Rational Subgroups Research Articles

Estimates for the number of rational points on simple abelian varieties over finite fields

Let A be a simple Abelian variety of dimension g over the field $$\mathbb {F}_q$$ . The paper provides improvements on the Weil estimates for the size of $$A(\mathbb {F}_q)$$ . For an arbitrary value of q we prove $$(\lfloor (\sqrt{q}-1)^2 \rfloor + 1)^g \le \#A(\mathbb {F}_q) \le (\lceil (\sqrt{q}+1)^2 \rceil - 1)^{g}$$ holds with finitely many exceptions. We compute improved bounds for various small values of q. For instance, the Weil bounds for $$q=3,4$$ give a trivial estimate $$\#A(\mathbb {F}_q) \ge 1$$ ; we prove $$\# A(\mathbb {F}_3) \ge 1.359^g$$ and $$\# A(\mathbb {F}_4) \ge 2.275^g$$ hold with finitely many exceptions. We use these results to give some estimates for the size of the rational 2-torsion subgroup $$A(\mathbb {F}_q)[2]$$ for small q. We also describe all abelian varieties over finite fields that have no new points in some finite field extension.

On monitoring the process mean of autocorrelated observations with measurement errors using the w‐of‐w runs‐rules scheme

AbstractRuns‐rules have been widely used since the 1950s in industrial and nonindustrial process monitoring applications to improve the performance of basic and other traditional monitoring schemes. However, none of the studies on runs‐rules have accounted for a process with a combined effect of measurement errors and autocorrelation. Hence, in this paper, the use of the w‐of‐w runs‐rules to improve the performance of the Shewhart scheme using an additive model with a constant variance and a first‐order autoregressive model is proposed. To reduce the combined negative effect of measurement errors and autocorrelation, we implement a sampling strategy based on rational subgroups in which (a) multiple measurements per item are taken (instead of a standard single measurement) and (b) non‐neighboring observations are gathered. Moreover, the latter sampling strategy is incorporated into the values of probability elements of a Markov chain matrix which is used to derive some closed‐form expressions for the zero‐ and steady‐state run‐length distribution. The main finding of this study is that, with respect to some overall performance measures, the proposed scheme outperforms the existing Shewhart scheme by a significant margin. A real‐life example is used to illustrate the practical implementation of the proposed scheme.

Rational subgroups and invariants of F4

It is known that simple algebraic groups of type F4, defined over a field k, are precisely the full automorphism groups of Albert algebras over k. For an Albert division algebra A over a field k of characteristic different from 2 and 3 and G = Aut(A), the full group of algebra automorphisms of A, it is known that any connected, simple k-subgroup of G is of type A2 or D4. In this paper, we classify all k-embeddings of connected, simple groups of type A2 and D4 in G and give Galois cohomological conditions for such embeddings to exist, along with some applications.

Shewhart-type monitoring schemes with supplementary w-of-w runs-rules to monitor the mean of autocorrelated samples

The simplicity of the Shewhart charts makes it popular in practice; however, it is insensitive to detecting small shifts. In an effort to preserve its simplicity but increase its detection ability, in this paper, we propose two Shewhart-type charts supplemented with w-of-w runs-rules to monitor the mean of autocorrelated samples using a first-order autoregressive model. It is shown that the higher the level of autocorrelation, the poor the proposed schemes perform. Hence, we implement the skipping sampling strategy which involves sampling of nonconsecutive observations to form the rational subgroups to compute the corresponding sample means. The Markov chain approach is used to derive zero- and steady-state closed-form expressions of the average run-length (ARL). To supplement the specific shift performance metric, i.e. ARL, we compute the overall performance metric so that these schemes can also be evaluated from a global point of view. A real-life example is provided to illustrate the implementation of the monitoring schemes proposed here.

One‐sided runs‐rules schemes to monitor autocorrelated time series data using a first‐order autoregressive model with skip sampling strategies

AbstractIn some industrial or health‐related processes, it makes more practical sense to monitor either an increase only or a decrease only in the quality characteristic of interest. Consequently, in this paper, we propose four one‐sided Shewhart charts supplemented with runs‐rules to monitor the mean of autocorrelated normally distributed samples using a stationary first‐order autoregressive model. To counteract the negative effect of autocorrelation, we implement a sampling strategy which involves sampling of non‐neighboring observations to form rational subgroups. The Markov chain technique is used to derive zero‐state and steady‐state closed‐form expressions of the specific shift performance metric, ie, average run‐length (ARL). Moreover, we compute the expected ARL metric which evaluates each monitoring scheme based on all specified range of possible values of the shift parameter, or more specifically, from a global point of view and thus gives a different perspective from the specific shift ARL metric. We observed that the steady‐state improved w‐of‐w and the improved 2‐of‐(H + 1) schemes yield a better overall performance than their corresponding basic counterparts for all different levels of autocorrelation. A real‐life example is provided to illustrate the implementation of the monitoring schemes proposed here.

