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Introduction to Sociology/Demography

Introduction

Demography is the study of human population dynamics. It encompasses the study of the size, structure and distribution of populations, and how populations change over time due to births, deaths, migration, and aging. Demographic analysis can relate to whole societies or to smaller groups defined by criteria such as education, religion, or ethnicity.

Why study demography?
Before proposing complex theories to explain sociological phenomena (e.g., World Systems Theory), especially at the macro and/or societal levels, sociologists should first turn to demographic indicators for possible explanations. Demographic analysis is a powerful tool that can explain a number of sociological phenomena.

For instance, in examining the elements that led to the first World War, most people turn to political and diplomatic conflicts but fail to consider the implications of expanding populations in the European countries involved. Expanding populations will result in increased competition for resources (i.e., food, land, access to trade routes and ports, etc.). Expanding populations may not have been the primary cause of World War I, but they may have played a role in the increased hostilities leading up to the war. In this fashion, demographic indicators are often informative in explaining world events and should be turned to first as explanations.

History
The study of human populations has its roots, like sociology generally, in the societal changes that accompanied both the scientific and industrial revolutions. Some early mathematicians developed primitive forms of life tables, which are tables of life expectancies, often used for life insurance and actuarial purposes. Censuses, another demographic tool, were instituted for primarily political purposes:
 * as a basis for taxation
 * as a basis for political representation

The development of demographic calculations started in the 18th century. Census taking, on the other hand, has a long history dating back close to 2,000 years among the Chinese and the Romans and even further back in history among some groups in the Middle East. Most modern censuses began in the late 18th century.

Data and Methods
Demography relies on large data sets that are primarily derived from censuses and registration statistics (i.e., birth, death, and marriage registrations). Large data sets over long periods of time (e.g., the U.S. census is conducted every 10 years) are required to develop trends in demographic indicators, like birth and death rates.

In many countries, particularly in developing nations, reliable demographic data are still difficult to obtain. In some locales this may be due to the association of census with taxation.

Demographic Indicators
Because demography is interested in changes in human populations, demographers focus on specific indicators of change. Two of the most important indicators are birth and death rates, which are also referred to as fertility (see also fecundity) and mortality. Additionally, demographers are interested in migration trends or the movement of people from one location to another. Some of the specific measures used to explore these elements of population change are discussed below. While demography often provides useful portraits of social patterns, it is important to note that - especially in relation to minority groups - accurate numerical values are often difficult to achieve, and thus demographic understandings of social structures and patterns are continuously shifting in relation to the availability of more accurate data and measurement techniques.

Fertility and Fecundity
Fertility, in demography, refers to the ability of females to produce healthy offspring. Fecundity is the potential reproductive capacity of a female. Some of the more common demographic measures used in relation to fertility and/or fecundity include:
 * crude birth rate: the annual number of live births per thousand people
 * general fertility rate: the annual number of live births per 1000 women of childbearing age (often taken to be from 15 to 49 years old, but sometimes from 15 to 44).
 * age-specific fertility rate: the annual number of live births per 1000 women in particular age groups (usually age 15-19, 20-24 etc.)
 * total fertility rate: the number of live births per woman completing her reproductive life if her childbearing at each age reflected current age-specific fertility rates
 * gross reproduction rate: the number of daughters who would be born to a woman completing her reproductive life at current age-specific fertility rates
 * net reproduction rate: the number of daughters who would be born to a woman according to current age-specific fertility and mortality rates

Another important demographic concept relating to fertility is replacement level. Replacement level fertility refers to the number of children that a woman (or monogamous couple) must have in order to replace the existing population. Sub-replacement fertility is a fertility rate that is not high enough to replace an existing population. Replacement level fertility is generally set at 2.1 children in a woman's lifetime (this number varies by geographic region given different mortality rates). Sub-replacement fertility is below approximately 2.1 children in a woman's life time. The reason the number is set to 2.1 children per woman is because two children are needed to replace the parents and an additional one-tenth of a child is needed to make up for the mortality of children and women who do not reach the end of their reproductive years. Of course, women don't have one-tenth of a child; this results from statistical averaging between women who have more than two children and those who have two or fewer children.

The chart below illustrates trends in childbearing by region of the world. Fertility rates dropped earlier in the more developed regions of the world, followed by Asia and Latin America. Fertility rates are just starting to decline in Africa.

The chart below highlights the varied fertility rates of specific countries as some have very low fertility rates, many have moderate rates, and some have very high rates.

The following chart illustrates the relationship between contraceptive use and the total fertility rate by regions of the world. Increased contraceptive use is associated with lower numbers of children per woman.

One of the strongest predictors of fertility rates is women's educational attainment. Almost universally, higher levels of educational attainment result in lower fertility rates. It is not, however, education itself that causes declines in fertility but rather its association with other factors that reduce fertility: women with higher levels of education delay marriage and are more likely to abstain from marriage and / or parenthood, have improved labor market opportunities, are more likely to use contraception during intercourse, and are less likely to adopt traditional childbearing roles.

Fertility rates are also closely related to a country's level of development, which influences other factors. For instance, women who have kids in developed countries have increased opportunity costs, meaning they will make less money because of time spent outside the workforce raising kids. This is true in developed countries because women are more likely to be highly skilled and well-paid (relative to women in developing countries). Additionally, delayed childbearing, probability of a child reaching adulthood, norms about ideal family sizes, and pervasiveness of contraceptives will all reduce fertility rates. But one of the biggest factors is the cost of children. In undeveloped and developing countries, children are often an economic asset to parents as they serve as cheap labor on the farm; they don't require pay, just food and shelter. That is not the case in developed countries, where very few people work in agriculture (roughly 2% in the US). Instead, children are an economic liability, meaning they cost money while not generating money for the parents. The cost of raising a child from birth to 18 in a middle-income home in 2022 was $310,000. The cost of raising children in developed countries reduces fertility rates in those countries. Further, mothers (but not fathers) in developed countries typically experience wage penalties (e.g., receive less pay than male and non-parental female workers), which can further exacerbate the financial cost of parenthood for women in developed countries.

Mortality
Mortality refers to the finite nature of humanity: people die. Mortality in demography is interested in the number of deaths in a given time or place or the proportion of deaths in relation to a population. Some of the more common demographic measures of mortality include:
 * crude death rate: the annual number of deaths per 1000 people
 * infant mortality rate: the annual number of deaths of children less than 1 year old per thousand live births
 * life expectancy: the number of years which an individual at a given age can expect to live at present mortality rates

Note that the crude death rate as defined above and applied to a whole population can give a misleading impression. For example, the number of deaths per 1000 people can be higher for developed nations than in less-developed countries, despite standards of health being better in developed countries. This is because developed countries have relatively more older people, who are more likely to die in a given year, so that the overall mortality rate can be higher even if the mortality rate at any given age is lower. A more complete picture of mortality is given by a life table which summarizes mortality separately at each age.

This chart depicts infant mortality by region of the world. The less developed regions of the world have higher infant mortality rates than the more developed regions. This chart depicts life expectancy by region of the world. Similar to infant mortality, life expectancies are higher in more developed regions of the world.

According to recent research, one of the best predictors of longevity (i.e., a long life) is education, even when other factors are controlled: the more educated you are, the longer you can expect to live. A few additional years of schooling can add several additional years to your life and vastly improve your health in old age. The mechanism through which this works is not the schooling itself, but schooling's influence on other health-related behaviors. The more education someone has, the lower his/her likelihood of smoking and engaging in unhealthy and high risk behaviors. Education also increases the probability of people engaging in healthy behaviors, like frequently exercising.

Other factors associated with greater longevity include:
 * wealth: money increases access to good healthcare, which improves health and increases longevity
 * race: whites live longer than blacks, though this is due to other social disparities, like income and education, and not to race itself
 * ability to delay gratification: with the ability to delay gratification people live healthier lives and engage in healthier behaviors (e.g., exercise)
 * larger social networks: having a large group of friends and close relationships with relatives increases your social support, which positively influences health
 * job satisfaction: people in more powerful and more satisfying jobs tend to be healthier than people in less satisfying jobs

The Demographic Transition

The demographic transition is a model and theory describing the transition from high birth rates and death rates to low birth and death rates that occurs as part of the economic development of a country. In pre-industrial societies, population growth is relatively slow because both birth and death rates are high. In most post-industrial societies, birth and death rates are both low. The transition from high rates to low rates is referred to as the demographic transition. This understanding of societal changes is based on the work of Thompson, Blacker, and Notestein, who derived the model based on changes in demographics over the preceding two hundred years or so.

The beginning of the demographic transition in a society is indicated when death rates drop without a corresponding fall in birth rates (usually the result of improved sanitation and advances in healthcare). Countries in the second stage of the demographic transition (see diagram) experience a large increase in population. This is depicted in the diagram when death rates fall in stage two but birth rates do not fall until stage three. The red line begins its rapid upward growth in stage two and begins to level off at the end of stage three.

By the end of stage three, birth rates drop to fall in line with the lower death rates. While there are several theories that attempt to explain why this occurs (e.g., Becker and Caldwell, who view children as economic commodities), why birth rates decline in post-industrial societies is still being evaluated. Recent evidence from Mongolia, which underwent a demographic and economic transition in the 1990s and early 2000s, suggests that during this transition women begin pursuing education in order to obtain greater wealth. Thus, fertility maximization (i.e., having lots of kids) was traded for the pursuit of status enhancement (i.e., having more money), suggesting that prior explanations for the demographic transition - namely, the cost of children - may have been at least partially correct. As a result of this transition, many developed countries now have a population that is static or, in some cases, shrinking.

As with all models, this is an idealized, composite picture of population change in these countries. The model is a generalization that applies to these countries as a group and may not accurately describe all individual cases. Whether or not it will accurately depict changes in developing societies today remains to be seen. For more information on the demographic transition, see here.

