes. By shift type, day shifts had a mean order to result time of 157 minutes, evening shifts 142 minutes and night shifts 317 minutes. Using rational subgrouping for disposition status, the order to result time for admitted patients was shorter at 190 minutes compared to discharged patients, which was 214 minutes.
 Advice and Lessons Learned: 
 1) This initiative highlights the benefits of using rational subgrouping for emergency department quality improvement projects. By separating our data based on sex, age, time of care, and disposition status, considerable variability was revealed between subgroups that would have otherwise been hidden.
 2) Groups more likely to experience longer wait times for abdominal or pelvic ultrasounds included females, patients arriving at night, and adolescents.
 3) Rational subgrouping with baseline data enables a targeted approach when designing plan-do-study-act cycles.

The rational cuspidal divisor class group of X0(N)

For any positive integer N, we completely determine the structure of the rational cuspidal divisor class group of X0(N), which is conjecturally equal to the rational torsion subgroup of J0(N). More specifically, for a given prime ℓ, we construct a rational cuspidal divisor Zℓ(d) for any non-trivial divisor d of N. Also, we compute the order of the linear equivalence class of Zℓ(d) and show that the ℓ-primary subgroup of the rational cuspidal divisor class group of X0(N) is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of Zℓ(d).

Evaluation subgroups of a map and the rationalized $G$-sequence

In this paper, we determine, in terms of the Sullivan models, rational evaluation subgroups of the inclusion $ \mathbb{C} P(n)\hookrightarrow \mathbb{C} P(n+k) $ between complex projective spaces and, more generally, the $ G $-sequence of the homotopy monomorphism $ \iota: X\hookrightarrow Y $ between simply connected formal homogeneous spaces for which $ \pi_{\ast}(Y)\otimes \mathbb{Q}$ is finite dimensional.

An unsupervised one-class-classifier support vector machine to simultaneously monitor location and scale of multivariate quality characteristics

PurposeThe purpose of this study is to propose an effective unsupervised one-class-classifier (OCC) support vector machine (SVM)-based single multivariate control chart (OCC-SVM) to simultaneously monitor “location” and “scale” shifts of a manufacturing process.Design/methodology/approachThe step-by-step approach to developing, implementing and fine-tuning the intrinsic parameters of the OCC-SVM chart is demonstrated based on simulation and two real-life case examples.FindingsA comparative study, considering varied known and unknown response distributions, indicates that the OCC-SVM is highly effective in detecting process shifts of samples with individual observations. OCC-SVM chart also shows promising results for samples with a rational subgroup of observations. In addition, the results also indicate that the performance of OCC-SVM is unaffected by the small reference sample size.Research limitations/implicationsThe sample responses are considered identically distributed with no significant multivariate autocorrelation between sample observations.Practical implicationsThe proposed easy-to-implement chart shows satisfactory performance to detect an out-of-control signal with known or unknown response distributions.Originality/valueVarious multivariate (e.g. parametric or nonparametric) control chart(s) are recommended to monitor the mean (e.g. location) and variance (e.g. scale) of multiple correlated responses in a manufacturing process. However, real-life implementation of a parametric control chart may be complex due to its restrictive response distribution assumptions. There is no evidence of work in the open literature that demonstrates the suitability of an unsupervised OCC-SVM chart to simultaneously monitor “location” and “scale” shifts of multivariate responses. Thus, a new efficient OCC-SVM single chart approach is proposed to address this gap to monitor a multivariate manufacturing process with unknown response distributions.

A New Robust Multivariate EWMA Dispersion Control Chart for Individual Observations

A multivariate control chart is proposed to detect changes in the process dispersion of multiple correlated quality characteristics. We focus on individual observations, where we monitor the data vector-by-vector rather than in (rational) subgroups. The proposed control chart is developed by applying the logarithm to the diagonal elements of the estimated covariance matrix. Then, this vector is incorporated in an exponentially weighted moving average (EWMA) statistic. This design makes the chart robust to non-normality in the underlying data. We compare the performance of the proposed control chart with popular alternatives. The simulation studies show that the proposed control chart outperforms the existing procedures when there is an overall decrease in the covariance matrix. In addition, the proposed chart is the most robust to changes in the data distribution, where we focus on small deviations which are difficult to detect. Finally, the compared control charts are applied to two case studies.

Derivations of a Sullivan Model and the Rationalized G -Sequence

Let G k , n ℂ for 2 ≤ k < n denote the Grassmann manifold of k -dimensional vector subspaces of ℂ n . In this paper, we compute, in terms of the Sullivan models, the rational evaluation subgroups and, more generally, the G -sequence of the inclusion G 2 , n ℂ ↣ G 2 , n + r ℂ for r ≥ 1 .

Model‐based process capability indices: The dry‐etching semiconductor case study

AbstractProcess capability indices are widely used to check quality standards both at the production level and for business activity. They consider the location and the deviation from specification limits and targets. The literature contains many contributions on this topic both in the univariate and the multivariate context. Motivated by a real semiconductor case study, we discuss the role of rational subgroups and the challenge they present in the computation of capability indices, especially when data refer to lots of products. In addition, our context involves a mix of problems: unilateral specification limit, nonsymmetric distribution of the data, evidence of data from a mixture of distributions, and the need to filter one component of the mixture. After solving the previous issues and because of the peculiar characteristics of semiconductor processes based on the so called “wafers,” we contribute to the literature a proposal on how to compute capability indices in the case of heteroscedastic spatial processes. With a generalized additive model, we show that it is possible to estimate a capability surface that allows the identification of regions expected to not be fully compliant with the desired quality standards.

Challenges for Child and Adolescent Psychiatric Research in the Era of COVID-19.

Challenges for Child and Adolescent Psychiatric Research in the Era of COVID-19.

A combined mixed-s-skip sampling strategy to reduce the effect of autocorrelation on the X̄ scheme with and without measurement errors

In order to reduce the effect of autocorrelation on the monitoring scheme, a new sampling strategy is proposed to form rational subgroup samples of size n. It requires sampling to be done such that: (i) observations from two consecutive samples are merged, and (ii) some consecutive observations are skipped before sampling. This technique which is a generalized version of the mixed samples strategy is shown to yield a better reduction of the negative effect of autocorrelation when monitoring the mean of processes with and without measurement errors. For processes subjected to a combined effect of autocorrelation and measurement errors, the proposed sampling technique, together with multiple measurement strategy, yields an uniformly better zero-state run-length performance than its two main existing competitors for any autocorrelation level. However, in steady-state mode, it yields the best performance only when the monitoring process is subject to a high level of autocorrelation, for any given level of measurement errors. A real life example is used to illustrate the implementation of the proposed sampling strategy.