text:
# Gini coefficient

File:Economics Gini coefficient.svg
Graphical representation of the Gini coefficient
(The area of the whole triangle is defined as 1.)

The Gini coefficient is a measure of statistical dispersion most prominently used as a measure of inequality of income distribution or inequality of wealth distribution. It is defined as a ratio with values between 0 and 1: the numerator is the area between the Lorenz curve of the distribution and the uniform distribution line; the denominator is the area under the uniform distribution line. Thus, a low Gini coefficient indicates more equal income or wealth distribution, while a high Gini coefficient indicates more unequal distribution. 0 corresponds to perfect equality (everyone having exactly the same income) and 1 corresponds to perfect inequality (where one person has all the income, while everyone else has zero income). The Gini coefficient requires that no one have a negative net income or wealth.

The Gini coefficient was developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variabilità e mutabilità" ("Variability and Mutability").

The Gini coefficient is also commonly used for the measurement of the discriminatory power of rating systems in credit risk management.

The Gini index is the Gini coefficient expressed as a percentage, and is equal to the Gini coefficient multiplied by 100. (The Gini coefficient is equal to half of the relative mean difference.)

## Calculation

The Gini coefficient is defined as a ratio of the areas on the Lorenz curve diagram. If the area between the line of perfect equality and Lorenz curve is A, and the area under the Lorenz curve is B, then the Gini coefficient is A/(A+B). Since A+B = 0.5, the Gini coefficient, G = A/(.5) = 2A = 1-2B. If the Lorenz curve is represented by the function Y = L(X), the value of B can be found with integration and:

${\displaystyle G=1-2\,\int _{0}^{1}L(X)dX}$

In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example:

• For a population uniform on the values yi, i = 1 to n, indexed in non-decreasing order ( yiyi+1):
${\displaystyle G={\frac {1}{n}}\left(n+1-2\left({\frac {\Sigma _{i=1}^{n}\;(n+1-i)y_{i}}{\Sigma _{i=1}^{n}y_{i}}}\right)\right)}$
• For a discrete probability function f(y), where yi, i = 1 to n, are the points with nonzero probabilities and which are indexed in increasing order ( yi < yi+1):
${\displaystyle G=1-{\frac {\Sigma _{i=1}^{n}\;f(y_{i})(S_{i-1}+S_{i})}{S_{n}}}}$
where:
${\displaystyle S_{i}=\Sigma _{j=1}^{i}\;f(y_{j})\,y_{j}\,}$ and ${\displaystyle S_{0}=0\,}$
${\displaystyle G=1-{\frac {1}{\mu }}\int _{0}^{\infty }(1-F(y))^{2}dy}$

Since the Gini coefficient is half the relative mean difference, it can also be calculated using formulas for the relative mean difference.

For a random sample S consisting of values yi, i = 1 to n, that are indexed in non-decreasing order ( yiyi+1), the statistic:

${\displaystyle G(S)={\frac {1}{n-1}}\left(n+1-2\left({\frac {\Sigma _{i=1}^{n}\;(n+1-i)y_{i}}{\Sigma _{i=1}^{n}y_{i}}}\right)\right)}$

is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like the relative mean difference, there does not exist a sample statistic that is in general an unbiased estimator of the population Gini coefficient. Confidence intervals for the population Gini coefficient can be calculated using bootstrap techniques.

Sometimes the entire Lorenz curve is not known, and only values at certain intervals are given. In that case, the Gini coefficient can be approximated by using various techniques for interpolating the missing values of the Lorenz curve. If ( X k , Yk ) are the known points on the Lorenz curve, with the X k indexed in increasing order ( X k - 1 < X k ), so that:

• Xk is the cumulated proportion of the population variable, for k = 0,...,n, with X0 = 0, Xn = 1.
• Yk is the cumulated proportion of the income variable, for k = 0,...,n, with Y0 = 0, Yn = 1.

If the Lorenz curve is approximated on each interval as a line between consecutive points, then the area B can be approximated with trapezoids and:

${\displaystyle G_{1}=1-\sum _{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})}$

is the resulting approximation for G. More accurate results can be obtained using other methods to approximate the area B, such as approximating the Lorenz curve with a quadratic function across pairs of intervals, or building an appropriately smooth approximation to the underlying distribution function that matches the known data. If the population mean and boundary values for each interval are also known, these can also often be used to improve the accuracy of the approximation.

## Income Gini coefficients in the world

A complete listing is in list of countries by income equality; the article economic inequality discusses the social and policy aspects of income and asset inequality.

File:World Map Gini coefficient.svg
Gini coefficient, income distribution by country.
 < 0.25   0.25–0.29 0.30–0.34   0.35–0.39   0.40–0.44 0.45–0.49   0.50–0.54   0.55–0.59   ≥ 0.60 N/A

While most developed European nations tend to have Gini coefficients between 0.24 and 0.36, the United States Gini coefficient is above 0.4, indicating that the United States has greater inequality. Using the Gini can help quantify differences in welfare and compensation policies and philosophies. However it should be borne in mind that the Gini coefficient can be misleading when used to make political comparisons between large and small countries (see criticisms section).

The Gini coefficient for the entire world has been estimated by various parties to be between 0.56 and 0.66.[1][2]

File:Gini since WWII.gif
Gini coefficients, income distribution over time for selected countries

### Correlation with per-capita GDP

Poor countries (those with low per-capita GDP) have Gini coefficients that fall over the whole range from low (0.25) to high (0.71), while rich countries have generally intermediate Gini coefficient (under 0.40). Generally, the lowest Gini coefficients can be found in the Scandinavian countries, in the recently ex-socialist countries of Eastern Europe and in Japan.

### US income gini coefficients over time

Gini coefficients for the United States at various times, according to the US Census Bureau:

• 1967: 0.397 (first year reported)
• 1968: 0.386 (lowest coefficient reported)
• 1970: 0.394
• 1980: 0.403
• 1990: 0.428
• 2000: 0.462
• 2005: 0.469 (most recent year reported; highest coefficient reported)[3]

Between 1968 and 2005, the Gini coefficient fell in only seven years. Some argue this rise corresponds to the lowering of the highest tax bracket, for example, from 70% in the 1960s to 35% by 2000. However, many other variables that could affect the Gini coefficient have changed during this period as well. For example, much technological progress has occurred, eliminating formerly middle-class factory jobs in favor of the service sector; additionally, the economy has shifted towards professions that require higher education.

## Advantages of Gini coefficient as a measure of inequality

• It can be used to compare income distributions across different population sectors as well as countries, for example the Gini coefficient for urban areas differs from that of rural areas in many countries (though the United States' urban and rural Gini coefficients are nearly identical).
• It is sufficiently simple that it can be compared across countries and be easily interpreted. GDP statistics are often criticised as they do not represent changes for the whole population; the Gini coefficient demonstrates how income has changed for poor and rich. If the Gini coefficient is rising as well as GDP, poverty may not be improving for the majority of the population.
• The Gini coefficient can be used to indicate how the distribution of income has changed within a country over a period of time, thus it is possible to see if inequality is increasing or decreasing.
• The Gini coefficient satisfies four important principles:
• Anonymity: it does not matter who the high and low earners are.
• Scale independence: the Gini coefficient does not consider the size of the economy, the way it is measured, or whether it is a rich or poor country on average.
• Population independence: it does not matter how large the population of the country is.
• Transfer principle: if income (less than the difference), is transferred from a rich person to a poor person the resulting distribution is more equal.

## Disadvantages of Gini coefficient as a measure of inequality

• The Gini coefficient of different sets of people cannot be averaged to obtain the Gini coefficient of all the people in the sets: if a Gini coefficient were to be calculated for each person it would always be zero. When measuring its value for a large, economically diverse country, a much higher coefficient than each of its regions has individually will result.
For this reason the scores calculated for individual countries within the EU are difficult to compare with the score of the entire US: the overall value for the EU should be used in that case, 31.3[4], which is still much lower than the United States', 45[5]. Using decomposable inequality measures (e.g. the Theil index ${\displaystyle T}$ converted by ${\displaystyle 1-{e^{-T}}}$ into a inequality coefficient) averts such problems.
• The Lorenz curve may understate the actual amount of inequality if richer households are able to use income more efficiently than lower income households. From another point of view, measured inequality may be the result of more or less efficient use of household incomes.
• Economies with similar incomes and Gini coefficients can still have very different income distributions. This is because the Lorenz curves can have different shapes and yet still yield the same Gini coefficient. As an extreme example, an economy where half the households have no income, and the other half share income equally has a Gini coefficient of ½; but an economy with complete income equality, except for one wealthy household that has half the total income, also has a Gini coefficient of ½. In practice, such distributions do not exist, and therefore, the impact of different but realistic curves is less obvious.

## Problems in using the Gini coefficient

• Gini coefficients do include income gained from wealth; however, the Gini coefficient is used to measure net income more than net worth, which can be misinterpreted. For example, Sweden has a low Gini coefficient for income distribution but a high Gini coefficient for wealth (5% of Swedish household shareholders hold 77% of the share value owned by households)[6]. In other words and as a normative statement: The Gini coefficient should be interpreted as measuring effective egalitarianism; and distribution of stock ownership does not appear to correlate to many recognized indicators of egalitarianism.
• Too often only the Gini coefficient is quoted without describing the proportions of the quantiles used for measurement. As with other inequality coefficients, the Gini coefficient is influenced by the granularity of the measurements. For example, five 20% quantiles (low granularity) will usually yield a lower Gini coefficient than twenty 5% quantiles (high granularity) taken from the same distribution. This is an often encountered problem with measurements.
• Care should be taken in using the Gini coefficient as a measure of egalitarianism, as it is properly a measure of income dispersion. Two equally egalitarian countries with different immigration policies may have different Gini coefficients.

## General problems of measurement

• Comparing income distributions among countries may be difficult because benefits systems may differ. For example, some countries give benefits in the form of money while others give food stamps, which may not be counted[citation needed] as income in the Lorenz curve and therefore not taken into account in the Gini coefficient.
• The measure will give different results when applied to individuals instead of households. When different populations are not measured with consistent definitions, comparison is not meaningful.
• As for all statistics, there may be systematic and random errors in the data. The meaning of the Gini coefficient decreases as the data become less accurate. Also, countries may collect data differently, making it difficult to compare statistics between countries.

