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## Introduction

Feminist theorists in general have argued that achieving gender equality requires not only favorable institutional and legal frameworks or changes in economic and political structures, but also changes in attitudes (Donovan, 2012; Herzog, 2007; United Nations, 2002). When women hold egalitarian attitudes, they are more likely to defend their rights and transform their resources into favorable outcomes for their lives. In contrast, women with a low support for egalitarianism are more likely to accept inequalities and transmit traditional ideas to their children (Klein, 1984). Promoting women’s adoption of egalitarian attitudes is crucial and a necessary condition for improving their wellbeing. Egalitarian attitudes challenge traditional practices that confine women into the domestic sphere and promote inequalities between women and men (Gerson, 1987).

Evidence points out that women’s adoption of egalitarian attitudes depends on micro and macro conditions. At the microlevel several individual level characteristics such as education level, number of children, and age, influence their attitudes toward gender equality (Baxter & Kane, 1995; Cassidy & Warren, 1996; Pampel, 2011; Thornton, Alwin & Camburn, 1983). Macro-social factors such as media, social programs, place of residence, religion, and institutions, also significantly influence women’s attitudes toward gender equality (Bolzendahl & Myers, 2004). In addition, macro-social factors have been found to moderate the effects of a number of determinants of women’s attitude toward gender equality (Fischer & Hout, 2006). Community characteristics are also found to shape women’s attitudes toward gender equality (Grown, Gupta & Pande, 2005). Even though community characteristics are likely to play a crucial role in shaping women’s attitudes toward gender equality, only a few studies have addressed the effect of residence community size on women’s support for gender equality. Many Mexican communities characterized by high levels of social and economic diversity make excellent research sites for testing research hypotheses related to the role of community characteristic on women’s attitudes toward gender equality.

In order to assess the role played by community characteristics in shaping women’s attitudes toward gender equality, we explore the effect of residential community size on women’s egalitarian attitudes using hierarchical linear modeling. This paper proceeds as follows. The next section explores how egalitarian attitudes are influenced through changes in socio-economic environments. Then, variables and methods are presented. The last section presents the results along with a discussion of our findings.

## Women’s attitudes toward egalitarianism

During the last century, Mexico experienced several socioeconomic and demographic transformations that increased people’s education level and altered individuals’ experiences and social interactions. These transformations were mainly a result of modernization (Parrado & Zenteno, 2005; Ribeiro, 1994). Industrialization brought about several compositional changes within households along side with profound socio-economic environmental changes influencing attitudes toward gender equality (Inglehart, 1990; Inglehart & Norris, 2003; Inglehart & Welzel, 2005).

Toward the end of the last century several Latin American countries turned away from the Import Substitution Industrialization model based on excessive government spending and rejection of laissez-faire economics (Vogel, 1974) to a new economic strategy for development labeled New Economic Model (NEM) (David, Dirven & Vogelgesang, 2000). NEM is based on the principles of Washington consensus that fully endorses the central role of free market forces in economic development polices (Reinhardt & Peres, 2000). Under NEM, Mexico experienced a rapid industrialization and urbanization that brought about massive transfer of labor to urban areas resulting in the incorporation of women into urban occupations (Cerrutti, 1997). During the last four decades most economies in Latin America have not only restructured but also increased their production capacity in response to higher levels of economic openness.

Strategies for economic development were based on trade liberalization. In 1985, Mexico joined the General Agreement on Tariffs and Trade (GATT) and in 1994 signed the North America Free Trade Agreement (NAFTA). As a consequence of economic liberalization there has been a significant proliferation of assembly manufacturing companies (the maquila sector) (Parrado & Zenteno, 2005). After the 1980s Mexican debt crisis (Buffie & Krause, 1989) the maquila sector rose into prominence as a source of female employment. In 1983, women represented 74.5% of the employees in the maquila sector (Parrado & Zenteno, 2005). The development of the service sector also promoted female participation in the labor market. In addition, the fertility decline that began in the 1960s gaining considerable momentum in the 1970s supported by strong state sponsored family planning facilitated women’s entry into the labor market (Rendón, 2008). In sum, prevailing economic policies that promoted industrialization and market liberalization alongside with ongoing sociodemographic changes saw the emergence of expansive job opportunities in urban areas triggering massive migration to large localities. Furthermore, the strong forces of urbanization in Mexico and Latin America contributed to the redefinition of women’s roles in society and career development (Ariza & De Oliveira, 2003).

According to modernization theories, in the face of changing situations people draw on new strategies for adaptation that alter their attitudes toward objects and situations (Chase, 2006; Chirot, 2012; Inglehart, 1990; Inglehart & Norris, 2003; Inglehart & Welzel, 2005). Two perspectives within modernization theories point to the process of development of women’s egalitarian attitudes. Modernization processes that bring about the transition from an agrarian economy to industrial production result in economic improvements. The first perspective postulates that improvements in material circumstances such as economic security influence individuals’ attitudes (Inglehart, 1990; Inglehart & Norris, 2003; Inglehart & Welzel, 2005). According to this perspective, people who experience high levels of insecurity with respect to basic needs are more likely to be supportive of traditional gender roles and conservative religious beliefs (Inglehart & Norris, 2003). Individuals vulnerable to unemployment risks, insecurity, natural disasters, or health problems are more likely to hold these views (Inglehart, 1990). Several studies support the argument that economic progress is highly related to egalitarian beliefs (Inglehart, 1990; Fernández, 2011; Goldin, 1990). As values of equality and equity become codified in employment rules and regulations, attitude toward gender equality appear to improve over time (Littrell & Bertsch, 2013; Deere & Doss, 2006).

The second perspective suggests that modernization brings about a number of changes in non-material, social environments including values and norms that promote a growing emphasis on equality (Bell, 1976). With respect to the second perspective, Bell (1976) argues that changes in people’s attitudes come about as a result of structural changes in sustenance activities, the strength of social institutions, and the emergence of new technology. According to Bell (1976), transformations in the socio-economic and technological environment produce profound modifications in individuals’ conceptions of the world. In the presence of technology, people experience improvements in terms of control of their environmental conditions brought about by rational approaches as well as skilled use of material and non-material resources (Chirot, 2012). Rationality is at the core of modernity and becomes a means through which social coexistence is made possible and regulated (Martinelli, 2005). The development of rational and utilitarian approaches in turn promotes new ideals such as equality with respect to improvements in life chances in a modern world (Inglehart, 1990).

Perhaps one of the most important components in the expansion and sustenance of modernization process is the development of modern educational systems. Increases in economic security along with appreciation of modern values such as gender equality often result from improvements in proportion of the population with high levels of education (Inglehart, 1990; Inglehart & Norris, 2003; Pampel, 2011). With higher levels of education, individuals are exposed to new knowledge and egalitarian views (Bolzendahl & Myers, 2004). Exposure to new values and ideas may also come about as result of exposure to high level of secondary contacts as well as internalization of messages and information carried through various communication channels such as new paper and mass media. Opportunities for engaging in secondary relationships and exposure to mass media tend more likely to be obtained in urban areas than in the rural. In general, improvement in educational levels as well as urban residence are likely to be positively related to women’s attitude toward gender equality in Mexico.

In Mexico, modernization brought about a large-scale migration to communities in volumes proportional to their size resulting in rapid growth of large cities and towns in the twentieth century (Sánchez & Pagán, 2001). The fast paced growth of large towns and cities considerably accelerated the spread of urbanism, altered the structure of traditional families, and facilitated the spread egalitarian norms (Ribeiro, 1994) resulting in wide spread variations in socio-economic contexts affecting levels of gender equality. People residing in urban areas are more exposed to modern egalitarian gender norms compared to individuals who live in smaller localities (Bolzendahl & Myers, 2004). Larger localities facilitate a diversity of human interactions that proliferate the number of autonomous choices individual have to make decisions with respect to their well being and life chances (Inglehart, 1990). Therefore, urban environments are more likely to encourage the development of egalitarian attitudes than rural areas. Variations in environments and socioeconomic development among communities in Mexico are likely associated with disparities in the levels of gender equality (Ariza & De Oliveira, 2003).

