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Association between Thyroid Hormone Levels and Diabetic Kidney Disease in Euthyroid Patients with Type 2 Diabetes

• Scientific Reportsvolume 8, Article number: 4728 (2018)
• doi:10.1038/s41598-018-22904-7
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Published online:

Abstract

The association between normal thyroid function and diabetic kidney disease (DKD) has gained increasing attention. The present study evaluated the relationship between normal thyroid hormone levels and DKD in type 2 diabetes mellitus (T2DM) patients. A total of 862 type 2 diabetes patients were enrolled in this cross-sectional study in Xian, Shaanxi Province, China. The subjects were evaluated for anthropometric measurements, thyroid function and DKD. Of 862 patients, 246 (28.5%) suffered from DKD, and the prevalence of DKD did not differ between men and women. The prevalence of DKD showed a significantly decreasing trend across the quartiles based on free triiodothyronine (FT3) levels (41.1%, 30.6%, 23.8%, and 18.9%, P < 0.001). In comparison with all participants categorized in the first FT3 quartile group (FT3-Q1) (<4.380), the adjusted odds ratio of DKD in the second FT3 quartile group (FT3-Q2), the third FT3 quartile group (FT3-Q3), and the fourth FT3 quartile group (FT3-Q4) were 0.655(95%CI: 0.406–1.057), 0.493(95%CI: 0.299–0.813), 0.406(0.237–0.697) (P < 0.05). Also, similar results were observed in men. Conversely, none of the FT3 groups was associated with DKD in women. The present study showed that FT3 within normal range was negatively correlated with DKD in T2DM patients.

Introduction

The prevalence of diabetes is rising rapidly worldwide, especially in developing countries. Diabetes mellitus is being one of the greatest health threats in the 21st century. Diabetic kidney disease (DKD) is one of the most common microvascular complications of diabetes mellitus and the leading cause of end-stage renal disease (ESRD) worldwide. About 20%–40% of patients with diabetes progress to DKD, and 40% also progress to ESRD1. Diabetic patients have a high prevalence of thyroid dysfunction as compared to healthy population; hypothyroidism is the most common disorder2. Several clinical studies showed that thyroid dysfunction was related to renal disease3,4. In addition, previous studies focused on the relationship between subclinical hypothyroidism (SCH) and DKD4,5,6, although the results were not consistent.

Rodacki et al.7 found that not only SCH increased the prevalence of DKD but also the high normal levels of thyroid stimulating hormone (TSH) increased the prevalence of DKD as compared to low normal levels in type 1 diabetes (T1DM) patients. Moreover, studies in patients with euthyroid general population showed that high normal levels of TSH and low levels of free triiodothyronine (FT3) were associated with chronic kidney disease (CKD) and albuminuria8,9,10,11. Therefore, these results indicated that the relationship between thyroid function and DKD had been extended to the areas of normal thyroid function. However, it is unclear whether the normal thyroid hormone levels and DKD are correlated in type 2 diabetes (T2DM) patients.

The present cross-sectional study investigated whether serum thyroid hormone concentrations within normal range, as well as thyroid peroxidase antibody (TPO-Ab) levels, are related to the prevalence of DKD among T2DM inpatients.

Methods

Subjects

The study was conducted at the First Affiliated Hospital of Fourth Military Medical University in Xian, Shaanxi Province, China from June 2014 to March 2016. The inclusion criteria were as follows: individuals with previously diagnosed T2DM and all patients were ≥18 years of age. The American Diabetes Association (ADA) criteria 201312 were used for the diagnosis of diabetes mellitus. T2DM was defined as an fasting plasma glucose (FPG) level ≥126 mg/dL (7.0 mmol/L), or oral glucose tolerance test ≥200 mg/dL (11.1 mmol/L), or HbA1c ≥48 mmol/mol (6.5%), or a history of T2DM based on the ADA 2013 criteria12. Duration of diabetes is defined in this presentation as time of diagnosis to entry into the trial13.

The following exclusion criteria were considered: (1) other types of diabetes mellitus; (2) acute intercurrent illness; (3) severe cardiac disease and other serious diseases; (4) urinary tract infections, hematuria (including menstrual period), and non-diabetic kidney disease; (5) chronic diseases that can affect the metabolic function, including hypothalamic disease, adrenal disease, history of thyroid disease, or any thyroid medication (levothyroxine or anti-thyroid drugs); (6) malignant tumors. Initially, 2689 inpatients were enrolled. After excluding the subjects that did not fulfill the inclusion criteria and those with incomplete laboratory results, a total of 862 euthyroid patients were included in the analysis. The study protocol was approved by the Ethics Committee of the First Affiliated Hospital of Fourth Military Medical University and conducted in accordance with the Declaration of Helsinki. All participants provided written informed consent.

Data collection

Patient data, including demographic characteristics, lifestyle habits (smoking and drinking), medical history, medication of diabetes, duration of diabetes, thyroid disease, and all results of tests and examinations were obtained from the discharged medical records. Body mass index (BMI) was calculated by the formula: body weight (kg)/height (m), and waist-hip ratio (WHtR) was calculated by waist (cm)/hip (cm). The blood pressure was measured with a cuff in the sitting position after a rest period of more than 10 min. Hypertension was defined by a systolic blood pressure (SBP) ≥140 mmHg or a diastolic blood pressure (DBP) ≥90 mmHg or both, or if the patient had already administered anti-hypertensive drugs. The subjects were requested to fast for a minimum of 8–12 h and avoid a high-fat diet or alcohol consumption for at least 24 h.

Also, the results were obtained for the following variables: FPG, serum creatinine (SCr), total serum cholesterol (TC), serum triglyceride (TG), serum high-density lipoprotein cholesterol (HDL-C), serum low-density lipoprotein cholesterol (LDL-C), and HbA1c. The thyroid function was confirmed by chemiluminescence immunoassay (ADVIA Centaur Siemens New York, USA). The measuring reference ranges of serum TSH, FT3, free thyroxine (FT4), and TPO-Ab were 0.35–5.5 μIU/mL, 3.5–6.5 pmol/L, 11.5–22.7 pmol/L, and 0–78 U/mL, respectively. As described in a previous publication14, the subjects were also divided into 4 groups according to the quartiles of FT3 and FT4 in the present study. Meanwhile, the National Academy of Clinical Biochemistry (NACB) recommended lowering the upper reference limit of TSH to 2.5 uIU/mL based on a large-scale epidemiological survey in 2003, which stated that >95% of the normal individuals presented TSH levels <2.5 uIU/mL, and those with high TSH levels were likely to experience various thyroid disorders15. In addition, several similar investigations7,16,17 divided the participants within the normal range of TSH into TSH <2.5μIU/mL and TSH ≥2.5 μIU/mL groups. Thus, we divided the subjects into 2 groups according to TSH <2.5μIU/mL and TSH ≥2.5 μIU/mL. To our knowledge, there is currently no study on the relationship between TPO-Ab and DKD. Moreover, as described in a previous investigation18, the subjects were divided into 2 groups according to negative of TPO-Ab and positive of TPO-Ab. The urine specimens were collected using the first urine sample in the morning. Before the examination, the patients were instructed to avoid exercise for 1 h. The urinary albumin-to-creatinine ratio (UACR), <3 mg/mmol is defined as normal19.

Assessment of eGFR, SCH, and DKD

Estimated glomerular filtration rate (eGFR) was calculated using a modified MDRD equation in the Chinese population20 as follows: $eGFR = 175 × SCr − 1.234 × age − 0.179 ( × 0.79 females )$. SCH was defined as normal FT4 and FT3 levels and an elevated TSH level21. According to the 2012 Kidney Disease Outcomes Quality Initiative (KDOQI) guidelines22 and Diabetic Kidney Disease Consensus in China-201419, DKD referred to CKD caused by diabetes that was defined as glomerular filtration rate(GFR) <60 mL/(min × 1.73 m2) or UACR >3 mg/mmol for more than 3 months, excluding non-DKD.

Statistical analysis

Statistical analyses were performed using SPSS 16.0 statistics software (SPSS Inc., Chicago, IL, USA). Continuous variables were expressed as mean ± standard deviation or median (interquartile range), and categorical variables were expressed as percentages. Continuous data were compared using Student’s t-test or Mann–Whitney U test, and categorical data were compared by chi-square test. The multiple logistic regression models were used to examine the relationships between TSH, FT3, FT4, and TPO-Ab and the prevalence of DKD with adjustment. Lots of similar investigations16,23,24,25,26 selected age, gender, BMI, smoking status, duration of T2DM, HbA1c,hypertension, TC, TG and angiotensin Receptor Blocker (ARB) or angiotensin converting enzyme inhibitors (ACEI) medication as the covariates. These studies16,23,24,25,26,27 showed that the above covariates do have a substantial impact on the dependent variable. In the present study, compared to the non-DKD patients, those with DKD had higher insulin medication rate (P < 0.05). In addition, metformin is known to cause a decrease in the TSH levels in T2DM patients28,29. Thus, the above covariates were adjusted in the logistic regression analysis. Odds ratios (ORs) with their corresponding 95% CI were calculated. All statistical assessments were two-sided, and a P-value < 0.05 was considered statistically significant.

Results

Characteristics of the study population

The general characteristics are shown in Tables 1 and 2. A total of 862 subjects were included in this study and 73.8% of the participants were men (n = 636). The mean age was 53.70 ± 11.56 years, and BMI was 25.92 ± 3.35 kg/m2. Among 862 patients, 246 (28.5%) were diagnosed with DKD. No significant differences were observed between non-DKD and DKD groups with respect to gender, BMI, TC, LDL-C, and HDL-C. Compared to the non-DKD patients, those with DKD were older with a prolonged duration of diabetes, higher prevalence of hypertension, higher ARB or ACEI medication, higher use of insulin, higher WHtR, SBP, DBP, HbA1c, FPG, TG and lower Hemoglobin (HGB) (all P < 0.05). Moreover, the differences in smoking and drinking status were significantly different among the groups (P = 0.002, P < 0.001, respectively). Subjects with DKD presented a significantly high UACR, uric acid (UA), SCr, and Cystatin C (Cysc) and low eGFR and FT3 (P < 0.05). However, the TSH levels (P = 0.386), FT4 levels (P = 0.302), and the rate of TPO-Ab positive (P = 0.781) were not significantly different between the DKD and non-DKD groups.