Population Growth and Overpopulation

Overpopulation indicates a scenario in which the population of a living species exceeds the carrying capacity of its ecological niche. Overpopulation is not a function of the number or density of the individuals, but rather the number of individuals compared to the resources they need to survive. In other words, it is a ratio: population over resources. If a given environment has a population of 10, but there is food and drinking water enough for only 9 people, then that environment is overpopulated, while if the population is 100 individuals but there are food and water enough for 200, then it is not overpopulated. Resources to be taken into account when estimating if an ecological niche is overpopulated include clean water, food, shelter, warmth, etc. In the case of human beings, there are others such as arable land and, for all but tribes with primitive lifestyles, lesser resources such as jobs, money, education, fuel, electricity, medicine, proper sewage and garbage management, and transportation.

Presently, every year the world's human population grows by approximately 80 million. About half the world lives in nations with sub-replacement fertility and population growth in those countries is due to immigration. The United Nations projects that the world human population will stabilize in 2075 at nine billion due to declining fertility rates. All the nations of East Asia, with the exceptions of Mongolia, the Philippines, and Laos, are below replacement level. Russia and Eastern Europe are dramatically below replacement fertility. Western Europe also is below replacement. In the Middle East Iran, Tunisia, Algeria, Turkey, and Lebanon are below replacement. Canada, Australia, and New Zealand are similar to Western Europe. In the United States, fertility is now at around 1.6 births per woman, which is below the replacement level. Many of these nations still have growing populations due to high rates of immigration.

Much of the projected growth is expected to come from African countries where birth rates remain high. While birth rates in most countries have fallen since 1990, in some parts of Africa birth rates have actually increased and the average woman has more than five children, well above the replacement rate.

Early Projections of Overpopulation
Early in the 19th century, Thomas Malthus argued in An Essay on the Principle of Population that, if left unrestricted, human populations would continue to grow until they would become too large to be supported by the food grown on available agricultural land. He proposed that, while resources tend to grow arithmetically, population grows exponentially. At that point, the population would be restrained through mass famine and starvation. Malthus argued for population control, through moral restraint, to avoid this happening.

The alternative to moral restraint, according to Malthus, is biological and natural population limitation. As the population exceeds the amount of available resources the population decreases through famine, disease, or war, since the lack of resources causes mortality to increase. This process keeps the population in check and ensures it does not exceed the amount of resources.

Over the two hundred years following Malthus's projections, famine has overtaken numerous individual regions. Proponents of this theory, Neo-Malthusians, state that these famines were examples of Malthusian catastrophes. On a global scale, however, food production has grown faster than population. It has often been argued that future pressures on food production, combined with threats to other aspects of the earth's habitat such as global warming, make overpopulation a still more serious threat in the future.

Population as a Function of Food Availability
There are some scholars who argue that human populations predictably grow and shrink according to their available food supply - populations grow when there is an abundance of food and shrink in times of scarcity. This idea is only slightly less problematic than the idea that human population growth is not guided by food production, as it suggests that that every time food production is intensified to feed a growing population, the population responds by increasing even more. Some human populations throughout history support this theory, as consistent population growth began with the agricultural revolution, when food supplies consistently increased.

Critics of this idea point out that birth rates are voluntarily the lowest in developed nations, which also have the highest access to food. In fact, the population is decreasing in some countries with abundant food supply. Thus, human populations do not always grow to match the available food supply. Critics cite other factors that contribute to declining birth rates in developed nations, including: increased access to contraception, later ages of marriage, the growing desire of many women in such settings to seek careers outside of child rearing and domestic work, and the decreased economic 'utility' of children in industrialized settings. The latter explanation stems from the fact that children perform a great deal of work in small-scale agricultural societies, and work less in industrial ones; this interpretation may seem callous, but it has been cited to explain the drop-off in birthrates worldwide in all industrializing regions.

For some, the concept that human populations behave in the same way as do populations of bears and fish is hard to believe; for others it indicates a feasible solution to population issues. In either case, since populations are tied to the food they consume, discussions of populations should not take place without considering the role played by food supply. There is a substantial body of research that has considered the ability of the planet to provide sufficient food for the world's growing population. This research suggests that the planet can potentially provide sufficient food for the projected peak population of humans of 9 billion people, but only if agriculture is carefully managed. Factors that must be considered include: genetically modified crops, employing agricultural tools in correct, context-specific ways, aquaculture, and simultaneously working to limit harm to the environment.

Effects of Overpopulation
Many proponents of population control have averred that famine is far from being the only problem attendant to overpopulation. These critics point out ultimate shortages of energy sources and other natural resources, as well as the importance of serious communicable diseases in dense populations and war over scarce resources such as land area. A shortage of arable land (where food crops will grow) is also a problem.

The world's current agricultural production, if it were distributed evenly, would be sufficient to feed everyone living on the Earth today. However, many critics hold that, in the absence of other measures, simply feeding the world's population well would only make matters worse, natural growth will cause the population to grow to unsustainable levels, and will directly result in famines and deforestation and indirectly in pandemic disease and war.

Some of the other characteristics of overpopulation include:
 * Child poverty
 * High birth rates
 * Lower life expectancies
 * Lower levels of literacy
 * Higher rates of unemployment, especially in urban areas
 * Insufficient arable land
 * Little surplus food
 * Poor diet with ill health and diet-deficiency diseases (e.g. rickets)
 * Low per capita GDP
 * Increasingly unhygienic conditions
 * Government is stretched economically
 * Increased crime rates resulting from people stealing resources to survive
 * Mass extinctions of plants and animals as habitat is used for farming and human settlements

Another point of view on population growth and how it effects the standard of living is that of Virginia Abernethy. In Population Politics she shows evidence that declining fertility following industrialization only holds true in nations where women enjoy a relatively high status. In strongly patriarchal nations, where women enjoy few rights, a higher standard of living tends to result in population growth. Abernathy argues that foreign aid to poor countries must include significant components designed to improve the education, human rights, political rights, political power, and economic status and power of women.

Possible Solutions to Overpopulation
Some approach overpopulation with a survival of the fittest, laissez-faire attitude, arguing that if the Earth's ecosystem becomes overtaxed, it will naturally regulate itself. In this mode of thought, disease or starvation are "natural" means of lessening population. Objections to this argument are:
 * 1) in the meantime, a huge number of plant and animal species become extinct
 * 2) this would result in terrible pollution in some areas that would be difficult to abate
 * 3) it creates certain moral problems, as this approach would result in great suffering in the people who die

Others argue that economic development is the best way to reduce population growth as economic development can spur demographic transitions that seem to naturally lead to reductions in fertility rates.

In either case, it is often held that the most productive approach is to provide a combination of help targeted towards population control and self-sufficiency. One of the most important measures proposed for this effort is the empowerment of women educationally, economically, politically, and in the family. The value of this philosophy has been substantially borne out in cases where great strides have been taken toward this goal. Where women's status has dramatically improved, there has generally been a drastic reduction in the birthrate to more sustainable levels. Other measures include effective family planning programs, local renewable energy systems, sustainable agriculture methods and supplies, reforestation, and measures to protect the local environment.

David Pimentel, a Cornell University professor of ecology and agricultural sciences, sees several possible scenarios for the 22nd century: Spreading awareness of the issues is an important first step in addressing it.
 * 1) a planet with 2 billion people thriving in harmony with the environment
 * or, at the other extreme, 12 billion miserable humans suffering a difficult life with limited resources and widespread famine

Underpopulation
Once countries pass through the demographic transition, some experience fertility rate decreases so substantial that they fall well below replacement level and their populations begin to shrink (as has Russia's in recent years, though emigration has also played a role in Russia's population decline). A new fear for many governments, particularly those in countries with very low fertility rates, is that a declining population will reduce the GDP and economic growth of the country, as population growth is often a driving force of economic expansion. To combat extremely low fertility rates, some of these governments have introduced pro-family policies, that include things like payments to parents for having children and extensive parental leave for parents. Such policies may reverse the low fertility rates, but they may also exacerbate overpopulation.

Migration
The likelihood of a given individual in the U.S. moving to another place in the U.S. in any given year has declined over the last 40 years. Only about 1 in 10 Americans have moved in the last year, which is about half the proportion that changed residences annually in the 1960s. The reduction in moves is attributable to aging populations (older people are less likely to move) and an increase in dual-career couples. Those who do move are generally driven by jobs.

Close to 37% of Americans have never moved from the community in which they were born. There are wide variations in native inhabitants, however: 76% of Texans were born in-state while only 14% of Nevadans were born in-state. Some states lose a large number of people who were born in the state as well, like Alaska, where only 28% of the people born in that state have remained there. Immigration is often a controversial topic, for a variety of reasons, though many have to do with competition between those already living in the destination location and those arriving in that location. One study found that one type of competition between immigrants and non-immigrants may be overstated. Some people have suggested that natives' opportunities to attend college are negatively impacted through competition with immigrants. Neymotin found that competition with immigrants does not harm the educational outcomes of U.S. natives and may in fact facilitate college attending. Additionally, recent research suggests that opposition to immigration is less about competition over jobs and more about how immigrants might change the identity of the society where the migrants move.

Urbanization
Urbanization is the physical growth of urban areas as a result of global change. Urbanization is also defined by the United Nations as movement of people from rural to urban areas with population growth equating to urban migration. Over half of the world's population lived in urban areas at the end of 2008 and that percentage continues to increase. Urbanization is closely linked to modernization, industrialization, and the sociological process of rationalization.

Movement

As more and more people leave villages and farms to live in cities, urban growth results. The rapid growth of cities like Chicago in the late 19th century and Mumbai a century later can be attributed largely to rural-urban migration and the demographic transition. This kind of growth is especially commonplace in developing countries.