## Data and Variables

The information for this study comes from the National Survey on the Dynamics of Households Relationships (ENDIREH) 2011, a national level representative survey conducted in Mexico by the National Institute of Statistics and Geography (INEGI) and the National Institute for Women. It contains information of Mexican women regarding their opinions about their sex roles, decision making, intra-household division of labor, and socio-demographic characteristics. In this study, women’s attitudes toward gender equality is the outcome variable and several socioeconomic characteristics, such as age, education, social interactions, and community size, are included as predictors.

Women’s attitudes toward gender equality. Women’s attitudes toward gender equality are measured through their opinions on female roles within and outside the household. This dimension comprises of 12 dichotomous variables representing agreement or disagreement stated in response to the following questions: (i) Does a good wife have to obey her husband in everything he orders? (ii) Can a woman choose her friends even if her husband does not like them? (iii) Does a woman have the same ability as a man to earn money? (iv) Is it a woman’s duty to have sexual intercourse with her husband even if she does not want to? (v) Are women free to decide if they work outside the home? (vi) Should men and women share caregiving tasks? (vii) Do you agree that men and women should have the same rights to make their own decisions? (viii) Do you agree that men and women should have the same freedoms? (ix) Do you agree that women have the right to defend themselves and denounce any physical harm or aggression? (x) Do you agree that women have the chance to decide over their own life? (xi) Do you agree that women have the right to live a life free of violence? (xii) Do you agree that women should have the right to decide when and the number of children they have? If women responded with an egalitarian position, the answer was coded as 1, and 0 otherwise. The twelve items were added to create a scale, which varied from 0 to 12. Larger numbers indicate higher levels of support for gender equality.

Women’s employment status. This is a dichotomous variable; 1 if they are employed and 0 otherwise.

Women’s education. Women’s education is indicated by the number of years of women’s formal education. Women were asked their highest grade obtained and years of education in the next grade. This information was coded as the number of years that are required in Mexico to complete the grade plus additional years of education completed in pursuit of the next grade.

Women’s age. This is a continuous variable measured in years.

Social interactions. To measure social interactions, respondents were asked six questions regarding their activities: if they (i) go out with friends?; (ii) talk with their neighbors?; (iii) meet their relatives?; (iv) attend meetings with organizations?; (v) practice a team sport? Other three questions if they had an emergency and had to borrow money, –excluding their partner– would you ask for money to your (vi) friends, (vii) neighbors, (viii) relatives. Responses of the eight items are dichotomous, with “yes” coded 1, and 0 otherwise. As religion is an institution that promotes traditional practices, the question (ix) do you attend religious ceremonies? This item is coded 1 if they responded “no”, and 0 otherwise. The nine items are added to create a composite scale.

Empowerment. This variable is a composite scale estimated using factor scores from principal components analysis. The empowerment variable is composed of four variables: housework, number of children, partner’s education, and age at marriage. Each of the four variables had a factor loading above 0.60. Partner’s education is indicated by the number of years of formal education. The variable ‘number of children’ is the total number of children alive. Regarding housework, respondents were asked about their and other household members’ participation in domestic and caregiving tasks. There were four questions related to intrahousehold division of labor: usually (i) who takes care of girls or boys? (ii) who takes care of elderly people? (iii) who does the domestic chores? (iv) who pays utilities and do daily expenses? Possible responses are: the respondent, partner, both (the respondent and her partner), daughters, sons, a domestic worker, another household member, or a person who is not a household member. When women identified themselves as the primary undertakers of each of the many household activities specified, it is coded as 1 and 0 otherwise. Because this variable intends to measure women’s burden inside the household, an additive scale is built.

Community size. The ENDIREH 2011 contains information regarding the state and population range of the community where women live. Community size is classified into six categories depending on the population size: i) less than 2,500 inhabitants; ii) between 2,500 and 5,000 inhabitants; iii) 5,000 or more, but less than 15,000 inhabitants; iv) 15,000 or more, but less than 50,000; v) between 50,000 and 100,000 inhabitants; vi) 100,000 or more inhabitants. The data set provided information on community size as well the state of residence of each respondent in the sample. There are 32 states and 6 population ranges. Because in the survey all states do not have communities with all population ranges, 177 groups were formed.

## Method

Hierarchical linear modeling technique is widely regarded as an appropriate method for to analyze data structures where women (level 1) are nested within communities (level 2). The primary focus of the analysis is on the relationship between ‘women’s attitudes toward gender equality’ and the level-1 predictors as well as community size (level 2). Level-1 predictors are women’s age, years of education, employment status, social interactions, and empowerment level. Hierarchical linear model analysis was conducted in four stages: unconstrained model (null), random coefficients model, means as outcome model, and intercepts and slopes as outcomes model. Level-1 predictors were entered as group centered variables in order to obtain more accurate estimates of the intercept and to analyze the effects of the level-1 and level-2 predictor variables independently. In Equation 2a and 2b, community size is added as a grand centered predictor. Effects were estimated using restricted maximum likelihood.

Level-1 model, level-2 model, and mixed model were estimated using Equations 1 and 2. Equations 1 and 2 present the intercepts and slopes as outcomes model.

Level-1 model:

(1)
${Y}_{\mathit{\text{ij}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\beta }_{\mathit{\text{oj}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\beta }_{\mathit{\text{kj}}}{X}_{\mathit{\text{ij}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{r}_{\mathit{\text{ij}}}$

where:

Yij = women’s attitudes toward gender equality for woman i (level-1 unit) in community j (level-2 unit).

Xij = values on the level-1 predictors for woman i in community j.

βoj = intercept for community j.

βkj = regression coefficients associated with predictor k for in community j.

rij = random error associated with woman i in community j.

Level-2 model:

(2)
$\begin{array}{cc}{\beta }_{0j}\text{\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}{\gamma }_{00}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\gamma }_{01}{G}_{j}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{U}_{0j}& \left(a\right)\\ {\beta }_{\mathit{\text{kj}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\gamma }_{k0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\gamma }_{k1}{G}_{j}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{U}_{\mathit{\text{kj}}}& \left(b\right)\end{array}$

where:

β0j = intercept for community j.

βkj = slope of predictor k for community j.

Gj = community size in community j.

γ00 = overall mean intercepted adjusted for values on the predictors.

γk0 = overall mean intercepted of predictor k adjusted for values on predictor k.

γ01 = regression coefficient associated with predictors relative to level-2 intercept.

γk1 = regression coefficient associated with predictor k relative to level-2 intercept.

U0j = random effects of community j adjusted for predictors on the intercept.

Ukj = random effects of community j adjusted for predictor k on the slope.

The unconstrained model:

Level-1 model:

(3)
${Y}_{\mathit{\text{ij}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\beta }_{\mathit{\text{oj}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{r}_{\mathit{\text{ij}}}$

Level-2 model:

(4)
${\beta }_{0j}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\gamma }_{00}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{U}_{0j}$

The random-intercepts model:

Level-1 model:

(5)
${Y}_{\mathit{\text{ij}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\beta }_{\mathit{\text{oj}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\beta }_{\mathit{\text{kj}}}{X}_{\mathit{\text{ij}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{r}_{\mathit{\text{ij}}}$

Level-2 model:

(6)
$\begin{array}{c}{\beta }_{0j}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\gamma }_{00}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{U}_{0j}\\ {\beta }_{\mathit{\text{kj}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\gamma }_{k0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{U}_{\mathit{\text{kj}}}\end{array}$

The means as outcome model:

Level 1 Model:

(7)
${Y}_{\mathit{\text{ij}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\beta }_{\mathit{\text{oj}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{r}_{\mathit{\text{ij}}}$

Level-2 model:

(8)
$\begin{array}{c}{\beta }_{0j}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\gamma }_{00}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\gamma }_{01}{G}_{j}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{U}_{0j}\\ {\beta }_{\mathit{\text{kj}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\gamma }_{k0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\gamma }_{k1}{G}_{j}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{U}_{\mathit{\text{kj}}}\end{array}$

## Results

A fully unconstrained model was first estimated to determine whether there is significant variability in women’s attitudes toward gender equality (see Equations 3 and 4). The chi-square value of the unconstrained model is statistically significant (χ2 = 10,382.73, p < 0.001). A significant chi-square reveals that there exists variance in women’s attitudes toward gender equality by the community size groupings. The intra-class correlation (ICC) obtained from the unconstrained model was 0.0847 (τ00/(τ00 + σ2)), suggesting that 8.47% of the variance in women’s attitudes toward gender equality is between-localities and the rest is among women within a community. Given the existence of variance at both levels, predictor variables are included at each level in the subsequent phases of analysis.