The prevalence of DKD did not differ between men and women (27.4% versus 31.9%, P > 0.05). Men were prone to excessive smoking and drinking, higher BMI, WHtR, DBP, HGB, UA, SCr, eGFR, and UACR, lower age, duration of T2DM, TC, and HDL-C (P < 0.05). Moreover, men used insulin to a lower extent than women (P = 0.001). Furthermore, men yielded significantly lower TSH levels and positive rate of TPO-Ab than women (P < 0.001). The former also showed higher FT3 levels than the latter (P < 0.001). However, the rest of the parameters were not statistically significant (Table 2).

Association of thyroid hormone levels and TPO-Ab with the prevalence of DKD

In order to explore the association of thyroid function with DKD, we divided the patients into four groups according to the quartiles of FT3 (<4.38, 4.38–4.73, 4.73–5.12 and ≥5.12pmol/L) and FT4 (<14.84, 14.84–16.13, 16.13–17.66 and ≥17.66 pmol/L) and divided two groups based on TSH (<2.5 and ≥2.50 mIU/L) and TPO-Ab levels (<78 U/mL, ≥78U/ml), respectively. The prevalence of DKD showed a significantly decreasing trend across the quartiles based on the FT3 levels (41.1%, 30.6%, 23.8 and 18.9%, p < 0.001 for the trend). The first FT3 quartile group (FT3-Q1) showed a significantly higher prevalence of DKD than the second FT3 quartile group (FT3-Q2), the third FT3 quartile group (FT3-Q3), and the fourth FT3 quartile group (FT3-Q4) (P = 0.027, P < 0.001 and P < 0.001; Fig. 1) (Fig. 1). The prevalence of DKD did not differ significantly among FT4, TSH, and TPO-Ab groups (Figs 1 and 2).

Relationship of thyroid hormone levels and TPO-Ab with the prevalence of DKD

The crude and adjusted relationships between FT3, FT4, TSH, TPO-Ab, and the prevalence of DKD in total population, women, and men are indicated in Tables 3, 4, and 5, respectively. In comparison to all participants categorized in FT3-Q1 (<4.380) (Table 2), the crude ORs for DKD of all participants in FT3-Q2, FT3-Q3, and FT3-Q4 (4.380–4.730, 4.730–5.120, ≥5.120)were 0.632 (95%CI: 0.425–0.939), 0.447 (95%CI: 0.295–0.680), 0.333 (95%CI: 0.215–0.516) (P = 0.023, P < 0.001, P < 0.001) respectively. After an adjustment of the potential factors, including age, gender, duration of T2DM, BMI, smoking, hypertension, ARB/ACEI medication, insulin medication, metformin medication, HbA1c, TG, and TC, the ORs for DKD in FT3-Q2, FT3-Q3, and FT3-Q4 were 0.605 (95%CI: (0.382–0.957), 0.516 (95%CI: 0.317–0.838), 0.471 (95%CI: 0.280–0.792) (P = 0.032, P = 0.008, P = 0.005). However, none of the ORs were statistically significant with respect to the TSH, FT4, and TPO-Ab levels.

In women, Table 4 showed that in the unadjusted model, compared to the FT3-Q1 (<4.15) (Table 3), the crude OR for DKD in FT3-Q4 (≥4.80) was 0.436 (0.197–0.967) (P = 0.041). However, after further adjustment of the potentially confounding factors, none of the FT3 quartiles was associated with DKD. In men, compared to the FT3-Q1 (<4.38) (Table 5), the crude ORs for DKD in FT3-Q2, FT3-Q3, and FT3-Q4 (4.38–4.69, 4.69–5.10, ≥ 5.10) were significantly different [0.577(0.362–0.920), 0.396(0.243–0.645), 0.273(0.162–0.459)]. After further adjusting the potential factors, the adjusted ORs for DKD were significant [0.569(0.333–0.973), 0.471(0.271–0.819), 0.384(0.209–0.708)]. The crude and adjusted ORs between FT4, TSH, TPO-Ab and DKD had no significant differences in men and in women, respectively.

Discussion

A total of 862 T2DM patients were enrolled in the cross-sectional study. The prevalence of DKD was 28.5% among T2DM patients, and no significance was observed between men and women. Thus, a higher prevalence of thyroid disease was observed in women as compared to men30. Therefore, we grouped men and women to discuss the association between thyroid hormones and DKD, respectively. Moreover, the prevalence of DKD decreased gradually as the quartiles of FT3 levels increased. The results of the current logistic regression analysis also revealed that FT3 levels were inversely associated with the prevalence of DKD in total population and men and not only in unadjusted models but also adjusted models. However, no significance was observed between FT3 and DKD in women. Therefore, FT3 levels exhibited a potentially significant association with DKD in euthyroid patients with type 2 diabetes.

Thyroid hormones participate in the physiological function of the kidney. Thyroid dysfunction can cause changes in renal blood flow, glomerular filtration rate, renal tubular absorption and secretion, and renal structure31. Furthermore, several studies showed that thyroid hormone levels within the normal range were associated with the risk of CKD9,10,11. A cohort study also indicated that low normal FT3 levels were associated with an increased risk of CKD in euthyroid function population9. Another study in middle-aged and elderly Chinese individuals found that low normal FT3 levels were associated with a high prevalence of microalbuminuria8. A small-scale study showed that the total triiodothyronine (TT3) level was significantly correlated with the levels of microalbuminuria. However, whether or not the thyroid hormone levels were within the normal range remained unclear in this study, while the FT3 levels had not been detected32. The above studies suggested that low normal FT3 levels were associated with CKD. However, only a few studies focused on the relationship between normal thyroid hormone levels and DKD in T2DM. Hitherto, only one study found a negative correlation between FT3 levels and DKD incidence in T2DM patients with normal thyroid function33. Our results were in agreement with the above study: FT3 levels were negatively correlated with the prevalence of DKD in T2DM patients with normal thyroid function. The current study failed to show a significant association between normal levels of FT3 and DKD in the women population. The potential reasons are illustrated as follows. Firstly, sex hormones (such as estrogen and testosterone) can regulate the thyroid function34. The gender difference in the relationship between FT3 and DKD can partially be explained by the difference in sex hormones. We did not measure levels of sex hormones in this study, which necessitates further research. Secondly, since the sample size of women was smaller than men, the precision and statistical power of the analysis might be lower for women. Additionally, the proportion of men was higher than that of women, which could also influence the correlation between thyroid hormone levels and DKD in women patients. Lastly, the FT3 levels of the women population were significantly lower than that in men (P < 0.001). A previous study23 in China based on the relationship between thyroid hormone and T2DM speculated that the prevalence of DKD depended on the concentrations of FT3 irrespective of the gender. Thus, further large-scale studies are required to clarify this hypothesis.

The present study had several characteristics. Firstly, men were prone to suffering from DKD than women3; however, the prevalence of DKD was higher in women than men without any statistical significance in the present study (31.9% vs 27.4%, P = 0.200). The possible explanations were as follows: the age of the women patients in the population was older than men (57.92 ± 11.51, women; 52.20 ± 11.21, men; P < 0.001), and the duration of T2DM of the former was longer than the latter (10.13 ± 7.03, women; 8.26 ± 6.39, men; P = 0.001). In addition, age and duration of diabetes are also risk factors for DKD3. Furthermore, the characteristics of the population in this study reflected the current situation of T2DM patients in Northwest China. Secondly, in this study, the “DKD” term was defined according to the guidelines of 2012 KDOQI22 and Diabetic Kidney Disease Consensus in China-201419: DKD was diagnosed with either GFR or UACR. Nevertheless, previous studies continue the usage of “dabetic nephropathy (DN)” term, and the value of UACR to diagnose DN, missing having normal UACR value and abnormal eGFR value patients.

The mechanism underlying the association between FT3 and DKD could be clearly demonstrated by the following: Firstly, T3 had been shown to influence the endothelial function by relaxation the vascular smooth muscle cells through direct or indirect effects in experimental models35,36,37. Low T3 was closely relevant to endothelial dysfunction in patients with advanced non-DKD38. The study confirmed that serum TSH levels in the upper reference range were also associated with impaired endothelial function measured via flow-mediated dilation39. The endothelial dysfunction was associated with albuminuria in diabetes40; therefore, low FT3 level and albuminuria may be related to endothelial dysfunction. Secondly, T3 could attenuate albuminuria and improve the renal structural damage in db/db diabetic mice by increasing the activity of phosphatidylinositol 3 kinase and decreasing the expression of transforming growth factor-β141. Finally, 3,5-diiodothyronine could protect the cells from renal damage in DKD by inhibiting the activation of NF-κβ and JNK42. These enzymes and pathways, mentioned above, were involved in the development of DKD.