High-dimensional data monitoring using support machines

Dealing with high-dimensional data is a major challenge primary in statistics and consequently in statistical process control (SPC). “Shewhart-type” charts are control charts using rational subgrouping. Shewhart-type charts are effective in the detection of large shifts. In high-dimensional settings, where the number of variables is nearly as large as or larger than the number of observations, it will be very hard to use Shewhart-type charts based on rational subgroups. An alternative is to use Shewhart-type charts based on individual observations. Also, for most conventional control charts, the design of the control limits is commonly based on the assumption that the quality characteristics follow a multivariate normal distribution. However, this may not be reasonable in many real-world problems. This paper addresses these issues and suggests a monitoring methodology motivated by statistical learning theory. The proposed multivariate control chart uses tensor space model to represent a high-dimensional vector. This chart makes use of information extracted from in-control preliminary samples. Simulation studies demonstrate that the proposed control chart has very good performances.

Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion

In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt produced a database of elliptic curves over $\mathbb{Q}$ ordered by height in which they computed the rank, the size of the $2$-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over $\mathbb{Q}$ whose rational torsion subgroup is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$. Conditional on GRH and BSD, we compute the rank of $92\%$ of the $202461$ curves with parameter height less than $10^3$. We also compute the size of the $2$-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.

New sampling strategies to reduce the effect of autocorrelation on the synthetic T2 chart to monitor bivariate process

AbstractIn this paper, synthetic T2 chart is developed to monitor bivariate process with correlated variables and autocorrelated observations. The proposed chart is a combination of the Hotelling's T2 chart and the conforming run length chart. The operation and design of the chart are described when observations are autocorrelated and cross correlated. The first‐order vector autoregressive process VAR (1) is used to model the bivariate data from an autocorrelated process of interest. Using an average run length as performance measure criterion in the VAR (1) model, it is observed that autocorrelation seriously impact the performance of the synthetic T2 chart. To reduce the effect of autocorrelation on the performance of the synthetic T2 chart, the skip and mixed sampling strategies are implemented to form rational subgroups in the construction of synthetic T2 chart. The average run length performance of the synthetic T2 chart implementing these strategies is compared with that of the standard strategy of formation of rational subgroups. It is observed that implementing skip and mixed sampling strategies within rational subgroup improves the performance of the synthetic T2 chart.

Rational torsion subgroups of modular Jacobian varieties

In this article, we study the Q-rational torsion subgroups of the Jacobian varieties of modular curves. The main result is that, for any positive integer N, J0(N)(Q)tor[q∞]=0 if q is a prime not dividing 6⋅N⋅∏p|N(p2−1). To prove the result, we explicitly construct a collection of Eisenstein series with rational Fourier expansions, and then determine their constant terms to control the size of the rational torsion subgroups.

Use of quality improvement (QI) methodology to decrease length of stay (LOS) for newborns with uncomplicated gastroschisis

PurposeGastroschisis is a congenital defect of the abdominal wall leading to considerable morbidity and long hospitalizations. The purpose of this study was to use quality improvement methodology to standardize care in the management of gastroschisis that may contribute to length of stay (LOS). MethodsA gastroschisis quality improvement team established a best-practice protocol in order to decrease LOS in infants with uncomplicated gastroschisis. The specific aim was to decrease median LOS from a baseline of 34days. We used statistical process control charts including rational subgroup analysis to monitor LOS. ResultsFrom December 2008 to December 2016, 119 patients with uncomplicated gastroschisis were evaluated. Retrospective data were obtained on 25 patients prior to protocol implementation. Ninety-four patients with uncomplicated gastroschisis comprised the prospective process stage. The median LOS for this retrospective cohort was 34days (IQR: 30.5–50.5), while the median LOS for the prospective cohort following implementation of the protocol decreased to 29days (IQR: 23–43). ConclusionsWith the use of quality improvement methodology, including standardization of care and a change in surgical approach, the median LOS for newborns with uncomplicated gastroschisis at our institution decreased from 34days to 29days. Level of evidence3.

