10.5: Inequality along the Margins
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- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
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Learning Objectives
By the end of this section, you will be able to:
- Identify contemporary types of migration.
- Describe the major migrant routes and some of the risks migrants face.
- Identify and give an example of circular migration.
- Describe the global impact of refugees.
- Give an example of a pandemic.
Contemporary Types of Migration
Because of emerging global forces of all kinds—social, economic, environmental, and political—there has been a recent rise in migration within geographical regions and across countries. Four of the most common types of contemporary migration are listed below. Each derives from different causes and is associated with different push and pull factors (Woldeab 2019). In some situations, these types of migration may overlap, such as in the aftermath of a natural disaster.
- Labor migration is the movement of people for the purpose of employment and/or economic stability. It may be an internal migration from one town to another within the same country of origin, or it may involve travel across countries in search of opportunities. In 2017, the United Nations International Labour Organization estimated that there were 164 million labor migrants worldwide (Global Migration Data Analysis Centre 2021).
- Forced migration or displacement , also called involuntary migration , is migration due to persecution, conflict, or violence and involves refugees and those seeking asylum. The UN Refugee Agency (UNHCR) estimated that as of the end of 2019, there were some 79.5 million forced migrants or displaced persons worldwide. More than two-thirds (68 percent) of those displaced persons came from just five countries: Syria, Venezuela, Afghanistan, South Sudan, and Myanmar (UNHCR 2020). One out of every 108 people was displaced in 2018 (UNHCR 2019).
- Forced labor , human trafficking , and modern slavery are a set of linked terms defined as the recruitment, transportation, transfer, and/or harboring of persons by means of threat or use of force or coercion for the purpose of exploitation (UN 2020). This includes sexual slavery and forced labor. As of 2016, some 25 million people were involved in forced labor and some 40.3 million in modern slavery worldwide, while an estimated 15.4 million were in forced marriages ( https://www.ilo.org/global/topics/fo...--en/index.htm ). While a large proportion of the victims are women, human trafficking involves men and children as well. The Counter Trafficking Data Collaborative ( https://www.ctdatacollaborative.org/ ) estimates that nearly 80 percent of international human trafficking journeys pass through airports and other official border control points.
- Environmental migration is displacement caused by natural disasters, such as earthquakes, hurricanes, or droughts. It can be permanent or temporary and is a rapidly growing area of migration due to global climate change. In 2018, 17.2 million people were displaced due to environmental conditions; by 2019, the number had risen to 24.9 million ( https://www.internal-displacement.or...port/grid2019/ ).
Anthropologists who study migration are often involved in multi-sited ethnographic research, exploring not only migrant populations but their communities of origin as well. Understanding the social and cultural attributes of communities of origin helps researchers gauge the level and types of adaptation caused by migration. Also, communities of origin typically remain part of migrants’ wider social networks and are vital to their well-being and success. It is not uncommon for relatives and other members of the migrants’ home communities to follow them to their new settlements and reestablish a sense of community and a set of self-help networks there. This process of serial migration from the same community of origin is known as chain migration .
Labor Migration and Migrant Routes
While migration, in its widest sense, is any movement that reestablishes a household, many migratory patterns are specifically associated with socioeconomic need, mainly shifting employment opportunities. Labor migration can be permanent or circular. Circular migration is a repeated pattern of movement between locations, usually mapped to the availability of work. One type of circular migration is seasonal migration, which is migratory movement that coincides with seasonal labor needs, such as planting, harvesting, service, and construction work. Some seasonal workers migrate, with or without their families, for temporary, often low-paid work. Other seasonal workers have long-term relationships with their employers and legal work permits (also called Employment Authorization Documents, or EADs, in the United States) and will return to the same work sites year after year, sometimes maintaining a joint household with other families at the work site. These individuals will often maintain a family household in their country of origin and send home remittances , or transfers of money from workers to their home countries, usually for their families. Today, one in nine people worldwide depends on remittances from migrants (Global Migration Data Analysis Centre 2021).