To estimate the random coefficients model, level-1 predictors were included in the analysis and the variance component of the intercepts was assessed (see Equations 5 and 6). The estimated variance among the means is 0.1659 with a significant chi-square value (χ2 = 12,404.26, p < 0.001). These results indicate the presence of significant differences in the mean levels of women’s attitudes toward gender equality among the 177 localities. The next step in the analysis involved evaluating the variance components of the slopes as specified by the random coefficients model. Table 1 presents the regression coefficient associated with each of the selected determinants of women’s attitudes toward gender equality.

Table 1

Regression coefficients of women’s attitudes toward gender equality (random coefficients model).

Fixed Effect Coefficient Standard Error t-ratio Approx. d.f. p-value

Intercept, π0
Intercept, β00 10.944451 0.031399 348.56 176 <0.001
Age slope, π1
Intercept, β10 –0.013020 0.000589 –22.09 176 <0.001
Education slope, π2
Intercept, β20 0.094892 0.002636 36.00 176 <0.001
Employment status slope, π3
Intercept, β30 0.140313 0.012595 11.14 176 <0.001
Social interaction slope, π4
Intercept, β40 0.080200 0.005335 15.03 176 <0.001
Empowerment slope, π5
Intercept, β50 0.090117 0.006900 13.06 176 <0.001

Source: Own elaboration with information from the ENDIREH 2011.

All the variables in the proposed model of women’s attitudes toward gender equality are statistically significant at the 0.01 level. Women’s age is negatively related to women’s attitudes toward gender equality (β = –0.013020, p < 0.001). Meanwhile, women’s education (β = 0.094892, p < 0.001), social interactions (β = 0.080200, p < 0.001), participation in the labor market (β = 0.140313, p < 0.001), and empowerment (β = 0.090117, p < 0.001) positively affects women’s egalitarian attitudes.

Table 2 presents the variance components of the intercept and slopes. The results suggest that 18.16% ((σ2null – σ2random)/σ2null) of the variance in women’s attitudes is due to level-1 predictors when community size is held constant. As shown in Table 2, the variance of the slopes are significant, which indicates that the relationship between the selected predictors and women’s attitudes vary significantly across communities in Mexico.

Table 2

Variance components of the intercept and slopes (random coefficients model).

Random Effect Standard
Deviation
Variance
Component
d.f. χ2 p-value

Intercept, r0 0.40777 0.16628 176 12,683.20 <0.001
Age slope, r1 0.00557 0.00003 176 381.71 <0.001
Education slope, r2 0.02902 0.00084 176 845.33 <0.001
Employment slope, r3 0.08770 0.00769 176 256.19 <0.001
Social interactions slope, r4 0.04450 0.00198 176 333.27 <0.001
Empowerment slope, r5 0.04290 0.00184 176 261.46 <0.001
level-1, e 1.20876 1.46109

Source: Own elaboration with information from the ENDIREH 2011.

In the next phase, community size variable was included in the level-2 model and level-1 predictors were removed (See Equations 7 and 8). In the random coefficients and means as outcomes model respectively, variables were entered independently. This model assesses whether the significant intercept variance found is related to community size. The regression coefficient relating community size to women’s attitudes toward gender equality are provided in Table 3. Community size coefficient was positive and significant (β = 0.171815, p < 0.001). Thus, result from the means as outcomes model indicate that women’s attitudes toward gender equality differ when women are grouped by the community size of their residence. Women in bigger localities are more likely to hold more egalitarian attitudes than women living in smaller localities. Community size explains 56.28% ((τ00– τMEANS)/τ00) of the between measure variance in women’s attitudes toward gender equality.

Table 3

Regression coefficients of women’s attitudes toward gender equality (means as outcomes model).

Fixed Effect Coefficient Standard Error t-ratio d.f. p-value

Intercept, π0
Intercept, β00 10.941924 0.021562 507.45 175 <0.001
Community size, β01 0.171815 0.012132 14.16 175 <0.001
σ2 = 1.78547
τ00 = 0.07227

Source: Own elaboration with information from the ENDIREH 2011.

Finally, the intercepts and slopes as outcomes model (see Equations 1 and 2) was estimated with all variables included simultaneously to test the interactions of the level-1 predictors with the level-2 community size variable. This model assesses whether the significant differences in intercepts and slopes of level-1 predictors are related to community size. Table 4 presents the fixed effects of the intercepts and slopes as outcome model.

Table 4

Regression coefficients of women’s attitudes toward gender equality (intercepts and slopes as outcomes model).

Fixed Effect Coefficient Standard Error t-ratio Approx. d.f. p-value

Intercept, π0
Intercept, β00 10.943069 0.021743 503.280 175 <0.001
Community size, β01 0.170896 0.012308 13.885 175 <0.001
Age slope, π1
Intercept, β10 –0.013035 0.000608 –21.443 175 <0.001
Community size, β11 0.000332 0.000304 1.090 175 0.277
Education slope, π2
Intercept, β20 0.095729 0.002085 45.909 175 <0.001
Community size, β21 –0.012335 0.001036 –11.910 175 <0.001
Employment status slope, π3
Intercept, β30 0.144674 0.013494 10.721 175 <0.001
Community size, β31 –0.022158 0.006192 –3.578 175 <0.001
Social interactions slope, π4
Intercept, β40 0.080339 0.005424 14.813 175 <0.001
Community size, β41 –0.004879 0.002622 –1.861 175 0.064
Empowerment slope, π5
Intercept, β50 0.086193 0.007520 11.461 175 <0.001
Community size, β51 –0.009236 0.003505 –2.635 175 0.009

Source: Own elaboration with information from the ENDIREH 2011.

Community size is positively related to women’s attitudes toward gender equality (β = 0.170896, p < 0.001). The interaction terms are significant only for women’s education (β = –0.012335, p < 0.001), participation in the labor market (β = –0.022158, p < 0.001), and empowerment (β = –0.009236, p < 0.009). The significance of the interaction terms provides empirical support to the presence of cross-level interaction of community size with women’s education, participation in the labor market, and empowerment. It can be concluded that community size influences the strength of the relationships of variables such as women’s education and participation in the labor market with their attitudes toward gender equality. The impact of female employment, women’s education and empowerment is higher in smaller localities.

Table 5 presents the residual variance of the slopes. There is a reduction when comparing results from Table 5 to the variance components in Table 2. The chi-square values of all variance components indicate that there is statistically significant residual variance in the slopes. In addition, results suggest that 81.45% (τ00(null)/(τ00(null) + τ00(fitted))) of the parameters variation in women’s attitudes toward gender equality is explained by the community size variable.

Table 5

Variance components of the intercept and slopes (intercepts and slopes as outcomes model).

Random Effect Standard
Deviation
Variance
Component
d.f. χ2 p-value

Intercept, r0 0.27505 0.07565 175 2903.74421 <0.001
Age slope, r1 0.00557 0.00003 175 374.44745 <0.001
Education slope, r2 0.01809 0.00033 175 314.37194 <0.001
Employment status slope, r3 0.07672 0.00589 175 228.60283 0.004
Social interactions slope, r4 0.04362 0.00190 175 323.63117 <0.001
Empowerment slope, r5 0.04848 0.00235 175 242.77608 <0.001
level-1, e 1.20873 1.46103

Source: Own elaboration with information from the ENDIREH 2011.

Figures 1, 2 and 3 show the effect of women’s education, employment status, and empowerment level, respectively, on their attitudes toward gender equality by community size. Women who live in large-sized localities hold in average more egalitarian attitudes than women who reside in smaller localities. The differences in the slopes show that the effect of women’s education, employment and empowerment levels is higher among women living in smaller localities.

Figure 1

Women’s attitudes toward gender equality, by community size and women’s education.

Source: Own elaboration with information from ENDIREH, 2011.

Figure 2

Women’s attitudes toward gender equality, by community size and employment status.

Source: Own elaboration with information from ENDIREH, 2011.

Figure 3

Women’s attitudes toward gender equality, by community size and Empowerment.

Source: Own elaboration with information from ENDIREH, 2011.

