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## Abstract

In this article, we report analyses of the effects of fertility and mortality trends on the mutual exposure of grandparents and grandchildren and their consequences for multigenerational processes of social mobility in the United States from 1900 to 2010. Using historical vital statistics and stable population models, we report systematic analyses of grandparent-grandchild exposures from both prospective (grandparent) and retrospective (grandchild) perspectives. We also estimate exposure levels and trends specific to education levels of grandparents and grandchildren and decompose the overall trend into the effect of changing mortality, fertility level, and fertility timing. We show that changes in mutual exposure of grandparent and grandchild generations may have contributed to an increasing association between grandparents’ and grandchildren’s educational attainments.

## Introduction

Demographic transitions in fertility, mortality, and family formation in the United States over the past century have changed the landscape of American family and kinship relationships. Kinship, or multigenerational relations, has long been the focus of formal demography in mathematical or simulation models that show the relationship between demographic behaviors and long-term population change (Bongaarts et al. 1987; Wachter et al. 1978; Wright 1929). Yet kinship relationships are also important from a sociological perspective: they govern the pattern of inheritance, intergenerational mobility, and family reproduction of social and cultural capital across generations (Mare 2011, 2014). In this article, we examine the demography of multigenerational social inequality, with a specific focus on how the mutual exposure of grandparents and grandchildren has evolved and potentially changed the transmission of educational (dis)advantages across generations.

Research on multigenerational social mobility and persistence of inequality has proliferated in recent years, with new studies leveraging the increased availability of longitudinal, genealogical, and linked administrative data that provide information about family members over three or more generations (reviewed in Ruggles 2012, 2014; Song and Campbell 2017). Many new studies also reflect the increasing academic and public concern about rising long-term social inequality in the United States and worldwide (Alvaredo et al. 2018; Piketty 2014) as well as the role of grandparents in reinforcing intergenerational correlations in social status (Chan and Boliver 2013; Ferrie et al. 2016; Hällsten and Pfeffer 2017; Knigge 2016; Mare and Song 2014; Pfeffer and Killewald 2017; Zeng and Xie 2014). Despite their importance, these studies have predominantly focused on socioeconomic determinants and outcomes of families while ignoring the interplay of demography and social inequality. Family members across generations are linked by not only socioeconomic statuses but also fertility, mortality, marriages, and other demographic behaviors (Maralani 2013; Mare and Maralani 2006; Mare and Schwartz 2006). These unequal demographic processes lead to variations in family size, kinship composition, and overlap between generations in a population, which in turn may influence the dynamics of family relationships in social mobility.

In this article, we focus on overlapping years between grandparents and grandchildren, also known as multigenerational exposure or shared lifetimes, a demographic outcome that has been ignored. Prior studies have analyzed some related measures, such as individuals’ years spent in different family roles as parent, child, or spouse (Watkins et al. 1987); the proportion of individuals who at various life stages would have living grandparents, parents, and offspring (Margolis and Wright 2016; Uhlenberg 1996); and years of grandparenthood and step-grandparenthood (Margolis 2016; Yahirun et al. 2018). These measures, however, are less relevant to changing multigenerational social mobility and inequality compared with the various forms of grandparent-grandchild exposure measures analyzed in the present study. We provide a more comprehensive analysis of demographic trends in grandparent-grandchild relationships than that in previous research and explicate the implications of the demographic trends for increasing multigenerational inequality. We illustrate the significance of grandparent-grandchild exposure in three steps. First, we examine the overall trend in exposure as well as the trends specific to socioeconomic groups defined by their levels of educational attainment. The education-specific trends identify subpopulations that have experienced the most (or least) exposure increase over time. Second, we define a new measure of family background: exposure-weighted family resources as measured by grandparents’ education. We compare inequality in educational distribution based on conventional unweighted measures and our exposure-weighted measure. Finally, we simulate implications of changing exposure for the effect of grandparents on grandchildren in the transmission of educational status. The results suggest that if effects of grandparents on grandchildren’s educational attainments depend on exposure, the effects may have increased over the past several decades.

The present study provides the first large-scale historical profile of changing multigenerational exposure in the United States during 1900–2010 using a vast amount of data drawn from historical vital registration, decennial censuses, contemporary social surveys, and demographic estimates from published work, model life tables, stable population models, and population simulations. The results complement several recent studies that examined grandparenthood from a cross-sectional or comparative perspective (e.g., Leopold and Skopek 2015; Margolis 2016; Skopek and Leopold 2017). Our systematic analysis of grandparent-grandchild exposure consists of two approaches—a prospective approach from the grandparent perspective and a retrospective approach from the grandchild perspective—each of which contains three more detailed measures: (1) years lived with any grandparent (grandchild), (2) years lived with all grandparents (grandchildren), and (3) years lived per grandparent (grandchild). Some of these measures are more closely related to changes in multigenerational social inequality than others. To obtain these exposure estimates, we refine the classic Goodman-Keyfitz-Pullum method of predicting kinship structure and relationships in a population and show how age-specific mortality and fertility determine the expected years that grandparent and grandchild generations overlap. The refined method also incorporates educational disparities in demographic rates and mobility probabilities into the exposure estimates, suggesting heterogeneity in multigenerational exposure across educational groups and its implications for the association of educational attainments between grandparents and grandchildren. By integrating theories of multigenerational social mobility into the demographic analyses of family relationships, our study sheds light on how demographic changes in the recent history of the United States have transformed grandparent (grandchild) availability, three-generation educational mobility, and lineage-based social inequality.

## Background

### Grandparent Effects and Multigenerational Exposure

With growing socioeconomic inequality and widening concerns about how families pass their socioeconomic advantages to their offspring, social scientists have increasingly focused on the potential role of multigenerational influences. The dominant approach to the study of social mobility and other intergenerational processes is to focus on two-generation nuclear family relationships alone. In this approach, connections between more remote kin derive from a chain of two-generational family connections that follow Markov chain–like processes. Yet individuals may also be affected by a wider set of kin, including grandparents and other, more distant relatives. Grandparents and others may directly affect subsequent generations by complementing or substituting for parents or via a variety of other institutional mechanisms embedded in families, the law, educational institutions, and wealth transmission (Coall and Hertwig 2010; Mare 2011).

Recent studies have begun to examine social mobility from the perspective of three or more generations and focused on whether grandparents’ characteristics are associated with those of their grandchildren, even when characteristics of parents are statistically controlled (e.g., Chan and Boliver 2013; Pfeffer 2014; Prix and Pfeffer 2017; Zeng and Xie 2014). Although these studies have considerable merit, their capacity to establish the importance of grandparent effects is incomplete. The importance of grandparents is, broadly speaking, proportional to the mutual availability or exposure of grandparents and grandchildren. To be sure, grandparents may influence their grandchildren and possibly subsequent generations of kin even after they die, through such mechanisms as wills and trusts, college legacy admission provisions, and wealth creation (e.g., Madoff 2010). But many of the most important mechanisms through which grandparents may influence grandchildren, especially when grandparents assist parents in raising children, require mutual exposure of grandparent and grandchild generations. A strong form of mutual exposure is actual coresidence in three-generation households, which may provide the most effective form of grandparental influence on grandchildren (Zeng and Xie 2014). Even when grandparents live apart from their adult offspring and their grandchildren, they are likely to exert a much stronger influence while alive than after they die. This suggests that estimated effects of grandparents’ resources on the well-being of their grandchildren may vary depending on the degree to which grandchildren are exposed to their grandchildren (and vice versa), whether through coresidence or shared lifetimes.

### Demographic Trends and Shared Lifetimes

Changes in fertility and mortality patterns have profound impacts on population-level distributions of kin relations and expected years spent in family roles as grandparents, parents, children, and grandchildren. With substantial increases in longevity over the twentieth century, children on average spend a larger proportion of their lives with living grandparents. Yet because of fertility decline or fertility postponement, members of the grandparent generation have grandchildren who are fewer in number and born later in their lives. If grandchildren are born to older parents (and to parents who themselves were born to older parents), they may have somewhat shorter exposure to grandparents than their counterparts who were born to younger parents (and parents who were themselves born to younger parents). Overall, the effect of mortality changes has outweighed that of fertility changes, leading to an increase in overlapped lifetimes of grandparents and grandchildren during the twentieth century in the United States and elsewhere (Bengtson 2001).