The rapid urbanization of the world’s population over the twentieth century is described in the 2005 Revision of the UN World Urbanization Prospects report. The global proportion of urban population rose dramatically from 13% (220 million) in 1900, to 29% (732 million) in 1950, to 49% (3.2 billion) in 2005 and to 56.2% in 2020 per the 2018 UN World Urbanization Prospects report. The same report projected that the figure is likely to rise to 60% (4.9 billion) by 2030 and 68.4% by 2050. Urbanization rates vary between countries. The United States and United Kingdom have a far higher urbanization level than China, India, Swaziland or Niger, but a far slower annual urbanization rate, since much less of the population is living in a rural area.

Causes

People move into cities to seek economic opportunities. A major contributing factor is known as "rural flight." In rural areas, often on small family farms, it is difficult to improve one's standard of living beyond basic sustenance. Farm living is dependent on unpredictable environmental conditions, and in times of drought, flood or pestilence, survival becomes extremely problematic. In modern times, industrialization of agriculture has negatively affected the economy of small and middle-sized farms and strongly reduced the size of the rural labour market. Cities, in contrast, are known to be places where money, services, and wealth are centralized. Cities are where fortunes are made and where social mobility is possible. Businesses, which generate jobs and capital, are usually located in urban areas. Whether the source is trade or tourism, it is also through the cities that foreign money flows into a country. As with immigration generally, there are factors that push people out of rural areas and pull them into urban areas.

There are also better basic services as well as other specialist services in urban areas that aren't found in rural areas. There are more job opportunities and a greater variety of jobs. Health is another major factor. People, especially the elderly, are often forced to move to cities where there are doctors and hospitals that can address their health needs. Other factors include a greater variety of entertainment (e.g., restaurants, movie theaters, theme parks, etc.) and better quality of education in the form of universities. Due to their high populations, urban areas can also have much more diverse social communities allowing others to find people like them when they might not be able to in rural areas. These conditions are heightened during times of change from a pre-industrial society to an industrial one.

Economic Effects
As cities develop, effects can include a dramatic increase in costs, often pricing the local working class out of the real estate market. For example, Eric Hobsbawm's wrote, "Urban development in our period [1789–1848] was a gigantic process of class segregation, which pushed the new labouring poor into great morasses of misery outside the centres of government and business and the newly specialised residential areas of the bourgeoisie. The almost universal European division into a 'good' west end and a 'poor' east end of large cities developed in this period." This is likely due the prevailing south-west wind which carries coal smoke and other airborne pollutants downwind, making the western edges of towns preferable to the eastern ones. Similar problems now affect the developing world; rising inequality results from rapid urbanization. The drive for growth and efficiency can lead to less equitable urban development.

Urbanization is often viewed as a negative trend, but can in fact, be perceived simply as a natural occurrence from individual and corporate efforts to reduce expense in commuting and transportation while improving opportunities for jobs, education, housing, and transportation. Living in cities permits individuals and families to take advantage of the opportunities of proximity, diversity, and marketplace competition.

Environmental Effects
One environmental concern associated with urbanization is the urban heat island. The urban heat island is formed when industrial and urban areas are developed and heat becomes more abundant. In rural areas, a large part of the incoming solar energy is used to evaporate water from vegetation and soil. In cities, where less vegetation and exposed soil exists, the majority of the sun’s energy is absorbed by urban structures and asphalt. Hence, during warm daylight hours, less evaporative cooling in cities allows surface temperatures to rise higher than in rural areas. Additional city heat is given off by vehicles and factories, as well as by industrial and domestic heating and cooling units. This effect causes the city to become 2 to 10o F (1 to 6o C) warmer than surrounding landscapes. Impacts also include reducing soil moisture and intensification of carbon dioxide emissions.

The effects of urbanization may be an overall positive for the environment. Birth rates of new urban dwellers fall quickly. This can prevent overpopulation. Additionally, it puts a stop to destructive subsistence farming techniques, like slash and burn agriculture. Finally, it minimizes land use by humans, leaving more for nature.

Other Effects
In addition to the effects noted above, urbanization can influence how people feel about their environment. For instance, one study found that people who live in large apartment buildings are less likely to know their neighbors. Even so, they do not feel greater fear when they are walking around alone than do individuals who live in single-family homes. However, when people in apartment buildings are home alone, they feel less fear than do people who are home alone in single-family homes. One possible explanation is a "fortress effect": individuals in large apartment buildings feel isolated from those outside the building. Another possible explanation is that the close presence of people around us, even if we do not know them, tends to afford some degree of security.

Changing Forms of Urbanization
Different forms of urbanization can be classified depending on the style of architecture and planning methods as well as historic growth of areas. In cities in the developed world, urbanization traditionally exhibited a concentration of human activities and settlements around the downtown area. Recent developments, such as inner-city redevelopment schemes, mean that new arrivals in cities no longer necessarily settle in the center. In some developed regions, the reverse effect, originally called counter urbanisation has occurred, with cities losing population to rural areas, and is particularly common for richer families. This has been possible because of improved communications and means of transportation, and has been caused by factors such as the fear of crime and poor urban environments. Later termed "white flight", the effect is not restricted to cities with a high ethnic minority population. When the residential area shifts outward, this is called suburbanization. Some research suggests that suburbanization has gone so far to form new points of concentration outside the downtown both in developed and developing countries such as India.

Urbanization can be planned or organic. Planned urbanization, (e.g., planned communities), is based on an advanced plan, which can be prepared for military, aesthetic, economic, or urban design reasons. Organic urbanization is not organized and happens haphazardly. Landscape planners are responsible for landscape infrastructure (e.g., public parks, sustainable urban drainage systems, greenways, etc.) which can be planned before urbanization takes place, or afterward to revitalize an area and create greater livability within a region.

Additional Reading
The International Handbook of the Demography of Race and Ethnicity Series: International Handbooks of Population, Vol. 4 Sáenz, Rogelio, Embrick, David G., Rodríguez, Néstor P. (Eds.) 2015, XXIII, 708 p. 91 illus., 41 illus. in color.

International Handbook of Rural Demography Series: International Handbooks of Population, Vol. 3 Kulcsár, László J., Curtis, Katherine J. (Eds.) 2012, XIV, 406 p.

International Handbook on the Demography of Sexuality Series: International Handbooks of Population, Vol. 5 Baumle, Amanda K. (Ed.) 2013, VI, 429 p. 46 illus.

Discussion Questions

 * Do you think the world is over-populated?
 * If you think the world is over-populated, what could be done about it?
 * How will your life decisions effect your life expectancy?
 * How will your life decisions effect your fertility rate?

title: Introduction to Sociology/Demography


text:
Cookbook talk:Salty (Namkin) Lassi

Sub-sections
There's also a pawpaw lassi. I don't know what people want to do with that. The mere usage of pawpaw makes it highly American, but maybe it belongs on the main Lassi page anyway. Probably it should at least have a link from here. —The preceding unsigned comment was added by AlbertCahalan (talk • contribs) 23:08, 9 May 2005 (UTC)

 * Lassi in India is made with various fruits and therefore i think it makes more sense to have them as sub sections on the main Lassi page instead of new pages. —The preceding unsigned comment was added by Poohneat (talk • contribs) 19:57, 27 August 2006 (UTC)

 * DO NOT SPLIT. I think it is great to have all these simple recipies together on one page, until it gets overloaded. The five that are there now look good together, and it seems like up to about a dozen would be OK. And I vote for taking the split tag off the article.  The Pawpaw Lassi article should be merged into here.  Maybe this article needs some new main divisions, one for India/Asia/authentic and one for Other/international/etc? 69.87.194.147 13:41, 25 December 2006 (UTC) —The preceding unsigned comment was added by 69.87.194.147 (talk • contribs) 13:41, 25 December 2006  (UTC)

 * Page has been included here. I'm deleting it. --Swift 10:46, 26 December 2006 (UTC)

 * Pawpaw refers to both the widely distributed tropical fruit Carica papaya, as well as the American fruit, Asimina triloba. In the US, "pawpaw" is A. triloba.  Outside of the US, pawpaw most likely would refer to C. papaya, although I believe "papaya" is the most common name (globally, for English speakers) for this fruit. I'm going to change "pawpaw" in this article to "papaya", as I really doubt that A. triloba is the fruit being used in this dish. If A. triloba is in fact the intended fruit, please change it back.  If this defies a regional usage of "pawpaw" for C. papaya, and you are tempted to change it back, please add a note making it clear which fruit is being referred to.

192.104.39.2 (talk) 21:23, 29 September 2008 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Cookbook talk:Salty (Namkin) Lassi


text:
User:Naseelah

=chapter one=

skill one
there is no skills really in the real life but we might say something about that you can link yahoo easily any time

$$square((12/3)+(34/2))$$

skill three
= chapter two =

skill three
chapter three ex1 ex2 ex3 conclusion

title: User:Naseelah


text:
User talk:Naseelah

when can i learn --Naseelah 03:26, 10 May 2005 (UTC)

title: User talk:Naseelah


text:
Talk:Latin/Phonology

Clarification
I'm a bit confused by this key. If you could, maybe add english words with the sounds. Are latin vowels constant? Does I always make the same sound? When is each sound used?

Inaccuracy
The table seems to be, grossly inaccurate. I cite http://www.jstor.org/view/00978507/ap020167/02a00150/0 or Allen's Vox Latina as evidence; the IPA representation of the long i is [i:] (triangle colon, of course, not an ordinary one), and similarly with the other Latin vowels; length, not tenseness is the issue. However, I am not sure what the current opinion in the field is, nor have I found any terribly explicit statement either way. I will edit the page when I am more certain.