## Conclusions

This paper examined the impact of several predictors and community size on women’s attitudes toward gender equality in Mexico. Results from this paper highlight the importance of contextual factors in women’s adoption of egalitarian ideas. Using hierarchical linear modeling we found that women’s attitudes toward gender equality are defined by women’s age, education, employment status, social interactions, and empowerment. Meanwhile, community size affects the extent to which women’s education, employment status, and empowerment affect their attitudes toward gender equality.

Women’s attitudes toward gender equality are crucial to promoting their wellbeing. When women hold egalitarian attitudes, they are more likely to effectively allocate available resources to their wellbeing (Gerson, 1987). However, mean levels of predictors of women’s egalitarian attitudes differ significantly among communities in Mexico. At the aggregate level, community size stimulates economic growth with a much larger share of their populations benefitting from higher levels of health and education. Similarly, a significantly higher proportion of women in small communities are more likely to have large proportion of children in poverty, and low levels of education than women residing in larger localities. In Mexican rural areas, only 28.5% of the economically active population are women. In contrast, in urban areas women represent 40.2% of the economically active population (INEGI, 2014). In both rural and urban areas, younger women are more likely to be economically active. Since age is negatively related to egalitarian ideas, it can be expected that as young cohorts replace older cohorts, women’s attitudes toward gender equality is likely to increase in Mexico. However, younger women living in larger localities tend to gain egalitarian attitudes at a much faster pace than those residing in smaller localities. As a consequence, attitudinal gap in terms of support for gender equality may significantly widen as community size differences increase.

This study proposed an ecological framework to analyze egalitarian attitudes among women. At the individual level, women’s characteristics determine their attitudes toward gender equality along with microsystem factors such as partner’ and households’ characteristics. At the macro-level, contexts such as urban significantly influence the extent to which individual characteristics and microsystem factors affect women’s egalitarian attitudes toward gender equality than those in rural localities. From a policy perspective, individual demographic characteristics such as age, number of children, and age at marriage are far less manipulable than social characteristics including education, participation in the labor market, and social interactions that can be improved by increasing access to social resources and enforcing labor laws. At the microsystem level, social policies should incorporate gender equality awareness programs that encourage the growth of egalitarianism among family members and coworkers. Regarding community characteristics, social policies should be geared toward ensuring all women’s access to education, health services, and public institutions. The quality of public services should remain similar across localities. It is important to highlight that interventions at one any level can have multiplicative effects on women’s attitudes toward gender equality and promote value and cultural change.

This paper analyzed the interaction between macro and micro determinants and increased the understanding on women’s attitudes toward gender equality in Mexico. A challenge for future research is to explore and identify other variables related to contextual factors that influence women’s egalitarian attitudes.


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# How much electrical power does one generator produce on a large turbofan?

From my related question, now I want to know just how much power does the generator produce? In other words, what is the max rating of the generator? It looks so small and that surprised me, but since it spins very fast it could get high power in a small package.

The generator from any large turbofan is fine. The answer will also help us figure how much electrical power a large commercial airliner could need at any given moment.

• This question is really too broad and dependent upon the engine in question. – Carlo Felicione Sep 5 '17 at 5:33
• Keep in mind there is a close correspondence between copper mass and kVA, and a close inverse correspondence between iron (magnetic) rotor mass and frequency. Hence the high frequencies of aircraft really reduce the iron mass. Nothing can reduce the copper mass. Meanwhile railways love 16.67 and 25 Hz because weight translates into traction, so they don't mind the iron mass. – Harper - Reinstate Monica Sep 5 '17 at 16:00

This is one of the four Variable Frequency Generator (VFG) of the Airbus A380. It's apparent power is 150 kVA.

Rotor, source: Safran

The total power available from the engines is 600 kVA.

Seen from the other side:

Stator, source: Thales

The A380 is a one of these "more electrical" planes, where hydraulics tends to be replaced by electric devices, and the electric circuits have been optimized by electronics and variable frequency. There are secondary sources for electricity: APU (auxiliary power unit) and RAT (ram air turbine). Overall view from the book Civil Avionics Systems (Moir, Seabridge, Jukes):

Source: Civil Avionics Systems

In case of total failure, the ultimate RAT backup is totally electric-based contrary to conventional aircraft where some hydraulics is also provided by the RAT to move the control surfaces.

• Interesting. Why do they measure it in apparent power? 150 kVA should be equivalent to 150 kW, since power is just volts times amps. – DrZ214 Sep 5 '17 at 5:28
• Power and apparent power are only equivalent when the load is a pure resistor. When it's a general impedance, that is when there is a capacity or inductance component, then voltage and current don't vary in phase, one is delayed (phase difference $\phi$), and instantaneous $u \times i$ is therefore different than when in phase (there is a real and an imaginary part appearing in the product). See Wikipedia. Actually the relationship is $P=U \times I \times cos \space \phi$ – mins Sep 5 '17 at 6:41
• @DrZ214 Or harmonics, or other non-linearity. Power factor is rarely equal to one in AC systems. – Someone Somewhere Sep 5 '17 at 8:09
• @aroth so do electricians. Unfortunately, lag and harmonics crashed the party, and all of us have to deal with it. If you draw 150kW from your 150kVA generator, expect magic smoke. – Harper - Reinstate Monica Sep 5 '17 at 16:04
• @DrZ214 because the actual limits of pretty much any kind of generating equipment are going to come at some number of VA. If you give it a rating in W, it has to come with a big fat asterisk, but if you rate it in VA, it will deliver rated VA under specified conditions. – hobbs Sep 5 '17 at 16:42

There are several max ratings of a particular generator, depending on the duty cycle:

• peak rating (5 sec);
• max. Continuous (no time limit);
• and often an intermediate value for 5 minutes is given.

The generator produces internal heat together with the electricity, which is not easy to remove by forced cooling and must dissipate through convection. The generator is rated in kVA, not kW, because the three phases consist of sine waves which are not in phase with each other. The kVA rating denotes the electrical power used, a kW rating is for the useful power output.

The Integrated Drive Units consist of a drive mechanism and the generator itself. The generator is a fast spinning device, rotating at a high, constant velocity. Since the rotational speed of the LP shaft is not constant, a constant speed mechanism is added between the generator and the LP axis. Classic technology is a hydraulic-mechanical solution with associated losses (like the automatic transmission in a motor car), newer technology such as presented here uses Constant Speed Variable Transmission.

The generator example in the picture is typical for single aisle and many twin aisle aircraft, and has the following power rating:

• Cont.Rating : 90kVA
• 5 min. Rating: 112.5kVA
• 5 sec. Rating : 150kVA
• Frequency:400±5Hz(3)
• Voltage:115/200V
• Size:375L×460W×330Hmm
• Weight:54.0kgf(dry)

Three IGTs of this class make up the AC power capability of a typical single isle aircraft such as the A320, the third generator being driven off of the APU and normally not operating during cruise. The A380 uses six generators of this same rating.

Power demand for some systems scale up with aircraft size. This site gives electrical power demand of a typical classic wide-body jet (pre-bleedless-air):

• In-flight entertainment and galley demand scales up with passenger count (33%).
• Air conditioning scales up with cabin volume (19%).
• Fuel system demand scales up with engine size (10%).
• Wing anti-icing scales up with wing span (8%).

Other electricity users such as the avionics are a relatively constant factor as a function of aircraft size. The same site gives the power demand as a function of flight time for a single aisle aircraft, showing that one 90 kVA generator can drive essential loads plus utility loads. The dotted lines in the graph are for continuous rating.

A380

The A380 has six AC generators on board: one driven by an engine each, and two APU generators. From the A380 FCOM:

Each engine has one generator. These engine-driven generators are the main source of electrical power. When an engine is running, its generator provides 115 V AC power at variable frequency. This frequency ranges from 360 Hz to 800 Hz, depending on the N3 rotation speed of the engine.

And

At least two engine generators (or APU generators) are necessary to supply the entire network.

B787

The B787 has a different arrangement. The B787 has no engine bleed air, and systems normally powered by bleed are now powered by AC electric power. A bleedless aircraft requires considerable extra generating capacity in order to power Cabin Air Compressors (about 19% of total power load for a wide-body) and electrical wing anti-icing systems (about 8% of power load), and the Boeing 787 incorporates four engine-driven Variable Frequency Starter-Generators (VFSGs) of 250 kVA apiece in order to accommodate these loads.