As one result of this criticism, in addition to or in competition with the Gini coefficient entropy measures are frequently used (e.g. the Atkinson and Theil indices). These measures attempt to compare the distribution of resources by intelligent agents in the market with a maximum entropy random distribution, which would occur if these agents acted like non-intelligent particles in a closed system following the laws of statistical physics.

## Notes

1. [1]
2. [2]
3. Note that the calculation of the index for the United States was changed in 1992, resulting in an upwards shift of about 0.02 in the coefficient. Comparisons before and after that period may be misleading. (Data from the US Census Bureau.)
4. [https://www.cia.gov/library/publications/the-world-factbook/geos/ee.html CIA World Factbook—The European Union
5. CIA World Factbook—The United States
6. (Data from the Statistics Sweden.)

## References

• Amiel, Y.; Cowell, F.A. (1999). Thinking about Inequality. Cambridge.
• Anand, Sudhir (1983). Inequality and Poverty in Malaysia. New York: Oxford University Press.
• Brown, Malcolm (1994). "Using Gini-Style Indices to Evaluate the Spatial Patterns of Health Practitioners: Theoretical Considerations and an Application Based on Alberta Data". Social Science Medicine. 38: 1243–1256.
• Chakravarty, S. R. (1990). Ethical Social Index Numbers. New York: Springer-Verlag.
• Dixon, PM, Weiner J., Mitchell-Olds T, Woodley R. (1987). "Bootstrapping the Gini coefficient of inequality". Ecology. 68: 1548–1551.
• Dorfman, Robert (1979). "A Formula for the Gini Coefficient". The Review of Economics and Statistics. 61: 146–149.
• Gastwirth, Joseph L. (1972). "The Estimation of the Lorenz Curve and Gini Index". The Review of Economics and Statistics. 54: 306–316.
• Gini, Corrado (1912). "Variabilità e mutabilità" Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Veschi (1955).
• Gini, Corrado (1921). "Measurement of Inequality and Incomes". The Economic Journal. 31: 124–126.
• Mills, Jeffrey A.; Zandvakili, Sourushe (1997). "Statistical Inference via Bootstrapping for Measures of Inequality". Journal of Applied Econometrics. 12: 133–150.
• Morgan, James (1962). "The Anatomy of Income Distribution". The Review of Economics and Statistics. 44: 270–283.
• Xu, Kuan (January, 2004). "How Has the Literature on Gini's Index Evolved in the Past 80 Years?". Department of Economics, Dalhousie University. Retrieved on 2006-06-01. The Chinese version of this paper appears in Xu, Kuan (2003). "How Has the Literature on Gini's Index Evolved in the Past 80 Years?". China Economic Quarterly. 2: 757–778.


text:
0
Research Papers

# Angular Velocity Estimation From the Angular Acceleration Matrix

[+] Author and Article Information
Philippe Cardou

Centre for Intelligent Machines, Department of Mechanical Engineering, McGill University, Room 461, Macdonald Engineering Building, 817 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canadapcardou@cim.mcgill.ca

Jorge Angeles

Centre for Intelligent Machines, Department of Mechanical Engineering, McGill University, Room 461, Macdonald Engineering Building, 817 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

$CPM(v)$ is defined as $∂(v×x)∕∂x$, for any $x,v∊R3$.

The axial vector $vect(A)$ of matrix $A∊R3×3$ is defined such that $vect(A)×u=(1∕2)(A−AT)u$, for any vector $u∊R3$.

It is common practice to find the null space of a matrix using the singular-value decomposition (SVD). However, algorithms that perform SVD are iterative, which is undesirable for angular-velocity estimation in real time, even for a relatively small matrix.

Note that gyroscopes are also subjected to misalignments, but not to position errors.

J. Appl. Mech 75(2), 021003 (Feb 20, 2008) (8 pages) doi:10.1115/1.2775495 History: Received April 17, 2006; Revised June 14, 2007; Published February 20, 2008

## Abstract

Computing the angular velocity $ω$ from the angular acceleration matrix is a nonlinear problem that arises when one wants to estimate the three-dimensional angular velocity of a rigid-body from point-acceleration measurements. In this paper, two new methods are proposed, which compute estimates of the angular velocity from the symmetric part $WS$ of the angular acceleration matrix. The first method uses a change of coordinate frame of $WS$ prior to performing the square-root operations. The new coordinate frame is an optimal representation of $WS$ with respect to the overall error amplification. In the second method, the eigenvector spanning the null space of $WS$ is estimated. As $ω$ lies in this space, and because its magnitude is proportional to the absolute value of the trace of $WS$, it is a simple matter to obtain $ω$. A simulation shows that, for this example, the proposed methods are more accurate than those existing methods that use only centripetal acceleration measurements. Moreover, their errors are comparable to other existing methods that combine tangential and centripetal acceleration measurements. In addition, errors of 2.15% in the accelerometer measurements result in errors of approximately 3% in the angular-velocity estimates. This shows that accelerometers are competitive with angular-rate sensors for motions of the type of the simulated example, provided that position and orientation errors of the accelerometers are accounted for.

<>

## Figures

Figure 1

Rotating uω* onto ûω

Figure 2

A brick rotating freely in space

Figure 3

Angular acceleration estimates

Figure 5

Angular-velocity estimates from the existing CA methods

Figure 6

Angular-velocity estimates from the proposed CA methods

Figure 7

Angular-velocity estimates from the existing TCA methods

Figure 8

Errors on the angular-velocity estimates

Figure 9

Errors on the angular-velocity estimates

Figure 10

Errors on the angular-velocity estimates

Figure 11

Norms of the errors on the angular-velocity estimates

Figure 4

Angular-velocity estimates from the existing TA method

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections

format: plain


text:
Question

# Solve the equation. Give the actual solution and approximate the solution to four decimal places. See the earlier example.$3^x=11$

Solution

Verified
Step 1
1 of 2

$3^{x} = 1$

$\log3^{x}=\log(11)$

$x\log3=\log(11)$

$x= \dfrac{\log(11)}{\log3}$

$x=2.1826583386441 \approx 2.1827$

Since both sides cannot have the same base, apply the logarithm.
Apply the law of exponent to break down the exponent on the left.
Simplify and solve for x for the exact value. Round to the fourth decimal place.


text:
# Are there chiral compounds that don't rotate plane-polarized light?

I know that meso compounds have chiral centers but don't rotate plane-polarized light, and I know that there can be non-traditionally chiral compounds (e.g. ones with large substituents that prohibit much rotation around single bonds) that can still rotate light, but I haven't been able to find any molecules that are chiral but don't rotate plane-polarized light (or rotate it impossibly little). I'm imagining something like an enantiomeric pair connected by one single bond with a slight modification to one that makes it have similar optical activity but not actually be identical (so the final molecule is indeed chiral), but I don't know exactly how to make it work.

• Achiral compounds don't rotate light. – Ivan Neretin Oct 16 '18 at 20:40
• @IvanNeretin My source for that claim was the first answer to this post on Quora. The reasoning seems OK to me, but I'm not knowledgeable enough about stereochemistry to accurately confirm or dispute it. – Carl Schildkraut Oct 16 '18 at 20:41
• @IvanNeretin More precisely (as I suspect the questioner intended) compounds with no chiral centres can be chiral because of restricted rotation or other geometric features eg hexahelicene. – matt_black Oct 16 '18 at 20:56
• Depends on what you mean by "do they rotate?" What if they don't rotate light by a measurable amount? I'm sure I can contrive a number of compounds where that is the case. – Zhe Oct 16 '18 at 21:13
• Take octane and replace one of the hydrogen atoms on the 2-carbon with a deuterium. – Zhe Oct 17 '18 at 11:47

Good question.
There's a phenomenon named cryptochirality[1] (meaning “hidden chirality”), when a compound, though chiral, has practically unmeasurable optical rotation activity.

It can happen to molecules with chiral center(s) bearing very similar substituents. (So, no tricks with bonded slightly modified enantiomeric pairs are needed.)

An example is 5-ethyl-5-propylundecane $$\ce{CH3[CH2]5-\overset{*}{C}(CH2CH3)(CH2CH2CH3)-[CH2]3CH3}$$,[2] don't call it “butyl(ethyl)hexyl(propyl)methane”, found e.g. in beans. Its specific rotation is $$[\alpha] < 0.001$$.

Another, more common example are fats, i.e. triglycerides, $$\ce{RCOOCH2\overset{*}{C}H(OCOR')CH2OCOR''}$$,[1] if containing e.g. only palmitic, oleic and similar long acyls, optical rotation is not demonstrable.[3]

(Related topic is chirality in polymers, see e.g. Q: Chirality on Carbon of PVC molecule.)