### Inequality, Attainment, and Multigenerational Exposure

Because fertility and mortality patterns vary among socioeconomic groups, levels and trends in mutual exposure of grandparents and grandchildren are also unequally distributed. In turn, trends in exposure may affect socioeconomic inequality and mobility. We consider five underlying mechanisms in the context of differences in educational attainment and mobility.

1. In any given period or generation, more highly educated grandparents provide greater potential benefits to their grandchildren.

2. Grandparent effects on grandchildren are likely to be greater when grandparents and grandchildren have shared lifetimes and mutual contact.

3. Secular increases in mutual exposure of grandparents and grandchildren create greater inequalities in overall family backgrounds of grandchildren because more-advantaged children experience more years of exposure to relatively highly educated grandparents, whereas less-advantaged children experience more years of exposure to poorly educated grandparents.

4. Trends in inequalities may be amplified or reduced by differences among education groups in mortality and fertility trends.

5. Secular increases in mutual exposure of grandparents and grandchildren are likely to increase the associations of the educational attainments of grandparents and grandchildren (both zero-order associations and associations net of parental characteristics) because of the growing capacity of grandparents to affect their grandchildren, whether in a favorable or unfavorable direction.

We report a systematic investigation of effects of mortality and fertility changes on exposure of grandchildren to grandparents and of grandparents to grandchildren in the United States over the twentieth century, on inequalities in exposure among families with varying levels of educational attainment, and on the associations of grandparents’ and grandchildren’s educational attainments. Our empirical analysis extends the work of others, notably Uhlenberg’s (1996, 2009) analyses of multigenerational relationships and some recent studies on grandparenthood and kin counts (Daw et al. 2016; Margolis 2016; Yahirun et al. 2018). A novelty of our analysis is that we examine the mutual exposure of grandparents and grandchildren taking both a prospective and a retrospective approach (Song and Mare 2015). Additionally, we obtain estimates of exposure and the effects of demographic trends for grandparents and grandchildren specific to the education levels of grandparents, which shows the degree to which trends and differences in exposure affect trends in both socioeconomic inequality and grandparent-grandchild educational mobility.

## Methods

Despite the growth of longitudinal and multigenerational microdata in recent decades, information on years of shared lifetimes of grandparents and grandchildren is often not directly asked for or available in social survey data. However, given that all kin relationships in a closed population result from birth, death, mating, and parenthood, we can extract kinship information from aggregate demographic rates if certain assumptions hold. The method we use here builds on the earlier development of kinship models by Lotka (1931) and Coale (1965) and later refined by Goodman et al. (1974). The details of our method are discussed in the online appendix.

### Period Estimates and Synthetic Cohorts

The present study provides estimates of multigenerational exposure based on period estimates of mortality and fertility. The difference between period and cohort estimates is that the period estimates show how long we would expect a person to live with his or her grandparents or grandchildren, on average, if everyone in the population experiences the age-specific fertility and mortality from a given year, whereas cohort estimates are based on demographic rates throughout the person’s lifetime. We use formulas derived from stationary and stable population assumptions to estimate grandparent-grandchild exposure for decades between 1900 and 2010 based on a synthetic cohort method. A synthetic cohort is defined as a year-specific hypothetical cohort of individuals whose age-specific vital rates characterize a stable population for that year. We create synthetic cohorts using decennial information on fertility and mortality to estimate trends in shared lifetimes of grandparents and grandchildren for the population as a whole and for grandparents with varying levels of educational attainment. We also obtain counterfactual estimates based on alternative assumptions about changes in fertility and mortality. In particular, we examine the effects of mortality, fertility levels (measured by the gross reproduction rate), mean age of childbearing, and variance of age at childbearing by varying each of these parameters while holding the others constant.

One limitation of the synthetic cohort estimate is that a person’s expected exposure with her grandchildren depends only on her own birth year and period estimates of mortality and fertility in that year. The estimates assume fixed demographic parameters across generations while ignoring possible changes associated with time. The limitation of this method is similar to that of population projection, in which the prediction for future generations is based on observed demographic behaviors in the present period and thus is not a real depiction of either the future or past trends of the population (Keyfitz 1981). In addition, the estimate does not consider heterogeneity (except in education) in multigenerational exposure among individuals who were born in the same year. Socioeconomic and demographic factors other than age and education may also influence the number of years that a person spends with her grandparents or grandchildren.

### Prospective and Retrospective Approaches

We measure grandparent-grandchild exposure both prospectively and retrospectively. Within each approach, we consider several measures taking into account that grandparents may have more than one grandchild and that grandchildren may have more than one living grandparent. These alternative measures are illustrated in Fig. 1. Prospectively and from the perspective of a person in the grandparent generation, we ask, What is a grandfather’s or grandmother’s exposure to grandchildren during the grandparent’s lifetime? We measure exposure in three ways: (1) years lived with any grandchild, (2) years lived with all grandchildren (grandchildren-years lived), and (3) years lived per grandchild. The first two of these measures are sensitive to fertility levels and timing in the grandparent and parent generations as well as grandparental mortality. The third measure conditions on a given grandchild and is sensitive only to fertility timing and grandparental mortality. The prospective approach focuses on the capacity of a grandparent of a given status to produce progeny of varying characteristics in subsequent generations. From the perspective of a grandparent, grandparenthood extends from the birth of the first grandchild to the grandparent’s end of life or, in rare cases, the death of the last grandchild.

Retrospectively and from the perspective of a grandchild, we ask, What is his or her exposure to grandparents during the grandchild’s lifetime? Again, we measure exposure in three ways: (1) years lived with any grandparent, (2) years lived with all grandparents (accumulated time the grandchild spent with the four grandparents), and (3) years lived per grandparent. The retrospective approach focuses on the “family background” of the grandchild. From the perspective of grandchildren, however, individuals with the same grandparents spent variable amounts of time with their grandparents; grandchildren born earlier live longer with their grandparents than do their later-born siblings or cousins.

### Measures of Grandparent-Grandchild Exposure

To estimate multigenerational mutual exposure, we rely on the method developed by Goodman, Keyfitz, and Pullum (Goodman et al. 1974; GKP hereafter). Specifically, the GKP method provides a general framework that shows how age-specific mortality and fertility determine the expected probability that a person at a given age has a living progenitor, a descendant, or an extended family member. We modify the GKP method to estimate multigenerational shared lifetimes by converting the probability of living kin to the years of overlapping lifespan between generations. For the sake of simplicity, we illustrate the method using females as an example. In the subsequent empirical analyses, we estimate exposure using both male and female mortality and fertility rates and by education groups.

For a given period, denote the probability that a woman survives to age x by l(x); this probability is the same as the number-surviving column of the life table when the radix l(0) is set to 1 and denotes the age-specific reproductive rate m(x), which is nonnegative between the limits of the childbearing ages α and β and 0 outside these limits. A woman gives birth to m(x)dx of a daughter in each dx of maternal age. We define three different types of prospective and retrospective grandparent-grandchild exposure measures, respectively (P1, P2, P3 and R1, R2, R3). In what follows, we denote the expected exposure years as an expectation E(∙) to emphasize the fact that all the estimates are based on aggregate-level age-specific fertility and mortality, and the estimate refers to the expected years of grandparenthood if a person survives to the age one can expect to live (i.e., life expectancy, e0).

#### Prospective Average Years Lived With Any Grandchild

According to the GKP method, the expected number of living granddaughters (daughter’s daughter) of a woman at age a can be estimated as
Assume a woman’s number of grandchildren follows a Poisson distribution with mean M2(a). The proportion with a positive number of grandchildren is $1−e−M2a$. The woman’s expected lifetime number of years lived with any granddaughter (daughter’s daughter) is

Similarly, we also estimate the expected number of other types of grandchildren (i.e., daughter’s son, son’s daughter, and son’s son) by substituting females’ demographic rates in the offspring or the grandchild generation with males’ rates and then combined these estimated using the method described in the online appendix.

#### Prospective Average Person-Years Lived With All Grandchildren

Using similar notations, for a woman at birth, her total expected number of years lived with all granddaughters (daughter’s daughters) during her expected lifetime e0 is
given that e0xy > 0. We can then estimate the expected number of years lived with other types of grandchildren and sum over these years to obtain the overall estimate.

