Curious Variables Experiment (CURVE). RZ LMi - the most active SU UMa star

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minitab 在质量管理中的英语-培训笔记

1.Pareto图得出关键少数（找出众多问题中的主要问题，优先排钱）2.用因果（鱼骨）图分析主要问题的原因（观点），末端原因应能一眼看出对策（员工未按标准作业执行，不能成为末端原因）3.用散点图（采集数据大于30个）验证因果是否具有相关性4.直方图（采集数据大于30个）显示数据频度分布，拟合的曲线为正态分布曲线分布的均值：衡量数据的准确性标准差：数据的精确性（标准差越小，精确度越好）6倍标准差（包含99.73%的数据）=过程能力（身高差超过6*15.47的可能性很小）5. 单值图，箱线图，时间序列图6.测量系统分析（需增加2张手机拍的图）A．交叉（不破坏）此图只具有参考意义测量值*测量员体现重复性来源标准差(SD) (6 * SD) 异 (%SV) (SV/Toler)合计量具 R&R 0.067596 0.40558 33.56 40.56（两者都小于10%，合格；10%~30%测量关键特性，任意一个大于30%，不合格）重复性 0.032592 0.19555 16.18 19.56再现性 0.059220 0.35532 29.40 35.53（再现性影响更大）测量员 0.028470 0.17082 14.13 17.08测量员*洗衣粉袋 0.051928 0.31157 25.78 31.16部件间 0.189745 1.13847 94.20 113.85合计变异 0.201426 1.20856 100.00 120.86产品过程可区分的类别数 = 3（可区分类别数>10,优秀；大于等于5，不合格；小于5，不及格）（衡量分辨力）综上红色字体，以上测量系统不合格B.嵌套（破坏性）嵌套式与交叉式相比，缺少再现性图过程公差 = 16研究变异 %研究变 %公差来源标准差(SD) (6 * SD) 异 (%SV) (SV/Toler)合计量具 R&R 1.59164 9.5499 42.86 59.69重复性 1.59164 9.5499 42.86 59.69 再现性 0.00000 0.0000 0.00 0.00 部件间 3.35534 20.1321 90.35 125.83 合计变异 3.71371 22.2823 100.00 139.26可区分的类别数 = 2分析同上，测量系统不合格7.测量线性研究偏倚——点线性——计数型测量系统分析评定值的属性一致性分析检验员自身（重复性）评估一致性# 检 # 相检验员验数符数百分比 95 % 置信区间钱 6 3 50.00 (11.81, 88.19)孙 6 4 66.67 (22.28, 95.67)赵 6 6 100.00 (60.70, 100.00)# 相符数: 检验员在多个试验之间，他/她自身标准一致。

python中准确率曲线函数

Python中准确率曲线函数一、概述在机器学习中，评估模型的准确率是非常重要的一环。

准确率曲线函数能够帮助我们分析模型的性能，找出最佳的阈值，并且是一种非常直观的评估方式。

本文将介绍Python中的准确率曲线函数，包括其原理、用法和实际应用。

二、准确率曲线函数的原理准确率曲线函数是一种衡量二分类模型性能的工具，它通过绘制不同阈值下的真阳性率（True Positive Rate, TPR）和假阳性率（False Positive Rate, FPR）的变化曲线来展现模型的表现。

TPR和FPR的定义如下：TPR = TP / (TP + FN)FPR = FP / (FP + TN)其中，TP代表真阳性，FN代表假阴性，FP代表假阳性，TN代表真阴性。

通过计算不同阈值下的TPR和FPR，我们可以绘制出准确率曲线，从而分析模型的性能。

三、准确率曲线函数的用法在Python中，我们可以利用scikit-learn库中的roc_curve函数来计算准确率曲线。

该函数的使用方法如下：```pythonfrom sklearn.metrics import roc_curvefpr, tpr, thresholds = roc_curve(y_true, y_score)```其中，y_true代表真实标签，y_score代表模型的得分。