Moved from Module Page
Note: this is very unfinished, but it's late and I'm tired. To do: clean this up, add diphthongs and digraphs. Redundancy (pointlessness?) in light of wikipedia article? Not that one's perfect, but it seems a waste to create two flawed documents rather than synthesize them into a better whole... Phthoggos 05:56, 20 Jul 2004 (UTC)

Also, why did my table move to the bottom of the screen? Phthoggos 05:57, 20 Jul 2004 (UTC) NOTE: This note should be in the discussion page, but this article is unfinished anyways and here i am sure you will see it. I put a link here with the Northern Europe pronunciation (the one from after the Middle Ages, in the times of the Great Geographical Discoveries to the Age of Reason) - SizarieldoR 10:53, 13 Jan 2007

title: Talk:Latin/Phonology


text:
Talk:How To Assemble A Desktop PC/Assembly

This is deadly: "If using thermal paste, apply it only to the CPU die (the square piece of silicon in the middle of the CPU)." First of all, there's no "if" about: thermal paste is compulsory for any CPU that takes a cooler and all modern ones do. Second, and much worse, the CPU surface on which paste goes is the flat metal face that shows after installation, NOT a "piece of silicon in the middle . . . ." The combination of this lingo and the pictures, which don't show the CPU, only the LGA grid, could cause a real disaster. Pictures should show socket with cap, socket open, cpu installed. Made preliminary change, someone with real expertise may want to repair. Also fixed a little grammar, added note on removing paste.

Bookaminor (discuss • contribs) 20:41, 27 October 2015 (UTC)

I changed small phillips to #2 phillips since that is more precise and the #2 is really what you want, not #1 which is too small. If you use #1 you will likely end up stripping the screws and the driver.

"and most of this can be averted by choosing a non-conductive floor surface such as floorboards, or tiles. Make sure you don't do this in an unfinished basement!"

I would think you would want a conductive floor surface to help ground out any static charges that might build up.
 * If you see a mistake JUST FIX IT! -CarbonUnit

Also, I've worked in an unfinished basement before... Never had any problems with static.

Some RAM chips do not go in either way

many faults
Many faults in this book:

Power Up: "Then press the power button..." should be double check everything (again) then press the power button... Expansions and Connections: "Next, close the case and take your computer to where you will be using it." Don't close the case until the system is confirmed working and that all fans are running, then you can close the case

In general the whole end of the guide is bad. PC building and testing... CPU (and heatsink/fan) -> mainboard... ram -> mainboard... Mainboard -> chassis... video card -> mainboard... power supply -> mainboard and fans... monitor -> video card... keyboard -> mainboard... DO NOT CONNECT ANYTHING ELSE!... double check everything... power on system... wait for POST, if system has hardware monitoring go into the BIOS and check the numbers... unplug system, note that I did not say to power it off with the power button and then unplug it, you need to unplug it to drain the PSU.... Now you can connect your other stuff and then double check everything again and POST the system.

"you should choose an area without carpet on the floor, as carpet tends to attract a lot of static, and most of this can be averted by choosing a non-conductive floor surface such as floorboards, or tiles. Make sure you don't do this in an unfinished basement!"....... non-conductive floor surface WTF are you talking about. YOU WANT A CONDUCTIVE (preferably earth grounded) SURFACE! to prevent any voltage differentials (static) from building up

I'm going to have to rewrite this book and show you how to do it the old school way. My first self built was a 286, I've been doing this as a hobby and professionally for many many years.

 * Yes, I agree. Please fix it. I fixed a couple of things ... --DavidCary 19:58, 3 October 2006 (UTC)

Grounding
I believe that regardless of an anti-static mat, that an anti-static strap should always be used, especially if the person is constructing a computer for the first time, I do not however want to change the artcle as I have no cited source to base my oppinion on. You can still damage components on an anti-static mat.

Tyler.szabo 04:21, 24 June 2006 (UTC)yu

 * It all depends on the type of anti-static mat, the static charge of the person and the conductivity of the workbench/surfaces in contact with the components. I have always built my own computers from components and never used any anti-static methods, at best I sometimes tried to make sure I gave the chance of any static to discharge before touching the components by contacting a grounded metal structure first. --Panic (discuss • contribs) 07:21, 15 July 2013 (UTC)

Why does it say under Safety Precautions "Never plug your computer in while you are connected to it by a wrist strap." - If the computer case is just sitting on a wooden table during assembly and installation, is that enough to dissipate any static electricity from the person, or does it need to be connected to the actual earth ground in the building? Oathed (discuss • contribs) 21:07, 10 May 2020 (UTC)

Sequence of actions
Connecting Power supply is said to be the first thing to do. In reality it would obstruct the rest of assembly with its numerous cables hanging around and closing off access to processor. In fact, even mounting motherboard with power supply sitting there is ridiculously hard. The power supply should be installed after motherboard and processor (and fan), but before installing optical disk drive - it may be impossible/hard to connect power cables to motherboard if it is there (not a huge problem, but still). I propose the following sequence:
 * 1) Remove the freaking power supply
 * 2) Motherboard
 * 3) Processor
 * Fan
 * 1) Memory, Expansion cards
 * 2) Power supply
 * 3) Connect cables to mb
 * HD, CD, FD
 * 1) Connect all other cables

That was Grue 17:16, 9 July 2006 (UTC)

Installing the power supply *first* may not be the best action, but I think *removing* the power supply can be more hassle than it is worth.

I have seen some PC chassis form factors where the power supply blocked access to everything else. On the other hand, I have been fortunate enough that most of the PC chassis I've worked on have been big enough that there was no overlap. One chassis I worked on has a hinge so that the motherboard would swing down, so you could install the CPU, RAM, etc. far away from the power supply. In those cases, removing the power supply would be an unnecessary hassle. --DavidCary 14:12, 3 October 2006 (UTC)

images
How do I get the pretty image of a "computer screw" in http://en.wikipedia.org/wiki/Screw to show up in this article?

--DavidCary 19:58, 3 October 2006 (UTC)

clean room
I removed

 * A clean room is a better place to work on your computer or to build a computer. a type 100 or above is perfict you could use a clean room type 50 for that matter but if oyu want to open your hard drive up use a type 100 clean room..

While yes, a clean room would be adequate, it is overkill for putting together a computer. Put a computer together in a room filled with cigaretter smoke, and it will work just fine. Also, opening up a hard drive is not something I would recommend, unless you *want* to destroy it.

--DavidCary 19:58, 3 October 2006 (UTC)

Account Only?
Perhaps this should be made account editing only? Unlike most articles, if a vandal got in here and wasn't caught he could do hundreds or thousands of dollars of damage. I will give no examples of type as your average vandal is to lazy to go into much depth and might get bored and turn away, and examples here would tell him how. On the other hand, he/she might get lucky or persistent and get away with something serious.--70.15.4.226 (talk) 22:05, 2 April 2008 (UTC)
 * I agree. I saw a tiny bit of random spam of something about "hitler" (it was barely noticable, but I deleted it). There might be some vandal lurking around...


text:
Cookbook:Cola Cake

| Dessert | Cake

Cake

 * 2 cups (275 g / 9.7 oz) flour
 * 2 cups (450 g / 16 oz) white granulated sugar
 * 1 cup (250 ml / 8.5 oz) cola
 * 2 sticks (250 g / 8.8 oz / 1 cup) margarine
 * 3 tablespoons cocoa
 * 2 eggs
 * ½ cup (125 ml / 4.2 oz) buttermilk
 * ½ teaspoon baking soda
 * 1 teaspoon vanilla extract
 * 1 ½ cup (125 g / 4.4 oz) miniature marshmallows

Frosting

 * 6 tablespoons cola
 * 2 tablespoons cocoa
 * 1 stick (125 g / 4.4 oz) margarine
 * 1 teaspoon vanilla
 * 4 cups (500 g / 1.1 lb) powdered sugar
 * 1 cup (80 g / 2.8 oz) chopped nuts (optional)

Cake

 * 1) Combine flour and sugar in a bowl.
 * 2) Mix cola, margarine, and cocoa in a saucepan, and bring to a boil.
 * 3) Pour over flour/sugar mixture and mix well.
 * 4) Beat the eggs and mix them in. Add milk, soda, vanilla and marshmallows to make thin batter. Marshmallows will float to the surface.
 * 5) Bake in a 9x13-inch (23x33 cm) pan at 350 F for 35 minutes. Make frosting before it cools.

Frosting

 * 1) Mix cola, cocoa, and margarine in a pan and bring to a boil.
 * 2) Add vanilla, sugar, and nuts. Mix until smooth.
 * 3) Spread the warm frosting on the cake.

title: Cookbook:Cola Cake


text:
GCSE Science/Current, voltage resistance and Ohm's law answers

1= 1.5 A 2= 0.5 ohm 3= 10 ohm


text:
Cookbook talk:Cola Cake

Bleh... this is like something out of the infamous "Cooking with 7-UP" recipe book from 1957, but it's Coke. AlbertCahalan 07:29, 10 May 2005 (UTC)
 * It's good, you should try it! Although, to be honest, I can't really taste any difference between this and a chocolate cake. It may just be a marketing ploy to get people to buy more coke, and it caught on because Coca-Cola is the "national drink". Marshmellows are a nice touch. PurplePieman 00:11, 12 May 2005 (UTC)

title: Cookbook talk:Cola Cake


text:
User talk:207.171.167.101

Cut it out now.

Cookbook:Grilling is a disambiguation page. This is necessary, because there are at least 3 meanings of this term.