• Three such IGTs make up the typical AC power capability of a typical single isle aircraft, such as the A320. Three generators for a twin-engine A320? Where does the 3rd generator go? – DrZ214 Sep 5 '17 at 8:51
• The third one is powered by the APU. – Koyovis Sep 5 '17 at 10:10

Variable Frequency Starter Generator

The variable frequency starter generator is a six-pole machine within an aluminum housing driven directly from the main engine gearbox. The generator is a brushless, three-phase, alternating current, and variable frequency synchronous machine. It has a nominal rating of 235 volts alternating current (VAC), 250 kVA, three phases, and 360–800 Hz output.

The 'all-electrical' Boeing 787 has two of these in each engine, for a total of four. Each is rated at 250 kVA. (For comparison, on an early 737 it was 45 kVA.)

The 787 has four times the potential electric generation of the 777 – 1.45 megawatts. The generators (four at 250 kVA, two per engine, and two at 225 kVA on the APU) produce 235 VAC for the large users on the bus – otherwise the traditional 115 VAC and 28 VDC are operational. Seventeen Remote Power Distribution Units (RPDU) power about 900 loads through the aircraft. The power distribution system is in the aft belly along with the Power Electronics Cooling System (PECS). It’s liquid cooled for the large motors, along with an Integrated Cooling System (ICS) for use by the galley carts, cabin air and IFE.

Total power generation potential is 1.45 MW.

Sources:

• and that 1.45 MW is enough for 150 homes running quite a range of electrical appliances including air conditioners – Hanky Panky Sep 5 '17 at 8:32

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Fibonacci Sequence | a Pattern of Natural Beauty| Binet formula

## Introduction

Fibonacci sequence $F_n$ is defined recursively in the following way: Let $F_0=0$ and $F_1=1$ we define $F_{i}=F_{i-2}+F_{i-1}$, for any integer $i > 2$.

Ted talk on the magic of Fibonacci numbers, by Prof. Arthur T. Benjamin

Consider the following numbers:

$\phi=\frac{1+\sqrt{5}}{2} \quad \mbox{and} \quad \psi=\frac{1-\sqrt{5}}{2}$

This two numbers are strongly related to Fibonacci numbers. The first formula to consider known as Binet Formula'':

#### Binet Formula

$$\tag{1} F_n=\frac{\phi^n-\psi^n}{\phi-\psi}$$

#### Proof:

It is fairly easy to prove the Binet formula by using mathematical induction. The base case

$F_0=\frac{\phi^0-\psi^0}{\phi-\psi}$ is readily verified.
Inductive Step:: Suppose that the Binet formula holds for integer $m$.

Notice that both $\psi$ and $\phi$ satisfy $x^2-x-1=0$. Thus,

$\phi^2= \phi+1$ and $\psi^2=\psi+1$. Using these two equalities we get $\begin{array}{l} \frac{\phi^{m+1}-\psi^{m+1}}{\phi-\psi} &= \frac{\phi^{m-1}\phi^2-\psi^{m-1}\psi^2}{\phi-\psi} \\ &= \frac{\phi^{m-1}(\phi+1)-\psi^{m-1}(\psi+1)}{\phi-\psi} \\ &= \frac{\phi^m-\psi^m}{\phi-\psi}+\frac{\phi^{m-1}-\psi^{m-1}}{\phi-\psi} \\ &= F_m+F_{m-1} \\ &= F_{m+1} \end{array}$ $\blacksquare$

Consider the sequence

$\frac{F_{n+1}}{F_n}$

What is the limit of this sequence, when $n$ goes to infinity? Does this sequence converge? Well, it turned out to be that this sequence converges to the number $\phi$. The number $\phi$, also known as the ''Golden Ratio'', is a very important number in mathematics appearing everywhere, you can find it in art, architecture, biology, nature and many many more.

Theorem $\frac{F_{n+1}}{F_n} = \phi$ Proof:

Multiplying $\phi$ and $\psi$ yields $\phi\psi=-1$ and from this we get

$\phi = -\psi^{-1}.$ Using the Binet formula we get $$\begin{array}{l} \tag{2} \frac{F_{n+1}}{F_n} &= \frac{\phi^{n+1}-\psi^{n+1}}{\phi^n-\psi^n} \\ &=\phi \left( \frac{\phi^n}{\phi^n-\psi^n} + \frac{\psi^{n+2}}{\phi^n-\psi^n}\right) \\ &=\phi \left( \frac{\phi^n-\psi^n}{\phi^n-\psi^n} + \frac{\psi^n+\psi^{n+2}}{\phi^n-\psi^n}\right) \\ &=\phi \left(1 + \frac{\psi^n+\psi^{n+2}}{\phi^n-\psi^n}\right) \end{array}$$ Since $|\psi|<1$, we get $$\tag{3} \lim_{n\to\infty} \psi^n = 0.$$ Since $\phi > 1$, we get $$\tag{4} \lim_{n\to\infty} \phi^n = \infty.$$ From equalities (3) and (4) we get $$\lim_{n\to\infty}\frac{\psi^n+\psi^{n+2}}{\phi^n-\psi^n}=0.$$ The last equality and equality (2) implies that $$\lim_{n\to\infty}\frac{F_{n+1}}{F^n}=\phi.$$ $\blacksquare$

Consider the sum $$\sum_{i=0}^n F_i$$

It is easy to see that $\phi-\psi=\sqrt{5}$ and that both $\phi=1-\psi$ and $\psi=1-\phi$. Using these equalities the Binet formula and the formula for geometric series we get

$$\begin{array}{l} \sum_{i=0}^{n} F_i &= \frac{1}{\sqrt{5}}\sum_{i=0}^n \phi^i-\psi^i \\ &= \frac{1}{\sqrt{5}}\left(\frac{\phi^{n+1}-1}{\phi-1}-\frac{\psi^{n+1}-1}{\psi-1}\right) \\ &= \frac{1}{\sqrt{5}}\left(\frac{(\phi^{n+1}-1)(\psi-1)-(\psi^{n+1}-1)(\phi-1)}{(\phi-1)(\psi-1)}\right) \\ &= \frac{1}{\sqrt{5}}\left(\frac{(\phi^{n+1}-1)(-\phi)- (\psi^{n+1}-1)(-\psi)}{-1}\right) \\ &= \frac{1}{\sqrt{5}}\left((\phi^{n+1}-1)\phi- (\psi^{n+1}-1)\psi \right) \\ &= \frac{1}{\sqrt{5}}\left(\phi^{n+2}-\phi- \psi^{n+2}+\psi \right) \\ &= \frac{1}{\sqrt{5}}\left(\phi^{n+2}-\psi^{n+2}-(\phi-\psi) \right) \\ &= \frac{1}{\sqrt{5}}\left(\phi^{n+2}-\psi^{n+2}-\sqrt{5}\right) \\ &= \frac{\phi^{n+2}-\psi^{n+2}}{\sqrt{5}}-1 \\ &= F_{n+2}-1. \end{array}$$ and from this we get $$\sum_{i=0}^n F_i = F_{n+2}-1.$$ Notice also that $F_n=F_{n+1}-F_{n-1}$, for any integer $n$ greater than $0$. We could have used this simple property to expand the above sum into telescopic series and get the same result. Have we chosen to go this path we would have got $$\begin{array}{ll} \sum_{i=0}^n F_i &= F_0 + (F_2- F_0)+( F_3-F_1)+ ( F_4- F_2)+(F_5- F_3) + \\ \; & \;\;\; \ldots + ( F_{n-1}- F_{n-3})+(F_n- F_{n-2}) +(F_{n+1}- F_{n-1}) \\ \ &= -F_1+F_n+F_{n+1} \\ \ &= F_{n+2}-1. \end{array}$$ This video (by Mark Eichenlaub) contains a visualized form of this telescopic series. $\blacksquare$

A Fibonacci number is any given element in the Fibonacci sequence. Now, given a positive number $x$ how do we know whether $x$ is a Fibonacci number or not?

### Theorem 1 (Successive Fibonacci Number Test)

Let $F_n$ be a Fibonacci sequence, let $x$ be any nonnegative integer and let $y$ be any positive integer.
The equality $y^2-xy-x^2=\pm 1$ if $\{ F_n, F_{n+1} \} = \{x,y\}$.

### Proof:

Without loosing generality, we assume that $F_n=x$ and let $F_{n+1}=y$, in the right part or the above if statement.