References:

1. Mislow K. & Bickart P.: An Epistemological Note on Chirality. Israel Journal of Chemistry 15, 1–6 (1976)
2. Wynberg H., Hekkert G.L., Houbiers J.P.M. & Bosch H.W.: The Optical Activity of Butylethylhexylpropylmethane. Journal of the American Chemical Society 87, 2635–2639 (1965)
3. Schlenk W.: Synthesis and analysis of optically active triglycerides. Journal of the American Oil Chemists’ Society 42, 945–957 (1965)
• Wow this answers a question I've mulled over for years! I had no idea there was a term for this. Thanks! – Nicolau Saker Neto Oct 16 '18 at 23:28
• @NicolauSakerNeto, slightly off-topic, but you might also be interested to know that these cryptochiral molecules are capable of asymmetric induction in the Soai reaction. That's one of the few reactions we know (I think it might be the only one) where a tiny enantiomeric excess in the system (which could theoretically be just stochastic) can be propagated, leading to significant levels of enantioenrichment in the product, so it's somewhat relevant to the origin of homochirality. – orthocresol Oct 16 '18 at 23:36


text:
NULL
Section
All sections
Countries | Regions
Countries | Regions
Article Types
Article Types
Year
Volume
Issue
Pages
IMR Press / RCM / Volume 23 / Issue 2 / DOI: 10.31083/j.rcm2302064
Open Access Original Research
Attitudes and behavioral intentions of aortic dissection survivors towards exercise: an application of the health action process approach framework
Show Less
1 Department of Emergency, Tongji Hospital Affiliated to Tongji Medical College, Huazhong University of Science and Technology, 430030 Wuhan, Hubei, China
2 School of Nursing, Tongji Medical College, Huazhong University of Science and Technology, 430030 Wuhan, Hubei, China
*Correspondence: huangsufang@tjh.tjmu.edu.cn (Sufang Huang)
Academic Editors: Francesco Onorati, Brian Tomlinson and Takatoshi Kasai
Rev. Cardiovasc. Med. 2022, 23(2), 64; https://doi.org/10.31083/j.rcm2302064
Submitted: 14 October 2021 | Revised: 12 November 2021 | Accepted: 28 December 2021 | Published: 15 February 2022
(This article belongs to the Special Issue State-of-the-Art Cardiovascular Medicine in Asia 2021)
This is an open access article under the CC BY 4.0 license.
Abstract

Background: Exercise can help patients with aortic dissection (AD) control blood pressure, prevent further progression of AD, promote mental health, improve the prognosis of cardiovascular disease, and improve their overall quality of life. However, it is not clear what the attitudes and behavioral intentions of AD survivors are towards exercise. Methods: This exploratory qualitative research was based on the Health Action Process Approach (HAPA) theory to explore the cognition, attitude, motivational factors, behavior intention, barriers and facilitators of exercise in patients with AD. Face-to-face and telephone semi-structured interviews were conducted in 24 AD patients from the Department of Cardio-Vascular Surgery of third-grade Class A hospitals in Wuhan, China from April 2021 to June 2021. The patient’s current stage of behavior was evaluated according to the Patient-Centered Assessment and Counseling for Exercise (PACE) questionnaire. Results: Among the 24 participants interviewed, 9 (37.5%) were in the pre-intention stage, 8 (33.3%) were in the intention stage, and 7 (29.2%) were in the action stage. The three groups were significantly different in terms of their history of previous cardiac surgery (Fisher’s exact test, p = 0.043) and in the type of interview conducted (Fisher’s exact test, p $<$ 0.001). In-patients with a history of cardiac surgery were more likely to be in the pre-intention stage (post-hoc test, p $<$ 0.05). Patients from different stages had different structures. AD patients in the pre-intention stage were more likely to express risk perception and negative results for exercise. The self-efficacy of this group was often low, and lacked exercise intention and clear action plans (post-hoc test, p $<$ 0.05). Conclusions: More attention should be paid to in-patients with a history of cardiac surgery, including health advice on exercise after discharge and promotion of changes in their health behavior. Guidelines based on the available exercise data in AD patients should be established to provide recommendations for individualized exercise programs. This should provide a basis for promoting patient rehabilitation and improving postoperative quality of life.

Keywords
aortic dissection
exercise
health action process approach
attitudes
behavioral intentions
1. Introduction

Aortic dissection (AD) is a pathological change where the blood in the aortic cavity enters the aortic media from the aortic intima tear and expands along the long axis of the aorta, resulting in separation of the true and false aortic cavities [1]. It is a serious cardiovascular emergency with the characteristics of sudden onset, rapid progression and high mortality.

In recent years, with the improvement of diagnostic methods, surgical techniques, anesthesia and drug treatment, the postoperative outcome of AD has steadily improved [2, 3]. Howard et al. [1, 4] reported the 5-year survival rate of patients with AD was 60%–80%. However, the medium-term prognosis is still poor and the risk of AD recurrence, rupture and cardiovascular disease is high in this group [5]. The quality of life (QoL) of AD survivors was not ideal [6, 7], and patients tend to have decreased physical function and some psychological problems such as anxiety and depression [8, 9, 10].

The average age of patients presenting with acute AD ranges from age 48 to 67 years (median age, 61) [11], with survivors now often being younger and more physically active [12]. As such, this patient group often desires a high quality of life. The most important factor following AD diagnosis is to control the patient’s blood pressure [1]. Moderate physical exercise is an important intervention to lower blood pressure, prevent the progression of AD, promote mental health, and improve the prognosis and overall quality of life of the AD patient [13, 14, 15]. A recent review established the safety and effectiveness of exercise-based cardiac rehabilitation for AD patients [16]. Moderate and low intensity aerobic exercise such as fast walking or cycling (3–5 Metabolic equivalents) for $\geq$30 minutes and for at least 150 minutes per week can help patients to better control arterial blood pressure and improve cardiovascular health [17].

Conceptual models or theories can provide a useful basis for exploratory research [18]. Schwarzer put forward the theory of Health Action Process Approach (HAPA) in 1992 [19]. In HAPA, the change of healthy behavior is regarded as a continuous change process that includes the beginning, maintenance and recovery of behavior. The theoretical framework of the model is shown in Fig. 1 (Ref. [19]). Some scholars have applied HAPA theory to patients with persistent musculoskeletal pain [20] or following lung cancer surgery [21] in order to explore the cognitive and behavioral characteristics of individuals in the process of exercise behavior change.

Fig. 1.

Health Action Process Approach (HAPA) mode. Schwarzer put forward the theory of HAPA in 1992 [19]. In HAPA, the change of healthy behavior is regarded as a continuous change process including the beginning, maintenance and recovery of behavior. Behavioral intention, as an important predictor of health behavior, is the core of the theoretical model.

In view of the sparse research published to date on the attitude to exercise in postoperative patients with AD, here we performed qualitative research on the exercise behavior of AD patients. Based on the HAPA theory, we explored the cognition, attitude, motivational factors, behavior intention, barriers and facilitators of exercise in patients with AD.

2. Methods
2.1 Study design

A qualitative study design and content analysis approach was implemented.

2.2 Participants

The method of purposive sampling was employed. When there were no new views in the structure of cognition, attitude, motivational factors, behavior intention, barriers and facilitators of exercise during the interview of patients with AD, this indicated the sample size had reached saturation and no further interviews were required [22]. Thirty survivors of AD were recruited from the Department of Cardio-Vascular Surgery of third-grade Class A hospitals in Wuhan, China. Six survivors refused to participate in the study because of a lack of time or interest. Of the remaining 24 patients, 12 in-patients were interviewed face-to-face and 12 discharged patients were interviewed by telephone. Inclusion criteria included: (1) patients were diagnosed according to the 2014 European Society of Cardiology guidelines; (2) age $\geq$18 years; (3) patients were in a stable condition and had stable vital signs with no serious complications; (4) voluntary participation; and (5) patients with the ability to understand and communicate in language or words. Exclusion criteria included: (1) patients with psychiatric illnesses, cognitive disorders or who were comatose; and (2) patients who were still in an unstable condition and needed further care.

2.3 Data collection

Data was collected from April 2021 to June 2021 by a graduate student with training in qualitative research methodology (DNF). Eligible patients were selected following the doctor’s ward rounds. Contact information for discharged patients was obtained from the doctors’ workstation. Informed consent from patients was obtained after fully explaining the purpose of the study. General information on patients with AD was obtained from the medical records according to their hospitalization number. The patient’s current stage of behavior was assessed according to the Patient-Centered Assessment and Counseling for Exercise (PACE) questionnaire [23]. The interview outline was drawn up based on the HAPA theory. After the pre-interview, the final version of the interview outline was modified and determined according to the interview results and the patient’s response, as shown in Fig. 2. The interview lasted 20–40 min and was conducted in a quiet, comfortable and closed location. The patients were in an office, while the telephone interviews were conducted in quiet rooms. Only the interviewer and interviewee participated. The interview was audio-recorded with field notes taken. No repeat interviews were conducted.

Fig. 2.

Interview outline. The current state of patients is divided into pre-intention stage, intention stage and action stage according to the PACE questionnaire. In each stage, different questions are asked according to HAPA theory.

2.4 Data analysis

Directed content analysis was used for data analysis [24]. After face-to-face and telephone interviews with 24 patients, the data reached saturation. Within 24 hours after the interview, the researcher gradually converted the recordings into written materials that were then checked by another researcher. After the initial transcription, accuracy was ensured by repeatedly referring to the recording or by further communication with the participants. The materials were coded, sorted, and analyzed using NVivo 11 (QSR International Pty Ltd., Melbourne, VIC, Australia). Principles of confidentiality were strictly abided, and the interviewees were replaced by codes in the data analysis (P: Pre-intention stage; I: Intention stage; A: Action stage). The preliminary analysis was carried out by DNF. The research assistant checked the survey results and interview recordings and regularly discussed and reported with the research team members until a final consensus was reached.

Continuous variables are presented as mean $\pm{}$ standard deviation or as median and interquartile range, depending on whether the variable conformed to a normal distribution. Data for independent samples were compared using the Student’s t-test, analysis of variance, Mann-Whitney U test or Kruskal–Wallis H test, as appropriate. Categorical data are presented as numbers and percentages and were compared using the Chi-square test or Fisher’s exact test. Post-hoc testing was used to evaluate differences between groups according to the adjusted standardized residual. Statistical significance was set at p $<$ 0.05 and data were analysed by SPSS version 24.0 for Windows (SPSS Inc., Chicago, IL, USA).

2.5 Ethics

Informed written consent was collected from all participants in the study. The study was conducted in accordance with the Declaration of Helsinki [25] and was approved by the Medical Ethics Committee of the Tongji Medical College, Huazhong University of Science & Technology, China (approval number: TJ-IRB20191221).

3 Results
3.1 Demographic characteristics

Table 1 shows the patient demographics and relevant factors for AD according to the different stages of behavior. A total of 24 AD patients who met the inclusion criteria were included in the study. The mean age of patients was 49.67 $\pm{}$ 12.42 years, with 75% men and 25% women. Interventional therapy was performed in 15 (62.5%) and surgical operation in 9 (37.5%). Of the 24 participants interviewed, 9 (37.5%) were in the pre-intention stage, 8 (33.3%) were in the intention stage, and 7 (29.2%) were in the action stage. Eleven patients (45.83%) had a history of previous cardiac surgery. Of these, 8 (63.6%) were re-entry patients in which subsequent physical examination showed the dissection had recurred. The remaining 3 patients had undergone previous stent surgery for coronary heart disease. The three groups were significantly different in terms of history of previous cardiac surgery (Fisher’s exact test, p = 0.043) and in the type of interview conducted (Fisher’s exact test, p $<$ 0.001). The patients were further divided into four groups according to whether they were in-patients or discharged, and whether or not they had a history of previous cardiac surgery. Significant differences were found between these groups (Fisher’s exact test, p $<$ 0.001): in-patients without a history of cardiac surgery showed a preference for the intention stage, in-patients with a history of cardiac surgery showed a preference for the pre-intention stage, while discharged patients without a history of cardiac surgery showed a preference for the action stage (post-hoc test, p $<$ 0.05). Table 2 shows the general population distribution for each structure.