#### Prospective Average Years Lived per Grandchild

Finally, we divide the person-years lived with all grandchildren by the total number of grandchildren to obtain the average years lived per grandchild. At a woman’s birth, her expected number of years lived with an average granddaughter (daughter’s daughter) during her lifetime e0 is
$EP3=EP2/NRRgm,$
where NRRgm is the grandmother’s net reproduction rate of granddaughters. A two-sex version of the formula used in our empirical analysis is shown in the online appendix.

#### Retrospective Average Years Lived With Any Grandparent by Age 25

From a retrospective perspective, we consider a person’s average years lived with any grandparent by age 25. The probability that a woman age a has a living maternal grandmother can be estimated following Keyfitz (1977:276):
$∫αβ∫αβBt−a−x−ylx+y+amylxmxladxdyBt−ala$
where B(t) is the total number of female births at time t, and the total number of women surviving to age a at time t is B(ta)l(a). Ages α and β refer to minimum and maximum ages of childbearing, respectively. In a stable population with the intrinsic growth rate r, B(taxy) = B(ta) × er(x + y), which implies that the expected years lived with the maternal grandmother by age 25 is

#### Retrospective Average Grandparent-Years Lived With All Grandparents

Similarly, we can estimate a person’s expected years lived with his or her maternal grandfather, paternal grandmother, and paternal grandfather by using male demographic rates. For example, if l(x + y + a) and m(y) refer to male age-specific mortality and fertility rates, respectively, E(R1) indicates the expected years lived with the maternal grandfather. Therefore, summing over the average years lived with each grandparent, we obtain one’s expected person-years lived with all grandparents up to age 25 as
$ER2=ER1mm+E(R1fm)+E(R1mf)+E(R1ff),$
where $ER1mm$, $E(R1fm)$, $E(R1mf)$, and $E(R1ff)$ refer to years lived with mother’s mother, father’s mother, mother’s father, and father’s father, respectively.

#### Retrospective Average Grandparent-Years Lived per Grandparent

Assuming that each woman has exactly four grandparents, the average number of years she will be exposed to each grandparent is$ER3=ER24$.

### The Effects of Mortality and Fertility

We use the period age-specific mortality and fertility rates to estimate the expected number of years that a person lived with grandchildren or grandparents from a synthetic cohort perspective. The overall trend in multigenerational exposure is driven by changes in mortality and fertility in the population. To assess the relative importance of mortality and fertility in shaping the trend, we simulate four counterfactual scenarios, each of which fixes three of the four demographic parameters at the level of 1900 and allows only one parameter to vary over time. These four parameters include level of mortality (l(x)), level of fertility (GRR), the mean age of childbearing (AR), and variances of the age of childbearing (σ2). The last three terms determine the age-specific maternity rates (m(x)) shown in the aforementioned exposure formulas. We describe how these parameters are estimated from the historical data in the Data section. For example, to evaluate the effect of mortality on multigenerational exposure, we fix the level of fertility and the timing of childbearing at the level of 1900 and allow the age-specific mortality rate to change from its level in 1900 to 2010. The counterfactual asks whether changing exposure is completely driven by changing mortality, and the difference between the actual trend and the counterfactual trend suggests the effect of mortality.

### Simulation on the Effect of Exposure on Social Mobility

The changing multigenerational exposure over historical periods also has implications for the transmission of educational inequality from grandparents to grandchildren. To illustrate this further, we assume that the effect of grandparents’ education on grandchildren’s education is associated with exposure, indicating that the association of educational attainments between grandparents and grandchildren is stronger when they have more years of shared lifetimes together, but this effect does not change over periods. Thus, the overall association of grandparent’s and grandchild’s educational attainment would change if the proportion of grandparents who spend more years with their grandchildren increases over time. Specifically, if we denote the expected grandparent effect on grandchildren’s education by E(βgp − gc, i), according to the law of iterated expectation,
$Eβgp−gc,i=EtEβgp−gc,iTi=∑tEβgp−gc,iTi·PTi,$
1
where Ti refers to the years of multigenerational exposure between grandparents and grandchildren, and βgp − gc, i refers to the association between grandparent and grandchild for the ith observation in the sample.
In the traditional mobility approach that omits exposure, βgp − gc is modeled by the following equation using standard statistical methods:
$yigc=β0+βgp−gc,i·yigp+εi,$
2
where ygc and ygp refer to years of schooling of grandchildren and grandparents, respectively, and ε is the error term. Assume that βgp − gc is a random parameter and associated with exposure (i.e., a greater grandparent effect where there is a longer exposure). In the simplest case, we model βgp − gc by a linear function:
$βgp−gc,i=θ0+θ1·Ti+vi,$
3
where vi is considered to be an individual-specific random parameter that is normally and independently distributed in the population. Combining Eqs. (2) and (3), we obtain
$yigc=β0+θ0·yigp+θ1·Ti·yigp+vi·yigp+εi,$
which can be viewed as a mixed-effect model with random slope effect, vygp, and fixed effects θ0 and θ1. By varying exposure from the level of 1900 to that of 2010, we observe the composition effect of changing multigenerational exposure on the association of educational status between grandparents and grandchildren. We call βgp − gc the gross effect of grandparents without controlling for the effect of parents. To model the net effect of grandparents E(βgp − gc | βp − gc), we modify Eq. (1) by adding the effects of parents:
$yigc=α0+αgp−gc·i·yigp+αp−gc,i·yip+εi,$
4
where $yip$ refers to parents’ years of schooling,1 and αgp − gc, i is identical to E(βgp − gc, i | βp − gc, i).

## Data

To obtain national mortality estimates, we rely on published decennial life tables for periods 1900 and 2010 (Arias 2014; Arias et al. 2008; Armstrong 1997; Armstrong and Curtin 1985; Carter et al. 2006; Greville 1947, 1975; Sirken 1964; Sirken and Carlson 1954; U.S. Census Bureau 1921). For years when only abridged life tables are available, we use the linear interpolation method described in Coale et al. (1983) to generate proportion of survivors to age x (i.e., l(x)) in the full decennial life tables. For age-specific fertility rates (ASFR), we use national vital registration data (Martin et al. 2013; U.S. Census Bureau 1975) for 1940 to 2010, vital registration data adjusted for geographic incompleteness for 1920 to 1940 (Linder and Grove 1947), and the estimates provided by Haines (1989) for 1900–1910.2 We estimate total fertility rates (TFR) based on ASFRs for five-year age groups and convert each TFR to the gross reproduction rate (GRR). For the age-specific maternity function, we assume a truncated normal distribution of m(x) within the range of reproductive ages 15 to 45 for women and 15 to 60 for men. The means (Ar) and variances (σ2) of the age of childbearing are estimated from the observed distributions of the ASFR by year.3 For the retrospective measures, we also need to estimate the intrinsic population growth rate for a stationary population with known fertility and mortality schedules (l(x) and m(x)) using the characteristic function.4

For education-specific trends in mortality, we rely on published estimates of life expectancy at age 25 (Centers for Disease Control and Prevention 2014; Elo and Preston 1996; Kitagawa and Hauser 1973; National Center for Health Statistics 2012; Rogot et al.1992) and construct period life tables for education groups based on regional-model life tables and unpublished United Nations life tables for populations with higher expectations of life than are included in the Coale-Demeny-Vaughan model tables (1983).

For fertility, we use the GRR for 1925–1990 computed by the own-children method published in Mare (1997: table 2). For the remaining periods, we estimate the education-specific GRR by using the overall GRR for all education groups and the proportions of educational groups in the population.5 These estimates are similar to those of Rindfuss and Sweet (1977), Rindfuss et al. (1996), and Yang and Morgan (2003).

For social mobility, we create period-specific intergenerational educational mobility tables by pooling data from the 1962 and 1973 Occupational Changes in a Generation surveys, the 1972–2014 General Social Surveys (GSS), and the 1968 wave of the Panel Study of Income Dynamics (PSID); Wave 2 data from the 1986, 1987, and 1988 Survey of Income and Program Participation (SIPP); and the 1987–1988 National Survey of Families and Households (NSFH). For historical changes in the educational attainment distribution, we rely on published aggregate data on years of school completed for men and women age 25 years and over by age and year (U.S. Census Bureau 2014). We summarize these demographic and mobility data in Tables A1A3 of the online appendix.