#### $\Rightarrow$

Let us denote $$\tag{5} y^2-xy-x^2=\pm 1.$$ Now, given equality (5) we need to prove that $F_n=x$ and $F_{n+1}=y$. If $x>y$, then $y^2-xy-x^2 < y^2-y^2-y^2 = -y^2 < -1$ and this contradict equality (5), thus $x\le y$. In addition if $x\ge 1$, then $y\ge 1$. Let us denote the set of all positive integers with $\mathcal{N}^{+}$ and the set of all nonnegative integers with $\mathcal{N}_0$. Now, consider the following set $A = \Big\{(x,y)\;\Big|\; y^2-xy-x^2=\pm 1, \mbox{ where } x\in \mathcal{N}_0 \mbox{ and } y \in \mathcal{N}^{+} \Big\}.$ Define $(x_0,y_0) < (x_1,y_1) \Longleftrightarrow y_0 < y_1$ for any $(x_0,y_0)$ and $(x_1,y_1)$ in $A$.
We need to prove that for any $(x,y)$ in $A$ there exists nonnegative integer $n$ such that $F_n=x$ and $F_{n+1}=y$. Namely $$\tag{6} \forall (x,y) \in A \big(\exists n \in \mathcal{N}_0( F_n=x \mbox{ and } F_{n+1}=y ) \big)$$ We prove (6) by induction. The base case, where $(x,y)=(0,1)$ is easily verified (we get $F_0=x=0$ and $F_1=y=1$). Let $(a,b)$ be any member of $A$. Suppose (this is the induction assumption) that for (6) holds for any $(u,v)$ that is less than $(a,b)$. If we prove (6) for $(a,b)$, then (by the axiom of induction) we done. Consider the pair $(b-a,a)$. Clearly, $b-a$ is nonnegative integer and, of course, $a$ is positive integer. Since $$\begin{array}{l} a^2-(b-a)a-(b-a)^2 &= a^2-ab+a^2-b^2+2ab-a^2 \\ &= -b^2+ab+a^2\\ &= -(b^2-ab-a^2) \\ &= -\pm 1 \\ &= \pm 1 \end{array}$$ the pair $(b-a,a)$ is in $A$. Given the fact that $a < b$ we get $(b-a,a) < (a,b)$, thus using the induction assumption we may infer the existence of positive integer $k$ such that $b-a=F_k$ and $a=F_{k+1}$. But, $$\begin{array}{l} b &=(b-a)+b \\ &=F_k+F_{k+1} \\ &=F_{k+2}, \end{array}$$ hence, $a=F_{k+1}$ and $b=F_{n+1}$, and since $(a,b)\in A$, the pair $(a,b)$ satisfies formula (6). $\square$

### $\Leftarrow$

Let us denote $x=F_n$ and $y=F_{n+1}$. It be easily verified that $$\begin{array}{l} \psi\phi=-1 \\ \phi+\psi=1 \\ \phi^2=\phi+1 \quad \mbox{and} \quad \psi^2=\psi+1 \end{array}$$ Using these equalities and Binet formula, we get $$\begin{array}{ll} y^2-xy-x^2 &=& (F_{n+1})^2-F_nF_{n+1}-(F_n)^2 \\ &=& \frac{\left(\phi^{n+1}-\psi^{n+1}\right)^2}{5}- \frac{\left(\phi^n-\psi^n\right)\left(\phi^{n+1}-\psi^{n+1}\right)}{5}-\frac{\left(\phi_n-\psi_n\right)^2}{5} \\ &=& \frac{1}{5}\Big(\phi^{2n}(\phi+1)-2(\phi\psi)^{n+1}+\psi^{2n}(\psi+1)-\phi^{2n+1}-\\ &=& \psi^{2n+1}+ \phi^n\psi^{n+1}+\psi^n\phi^{n+1}-\phi^{2n}+2\phi^n\psi^n-\psi^{2n}\Big) \\ &=& \frac{1}{5}\Big(\phi^{2n+1}+\phi^{2n}+2\cdot(-1)^n+\psi^{2n+1}+\psi^{2n}-\\ &=& \phi^{2n+1}-\psi^{2n+1}+ (-1)^n\psi+(-1)^n\phi-\phi^{2n}+2(-1)^n-\psi^{2n}\Big) \\ &=& \frac{1}{5}\left(2\cdot (-1)^n+ (-1)^n(\psi+\phi)+2(-1)^n\right) \\ &=& \frac{1}{5}\left(2\cdot (-1)^n+ (-1)^n+2(-1)^n\right) \\ &=& \frac{1}{5}\left(5\cdot (-1)^n\right) \\ &=& (-1)^n \\ &=& \pm 1 \\ \end{array}$$ $\blacksquare$

Still we need to know how to determine whether a given number is a Fibonacci number or not?

### Theorem 2 (Fibonacci Number Test)

Let $x$ be any positive integer. $x$ is a Fibonacci number if either one of $5x^2+4$ or $5x^2-4$ is a perfect square.

### Proof:

Recall that any integer $q$ is a perfect square if both $q$ and $\sqrt{q}$ are integers. From the equation $$y^2-xy-x^2=\pm 1 \tag{7}$$ we get $y=\frac{x\pm \sqrt{x^2+4(x^2\pm 1)}}{2}$ Since we need the positive part of $y$, we may write the last equality in the following form $$y=\frac{1}{2}\left( x+\sqrt{5x^2\pm 4} \right). \tag{8}$$

### $\Rightarrow$

Since $x$ is a Fibonacci number, there must be nonegative integer $n$ such that $x=F_n$. We can thus use the Successive Fibonacci Number Test to conclude that $y=F_{n+1}$ is a solution to equality (7). Hence the right part of equality (8) is a Fibonacci number and this implies that $\sqrt{5x^2\pm 4}$ is an integer.

### $\Leftarrow$

Since $5x^2\pm 4$ is a perfect square, $\sqrt{5x^2\pm 4}$ is an integer. We know that $x$ is any integer, hence only one of the following cases hold:
1. The number $x$ is even.
2. The number $x$ is odd.
In the first case we can see that $x^2$ is even, hence $5x^2\pm 4$ is even. This implies that $\sqrt{5x^2\pm 4}$ is even. The sum of two even numbers must be even, therefor $x+\sqrt{ 5x^2\pm 4}$ is even. This implies that the right side of equality (8) is an integer, thus $y$ is an integer.
In the second case one can easily see that $x^2$ must be odd, therefore $5x^2\pm 4$ must be odd. This implies that $\sqrt{5x^2\pm 4}$ is odd, thus $x+\sqrt{5x^2\pm 4}$ is even, therefor the right side of equality (8) must be an integer, hence $y$ is an integer.

In any case we get that the integers $x$ and $y$ satisfy equality (1), hence we can use the Successive Fibonacci Number Test to deduce the existence of a natural number $n$ such that $x=F_n$ and $y=F_{n+1}$. Hence, $x$ is a Fibonacci number. $\blacksquare$

Tags Math


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# Is the group von Neumann algebra construction functorial?

Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set of all bounded linear operators on Hilbert space $l^2(G)$. Let $f:G \to H$ be any homomorphism of groups. My question is: is there a homomorphism of the group von Neumann algebra $NG \to NH$ induced from $f$?.

If $NG$ is replaced with $CG$, it's obvious true. If $NG$ is replaced with $C^\ast_r(G)$, the reduced group $C^\ast$ algebra, it's not necessary true.

-
Didn't this use to have a fa.functional-analysis tag? Is that somehow less relevant than the NCG tag? (If anything, this should have a ct.category-theory tag, IMHO) –  Yemon Choi Feb 12 '10 at 8:14

By the way, here's the "correct" functorial property. If G and H are abelian, and $f:G\rightarrow H$ is a continuous group homomorphism, then we get a continuous group homomorphism $\hat f:\hat H\rightarrow \hat G$ between the dual groups. By the pull-back, we get a *-homomorphism $\hat f_*:C_0(\hat G) \rightarrow C^b(\hat H)$. We should think of $C^b(\hat H)$ as the multiplier algebra of $C_0(\hat G)$. Then $C_0(\hat G) \cong C^*_r(G)$, and so we do get a *-homomorphism $C^*_r(G) \rightarrow M(C^*_r(H))$; the strict-continuity extension of this is a *-homomorphism $M(C^*_r(G)) \rightarrow M(C^*_r(H))$ which does indeed send $\lambda(s)$ to $\lambda(f(s))$.