该函数将返回不同阈值下的FPR、TPR和阈值，我们可以利用这些数据来绘制准确率曲线。

四、准确率曲线函数的实际应用下面以一个实际案例来展示准确率曲线函数的应用。

假设我们有一个二分类模型，我们可以先使用该模型对测试集进行预测，然后利用roc_curve函数计算准确率曲线。

我们可以利用matplotlib库来绘制该曲线，并找出最佳阈值。

```pythonimport matplotlib.pyplot as pltplt.plot(fpr, tpr, label='ROC curve (area = 0.2f)' roc_auc)plt.plot([0, 1], [0, 1], 'k--')plt.xlim([0.0, 1.0])plt.ylim([0.0, 1.05])plt.xlabel('False Positive Rate')plt.ylabel('True Positive Rate')plt.title('Receiver Operating Characteristic')plt.legend(loc="lower right")plt.show()```通过观察准确率曲线，我们可以找出最佳阈值，从而提高模型的性能。

分类变量的重复测量

分类变量的重复测量资料分析
流行病与卫生统计学教研室沈毅
2019.3.15
浙江大学医学院流行病与卫生统计学教研室沈毅
分类变量（categorical variable）又称为定性变量（qualitative variable），在工作中应用甚广。根据其不同的取值性质，又可分为3种类型：第一种是名义刻度（nominal scale）的分类变量，它是按事物属性分类的变量，如性别、职业等。在统计学上为了计算方便，将这些不同的属性进行数量化处理，如男性赋值为1，女性赋值为2。这种数值只是作为属性的代码，其间并无大小之分。
浙江大学医学院流行病与卫生统计学教研室沈毅
把分类变量作为反应变量进行重复观察的情形在工作中应用较广。在本书第九章第五节中介绍了二分类反应变量的重复测量资料分析方法。
本章将介绍分类反应变量重复测量资料的一般分析方法。主要介绍加权最小二乘法分析方法。第一节一个总体的二分类反应重复测量资料的分析。
PROC CATMOD ORDER=DATA; WEIGHT count; RESPONSE marginals; MODEL year0*year3*year6=gender| _RESPONSE_/PRED=FREQ COV; REPEATED year;
浙江大学医学院流行病与卫生统计学教研室沈毅
三、配合线性模型的步骤
表11.2为资料的原始记录形式，需要将其整理成边际频数表的格式后再配合模型。计算步骤介绍如下。 1.首先用下列SAS程序计算边际合计数程序中的subj 为受试者号，time1、time2、time3代表3个疗程。
浙江大学医学院流行病与卫生统计学教研室沈毅
第二种为有序刻度(ordinal scale)的分类变量，它是根据事物呈现出的程度或水平不同进行赋值。如临床化验结果用符号“-、+、++、+++”，文化程度用 “文盲、小学、中学、大学、研究生”来划分等级，在进行数量化处理时赋值1、2、3、…。这里需要注意的是，1与2之差不一定等于2与3之差。第三种是区间刻度（interval scale)，如人口学统计中的年龄分组，“0-，10-， 20-，…”就是典型的例子。根据资料的性质，区间跨度有等距的，也有不等距的。

python的科赫曲线代码绘制三阶六边形

python的科赫曲线代码绘制三阶六边形科赫曲线是一种分形。

其形态似雪花，又称科赫雪花、雪花曲线。

它最早《关于一条连续而无切线，可由初等几何构作的曲线》科赫曲线是deRham曲线的特例。

1.给定线段AB，科赫曲线可以由以下步骤生成：
2.将线段分成三等份（AC,CD,DB）
3.以CD为底，向外（内外随意）画一个等边三角形DMC
4.将线段CD移去
分别对AC,CM,MD,DB重复1~3。

科赫雪花是以等边三角形三边生成的科赫曲线组成的。

科赫雪
花的面积是
，其中S是原来三角形的边长。

每条科赫曲线的长度是无限大，它是连续而无处可微的曲线。

画法：
1、任意画一个正三角形，并把每一边三等分；
2、取三等分后的一边中间一段为边向外作正三角形，并把这“中间一段”擦掉；
3、重复上述两步，画出更小的三角形。

4、一直重复，直到无穷，所画出的曲线叫做科赫曲线。

和皮亚诺类似：
1、曲线任何处不可导，即任何地点都是不平滑的
2、总长度趋向无穷大
3、曲线上任意两点沿边界路程无穷大
4、面积是有限的
5、产生一个匪夷所思的悖论："无穷大"的边界，包围着有限的面积。

三分类roc曲线 python 代码

三分类roc曲线 python 代码三分类ROC曲线Python代码ROC曲线是一种用于评估分类模型性能的常用工具。

在二元分类中，ROC曲线可以帮助我们评估模型的灵敏度和特异性。

但是，在多分类问题中，我们需要使用三分类ROC曲线来评估模型的性能。

本文将介绍如何使用Python编写三分类ROC曲线代码，并提供完整的示例代码和详细说明。

1. 什么是三分类ROC曲线？在多分类问题中，每个类别都有一个对应的真正例率（TPR）和假正例率（FPR）。

三分类ROC曲线展示了这些TPR和FPR值之间的关系。

2. 数据集为了演示如何使用Python编写三分类ROC曲线代码，我们将使用Iris数据集。

这个数据集包含150个样本，其中每个样本都有四个特征：花萼长度、花萼宽度、花瓣长度和花瓣宽度。

每个样本都属于三种不同类型之一：Iris Setosa、Iris Versicolour或Iris Virginica。

首先，我们需要导入所需的库并加载数据集：```pythonimport pandas as pdimport numpy as npfrom sklearn.datasets import load_irisiris = load_iris()X = iris.datay = iris.target```3. 模型训练接下来，我们需要训练一个分类模型。

在这个例子中，我们将使用逻辑回归模型。

```pythonfrom sklearn.linear_model import LogisticRegressionfrom sklearn.model_selection import train_test_splitX_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)model = LogisticRegression()model.fit(X_train, y_train)```4. 绘制三分类ROC曲线现在，我们可以使用sklearn.metrics库中的roc_curve函数来计算每个类别的TPR和FPR值，并绘制三分类ROC曲线。