However, the present study failed to find a correlation between TSH and DKD. The current results were in agreement with the study by Wu et al.33; however, the results differed from those reported by Qi et al.16 demonstrated an association between high TSH levels and an increased risk of DKD. Currently, the reasons for this discrepancy are yet to be elucidated, which might be related to differences in demographic and clinical characteristics, ethnicity, and research design. These phenomena might be attributed to the following: Patients with positive TPO-Ab may not be excluded in the present study. The population with positive TPO-Ab has a higher TSH level than the negative TPO-Ab in euthyroid population, and the TPO-Ab levels may be responsible for the endothelial dysfunction and subsequent microalbuminuria43. On the other hand, metformin is known to cause a decrease in the TSH levels in T2DM patients28,29. Thus, metformin is widely used in T2DM patients as a first-line treatment. In the current analysis, we adjusted the rate of metformin usage; however, these factors may influence the association between TSH and DKD. None of the ORs were found to be significant in FT4 quartiles, and this result was consistent with the previous study33, which might be attributed to the role of thyroid hormones that is achieved by FT3 binding to receptors and other related proteins to regulate the transcription of target genes and the expression of proteins. Similarly, no relationship between TPO-Ab and DKD was found in this study, which might be associated with a low positive TPO-Ab rate (8.0%). Moreover, previous studies indicated that T1DM with thyroid dysfunction was mostly due to the presence of autoimmune diseases and T2DM with thyroid dysfunction was primarily due to insulin resistance44,45.

In conclusion, the present study showed that FT3 in normal range was negatively correlated with DKD in T2DM with normal thyroid function. These results could aid in predicting the risk of DKD development, as well as the basis for a large-scale cohort study.

Nevertheless, the present study has some limitations. Firstly, this was a cross-sectional study with lack of long-term follow-up; thus, the relative risk could not be assessed. Therefore, further prospective and longitudinal studies should be conducted to confirm the relative risk between FT3 levels within normal range and DKD in T2DM patients. Secondly, UACR and thyroid function were estimated only once, which could result in misleading classifications. Thirdly, we did not assess the reverse triiodothyronine (rT3) of the population, and diabetes may have a low triiodothyronine (T3) syndrome, which is prone to hypothyroidism46, and thus, the relationship between hypothyroidism and some factors may have been underestimated in this study. Furthermore, the proportion of men was significantly higher than that of women, which might affect the relationship between the levels of thyroid hormone and DKD in women patients.

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35. 35.

Napoli, R. et al. Impact of hyperthyroidism and its correction on vascular reactivity in humans. Circulation 104, 3076–3080 (2001).

36. 36.

Isumi, Y. et al. Regulation of adrenomedullin production in rat endothelial cells. Endocrinology 139, 838–846, https://doi.org/10.1210/endo.139.3.5789 (1998).

37. 37.

Ojamaa, K., Klemperer, J. D. & Klein, I. Acute effects of thyroid hormone on vascular smooth muscle. Thyroid: official journal of the American Thyroid Association 6, 505–512, https://doi.org/10.1089/thy.1996.6.505 (1996).

38. 38.

Yilmaz, M. I. et al. Low triiodothyronine alters flow-mediated vasodilatation in advanced nondiabetic kidney disease. American journal of nephrology 33, 25–32, https://doi.org/10.1159/000322581 (2011).

39. 39.

Volzke, H. et al. Are serum thyrotropin levels within the reference range associated with endothelial function? European heart journal 30, 217–224, https://doi.org/10.1093/eurheartj/ehn508 (2009).

40. 40.

Siddiqi, F. S. & Advani, A. Endothelial-podocyte crosstalk: the missing link between endothelial dysfunction and albuminuria in diabetes. Diabetes 62, 3647–3655, https://doi.org/10.2337/db13-0795 (2013).

41. 41.

Lin, Y. & Sun, Z. Thyroid hormone ameliorates diabetic nephropathy in a mouse model of type II diabetes. The Journal of endocrinology 209, 185–191, https://doi.org/10.1530/JOE-10-0340 (2011).

42. 42.

Shang, G. et al. 3,5-Diiodo-l-thyronine ameliorates diabetic nephropathy in streptozotocin-induced diabetic rats. Biochimica et biophysica acta 1832, 674–684, https://doi.org/10.1016/j.bbadis.2013.01.023 (2013).

43. 43.

Xiang, G. D. et al. Impairment of endothelium-dependent arterial dilation in Hashimoto’s thyroiditis patients with euthyroidism. Clinical endocrinology 64, 698–702, https://doi.org/10.1111/j.1365-2265.2006.02531.x (2006).

44. 44.

American Diabetes, A.. 2. Classification and Diagnosis of Diabetes. Diabetes care 39(Suppl 1), S13–22, https://doi.org/10.2337/dc16-S005 (2016).

45. 45.

Sobel, R. Screening for thyroid disease. Israel journal of medical sciences 26, 516–517 (1990).

46. 46.

Moura Neto, A. et al. Relation of thyroid hormone abnormalities with subclinical inflammatory activity in patients with type 1 and type 2 diabetes mellitus. Endocrine 51, 63–71, https://doi.org/10.1007/s12020-015-0651-5 (2016).

Acknowledgements

The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (Nos 81573746, 31600681), Military medical science and technology youth training program (17QNP029), Beijing Nova Program of Science and Technology (xx2017103) and the Translational Medicine Project of the Chinese PLA General Hospital (2016TM-012).

Author information

Author notes

1. Jian Zou, Feng Tian and Yi Zhang contributed equally to this work.

Affiliations

1. Department of Medicine, The 522 Hospital of the Chinese PLA, Luoyang, Henan Province, China

• Jian Zou
• , Feng Tian
•  & Chao Xu
2. Department of Endocrinology and Metabolism, The First Affiliated Hospital of Fourth Military Medical University, Xi’an, Shaanxi Province, China

• Jian Zou
• , Yi Zhang
• , Jiajia Zhai
• , Min Shi
• , Wenjuan Li
•  & Xiaomiao Li

• Yi Zhang

• Zeping Li

• Chao Yang

• Haixu Chen

• Juan Zhang
8. Department of Respiratory, The Second Affiliated Hospital of Fourth Military Medical University, Xi’an, Shaanxi Province, China

• Yonghong Xie

Contributions

Conceived and designed the experiments: X.M.L., Y.H.X., J.Z., F.T., Y.Z. Performed the experiments: J.Z., F.T., Y.Z., J.J.Z., M.S., J.Z., F.T., W.J.L. Analyzed the data: J.Z., F.T., Y.Z., J.J.Z., X.M.L., Y.H.X.. Contributed reagents/materials/analysistools: J.Z., F.T., Y.Z., J.J.Z., M.S., J.Z., F.T., W.J.L. Wrote the paper: J.Z., Y.Z., J.J.Z., M.S., J.Z., F.T., W.J.L., X.M.L., Y.H.X. Revised the manuscript: J.Z., Z.P.L., C.Y., H.X.C., C.X.

Competing Interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Yonghong Xie or Xiaomiao Li.


text:
# How do you integrate int xsqrt(3x+ 1) dx?

##### 1 Answer
Apr 15, 2017

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {\left(3 x + 1\right)}^{\frac{3}{2}} - \frac{4}{135} {\left(3 x + 1\right)}^{\frac{5}{2}} + C$

#### Explanation:

You will need substitution in addition to integration by parts. Let $u = x$ and $\mathrm{dv} = \sqrt{3 x + 1}$. By the power rule, $\mathrm{du} = \mathrm{dx}$. But we will need to make a substitution to find $v$. Let $t = 3 x + 1$. Then $\mathrm{dt} = 3 \mathrm{dx}$ and $\mathrm{dx} = \frac{\mathrm{dt}}{3}$.

$\int \sqrt{3 x + 1} = \sqrt{t} \cdot \frac{\mathrm{dt}}{3} = \frac{1}{3} \sqrt{t} \mathrm{dt} = \frac{2}{9} {t}^{\frac{3}{2}} = \frac{2}{9} {\left(3 x + 1\right)}^{\frac{3}{2}}$

Apply integration by parts now.

$\int u \mathrm{dv} = u v - \int v \mathrm{du}$

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {\left(3 x + 1\right)}^{\frac{3}{2}} - \int \frac{2}{9} {\left(3 x + 1\right)}^{\frac{3}{2}} \mathrm{dx}$

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {\left(3 x + 1\right)}^{\frac{3}{2}} - \frac{2}{9} \int {\left(3 x + 1\right)}^{\frac{3}{2}} \mathrm{dx}$

Now let $n = 3 x + 1$. Then $\mathrm{dn} = 3 \mathrm{dx}$ and $\mathrm{dx} = \frac{\mathrm{dn}}{3}$.

$\int {\left(3 x + 1\right)}^{\frac{3}{2}} = \frac{1}{3} \int {n}^{\frac{3}{2}} \mathrm{dn} = \frac{2}{15} {n}^{\frac{5}{2}} = \frac{2}{15} {\left(3 x + 1\right)}^{\frac{5}{2}}$

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {\left(3 x + 1\right)}^{\frac{3}{2}} - \frac{2}{9} \left(\frac{2}{15}\right) {\left(3 x + 1\right)}^{\frac{5}{2}} + C$

$\int x \sqrt{3 x + 1} = \frac{2}{9} x {\left(3 x + 1\right)}^{\frac{3}{2}} - \frac{4}{135} {\left(3 x + 1\right)}^{\frac{5}{2}} + C$

Hopefully this helps!

format: plain


text:
All the solutions provided in McGraw Hill My Math Grade 5 Answer Key PDF Chapter 1 Review will give you a clear idea of the concepts.

Review

Vocabulãry Check

choose the correct word(s) to complete each sentence.

decimal decimal point equivalent decimals expanded form
period place value place-value chart standard form

Question 1.
Decimals that have the same value are ____.
Definition: When two decimals are representing the same fraction, they are called the decimal equivalent. In other words, equivalent decimals are decimals that are equal. Hence they are also called equal decimals.

Question 2.
____ is a system for writing numbers. In this system, the position of a digit determines its value.
Definition: Place value describes the position or place of a digit in a number. Each digit has a place in a number. When we represent the number in general form, the position of each digit will be expanded. Those positions start from a unit place or we also call it one’s position. The order of place value of digits of a number of right to left is units, tens, hundreds, thousands, ten thousand, a hundred thousand, and so on.

Question 3.
The usual or common way to write a number is called _____
A standard form of a number is basically mentioned for the representation of large numbers or small numbers. We use exponents to represent such numbers in standard form.