For non-abelian group (in fact, non-amenable groups) it's necessary to work with $C^*(G)$ instead.

We cannot ensure a map to $C^*_r(H)$ itself, as we cannot ensure a map from $C_0(\hat G)$ to $C_0(\hat H)$; indeed, this would only happen when $\hat f$ were a proper map.

Similarly, we don't get maps at the von Neumann algebra level, as we don't get a map $L^\infty(\hat G) \rightarrow L^\infty(\hat H)$: we would need that $\hat f$ pulled-back null sets in $\hat G$ to null sets in $\hat H$.

-

Let $f : G \to H$ be a homomorphism of discrete groups.

The homomorphism $f$ extends to a homomorphism of reduced group $C^{\ast}$-algebras if and only of $\ker(f)$ is amenable, and extends to a homomorphism of group von Neumann algebras if and only if $\ker(f)$ is finite.

-
Nice to know, but how to use amenability or finitness of \kerf? Can you give some hints or references? –  m07kl Oct 23 '10 at 14:35
If $\ker(f)$ is finite, then the normalized sum over all elements in $\ker(f)$ is a central projection $p \in N(G)$ and the extension is given by the cut-down composed with the inclusion of $G/\ker(f) \subset H$, i.e. $N(G) \to pN(G)p = N(G/\ker(f)) \subset N(H)$. –  Andreas Thom Oct 23 '10 at 16:03
If $\ker(f)$ is amenable, then take a sequence of Foelner sets $F_n \subset \ker(f)$ and define vectors $$\xi_n = |F_n|^{-1} \cdot \sum_{g \in F_n} \delta_g.$$ Then, the states $\phi_n(a) = \langle a \xi_n, \xi_n \rangle$ on $C^{\ast}_{red}(G)$ will have a weak limit and the associated GNS-representation gives a $\ast$-homomorphism $C^{\ast}_{red}(G) \to C^{\ast}_{red}(G/\ker(f))$. Again, this can be composed with the inclusion $C^{\ast}_{red}(G/\ker(f)) \subset C^{\ast}_{red}(H). – Andreas Thom Oct 23 '10 at 16:09 Compare with mathoverflow.net/questions/28502/… – Alain Valette May 1 '11 at 18:19 Well, for$s\in G$let$\lambda(s)$be the left-translation operator by$s$; all such operators are in the group von Neumann algebra. I guess that the hoped for homomorphism$F:NG \rightarrow NH$should satisfy$F(\lambda(s)) = \lambda(f(s))$for$s\in G$, and we should have that$F$is an (ultraweakly) continuous$*$-homomorphism. In particular,$F$is contractive. Then$F$need not exist. Let$G=\mathbb Z$and$H=\mathbb Z/n\mathbb Z$. Then$NH = CH$, so we have a trace on$NH$which sends$\lambda(0)$to$1$. So if we apply$F$, and then take the trace, we should get an ultraweakly continuous functional$\phi$on$NG$which satisfies$\phi(\lambda(ns)) = 1$for all$s\in\mathbb Z$. But this can't happen: maybe we can see this via the Fourier transform. Then$NG \cong L^\infty(\mathbb T)$and$\phi$induces$h\in L^1(\mathbb T)$which satisfies$\int h(\theta) e^{ins\theta} d\theta = 1$for all$s\in\mathbb Z$. This violated Reimann-Lebesgue. On the other hand, if$G \subseteq H$is an inclusion (of discrete groups, to avoid topology) then we do get an inclusion$NG \rightarrow NH$. Here's a construction. Find an index set$I$and$(h_i)$in$H$such that$H$is the disjoint union of$\{Gh_i\}$. Then define$V:l^2(H)\rightarrow l^2(G)\otimes l^2(I)$by$V(\delta_h) = \delta_g\otimes\delta_i$if$h=gh_i$(so defined on point-masses, and extend by linearity). So$V$is unitary, and$\theta:x\mapsto V^*(x\otimes 1)V$is a normal$*$-homomorphism$NG\rightarrow B(l^2(H))$. Then, for$r\in G$,$V^*(\lambda(r)\otimes 1)V(\delta_h) = V^*(\delta_{rg}\otimes\delta_i) = \delta_{rh}$as$rg\in G$. So$\theta$maps into$NH$, and does what we want. Surely there is some general result, but I'm not sure of it... - Is this somehow connected to the fact that, since the inverse image of an element of$H$may be infinite, the map $G\to H$ does not induce a map $\ell^2(H)\to\ell^2(G)$? – Harald Hanche-Olsen Feb 11 '10 at 16:39 Matt: what was wrong with the statement you struck through? If G is a closed subgroup of H then the sub-vN-algebra of N(H) that is generated by "left translation by elements of G" is isomorphic to N(G), is it not? I thought this was in Takesaki & Tatsuuma, for instance. – Yemon Choi Feb 11 '10 at 18:41 Takesaki & Tatsuuma certainly study how one can "see" closed subgroups of G as certain subalgebras of NG, but they seem to studiously avoid claiming that the subalgebras are isomorphic to NH, at least on my reading. I struck through the sentence, as the proof I had was (obviously) wrong. – Matthew Daws Feb 11 '10 at 21:11 Fixed my proof (modulo not getting the LaTeX to render correctly). – Matthew Daws Feb 11 '10 at 22:17 Just to note (mostly for amusement) that in Matt's counterexample, one could take$n=1$. (Thus: the augmentation character is well-defined as a functional on the$\ell^1$-group algebra, here being evaluation of the Fourier transform at$1$; but it doesn't extend in the right way to VN(Z), because one can't evaluate an element of$L^\infty(T)$at the point$1\$. –  Yemon Choi Feb 12 '10 at 6:26


text:
# zbMATH — the first resource for mathematics

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Errata on: “Banach-Saks properties of ${C}^{*}$-algebras and Hilbert ${C}^{*}$-modules”. (English) Zbl 1206.46010

Summary: Due to an example indicated to us in September 2009, we have to add one more restriction to the suppositions on the imprimitivity bimodules treated in Proposition 4.1, Theorem 5.1, Theorem 6.2 and Proposition 6.3 of [ibid. 3, No. 2, 91–102 (2009; Zbl 1206.46009)].

In the situation when the Banach-Saks property holds for the imprimitivity bimodule, we can describe all possible additional examples violating the newly invented supposition, so the classification of Hilbert ${C}^{*}$-modules with the Banach-Saks property is complete. Beyond that, there is still an open problem for a certain class of imprimitivity bimodules with the weak or uniform weak Banach-Saks property which might violate the additional condition.

##### MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces 46L08 ${C}^{*}$-modules 46L05 General theory of ${C}^{*}$-algebras


text:
# find_MAP error: setting an array element with a sequence

I am trying to fit a model according to some data.

The data I have are supposed to obey the following formula:

\mu_{g, k} = (\mu_0 + \frac{\alpha_g}{\delta_g})\exp(-\delta_g t_k) + \frac{\alpha_g}{\delta_g} + \beta_g \exp(-\delta_g t_k) \sum_{i=0}^{k-1} \exp(\delta_g t_i) \frac{\eta_i}{\gamma_g+\eta_i} (t_{i+1}-t_i)

The parameters are as follows: (\alpha_g, \beta_g, \gamma_g, \delta_g) for g \in \{1, \cdots, 12\} and \eta_k for k \in \{0, \cdots, 9\}. The t_k for k \in \{0, \cdots, 10\} are given (time points where the data was collected).

I tried to implement the procedure in Python using the pymc library. I’m totally new to the subject. I read a good part of the documentation, and totally the get started page.