我的一次实验记忆巧克力英语作文

我的一次实验记忆巧克力英语作文Chocolate, a delectable confectionery delight, has captivated the palates of countless individuals across the globe for centuries. Its rich, complex flavor and velvety texture evoke a symphony of sensations, leaving an unforgettable impression on the human gustatory experience.As a curious and eager student, I embarked on an experiment to delve into the intricate world of chocolate, unraveling its secrets and exploring its captivating allure. My objective was to conduct a memory experiment toascertain the extent to which the consumption of chocolate influenced my ability to recall information.I designed a rigorous experimental protocol, meticulously controlling all variables that couldpotentially confound the results. The experiment consistedof two phases: a study phase and a test phase. During the study phase, I presented participants with a series of unfamiliar words and paired them with either chocolate or aneutral control substance. The chocolate condition involved consuming a small piece of dark chocolate, approximately 10 grams, while the control condition involved consuming a piece of unsweetened cracker.After a brief interval, participants entered the test phase, where they were asked to recall as many words as possible from the study phase. The number of wordscorrectly recalled was recorded for each participant in both the chocolate and control conditions.The results of my experiment yielded intriguinginsights into the relationship between chocolate consumption and memory. Statistical analysis revealed that participants who consumed chocolate during the study phase exhibited a significant improvement in their ability to recall words compared to those who consumed the neutral control substance. This finding suggests that chocolate may possess memory-enhancing properties, potentially due to its high concentration of flavonoids, which have been associated with improved cognitive function.To further validate my findings, I conducted a comprehensive literature review, delving into the existing body of research on the effects of chocolate on memory. Numerous studies have reported similar positive outcomes, indicating that chocolate consumption can enhance both short-term and long-term memory performance.One particularly compelling study, published in the journal "Appetite," investigated the impact of chocolate on episodic memory, which refers to the ability to recall specific events and experiences. The researchers found that participants who consumed chocolate prior to a memory task performed significantly better than those who consumed a placebo. This study suggests that chocolate may improve the encoding and consolidation of memories, leading to enhanced recall later on.Another study, published in the journal "Neurology," examined the relationship between chocolate consumption and cognitive decline in older adults. The researchers followed a group of elderly participants for several years, tracking their chocolate consumption and cognitive function. Theyfound that participants who consumed chocolate regularly had a lower risk of developing cognitive impairment and dementia compared to those who did not consume chocolate. This finding suggests that chocolate may have neuroprotective properties that help to preserve cognitive function as we age.In addition to its potential memory-enhancing effects, chocolate has also been linked to a number of other health benefits. For instance, chocolate contains high levels of antioxidants, which can help to protect cells from damage caused by free radicals. Some studies have also suggested that chocolate may improve cardiovascular health by lowering blood pressure and reducing the risk of blood clots.Overall, the evidence suggests that chocolate, particularly dark chocolate with a high cocoa content, offers a unique combination of culinary delight and potential health benefits. While further research is needed to fully understand the mechanisms by which chocolate exerts its effects on memory and other cognitive functions,the current findings provide a tantalizing glimpse into the potential of this delectable treat to enhance our cognitive abilities.In conclusion, my experiment, coupled with the wider body of research, provides compelling evidence that chocolate consumption can enhance memory performance. Whether enjoyed as a sweet indulgence or incorporated into a healthy diet, chocolate appears to possess cognitive-boosting properties that make it a delectable and potentially beneficial addition to our daily lives.。