Many people migrate in search of work and a better life without legal permits or assurance of employment. The migration journey made in search of opportunities can be filled with dangers, hardships, and even death. Some regions of the world have well-established migrant trails, which are the routes of most worldwide migration. The most congested migration routes are:
- the eastern Mediterranean route, with a flow of migrants from the Middle East and North Africa to Europe, crossing through Turkey;
- the Mediterranean Sea route, with migration from the Middle East and North Africa to Europe, across the Mediterranean Sea;
- the Southeast Asian route, with migrants primarily moving southward from the Asian mainland into Indonesia and Malaysia; and
- the Central American route, which brings migrants from South and Central America into North America.
These migrant trails have a huge impact on the social, political, and economic life of all of the countries that are a part of the route, bringing both benefits and challenges. Those in the United States are most familiar with the Central American route, which begins as far south as South America and extends as far north as Canada. The most contested part of the “trail,” however, is the portion along the Rio Grande, the river that separates Mexico and the United States.
In his remarkable four-field study of undocumented migrants entering the United States across the border with Mexico, The Land of Open Graves (2015), Chicano anthropologist Jason De León reveals a less visible side of undocumented migration. He describes a type of cat-and-mouse game between migrants and those attempting to stop them, resulting in widespread suffering and high human and financial costs. De León conducted a multi-sited ethnography, doing research in various locations in both Mexico and the United States and consulting various groups along the migration route, including illegal migrants and border patrol agents as well as smuggling groups and drug traffickers.
Profiles in Anthropology
Jason De León (1977-)
Personal History: Jason De León is a U.S. anthropologist and Mexican-Filipino American who grew up in several cities in the United States, including McAllen, Texas, near the U.S.-Mexico border; and Long Beach, California, where he graduated from Wilson High School. He earned his bachelor’s degree in anthropology from the University of California, Los Angeles and his master’s and doctoral degrees from Pennsylvania State University. His doctoral work focused on ancient tool production and trade in the Valley of Mexico.
Area of Anthropology: Although De León’s training includes a specialization in archaeology, his holistic research approach is four-field, combining archaeology with ethnographic research, physical anthropology analyses, and linguistic anthropology. His work is multi-disciplinary in nature and multi-sited, involving not just Mexico and the United States but also numerous other countries of migrant origin. His interests include undocumented migration, photo-ethnography, and human smuggling. He seeks out the stories not only of people, such as migrants and their families, smugglers, and border guards, but also of their material artifacts—the items they bring, wear, and use to survive their dangerous journeys.
Accomplishments in the Field: De León is the executive director of the Undocumented Migration Project (UMP), a nonprofit organization founded in 2009 that focuses on the long-term anthropological study of clandestine movements between Latin America and the United States. UMP sponsors an educational exhibit called Hostile Terrain 94 (HT94), a pop-up participatory art project that displays the handwritten toe tags of some 3,200 migrants who have died while trying to cross the Sonoran Desert in the southwestern United States since the mid-1990s, showing the locations where each of the individuals died along their journey. It is a poignant reminder of the many dangers of migration, both human and environmental.
De León received the prestigious five-year MacArthur Foundation Fellowship (2017–2022) for his work on undocumented migrants. This award, given for talent, creativity, contribution to one’s field, and potential, allows scholars to focus on future research in an area of great importance. In addition, De León’s 2015 book, The Land of the Open Graves: Living and Dying on the Migrant Trail , has received various awards and commendations.
Importance of Their Work: Anthropologists often work in specific places and more geographically bounded settings. The research of Jason De León expands our understanding of the lives of those who migrate and the various ways that movement ties together people, places, and cultures.