Cookbook:Barbecue follows the dictionary definition. Maybe you'd like to change the English language, but this isn't the place to do it. Go to http://m-w.com/ and look up barbecue please. AlbertCahalan 07:45, 10 May 2005 (UTC)

We wouldn't need a disambiguation page if you didn't keep posting the wrong information. Also, aren't disambiguation pages are marked 'Disambigusation Page'. Dictionaries are inferior to encyclopedias, and since we're using Wikibooks, lets use Wikipedia's definition. Unless you'd like to rewrite that. Seriously, check out Barbecue and stop using an incorrect definition. 207.171.167.101 10 May 2005

I checked it out, and you're still wrong. Why do you believe that "The term barbecue is incorrectly used interchangably with grilling"? Who are you to say that the most people are incorrect? Language is defined by usage, and so you are incorrect. If everybody else thinks that barbecuing is cooking with fire, then that's what it means. I'll use a dictionary to determine this, because maybe you wrote the Wikipedia entry. The dictionary agrees with what I've always thought, and what everybody I'd ever met used as the definition. You're the first person I've ever encountered who would defy the dictionary on this. AlbertCahalan 08:21, 10 May 2005 (UTC)

title: User talk:207.171.167.101


text:
User:GauravV

Hi,

I go by the name Gaurav on Wikipedia, so I'd appreciate if you'd leave me messages of my talk messages on my Wikipedia talk page.

cheers.

title: User:GauravV


text:
Introduction to Sociology/Collective Behavior

Introduction
The term Collective behavior refers to social processes and events which do not reflect existing social structure (laws, conventions, and institutions), but which emerge in a "spontaneous" way. Collective behavior might also be defined as action which is neither conforming (in which actors follow prevailing norms) nor deviant (in which actors violate those norms). Collective behavior, a third form of action, takes place when norms are absent or unclear, or when they contradict each other. Scholars have devoted far less attention to collective behavior than they have to either conformity or deviance.

Examples of collective behavior include: religious revival meetings (like those depicted in the documentary Marjoe), a panic in a burning theater (e.g., the Kentucky Beverly Hills Supper Club fire), a sudden widespread interest in a website (e.g., MySpace) or clothing item (e.g., wriststrong bracelets), a collective social movement to improve the environment (e.g., Greenpeace), or the rapid spread of rumors (e.g., that Barack Obama is Muslim or not a US citizen). These diverse actions fall within the area sociologists call collective behavior.

Collective behavior differs from group behavior in three ways:
 * 1) collective behavior involves limited and short-lived social interaction while groups tend to remain together longer
 * 2) collective behavior has no clear social boundaries; anyone can be a member of the collective while group membership is usually more discriminating
 * 3) collective behavior generates weak and unconventional norms while groups tend to have stronger and more conventional norms

Traditionally, collective behavior in sociology includes four forms : the crowd, the public, the mass, and the social movement. While there is a degree of debate over what should be included under the label of "collective behavior" among sociologists today, often included are additional behaviors like: rumors, riots, trends, and fads.

Why Study Collective Behavior?
Aside from the intrinsic interest of understanding why large groups of people behave the way they do, there are practical reasons why the study of collective behavior is important. Two examples might illustrate the practical importance:

Better architectural design and crowd management might have avoided this tragedy. How to redesign buildings and manage crowds are two types of knowledge that can result from the study of collective behavior. Understanding how people behave in riots, what sets them off, and how they can be rapidly concluded is also knowledge that can result from the study of collective behavior. Additionally, understanding how humans react during natural disasters and ensure that the damage that occurs is entirely a result of the disaster and not the human response to it.

Another motivation for studying collective behavior is in order to actually change elements of society. This is the component of collective behavior known as "social movements." Again, an example may help illustrate this point:

Understanding how to organize a social movement to pursue social change is one of the areas studied by sociologists. Better understanding how to organize such a movement can provide movement members the tools they need to succeed.

Various forms of collective behavior are examined in detail in the following sections.

Crowds
A crowd is a gathering of people who share a purpose or intent and influence one another. Crowds are a common occurrence in modern life. Most sporting events, concerts, and other performances result in the gathering of crowds. Blumer (1951) differentiated four types of crowds:
 * casual - loose collection of people with no real interaction (e.g, people at the mall)
 * conventional - deliberately planned meeting (e.g., community meeting organized by political leaders)
 * expressive - depicts a crowd at an emotionally charged event (e.g., a political rally or soccer game in Europe or Latin America)
 * acting - a crowd intent on accomplishing something (e.g., fans rushing a stage during or after a concert)

When crowd behavior is directed toward a specific, violent end, the result is a mob. Mobs tend to be highly emotional. Examples of mob violence include the lynchings in the Southern U.S. during the 19th and 20th centuries. Violent crowd behavior without a specific goal is a riot. Because riots do not have a specific end, it is assumed that their intention is to express general dissatisfaction.

Diffuse Crowds
Collective behavior can also refer to behavior that is diffused or dispersed over large distances. Not all collective behavior has to occur in the immediate vicinity of others (compact crowds). This is especially true with the advent of mass media, which allows for the rapid distribution of information around the world.

Contagion Theory
Originally proposed by Gustave LeBon (1896), contagion theory proposes that crowds exert a hypnotic influence on their members. The hypnotic influence, combined with the anonymity of belonging to a large group of people, results in irrational, emotionally charged behavior. Or, as the name implies, the frenzy of the crowd is somehow contagious, like a disease, and the contagion feeds upon itself, growing with time. This also implies that the behavior of a crowd is an emergent property of the people coming together and not a property of the people themselves.

There are several problems with LeBon's theory. First, contagion theory presents members of crowds as irrational. Much crowd behavior, however, is actually the result of rational fear (e.g., being trapped in a burning theater) or a rational sense of injustice (e.g., the Cincinnati race riots). Second, crowd behavior is often instigated by and guided by individuals. That the crowd seems to take on a life of its own is certainly true, but the influence of the individual should not be overlooked.

It is also worth noting that LeBon's book is from the perspective of a frightened aristocrat. He interprets the crowd episodes of the French Revolution as irrational reversions to animal emotion, which he sees as characteristic of crowds in general. Blumer sees crowds as emotional, but as capable of any emotion, not only the negative ones of anger and fear.

Convergence Theory
Convergence theory argues that the behavior of a crowd is not an emergent property of the crowd but is a result of like-minded individuals coming together. In other words, if a crowd becomes violent (a mob or riot), convergence theory would argue that this is not because the crowd encouraged violence but rather because people who wanted to become violent came together in the crowd.

The primary criticism of convergence theory is that there is a tendency for people to do things in a crowd that they would not do on their own. Crowds have an anonymizing effect on people, leading them to engage in sometimes outlandish behavior. Thus, while some crowds may result from like-minded individuals coming together to act collectively (e.g., political rally), some crowds actually spur individuals into behavior that they would otherwise not engage in.

Emergent-Norm Theory
Emergent-Norm Theory combines the above two theories, arguing that it is a combination of like-minded individuals, anonymity, and shared emotion that leads to crowd behavior. This theory takes a symbolic interactionist approach to understanding crowd behavior. It argues that people come together with specific expectations and norms, but in the interactions that follow the development of the crowd, new expectations and norms can emerge, allowing for behavior that normally would not take place.

Crowds as "Gatherings"
More recent research into collective behavior has begun to change sociological thinking on crowds. This new approach distinguishes what brings people together as a gathering from what they do once gathered. Most gatherings are temporary and are formed by an assembling process. Individuals who gather are often acquainted and invited or informed about the gathering. Once gathered, the subgroups of individuals who already know one another tend to stay together throughout the gathering. There are, of course, some solitary individuals as well.

This line of research also dispenses with the idea that crowds impair judgment. Alcohol and drugs, which can contribute to deviant behavior, certainly can impair judgment and influence the actions of crowds, but crowds themselves do not necessarily impair judgment. The actions of individuals at gatherings also illustrate that individuals remain independent, sometimes responding to solicitations, sometimes ignoring them, sometimes interacting with their subgroup, and sometimes acting spontaneously.

Gatherings also exhibit dispersing processes that end the gatherings. Sometimes these are emergency dispersal, as when authorities arrive and try to end the gathering. Sometimes they are planned or the enthusiasm of the gathering wanes and people simply leave. Perhaps the key point of the "crowds as gatherings" approach is that there is a great deal of variation in gatherings.

Panic
Panic is a sudden terror which dominates thinking and often affects groups of people. Panics typically occur in disaster situations, such as during a fire, and may endanger the overall health of the affected group. Architects and city planners try to accommodate the symptoms of panic, such as herd behavior, during design and planning, often using simulations to determine the best way to lead people to a safe exit.

Moral Panic
A moral panic is a mass movement based on the perception that some individual or group, frequently a minority group or a subculture, poses a menace to society. These panics are generally fuelled by media coverage of social issues (although semi-spontaneous moral panics do occur and some moral panics have historically been fueled by religious missions, governmental campaigns, and scientific mobilizing against minority groups that used media outlets to further their claims), and often include a large element of mass hysteria. A moral panic is specifically framed in terms of morality, and usually expressed as outrage rather than unadulterated fear. Though not always, very often moral panics revolve around issues of sex and sexuality. A widely circulated and new-seeming urban legend is frequently involved. These panics can sometimes lead to mob violence. The term was coined by Stanley Cohen in 1972 to describe media coverage of Mods and Rockers in the United Kingdom in the 1960s.

Recent moral panics in the UK have included the ongoing tabloid newspaper campaign against pedophiles, which led to the assault and persecution of a pediatrician by an angry, if semi-literate, mob in August 2000, and that surrounding the murder of James Bulger in Liverpool, England in 1993. (See this page for examples of moral panic.)

Riots

A riot is a form of civil disorder characterized by disorganized groups lashing out in a sudden and intense rash of violence, vandalism or other crime. While individuals may attempt to lead or control a riot, riots are typically chaotic and exhibit herd-like behavior. Riots often occur in reaction to a perceived grievance or out of dissent. Historically, riots have occurred due to poor working or living conditions, government oppression, taxation or conscription, conflicts between races or religions, the outcome of a sporting event, or frustration with legal channels through which to air grievances. Riots typically involve vandalism and the destruction of private and public property. The specific property to be targeted varies depending on the cause of the riot and the inclinations of those involved. Targets can include shops, cars, restaurants, state-owned institutions, and religious buildings.

Mass Hysteria
Hysteria is a diagnostic label applied to a state of mind, one of unmanageable fear or emotional excesses. People who are "hysterical" often lose self-control due to the overwhelming fear.