Question 4.
The way of writing a number as the sum of the values of its digits is called ___.
The expanded form of the numbers helps to determine the place value of each digit in the given number. It means that the expansion of numbers is based on the place value. The expanded form splits the number, and it represents the number in units, tens, hundreds and thousands form. For example, the expanded form of the number 1682 is 1000+600+80+2.

Question 5.
A ____ is a number that has a digit in the tenth place, hundredth place, and/or beyond.
decimals are one the types of numbers, which has a whole number and the fractional part separated by a decimal point.

Question 6.
The ____ is a period separating the ones and the tenths in a decimal number.
The dot present between the whole number and fractions part is called the decimal point. For example, 34.5 is a decimal number.

Question 7.
A ____ is a chart that shows the value of the digits in a number.
Place value charts help us to make sure that the digits are aligned in the correct places. A place value chart displays the correct position or place of a digit in a number. To identify the positional values or worth of different digits in a number accurately, we first write the given digits in the place value chart to check their position. In order to make the process easier, the larger numbers are divided into periods that are separated with the help of commas.

Concept Check

Name the place of the underlined digit. Then write the value of the digit.

Question 8.
195,489
Place value describes the position or place of a digit in a number. Each digit has a place in a number. When we represent the number in general form, the position of each digit will be expanded. Those positions start from a unit place or we also call it one’s position. The order of place value of digits of a number of right to left is units, tens, hundreds, thousands, ten thousand, a hundred thousand, and so on.

The underlined number is 5
5 is in the thousand’s place.

Question 9.
6,720,341
Place value describes the position or place of a digit in a number. Each digit has a place in a number. When we represent the number in general form, the position of each digit will be expanded. Those positions start from a unit place or we also call it one’s position. The order of place value of digits of a number of right to left is units, tens, hundreds, thousands, ten thousand, a hundred thousand, and so on.

The underlined number is 6
6 is in the hundred thousand places.

Write each number in standard form.

Question 10.
94 million, 237 thousand, 108 ____
A Standard Form is a process of formulating a given mathematical concept such as a number, equation, polynomial, etc. in the standard form by following certain basic rules.
Standard Form of a Number:
The definition of the standard form of a number is representing the very large expanded number in a small number.
The standard form of a number is 94,237,108

Question 11.
8 × 1,000,000 + 5 × 10,000 + 2 × 1,000 + 6 × 100 ____
8,000,000 + 50,000 + 2,000 + 600
Place value charts help us to make sure that the digits are aligned in the correct places. A place value chart displays the correct position or place of a digit in a number. To identify the positional values or worth of different digits in a number accurately, we first write the given digits in the place value chart to check their position. In order to make the process easier, the larger numbers are divided into periods that are separated with the help of commas.

In standard form, we can write as 8,052,600

Shade the model to show each fraction. Write each fraction as a decimal.

Question 12.
$$\frac{19}{100}$$

The above-given model is the hundredth cube.
In fractions, we can write as 19/100
In decimals, we can write it as 0.19

Question 13.
$$\frac{8}{10}$$

The above-given model is the tenth cube
In fractions, we can write as 8/10
In decimals, we can write it as 0.8

Write each number ¡n standard form and expanded form.

Question 14.
five and nine tenths
The expanded form of the numbers helps to determine the place value of each digit in the given number. It means that the expansion of numbers is based on the place value. The expanded form splits the number, and it represents the number in units, tens, hundreds and thousands form.
(5 x 1) + (9 x 1/10)
Standard form:
5.90
5 is in the one’s place
‘.’ represents “and”
9 is in the tenth place.

Question 15.
seven hundred twelve thousandths
The expanded form of the numbers helps to determine the place value of each digit in the given number. It means that the expansion of numbers is based on the place value. The expanded form splits the number, and it represents the number in units, tens, hundreds and thousands form.
(0 x 100) + (7 x 1/10) + (1 x 1/100) + (2 x 1/1000)
The standard form is 0.712

Write <, >, or = in each to make a true sentence.

Question 16.
14,589  14,985
Explanation:
There are certain rules, based on which it becomes easier to compare numbers. These rules are:
– Numbers with more digits
– Numbers starting with a larger digit
Since, both the numbers, 14589 and 14985 are having five digits thus we will compare the left-most digit of both numbers.
– As we can see the first two digits are the same for both numbers, thus we need to compare the next left-most digit of both numbers.
5 < 9
Therefore, 14,589 < 14,985
Hence, 14,589 is less than 14,985

Question 17.
506,789  505,789
Explanation:
Since, both the numbers, 506,789 and 505,789 are having six digits thus we will compare the left-most digit of both numbers.
– As we can see the first two digits are the same for both numbers, thus we need to compare the next left-most digit of both numbers.
6 > 5
Therefore, 506,789 > 505,789
Hence, 506,789 is greater than 505,789

Question 18.
8,913  8,931
Explanation:
Since, both the numbers, 8,913 and 8,931 are having four digits thus we will compare the left-most digit of both numbers.
– As we can see the first two digits are the same for both numbers, thus we need to compare the next left-most digit of both numbers.
1 < 3
Therefore, 8,913 < 8,931
Hence, 8,913 is less than 8,931

Question 19.
0.49  0.71
Explanation:
Since, both the numbers, 0.49 and 0.71 are having three digits thus we will compare the left-most digit of both numbers.
– As we can see the first digit is the same for both numbers, thus we need to compare the next left-most digit of both numbers.
4 < 7
Therefore, 0.49 < 0.71
Hence, 0.49 is less than 0.71

Question 20.
9.02  9.020
Explanation:
They both are equivalent decimals. Because all the place values of digits are the same.
If we put ‘0’ in the thousandth place then the first number becomes 9.020
or else 9.020 is nothing but 9.02
Finally, we can write as 9.02 is equal to 9.020
9.02 = 9.020

Question 21.
0.843  0.846
Explanation:
Since, both the numbers, 0.843 and 0.846 are having four digits thus we will compare the left-most digit of both numbers.
– As we can see the first three digits are the same for both numbers, thus we need to compare the next left-most digit of both numbers.
3 < 6
Therefore, 0.843 < 0.846
Hence, 0.843 is less than 0.846

Question 22.
Order the set of numbers from least to greatest.
13.84, 13.097, 12.655, 13.6
Ascending order is a method of arranging numbers from smallest value to largest value. The order goes from left to right. It is also named increasing order.
The above-given numbers: 13.84, 13.097, 12.655, 13.6
compare the numbers, after sorting out the numbers, the answer is:
12.655, 13.097, 13.6, 13.84

Problem Solving

Question 23.
One cup is equal to $$\frac{5}{10}$$ pint. Write this fraction as a decimal.
The above-given fraction is 5/10
We have to write in decimal form.
The decimal form can write as 0.5

Question 24.
Charles is moving from Springfield, which has 482,653 people, to Greenville, which has 362,987 people. Is he moving to a city with a greater or smaller population? Explain.
The number of people in Springfield = 482,653
The number of people in Greenville = 362,987
Now we have to find out whether Charles is moving to a city with the greatest population or a smaller population.
Now compare the numbers.
Charles is moving to Greenville.
both are having the same number of digits. So compare the left-most digit.
3 < 4
Therefore, 362,987 is less than 482,653
Hence, Charles is moving to a city with a smaller population.

Question 25.
The Pacific Ocean has sixty-four million, one hundred eighty-six thousand, three hundred square miles. How many square miles is the Pacific Ocean in standard form?
A Standard Form is a process of formulating a given mathematical concept such as a number, equation, polynomial, etc. in the standard form by following certain basic rules.
Standard Form of a Number:
The definition of the standard form of a number is representing the very large expanded number in a small number.
In the standard form, we can write a 64,186,300
Therefore, the square miles is the pacific ocean is 64,186,300 miles.

Test Practice

Question 26.
Which number results in a true sentence in 475 < ___?
A. 473
B. 474
C. 475
D. 476
You can go for option verification.
Check option D holding with number 476.
Compare 475 and 476
The first two digits are the same. So compare the next left-most digit.
5 < 6
Therefore, 475 < 476.

Reflect

Use what you learned about place value to complete the graphic organizer.


text:
2-13-1.1-c3-0-2 Degree 2, weight $\gt 1$, Rank 1 L-function of smallest conductor
2-127-1.1-c3-0-30 Degree 2, weight $\gt 1$, Rank 2 L-function of smallest conductor


text:
This twiki page describes the setup used as of 2016 to scan the LHCb software using the Coverity static code analyzer. See also JIRA issue LBCORE-470.

Coverity is a commercial tool and can only be run on specific (licensed) machines. The previous setup used by LHCb in 2011, described here, was based on a private LHCb installation of a Coverity server instance. This is no longer needed, as the 2016 setup is based on the existing infrastructure that is maintained outside LHCb, thanks to the EP-SFT group. This consists of a Coverity Connect server on coverity.cern.ch and a build node on buildcoverity.cern.ch.

Note that Coverity itself hosts and offers a free Coverity Scan service for open source software. Using a custom privately maintained service, however, offers more felxibility in the build environment and in the triage of defects.

This page contains four sections, describing: how to prepare the Coverity Connect server, prior to Coverity scans (admin operations); how to prepare Coverity scans while building the software, and commit the results to the Coverity Connect server (librarian operations); how to interact with the Coverity Connect server to triage defects (developer operations); the current status of Coverity defects for the LHCb software projects.

# 1. Prepare the Coverity Connect server, prior to Coverity scans (admin operations)

The results of the Coverity scans for the LHCb software can be analyzed by logging on the Coverity Connect instance on coverity.cern.ch.

Before using Coverity, your admin must have configured several server-side entities for you, including users and user groups, projects, streams, (optionally) triage stores and (optionally) component maps. For help on how to configure the Coverity Connect server, refer to the Coverity Usage and Administration Guide (authentication required). The following sub-sections describe the setup that was adopted for the LHCb software.