Table 1.Patient demographic information.
 Demographic information Pre-intention stage (n = 9) Intention stage (n = 7) Action stage (n = 8) F/t/$\chi^{2}$ p Age (years), mean $\pm$ SD 55.56 $\pm$ 11.24 50.14 $\pm$ 12.66 42.63 $\pm$ 11.14 2.627 0.096 Interview time interval (days), mean $\pm$ SD Face-to-face interview 14.80 $\pm$ 16.96 12.57 $\pm$ 7.37 / 0.276 0.794 Telephone interview 405.25 $\pm$ 165.23 / 495.75 $\pm$ 117.83 –1.104 0.295 Sex, n (%) 1.138 0.720 Male 7 (77.8) 6 (85.7) 5 (62.5) Female 2 (22.2) 1 (14.3) 3 (37.5) Marital Status, n (%) / / Single 0 (0.0) 0 (0.0) 0 (0.0) Married 9 (100.0) 7 (100.0) 8 (100.0) Divorced or widowed 0 (0.0) 0 (0.0) 0 (0.0) Occupational status, n (%) 5.471 0.124 Full-time work 5 (55.6) 7 (100.0) 7 (87.5) Unemployed 1 (11.1) 0 (0.0) 1 (12.5) Retired 3 (33.3) 0 (0.0) 0 (0.0) Highest level of education completed, n (%) 7.156 0.315 Primary school 3 (33.3) 1 (14.3) 1 (12.5) Junior high school 4 (44.4) 4 (57.1) 1 (12.5) High school 1 (11.1) 2 (28.6) 3 (37.5) College and above 1 (11.1) 0 (0.0) 3 (37.5) Living alone, n (%) 1 (11.1) 0 (0.0) 0 (0.0) 1.683 1.000 Residential address, n (%) 5.250 0.295 Urban district 1 (11.1) 1 (14.3) 4 (50.0) Town 3 (33.3) 2 (28.6) 3 (37.5) Countryside 5 (55.6) 4 (57.1) 1 (12.5) Length of stay (days), mean $\pm$ SD 25.67 $\pm$ 22.53 17.00 $\pm$ 4.36 16.88 $\pm$ 10.99 11.841 0.560 History of previous cardiac surgery, n (%) 7 (77.8) 1 (14.3) 3 (37.5) 6.417 0.043 Classification of aortic dissection, n (%) 0.393 1.000 Type A aortic dissection 4 (44.4) 4 (57.1) 4 (50.0) Type B aortic dissection 5 (55.6) 3 (42.9) 4 (50.0) Type of operation, n (%) 1.712 0.504 Interventional therapy 6 (66.7) 3 (42.9) 6 (75.0) Surgical operation 3 (33.3) 4 (57.1) 2 (25.0) Past history, n (%) Hypertension 6 (66.7) 5 (71.4) 5 (62.5) 0.303 1.000 Heart disease 2 (22.2) 0 (0.0) 2 (25.0) 1.939 0.502 Kidney disease 1 (11.1) 0 (0.0) 0 (0.0) 1.683 1.000 Lung disease 2 (22.2) 1 (14.3) 0 (0.0) 1.872 0.609 Digestive system diseases 2 (22.2) 1 (14.3) 0 (0.0) 1.872 0.609 Obesity 0 (0.00) 1 (14.3) 0 (0.0) 2.185 0.292 Other 0 (0.00) 1 (14.3) 0 (0.0) 2.185 0.292 Medications, n (%) Statin 4 (44.4) 4 (57.1) 6 (75.0) 1.639 0.515 Aspirin 2 (22.2) 2 (28.6) 2 (25.0) 0.327 1.000 Warfarin 1 (11.1) 1 (14.3) 2 (25.0) 0.819 0.810 Antihypertensives 9 (100.0) 6 (85.7) 7 (87.5) 1.614 0.511 Other antiplatelet agents 1 (11.1) 0 (0.0) 0 (0.0) 1.683 1.000 Type of interview, n (%) 16.000 $<$0.001 Face-to-face 5 (55.6) 7 (100.0) 0 (0.0) Telephone 4 (44.4) 0 (0.0) 8 (100.0) Patient classification, n (%) 22.690 $<$0.001 In-patients without a history of cardiac surgery 1 (11.1) 6 (85.7) 0 (0.0) In-patients with a history of cardiac surgery 6 (66.7) 1 (14.3) 0 (0.0) Discharged patients without a history of cardiac surgery 1 (11.1) 0 (0.0) 5 (62.5) Discharged patients with a history of cardiac surgery 1 (11.1) 0 (0.0) 3 (37.5) Note: the interview time interval was classified into two groups according to the type of interview. In-patients (i.e., face-to-face interviewers) were analysed according to the difference between the interview date and operation date, while discharged patients (i.e., telephone interviews) were analysed according to the difference between the interview date and the discharge date.
Table 2.Structural distribution characteristics.
 Total number (n = 24) Pre-intention stage (n = 9) Intention stage (n = 7) Action stage (n = 8) $\chi^{2}$ p Risk perception, n (%) 11 (45.8) 7 (77.8) 4 (57.1) 0 (0.0) 11.249 0.003 Positive outcome expectations, n (%) 18 (75.0) 3 (33.3) 7 (100.0) 8 (100.0) 11.386 0.001 Negative outcome expectations, n (%) 5 (20.8) 5 (55.6) 0 (0.0) 0 (0.0) 8.526 0.005 Action self-efficacy, n (%) 9 (37.5) 0 (0.0) 4 (57.1) 5 (62.5) 9.187 0.008 Intentions, n (%) 11 (45.8) 0 (0.0) 5 (71.4) 6 (75.0) 12.773 0.001 Implementation intention Action planning, n (%) 12 (50.0) 0 (0.0) 4 (57.1) 8 (100.0) 18.562 $<$0.001 Coping planning, n (%) 2 (8.3) 0 (0.0) 0 (0.0) 2 (25.0) 3.000 0.178 Maintenance self-efficacy, n (%) 13 (54.2) 0 (0.0) 5 (71.4) 8 (100.0) 19.438 $<$0.001 Recovery self-efficacy, n (%) 1 (4.2) 0 (0.0) 0 (0.0) 1 (12.5) 1.918 0.625 Barriers, n (%) 11 (45.8) 4 (44.4) 3 (42.9) 4 (50.0) 0.247 1.000 Promoting factors, n (%) 12 (50.0) 3 (33.3) 4 (57.1) 5 (62.5) 1.650 0.589
3.2 Motivational phase constructs

In the motivation phase constructs, individuals move from no behavior intention to the formation of behavioral intention and finally to take action. Table 3 shows some of the views of patients at different stages with regard to risk perception, outcome expectation, action self-efficacy and intention. These all appear in patients at the pre-intention, intention and action stages.

Table 3.Some patient views on the motivational phase constructs.
3.2.1 Risk perception

Risk perception refers to the overall evaluation of the potential severity and controllability of the disease, as well as the possibility that patients with AD are at risk for recurring events upon exercise participation [26]. The results showed that AD patients in the pre-intention stage were more likely to express risk perception, and that patients in the action stage had less risk perception (post-hoc test, p $<$ 0.05). The pre-intention and intention stage AD patients were more concerned about the adverse effects of disease compared to the action stage AD patients who already exercised. The patients in the pre-intention stage were more worried about postoperative pain and the impact on life and finances. Four of the 7 patients in the intention stage were more worried about their recovery after operation and expressed some fear of accidents during exercise. However, AD patients who had experienced AD events previously (i.e., re-entry patients) were most concerned about a recurrence of dissection, independently of exercise.

3.2.2 Outcome expectations

Outcome expectations indicate a comprehensive balance of the possible consequences of exercise in patients with AD, including positive outcome expectations as well as negative outcome expectations [27]. Patients in the pre-intention stage were more likely to express negative results for exercise, while patients in the action stage tended to express positive results (post-hoc test, p $<$ 0.05). Patients in both the intention and action stages believed that physical activity and exercise were beneficial for them. Compared to patients in the intention stage, patients in the action stage identified different and more specific positive outcome expectations, including: (1) positive emotions, (2) self-confidence, and (3) family benefits. However, 5 out of 9 patients in the pre-intention stage also reported some negative outcome expectations. These included requiring great effort, worrying about accidents, increased family concerns, and feeling a psychological loss when the expected goal of exercise was not achieved.

3.2.3 Action self-efficacy

Action self-efficacy refers to the patients’ confidence in their own ability when they begin exercise [28]. There was a large difference in action self-efficacy between patients in the pre-intention stage compared to the other two groups. The action self-efficacy of patients in the pre-intention stage was often low (post-hoc test, p $<$ 0.05). Patients in both the intention and action stages had a high level of action self-efficacy when they decided to start exercising. However, the self-efficacy of action was low in patients in the pre-intention stage due to an unwillingness to try, poor compliance, and for other reasons.

3.2.4 Intentions

Patients in the pre-intention stage were found to often lack the intention of exercise, while patients in the action stage have a clear intention to participate in physical activities and to exercise (post-hoc test, p $<$ 0.05). Reasons for the lack of exercise intention in pre-intention stage patients was mainly due to: (1) symptoms of discomfort, (2) lack of confidence, (3) poor compliance, and (4) their family’s overprotection.

3.3 Volitional phase constructs

In the volitional phase constructs, individuals make plans, overcome the difficulties encountered in action and adjust themselves, so as to adhere to action and finally develop healthy behavior. Patients in the pre-intention stage had not yet experienced the volitional phase. Therefore, this stage only focused on patients in the intention stage and action stage. Table 4 shows some patient views with regard to implementation intention, maintenance self-efficacy and recovery self-efficacy.