To simulate how trends in multigenerational exposure may change the associations between grandparents’ and grandchildren’s educational attainments, we use microdata from all waves of the PSID from 1968 to 2015. The PSID project began with more than 18,000 individuals from roughly 4,800 families in the United States and followed targeted respondents over time according to a genealogical design. Specifically, only household members of the 1968 original sample and their progeny are followed, thereby excluding spouses who later marry into a PSID household or their family of origin. The PSID project provides a Family Identification Mapping System (FIMS) tool designed to link family members across generations. Education is asked in each wave of the individual file as well as in most waves of the household file for household heads and their wives. We use the maximum years of schooling reported by PSID respondents by pooling data from all years with nonmissing data. During some waves, household heads and wives were also asked to report the education information of their parents, who may not be PSID respondents or ever show up in a PSID household. This survey design allows us to impute the missing education information for some grandparents. Multigenerational exposure is measured as the years of overlapping lifetimes between a grandparent and a grandchild during the first 25 years of the grandchild’s life and is calculated by the birth and death years of the two generations. In most cases, the birth and death information is available only for individuals who are PSID respondents. We define five 10-year birth cohorts of grandparents and grandchildren, respectively, using information from their ages. Because of the growth of single-parent families, we have more information from mothers available than from fathers. We include a dummy variable to indicate whether father’s education is missing. The final sample is restricted to grandchildren who were born before 1990 and thus are aged 25 and older in 2015. Because of the sampling design of the PSID, we often have only either paternal or maternal grandparents available in the data. If a person has more than one grandparent available, we include all grandparents in the analysis, and each case refers to a grandparent-grandchild dyad. Descriptive statistics for all measures are displayed in the Table A8 (online appendix).

## Results

### Overall Trends in Mutual Exposure

Table 1 summarizes, in qualitative terms, the effects of mortality and fertility trends on various measures of exposure. Panels a–f of Fig. 2 highlight several patterns contained in the estimates in Table 1. The solid lines, labeled as “overall,” show trends in exposure estimated from four demographic parameters—age-specific mortality, mean age of childbearing (fertility), variance in the age of childbearing, and fertility level—at their observed levels between 1900 and 2010. All the other lines labeled as counterfactual effects (e.g., “mortality effect”) refer to hypothetical estimates when three of the four parameters are fixed at their values for 1900 and the fourth parameter (e.g., age-specific mortality rate) is allowed to change with its observed levels between 1900 and 2010.

Panel a of Fig. 2 shows that over the twentieth century, the expected number of years a person could expect to live with any grandchild increased markedly from less than 5 years to almost 35 years. This trend is overwhelmingly due to increased longevity, as suggested by the line labeled “mortality effect.” Absent changes in fertility levels and timing, the trend in grandparent-grandchild exposure would be much the same, even somewhat more in the direction of increased exposure. All fertility effects on this exposure outcome were relatively modest, although changes in average age of childbearing (in both grandparent and parent generations) tended to increase exposure through the Baby Boom years and then reduce it somewhat as average childbearing age increased.6

Panel b of Fig. 2 shows that the trend in the expected number of years that a person could expect to live with all grandchildren reached a peak in the 1950s and 1960s because of high fertility levels during that period. The counterfactual simulation, however, does not suggest a strong fertility effect if we fix mortality at the 1900 level.7 Panel c of Fig. 2 shows an almost monotonically increasing trend in the years of exposure per grandchild when we remove variations in the number of grandchildren among grandparents from the exposure measure used in panel b.

The results shown in panels d–f of Fig. 2 suggest that from the (retrospective) standpoint of grandchildren, exposure to grandparents increased as well, albeit much less dramatically than the prospective trends. The trend in panel d suggests only about a four-year increase in exposure to any grandparent over the course of the century. One reason for this is that we measure the exposure between a grandchild and his or her grandparents before the grandchild reaches age 25. A substantial proportion of people, even at the beginning of the twentieth century, had at least one grandparent alive during their first 25 years of life. However, because most families have experienced an increase in the number of living grandparents, the increase in exposure is much greater when we focus on all grandparents (Fig. 2, panel e). The trend shown in panel f shows the corresponding pattern for the average grandparent within a family. It is much the same as the trend in panel e except that the range of the trend is approximately one-fourth of that in panel e (inasmuch as every person is assumed to have four grandparents). Increases in longevity are primarily responsible for these retrospective trends, although again, grandchildren’s exposure to grandparents also varies inversely with average age at childbearing over this period.

### Education-Specific Trends in Mutual Exposure

Panels a–c of Fig. 3 illustrate how trends in mutual exposure vary by the educational attainment of grandparents, as measured prospectively by the years a grandparent shares with any grandchild, with all grandchildren, and per grandchild. For the most part, the education-specific trends mirror the trends for all groups combined. However, in the early twentieth century, mutual exposure of grandparents and grandchildren was greater for the least-educated grandparents than for families in which grandparents had more schooling. This pattern of differences results from the higher and earlier fertility in less-educated families during this period and, compared with later periods, a somewhat smaller mortality difference among education groups. In contrast, 100 years later, exposure is greatest for grandparents with 12 years of schooling and least for grandparents with 0–8 years of schooling (in panels a–c of Fig. 3). These changes reflect a compression of fertility differences among education groups combined with somewhat higher mortality differences in the recent period. However, we still observe the greatest exposure for the least-educated group when we consider exposure to all grandchildren (Fig. 3, panel b) inasmuch as the higher fertility level for this group outweighs their mortality disadvantages in the exposure estimation.

Panels d–f of Fig. 3 illustrate how trends in mutual exposure vary by the grandparent’s education based on retrospective measures. We present the retrospective years of mutual exposure between a grandchild and maternal grandmother rather than any grandparent. This measure allows us to control the educational attainment of one grandparent without considering the educational attainments of all the other grandparents and the patterns of educational assortative mating. Exposure estimates in panel e combine the mutual exposure between a grandchild and each of the four grandparents who are assumed to have the same level of educational attainment.8 The trends in panels e and f are identical except that the scale used to measure mutual exposure per grandparent in panel e is one-fourth of the mutual exposure with all grandparents shown in panel f. Panels d and f are similar but not identical because a person’s exposure to grandparents may vary between grandfathers and grandmothers and between paternal and maternal grandparents. Panels d–f yield the same conclusions: education-specific exposure trends broadly mirror those for all education groups combined; the relative ranks of the education groups have changed over the century; and for the most part, children are more exposed to grandparents who have lower levels of educational attainment.

### Mutual Exposure and Family Inequality

Average levels of educational attainment have increased massively throughout the twentieth century along with the substantial upward trend in mutual exposure of grandparents and grandchildren documented in this study. As Menken (1985:469) pointed out, these demographic changes may have altered “the boundaries of expectations and obligations” within families. For example, the number of years that one has living elderly parents has increased, but the number of children has declined due to low fertility. Yet, most stratification research tends to treat the socioeconomic resources of families as fixed stocks that are unaffected by the duration of time that family members spend with one another. This approach is typical regardless of whether family background effects on children are viewed as two-generational (parent to child) or multigenerational (parent and grandparent to child). Unfortunately, this assumption ignores the considerable heterogeneity across families in mutual exposures of family members who are in different generations and the large variation in the distribution of exposures across time and place. Therefore, it may be fruitful in documenting socioeconomic inequality to investigate how the unequal resources that family members contribute to the welfare of subsequent generations combine with differences in their exposures to their children and grandchildren. Before turning to the ways that exposure trends may modify the estimated associations between grandparents’ and grandchildren’s educational attainments, we first show trends in the distribution of educational attainment weighted by the average mutual exposures of grandparent education groups.

Figure 4 illustrates how grandparent exposure and educational attainment combine to reveal greater inequality and greater growth in inequality of family backgrounds than one can see from the distribution of grandparents’ educational attainments alone. We weight grandparent’s years of school completed by the expected number of years of mutual exposure between a grandparent and any grandchild in the prospective measure (panel a) and between a grandchild and maternal grandmother in the retrospective measure (panel b). We do not present the trends weighted by our other four measures because the trends are similar across all six exposure measures. Prospectively, exposure-weighted educational attainment has grown for all grandparent education groups over the past 100 years, but it has grown much more for highly educated grandparents. The expected gap between the highest and lowest education groups is approximately 75 years at the beginning of the twentieth century but grows to almost 425 years over the subsequent 100 years. The retrospective measure shows a similar trend but with less growth in the overall level of the exposure and relative gaps in the exposure among education groups.