python 误差曲线置信区间

题目：探究Python误差曲线与置信区间的相关性一、概述Python作为一种广泛应用的编程语言，在数据分析和统计学领域也有着重要的地位。

误差曲线和置信区间是统计学中常见的概念，对于数据分析和结果解释具有重要意义。

本文将探讨Python中误差曲线与置信区间的相关性，希望能够为相关领域的研究者和实践者提供参考。

二、Python中的误差曲线1. 误差曲线的定义误差曲线是指在统计数据中，用来表示平均值附近的变化范围的一条曲线。

在Python中，我们可以使用matplotlib库来绘制误差曲线，通过展示数据的波动范围，能够更直观地理解数据的分布特征。

2. Python绘制误差曲线的方法在Python中，我们可以使用matplotlib库的errorbar函数来绘制误差曲线，该函数能够展示数据点与其对应的误差范围，使得数据的波动情况一目了然。

3. 误差曲线的应用误差曲线在数据分析和统计学中具有重要的应用，能够帮助研究者对数据进行更全面的分析和解释。

在科研领域和实际应用中，误差曲线能够有效地辅助决策和结果评估。

三、Python中的置信区间1. 置信区间的定义置信区间是指用来估计总体参数的区间估计方法，它表示了参数估计的不确定性范围。

在Python中，我们可以使用scipy库来计算数据的置信区间，从而对数据的总体特征进行推断。

2. Python计算置信区间的方法使用scipy库中的stats模块，可以方便地计算数据的置信区间。

通过指定置信水平和样本数据，即可得到数据的置信区间范围，从而更准确地评估统计结论的可靠性。

3. 置信区间的应用置信区间是统计学中常用的工具，能够帮助研究者对样本数据进行推断，并对总体特征进行估计。

在实际应用中，置信区间的计算结果能够有效地指导决策和结论的推断。

四、Python中的误差曲线与置信区间的关联1. 误差曲线与置信区间的概念通联误差曲线和置信区间在统计学中都与数据的不确定性和可靠性有关。

python nrubs 曲线拟合-概述说明以及解释

python nrubs 曲线拟合-概述说明以及解释1.引言1.1 概述概述本文将介绍Python中NRubs曲线拟合的概念和应用。

在现实生活和工作中，我们经常需要通过一系列数据点来近似表示一个曲线。

NRubs 曲线拟合是一种数学方法，可用于找到一个平滑的曲线，以最佳地逼近给定的数据点。

Python作为一种高级编程语言，提供了许多强大的工具和资源，使我们能够轻松地进行数据处理和曲线拟合。

本文将首先介绍Python的基础知识，包括数据结构、变量和函数等方面的内容。

然后，我们将深入探讨NRubs曲线拟合的概念。

NRubs曲线拟合是一种基于样条函数的方法，通过将给定的数据点与多项式函数相连，生成一条平滑的曲线。

在理解NRubs曲线拟合的原理和数学模型之后，我们将学习如何在Python中应用这种方法。

在正文部分，我们将详细介绍Python中的NRubs曲线拟合的实现步骤和技巧。

通过使用Python的相关库和函数，我们可以轻松地进行数据处理、拟合曲线并可视化结果。

在结论部分，我们将总结本文的主要内容，并探讨NRubs曲线拟合在实际应用中的潜力和局限性。

我们将指出NRubs曲线拟合的优点和不足之处，并提出如何进一步改进和应用这种方法的建议。

通过本文的学习，读者将掌握Python中NRubs曲线拟合的基本原理和实践技巧。

有了这些知识，读者可以更好地应用NRubs曲线拟合解决实际问题，并在数据分析和科学研究领域中发挥更大的作用。

1.2 文章结构文章结构部分的内容可以包括以下内容：文章结构部分旨在介绍整篇文章的组织架构和内容安排，以便读者能够更好地理解文章的组成和流程。

下面将详细介绍本文的结构。

本文分为引言、正文和结论三个部分。

1. 引言部分（Introduction）：主要对本文的主题进行概述，简要介绍Python和NRubs曲线拟合的基本概念和应用。

首先，通过引入Python 的基础知识，为读者提供了解Python编程语言的必要背景。

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a r X i v :0806.1657v 1 [a s t r o -p h ] 10 J u n 2008Curious Variables Experiment (CURVE).RZ LMi -the most active SU UMa star.A.O l e c h 1,M.W i ´sn i e w s k i 1,K.Z ło c z e w s k i 1,L.M.C o o k 2,K.M u l a r c z y k 3,and P.K e ¸d z i e r s k i 31Nicolaus Copernicus Astronomical Center,Polish Academy of Sciences,ul.Bartycka 18,00-716Warszawa,Polande-mail:(olech,mwisniew,kzlocz)@.pl2Center for Backyard Astrophysics (Concord),1730Helix Court,Concord,CA 94518,USAe-mail:lcoo@3Warsaw University Observatory,Al.Ujazdowskie 4,00-476Warszawa,Polande-mail:(kmularcz,pkedzier)@.plAbstractWe report extensive photometry of the frequently outbursting dwarf nova RZ Leo Minoris.During two seasons of observations we detected 12superoutbursts and 7normal outbursts.The V magnitude of the star varied in range from 16.5to 13.9mag.The superoutbursts occur quite regu-larly ﬂashing every 19.07(4)days and lasting slightly over 10days.The average interval between two successive normal outbursts is 4.027(3)days.The mean superhump period observed during the superoutbursts is P sh =0.059396(4)days (85.530±0.006min).The period of the superhumpswas constant except for one superoutburst when it increased with a rate of ˙P/P sh =7.6(1.9)·10−5.Our observations indicate that RZ LMi goes into long intervals of showing permanent superhumps which are observed both in superoutbursts and quiescence.This may indicate that decoupling of thermal and tidal instabilities play important role in ER UMa systems.No periodic light varia-tions which can be connected with orbital period of the binary were seen,thus the mass ratio and evolutionary status of RZ LMi are still unknown.Key words:Stars:individual:RZ LMi –binaries:close –novae,cataclysmic variables1IntroductionDwarf novae are believed to be unmagnetized close binary systems containing white dwarf primary and low mass main sequence secondary.The secondary ﬁlls its Roche lobe and looses the material through the inner Lagrangian point.This matter forms an accretion disc around the white dwarf.One of the most intriguing classes of dwarf novae are SU UMa stars which have short orbital periods (less than 2.5hours)and show two types of outbursts:normal outbursts and superoutbursts.Superoutbursts are typically about one magnitude brighter than normal outbursts,occur about ten times less frequently and display characteristic tooth-shape light modulations i.e.so called super-humps(see Warner1995for review).The behavior of SU UMa stars in now quite well understood within the frame of the thermal-tidal instability model(see Osaki1996for review).Superhumps occur at a period slightly longer than the orbital period of the binary system.They are most probably the result of accretion disk precession caused by gravitational perturbations from the secondary.These perturbations are most effective when disk particles moving in eccentric orbits enter the3:1resonance.Then the superhump period is simply the beat period between orbital and precession rate periods.Although in the last decades signiﬁcant progress has been made in explaining the behaviour of dwarf novae light curves,some physical processes ongoing in these systems are still not fully understood(see for example Smak 2000,Schreiber and Lasota2007).In the beginning of90ties of XX century,SU UMa stars were believed to be quite uniform group of variables with common properties.These objects went into superoutburst every year or so and between two successive superoutbursts showed∼10ordinary outbursts.However,there were some exceptions like WZ Sge,which show infrequent and large amplitude superoutburst followed by the period of quiescence with no single eruption lasting even30years.In1995astronomical community was alerted about the presence of stars characterized by com-plete opposite behavior.First,Misslet and Shafter(1995)reported observations of PG0943+521 (later called ER UMa),which allowed to detect superhumps with period of0.0656days and include this object into the SU UMa group of variables.The most intriguing feature of the long term light curve of ER UMa was very short interval between two successive superoutbursts(so called supercy-cle)reaching only44days.This value was about three times shorter than shortest previously known supercycles.This work was quickly followed by paper of Robertson et al.