In his article “On Not Looking Away,” digital and multimedia advisor Arran Skinner (2019) reports on the tragic deaths of Mexican migrants Óscar Martínez Ramírez and Angie Valeria, his 23-month-old daughter, both of whom drowned and washed up on the shores of the Rio Grande. “We are choosing to ignore this evidence [of atrocity], to actively look away,” Skinner writes. But De León is not looking away. Through his research, he is bringing to light the stories of those who migrate in search of hope and better lives. As global movements become more common because of political, economic, and environmental challenges, studies such as De León’s illustrate the growing importance of migration for our species.
Since 1994, the US Border Patrol has had a policy of “prevention through deterrence” that attempts to prevent undocumented migrants from reaching the U.S. border. Legal international entryways in cities such as Tucson, Arizona, and El Paso, Texas, were heavily fortified with fencing and additional patrol agents to make undocumented crossing exceptionally difficult. As a result, migrant entry points shifted away from urban areas and into more hostile terrain, such as the Sonoran Desert region of Arizona. While this has not significantly lowered the frequency of these crossings, it has made the journey much more dangerous and far less visible to residential populations and humanitarian groups. In addition to the threat of harsh and rugged landscapes, there are the dangers of extreme weather, dehydration, bandits, and even wild animals. De León concludes, “The Border Patrol has intentionally set the stage so that other actants [agents of deterrence] can do most of the brutal work” (61).
During his study, De León and his team located the body of Maricela Zhagüi Puyas, a woman originally from Cuenca, Ecuador. She had left her family, including her children, in Ecuador in order to seek employment in the United States, hoping to send money home to them. She was in debt for more than $10,000, most of it to the trail guide (called a coyote ) who was supposed to guide her on her journey.  Maricela had made a journey of more than 5,000 miles from Cuenca, Ecuador, all the way to the Sonoran Desert in Arizona, when she died of exhaustion and exposure, technically having reached the United States. In the 14-year period between 2000 and 2014, 2,721 migrants were found dead in Arizona’s Sonoran Desert, approximately 800 of whom remain unidentified today. In 2020, there were an estimated 227 migrant deaths in the Sonoran graveyard, making it the deadliest year on record for that corridor trail (Snow 2021). De León’s work continues today through a series of pop-up exhibitions and workshops entitled Hostile Terrain 94 .
This humanitarian crisis is far from being resolved. In 2020, 400,651 undocumented migrants were apprehended and expelled by the U.S. Border Patrol (U.S. Customs and Border Protection 2020). Immigrants, both documented and undocumented, make up a majority of the farmworkers and meatpacking workforce in the United States today. Once employed, these immigrants, who are frequently separated from their families, face hazardous working conditions, language barriers, long hours, low pay, and substandard housing. Because of their legal status, many also struggle with inadequate access to health care and rising discrimination.
Biocultural anthropologist Shedra Snipes and her team (Snipes et al. 2007) conducted focus group interviews among 69 male and female Mexican immigrant farmworkers in the Yakima Valley of Washington State. They were particularly interested in the ways the farmworkers defined and experienced stress. Their interviewees distinguished between physical and mental stressors and cited the most common causes of stress as work, personal illness, lack of work, family illness, and family stress. Snipes et al. noted that many stressors were linked by a common theme of inconsistent work and the injusticia (injustice and unfairness) of low pay and poor working conditions. One farmworker noted, “Sometimes there are many people wanting to work in the field.  Another common theme was the stress of living in a different culture. Several farmworkers commented that cultural differences, such as language barriers, communication from schools regarding their children, or complaints from neighbors when they had rowdy family get-togethers, contributed to their experience of stress. As this example shows, at the intersection of culture and migration, many factors affect an individual’s ability to adapt to new living conditions.