We calculate several simple inequality indices using the education-reweighted exposure estimates shown in Fig. 4 and report the trend in inequality in Table 2. These inequality measures—including the Gini coefficient, the standard deviation of log exposure, and the Atkinson index—have been widely used in the study of income inequality. All inequality measures were stable up until 1960, declined during the 1970s, and increased sharply after 1980. In fact, all inequality measures suggest the same upward trend after the 1990s, indicating that children are born into families with more inequality in the grandparent generation.

### Exposure and Multigenerational Transmission of Educational Status

Having shown the substantial increases in mutual exposure of grandparents and grandchildren in the twentieth century United States and rising inequality in adult exposure-weighted educational attainment during the past 30 years, we turn next to the implications of these trends for the gross and net associations of grandparents’ educational attainments and those of their grandchildren. The gross effect of grandparents refers to the zero-order association of education between grandparents and grandchildren, also known as the unconditional effect, and the net effect of grandparents refers to the partial or conditional association of education between grandparents and grandchildren that controls for parents’ educational attainments. The net effect of grandparents has been the focus of most recent studies on multigenerational social mobility, with a positive effect indicating greater social inequality than that in a Markovian world wherein each generation influences only the immediately following generation without lagged influences from grandparents and beyond (Mare 2011).

We hypothesize that if the effects of grandparents on grandchildren vary based on whether grandchildren are born while their grandparents are still alive and whether and how long they live together, the effects of grandparents may have grown over the past century as a result of the substantial increases in overlapping lifetimes of grandparents and grandchildren. Unfortunately, microdata are unavailable to study historical variation in grandparent effects and their variations among families with different configurations of living kin. Thus, we take an indirect approach, using the PSID to estimate associations between the educational attainments of grandparents and grandchildren, separately for grandparents who have overlapping lifetimes with their grandchildren and for those who die before their grandchildren are born. Under the assumption that the interaction effect of grandparents’ survival and grandparents’ educational attainments on grandchildren’s education is stable over time, we extrapolate how the average effects of grandparents’ educational attainments have changed over the twentieth century.

Table 3 presents model estimates from the PSID sample for the analysis described in Eqs. (1)–(3). Education in each generation is measured by the years of school completed. Models 1a–1d show the gross association between grandparents’ and grandchildren’s education. Without controlling for years of exposure in Model 1a, each 1-year increase in grandparents’ years of schooling leads to a 0.1-year increase in grandchildren’s schooling. Model 1d further shows that the effect of grandparents also varies by the duration of exposure between grandparents and grandchildren, suggesting that the effect is stronger among grandparents who spend more years with their grandchildren. Similar effects are also observed in Models 2a–d except that the net effect of grandparents is much smaller than the gross effect and becomes insignificant when the effect of exposure is included.

For the purposes of the simulation, we assume that the association of grandparent’s and grandchild’s educational attainment, conditional on the years of their mutual exposure, is fixed at the level observed in Table 3. The overall association of grandparent’s and grandchild’s educational attainment changes only as a function of the distribution of grandparent-grandchild exposure in the population (see Eq. (1)). Specifically, when grandparent effects are stronger for grandchildren who have more shared lifetimes with their grandparents, the increasing average exposure documented earlier would imply stronger expected grandparent effects in the population.

The left two graphs in Fig. 5 summarize our earlier estimates of changes in years lived per grandchild and the effect of exposure on the gross and net effects of grandparents estimated from Table 3. The simulation results shown in the main graph suggest that the zero-order regression between grandparent’s and grandchild’s schooling is expected to increase from less than 0.1 of a year to higher than 0.12 of a year between 1900 and 2010, indicating that grandparent effects are stronger in recent periods when grandparents live long enough to influence their grandchildren. The net association shows a similar trend, but the amount of increase is greater than the zero-order association (from approximately 0.015 in 1900 to about 0.035 in 2010). Although these calculations are an extrapolation based on aggregate data and rely on oversimplified model specifications, they do support the conjecture that increased mutual exposure of grandparents and grandchildren leads to stronger multigenerational effects.

## Conclusion

In this article, we examine the effects of demographic trends on the mutual exposure of grandparents and grandchildren and their consequences for multigenerational processes of social mobility in the United States from 1900 to 2010. Using historical vital statistics and census data, in combination with stable population models, we analyze grandparent-grandchild exposures from both prospective (grandparent) and retrospective (grandchild) perspectives and document very large increases in average shared lifetimes of grandparents and grandchildren. For example, the expected average years that a grandparent lived with any grandchild has increased from roughly 5 years in 1900 to almost 35 years in 2010. We show that these demographic changes also beget changes in social stratification in at least three ways. First, whereas all education groups have experienced an increase in the grandparent-grandchild exposure, their differences have changed, with families in which grandparents received 12 years of education showing the largest increase in mutual exposure between grandparents and grandchildren by most measures.

Second, we examine a new measure of family background—education-reweighted years of exposure to grandparents—which takes into account the effect of exposure on grandchildren’s socioeconomic attainments. Several social inequality indices calculated from prospective and retrospective forms of this measure suggest stable weighted educational inequality from 1900 to approximately 1970, a brief decline in the 1970s and 1980s, and a sharp increase after 1990. Although variation in years of school completed by parents or grandparents are not typical indices of family or individual inequality, our exposure-weighted estimates show yet another way that childhood inequality has grown since the 1980s.

Finally, we show that changing exposure has important implications for the transmission of educational status between grandparents and grandchildren. A simulation with parameters of educational mobility estimated from PSID and exposure trends derived from aggregate historical data suggests that grandparent effects on educational attainment may have increased over the past 110 years as a result of the large increase in average shared lifetimes of grandparents and grandchildren. Whereas estimates of grandparent-grandchild associations are small in many populations, whether these effects vary across time and place and what mechanisms may govern such variation are open questions. In societies where grandparent effects are more strongly associated with years of grandparent-grandchild exposure or changes in exposure over time are more dramatic than in the United States, the overall changes in grandparent effects would be more significant than those shown here. However, in societies or subpopulations where grandparent effects are insignificant (see a review in Solon 2018), the composition effect of exposure on changes in grandparent effects would be minimal.

We acknowledge two potential limitations in our estimates. First, we rely on a single socioeconomic indicator, educational attainment, which is central to processes of social mobility but may not respond to changes in family demography in the same way as other dimensions of inequality, including income, wealth, homeownership, and occupational status. Although data constraints make the type of analysis presented here quite difficult for these other measures, future work should try to extend these analyses beyond educational attainment. Second, we focus on one type of mutual exposure—shared lifetimes—and owing to data limitations, we do not consider stronger forms of exposure created by coresidence of grandparents and their grandchildren. Educational stratification patterns are likely to vary more dramatically with coresidential status than shared lifetime exposure, and future research should address this issue, even if it is not possible to consider the impact of coresidential trends for the long timespan examined in the analysis reported here. Coresidential patterns fluctuate over time in response to a variety of social and economic trends. From the standpoint of overall change in exposure, however, it is likely that the basic demographic mechanisms emphasized in this study, working through shared lifetimes, are the most important sources of long-run change in grandparent-grandchild educational mobility patterns over the last century.

## Acknowledgments

This research was supported by the National Science Foundation (SES-1260456). The authors also benefited from facilities and resources provided by the California Center for Population Research (CCPR) at UCLA, which receives core support (P2C-HD041022) from the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD). Earlier versions of this article have been presented at the 2015 RC28 Conference at the University of Pennsylvania; the 2016 ASA annual meeting in Seattle, Washington; the 2016 PAA annual meeting in Washington, DC; the 2017 ISA-RC28 Spring Meeting in Cologne, Germany; the Demography Workshop at the University of Chicago; the Demography and Inequality workshop organized by the European Consortium for Sociological Research in Berlin, Germany; and the Office of Population Research at Princeton University. We thank Cameron Campbell, Hal Caswell, Irma Elo, Anette Fasang, Michael Hout, Giovanna Merli, John Murphy, Judith Seltzer, Kazuo Yamaguchi, and numerous seminar participants for helpful discussions and comments, and Xia Zheng for outstanding research assistance.