(1995),who conﬁrmed all ﬁndings of Misslet and Shafter(1995)and precisely determined the value of supercycle of ER UMa to be equal to42.95days.Moreover,they found two more objects with similar properies-V1159 Ori with supercycle of44.5days and RZ LMi with supercycle as short as18.87days!In the same year Nogami et al.(1995)published paper which conﬁrmed extremely short supercycle of RZ LMi and showing that it belongs to SU UMa variables exhibiting clear superhumps with period of0.05946 days.One year later the number of these unusual variables increased to four objects.Kato et al.(1996) reported the discovery that DI UMa has a supercycle of25days and shows clear superhumps with period of0.0555days.Theﬁfth ER UMa-type variable-IX Dra-was discovered by Ishioka et al.(2001).Their ob-servations revealed a supercycle length of53days and an interval between normal outbursts of3-4 days.Olech et al.(2004)determined precisely both superhump and orbital periods of the binary and estimated the supercycle length to54days.The basic properies ofﬁve known up-today members of ER UMa group are summarized in Table 1.It is clear that ER UMa stars consist a group of variables with common properties such as ex-tremely short supercycles,small amplitudes of eruptions and relatively long superoutbursts lasting even longer than half of the supercycle.However,the period excessεdeﬁned as P sh/P orb−1,which is connected with mass ratio by the following relation:0.23qε≈Table1:B ASIC PROPERTIES OF ER UM A VARIABLES.P orb AND P sh DENOTE ORBITAL AND SU-PERHUMP PERIODS,εIS A PERIOD EXCESS,T s AND T n ARE SUPERCYCLE AND CYCLE PERIODS,T sup IS DURATION OF THE SUPEROUTBURST,A sup AND A n ARE AMPLITUDES OF SUPEROUTBURST AND NORMAL OUTBURST.P orbεT n A sup Ref[days][days][days][mag]?? 3.8 2.5(1,2) DI UMa0.055525.012 2.10.06366 2.97 4.4 2.6(2,5,6) V1159Ori0.06428444.6-53.316 1.40.066460.76 3.1 2.2(9,10)Table2:J OURNAL OF THE CCD OBSERVATIONS OF RZ LM I.No.of Endframes2453000.+2004Jan22/2327.491640.0592004Jan25/2630.39285 2.5792004Jan26/2731.59037 1.8532004Jan29/3034.35335 4.1922004Jan30/3135.29448 4.5782004Feb01/0237.255960.9182004Feb11/1247.32035 4.6642004Feb12/1348.271768.1462004Feb16/1752.665420.7152004Feb19/2055.265139.7482004Feb20/2156.2664710.2882004Feb21/2257.25898 5.4232004Feb24/2560.45026 5.7232004Feb25/2661.57348 1.9192004Feb26/2762.27642 5.6012004Mar10/1175.40324 5.2042004Mar11/1276.26170 6.3712004Mar12/1377.26967 6.8582004Mar13/1478.246687.2722004Mar17/1882.26416 3.2052004Mar18/1983.31747 2.5972004Mar19/2084.29448 3.2342004Mar21/2286.27288 2.8362004Mar22/2387.26646 2.5962004Mar29/3094.452880.9832004Mar30/3195.31690 2.0022004Apr13/14109.34651 3.8972004Apr14/15110.28399 5.1522004Apr15/16111.27972 3.0272004Apr16/17112.33495 2.5142004Apr18/19114.42736 1.0282004Apr19/20115.28831 3.2942004Apr20/21116.27826 4.0822004Apr21/22117.28336 3.3112004Apr22/23118.28635 4.1292004Apr23/24119.28807 1.0142004Apr25/26121.285080.5682004May03/04129.33305 3.1442004May04/05130.33826 2.2842004May05/06131.32664 2.9492004May10/11136.433890.0552004May11/12137.33082 1.9012004May12/13138.32363 2.4602004May14/15140.33032 1.6462004May16/17142.33325 1.1792004May17/18143.33488 1.4222004May23/24149.35765 1.3072004May24/25150.389730.0382005Jan06/07377.551440.0882005Jan10/11381.620640.0792005Jan16/17387.650400.0752005Jan31/01402.476270.0902005Feb07/08409.370520.2952005Feb08/09410.408270.1552005Feb09/10411.341970.3462005Feb10/11412.233830.1252005Feb11/12413.511030.1752005Feb28/01430.360680.0382005Mar03/04433.301740.2142005Mar19/20449.345250.0942005Mar29/30459.302300.2402005Mar30/31460.383660.1502005Mar31/01461.290920.2442005Apr01/02462.270250.3092005Apr02/03463.310980.2252005Apr03/04464.274730.1452005Apr04/05465.262300.2222005Apr04/05465.796990.0922005Apr05/06466.278720.0902005Apr05/06466.719480.1442005Apr06/07467.334960.0592005Apr09/10470.704110.1592005Apr12/13473.660640.1972005Apr13/14474.276360.2172005Apr14/15475.652470.2042005Apr15/16476.649400.2312005Apr17/18478.665220.1722005Apr19/20480.670210.1692005Apr20/21481.658460.1442005Apr28/29489.456820.0162005May07/08498.323020.0902005May10/11501.314750.0982005May11/12502.360840.0052005May20/21511.349240.0622005May21/22512.349930.0712005May28/29519.383380.01655523Observations and Data ReductionObservations of RZ LMi reported in present paper were obtained during46nights between January 22,2004and May28,2005at the Ostrowik station of the Warsaw University Observatory and at CBA Concord at the San Francisco suburb of Concord,approximately50km from East of the City. The Ostrowik data were collected using the60-cm Cassegrain telescope equipped with a Tektron-ics TK512CB back-illuminated CCD camera.The scale of the camera was0.76"/pixel providing a 6.5’×6.5’ﬁeld of view.The full description of the telescope and camera was given by Udalski and Pych(1992).The Ostrowik data reductions were performed using a standard procedure based on the IRAF1 package and proﬁle photometry was derived using the DAOphotII package(Stetson1987).The CBA data were collected using an f/4.573-cm reﬂector operated at prime focus on an English cradle mount.Images were collected with a Genesis G16camera using a KAF1602e chip giving a ﬁeld of view of14.3′×9.5′.Images were reduced using AIP4WIN software(Berry&Burnell2000).In both sites we monitored the star in“white light”in order to be able to observe it with good precision also at minimum light of around17mag.A full journal of our CCD observations of RZ LMi is given in Table2.In total,we monitored the star for165.5hours and obtained5552exposures.Relative unﬁltered magnitudes of RZ LMi were determined as the difference between the mag-nitude of the variable and the intensity averaged magnitude of two nearby comparison stars.The magnitudes and colors of our comparison stars were taken from Henden and Honeycutt(1995).Trans-formation to Johnson V magnitudes was done using BV R photometry of theﬁeld of variable obtained on2004Apr21.The accuracy of our measurements varied between0.004and0.119mag depending on the bright-ness of the object and atmospheric conditions.The median value of the photometric errors was0.012 mag.4General light curveThe global light curve spanning whole period of our observations is shown in Fig.1.In total we de-tected12long eruptions and7short outbursts.The superoutburst are labeled by corresponding roman numbers.In quiescence the star fades to V≈16.5mag and during the highest phase of superoutburst reaches13.9mag giving the full amplitude of variability equal to A s=2.6mag.It is only slightly larger than the value of2.5mag determined by Robertson et al.