Refugees Beyond the Nation-State
Refugees are persons who are forced to cross international boundaries to seek residence. Pushed out of their countries, most commonly because of war, famine, or persecution, they typically arrive under extreme circumstances with little food, clothing, or material possessions. They are frequently separated from their relatives and have little chance of finding employment or reestablishing their household. Because of their status as stateless persons (persons forced to leave their countries) and their inability to procure proper travel documentation, refugees are protected under the 1951 United Nations Refugee Convention, which derives from Article 14 of the Universal Declaration of Human Rights, passed in1948. The Universal Declaration of Human Rights establishes an international legal right for people to seek asylum , which is legal protection extended by one country to citizens of another. The United Nations High Commissioner for Refugees heads the UN Refugee Agency, a global organization that directs troops and aid workers to set up refugee camps and organizes international efforts to ease the suffering of refugees.
In her ethnographic study of Congolese refugees in the Ugandan capital city of Kampala, cultural anthropologist Georgina Ramsay (2016) focuses on the ways in which refugees protect themselves, both physically and psychologically, by what they call “avoiding poison.” In 2012, there were approximately 50,000 refugees living in Kampala as a result of ongoing political instability, warfare, and corruption in the Democratic Republic of the Congo. Housed initially in a refugee settlement away from urban areas, the group of refugees interviewed by Ramsay opted to move to Kampala for greater opportunities and more security, as the refugee settlements were troubled by crime and violence. As one informant told Ramsay, “There are bad people everywhere in the camp” (115). The government of the Democratic Republic of the Congo allowed resettlement in Kampala if the refugees procured a legal permit and a way to make a living independent of government funding or humanitarian aid. Given their displacement from their own ethnic communities and social networks, refugees faced unreliable social communities, in which their relationships were recently formed, as well as fear and the looming threat of having to return to the settlements if they lost their jobs or housing arrangements. Many either relied on or supplemented their wages with remittances from relatives living elsewhere in an effort to create greater security in the urban environment.
The “poison” feared by this group of refugees is a symbolic agent administered by “unknown assailants” (113), most often sprinkled into the food they prepare, and capable of making them sick both physically and psychologically. The administering of this poison is not always intended as a personal attack; rather, the refugees believe that their day-to-day life outside of their cultural homelands makes them vulnerable. They believe that they are most vulnerable during cooking and eating. In their home communities, cooking and eating were normally times of social interaction and sharing, but cooking and eating are now highly privatized acts for them. Families eat only with each other, within their own homes, and do not accept any shared food, even when they are hungry. The result is an intentional physical distancing from each other and a strengthening of family-only social bonds. While this approach clearly weakens the refugees’ ability to build a large and sustainable self-help community in Kampala, it does afford them a sense of positive control (agency) over their day-to-day lives. This sense of social agency over the threat of “poison,” giving the refugees an ability to control some aspects of their day-to-day lives, is an example of the adaptive nature of culture under very challenging circumstances.
Pandemic as a Global Migration
People and goods are not the only things that migrate. Along with human migration, there is a host of secondary movements that can affect the human population globally. Diseases are a prime example. Diseases that may have once been contained in a single region can move, along with their human and animal hosts, into new geographic areas, where they can become even more virulent. When diseases spread more than expected among a given group of people, they are referred to as epidemics . An outbreak of a disease over a very broad area, typically crossing international boundaries, is called a pandemic . Some early pandemics in Europe were the plague of Athens in 430 BCE (possibly typhus or typhoid fever or Ebola), the Antonine plague from 165 to 180 CE (possibly smallpox), and the Black Death from 1347 to 1351 (caused by a bacterium carried by fleas and infected rodents). In the Americas, Mexico and Central America suffered from various documented pandemics, starting with the arrival of the Spanish in Mexico in 1519, which set off a widespread smallpox outbreak that extended into South America. There have been other pandemics, including the cocoliztli epidemic from 1545 to 1548, likely a form of enteric fever, and the so-called Spanish flu, first detected in the United States in 1918 (Alchon 2003; Vågene et al. 2018). The most serious recent pandemic in the United Stated had been the swine flu pandemic of 2009–2010 ... until 2019–2020.