## Notes

1

Parents’ education refers to father’s years of schooling if grandparents are from the paternal side; otherwise, it refers to mother’s years of schooling.

2

Haines (1989) did not provide separate estimates for 1900 and 1910. We use the midpoint method to estimate TFR in 1900 and 1910.

3

Specifically, we estimate m(x) ∝ N(Ar, σ2), within the range of x ∈ [15,45] for woman and [15,60] for men. We assume that the mean age of childbearing (Ar) and the variance of age of childbearing (σ2) are the same for men and women given that the Census Bureau provides these estimates only for females. Specifically, because $GRR=∫αβmxdx$, we estimate m(x) = GRR ∙ ψ(Ar, σ2, α, β; x), where $ψArσ2αβx=ϕArσ2xΦArσ2β−ΦArσ2α$ if α < x < β, and ϕ(⋅) and Φ(⋅) refer to the probability density function and the cumulative distribution function of a normal distribution, respectively.

4

Specifically, the relationship of the intrinsic growth rate and the age-specific fertility and mortality schedules follows $∫0te−rxlxmxdx=1$, where r is the intrinsic growth rate, and t is the maximum age in the population.

5

We assume that the GRR for the first education group in 1900 is proportional to the GRR for the second education group in 1920 with a constant c, where c = GRR1900 / ∑iGRRedu = i ∣ year = 1920 ∙ P(edu = i | year = 1900).

6

Because of interaction between mortality and fertility levels, the fertility effect would be bigger if we fixed the mortality level at a lower level (e.g., the level of 1950) than observed in Fig. 2, panel a. The results are included in Tables A6 and A7 of the online appendix.

7

The estimated fertility effect becomes larger when we fix mortality at a lower level (e.g., the level of 1950) than that shown in Fig. 2, panel b. Again, this implies that the effects of fertility and mortality are not additive; instead, their interaction affects the trend in exposure.

8

It is also possible to obtain estimates for grandchildren whose grandparents differ in their levels of education.

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text:
# Math Help - group theory abstract algebra

1. ## group theory abstract algebra

consider all the rationals except 1 with the binary operation a*b=a+b-ab for any a,b elements in Q\ {1}.
show that this is a group.
why is 1 excluded

2. Verify that $\left\langle\mathbb Q\setminus\{1\},\star\right\rangle$ satisfies all the group axioms. (Do you know what the group axioms are?) Hint for identity: $a\star0=0\star a=\ldots\,?$

This group is isomorphic to $\left\langle\mathbb Q\setminus\{0\},\times\right\rangle$ (the group of nonzero rationals under ordinary multiplication) via the mapping $a\mapsto\frac1{1-a}$ for $a\in\mathbb Q\setminus\{1\}.$

3. Originally Posted by bettywhit
consider all the rationals except 1 with the binary operation a*b=a+b-ab for any a,b elements in Q\ {1}.
show that this is a group.
why is 1 excluded
thanks i figured out how to show it is a group but i am not sure why 1 is excluded

4. Originally Posted by bettywhit
thanks i figured out how to show it is a group but i am not sure why 1 is excluded

The group's unit is 0, since $a*0=a+0-a \cdot 0=a$. If 1 were an element of this group, then you'd get $a*1=a+1-a\cdot 1=a+1-1=a$ and you have ANOTHER unit, which cannot be in a group.

Tonio

5. Something is wrong here.

$a\star1\ =\ a+1-a\cdot1=a+1-\color{red}a\color{black}=\color{red}1$

You just cannot have two identities, even in a semigroup! Including $1$ would make $\left\langle\mathbb Q,\star\right\rangle$ a semigroup rather than a group (the element $1$ will not have an inverse). If a semigroup has an identity, then that identity is always unique.

6. If 1 was included, the inverse would also be undefined,

format: plain


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# Technical question regarding lagrangian and action variation in QFT

In my textbook the lagrangian density of the system is taken to be $\mathcal{L} = -\frac{1}{2} \partial^{\mu} \phi \partial_{\mu} \phi - \frac{1}{2}m^2 \phi^2$ where $\phi \equiv \phi(x)$ is a complex scalar field.

We then try to determine the equation of motion by using that the variation of action must vanish. The author then introduce a variation $\delta \phi$ and quickly writes:

$$0 = \delta s = \int d^4x\ \left[-\frac{1}{2} \partial ^{\mu} \delta \phi \partial_{\mu} \phi -\frac{1}{2} \partial ^{\mu} \phi \partial_{\mu} \delta\phi -m^2\phi\delta \phi\right]$$

So my question is simply, what is going on here? How do you obtain the expression in the brackets? How does it relate to $\mathcal{L}$.

For the record, the tensor notation is still new to me, so my confusion might be related to this.

• you must regard $\delta$ as a differential operator (aka, linearity+Leibnitz rule). The justification of this rule lies in rigorous the definition of $\delta$. – AccidentalFourierTransform Nov 30 '16 at 21:05
• I think it is using $\delta(ab) = a\delta (b)+\delta(a) b$. – user130529 Nov 30 '16 at 21:06
• so $\delta(\phi \partial_{\mu} \phi) = \phi \delta \partial_{\mu} \phi + \delta \phi \partial_{\mu} \phi$ ? – Fabric Nov 30 '16 at 21:21
• Yes, if $\delta$ works like a differential operator as mentioned by AccidentalFourierTransform (and I see no reason why it wouldn't). – user130529 Nov 30 '16 at 21:36

Formally you can define your variation as $\delta \mathcal S = \int \mathrm{d}^d x \ \mathcal L [\phi + \delta \phi] - \mathcal L [\phi]$, where you treat $\delta \phi$ as small, i.e. terms which contain higher orders than just linear are to be neglected.
This is equivalent to taking $\delta$ as a differential operator, e.g.
$\delta( \phi^3) = (\phi + \delta \phi)^3 - (\phi)^3 = \phi^3 + 3 \phi^2 \delta \phi + \mathcal{O}(\delta \phi^2) - \phi^3 = 3 \phi^2 \delta \phi$, as expected by simple differentiation of $\phi^3$ wrto $\phi$.


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# How do microwaves heat moisture-free items?

Today I learnt that microwaves heat food by blasting electromagnetic waves through the water molecules found in the food.

Does that mean food with 0% moisture (if such a thing exists - dried spices?) will never receive heat from a microwave oven? And how in that case is a microwave able to melt plastics etc., which contain no obvious water?

• I won't leave an answer, as I'm only guessing, but one might ask the following questions: is it really possible for a food item to be moisture free? If a plastic is porous, might it absorb some water to its surface while in the air? Most importantly: what is special about water that makes it heat up under microwave irradiation? I think the answer is that $H_2O$ has some vibrational modes that correspond to the frequency of the microwaves. If this is the case: could the polymers in a plastic also have modes which correspond to microwave frequencies? – Lachy Dec 3 '14 at 1:46
• This is why if you sprinkle a little water on top of a cookie and put it in the microwave, it will taste amazing. – Señor O Dec 3 '14 at 15:42
• @SeñorO I'm not following your line of reasoning... – Michael Dec 3 '14 at 20:07
• @Michael Normally if you put a cookie in the microwave, the only part that really heats up are the chocolate chips. If you sprinkle some water on the whole cookie, it heats up the doughy parts and makes it warm and gooey like it just came out of the oven. Any excess water on the cookie will vaporize in the process. – Señor O Dec 3 '14 at 21:01
• @SeñorO I see. So the taste of the cookie is affected by its temperature and viscosity. – Michael Dec 3 '14 at 22:14

Microwave heating is largely caused by the changing electric and magnetic fields (i.e. the "microwaves") which are emitted by your microwave oven affecting polar molecules. As the direction of the electric field changes over time, the polar molecules (often, of water) attempt to follow the field by changing their orientation inside the material to line up along the field lines in an energetically favorable configuration (namely, with the positive side pointing in the same direction as the field lines). As these molecules change direction rapidly (millions of times per second at least), they gain energy - which increases the temperature of the material. This process is called dielectric heating.