Table 4.Some patient views on volitional phase constructs.
 Theme Codes Sample quotes Implementation intention Action planning “I plan to go for a one hour walk each morning and evening. Next to my house is the road. I’m going to be there.” (Participant I2) “I jog every morning for about 30–40 minutes (…).” (Participant A2) “The doctor said that walking was good for cardiopulmonary function. I definitely want to go for a walk, but I have no plan at present because I’m not sure about the time, frequency and intensity (…).” (Participant I4) Coping and planning: recording, finding new ways “I’m busy with my work (…) There is a health log APP. I record it every day, so I won’t forget.” (Participant A4) “Sometimes when the weather is bad, I will go to the indoor stadium for a walk.” (Participant A2) Maintenance self-efficacy Positive feedback “I don’t have to do farm work after discharge, so I have plenty of time to go for a walk every day (…) I am very confident that I can stick to this.” (Participant I7) “I feel very comfortable going for a jog every morning. If I stop one day, I may not adapt.” (Participant A2) Worry they can’t stick to it “The doctor said that I should lose weight, so I should start to exercise slowly after discharge. But I think it will be hard to lose weight (…).” (Participant I6) Recovery self-efficacy Only one patient experienced a period of interrupted exercise due to a business trip “I went on a business trip for a while and didn’t exercise after coming back, so I sat almost all the time, including at work (…) My wife would often remind me that it was time to exercise. Now I’m going back to my regular activities, and I feel good (…).” (Participant A4) The patients in pre-intention stage: P1, P2, P3 … The patients in intention stage: I1, I2, I3 … The patients in action stage: A1, A2, A3 …
3.3.1 Implementation intention

Implementation intention includes action planning and coping planning [29]. Action planning for patients in the pre-intention stage had not been developed, whereas patients in the action stage had a clear action plan (post-hoc test, p $<$ 0.05). Four of the 7 patients in intention stage and all patients in action stage had clear action planning. Walking was found to be the most popular exercise for AD patients (50%). Other activities included square dancing, fast walking, fitness and jogging. Three of the 7 patients in intention stage had not yet thought about action planning. Only two of the 8 patients in action stage said they had developed coping strategies. Coping strategies include: (1) recording, and (2) finding new ways.

3.3.2 Maintenance self-efficacy

Maintenance self-efficacy refers to the optimistic belief of patients with AD in coping with difficulties during maintenance exercise [19]. The maintenance efficacy of patients in the pre-intention stage was found to be lower, while patients in the action stage had higher maintenance self-efficacy (post-hoc test, p $<$ 0.05). Five of the 7 patients in intention stage and all patients in action stage felt confident to maintain exercise participation. Positive feedback was the most important factor for maintaining self-efficacy. Just 2 of the 7 patients in intention stage worried that they could not adhere to an exercise program.

3.3.3 Recovery self-efficacy

Recovery self-efficacy refers to the ability of AD patients to overcome obstacles when they do not carry out the exercise plan on schedule [30]. Most patients in the intention and action stages claimed that the biggest task after discharge was to take care of their body and they would always adhere to appropriate activities. Only one of the 24 patients experienced a period of interrupted exercise, but they then overcame the obstacles and resumed regular exercise.

In addition to the above constructs, HAPA also recognizes that barriers and facilitators can also affect an individual’s behavioral intention [31]. Table 5 shows some views of patients at different stages on barriers and promoting factors of exercise.

Table 5.Some views of patients on barriers and on promoting factors.
 Theme Codes Sample quotes Barriers Finances “This operation cost our family a lot of money. Originally, my family was from the countryside (…) Now that I have time to do some farm work and earn more money, there is no time to exercise.” (Participant P7) Attitudes of family members “I am usually at home alone, and my children work outside. They do not like that I go out alone for a walk, for fear of accidents (…).” (Participant P9) Environmental factor “Sometimes when it rains or snows, I don’t want to go out and I prefer to watch TV at home (…).” (Participant A3) Promoting factors Personal interest “I usually like to exercise because it can improve my mood. Sometimes I feel like going for a long walk and even though I feel a little uncomfortable, I believe I can do it.” (Participant A5) Support from others “I also have friends around me who have this disease. We have arranged to go for a walk after I leave the hospital.” (Participant I5) “I have been hospitalized three times. Each time my husband has given me a lot of support. He encouraged me to walk slowly until I could walk on my own (…) Now my situation is getting better each day and so I really appreciate his support.” (Participant A6) Encouragement of medical staff “The doctor encouraged me to get out of bed and to walk as soon as I felt better (…) He also suggested that I do some low to moderate intensity exercise after I leave the hospital and he encouraged me to lose weight.” (Participant I6) “I added the doctor’s WeChat contact information, so that he could give me some guidance.” (Participant P4) The patients in pre-intention stage: P1, P2, P3 … The patients in intention stage: I1, I2, I3 … The patients in action stage: A1, A2, A3 …
3.4.1 Barriers

Almost half of patients (11/24, 45.8%) described the presence of barriers to exercise: (1) financial problems, (2) attitudes of family members, and (3) environmental factors. Three out of 11 patients expressed great concern about financial problems. The families of three out of 11 patients thought that accidents might occur during exercise and thus did not support it. Weather conditions were also factors that stopped patients from carrying out exercise.

3.4.2 Promoting factors

Half the patients (12/24) expressed that various promoting factors increased their ability to support exercise participation, including (1) personal interest, (2) support from other patients and from their families, and (3) the encouragement of medical staff. The patients’ enthusiasm and mental state also affected their desire to participate in exercise. Two out of 12 patients indicated that the support of other patients could promote their compliance for physical activities. Three out of 12 patients expressed that family support is very important. The professional advice of medical staff also directly affected the enthusiasm and compliance of patients to participate in sports.

4. Discussion

This is the first qualitative study to explore the attitudes and behavioral intentions of AD survivors towards exercise. The following aspects were addressed: (1) use of the PACE questionnaire to classify the current stage of patients, (2) the use of HAPA theory to outline and conduct qualitative interviews, (3) comparison of the differences in HAPA constructs between patients from different stages.

Among the 24 participants interviewed, 9 (37.5%) were in the pre-intention stage, 8 (33.3%) were in the intention stage, and 7 (29.2%) were in the action stage. In a study by Chaddha et al. [14], 62/82 (76%) of AD survivors participated in structured physical activity, which was a higher rate than in our study. Our current study included both in-hospital patients and discharged patients. Most of the hospitalized patients were still in the recovery stage and therefore rarely able to participate in exercise behavior. This may explain the difference in exercise participation rate between our study and that of Chaddha et al. [14]. Interestingly, we found that in-patients without a history of cardiac surgery showed a preference for the intention stage, whereas in-patients with a history of cardiac surgery showed a preference for the pre-intention stage. Therefore, medical staff should pay close attention to in-patients with a history of cardiac surgery. They should carefully explain health education to them, explain the benefits of exercise on the cardiovascular system, inform them of any precautions for exercise after discharge, and promote behavior intention and progress to the intention and action stages.

The present study found that AD survivors in the pre-intention and intention stages expressed more risk perception and concerns about the adverse effects of disease, including recurrence, pain, the impact on life and finances, postoperative rehabilitation and accidents during exercise. This was consistent with the findings of Chadha’s study, in which 76% of AD patients believed the disease had a negative impact on their lives [9]. New dissection, organ ischemia, aneurysm formation or rupture may occur after AD operation [1, 32, 33]. Depression and anxiety were more common in AD survivors than in other heart disease survivors [9]. Chaddha et al. [14] also pointed out that fear was the main reason that most survivors limited their activities.

Of interest, the results also showed that outcome expectations were different for different stages. Action stage patients often had more positive outcome expectations, while pre-intention stage patients had more negative outcome expectations. Previous studies have reported that outcome expectations play a significant role in predicting behavioral intention, mainly in the intention formation stage. The more positive expectations an individual has about taking a certain health promotion behavior or giving up a certain health hazard behavior, the more favorable the formation of behavior intention [27].

We also explored the reasons underlying different intention performance for different stages. The reasons why exercise was often not considered in the pre-intention stage included discomfort, lack of self-confidence, poor compliance and family overprotection. This was consistent with the findings of other related studies [34, 35]. Implementation intention is the further concretization of intention [36] and can promote the effective transformation from intention to actual behavior [29]. In the intention and action stages of AD patients, the main exercise methods are walking, fitness, square dancing and jogging. However, some patients in the intention stage were unable to organize the frequency, time and intensity of exercise. Surprisingly, only 2 of the 24 patients in this study had coping plans. Previous studies have shown that exercise is safe for patients with AD and there are no serious complications [12, 37, 38, 39]. However, there is still a lack of guidance regarding the appropriate level, type, frequency, intensity, and duration of physical activity. Therefore, it seems timely to establish guidelines based on the available data for exercise in AD patients. This should provide some guidance to physicians and patients for organizing and performing exercise after an AD event.

Self-efficacy can be divided into action self-efficacy, maintenance self-efficacy and recovery self-efficacy [28]. Self-efficacy can predict behavioral intention, as well as predicting behavior change in the next stage [30]. This study showed that action self-efficacy was present in all three stages, whereas the structure of maintaining self-efficacy and recovery self-efficacy existed only in the intention and action stages. A previous study found that self-efficacy can affect the choice of activities, the level of effort and perseverance in the face of difficulties [40]. The current finding also suggests that regardless of the type of self-efficacy, the action stage tended to show more positivity and thus higher exercise compliance.

Various barriers were expressed in this study, including the lack of time and opportunity to exercise, concern about accidents, and bad weather preventing outdoor exercise. Previous studies have reported that illness or injury, work commitments, and poor weather are key barriers to exercise [41, 42]. In contrast, favorable factors for the promotion of exercise behavior included personal interest, support from others, and encouragement by medical staff. The recommendation by medical staff was previously identified as an important factor to improve the behavior compliance of patients [43, 44]. Our research also found that the attitude of family members can have a significant positive impact on the enthusiasm and compliance of patients to participate in exercise. Conversely, however, family members may also hinder exercise behavior.