I came up with the following code

# Estimate
# 1. for each of the 12 genes under consideration the 4 parameters encoding their expression level
# 2. and for the (putative) transcription factor the 10 values describing its activity over time
# using the find_MAP procedure of the pymc3 library.

from io import BytesIO
import matplotlib.pyplot as plt
plt.style.use('seaborn-darkgrid')
import numpy as np
import pymc3 as pm

print('Running on NumPy v{}'.format(np.__version__))
print('Running on PyMC3 v{}'.format(pm.__version__))

expr = np.genfromtxt(BytesIO(Binary), missing_values = 'NA', filling_values = np.nan, skip_header = 1)
C, R, T, G = 2, 4, 11, 12 # condition, replicat, time, gene
exprcrtg = expr[ : , 3 : ].reshape(C, R, T, G)

time = np.array([0., 0.5, 1., 3.5, 6.5, 9.25, 12.5, 18.5, 22.5, 26.5, 30.5])

with pm.Model() as Basic_model:

# Priors for unknown model parameters
alpha = pm.HalfNormal('alpha', sd = 10, shape = G)
beta  = pm.HalfNormal('beta' , sd = 10, shape = G)
gamma = pm.HalfNormal('gamma', sd = 10, shape = G)
delta = pm.HalfNormal('delta', sd = 10, shape = G)
eta   = pm.HalfNormal('eta'  , sd = 10, shape = T-1)
sigma = pm.HalfNormal('sigma', sd = 10, shape = G)

# Expected value of outcome
alpha = alpha[np.newaxis, : ]
beta  = beta [np.newaxis, : ]
gamma = gamma[np.newaxis, : ]
delta = delta[np.newaxis, : ]
eta   = eta  [ : ,np.newaxis]
time  = time [ : ,np.newaxis]
mu0      = np.nanmean(exprcrtg[ : , : , 0, : ], axis = (0, 1))

integral = np.cumsum(np.exp(delta*time[:-1]) * eta / (gamma + eta) * (time[1:] - time[:-1]))
mu       = (mu0 - alpha/delta)*np.exp(-delta*time[1:]) + alpha/delta + beta*np.exp(-delta*time[1:])*integral

# Likelihood (sampling distribution) of observations
Y_obs = pm.Normal('Y_obs', mu = mu, sd = sigma, observed = exprcrtg[ : , : , 1: , : ])

# Maximum a posteriori estimate
map_estimate = pm.find_MAP(model = Basic_model)

print(map_estimate)

I pasted the data here.

I get the following error:
ValueError: setting an array element with a sequence.

To eliminate the possibility of a trival error, I’ve tried this bit of code outside of the pm.Model() and it shows no error.

So my guess is that I am making a wrong use of pm.Model() but I have no clue as to what it could be.

I am very willing to share more information about the model if it can help to solve my issue.


text:
# Measurement of the tt production cross section in pp collisions at \sqrt{s} = 7 TeV with lepton + jets final states

CMS Collaboration; Chatrchyan, S; Khachatryan, V; Amsler, C; Chiochia, V; De Visscher, S; Favaro, C; Ivova Rikova, M; Millan Mejias, B; Otiougova, P; Robmann, P; Snoek, H; Tupputi, S; Verzetti, M (2013). Measurement of the tt production cross section in pp collisions at \sqrt{s} = 7 TeV with lepton + jets final states. Physics Letters B, 720(1-3):83-104.

## Abstract

A measurement of the t t-bar production cross section in pp collisions at sqrt(s) = 7 TeV is presented. The results are based on data corresponding to an integrated luminosity of 2.3 inverse femtobarns collected by the CMS detector at the LHC. Selected events are required to have one isolated, high transverse momentum electron or muon, large missing transverse energy, and hadronic jets, at least one of which must be consistent with origination from a b quark. The measured cross section is 158.1 +/- 2.1 (stat.) +/- 10.2 (syst.) +/- 3.5 (lum.) pb, in agreement with standard model predictions.

## Abstract

A measurement of the t t-bar production cross section in pp collisions at sqrt(s) = 7 TeV is presented. The results are based on data corresponding to an integrated luminosity of 2.3 inverse femtobarns collected by the CMS detector at the LHC. Selected events are required to have one isolated, high transverse momentum electron or muon, large missing transverse energy, and hadronic jets, at least one of which must be consistent with origination from a b quark. The measured cross section is 158.1 +/- 2.1 (stat.) +/- 10.2 (syst.) +/- 3.5 (lum.) pb, in agreement with standard model predictions.

## Statistics

### Citations

Dimensions.ai Metrics
57 citations in Web of Science®
37 citations in Scopus®


text:
# Plz help fast!!Speech on the Constitution Lesson 2.06Directions: Write a 5-7 Sentence paragraph.Discuss the separation

###### Question:

Plz help fast!!Speech on the Constitution Lesson 2.06 Directions: Write a 5-7 Sentence paragraph.
Discuss the separation of powers and checks and balances. Use examples to explain how these
work in our branches of government.

### A 29 gram sample of a substance that's used to preserve fruit and vegetables has a k-value of 0.1359. N = Noe -kt No = initial

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text:
A capacitor is charged by using a battery which is then disconnected. A dielectric slab is then slipped between the plates, which results in -

(1) Reduction of charge on the plates and increase of potential difference across the plates

(2) Increase in the potential difference across the plate, reduction in stored energy, but no change in the charge on the plates

(3) Decrease in the potential difference across the plates, reduction in the stored energy, but no change in the charge on the plates

(4) None of the above

Concept Questions :-

Dielectrics
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

The capacity of a condenser is 4 × 10–6 farad and its potential is 100 volts. The energy released on discharging it fully will be -

(1) 0.02 Joule

(2) 0.04 Joule

(3) 0.025 Joule

(4) 0.05 Joule

Concept Questions :-

Energy stored in capacitor
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

The insulated spheres of radii R1 and R2 having charges Q1 and Q2 respectively are connected to each other. There is -

(1) No change in the energy of the system

(2) An increase in the energy of the system

(3) Always a decrease in the energy of the system

(4) A decrease in the energy of the system unless Q1R2 = Q2R1

Concept Questions :-

Electric potential
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

The energy stored in a condenser of capacity C which has been raised to a potential V is given by -

(1) $\frac{1}{2}CV$

(2) $\frac{1}{2}C{V}^{2}$

(3) CV

(4) $\frac{1}{2VC}$

Concept Questions :-

Energy stored in capacitor
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

If two conducting spheres are separately charged and then brought in contact -

(1) The total energy of the two spheres is conserved

(2) The total charge on the two spheres is conserved

(3) Both the total energy and charge are conserved

(4) The final potential is always the mean of the original potentials of the two spheres

Concept Questions :-

Electric potential
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

Eight drops of mercury of equal radii possessing equal charges combine to form a big drop. Then the capacitance of bigger drop compared to each individual small drop is

(1) 8 times

(2) 4 times

(3) 2 times

(4) 32 times

Concept Questions :-

Capacitance
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

A condenser of capacity 50 μF is charged to 10 volts. Its energy is equal to

(1) 2.5 × 10–3 Joule

(2) 2.5 × 10–4 Joule

(3) 5 × 10–2 Joule

(4) 1.2 × 10–8 Joule

Concept Questions :-

Energy stored in capacitor
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

A parallel plate condenser has a capacitance 50 μF in air and 110 μF when immersed in an oil. The dielectric constant ‘k’ of the oil is

(1) 0.45

(2) 0.55

(3) 1.10

(4) 2.20

Concept Questions :-

Dielectrics
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

Separation between the plates of a parallel plate capacitor is d and the area of each plate is A. When a slab of material of dielectric constant k and thickness t(t < d) is introduced between the plates, its capacitance becomes -

(1) $\frac{{\epsilon }_{0}A}{d+t\left(1-\frac{1}{k}\right)}$

(2) $\frac{{\epsilon }_{0}A}{d+t\left(1+\frac{1}{k}\right)}$

(3) $\frac{{\epsilon }_{0}A}{d-t\left(1-\frac{1}{k}\right)}$

(4) $\frac{{\epsilon }_{0}A}{d-t\left(1+\frac{1}{k}\right)}$

Concept Questions :-

Dielectrics
High Yielding Test Series + Question Bank - NEET 2020

Difficulty Level:

The capacity of parallel plate condenser depends on

(1) The type of metal used

(2) The thickness of plates

(3) The potential applied across the plates

(4) The separation between the plates

Concept Questions :-

Capacitance


text:
# Non-linear conductivity of ZnO

1. Oct 12, 2012

### eric336

I want to understand the conductivity of ZnO. I know it is non-linear (that's why it is used in surge arrester to begin with), but can anyone enlighten me on why it is non-linear? Is it the function of electric field, current, or anything? I'd appreciate if you can provide the expression of conductivity.

2. Oct 13, 2012

### ChaseRLewis73

ZnO is a semiconductor material that naturally is n-type without doping. Thus it will have a good conductivity because it allows easy transitions to the conduction band (what it means to be n-type).

These slides go over it