However, water is not the only polar molecule in the world. You can test for yourself that most plastics don't heat in a microwave while most glass and ceramic objects do. So, a microwave oven melting your plastic bowl has more to do with it over-heating your food than over-heating that food's container.

EDIT: After doing some research to address some questions brought up in the comments to this post, I've found some very interesting information about why glass and ceramics heat up in the microwave which I will share here.

First of all, according to this article from the Royal Society of Chemistry so-called "earthenware" ceramics are fired at categorically lower temperatures than "stoneware." As a result, a non-negligible quantity of water molecules remain inside the now-seemingly-dry "earthenware," while the vast preponderance of water molecules in "stoneware" have been removed as a result of the higher firing temperature. The conclusion is that earthenware ceramics heat up in the microwave because they have the polar water molecules in them which undergo dielectric heating. On the other hand, stoneware (and apparently porcelain) will not heat in the microwave due to their respective lack of water molecules. Either way, I still wouldn't recommend microwaving your grandmother's porcelain china to find out.

Second, glass' molecular structure is apparently locally tetrahedral but without long-range order (i.e. it is an amorphous solid) which means that there tend to be spaces in the molecular structure of the glass to accommodate ionic impurities (mostly sodium, see this explanation of how glass is made to get an idea of the chemicals that go into the final product). These impurities are only loosely bound and are able to move around within the amorphous structure of the glass. These ions of sodium or other elements have a net charge (they are ions after all) which means that the oscillating electric field produced by the microwave oven causes the ions to jostle back and forth, gaining energy. The idea is very similar to the rotations of polar molecules (which have an electric dipole but no net charge), but the mechanism is different (namely, translational energy rather than rotational energy).

So in summary, ceramics apparently heat up because they still contain some water, while glass heats up mostly because of the presence of semi-free, charged ions.

• I'm not sure how well the molecular model applies to glass. The core mechanism is that the matter to be heated has internal electric fields due to the non-uniform charge density. If the external electric field alternates, the internal field will never have the ability to align. – MSalters Dec 3 '14 at 14:57
• I don't really like this kind of answer : I find it misleading because I don't think that this explains the dielectric losses. The proof is that the magnetic losses are maximum when the angle between the field and the polarization is the highest. – TZDZ Dec 3 '14 at 15:55
• I guess ceramics can be heated in the oven because they contain water, not because they are polar. Also, I believe glass is not heated in the oven (didn't do a controlled experiment, but I remember heating a glass of tea, and the upper part of the glass was colder). – user27542 Dec 3 '14 at 18:35
• A good piece of ceramics that has been fired correctly ought to be entirely free of moisture. Now maybe the material being heated is the glaze and not the clay, but that's a different issue. Glass is another matter. There are an enormous number of different glass formulae, some of which will heat in a microwave and some of which won't. I'm not an expert on the chemistry of all glass and all ceramics, but I'll try to do a little research once I can get on a computer (not just my phone). In the meantime, feel free to put an empty glass or coffee mug into your microwave and see for yourself. – Geoffrey Dec 3 '14 at 20:44
• If you are going to do this experiment, be careful as to the plastic you use; some (e.g. the melamine used in "unbreakable" plates and cookware) will not only heat up considerably but can also crack explosively and/or release formaldehyde. Stinks up your kitchen for ages. – Jon Hanna Dec 3 '14 at 23:44

Microwave heating actually has nothing to do with the moisture content of items. It has everything to do with the amount of electric dipoles (polar molecules) in the item of concern. Water molecules (with many other organic molecules) happen to be electric dipoles. (That is, one side of the molecule has a positive charge and the other side has a negative charge.)

When the oven uses a microwave to make an electric field, all electric dipoles move to align with that field. If the direction of that field quickly flips, you give these dipoles kinetic energy. As you increase a group of molecules' kinetic energy, you increase the temperature of that group.

Any material containing significant amounts of electric dipoles will heat in a microwave. Also, resonance of water molecules have nothing to do with heating food in microwave ovens. The oscillations of the waves in microwave ovens is too slow for resonance to play a role.

For further reading, check out wikipedia's articles on Microwave Ovens and Dielectric Heating, which should answer your questions more throughly.

• The question that the otherwise fine answers of both PipperChip and of @Geoffrey miss is that why the motion of polar molecules in a microwave field is absorptive? After all, they move in sync along the electric field lines that can be quite well modeled as harmonic motion with friction. What causes the friction? – hyportnex Dec 8 '14 at 0:58
• @user31748 Actually, the temperature of a system is just a statistical measure of the overall internal energy. The molecules become excited rotationaly which increases the internal energy thus increasing the temperature. You should probably take that friction analogy with a grain of salt. – Geoffrey Dec 12 '14 at 1:07
• I was told to cut down on salt... Ok the internal energy increases, but the question remains: granted the increase of internal energy what is the cause dissipation. Somehow the normally and coherently oscillating dipoles become disordered. How? In the case of ferromagnetism the "friction" is between neighboring domains, but what about dielectrics? I think, and this is just an uninformed guess, the neighboring dipoles interact electrically and that interaction propagates. What is the real dissipation mechanism? – hyportnex Dec 12 '14 at 1:17
• There is not just perfectly coherent oscillatory motion of the molecules going on - there is always some superpower random motion. So there is a chance of two molecules hitting each other - the faster they rotate the larger the transfer of energy. This in turn increases the incoherence which sustains the dissipation mechanism. – Floris Dec 12 '14 at 3:57
• As I said, I do not understand, but I do know that the microwave oven's operating frequency has absolutely nothing to do with water or any other material resonant frequency whatsoever. Any coincidence is coincidental. The ISM (industrial-scientific-medical) band frequencies given out by the FCC were determined by regulatory/bureaucratic/interference considerations not by physics. – hyportnex Jan 1 '15 at 17:27

I don't think it's dielectric heating .. it's more akin to inductive heating .. the establishment of eddy currents (largely surficial) which heat through resistance heating. Best material would be moderately electrically conductive substance .. hence ceramics containing metal atoms heat - where ceramics containing exclusively metal-oxide atoms - don't!

It's an AC environment so it would be more accurate to refer to the object's reactive impedance (and not it's ohmic resistance).


text: √121 = 11. will have an infinite number of decimals. ... Is the square root of 98 a rational or irrational number? Since 1582 is not a perfect square, it is an irrational number. It is an irrational number if it is not a perfect square. Is the square root of 1582 rational or irrational? The square root of 52 is a rational number if 52 is a perfect square. 2. It is an irrational number if it is not a perfect square. Rational numbers refer to the numbers that can be written as a simple fraction. geno3141 Oct 7, 2014 #2 +111868 +8 . The Square root of -121 is imaginary. This means that the answer to "the square root of 52?" Best Answer. It is a rational number; not an irrational number. Identify square root of 2 as either rational or irrational, and approximate to the tenths place. The square root of 1582 is a rational number if 1582 is a perfect square. Rational and Irrational Numbers. Square root of 121 = 11. square root of 1.21 = 1.1. 1. The numbers 1, 2 and 5 are rational because their square roots can be written as a simple fraction. √81 = 9. It has value of i11.It is not rational, irrational, nor real. ★★★ Correct answer to the question: Simplify each square root expression. irrational. 1/root (121) can be 1/11 OR - 1/11 since 121 has TWO square roots. - 23097961 There is a simple proof for the irrationality of $\sqrt 3$. Since 52 is not a perfect square, it is an irrational number. The radical answer (yes this is irrational, but we are not done yet.) Regarding root (384), we can simplify 384 into 4 X 96 (not 94). in your final answer, include all of your work. √121 √48 - … will have an infinite number of decimals. This means that the answer to "the square root of 1582?" describe the simplified form of the expression as rational or irrational. 9 is a rational number because it can be written as 9⁄1. Is the square root of 52 rational or irrational? Yes, $\sqrt{12}=2\sqrt 3$ is irrational. It is an irrational number


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# Тексты для чтения и грамматические тесты. Антонишкис А.А - 8 стр.