(1995).During the brightest normal outburst the star reaches14.4mag.First,from global light curve we selected only nights during which the star was in superoutburst(it means that we detected clear superhumps).Then we computed ANOVA statistics with two harmonic Fourier series(Schwarzenberg-Czerny1996).The resulting periodogram,for the frequency range 0÷0.15c/d,is shown in Fig.2.The dominant peak is detected at frequency f0=0.05245(10)c/d, which corresponds to the period of19.07(4)days.This value is interpreted as supercycle length i.e. mean interval between two successive superoutbursts.It is in quite good agreement with value of 18.87obtained by Robertson et al.(1995).Next,weﬁtted analytical light curve to the superoutburst number V(solid line in Fig.1),which has very good coverage,and repeated it every19.07days.The stability of the supercycle periodis very interesting.The analytical light curve has no problems with hitting precisely superoutbursts numbers I,II,III,VI and VII in2004and even superoutbursts numbers XXIV and XXVI occurring one year later.Figure1:The general photometric behavior of RZ LMi during our campaign.Dots and open circles correspond to our and AAVSO observations.The solid lineﬁtted to eruption no.V is repeated every 19days.Figure2:The ANOVA spectrum of RZ LMi global light curve after removing the data from quiescence and normal outbursts.Additionally,Fig.3shows the light curve consisting of only superoutburst data and phased with period19.07days.One can clearly see that superoutburst lasts slightly over half of the supercycle i.e. over10days.It consists of:initial rise,which takes about1.3days,plateau phase with linear decreaseof brightness at rate of0.063mag/day and lasting7.5days andﬁnal decline which takes about1.5 days.Figure3:The light curve of RZ LMi in superoutbursts obtained by folding the general light curve with supercycle period of19.07days.Figure4:The ANOVA spectrum of RZ LMi global light curve after removing the data from superout-bursts.Figure5:The light curve of RZ LMi in normal outbursts and quiescence obtained by folding the general light curve with cycle period of4.027days.Now we can make opposite operation i.e.remove from the light curve all superoutbursts and leave intervals when star is in quiescence and goes into normal outbursts.Again,for the resulting light curve,we computed the ANOVA statistics and showed the result in Fig.4.The dominant peak has a double structure with maxima at frequencies f1=0.2483(2)and f2=0.2509c/d.The phased light curve looks better for theﬁrst frequency,and we choose it as correct value.The corresponding period of4.027(3)days is interpreted as normal cycle i.e.interval between two successive normal outbursts.The light curve phased with this period is shown in Fig.5.Normal outburst lasts2.8days and consists of quick initial rise lasting only half a day,narrow maximum and slower decline.Taking into account the fact that every supercycle we observe two normal outbursts,RZ LMi is in the quiescence only for3days in each supercycle.5SuperhumpsThe superhumps of RZ LMi were observed on several occasions.Fig.6shows data from three consecutive nights of superoutburst no.II which occurred in February2004.Periodic,tooth-shape light variations with amplitude of0.1-0.2mag are clearly visible.Figure6:Superhumps of RZ LMi from three consecutive nights of February2004.Additionally,Fig.7shows global light curve of superoutburst no.V,which has the best observa-tional coverage.The observing runs,due to the geometric conditions,are not as long as in February, but the star was observed on almost every night of the superoutburst.We were able to see the initial rise(Apr13),the birth of superhumps before the maximum brightness(Apr14),full amplitude vari-ations which occurred one night later,slow evolution towards smaller amplitudes occurring during nextﬁve days of plateau phase and trace of superhumps duringﬁnal decline.Figure7:Nightly light curves of RZ LMi from its April2004superoutburst.5.1ANOVA analysisThe data from each night containing superhumps wereﬁtted with straight line or parabola.In purpose of detrending,this analytic curve was subtracted from real light curve.As a result we obtained a set of data with average brightness equal to zero and consisting only short term modulations.For these sets we computed ANOVA statistics and showed corresponding periodograms in Fig.8. Additionally,the frequencies and periods determined using these periodograms are summarized in Table3.The main frequencies detected in each superoutburst are consistent within the errors with each other and power spectrum computed for all superoutbursts returns the mean frequency f sh=16.8363±Table3:F REQUENCIES AND PERIODS OF SUPERHUMPS FOUND IN THE PERIODOGRAMS COM-PUTED FOR DETRENDED DATA OF SIX SUPEROUTBURSTS.Date P sh[d]2004,Jan29-Feb010.0595(4)No.II16.824±0.0202004,Mar10-Mar130.05941(9)No.V16.823±0.0102004,May04-May120.05942(7)No.XXIV16.836±0.025Mean16.8363±0.0010.001,which corresponds to the period of P sh=0.059396(4)days(85.530±0.006min),conﬁrming that RZ LMi is one of the shortest period SU UMa,and particularly ER UMa,stars.Figure8:ANOVA power spectra for superhumps observed in six superoutbursts of RZ LMi and com-posite spectrum obtained from all data from supermaxima.Table4:C YCLE NUMBER E,O−C VALUES AND TIMES OF MAXIMA FOR SUPERHUMPS OBSERVED IN SIX SUPEROUTBURSTS.N OTE THAT THE FIRST THREE SUPEROUTBURSTS HAVE COMMON E NUMBERING.HJD max−2453000O−C E Error00.0040110.34400.031610.0050110.40330.0302130.0020110.46100.0018 140.0025111.2905-0.030355.28300.0422170.002755.34180.0320340.002555.3992-0.0018350.003555.45920.0082690.002555.51980.0284840.003055.5745-0.0508850.002555.63900.03491010.002556.2900-0.00641020.003056.35000.00361170.002556.4065-0.04531180.003056.4680-0.01001190.00303690.0025130.35600.0129 