The term also occurs in the phrase mass hysteria to describe mass public near-panic reactions. It is commonly applied to the waves of popular medical problems that everyone gets in response to news articles, such as the yuppy flu of the late 1980s. A similar usage refers to any sort of public wave phenomenon, and has been used to describe the periodic widespread reappearance and public interest in UFO reports, crop circles, and similar examples.

Hysteria is often associated with movements like the Salem Witch Trials, the Red Scare, McCarthyism, and Satanic ritual abuse, where it is better understood through the related sociological term of moral panic.

Mass hysterias can also exhibit themselves in the sudden onset of psychogenic illnesses, or illnesses that are the result of psychology and not an external source (e.g., like a pollutant or infectious agent). A recent example of psychogenic illness resulting from mass hysteria occurred in Jilin, China in 2009 when hundreds of workers at an acrylic yarn factory began to fall ill. Doctors in China determined that, for most of those who fell ill, there were no physical indications of poisoning, which is what the workers claimed caused the illness.

Fads
A fad, also known as a craze, refers to a fashion that becomes popular in a culture (or subcultures) relatively quickly, remains popular, often for a rather brief period, then loses popularity dramatically. (See this page for a list of fads.)

Rumors
A rumor is often viewed as "an unverified account or explanation of events circulating from person to person and pertaining to an object, event, or issue in public concern" (p. 33), though the definition can vary. Rumors generally involve some kind of a statement the veracity of which is not quickly or ever confirmed.

Rumors have three basic characteristics
 * 1) they're transmitted by word of mouth
 * 2) they provide "information" about a person, happening, or condition
 * 3) they express and gratify the emotional needs of the community

There are also various types of rumors, including:
 * 1) Pipe dream rumors, which reflect public desires and wished-for outcomes
 * 2) Bogie or fear rumors that reflect feared outcomes
 * 3) Wedge-driving rumors that intend to undermine group loyalty or interpersonal relations

As rumors travel they grow shorter, more concise, and more easily grasped. About 70% of details in a message are lost in the first 5 to 6 transmissions. Negative rumors are more likely to be disseminated than positive rumors.

Rumors may be part of a collective explanation process. Bordia and DiFonzo found that 29.4% of statements on archived internet message boards are sensemaking statements that attempt to solve problems. The rest of the discussion revolves around these statements.

Rumors may also be political communication strategies. Media and particular cultural-historical conditions may facilitate a rumor's diffusion. In 2006, Jayson Harsin introduced the concept of the "rumor bomb" to describe the widespread phenomenon of rumoresque communication in contemporary relations between media and politics, especially within the complex convergence of multiple forms of media, from cell phones and internet, to radio, TV, and print. Harsin treats rumor as a particular rhetorical strategy. For Harsin a "rumor bomb" extends the definition of rumor into a political communication concept with the following features:
 * 1) A crisis of verification. A crisis of verification is perhaps the most salient and politically dangerous aspect of rumor.
 * 2) A context of public uncertainty or anxiety about a political group, figure, or cause, which the rumor bomb overcomes or transfers onto an opponent.
 * 3) A clearly partisan even if an anonymous source (eg. "an unnamed advisor to the president"), which seeks to profit politically from the rumor bomb’s diffusion.
 * 4) A rapid diffusion via highly developed electronically mediated societies where news travels fast.

Examples of rumors include:
 * the alleged removal of weapons of mass destruction from Iraq to Syria
 * sexual innuendo in The Little Mermaid adaptation by Disney
 * sexual innuendo in The Lion King, another Disney film
 * sexual innuendo in Aladdin, another Disney film
 * that John McCain had an illegitimate black child
 * that Barack Obama is a Muslim and was not born in the U.S.

There are several good online resources for checking rumors and urban legends, including:
 * snopes.com, particularly useful for checking email rumors
 * urbanlegend.com
 * factcheck.org, particularly useful for checking political rumors

Research Examples
Berk (1974) uses game theory to suggest that even a panic in a burning theater can reflect rational calculation: If members of the audience decide that it is more rational to run to the exits than to walk, the result may look like an animal-like stampede without in fact being irrational. In a series of empirical studies of assemblies of people, McPhail (1991) argues that crowds vary along a number of dimensions, and that traditional stereotypes of emotionality and unanimity often do not describe what happens in crowds.

Suggested Multimedia Resources

 * The Final Report: LA Riots. http://channel.nationalgeographic.com/series/final-report/2770/Overview (Accessed May 5, 2008).

Additional Reading
McPhail, Clark and Ronald Wohlstein. 1983. “Individual and Collective Behavior WithinGatherings, Demonstrations, and Riots.” Annual Review of Sociology 9: 579-600. Granovetter, Mark. 1978. “Threshold Models of Collective Behavior.” American Journal of Sociology 83(6): 1420-1443. Snow, David, Louis Zurcher, and Robert Peters. 1981. "Victory Celebrations as Theater: A Dramaturgical Approach to Crowd Behavior." Symbolic Interaction 4:21-42. Vider, Stephen. 2004. "Rethinking Crowd Violence: Self-Categorization Theory and the Woodstock 1999 Riot." Journal for the Theory of Social Behaviour 34:141-166. Cress, Daniel and David Snow. 1996. “Mobilization at the Margins: Resources, Benefactors, and the Viability of Homeless Social Movement Organizations.” American Sociological Review, 61:1098-1109. Staggenborg, Suzanne. 1988. “The Consequences of Professionalization and Formalization in the Pro- Choice Movement.” American Sociological Review, vol. 53(4), pp. 585-605. Piven, Frances Fox. 2006. Challenging Authority: How Ordinary People Change America. New York: Rowman & Littlefield Publishers, Inc. McAdam, Doug. 1999. Political Process and the Development of Black Insurgency, 1930 – 1970. Chicago: University of Chicago Press. Meyer, David. 2004. “Protest and Political Opportunities.” Annual Review of Sociology, 30: 125-145.

Discussion Questions

 * Have you ever been in a crowd and done something you would not have done otherwise?
 * Do people typically panic in emergencies?
 * Why do so many movies and TV shows suggest that people would panic if they "knew the truth"?
 * Do you pass on rumors or chain emails without looking into them? Why or why not?


text:
Network Plus Certification/Introduction

The Network+ Exam is just one exam, and for those who took it before January 1st, 2011, it never expires. (Those taking the test after that date must either retake it at a set interval or submit proof of their continuing education in networking. The COMPTIA website has more details.) It has 100 multiple-choice "situational and identification" questions. Basically, if you know your hardware and software, as well as know what to do in a given situation, you should pass. You are given 90 minutes to take the exam, which should be plenty if you are fully prepared. It's recommended by CompTIA that you already have an A+ Certification under your belt and at least nine months networking experience, but the only prerequisite for taking the exam is paying the exam fee ($240 US). Minimum passing score is 720 on a scale of 100-900, but that doesn't really mean what you think it means: some questions are worth more than others. The test is available in English, German, Japanese and Korean.

While the A+ certification is definitely an industry standard and a prerequisite for many jobs, the Network+ is less so, but gaining in popularity.

The goals of the Network+ Certification is to ensure that employers hire people who will not need to be trained in network fundamentals before being hired. A Network+ Certified tech should be able to make recommendations for installing or expanding a network, document and perform preventative maintenance on a network, and troubleshoot network problems as they arise. Having an A+ and Network+ Certification should mean that you are fully capable of being an on-site technical support person, a network administrator of a local area network, or a valuable asset to an enterprise-level networking team, but not exactly be running it.

Scope of this Wikibook
The scope of this book is to provide a solid foundation for network administration. It should include all of the information that is needed to pass the CompTIA Network+ Exam, and match its scope. In fact, the finished product should look like the official CompTIA Network+ exam objectives, but expanded with more information and explanation.

Structure of this Wikibook
This "book" is not a "book" per se, but rather a few introductory pages and, more importantly, a study guide with a collection of links to articles in Wikipedia and the most relevant information summarized for each concept. The reason for this is twofold: there is no need to "re-invent the wheel" when there are so many great articles that serve the purpose of explaining, say, the OSI model. Two, a person interested in this "book" is one that wants to take and pass the Network+ Exam, and this method may be better suited to them than a traditional book, allowing them to quickly review concepts just before the exam or delve into the details of anything not immediately familiar.

Exam overview
The objectives of the Network+ exam are broken down into 6 categories:

 * Network Technologies
 * Understand the protocols used over a network and the ports they use, the addressing systems used on a network, and how a network can be implemented via wireless technologies.

 * Network Media and Topologies
 * Identify cables and connectors used to network computers together, understand the various manners in which a network can be physically laid out, and have knowledge of the standards used for communication within a network and between networks.

 * Network Devices
 * Differentiate between the various devices used on a network and understand their function, installation, and configuration. Have deeper knowledge of the functions of a switch and the factors to consider when setting up a wireless network.

 * Network Management
 * Understand the different conceptual layers in the networking model, the need for proper network documentation, the process of troubleshooting network problems, and the options for increasing network performance.

 * Network Tools
 * Be aware of the command line, software, and hardware tools available for installation, testing, and troubleshooting networks.

 * Network Security
 * Know what hardware devices, software, and policies will ensure the security of the data and devices on a network and be aware of the corresponding threats they are designed to guard against. Have knowledge of the methods users can use to securely access a network and how their identities can be verified before granting access in the first place.

title: Network Plus Certification/Introduction


text:
Talk:World History/Causes and course of the Second World War

okay, i daredn't change this myself before being certain, as the author has clearly made a lot of effort with this piece. three things:
 * 1) mussolini's agression was not necessarily 'foolish' - in principle it could have been bold and daring - he just didn't have the calibre and quantity of troops, mechanized divisions etc.
 * 2) battle of britian and russia - hitler had always been planning agression against the soviets. The whole idea of invading france after a pact with stalin was to avoid a war on two fronts. His plan was to sweep over western europe, including britain if he had to. His long term goal was always to invade russia, he didn't 'then turn to russia' after failing to conquer britain. After all, they were idealogical enemies. He greatly respected the british (a dubious honour for us), and didn't have any huge desire to invade them - that was arguably to prove his downfall. He never turned the full might of germany's war machine against the island. God knows what would have happened had he done so.
 * 3) "The Worldwide Great Depression" was naturally worldwide - that, in my experience, has always been taken as given. Is 'worldwide' really necessary? I have always read it simply as "The Great Depression"

keep up the great work, --Mark Lewis 20:07, 10 May 2005 (UTC)

Page orginization
What do most people find a more preferable way to break up sections of WWII on a page; by Theatre/Campaign or Year?