Orbit equivalence of Cantor minimal systems and their continuous spectra

To any continuous eigenvalue of a Cantor minimal system $$(X,\,T)$$ , we associate an element of the dimension group $$K^0(X,\,T)$$ associated to $$(X,\,T)$$ . We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by Cantor minimal systems. The strong orbit equivalence (respectively orbit equivalence) class of a Cantor minimal system associated to an irrationally miscible dimension group $$(G,\,u)$$ (resp. with trivial infinitesimal subgroup) with trivial rational subgroup, have no non-trivial continuous eigenvalues.

Shewhart dispersion charts made easy for mild to moderately autocorrelated normally distributed data

ABSTRACTThe aim is to make it easier for users to implement dispersion monitoring plans when the data are autocorrelated. If the data follow a low-order ARMA model then the same threshold for the monitoring plan is applicable by revising the rational subgroup size. A robust plan for non-normal data is achieved in most cases by first transforming the data to an approximate normal distribution, and then applying the plan for normally distributed data. The dispersion plan is based on the robust regression estimator of the slope coefficient when regressing the ordered rational sub-group values against their “expected” in-control values.

Analysis of the Effect of Subgroup Size on the X-Bar Control Chart Using Forensic Science Laboratory Sample Influx Data

This paper analyzes the effect of subgroup size on the x-bar chart characteristics using sample influx (SIF) into forensic science laboratory (FSL). The characteristics studied include changes in out-or-control points (OCP), upper control limit UCLx, and zonal demarcations. Multi-rules were used to identify the number of out-of-control-points, Nocp as violations using five control chart rules applied separately. A sensitivity analysis on the Nocp was applied for subgroup size, k, and number of sigma above the mean value to determine the upper control limit, UCLx. A computer code was implemented using a FORTRAN code to create x-bar control-charts and capture OCP and other control-chart characteristics with increasing k from 2 to 25. For each value of k, a complete series of average values, Q(p), of specific length, Nsg, was created from which statistical analysis was conducted and compared to the original SIF data, S(t). The variation of number of out-of-control points or violations, Nocp, for different control-charts rules with increasing k was determined to follow a decaying exponential function, Nocp = Ae–α, for which, the goodness of fit was established, and the R2 value approached unity for Rule #4 and #5 only. The goodness of fit was established to be the new criteria for rational subgroup-size range, for Rules #5 and #4 only, which involve a count of 6 consecutive points decreasing and 8 consecutive points above the selected control limit (σ/3 above the grand mean), respectively. Using this criterion, the rational subgroup range was established to be 4 ≤ k ≤ 20 for the two x-bar control chart rules.

The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains

Let \(\mathfrak n\) be a square-free ideal of \(\mathbb {F}_q[T]\). We study the rational torsion subgroup of the Jacobian variety \(J_0(\mathfrak n)\) of the Drinfeld modular curve \(X_0(\mathfrak n)\). We prove that for any prime number \(\ell \) not dividing \(q(q-1)\), the \(\ell \)-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the \(\ell \)-primary part of the cuspidal divisor class group for any prime \(\ell \) not dividing \(q-1\).

Nonparametric partially random sequential test under Phase II sampling: An illustration to monitor water samples for arsenic contamination

ABSTRACTWe introduce a class of nonparametric two-sample tests based on a new partially sequential sampling scheme. Existing partially sequential procedures based on inverse sampling schemes, pioneered by Wolfe (1977) and Orban and Wolfe (1980), are updated in the light of random sequential sampling techniques, proposed by Mukhopadhyay and de Silva (2008). In a quality control setup, the present procedure can be looked upon as a Phase II on-line monitoring with rational subgroups of variable sizes where standards are unknown. We consider a training sample of prefixed size m as Phase I observations and adopt a random sequential sampling in Phase II. We discuss statistical methodologies in detail and provide some asymptotic results. Numerical results based on Monte Carlo are presented to justify asymptotic theory. We computationally investigate the power performances of the proposed test against some fixed alternative. We illustrate our procedure with real data related to water samples for monitoring arsenic contamination. Some concluding remarks along with possible future research problems are offered

Monitoring the mean vector with Mahalanobis kernels

Statistical process control (SPC) applies the science of statistics to various process controls in order to provide higher-quality products and better services. Multivariate control charts are essential tools in multivariate SPC. Hotelling’s charts based on rational subgroups of sample sizes larger than one are very sensitive for detecting relatively large shifts in the process mean vectors. However, it makes some very restrictive assumptions (multivariate normal distribution) that are usually difficult to be satisfied in real applications. Modern processes do not satisfy classical methods assumptions, such as normality or linearity. To overcome this issue, introduction of new techniques from statistical machine learning theory has been applied. Control charts based on Support Vector Data Description (SVDD), a popular data classifier method inspired by Support Vector Machines, benefit from a wide variety of choices of kernels, which determine the effectiveness of the whole model. Among the most popular choices of kernels is the Euclidean distance-based Gaussian kernel, which enables SVDD to obtain a flexible data description, thus enhances its overall predictive capability. This paper explores an even more robust approach by incorporating the Mahalanobis distance-based kernel (hereinafter referred to as Mahalanobis kernel) to SVDD and compares it with SVDD using the traditional Gaussian kernel.