## Users and user groups

Only users belonging to the ph-sft-coverity-users egroup are allowed to use the coverity.cern.ch server. Users in this egroup are automatically imported (every night?) to the server via LDAP. They can authenticate with their CERN SSO credentials.

At the moment, around 20 LHCb accounts have been authorized to use the server by including them in this egroup. The list includes all application managers and a few power developers.

A local group lhcb-coverity-users-local has been created on coverity.cern.ch, including all LHCb users so far existing on the server. To simplify email communication with these users, a CERN egroup lhcb-coverity-users including the same users has also been created. The local group on the Coverity server and the CERN egroup are kept in sync manually for the moment. Eventually, the local egroup on the Coverity server may be replaced by the CERN egroup, imported via LDAP (see LBCORE-1063 - but this does not seem technically possible through the present interface?).

For the moment, only a few users have been individually granted admin/owner privileges over the relevant LHCb project, streams and triage stores. Unlike projects, streams and triage stores, component maps are all owned by the admin user and their ownership may not be transferred; managing component maps requires the global (not per-project) role "project admin" (see "managing custom roles").

A few "user views" have also been created in the user space of one user and have been shared with the lhcb-coverity-users-local. This is described more in detail in section 3 below.

## Projects

A single Coverity "project" LHCb has been created to hold all defects from all relevant LHCb software projects.

From a user perspective, a Coverity project provides the top-level context that can be changed with a drop-down menu. This means, for instance, that defects from the builds of GAUDI and BOOLE will normally both appear in the same defect list (unless one selects them by stream and/or component, as described below).

## Streams

Twelve separate Coverity "streams" have been created for LHCb, one for each of the LHCb software projects that are so far included in Coverity scans: GAUDI, LHCB, LBCOM, REC, BOOLE, PHYS, HLT, ANALYSIS, STRIPPING, DAVINCI, GEANT4, GAUSS.

A Coverity stream belongs to a single project. When defects are committed to the database of the Coverity Connect server from a software build (see section 2), they are sent to a single stream; in turn, this automatically defines the project they belong to.

From a user perspective, it is possible to select and analyze only the defects committed to a given stream. This is what has been done in the "user views" described above, created by one user and shared with all LHCb users. For instance, the LHCb-05-Boole view shows all outstanding defects in BOOLE.

## Triage stores

Coverity "triage stores" describe the storage space on the Coverity Connect server where defects are stored. One stream uses one and only one triage store, but the same triage store may be used by different streams. Creating triage stores is optional: the Default triage store is used by all streams that have no specific triage store defined.

A single triage store has been created for all LHCb defects. It is used by all twelve LHCb streams.

## Component maps

A Coverity "component map" is an optional entity allowing the categorization of defects by component within each project. One stream uses one and only one component map, but the same component map may be used by different streams (e.g. the Default component map is used by all streams for which no dedicated component map was created).

A single component map has been created for all LHCb defects. It is used by all twelve LHCb streams.

The LHCb component map defines several components, listed below.

• One component for each of the twelve LHCb software projects: GAUDI, LHCB, LBCOM, REC, BOOLE, PHYS, HLT, ANALYSIS, STRIPPING, DAVINCI, GEANT4, GAUSS.
• Several components for defects coming from external software, "external (<pkg>)" where <pkg> is one of Boost, clhep, fastjet, gcc, HepMC, MCGenerators, Python, ROOT, system, tbb.
• A default component, Other, for defects coming from a not-yet-identified source. Currently no defects are assigned to the Other component.

The LHCb component map also defines several file rules, allowing the automatic categorization of a defect by component, depending on the name of the file where that defect was found. The "system" component refers to headers in /usr, for instance.

Note that selecting defects by component or by stream is not the same thing. A defect can belong to the BOOLE stream and to the GAUDI component, if it was found in a Gaudi header used by a Boole class.

# 2. Prepare Coverity scans and commit results to the Coverity Connect server (librarian operations)

Coverity scans for the LHCb software are prepared on a specific SLC6 node buildcoverity.cern.ch, maintained by EP-SFT, where the Coverity client software is installed. The new infrastructure allows software builds with c++11, which could not be used on the previous EP-SFT infrastructure (before 2015).

To access buildcoverity.cern.ch and use the Coverity CLI, log in using your CERN SSO credentials:

    ssh buildcoverity.cern.ch [-l username]

For the moment, only a few one-off manual scans of the full LHCb stacks have been performed using a private account avalassi. Eventually, the plan is to configure automatic nightly (or weekly?) scans using LHCb service accounts. (see LBCORE-1058).

The scans of the LHCb software are being performed in directory /builda/LHCb. Extended attribute ACLs (via getfacl and setfacl) are used to make sure that all relevant users can read and write the same files there. You can use getfacl to check the existing ACLs (note that ls -l lists a + at the end of the standard Unix permissions). This is the command that was last used to prepare the directory:

    setfacl -b -R -m u:avalassi:rwx -m u:lhcbsoft:rwx -m u:marcocle:rwx -m g:z5:rwx -dm u:avalassi:rwx -dm u:lhcbsoft:rwx -dm u:marcocle:rwx -dm g:z5:rwx /builda/LHCb

The full process of preparing Coverity scans includes three separate steps that involve three different commands from the Coverity client CLI:

• first, checkout the code and build it through cov-build;
• then, analyze the build results using cov-anayze;
• finally, commit the analysis results to the Coverity Connect server using cov-commit-defects.

All scripts used to perform Coverity scans are stored locally on buildcoverity for the moment (and documented in thiw twiki). Eventually, it may be useful to store them in a gitlab repositiry ((see LBCORE-1059).

A full Coverity scan for the LHCb software is triggered using the following script (download-build-analyze-commit.sh), including the three steps described above. These steps are described in more details in the following subsections:

    1#!/bin/bash -e
2cd /builda/LHCb
3date=date
4echo "Full Coverity scan started at "$date 5export CMTCONFIG=x86_64-slc6-gcc49-dbg # Geant4 now supports c++1y 6unset BINARY_TAG 7. /cvmfs/lhcb.cern.ch/lib/LbLogin.sh 8\rm -rf LbNightlyTools 9git clone https://gitlab.cern.ch/lhcb-core/LbNightlyTools.git 10cd LbNightlyTools 11mkdir logs 12touch logs/log-download-started 13. setup.sh 14lbn-get-configs 15lbn-checkout lhcb-gaudi-head 16touch logs/log-build-started 17for project in Gaudi LHCb Lbcom Rec Boole Phys Hlt Analysis Stripping DaVinci Geant4 Gauss ; do ../scripts/build.sh$project; done
18touch logs/log-analyze-started
19for project in Gaudi LHCb Lbcom Rec Boole Phys Hlt Analysis Stripping DaVinci Geant4 Gauss ; do ../scripts/analyze.sh $project; done 20touch logs/log-commit-started 21for project in Gaudi LHCb Lbcom Rec Boole Phys Hlt Analysis Stripping DaVinci Geant4 Gauss; do ../scripts/commit.sh$project; done
22touch logs/log-all-done
23echo "Full Coverity scan started at "$date 24echo "Full Coverity scan ended at "date 25arch=../archive/date +%Y""%m""%d""_%H""h%""M 26mkdir -p$arch/scripts
27\cp -dpr logs/* $arch/ 28\cp -dpr ../scripts/*sh$arch/scripts/
29\mv $arch/scripts/setup_dummy.sh$arch/scripts/setup.sh

For reference, the first full scan performed using this script took approximately 10h50m (approximately 20m for checkouts, 4h40m for builds, 2h00m for analysis, 3h50m for commits).

Full Coverity scan started at Sun Feb 28 19:15:57 CET 2016
Full Coverity scan ended   at Mon Feb 29 06:03:09 CET 2016