There are still many cases of cold war which can lead to III world war. The whole world is on depression. This could be the most dangerous n unforgetable war. Nitesh dhital (discuss • contribs) 11:57, 3 July 2016 (UTC)

Pacific theatre has US bias - no Burma campaign
The pacific theatre headings don't seem to cover the Burma campaign at all. Although I say US bias, I believe that there was US involvment there as well. In the UK it is often called the forgotten war. --David Woolley 13:53, 29 November 2005 (UTC)

order of events
Following the conquest of the Low Countries, Hitler occupied Denmark and Norway, beginning on April 9, 1940. Thats not true. It was the other way round. --NeuerNutzer2009 (discuss • contribs) 22:12, 14 August 2011 (UTC)

European Theatre
Information was added up to the invasion plans for France.


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Cookbook:Spanish Flan

| Spanish Cuisine

Ingredients

 * 5 eggs
 * 1 tsp vanilla
 * 1 can of sweetened condensed milk
 * 1 can of evaporated milk (large can)
 * Caramel ice cream topping

Instructions

 * 1) Heat oven to 375°F.
 * 2) Combine all ingredients except caramel topping in blender. Blend for 2 minutes.
 * 3) Place a thin layer of caramel topping on bottom of a round pie pan. Pour egg mixture on top of caramel topping.
 * 4) Place the smaller pie pan in a larger pan filled with enough water so that it comes about 1 inch up the side of the smaller pan. This will act as a bain marie and stop the flan from burning.
 * 5) Place both pans in oven. Cook for 1 hour or until center is firm.
 * 6) Remove from oven and let cool.
 * 7) Place a round dish over the top of the smaller pan, and flip it over to unmold it. The flan can be served on the dish.

title: Cookbook:Spanish Flan


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User:Stoli~enwikibooks

I just registered so I could edit a few grammatical mistakes in Blender_3D:_Noob_to_Pro which initially led me to WikiBooks.

title: User:Stoli~enwikibooks


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Cookbook:Red Pepper

Red Pepper is an ambiguous term. It could be referring to...

 * ripe bell peppers
 * powdered dried Cayenne peppers
 * whole red chili peppers (generally hot ones)
 * pizza pepper, a seasoning made from large flakes of chili peppers and other peppers and typically available at pizza restaurants
 * crushed red pepper flakes

See also: green pepper

title: Cookbook:Red Pepper


text:
Cookbook:Green Pepper

Green Pepper is an ambiguous term. Certainly, it involves a green-colored chili pepper. Beyond that though...

 * unripe bell peppers
 * whole green chili peppers (generally hot ones)

See also: red pepper

title: Cookbook:Green Pepper


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Talk:Dutch/Lesson 4

/Archive/

title: Talk:Dutch/Lesson 4


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User:Kenyon

See my Wikipedia page.

title: User:Kenyon


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Talk:Real Analysis/Sequences

A minor problem with 'Algebraic Operations on Sequences'

I believe that there is a problem with the 4th result (about quotient of 2 limits) I reckon it is not enough that y_n is non-zero for all n, but believe that we also need to assume that the limit of the y's is also nonzero. Otherwise there would exist a counter-example where the y_n are alternatively positive and negative, while tending to zero.

Does anyone agree?

82.27.163.136 10:07, 18 February 2006 (UTC)madmath789

Yes, your correct, and now the article assumes the limit is not zero, but allows some of the y_n's to be zero. Thenub314 (talk) 15:41, 5 January 2009 (UTC)

Completeness
There seems to be a general confusion here about the differences between the least upper bound property and completeness. In particular some of the theorems stated are not equivalent to the least upper bound axiom, and hence could not be used as an axiom defining the real numbers. I think we should be a bit more careful about how we use the term completeness. Thenub314 (talk) 15:45, 5 January 2009 (UTC)

limit superior
How this definition define limit superior and for what kind of sequence we can define limit superior? Anuvabd1 (discuss • contribs) 06:03, 7 May 2019 (UTC)

title: Talk:Real Analysis/Sequences


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Real Analysis/Limits

The challenge in understanding limits is not in its definition, but rather in its execution. Successfully completing a limit proof, using the epsilon-delta definition, involves learning many different concepts at once—most of which will be unfamiliar coming out from earlier mathematics. This chapter will serve as a guide in navigating these proofs, as the skills here will serve you well in higher mathematics.

Definition
The definition of a limit, in ordinary real analysis, is notated as:

 * $$\lim_{x \rightarrow c}f(x) = L$$

One way to conceptualize the definition of a limit, and one which you may have been taught, is this: $$\lim_{x \rightarrow c}f(x) = L$$ means that we can make f(x) as close as we like to L by making x close to c. However, in real analysis, you will need to be rigorous with your definition—and we have a standard definition for a limit.

The notation of a limit is actually a shorthand for this expression:

This definition gives a lot of people a lot of trouble, but since it is so fundamental to higher mathematics, there are many ways to help solidify the definition down. This chapter will be a guide in solidifying the behavior of this definition and provide necessary insight into working with the definition, whereas the Exercises will help unravel the puzzle, solidify the concept, and enable you to execute this definition properly.

Corollaries of Limits

It is very common, given limits, to work with the concept of infinity. However, the concept of infinity has yet to be well defined. Intuitively, we know that infinity represents endlessness and it is represented as $∞$. Yet, infinity itself is not a number. The current limit definition will fail if we use infinity like a number. If you suppose some limit where $c = ∞$ and we use our original definition, it would mean that

 * $$\lim_{x \rightarrow \infty}f(x) = L$$ means that  $$\forall M>0: \exists \delta : 0 < |x - \infty| < \delta \implies |f(x) - L| < \epsilon $$

Which is clearly nonsense!
 * 1) You cannot "subtract by infinity" - infinity isn't a number nor is it really a variable.
 * 2) Infinity cannot be bounded, yet by putting infinity in a $$|a - b| < x$$ format, it implies boundedness.

So, the definition needs to be rewritten, which is done in the following chart. The definitions for when either the limit as x approaches positive or negative infinity; or the limit as ƒ(x) converges to positive or negative infinity are as follows:

Take a note of the following variables: We only use big N and M because the connotation associated with ε and δ is that they are small numbers. Big N and M has the opposite connotation.
 * 1) N usually notates a limit with infinity and is analogous to ε.
 * 2) M usually notates a limit with infinity and is analogous to δ.

Conceptualization

 * 1) For all ε, only it, the ε variable, will be used to derive a δ. This powerful statement basically states that δ is related to ε. To excuse mathematically rigorous language for a moment, δ can be imagined of like a function which outputs ε. This is actually important, as neither δ or ε are allowed to have variables, such as x, be part of their formulation.
 * 2) ε and δ are supposed to represent bounds. Hence the absolute value signs. They are mathematically equivalent to writing $$-\delta < x - c < \delta$$ and $$-\epsilon< f(x) - L < \epsilon$$, which exemplifies their bound-like nature a lot more.
 * 3) This limit definition is designed to ignore the value of f(c) and whether or not c is even in the domain of ƒ. The requirement $$|x-c|>0$$ provides the appeal in studying calculus, by removing the technicality of having to analyze the behavior at the point (which is usually undefined to begin with). It is the mathematical implementation of the idea that the behavior of a function near a point shouldn't be affected by its behavior at the point. Thus, f(x) need not be defined at c to have a limit there.

Properties
Given that limits are such a fundamental concept of calculus, it should be reasonable to expect that limits should have some intriguing properties to both warrant analysis as well as be a staple mathematical topic in both elementary, applied, and higher mathematics.

Uniqueness
A limit is unique, in that there is always one and only one answer if the input is the same. This is commonly rephrased as "a function cannot approach two different limits at c". Limits having unique answers is very important, since if they don't, the use of limits will grow so complex that it will simply become unusable.

Algebraic Operations
If $$\lim_{x \rightarrow c}f(x) = L$$, and $$\lim_{x \rightarrow c}g(x) = M$$, then:

By applying the corresponding theorems for sequential limits, we find that functional limits are both unique—they preserve algebraic operations and ordering—and that a corresponding "Squeeze Theorem" holds.

 * If $$\exists \delta: f(x) > g(x) \forall x \in A$$, then $$L > M$$.

 * $$L=M, f(x) \leq h(x) \leq g(x) \implies \lim_{x \rightarrow c}h(x) = L$$

 * If L = 0 and h(x) is bounded, then $$\lim_{x \rightarrow c}f(x)h(x) = 0$$.

Proof
For the various operations, the following proofs may require more or less knowledge of algebraic inequality manipulation.

Addition
Of the operations, the proof for addition is the most simple, as it relies on the least amount of inequality algebra.

Subtraction
Subtraction follows from the addition proof by imagining a function h that is the negation of the function g beforehand. In other words, imagine the function g in the proof as a variable for a negated function.

$$\blacksquare$$

Multiplication
Of the operations, the proof for multiplication is the most complex, as it relies on the greatest amount of inequality algebra. It also requires a seemingly contrived lemma to operate. We will start by proving the lemma, which is simply an algebraic relationship between inequalities, similar to that of the binomial theorem relates a summation of terms and a product.