5. Conclusions

This study explored the attitude and behavior intention of AD patients in different behavior stages based on HAPA theory. In-patients with a history of cardiac surgery showed a preference for the pre-intention stage. Moreover, patients from different stages had different structures. For example, AD patients in the pre-intention stage were more likely to express risk perception and have negative results for exercise. The self-efficacy of this group was often low and lacking in exercise intention and clear action plans (post-hoc test, p $<$ 0.05). Therefore, more attention should be given to in-patients with a history of cardiac surgery. This could include more health education regarding exercise after discharge and the promotion of changes in their health behavior. This study provides direction for understanding the relationship between different variables and for predicting the process of exercise behavior change. The results also provide supporting evidence and effective evaluation tools for exercise rehabilitation intervention in AD patients. However, this study was conducted in a single center only and with a relatively small sample size. Its purpose was to understand the cognition, attitude and intention of AD patients to exercise through qualitative research. The variables of “history of previous cardiac surgery” and “type of interview conducted (i.e., in-patient or discharged patient)” were statistically significant in each of the three different stages. Future work should use a quantitative questionnaire designed for a study involving multiple centers to predict the exercise intention of AD patients. This should allow a more comprehensive understanding of the factors influencing behavior intention, thus providing a basis for promoting patient rehabilitation and improving their postoperative quality of life.

Author contributions

DNF—conception of the study, major drafting of the work, final approval and agreeing to the accuracy of the work. SFH—conception of the study, help in design of the study, drafting of the work, final approval and agreeing to the accuracy of the work. XRL—supervision, critical revision of the manuscript, final approval and agreeing to the accuracy of the work. All authors contributed to editorial changes in the manuscript. All authors read and approved the final manuscript.

Ethics approval and consent to participate

Informed written consent was collected from all participants in the study. The study was conducted in accordance with the Declaration of Helsinki and was approved by the Medical Ethics Committee of the Tongji Medical College, Huazhong University of Science & Technology, China (approval number: TJ-IRB20191221).

Acknowledgment

Thanks to all the peer reviewers for their opinions and suggestions.

Funding

This research was funded by National Natural Science Foundation of China, grant number: Project 71874063.

Conflict of interest

The authors declare no conflict of interest.

Publisher’s Note: IMR Press stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Share


text:
## Nali Group Title When is Emf= vBl negative like (Emf=- vBl ) and when is it positive? 7 months ago 7 months ago

1. theEric

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html The emf would be negative if the vector $$\vec v\times\vec B$$ is negative. It depends on your coordinate system, how you allocate the signs. But you can always look at the velocity and magnetic field vectors. In this drawing, the current will be out of the screen.|dw:1397363101158:dw| Swap the vectors, and it will be into the screen. Check out the right hand rule! :)

2. Nali

thanks!

3. theEric

You're welcome! When I answered this, I was in a rush. Let me know if you have any questions!


text:
# In which two market structures are prices set mainly by the supply and demand of many buyers and sellers? a.monopoly and oligopoly b.oligopoly and perfect competition c.monopoly and monopolistic competition d.perfect competition and monopolistic competition

###### Question:

In which two market structures are prices set mainly by the supply and demand of many buyers and sellers?
a.monopoly and oligopoly
b.oligopoly and perfect competition
c.monopoly and monopolistic competition
d.perfect competition and monopolistic competition

### Do the data in the table represent a direct variation or inverse variation? Write an equation to model the data in the table? x 1 3 5 10 y 4 12 20 40

Do the data in the table represent a direct variation or inverse variation? Write an equation to model the data in the table? x 1 3 5 10 y 4 12 20 40...

### What is Zionism? * 10 points the plan to let Arabs and Jews share the same land in Palestine equally. the idea that all religious groups should have their own states. the hope that all governments will be based on religious principals the belief that Jews deserved to return to a homeland in Zion Which of the following is an example of Anti-Semitism? 10 points Sykes-Picot Agreement The Holocaust Creation of Israel The break-up of the Ottoman Empire During and after World War II many surviving Jew

What is Zionism? * 10 points the plan to let Arabs and Jews share the same land in Palestine equally. the idea that all religious groups should have their own states. the hope that all governments will be based on religious principals the belief that Jews deserved to return to a homeland in Zion Whi...

### Which of the following describes an important difference between the Muslim calendar and the Gregorian calendar?

Which of the following describes an important difference between the Muslim calendar and the Gregorian calendar?...

### The shadow of the moon during a solar eclipse travels 2300 miles in one hour in the first 20 minutes the Shoadow traveled seven 66 2/3 miles how long does it take for the Shadow to reach 1150 miles

The shadow of the moon during a solar eclipse travels 2300 miles in one hour in the first 20 minutes the Shoadow traveled seven 66 2/3 miles how long does it take for the Shadow to reach 1150 miles...

### Acellus Find the standard deviation of the given data rounded to the nearest hundredth. 3,4,5,7,10,12,15

Acellus Find the standard deviation of the given data rounded to the nearest hundredth. 3,4,5,7,10,12,15...

### Tengo una lesión grave en la rodilla y el doctor me dijo que necesito _____________ y sesiones de fisioterapia.

Tengo una lesión grave en la rodilla y el doctor me dijo que necesito _____________ y sesiones de fisioterapia....

### Two-Way Tables Laura wants to know whether the top golfers in the world tend to be American. To do this, she searches the Internet to find the current top-20 male golfers and top-20 female golfers and records whether they are American. This two-way table shows her data. Keep the two way table window open as you work on the tasks. Part A Find the relative frequencies by rows.

Two-Way Tables Laura wants to know whether the top golfers in the world tend to be American. To do this, she searches the Internet to find the current top-20 male golfers and top-20 female golfers and records whether they are American. This two-way table shows her data. Keep the two way table window...

### ASAP ANSWER PLEASE!!!! Read the following sentence, then decide if it is a simple, compound, or complex sentence. I stayed up all night studying for my chemistry test, so I felt really happy when I got an ‘A’ on the test. A. simple sentence B. compound sentence C. complex sentence

ASAP ANSWER PLEASE!!!! Read the following sentence, then decide if it is a simple, compound, or complex sentence. I stayed up all night studying for my chemistry test, so I felt really happy when I got an ‘A’ on the test. A. simple sentence B. compound sentence C. complex sentence...

### What element do Rosh Hashanah and Yom Kippur share?

What element do Rosh Hashanah and Yom Kippur share?...

### These clouds appear fibrous or curly. They are high, thin, white, feathery clouds made of ice crystals. They are associated with fair weather, but they can indicate approaching storms too. Which type of cloud is this?

These clouds appear fibrous or curly. They are high, thin, white, feathery clouds made of ice crystals. They are associated with fair weather, but they can indicate approaching storms too. Which type of cloud is this?...

### There are six red marbles and 11 orange marbles in a bag. ￼ you randomly choose one of the marvelous. What is the theoretical probability of choosing a red marble?￼ ( on a time limit so PLEASEE hurry)

There are six red marbles and 11 orange marbles in a bag. ￼ you randomly choose one of the marvelous. What is the theoretical probability of choosing a red marble?￼ ( on a time limit so PLEASEE hurry)...

### A bakery packages 868 muffins into a 31 boxes. The same number of muffins are put into each box. How many muffins are in each box?

A bakery packages 868 muffins into a 31 boxes. The same number of muffins are put into each box. How many muffins are in each box?...

### Which statements are true regarding the rotational symmetry of the shape of a cross? Select three options. The order of rotational symmetry is 2. The shape directly maps onto itself after a rotation of 270 degrees. The smallest angle of rotational symmetry is 90 degrees. The shape directly maps onto itself after a rotation of 120degrees. The shape directly maps onto itself after a rotation of 180degrees.

Which statements are true regarding the rotational symmetry of the shape of a cross? Select three options. The order of rotational symmetry is 2. The shape directly maps onto itself after a rotation of 270 degrees. The smallest angle of rotational symmetry is 90 degrees. The shape directly maps ont...

### There are many mitochondria in muscle cells. Which best explains how this is important to cellular respiration? Mitochondria are responsible for DNA replication, which allows more energy to be generated. Mitochondria are plentiful in muscles so that new muscle can be created. Muscles need lots of energy, and glycolysis delivers this energy to mitochondria. Muscles need lots of energy, and the final stage of cellular respiration takes place in the mitochondria.

There are many mitochondria in muscle cells. Which best explains how this is important to cellular respiration? Mitochondria are responsible for DNA replication, which allows more energy to be generated. Mitochondria are plentiful in muscles so that new muscle can be created. Muscles need lots of en...

### The sum of the ages of the father and son is 44 years .After 8 years the age of the father will be twice the age of the son . Find their present ages.

the sum of the ages of the father and son is 44 years .After 8 years the age of the father will be twice the age of the son . Find their present ages....

### How to make a business title

how to make a business title...

### 18÷1 put it in long division

18÷1 put it in long division...

### $(\frac{7}{4} )^2$ simplify

$(\frac{7}{4} )^2$ simplify...

### Solve the exponential equation. 3x = 1/27 -3 -1/3 1/3

Solve the exponential equation. 3x = 1/27 -3 -1/3 1/3...

### Read the paragraph. Then answer the question that follows. Perhaps you wanted pizza for dinner, but were out voted by the rest of the family who wanted chili. This is similar to what happens in a community. One person has to give up a right for the good of the group. Sometimes citizens' duties and rights conflict with each other. A good example is a public protest. People have the right to meet in groups and share ideas. However, a protest can disrupt traffic or other normal activities. A city m

Read the paragraph. Then answer the question that follows. Perhaps you wanted pizza for dinner, but were out voted by the rest of the family who wanted chili. This is similar to what happens in a community. One person has to give up a right for the good of the group. Sometimes citizens' duties and r...