-rw-r--r--.  1 avalassi zg    0 Feb 28 19:16 log-download-started
-rw-r--r--.  1 avalassi zg    0 Feb 28 19:31 log-build-started
drwxr-xr-x. 18 avalassi zg 4096 Feb 28 19:31 ../
-rw-r--r--.  1 avalassi zg  546 Feb 28 19:51 log-Gaudi-build.txt
-rw-r--r--.  1 avalassi zg  546 Feb 28 20:31 log-LHCb-build.txt
-rw-r--r--.  1 avalassi zg  545 Feb 28 20:40 log-Lbcom-build.txt
-rw-r--r--.  1 avalassi zg  599 Feb 28 21:22 log-Rec-build.txt
-rw-r--r--.  1 avalassi zg  602 Feb 28 21:27 log-Boole-build.txt
-rw-r--r--.  1 avalassi zg  545 Feb 28 22:08 log-Phys-build.txt
-rw-r--r--.  1 avalassi zg  541 Feb 28 22:25 log-Hlt-build.txt
-rw-r--r--.  1 avalassi zg  557 Feb 28 22:50 log-Analysis-build.txt
-rw-r--r--.  1 avalassi zg  556 Feb 28 22:58 log-Stripping-build.txt
-rw-r--r--.  1 avalassi zg  555 Feb 28 23:29 log-DaVinci-build.txt
-rw-r--r--.  1 avalassi zg  552 Feb 28 23:55 log-Geant4-build.txt
-rw-r--r--.  1 avalassi zg  606 Feb 29 00:07 log-Gauss-build.txt
-rw-r--r--.  1 avalassi zg    0 Feb 29 00:07 log-analyze-started
-rw-r--r--.  1 avalassi zg 4877 Feb 29 00:28 log-Gaudi-analyze.txt
-rw-r--r--.  1 avalassi zg 5273 Feb 29 00:59 log-LHCb-analyze.txt
-rw-r--r--.  1 avalassi zg 3288 Feb 29 01:02 log-Lbcom-analyze.txt
-rw-r--r--.  1 avalassi zg 4338 Feb 29 01:19 log-Rec-analyze.txt
-rw-r--r--.  1 avalassi zg 3128 Feb 29 01:21 log-Boole-analyze.txt
-rw-r--r--.  1 avalassi zg 4252 Feb 29 01:32 log-Phys-analyze.txt
-rw-r--r--.  1 avalassi zg 3853 Feb 29 01:39 log-Hlt-analyze.txt
-rw-r--r--.  1 avalassi zg 3645 Feb 29 01:45 log-Analysis-analyze.txt
-rw-r--r--.  1 avalassi zg 3167 Feb 29 01:47 log-Stripping-analyze.txt
-rw-r--r--.  1 avalassi zg 2931 Feb 29 01:53 log-DaVinci-analyze.txt
-rw-r--r--.  1 avalassi zg 4078 Feb 29 02:06 log-Geant4-analyze.txt
-rw-r--r--.  1 avalassi zg 4337 Feb 29 02:10 log-Gauss-analyze.txt
-rw-r--r--.  1 avalassi zg    0 Feb 29 02:10 log-commit-started
-rw-r--r--.  1 avalassi zg 2550 Feb 29 02:30 log-Gaudi-commit.txt
-rw-r--r--.  1 avalassi zg 2548 Feb 29 02:51 log-LHCb-commit.txt
-rw-r--r--.  1 avalassi zg 2262 Feb 29 03:02 log-Lbcom-commit.txt
-rw-r--r--.  1 avalassi zg 2258 Feb 29 03:13 log-Rec-commit.txt
-rw-r--r--.  1 avalassi zg 2259 Feb 29 03:16 log-Boole-commit.txt
-rw-r--r--.  1 avalassi zg 2257 Feb 29 03:22 log-Phys-commit.txt
-rw-r--r--.  1 avalassi zg 2254 Feb 29 03:28 log-Hlt-commit.txt
-rw-r--r--.  1 avalassi zg 2267 Feb 29 03:33 log-Analysis-commit.txt
-rw-r--r--.  1 avalassi zg 2102 Feb 29 05:40 log-Stripping-commit.txt
-rw-r--r--.  1 avalassi zg 2266 Feb 29 05:49 log-DaVinci-commit.txt
-rw-r--r--.  1 avalassi zg 2091 Feb 29 05:56 log-Geant4-commit.txt
-rw-r--r--.  1 avalassi zg    0 Feb 29 06:03 log-all-done
drwxr-xr-x.  2 avalassi zg 4096 Feb 29 06:03 ./
-rw-r--r--.  1 avalassi zg 2088 Feb 29 06:03 log-Gauss-commit.txt

## Download and build the software (cov-build)

As described in the script above, LHCb software builds for Coverity scans are executed within the LbNightlyTools framework, the same that is used for the LHCb nightly builds.

The x86_64-slc6-gcc49-dbg platform is chosen as this is the most recent compiler supported by LHCb on SLC6 platforms.

To avoid internal inconsistencies, all LHCb software projects are downloaded in one go: first they are checked out from SVN into tmp/checkout, then tar.bz2 archives are created in artifacts.

The code is then built using the build.sh script, a wrapper over the lbn-build command, invoked through cov-build:

    1#!/bin/bash -e
2dir=basename $PWD 3if [ "$dir" != "LbNightlyTools" ]; then
4  echo "ERROR! Please run this script from LbNightlyTools"; exit 1
5fi
6if [ "$1" == "" ] || [ "$2" != "" ]; then
7  echo "Usage: $0 <project>" 8 echo "Example:$0 Gaudi"
9  exit 1
10fi
11proj=$1 12mkdir -p cov-out 13\rm -rf cov-out/$proj
14mkdir -p logs
15out=logs/log-${proj}-build.txt 16\rm -rf$out
17if [ "$proj" == "Gaudi" ]; then 18 nounpack= 19else 20 nounpack="--no-unpack" 21fi 22echo "======================================================================" 23# See http://stackoverflow.com/questions/3173131 24echo "Output copied to logfile$out"
25exec >  >(tee -a $out) 26exec 2> >(tee -a$out >&2)
27date=date
28echo "Build of $proj started at "$date
29time /coverity/cov-analysis-linux64-7.6.1/bin/cov-build --dir cov-out/$proj \ 30 lbn-build$nounpack -j32 --projects $proj lhcb-gaudi-head 31echo "Build of$proj started at "$date 32echo "Build of$proj ended   at "date

The lbn-build command unpacks all tar.bz2 archives into the build directory. To avoid unpacking more than once, all builds after the first build (that of Gaudi) are started with the --no-unpack option.

The output of the cov-build command is stored in per-project subdirectories of cov-out.

Note that ccache should NOT be enabled in the build procedure for Coverity scans. If ccache is used and a source code file is not rebuilt because it is already cached, Coverity will handle this as if the source code had been fully removed or, equivalently, as if any defects in that file had been fixed.

Note also that you will need an AFS token to execute the above script, otherwise you will several errors at various points. This issue should be kept in mind when implementing automatic nightly Coverity scans (see LBCORE-1058); one possibilty could be to make sure that the script above does not require AFS tokens.

Note finally that the cov-build step does not perform any static code analysis of defects: this step only performs the full build of the software and keeps track of all build commands executed, so that the relevant build environment and options can be used in the following cov-analyze step. As the builds of the LHCb software are performed using cmake, it would be useful to investigate if the cov-build step may be bypassed and replaced by an equivalent step based on CMAKE_EXPORT_COMPILE_COMMANDS (see LBCORE-1057).

## Analyze build results (cov-analyze)

The results of the builds are then analyzed using the analyze.sh script, a wrapper over cov-analyze:

    1#!/bin/bash -e
2dir=basename $PWD 3if [ "$dir" != "LbNightlyTools" ]; then
4  echo "ERROR! Please run this script from LbNightlyTools"; exit 1
5fi
6if [ "$1" == "" ] || [ "$2" != "" ]; then
7  echo "Usage: $0 <project>" 8 echo "Example:$0 Gaudi"
9  exit 1
10fi
11proj=$1 12if [ ! -d$PWD/build ]; then
13  echo "ERROR! Directory build not found"; exit 1
14fi
15if [ ! -d cov-out/$proj ]; then 16 echo "ERROR! Please build the code first!"; exit 1 17 exit 1 18fi 19\rm -rf cov-out/$proj/output;
20mkdir -p logs
21out=logs/log-${proj}-analyze.txt 22\rm -rf$out
23echo "======================================================================"
24# See http://stackoverflow.com/questions/3173131
25echo "Output copied to logfile $out" 26exec > >(tee -a$out)
27exec 2> >(tee -a $out >&2) 28date=date 29echo "Analyze of$proj started at "$date 30time /coverity/cov-analysis-linux64-7.6.1/bin/cov-analyze --dir cov-out/$proj \
31  --all --enable-constraint-fpp --enable-fnptr --enable-single-virtual --force
32echo "Analyze of $proj started at "$date
33echo "Analyze of $proj ended at "date Note that, to avoid analyzing code repeatedly, Coverity can build models of already analyzed code (using the cov-collect-models), i.e. dump the information for already analyzed code fragments in xmldb files. Eventually, it may be useful to implement analysis models in the LHCb Coverity scans, to speed them up further (see LBCORE-1060). ## Commit analysis results to the Coverity Connect server (cov-commit-defects) Finally, the results of the analysis are committed to the Coverity Connect server using the commit.sh script, a wrapper over cov-commit-defects:  1#!/bin/bash -e 2dir=basename$PWD
3if [ "$dir" != "LbNightlyTools" ]; then 4 echo "ERROR! Please run this script from LbNightlyTools"; exit 1 5fi 6if [ "$1" == "" ] || [ "$2" != "" ]; then 7 echo "Usage:$0 <project>"
8  echo "Example: $0 Gaudi" 9 exit 1 10fi 11proj=$1
12stream=LHCb-${proj}-Stream # eg LHCb-Gaudi-Stream 13stripall= 14for proj1 in Gaudi LHCb Lbcom Rec Boole Phys Hlt Analysis Stripping DaVinci Gean 15t4 Gauss; do 16 PROJ1=echo$proj1 | awk '{print toupper($0)}' 17 # NB stripped paths will start by "/" even if strip prefixes end in "/" 18 # e.g. add /builda/LHCb/LbNightlyTools/build/GAUDI 19 stripall="$stripall --strip-path $PWD/build/$PROJ1"
20done
21if [ ! -d cov-out/$proj ]; then 22 echo "ERROR! Please build the code first!"; exit 1 23 exit 1 24fi 25if [ ! -d cov-out/$proj/output ]; then
26  echo "ERROR! Please analyze the code first!"; exit 1
27  exit 1
28fi
29# Set environment variable COVERITY_PASSPHRASE to authenticate
30# See /coverity/cov-analysis-linux64-7.6.1/bin/cov-commit-defects --help
31if [ -f ../scripts/setup.sh ]; then . ../scripts/setup.sh; fi
32mkdir -p logs
33out=logs/log-${proj}-commit.txt 34\rm -rf$out
35echo "======================================================================"
36# See http://stackoverflow.com/questions/3173131
37echo "Output copied to logfile $out" 38exec > >(tee -a$out)
39exec 2> >(tee -a $out >&2) 40date=date 41echo "Commit of$proj started at "$date 42time /coverity/cov-analysis-linux64-7.6.1/bin/cov-commit-defects \ 43 --dir cov-out/$proj --host lcgapp10.cern.ch --port 8080 --user admin \
44  --stream $stream$stripall --strip-path /afs/cern.ch/sw/lcg/releases \
45  --strip-path /cvmfs/sft.cern.ch/lcg/releases \
46  --strip-path /cvmfs/lhcb.cern.ch/lib/lcg/releases
47date
48echo "Commit of $proj started at "$date
49echo "Commit of \$proj ended   at "date

# 3. Interact with the Coverity Connect server to triage defects (developer operations)

From a user (application manager and/or power developer) perspective, interacting with the Coverity Connect server to analyze and triage defects is the most interesting aspect of the Coverity infrastructure.