ROBUSTIFICACIÓN DE LA CARTA DE CONTROL MULTIVARIADA √(|S|) EN LA FASE I DE CONTROL

En este artículo se estudia la robustificación de la carta basada en la raiz cuadrada de la varianza muestral generalizada √|S| para el control de la variabilidad de un proceso normal bivariado, en la etapa 1 de la Fase I de control, construida con observaciones sobre subgrupos racionales y utilizando los estimadores robustos MVE, MCD, estimador S. Estas cartas se comparan con la carta usual basada en el estimador insesgado muestral S de la matriz de covarianza ∑_0, en presencia de outliers provenientes de esquemas de perturbación del tipo contaminación con inflación de ∑_0 y contaminación perturbando sólo la correlación. Como medida de desempeño se usa el error cuadrático medio en la estimación de ∑_0 y el sesgo absoluto en la estimación de √(|∑_0 | ), sobre los estimadores insesgados para cada uno de estos parámetros, respectivamente, construidos con los subgrupos racionales que quedan después del proceso de depuración realizado en la Fase I y que se consideran como el conjunto de datos que representa el estado de variación estable del proceso.

A tutorial on an iterative approach for generating Shewhart control limits

ABSTRACTStandard statistical quality control text books (e.g., Montgomery 2015) discuss the differences between Phase I and Phase II control charts. In Phase I, control charts are more of an exploratory data analytical tool whose purpose is to generate the control limits for Phase II. Phase II is the true control procedure, which may be viewed as a series of hypothesis tests. The null hypothesis is that the process is in-control, and the alternative is that it is out-of-control. The typical presentation of Phase I is a preset number of rational subgroups. Such an approach fails to address the tension that exists between having enough information to generate reliable control limits and being able to start active control of the process. Vining (1998; 2009) outline an iterative approach that he developed in the early 1980s when he was employed by the Faber-Castell Corp. This article discusses the differences between Phase I and Phase II control charts and the impact of the number of rational subgroups in the Phase I study upon the quality of the resulting control limits. It then presents two brief case studies that illustrate the iterative approach for the basic Shewhart and R charts.

Developing a system that can automatically detect health changes using transfer times of older adults.

BackgroundAs gait speed and transfer times are considered to be an important measure of functional ability in older adults, several systems are currently being researched to measure this parameter in the home environment of older adults. The data resulting from these systems, however, still needs to be reviewed by healthcare workers which is a time-consuming process.MethodsThis paper presents a system that employs statistical process control techniques (SPC) to automatically detect both positive and negative trends in transfer times. Several SPC techniques, Tabular cumulative sum (CUSUM) chart, Standardized CUSUM and Exponentially Weighted Moving Average (EWMA) chart were evaluated. The best performing method was further optimized for the desired application. After this, it was validated on both simulated data and real-life data.ResultsThe best performing method was the Exponentially Weighted Moving Average control chart with the use of rational subgroups and a reinitialization after three alarm days. The results from the simulated data showed that positive and negative trends are detected within 14 days after the start of the trend when a trend is 28 days long. When the transition period is shorter, the number of days before an alert is triggered also diminishes. If for instance an abrupt change is present in the transfer time an alert is triggered within two days after this change. On average, only one false alarm is triggered every five weeks. The results from the real-life dataset confirm those of the simulated dataset.ConclusionsThe system presented in this paper is able to detect both positive and negative trends in the transfer times of older adults, therefore automatically triggering an alarm when changes in transfer times occur. These changes can be gradual as well as abrupt.

The T2 chart with mixed samples to control bivariate autocorrelated processes

In this paper, we propose the use of the T2 chart with the mixed sampling strategy (MS) to monitor the mean vector of bivariate processes with observations that fit to a first-order vector autoregressive model. With the MS, rational subgroups of size n are taken from the process and the selected units are regrouped to form the mixed samples. The units of the mixed samples are units selected from the last two rational subgroups. The aim of the proposed sampling strategy is to reduce the negative effect of the autocorrelation on the performance of the T2 chart. When the two variables are autocorrelated, the MS always enhances the T2 chart performance, however, the mixed samples are not recommended for bivariate processes with only one autocorrelated variable which is rarely affected by the assignable cause.