-- 0.014319--


text:
# [OS X TeX] Equation alignment in enumerated list

Vincent McGarry vmcgarry at austin.rr.com
Fri Dec 5 00:12:35 CET 2008

Try the following:

\begin{enumerate}
\item
\begin{aligned}[t] Equations go here! \end{aligned}
\item
\begin{aligned}[t] Equations go here! \end{aligned}
\end{enumerate}

Vince McGarry

On Dec 4, 2008, at 3:18 PM, Randy Scott wrote:

> Hi,
>
> I'm trying to create an enumerated list, where the items listed are
> equations. So far so good. I've been using \displaystyle . . . to > get my > equations to line up with the the text that accompanies the equations. > > However, I need to make an entry that has the top line of a series of > equations aligned with the item marker, and then the subsequent lines > aligned at the equal sign with the top equation. For example: > > 1. f(x) = x^2 + 1/x > = x^3/x + 1/x > = (x^3+1)/x > > I've tried a \parbox[t] but that doesn't align the baseline of the > equation > with the item marker. I've tried \begin{align*} but that won't operate > inside a\displaystyle arrangement.
>
> Any ideas that you all could send my way would be appreciated.
>
> Thanks,
>
> Randy Scott
> Santiago Canyon College
> Orange, CA
>
> ----------- Please Consult the Following Before Posting -----------
> TeX FAQ: http://www.tex.ac.uk/faq
> List Reminders and Etiquette: http://www.esm.psu.edu/mac-tex/list/
> List Archive: http://tug.org/pipermail/macostex-archives/
> TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> List Info: http://email.esm.psu.edu/mailman/listinfo/macosx-tex
>


text:
# Protoplanetary disc evolution affected by star-disc interactions in young stellar clusters [EPA]

Most stars form in a clustered environment. Therefore, it is important to assess how this environment influences the evolution of protoplanetary discs around young stars. In turn, this affects their ability to produce planets and ultimately life. We present here for the first time 3D SPH/N-body simulations that include both the hydrodynamical evolution of the discs around their natal stars, as well as the dynamics of the stars themselves. The discs are viscously evolving, accreting mass onto the central star and spreading. We find penetrating encounters to be very destructive for the discs as in previous studies, although the frequency of such encounters is low. We also find, however, that encounter influence the disc radii more strongly than other disc properties such as the disc mass. The disc sizes are set by the competition between viscous spreading and the disruptive effect of encounters. As discs spread, encounters become more and more important. In the regime of rapid spreading encounters simply truncate the discs, stripping the outer portions. In the opposite regime, we find that the effect of many distant encounters is able to limit the disc size. Finally, we predict from our simulations that disc sizes are limited by encounters at stellar densities exceeding $\sim 2-3 \times 10^3 \ \mathrm{pc}^{-2}$.

G. Rosotti, J. Dale, M. Ovelar, et. al.
Wed, 9 Apr 14
64/67


text:
# How does the JWST change focus when it goes from looking at a near subject to looking at a far subject? (Does it even change focus?)

After following the extensive steps in the process of aligning and phasing the optics and instruments of JWST... I now wonder if further focus adjustments would be made based on distance to subject. Perhaps that focus is already factored into the specific instruments. Perhaps big redshifts don't care about focus. Perhaps 'focused at infinity' is all that is needed.

• The JWST will only ever focus on objects that are at least hundreds of millions of km away. Most objects will be many lightyears (hundreds of trillions of km). So no need to change focus. Apr 5 at 1:24
• If you look at a camera with manually adjustable focus, you'll see the distance from the subject (in meters) printed as 0.3..............1.........2.....8..20∞ The difference between focus 100m away and 10km away is negligible. The difference in focus between 1AU and 1 parsec is none.
– SF.
Apr 5 at 9:47
• @hdhondt: Are you sure the JWST will never focus on Mars? Apr 5 at 13:10
• @TonyK A quick calculation shows that the image of the Mars would be around $100\rm\, nm$ out of the focal plane. Visible light has a wavelength of around $500\rm\, nm$, infrared even more. I don't know if JWST will even bother. Apr 5 at 15:06
• À propos Mars, Mark McCaughrean, senior advisor for science and exploration at the European Space Agency, says in an article about JWST: "We can look at planets like Mars, Jupiter, Saturn, Uranus and Neptune but also into the Kuiper Belt." Apr 5 at 15:48

It doesn't change focus. Not even to correct for which wavelengths are being measured, because that's needed by only refractors, not reflectors like JWST. In the terminology of handheld terrestrial cameras and lenses, everything that JWST or even a backyard Celestron sees is "at infinity."

Generally, focusing means aiming light rays from a subject, gathered by a big mirror or a big lens, to a point on the focal plane. (A pinhole camera is always in focus, because its "lens" is so small.) For a handheld camera pointing at a subject at arm's length, those rays diverge by the time they reach the face of the lens. A subject across the room has rays that diverge less, so you have to "change the focus." A subject in the next solar system has rays whose divergence is unmeasurably small: they're exactly parallel, for Alpha Centauri or for something at the edge of the observable universe. (For nearby Alpha Centauri, a bit of trig shows that across the 6.5 m width of JWST's mirror, its rays diverge by about a femtodegree.)

• So even looking a solar system objects like asteroids (which JWST will most certainly do) there is no need to refocus? Cool! Apr 5 at 6:54
• The rays from something in the asteroid belt, by the time they reach JWST, would diverge by - brace yourself - almost a nanodegree! Apr 5 at 14:52
• @TrySCE2AUX if my math is right, an object would have to be within about 3km of JWST to have a focal blur radius on the order of one NIRCAM pixel. Even if I messed that up, I didn't do so by enough orders of magnitude for solar system objects to be any worry. Anything at a distance that's millions of times the diameter of the mirror is effectively at infinity. Apr 5 at 16:56
• I assume they have some sort of a focusing-adjustment mechanism so that if the mirrors drift away from "infinity" it can be re-focuses, but that said mechanism does not depend on the distance away it's target is because even one AU is basically infinite. Apr 5 at 22:05
• "Basically infinite" starts at only 3 km, not 1 AU, hobbs calculated. Also, if the mirrors wander around and need to be shepherded back, that's called alignment, not focusing. Apr 5 at 22:12

The JWST focal length is 131.4 m. There is an equation for object distance $$u$$, image distance $$v$$ and focal length $$f$$:

$$\frac{1} { f} = \frac{1} {u} + \frac{1} {v}$$

If the object is in infinite distance then

$$\frac{1} { u} = 0$$

the equation simplifies to:

$$\frac{1} { f} = 0 + \frac{1} {v}$$

$$\frac{1} { f} = \frac{1} {v}$$

$$v = f$$

and $$v$$ is equal to $$f$$.

I set the image distance $$v$$ to 131.4 m + 10 nm (10 nanometers or 10-8 m or 0.00000001 m) and calculate the distance of the object $$u$$.

$$\frac{1} { 131.4 m} = \frac{1} {u} + \frac{1} { 131.4 m + 10 nm}$$

$$\frac{1} { u} = \frac{131.4 m} {u} - \frac{1} { 131.4 m + 10 nm}$$

$$\frac{1} { u} = \frac{10 nm} {131.4 m * (131.4 m + 10 nm)}$$

$$u = \frac{131.4 m * (131.4 m + 10 nm)} {10 nm}$$

$$u = 131.4 * (131.4 m + 10 nm) * 10^8$$

$$u = 131.4^2 * 10^8 m$$

$$u = 17.26 * 10^{12} m$$

$$u = 11.5 AU$$

The distance of Saturn to Sun is 9.5 AU.

So if the object is not at infinite distance but at 11.5 AU, the change of image distance (the distance to refocus properly) is 10 nm, at least sixty times smaller than the used IR wavelengths, which are 600 to 2500 nm.

The stars imaged by the JWST are much farther than Saturn, so the change of image distance is much smaller than 10 nm and very, very close to zero.


text:
English

# Item

ITEM ACTIONSEXPORT
An OSp extension of Canonical Tensor Model

Narain, G., & Sasakura, N. (2015). An OSp extension of Canonical Tensor Model. Progress of Theoretical & Experimental Physics, 2015(123): A05. doi: 10.1093/ptep/ptv169.

Item is

### Basic

show hide
Genre: Journal Article

### Files

show Files
hide Files
:
1509.01432.pdf (Preprint), 779KB
Name:
1509.01432.pdf
Description:
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
-
-
:
Prog. Theor. Exp. Phys.-2015-Narain-.pdf (Publisher version), 2MB
Name:
Prog. Theor. Exp. Phys.-2015-Narain-.pdf
Description:
Open access
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
-
-

show

### Creators

show
hide
Creators:
Narain, Gaurav1, Author
Sasakura, Naoki, Author
Affiliations:
1AEI-Golm, MPI for Gravitational Physics, Max Planck Society, Golm, DE, ou_24008

### Content

show
hide
Free keywords: High Energy Physics - Theory, hep-th,General Relativity and Quantum Cosmology, gr-qc
Abstract: Tensor models are generalizations of matrix models, and are studied as discrete models of quantum gravity for arbitrary dimensions. Among them, the canonical tensor model (CTM for short) is a rank-three tensor model formulated as a totally constrained system with a number of first-class constraints, which have a similar algebraic structure as the constraints of the ADM formalism of general relativity. In this paper, we formulate a super-extension of CTM as an attempt to incorporate fermionic degrees of freedom. The kinematical symmetry group is extended from $O(N)$ to $OSp(N,\tilde N)$, and the constraints are constructed so that they form a first-class constraint super-Poisson algebra. This is a straightforward super-extension, and the constraints and their algebraic structure are formally unchanged from the purely bosonic case, except for the additional signs associated to the order of the fermionic indices and dynamical variables. However, this extension of CTM leads to the existence of negative norm states in the quantized case, and requires some future improvements as quantum gravity with fermions. On the other hand, since this is a straightforward super-extension, various results obtained so far for the purely bosonic case are expected to have parallels also in the super-extended case, such as the exact physical wave functions and the connection to the dual statistical systems, i.e. randomly connected tensor networks.