## Select which defects to view

To use the Coverity Connect server, simply connect to https://coverity.cern.ch.

In the red project pulldown menu in the top left corner of the GUI, select LHCb as the Coverity project you want to work with.

Click on the three horizontal white lines in the top left corner of the GUI, just below the Coverity log and the project pulldown menu: the view panel will now slide into the GUI from the left side.

For LHCb users belonging to the lhcb-coverity-users-local group, the view panel will look like the following (in the meantime, Geant4 and Gauss views have also been added):

The view panel allows you to choose the specific view you want to work with, within the selected LHCb project. The view that is currently used (in the example above, the "LHCb-05-Boole" view within the "Issues: By snapshot" view type) is displayed in white bold characters in the view panel and is also printed on the top left corner, below the project pulldown menu.

For LHCb users, the view panel will normallly display both standard views and LHCb-specific views: the latter are displayed in italics, because they are private views of one user, shared with the LHCb group.

The view panel is largely configurable by each user. If the LHCb specific shared views do not appear in your interface, click on "Layout Preferences" in the menu next to "Issues by Snapshot", and in the "Layout" tab select "Display all views shared with you".

Some of the views you may find most relevant for your work include:

• the standard "High impact outstanding" view, showing only defects classified as High-impact (and not yet fixed or dismissed), from all streams in the LHCb project
• the shared "All issues in last snapshot" view, showing all defects from all LHCb streams (irrespective of impact and including also defects triaged and dismissed as fake or irrelevant)
• the shared "LHCb-05-Boole" view and its siblings, showing all defects from one of the twelve LHCb streams (again, irrespective of impact and including also dismissed defects)

You can also add your own view. For instance, you may be interested in defects in the Boole stream, but only want to list outstanding ones, excluding those that have been dismissed as irrelevant. To create a new view, click on "Add New View" in the menu next to "Issues by Snapshot"; alternatively, you may also click on "Edit Settings" for an existing view, and then select "Save as a Copy" to create a clone of that view, which you may later modify.

As mentioned above, note that streams and components are not the same thing. A defect appearing in the "LHCb-05-Boole" view belongs to the BOOLE stream, but it can belong to the GAUDI component, if it was found in a Gaudi header used by a Boole class.

The shared "LHCb-OtherComponent" view is a control view, which should normally be empty if every defect is assigned to an LHCb or external component. If it is not empty, please report it as this means that the LHCb component map must be modified.

## Triage defects

Selecting the appropriate view will allow you to list and start working on individual defects in the LHCb software.

Clicking on an individual defect in the top left window will pop up the source code for that defect in the bottom left window and the metadata for that defect in the window to the right of the GUI.

The following is an example from the Gaudi stream:

In this example, defect 64252 in the build of Gaudi originates from the gcc hashtable.h header, and has therefore been assigned to the "external (gcc)" component during the Coverity scan. As this defect cannot be fixed in the Gaudi source code, a Gaudi developer has dismissed this defect as "Ignore".

For another defect that points to a real bug in the Gaudi code, the workflow could for instance be the following: triage the defect as "Fix required" initially; later on, commit a bug fix and triage the defect as "Fix submitted"; finally, cross check that the defect has disappeared from the Coverity server after the next scan.

# 4. Status of Coverity defects for the LHCb software projects

For reference, the total number of defects found during the first full scan on 29 Feb 2016 was the following:

grep Defect ls -tr log-*-analyze.txt
log-Gaudi-analyze.txt:Defect occurrences found       : 1352 Total
log-LHCb-analyze.txt:Defect occurrences found       : 2272 Total
log-Lbcom-analyze.txt:Defect occurrences found       : 626 Total
log-Rec-analyze.txt:Defect occurrences found       : 1604 Total
log-Boole-analyze.txt:Defect occurrences found       : 520 Total
log-Phys-analyze.txt:Defect occurrences found       : 895 Total
log-Hlt-analyze.txt:Defect occurrences found       : 973 Total
log-Analysis-analyze.txt:Defect occurrences found       : 920 Total
log-Stripping-analyze.txt:Defect occurrences found       : 518 Total
log-DaVinci-analyze.txt:Defect occurrences found       : 701 Total
log-Geant4-analyze.txt:Defect occurrences found       : 1132 Total
log-Gauss-analyze.txt:Defect occurrences found       : 1487 Total

-- AndreaValassi - 2016-02-18

Topic revision: r14 - 2017-06-29 - AndreaValassi

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# Workshop: Geometric Aspects of Flexible Stability

2022.03.21. - 2022.03.25.
Erdős Center

### Description

The flexible stability is a relatively new notion of rigidity for countable sofic groups. Roughly speaking, a sofic group is flexibly stable if all its sofic approximations are constructed by taking unions of finite Schreier graphs and adding a little bit of noise. The subject of the workshop will be centered around the following question: Suppose G is a fundamental group of a Riemannian manifold M. How can we use the geometry of M to study the flexible stability of G?

Each participant is encouraged to give a lecture on a topic of his/her choice. We plan to devote at least half of the workshop to discussions and joint research.

• Oren Becker
• Michael Chapman
• Alon Dogon
• Mikael de la Salle
• Mikolaj Fraczyk
• Maria Gerasimova
• Lukas Gohla
• Lukasz Grabowski
• Izhar Oppenheim
• Arie Levit
• Alex Lubotzky
• Bharatram Rangarajan
• Héctor Jardón Sánchez
• Raz Slutsky
• Itamar Vigdorovich

### Schedule

Monday 21.03
10:00–11:00    Arie Levit                   Surface groups are flexibly stable
11:10–12:10    Mikolaj Fraczyk        On certain branched covers of higher rank locally symmetric spaces
12:10–14:10    Lunch break
14:10-15:10     Mikael de la Salle     Spectral gap and Hilbert-Schmidt (flexible) stability
15:40-∞            discussion time

Tuesday 22.03
10:00–11:00    Michael Chapman    Efficient stability for finite groups
11:10–12:10    Lukasz Grabowski   Stability with respect to the rank metric
12:10-∞           free time

Wednesday 23.03
10:00–11:00    Raz Slutsky              The space of traces of the free group and other free products
11:30–12:30    Itamar Vigdorovich  Studying HS-stability through characters and dynamics
12:30–14:10    Lunch Break
14:10–15:10    Alon Dogon              Flexible HS-stability, projective representations and Connes-embeddable groups
15:40–∞          discussion time

Thursday 24.03
10:00–11:00    Alex Lubotzky           Good locally testable codes
11:10–12:10    Bharatram Rangarajan    Cohomological Techniques in Uniform Stability
12:10–14:10    Lunch Break
14:10–15:10    Izhar Oppenheim     Local criteria for vanishing of Banach cohomology
15:40–∞          discussion time

Friday 25.03
10:00–11:00    Lukas Gohla            TBA
11:10–12:10    Oren Becker            Testability of relations between permutations
12:10–∞

### Abstracts

Oren Becker Testability of relations between permutations

Let $$A$$ and $$B$$ be permutations in $$S_n$$, such that either (*) $$AB=BA$$, or (**) the pair $$(A,B)$$ is far from every pair of permutations $$(A',B')$$ satisfying $$A'B'=B'A'$$. Is there a probabilistic algorithm that distinguishes, with high probability of success, between Case (*) and Case (**) by reading only $$k$$ entries of $$A$$ and $$B$$, for $$k$$ independent of $$n$$? In other words, is the equation $$XY=YX$$ testable in permutations? What about other equations, such as $$XY^2=Y^2 X$$ or $$XY^2=Y^3 X$$? What about simultaneous systems of equations? Problems of this sort belong to the field of Property Testing. I will explain the connection to stability, and introduce a relaxation of stability, called Benjamini-Schramm rigidity, and its applications to the algorithmic problem described above. Based on joint work with Alex Lubotzky and Jonathan Mosheiff.

Michael Chapman Efficient stability for finite groups

We define quantitative notions of stability for finite groups and relate them to property testing algorithms. We then discuss some results in this direction and possible applications of such results for group hardness of approximation.

Michael de la Salle Spectral gap and Hilbert-Schmidt (flexible) stability

I will explain a very simple idea to use spectral gap estimates for questions of stability in normalized Hilbert-Schmidt norm. In its most elementary form, this idea allows us to prove that the direct product of two (flexibly) stable groups is again (flexibly) stable, provided that one of the groups has Kazhdan's property (T). This property (T) assumption is necessary, as Adrian Ioana has proved that the direct product of two non-abelian free groups is not stable. This idea is also applicable for finite groups, and this allow to greatly simplify some of the stability results that have been proved by Ji-Natarajan-Vidick-Yuen-Wright in their announced resolution of the Connes embedding problem.

Alon Dogon Flexible HS-stability, projective representations and Connes-embeddable groups

Flexible HS-stability is a rigidity property for sequences of approximate unitary representations of a group, asking that all such are close to sequences of compressions of genuine representations (w.r.t. the Hilbert Schmidt norm). Recently, in the work of Ioana, Spaas and Wiersma, a novel way of refuting this property was discovered. The method is based on (relative) property (T) being an obstruction to having certain non-trivial approximately multiplicative projective unitary representations. We will present their work, using $${\rm SL}_2(\mathbb Z)$$ semidirect product $$\mathbb Z^2$$ as an example. We will comment on ongoing work of the author, which utilizes this method to show flexible HS-stability implies the existence of non Connes-embeddable groups in certain situations.