As you can see, the lemma itself describes a simple to prove and valid, yet very contrived and unnatural-looking relationship between numbers. But, this relationship is very attractive to be applied blindly for limits, because any value of a, b, c, and d inputted (even 0's) works, and that x > 0 is a condition that matches the ε variable.

As you will see below, we will apply this lemma for multiplication.

Multiple
The proof for multiples of some function ƒ follow from the proof on multiplication. It however relies on the limit of a constant proof. Because of the proofs reliance on two previous proofs and that those proofs this one relies on are robust (they account for things like 0's), this proof is just as robust, even working when a = 0.

Reciprocal
Of the operations, the proof for the reciprocal is similar to that of multiplication. It too requires a seemingly contrived relationship between some mathematical statements in order to function, and relies on the argument that the formula or concepts attached to assuring that epsilon and delta's boundedness is maintained is what defines a valid limit. Anyways, let us begin with the "contrived relationship".

As you can see, the lemma itself describes a simple to prove and valid, yet very contrived and unnatural-looking relationship between numbers. But, this relationship is very attractive to be applied blindly for limits, because any value of a, and b inputted (not including 0's) works, and that x > 0 is a condition that matches the ε variable.

As you will see below, we will apply this lemma for the reciprocal. Note that the proof is a simple assertion statement.

Division
The proof for division of the function ƒ by g is a corollary based on the proof done for limits of multiplication and limits of reciprocals.

As always, this proof has the obvious restriction that M cannot be 0.

Limits of Functions
Here, we will prove the answers to many of the functions you may commonly see. As always, the following chart for quick recall is provided below.

Note that for the linear function, we used z instead of the usual x because the variable name x is already defined and is being used by the limit notation.

Limits of Sequences
Consider the sequences $$(x_n) = (\frac{1}{n}), (y_n = (\frac{-1}{n})$$. Each converges to zero, but $$(f(x_n)) = 1$$ and $$(f(y_n)) = 0$$, and these have different limits as $$n \rightarrow \infty$$. Thus the limit does not exist.

Limits of Discontinuity
We'll be giving many more examples in the section on continuity. Although discontinuity is more sensibly important using continuity (which is covered in the next chapter), the definition of discontinuity is actually defined in regards to limits.

Point Discontinuity
An example of point discontinuity would be the functions

 * Example 1 $$f(x) =

\begin{cases} 1 & \mbox{if }x = 0 \\ 0 & \mbox{if }x \not= 0 \\ \end{cases} $$
 * Example 2 $$g(x) = \frac{x(x-1)}{x-1}$$

For the following functions,

 * Example 1 $$\lim_{x \rightarrow 0} f(x) = 0$$
 * Example 2 $$\lim_{x \rightarrow 1} f(x) = 1$$

The proof of which is the following:

Jump Discontinuity
0 & \mbox{if }x\leq 0 \\ 1 & \mbox{if }x > 0 \\ \end{cases}$$. Then $$\lim_{x \rightarrow 0} f(x)$$ does not exist.
 * Let $$f(x) = \begin{cases}

Limits of Unusual Functions
Many of the examples here may seem a bit contrived and appear quite nasty, with even more nastier proofs, but if done correctly, these examples (and the associating exercises) will solidify not only the methodology of a limit proof, but of how mathematics can, using verified theorems and behaviors, solve some seemingly unsolvable problems.

Our first example, often given as a demonstration of just how nasty functions can get (and how far a definition can take you), is

 * $$f(x) = \begin{cases}

1/q & \mbox{if }x \text{ is rational, and thus } x = p/q \\ 0 & \mbox{else} \\ \end{cases}, \forall x \in (0, 1) $$

For the function ƒ, $$\lim_{x \rightarrow c} f(x) = 0$$ for all numbers in the domain. Yes, really.

The first step in understanding the proof of this statement is to stop imagining limits and continuity as the same - that is, if the first step of this problem is to imagine the graph of this function and in a sense, zoom in until an answer can be deduced graphically. Do not be saddened if this is how you thought about how to work out this problem; this method is a simplified explanation of limits commonly taught in elementary mathematics and would thus be ingrained in you anyways.

This proof demonstrates a method of mathematical proof through manipulating theorems instead of manipulating numbers or variables to form the epsilon-delta model, which in turn implies the limit's validity; the existence of a limit. It also shows how a limit proof is actually an exercise in trying to relate two easily malleable inequalities together using valid theorems.

The next example, of a similar vein, is

 * $$g(x) = \begin{cases}

1 & \mbox{if }x \in \mathbb{Q} \\ 0 & \mbox{else} \\ \end{cases} $$

For the function g, $$\lim_{x \rightarrow c} g(x)$$ does not exist for any $$c \in \mathbb{R}$$.

Given $$x \in R$$, let $$x_n$$ be any rational number in the interval $$(\frac{-1}{n},\frac{1}{n})$$, and let $$y_n$$ be any irrational number in the same interval ($$x_n$$ and $$y_n$$ are guaranteed to exist by density of the rationals and irrationals). Given any $$\epsilon > 0, |x_n - 0| < \frac{1}{n}$$ and $$|y_n - 0| < \frac{1}{n}$$, so $$(x_n),(y_n) \rightarrow 0$$. However, (g(x_n)) = 1 and (g(y_n)) = 0, so their limits are 1 and 0. Since these are not equal, $$\lim_{y\rightarrow x} g(y)$$ does not exist.

Appendix
Here, we will expose more topics in regards to limits. First, we will give a review on the nature of functions. Recall that a function from a set X to a set Y is a mapping $$f: X \rightarrow Y$$ such that f(x) is a unique element of Y for every $$x\in X$$. In analysis, we tend to talk about functions from subsets $$A \subseteq \mathbb{R}$$ to $$\mathbb{R}$$.

The definition for the limit of a function is much the same as the definition for a sequence. In fact, as we will see later, it is possible to define functional limits in terms of sequential limits. For the moment, however, let us reevaluate the definition of a limit for a function ƒ given a generalized-enabled function:

Given a subset $$A \subset \mathbb{R}$$ and a function $$f:A\rightarrow \mathbb{R}$$, we say $$\lim_{x \rightarrow c}f(x) = L$$ if $$\forall \epsilon > 0: \exists \delta: 0<|x-c|<\delta \implies |f(x)-L|<\epsilon$$

Sequential Limits
One curious result of thinking about real numbers as built upon natural numbers and the like (as we have structured our section on numbers in this wikibook) is that the definition of a limit, which we have used the real number version for real functions all this time, can be derived using sequential limits instead of axiomatically, as so:

Given a subset $$A \subset \mathbb{R}$$ and a function $$f:A\rightarrow \mathbb{R}$$, we say $$\lim_{x \rightarrow c}f(x) = L$$ if $$\forall (x_n)_{n=1}^{\infty}$$ such that $$x_n \not= c, \lim_{n \rightarrow \infty}(x_n) = c$$, and $$\lim_{n \rightarrow \infty}(f(x_n)) = L $$

Note that the requirement $$x_n \not= c$$ corresponds with the requirement $$|x - c| > 0$$.

As an exercise to test your understanding, prove that these two definitions are equivelant. Note that taking the contrapositive gives a good criterion for determining whether or not a function diverges:

If $$\exists (x_n), (y_n): (x_n)\rightarrow c, (y_n)\rightarrow c$$, and $$ \lim_{n \rightarrow \infty}(f(x_n)) \not= \lim_{n \rightarrow \infty}(f(y_n))$$, then $$\lim_{x \rightarrow c}f(x)$$ does not exist.

Definition on an Arbitrary Metric Space
Let $$(X,d_1)$$, and $$(Y,d_2)$$ be metric spaces. And let $$f:X \to Y$$

The limit as $$x \in X$$ approaches $$a \in X$$ of $$f$$ is equal to $$L \in Y$$ if $$\forall \epsilon > 0 \text{ }\exists \delta \text{ such that if } 0<d_1(x,a)< \delta \text{, } d_2(f(x),L) < \epsilon$$

This is denoted $$\lim_{x \to a} f = L$$

title: Real Analysis/Limits


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title: Template:Ada/2005/RMA2


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Cookbook:Chocolate Truffles

| Desserts | Confections

Chocolate truffles are soft chocolate balls. This recipe makes about 24.

Ingredients

 * 8 oz (227 g) bittersweet/plain chocolate, chopped into small pieces
 * 6 imperial tablespoons (7.2 US tablespoons / 108 ml) double cream (48% fat)
 * Cocoa powder
 * Icing sugar

Procedure

 * 1) Put the chocolate in a bowl and set over (a double boiler).
 * 2) Heat until melted and smooth, stirring occasionally.
 * 3) Remove from the pot of hot water.
 * 4) Add the cream and stir well. Leave to cool.
 * 5) Cover and refrigerate for 35 minutes or until the mixture is just firm enough to hold its shape.
 * 6) Sift a layer of cocoa powder and icing sugar together onto a plate. The icing sugar is to make sure the cocoa powder isn't too bitter.
 * 7) Take a heaped teaspoonful of the chocolate mixture and shape it into a ball.
 * 8) Roll each truffle in the cocoa powder/icing sugar to coat all over in an even layer. Shake off the excess and set on a tray.
 * 9) Continue shaping and coating the remaining truffles. Set on a tray, making sure they don't touch each other.
 * 10) Cover and refrigerate. Remove 15 minutes before serving.

Notes, tips, and variations

 * Vanilla truffles: Use white chocolate, reducing the icing sugar to preference. Add about ½ tablespoon of vanilla extract.
 * Lime: Use white chocolate, reducing icing sugar content to preference. Add about ¾–1 of a tablespoon of diluted lime cordial, and a couple of pinches of ground lime zest.
 * In the USA, try using 2 tablespoons unsalted butter with 5 ⅓ tablespoons heavy cream to replace the double cream.

title: Cookbook:Chocolate Truffles


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User:Kevin Rector

I am an active Wikipedian and Wiktionarian. I am an administrator at Wiktionary.

title: User:Kevin Rector