### Details

show
hide
Language(s):
Dates: 2015-09-042015
Publication Status: Published in print
Pages: 27pages, 27 figures
Publishing info: -
Rev. Method: -
Identifiers: arXiv: 1509.01432
URI: http://arxiv.org/abs/1509.01432
DOI: 10.1093/ptep/ptv169
Degree: -

show

show

show

### Source 1

show
hide
Title: Progress of Theoretical & Experimental Physics
Source Genre: Journal
Creator(s):
Affiliations:
Publ. Info: -
Pages: - Volume / Issue: 2015 (123) Sequence Number: A05 Start / End Page: - Identifier: -


text:
このページ(論文)をリンクする場合は次のURLを使用してください: http://hdl.handle.net/2261/1565

 タイトル: The Andreotti Grauert vanishing theorem for dihedrons of $\Bbb C^n$ 著者: Zampieri, Giuseppe 発行日: 1995年 出版者: Graduate School of Mathematical Sciences, The University of Tokyo 掲載誌情報: Journal of Mathematical Sciences, The University of Tokyo. Vol. 2 (1995), No. 1, Page 233-246 抄録: Let $X$ be a complex manifold, $\OX$ the sheaf of analytic functions on $X$, $W$ an open set of $X$ with $C^2$-boundary $M=\partial W$ ($W$ locally on one side of $M$), $z_o$ a point of $M$, $p_o$ the exterior conormal to $W$ at $z_o\,$. If the number of negative eigenvalues for the Levi form of $M$ in a neighborhood of $p_o$ is $\geq s^-$ (resp. $\equiv s^-$), then vanishing of local cohomology groups of $\OX$ over $W$ in degree \$ URI: http://hdl.handle.net/2261/1565 ISSN: 13405705 出現カテゴリ: Journal of Mathematical Sciences, the University of TokyoJournal of Mathematical Sciences, the University of Tokyo

この論文のファイル:

ファイル 記述 サイズフォーマット


text:
Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 25 May 2017, 00:23

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Seven family members are seated around their circular dinner

Author Message
TAGS:

### Hide Tags

Intern
Joined: 01 Sep 2010
Posts: 23
Followers: 1

Kudos [?]: 80 [3] , given: 8

Seven family members are seated around their circular dinner [#permalink]

### Show Tags

04 Oct 2010, 06:30
3
KUDOS
3
This post was
BOOKMARKED
00:00

Difficulty:

35% (medium)

Question Stats:

59% (01:39) correct 41% (00:53) wrong based on 386 sessions

### HideShow timer Statistics

Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Jul 2013, 08:43, edited 1 time in total.
Edited the OA.
Retired Moderator
Joined: 02 Sep 2010
Posts: 803
Location: London
Followers: 111

Kudos [?]: 1020 [2] , given: 25

Re: Arrangements around the table [#permalink]

### Show Tags

04 Oct 2010, 09:01
2
KUDOS
Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Are you sure you got the right OA on this one ?

As I see it : 7 members, and Michael & Bobby sit together. So treat those two like a single person. Now you have to arrange 6 people around a table. This may be done in 5! or 120 ways

The number of ways to arrange Michael & Bobby is 2 (MB or BM).

So net there are 240 ways or B

I don't think this is a GMAT level question, may be a tad too hard
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 38859
Followers: 7728

Kudos [?]: 106069 [2] , given: 11607

Re: Arrangements around the table [#permalink]

### Show Tags

04 Oct 2010, 09:06
2
KUDOS
Expert's post
3
This post was
BOOKMARKED
Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Glue Michael and Bobby so that they create one unit, so we would have total of 6 units: {1}{2}{3}{4}{5}{MB} --> # of different arrangements of $$n$$ objects around the table (circular arrangements) is is $$(n-1)!$$, so our 6 objects can be arranged in $$(6-1)!=5!$$.

On the other hand Michael and Bobby in 2! ways --> total $$5!*2!=240$$.

Similar question also posted by you: ways-to-sit-around-the-table-102187.html
_________________
Intern
Joined: 01 Sep 2010
Posts: 23
Followers: 1

Kudos [?]: 80 [0], given: 8

Re: Arrangements around the table [#permalink]

### Show Tags

04 Oct 2010, 09:47
D is the OA for this question (I double checked), but I suspected it's wrong.

I also got 240 ways to arrange them around the table...
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15434
Followers: 649

Kudos [?]: 207 [0], given: 0

Re: Seven family members are seated around their circular dinner [#permalink]

### Show Tags

21 Sep 2013, 13:27
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15434
Followers: 649

Kudos [?]: 207 [0], given: 0

Re: Seven family members are seated around their circular dinner [#permalink]

### Show Tags

02 Mar 2015, 10:17
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15434
Followers: 649

Kudos [?]: 207 [0], given: 0

Re: Seven family members are seated around their circular dinner [#permalink]

### Show Tags

29 Jun 2016, 00:42
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
SVP
Joined: 08 Jul 2010
Posts: 1635
Location: India
GMAT: INSIGHT
WE: Education (Education)
Followers: 90

Kudos [?]: 1757 [0], given: 45

Re: Seven family members are seated around their circular dinner [#permalink]

### Show Tags

29 Jun 2016, 02:25
Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Lets consider Michael and Bobby as one individual and fix their position so that all the members do NOT move together while they remain in same order relatively

Now after fixing Michael and Bobby we have 5 other member left to change their positions among themselves which can change positions in 5! ways

but Michael and Bobby and exchange positions between the two in 2! ways'

Hence, Total ways of different arrangements = 5!*2! = 120*2 = 240

_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

Re: Seven family members are seated around their circular dinner   [#permalink] 29 Jun 2016, 02:25
Similar topics Replies Last post
Similar
Topics:
At a dinner party, 5 people are to be seated around a circular table. 5 21 Dec 2011, 13:46
At a dinner party, 5 people are to be seated around a 6 13 Jun 2011, 23:04
8 At a dinner party, 5 people are to be seated around a 11 22 Jul 2011, 14:10
10 At a dinner party, 5 people are to be seated around a circul 7 30 Dec 2013, 08:48
78 At a dinner party, 5 people are to be seated around a 12 08 Jul 2013, 07:41
Display posts from previous: Sort by


text:
# Reconstructing the EFT of Inflation from Cosmological Data

### Submission summary

 As Contributors: Subodh Patil · Subir Sarkar Arxiv Link: https://arxiv.org/abs/1904.00991v1 (pdf) Date submitted: 2019-04-02 02:00 Submitted by: Patil, Subodh Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory High-Energy Physics - Phenomenology Cosmology and Nongalactic Astrophysics Approach: Theoretical

### Abstract

Reconstructions of the primordial power spectrum (PPS) of curvature perturbations from cosmic microwave background anisotropies and large-scale structure data suggest that the usually assumed power-law PPS has localised features (up to $\sim 10\%$ in amplitude), although of only marginal significance in the framework of $\Lambda$CDM cosmology. On the other hand if the underlying cosmology is assumed to be Einstein-de Sitter, larger features in the PPS (up to $\sim 20\%$) are required to accurately fit the observed acoustic peaks. Within the context of single clock inflation, we show that any given reconstruction of the PPS can be mapped on to functional parameters of the underlying effective theory of the adiabatic mode within a 2nd-order formalism, provided the best fit fractional change of the PPS, $\Delta\mathcal{P}_\mathcal{R}/\mathcal{P}_\mathcal{R}$ is such that $(\Delta\mathcal{P}_\mathcal{R}/\mathcal{P}_\mathcal{R})^3$ falls within the $1\,\sigma$ confidence interval of the reconstruction for features induced by variations of either the sound speed $c_\mathrm{s}$ or the slow-roll parameter $\epsilon$. Although there is a degeneracy amongst these functional parameters (and the models that project onto them), we can identify simple representative inflationary models that yield such features in the PPS. Thus we provide a dictionary (more accurately, a thesaurus) to go from observational data, via the reconstructed PPS, to models that reproduce them to per cent level precision.

### Ontology / Topics

See full Ontology or Topics database.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

#### Published as SciPost Phys. 7, 049 (2019)

Resubmission 1904.00991v2 on 16 July 2019

Submission 1904.00991v1 on 2 April 2019

## Reports on this Submission

### Anonymous Report 1 on 2019-5-29 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1904.00991v1, delivered 2019-05-29, doi: 10.21468/SciPost.Report.981

### Strengths

1. Well-written and explained
2. Valuable result allowing accurate (percent level) reconstruction of potential features in the primordial power spectrum from (single-clock) EFT models of inflation
3. Good and clear analytic toy-model examples providing proof of principle

### Weaknesses

1. Discussion of the phase transitions and in particular the additional sharp oscillations that need to be introduced for the Einstein-de Sitter background results.
2. Explanation and discussion of the leading order corrections to the potential

### Report

I enjoyed reading this paper, which is well-written and very clear. The authors show convincingly that they can reconstruct, to second order, estimated features in the primordial power spectrum from time dependent parameters in and EFT description of single-clock models of inflation. First establishing the relations to second order in the features, they apply their method to an analytic toy model example, providing a proof of principle. They then apply their method to a Lambda-CDM and Einstein-de Sitter background model, resulting in 2nd-order reconstructed slow-roll epsilon parameters and the inflationary potential. Their method is a valuable contribution that will allow a more precise reconstruction of (and further constrain the presence of) features, assuming a particular background model. I would have appreciated a more elaborate discussion on the meaning and implications of the reconstruction in the Einstein-de Sitter case, where the result for the slow-roll parameter has more fine-structure and in particular seems to require a sharp oscillation over less than a single e-fold. To explain better how the (claimed) significance of this sharp oscillation compares to the (larger) bumps would be useful. In the discussion, which is very brief, the authors introduce an example of leading order adiabatic corrections to the effective potential that might give rise to the reconstructed features. This paragraph would benefit from a bit more context (instead of just referring to the appendix) and a more specific discussion with respect to the two background scenario's discussed (\Lambda CDM versus Einstein-de Sitter).

Besides these relatively minor points, which I would like to see addressed, I would recommend the manuscript to be published.

### Requested changes

1. Address the significance of the sharp oscillation in the Einstein-de Sitter scenario
2. Expand the discussion on the leading corrections to the effective potential and in particular what that means for the \Lambda CDM and Einstein-de Sitter background results

• validity: high
• significance: good
• originality: ok
• clarity: high
• formatting: good
• grammar: excellent

### Author:  Subodh Patil  on 2019-07-16  [id 563]

(in reply to Report 1 on 2019-05-29)
Category:


text:
# Thread: Generating a subspace (dimension, basis)

1. ## Generating a subspace (dimension, basis)

The following system of vectors generates a subspace. What is the dimension of the generated subspace? Let's show a basis of the subspace. Choose a basis of the subspace from the given generating elements!

[1,1,0,...,0], [0,1,1,0,...,0], ..., [0,0,...1,1], [1,0,...,0,1]

I would be grateful if you could help me!