Mikolaj Fraczyk On certain branched covers of higher rank locally symmetric spaces

A finitely generated group $$\Gamma$$ is flexibly stable if all its sofic approximations are, up to a small modification, disjoint unions of coset graphs of $$\Gamma$$. There is a Riemannian geometric analogue of this property for a symmetric space $$X$$: suppose that $$M$$ is a Riemannian manifold which locally looks like $$X$$ in most places. Then there exists a disjoint union $$Y$$ of finite volume quotients of $$X$$, such that after cutting out small subsets of $$M$$ and $$Y$$, they become isometric. Such a property is probably very hard to prove any higher rank symmetric space. A good place to start is the case when $$M$$ is a branched cover of a locally symmetric space with very small branching locus. In my talk I will explain a work in progress, where I show that at least certain very specific constructions of cyclic branched covers of higher rank locally symmetric spaces will never yield a counter-example to flexible stability in higher rank.

Lukasz Grabowski Stability with respect to the rank metric

This talk is concerned by stability with respect to the rank metric. This topic is motivated by the stability with respect to the Hamming distance, as studied by e.g. Arzhantseva-Paunescu and Becker-Lubotzky-Thom. In the talk, I will review some parts of the work by myself and Elek about rank stability for abelian groups, and by Becker-Lubotzky-Thom on stability with respect to the Hamming distance for polycyclic groups. This will allow me to explain what are the obstacles to obtain similar results for the rank distance for polycyclic groups, and the progress towards solving them (joint work with Joan Claramunt). Those latter resutls are essentially about analogues of effective Nullstellensatz and effective ideal membership problems for the group rings of polycyclic groups.

Arie Levit Surface groups are flexibly stable

I will describe a geometric proof that surface groups are flexibly stable relying on branched covers and negative curvature. I will attempt to survey the main ideas used in the proof. This talk is based on a joint work with Nir Lazarovich and Yair Minsky.

Alex Lubotzky Good locally testable codes

An error-correcting code is locally testable (LTC)  if there is a random tester that reads only a small number of bits of a given word and decides whether the word is in the code, or at least close to it.  A long-standing problem asks if there exists such a code that also satisfies the golden standards of coding theory: constant rate and constant distance.
Unlike the classical situation in coding theory, random codes are not LTC, so this problem is a challenge of a new kind. We construct such codes based on what we call (Ramanujan) Left/Right Cayley square complexes. These are 2-dimensional versions of the expander codes constructed by Sipser and Spielman (1996). The main result and lecture will be self-contained. But we hope also to explain how the seminal work Howard Garland ( 1972) on the cohomology of quotients of the Bruhat-Tits buildings of p-adic Lie group has led to this construction ( even though, it is not used at the end). Based on joint work with I. Dinur, S. Evra, R. Livne, and S. Mozes

Izhar Oppenheim Local criteria for vanishing of Banach cohomology

The aim of this lecture is to introduce local criteria for vanishing of Banach cohomology for groups acting on simplicial complexes. The main idea is to generalize the machinery of Garland to Banach spaces. Such generalizations are available in two flavors depending whether the simplicial complex is colorable or not. If time (and interest) permit, we will also discuss applications for $$L^p$$ fixed point spectrum of random groups.

Bharatram Rangarajan Cohomological Techniques in Uniform Stability

In an ongoing joint work with Glebsky, Lubotzky and Monod, we construct an analogue of bounded cohomology in an asymptotic setting in order to prove uniform stability of lattices in Lie groups (of rank at least two) with respect to unitary groups equipped with a metric induced by a submultiplicative norm. The main idea is the notion of "defect diminishing", which allows us to reduce stability as a homomorphism lifting problem with abelian kernel, and relates to an asymptotic bounded cohomology $$H_a^2$$ whose vanishing implies uniform stability. If time permits, I shall sketch a proof of the main result: namely, the vanishing of $$H_a^2(G,V)$$  for lattices in high rank Lie groups (along the lines of the proof in Burger-Monod that bounded cohomology vanishes for such groups), implying uniform stability for this class of groups.

Raz Slutsky The space of traces of the free group and other free products

After introducing the notion of traces and characters on groups and their relation to classical rigidity results, Invariant Random Subgroups (IRS) and Hilbert-Schmidt Stability, we will show that the space of traces of the free group $$F_n$$ is a Poulsen simplex for every $$2 \leq n \leq \infty$$, that is, the set of extreme points is dense. We will then discuss the space of traces of other free products. This is based on joint work with Joav Orovitz and Itamar Vigdorovich.

Itamar Vigdorovich Studying HS-stability through characters and dynamics

We study the relation between Hilbert-Schmidt stability of amenable groups and a certain finite approximation property of dynamical systems that are naturally associated with the group.  For example, we find that HS-stability of polycyclic groups happens to be equivalent to a purely dynamical question regarding toral dynamics, and is thus related to classical dynamical conjectures; another example, is that HS-stability of wreath products $$\mathbb Z\wr \mathbb A$$ boils down to standard dynamical properties of Bernoulli shifts. In this talk I'll elaborate on this connection which is based on the work of Hadwin and Shulman regarding group characters, and explain how it can be used to derive new positive and negative results in stability. The talk is based on a joint work with Arie Levit

Mikolaj Fraczyk

### Participants

Lukasz Grabowski
Oren Becker
Michael Chapman
Alon Dogon
Mikael de la Salle
Maria Gerasimova
Lukas Gohla
Izhar Oppenheim
Arie Levit
Alex Lubotzky
Bharatram Rangarajan
Héctor Jardón Sánchez
Raz Slutsky
Itamar Vigdorovich


text:
Documentation

# boxcox

Box-Cox transformation

`boxcox` has been partially removed and will no longer accept a `fints` object (`tsobj`).

Replace all instances of a `fints` object (`tsobj`) for input with an array by using `fts2timetable` to convert a `fints` object to a `timetable` object and then use `timetable2table` and `table2array`.

## Syntax

```[transdat,lambda] = boxcox(data)
[transfts,lambda] = boxcox(tsobj)
transdat = boxcox(lambda,data)
transfts = boxcox(lambda,tsobj)
```

## Arguments

 `data` Data vector. Must be positive and specified as a column data vector. `tsobj` Financial time series object.

## Description

`boxcox` transforms nonnormally distributed data to a set of data that has approximately normal distribution. The Box-Cox transformation is a family of power transformations.

If λ is not = `0`, then

`$data\left(\lambda \right)=\frac{dat{a}^{\lambda }-1}{\lambda }$`

If λ is = `0`, then

`$data\left(\lambda \right)=\mathrm{log}\left(data\right)$`

The logarithm is the natural logarithm (log base e). The algorithm calls for finding the λ value that maximizes the Log-Likelihood Function (LLF). The search is conducted using `fminsearch`.

`[transdat,lambda] = boxcox(data)` transforms the data vector `data` using the Box-Cox transformation method into `transdat`. It also estimates the transformation parameter λ.

`[transfts,lambda] = boxcox(tsojb)` transforms the financial time series object `tsobj` using the Box-Cox transformation method into `transfts`. It also estimates the transformation parameter λ.

If the input data is a vector, `lambda` is a scalar. If the input is a financial time series object, `lambda` is a structure with fields similar to the components of the object; for example, if the object contains series names `Open` and `Close`, `lambda` has fields `lambda.Open` and `lambda.Close`.

`transdat = boxcox(lambda, data)` and ```transfts = boxcox(lambda, tsobj)``` transform the data using a certain specified λ for the Box-Cox transformation. This syntax does not find the optimum λ that maximizes the LLF.

## Examples

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Use `boxcox` to transform the data series contained in a financial time series object into another set of data series with relatively normal distributions.

Create a financial time series object from the supplied `whirlpool.dat` data file.

`whrl = ascii2fts('whirlpool.dat', 1, 2, []);`
```Warning: FINTS will be removed in a future release. Use TIMETABLE instead. For more information, see <a href="matlab:web(fullfile(docroot, 'finance/convert-from-fints-to-timetables.html'))">Convert Financial Time Series Objects (fints) to Timetables</a>. ```

Fill any missing values denoted with `NaN`'s in whrl with values calculated using the linear method.

`f_whrl = fillts(whrl);`
```Warning: FINTS will be removed in a future release. Use TIMETABLE instead. For more information, see <a href="matlab:web(fullfile(docroot, 'finance/convert-from-fints-to-timetables.html'))">Convert Financial Time Series Objects (fints) to Timetables</a>. ```

Transform the nonnormally distributed filled data series `f_whrl` into a normally distributed one using Box-Cox transformation.

`bc_whrl = boxcox(f_whrl);`
```Warning: FINTS will be removed in a future release. Use TIMETABLE instead. For more information, see <a href="matlab:web(fullfile(docroot, 'finance/convert-from-fints-to-timetables.html'))">Convert Financial Time Series Objects (fints) to Timetables</a>. ```

Compare the result of the `Close` data series with a normal (Gaussian) probability distribution function and the nonnormally distributed `f_whrl`.

```subplot(2, 1, 1); hist(f_whrl.Close);```
```Warning: FINTS will be removed in a future release. Use TIMETABLE instead. For more information, see <a href="matlab:web(fullfile(docroot, 'finance/convert-from-fints-to-timetables.html'))">Convert Financial Time Series Objects (fints) to Timetables</a>. ```
```grid; title('Nonnormally Distributed Data'); subplot(2, 1, 2); hist(bc_whrl.Close);```
```Warning: FINTS will be removed in a future release. Use TIMETABLE instead. For more information, see <a href="matlab:web(fullfile(docroot, 'finance/convert-from-fints-to-timetables.html'))">Convert Financial Time Series Objects (fints) to Timetables</a>. ```
`grid; title('Box-Cox Transformed Data');`

The bar chart on the top represents the probability distribution function of the filled data series, `f_whrl`, which is the original data series `whrl` with the missing values interpolated using the linear method. The distribution is skewed toward the left (not normally distributed). The bar chart on the bottom is less skewed to the left. If you plot a Gaussian probability distribution function (PDF) with similar mean and standard deviation, the distribution of the transformed data is very close to normal (Gaussian). When you examine the contents of the resulting object `bc_whrl`, you find an identical object to the original object `whrl` but the contents are the transformed data series.