Составители:

Рубрика:

• ## Иностранный язык

8
contained in the first paragraph. The rest of the paragraph supports and develops the
main idea. Sentences which actually contain the main idea are called topic
sentences. It is a help to be able to recognize the indicators which show the function
of the other sentences that develop the topic sentences. Some indicators are: for
example, an example/illustration/instance of this is Thus/so/consequently/as a
result,
In addition/furthermore/moreover/also The reason for this is that Because of
This is because Due to Owing to
Text A. A recent investigation by scientists at the US Geological Survey
shows that strange animal behaviour might help predict future earthquakes.
Investigators found such occurrences in a ten-kilometer radius of the epicenter of a
fairly recent quake. Some birds screeched and flew about wildly; dogs yelled and
ran around uncontrollably.
Scientists believe that animals can perceive these environmental changes as
early as several days before the mishap.
In 1975 after observing animal behaviour, the Chinese were able to predict a
devastating quake. Although hundreds of thousands of people were killed, the
government was able to evacuate millions of other people and thus the death toll at
a lower level.
1. What is the main idea of the passage?
a. Earthquakes cause changes in the environment.
b. Animals go mad before a quake.
c. Animal behaviour can be used to predict future earthquakes.
d. The Chinese can predict future earthquakes.
2.What predictions can be made by observing animal behaviour?
a. an impending earthquake
b. the number of people who will die
c. the ten-kilometer radius of the epicenter
d. environmental changes
3. Why can animals perceive these changes when humans cannot?
a. Animals are smarter than humans.
b. Animals have certain instincts that human dont possess.
c. By running around the house, they can feel the vibrations.
d. Humans dont know where to look.
4. Which of the following is not true?
a. Some animals may be able to sense an approaching earthquake.
b. By observing animal behaviour scientists perhaps can predict
earthquakes.
c. The Chinese have successfully predicted an earthquake and saved
many lives.
                                          8
contained in the first paragraph. The rest of the paragraph supports and develops the
main idea. Sentences which actually contain the main idea are called topic
sentences. It is a help to be able to recognize the indicators which show the function
of the other sentences that develop the topic sentences. Some indicators are: for
example, an example/illustration/instance of this is… Thus/so/consequently/as a
result, …
In addition/furthermore/moreover/also… The reason for this is that … Because of…
This is because… Due to … Owing to …

Text A. A recent investigation by scientists at the US Geological Survey
shows that strange animal behaviour might help predict future earthquakes.
Investigators found such occurrences in a ten-kilometer radius of the epicenter of a
fairly recent quake. Some birds screeched and flew about wildly; dogs yelled and
ran around uncontrollably.
Scientists believe that animals can perceive these environmental changes as
early as several days before the mishap.
In 1975 after observing animal behaviour, the Chinese were able to predict a
devastating quake. Although hundreds of thousands of people were killed, the
government was able to evacuate millions of other people and thus the death toll at
a lower level.

1.   What is the main idea of the passage?
a.   Earthquakes cause changes in the environment.
b.   Animals go mad before a quake.
c.   Animal behaviour can be used to predict future earthquakes.
d.   The Chinese can predict future earthquakes.

2.What predictions can be made by observing animal behaviour?
a. an impending earthquake
b. the number of people who will die
c. the ten-kilometer radius of the epicenter
d. environmental changes

3. Why can animals perceive these changes when humans cannot?
a. Animals are smarter than humans.
b. Animals have certain instincts that human don’t possess.
c. By running around the house, they can feel the vibrations.
d. Humans don’t know where to look.

4. Which of the following is not true?
a. Some animals may be able to sense an approaching earthquake.
b. By observing animal behaviour scientists perhaps can predict
earthquakes.
c. The Chinese have successfully predicted an earthquake and saved
many lives.


text:
zbMATH — the first resource for mathematics

Global 2-forms on regular 3-folds of general type. (English) Zbl 0838.14032
Let $$V$$ be a smooth projective variety of $$\mathbb{C}$$-dimension = 3. The author is motivated by the following problem: Assume that $$V$$ is of general type. Then find some universal integer $$N$$ such that $$|NK |$$ defines a birational map where $$K$$ is the canonical bundle of $$V$$. For irregular 3-folds $$V$$ of general type, the problem was settled by Kollár. Therefore, in this paper, the author concentrates on regular 3-folds $$V$$ of general type, i.e. $$h^1 (V,{\mathcal O}_V) = 0$$. Let $${\mathcal H}^2 : = H^0 (V, \Omega^2_V) \otimes_\mathbb{C} {\mathcal O}_V \subset \Omega^2_V$$ be the subsheaf of $$\Omega^2_V$$ spanned by $$H^0 (V, \Omega^2_V)$$. It is clear that $\text{rank } {\mathcal H}^2 \leq 3.$ The main result of this paper is the following theorem:
The above problem admits a positive answer for regular 3-folds $$V$$ of general type, provided rank $${\mathcal H}^2 = 1$$ or 3. Some discussion when rank $${\mathcal H}^2 = 2$$ is carried out.

MSC:
 14J30 $$3$$-folds
Full Text:
References:
 [1] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. · Zbl 0718.14023 [2] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties , Math. USSR-Izv. 13 (1979), 499-555. · Zbl 0439.14002 [3] A. R. Fletcher, Contributions to Riemann-Roch on projective $$3$$-folds with only canonical singularities and applications , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 221-231. · Zbl 0662.14026 [4] M. Hanamura, Pluricanonical maps of minimal $$3$$-folds , Proc. Japan Acad. 61 (1985), no. 4, 116-118. · Zbl 0598.14032 [5] J. Kollár, Higher direct images of dualizing sheaves, I , Ann. of Math. (2) 123 (1986), no. 1, 11-42. JSTOR: · Zbl 0598.14015 [6] T. Mabuchi, Invariant $$\beta$$ and uniruled threefolds , J. Math. Kyoto Univ. 22 (1982), no. 3, 503-554. · Zbl 0511.32016 [7] S. Mori, Classification of higher-dimensional varieties , Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., Part 1, vol. 46, Amer. Math. Soc., Providence, 1987, pp. 269-331. · Zbl 0656.14022 [8] M. Reid, Young person’s guide to canonical singularities , Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., Part 1, vol. 46, Amer. Math. Soc., Providence, 1987, pp. 345-414. · Zbl 0634.14003 [9] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces , Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329-353. · Zbl 0513.14019 [10] S. Zucker, Remarks on a theorem of Fujita , J. Math. Soc. Japan 34 (1982), no. 1, 47-54. · Zbl 0503.14002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Electrical physic

1. Jan 27, 2012

### vtran0703

My brother asked me this question and i completely have no idea how to do cause it is a very long time. Can anyone help, or give me an idea on how to do

A rectangular insulating sheet carrying a negative charge Q evenly distributed over its surface is formed into a cylinder, producing a thin walled hollow object of length L and radius R.

• We wish to find the electric force on a positive point charge q, located on the axis of the cylinder at one end.

a. Draw a sketch showing the tube, the victim charge q, and an appropriate coordinate system for describing the problem. In your sketch clearly show E at the location of q.

b. Choose and then clearly indicate the charge element dq you will use for the integration in your sketch.

c. Set up the integration. That is express F as a constant times an integral over a single variable.

Thank you

2. Jan 27, 2012

### tiny-tim

welcome to pf!

hi vtran0703! welcome to pf!

it's a fairly straightforward integration problem …

find the charge on a small area dzdθ of the cylinder, find the force at E from that charge, and then integrate over the whole cylinder

show us how far you get, and where you're stuck, and then we'll know how to help!

3. Jan 27, 2012

### vtran0703

It's been quite a while since I saw these problem. To be honest, I wasn't good with them back then either, but I'll try, any help would be great

4. Jan 27, 2012

### Mindscrape

Remember that
$$\mathbf{E} = \int \frac{\lambda(\mathbf{r'}) d\mathbf{l}}{|\mathbf{r}-\mathbf{r'}|^2}\hat{(\mathbf{r}-\mathbf{r'})}$$
or alternatively
$$\mathbf{E} = \int \frac{\lambda(\mathbf{r'}) \mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3 }d\mathbf{l}$$
where r is the vector from the origin to a point in space and r' is a vector from the origin to the charge. So you make all the these infinitesimally different vectors as you sweep along the charge edge, and sum them up. A good idea for an origin is one where you can exploit symmetry in the problem, what's the most symmetric origin? What is the electric field from there?

Last edited: Jan 27, 2012