3700.0030130.41700.0401 3710.0025131.3630-0.0303 3820.0030131.4207-0.058757.36000.005600.002057.42000.015610.002060.5060-0.035820.001560.6235-0.0579100.003561.6340-0.0475110.00307020.0035474.42200.0760 7050.0035475.6640-0.0207 7060.0020475.7228-0.0311 7190.0025475.7825-0.0263 7200.0030475.84400.0088 7210.0025476.6750-0.0052 7220.0030476.7325-0.0374 7370.0030476.7924-0.0293 7380.0020476.8515-0.0346 7390.0023478.69830.0478478.75800.0525478.81600.02875.2The O−C analysisIn the light curve of RZ LMi from all superoutbursts we detected70maxima of superhumps.Their times are listed in Table4together with the errors,cycle number E and O−C values computed according to the ephemeris which will be described further.The O−C values fromﬁrst three superoutbursts shows no signs of signiﬁcant trend indicating that the period of superhumps was roughly constant.There are observational evidences that ER UMa stars shows ordinary superhumps also in quies-cence indicating that in these systems the disk is elliptical and tidally unstable all the time.It might suggest that the star should remember the phase of the superhumps from one superoutburst to another. The O−C data from our superoutbursts number I,II and III seem to conﬁrm this hypothesis.They can beﬁtted with common linear ephemeris in the form:HJD max=2453034.6076(10)+0.059405(2)·E(2) The corresponding O−C diagram is shown in Fig.9.Figure9:The O−C diagram for superhumps maxima of RZ LMi detected during its superoutbursts number I,II and III.Black dots correspond to possible late superhumps described in Sect.6.Moreover,the detrended light curve containing superhumps from all superoutbursts might be phased with one period and shows no traces of phase shifts between superhumps from different su-peroutburst.Such a light curve is plotted in Fig.10.Something strange happened to RZ LMi during superoutburst number IV.We detected then a clear eruption,which has properies of superoutburst i.e.is brighter that ordinary outburst and showsFigure10:The detrended light curve from data collected during all superoutbursts observed in2004 folded on superhump period.decline typical for plateau phase but during two nights of this bright state we have not detected any superhumps.The superoutburst no.V occurred in right time but with slightly different behaviour of super-humps.Their maxima can beﬁtted with following linear ephemeris:HJD max=2453110.3408(11)+0.059414(15)·E(3) but from the O−C values computed according to this ephemeris and shown in Table4and in Fig.11 it is evident that the period of superhumps was quickly increasing.Thus the moments of maxima can beﬁtted with the following parabola:HJD max=2453110.3436(13)+0.059152(65)·E+2.27(55)·10−6·E(4) indicating that the period was increasing with the rate of˙P/P sh=7.6(1.9)·10−5.Such a period derivative is typical for SU UMa stars with superhump periods of around0.06days(for example see Fig.5in Rutkowski et al.2007).Figure11:The O−C diagram for superhumps maxima of RZ LMi detected during its superoutburst number V.There are insufﬁcient number of data to investigate possible period changes during superoutburst no.VI,thus the corresponding moments of maxima wereﬁtted only with the linear ephemeris:Figure12:The O−C diagram for superhumps maxima of RZ LMi detected during its superoutburst number XXIV.HJD max=2453130.3566(17)+0.05918(14)·E(5) There was only one superoutburst with sufﬁcient amount of data for O−C analysis in2005season. It was superoutburst no.XXIV and its maxima can beﬁtted with the following linear ephemeris:HJD max=2453473.7045(9)+0.059416(21)·E(6) However,the data collected in Table4and shown in Fig.12might suggest slight increasing trend with rate of˙P/P sh=4.5(2.5)·10−5.On the other hand,the error of this determination is large,and within2σit is consistent with constant value of period.6Quiescence and normal outburstsAs we wrote earlier RZ LMi is so active that it is difﬁcult toﬁnd it in quiescence.However,on three occasions,we collected sufﬁcient amount of data to make the analysis of behaviour of the star in minimum light and in ordinary outbursts.Figure13:Sample light curves of RZ LMi from quiescence.Theﬁrst interval of data comes from2004,Mar17-22when we observed RZ LMi on four nights of minimum light and one night of the normal outburst.Sample light curves from these period are shown in Fig.13and display clear and periodic light variations of amplitude around0.3-0.4mag. Taking into account that these data were collected just after theﬁnal decline of superoutburst no.III, one can suspect that we observe so called late superhumps-the phenomenon occurring at the endof superoutburst with period roughly equal to period of ordinary superhumps but with phase shift reaching up to0.5cycle.O−C diagram from Fig.9shows the moments of the maxima observed on 2004Mar17as black dots suggesting that they are shifted in phase by about0.3cycle i.e.signiﬁcantly less than typical value of0.5cycle.Figure14:ANOVA power spectra for three long runs covering the quiescence and normal outbursts.Toﬁnd a period of these variations,weﬁrst transformed our light curves to the intensity units, next we detrended them removing long scale behaviour.The resulting ANOVA periodogram is shown in upper panel of Fig.14.The highest peak occurs at frequency f0=16.778±0.02c/d corresponding to the period of0.05960(7)days.This is only0.3%longer than mean superhump period and the two periods differ by the value which is about three times larger that the error of the period determina-tion.It is also possible that true value of frequency appears as1-day alias at f0=17.778±0.02c/d corresponding to the period of0.05625(7)days which is signiﬁcantly shorter than superhump period and might be also shorter than unknown orbital period of the system.In this case this period might be assumed as period of negative superhumps.However,it is known that negative superhump,orbital and positive superhump periods correlate with each other(Retter et al.2002,Olech et al.2007). This correlation indicates that the orbital period should be around0.0574days and superhump period excessεshould be as large as3.5%i.e.about three times too high for star with such a superhump period.Thus theﬁnal conclusion is that in quiescence RZ LMi showed modulations with period roughly equal to superhump period and indicating that in this interval the disc could be still eccentric and precessing.Two other long intervals when the star was observed in quiescence occurred on2005,Feb07-11and2005,Mar19-Apr06.From two lower periodograms shown in Fig.14it is clear that no periodic modulations were observed at that time.7Discussion7.1Evolutionary status of RZ LMiFrom our Table 1summarizing main properies of ER UMa stars,it is clear that these objects have many common properties but may be divided into two subgroups probably with different evolutionary status.Fig.15,repeated after Patterson (1998,2001)and Olech et al.(2004),shows correlation between period excess (i.e.mass ratio)and orbital period of the system.The solid line shows the evolutionary track of a dwarf nova with a white dwarf of mass 0.75M ⊙and secondary component with effective radius 6%larger than that of single main sequence star.The nova evolves towards the shorter periods ﬁrst due to the magnetic braking,next due to the emission of gravitational waves.After reaching the period minimum,the secondary becomes degenerate brown dwarf and system starts to increase its orbital period.-1.4-1.3-1.2-1.1-1-.9-.8-2-1.5-1log P orb (days)l o g εqFigure 15:The relation between the period excess and orbital period of the system.The solid line corresponds to the evolutionary track of a binary with a white dwarf of 0.75M ⊙and a secondary witheffective radius 6%larger than in the case of an ordinary main sequence star.Calculations were made under the assumption that below the orbital period of two hours the angular momentum loss in only due to gravitational radiation.T riangles denote the positions of ER UMa and V1159Ori.It seems that DI UMa and IX Dra (both belonging to ER UMa stars)are such evolved period bouncers,which in fact should be similar to old and inactive WZ Sge stars (WZ Sge,AL Com and EG Cnc showed in the plot).On the other hand,ER UMa and V1159Ori,shown as ﬁlled triangles,seem to be much younger objects still evolving towards shorter periods.Where is the place of RZ LMi?It is difﬁcult to answer this question without knowledge about the orbital period of the system.Our photometric data showed no other short term modulations than these corresponding to the ordinary superhumps.It would be very tempting to make the spectroscopicobservations of the star in quiescence.With minimum brightness of16.5mag it can be done with2-3-meter class telescope.7.2Stability of the supercycleThe comprehensive analysis of the global light curve of RZ LMi made by Robertson et al.(1995)and based on almost three years observing period showed that supercycle of RZ LMi is not stable.Their O−C diagram for supermaxima was characterized by clear decreasing trend with˙P=−1.7·10−3. However graph shows also occasional jumps where particular superoutburst occur even5days before or after the predicted moment.If this decreasing trend would continue to the epoch of our observations the supercycle should be then around18.5days,which is in disagreement with determined value of 19.07days.Our global light curve spans only two seasons and has no enough data to construct reliable O−C diagram for supermaxima.However,quick look at Fig.1,could draw some valuable conclusions. In2004the19-day periodicity is preserved through all superoutbursts except eruption number IV. In this case,we,in fact,are not certain whether we deal with superoutburst which occurred slightly before predicted moment or exceptionaly bright normal outburst lasting longer than usual.Vicinity of eruption number IV is also the time when disk could loose its eccentricity,expel the matter via this long outburst and rebuilt eccentricity again in superoutburst no.V.Data from2005seem to conﬁrm stability of19-day supercycle.The superoutburst no.XXIV, which has the best observational coverage,occurs at right time according to19-day ephemeris.The problem is with superoutburst no.XXIII,where instead of supermaximum we noted two ordinary outbursts.Our light curve,however,does not exclude possibility that supermaximum occurred a few days earlier according to the ephemeris.Mass transfer from the secondary to the disk,building the eccentricity,ignition of the outbursts and superoutbursts due to the thermal and tidal instabilities are stochastic processes,which are far for regularity.The question is why RZ LMi is so regular?Even if we observe some shifts in time of the start of particular supermaximum,the clock returns to stability without shift of the phase of whole pattern.This is hard to explain from the point of view of standard thermal-tidal instability model and might need some help from,for example,external force.The present number of known SU UMa systems reached the level for which the statistics tells us that some of these close binaries might be orbited by a third body.Is this in case of RZ LMi?We do not know.But the hypothesis that19-day period is the orbital period of the third body(or some kind resonant value)and cause of both the stability of supercycle and high activity of the star,which without this body would be quiet WZ Sge object,is tempting.7.3Permanent superhumper?The standard thermal-tidal instability model is unable to produce supercycles shorter that40days. Activity of the ordinary SU UMa variable can be increased by increasing a mass transfer rate.But when it reaches˙M≈3·1016g/s the supercycle starts to lenghten again due to the fact that super-outburst lasts longer.Further increasing of mass transfer causes transition of the star to the group of permanent superhumpers which are in permanent state of supermaximum and show inﬁnite value of supercycle.Osaki(1995)tried to explain properties of RZ LMi by artiﬁcial ending the superoutburst at the moment,when the disk had shrunk from0.46a to only0.42a,whereas a typical value used for ordinary SU UMa stars is0.35a.。