In the last few months of 2019, the viral coronavirus SARS-CoV-2, known as COVID-19, began a global migration from Wuhan, Hubei Province, China, to every continent of the world. Carried between geographically distant locations by human hosts traveling for all sorts of reasons—including work, study, tourism, visitation, and displacement—as well as within towns and communities by people shopping, attending religious services and schools, or even visiting friends and families, COVID-19 quickly became a global emergency. First reported to the World Health Organization (WHO) on December 31, 2019, COVID-19 was officially declared a global pandemic on March 11, 2020. Throughout 2020, the disease continued to spread rapidly, overwhelming medical facilities, ravaging countries’ economies, and forcing people to alter the structures of most social institutions, including schools, churches, weddings, and even funerals. By October 2021, some 248 million people had been infected, including several world leaders, and more than 5 million people had died from the disease.
The COVID-19 virus spreads through airborne transmission when someone inhales droplets expelled by an infected person coughing, sneezing, or even exhaling. As with measles and tuberculosis, the only fully effective form of containment outside of a vaccine and the development of antibodies is quarantine. When the WHO declared COVID-19 a global pandemic, the most important advice was to limit all unnecessary movements and gatherings, wear masks, and practice physical distancing. But given the global nature of our lives today, it was very difficult to halt either the movement of people or the spread of the disease. On January 20, 2020, the first reported case in the United States was diagnosed in Washington State, in a man in his thirties who had just returned from Wuhan. By that point, the virus had already spread to Taiwan, Japan, Thailand, and South Korea. On January 24, the first European cases were reported in France. The disease continued to quickly spread all over the world, including on international transport, such as cruise and cargo ships. In December 2020, there were several cases reported in Antarctica. Only 14 countries reported no COVID-19 cases as of April 2021, all except two of them island nations or territories in the Pacific and Atlantic Oceans with strict travel policies: Tuvalu, Tonga, Tokelau, St. Helena, the Pitcairn Islands, Palau, Niue, Nauru, Kiribati, the Federated States of Micronesia, the Cook Islands, and American Samoa. (The two non-island nations, North Korea and Turkmenistan, are believed to have unreliable data.) As a result of migration, the disease transformed peoples’ lives everywhere.
But migration can also bring relief from pandemics. The same conveyances that led to the initial spread of the disease have also brought relief workers, food, medical supplies, and life-saving vaccines to communities worldwide. In addition, scientists and researchers worked tirelessly in multinational efforts to sequence the COVID-19 genome so that vaccine development could proceed rapidly. Globally, several countries developed lifesaving vaccines and began working together to disperse them to communities in need. As our world becomes increasingly interdependent, it is critical that we understand the important role of migration in so many aspects of our survival.
Ethnographic Sketches
Migration in El Angosto
Experience of Marjorie Snipes, chapter author
We often think of rural communities as being separate from global forces, but this is not always true. In El Angosto, a small Indigenous community in the northwestern Andes of Argentina, diverse forms of migration, dependent on internal and external factors, are part of people’s day-to-day lives.
I conducted fieldwork in El Angosto, Argentina, during the 1990s and early 2000s (Snipes 1996). This small highland community is located at about 11,000 feet above sea level and nestled in a rugged river valley along the Río Grande de San Juan, the international boundary between Argentina and Bolivia. At that time, the community had a population of about 200 people, most practicing agropastoralism, with each family raising corn, wheat, alfalfa, and broad beans and tending herds of goats and sheep. In order to provide ample pastureland and keep animals away from their gardens, they were transhumant, moving their herds to higher altitudes during the spring and summer seasons, away from the primary households with their gardens and accompanied by seasonal shepherds. After the herds moved from their winter corrals, families cleaned them out and used the manure to fertilize the gardens. Through transhumance, families benefit from this dual subsistence system, producing most of their daily food needs.
Although not dependent on money for their daily food, Angosteños participate in the global economy in various ways. Historically, the community is part of a vast Andean trade network that connects small highland communities of northern Argentina and southern Bolivia through itinerant trade. Extensive long-distance trade networks have been an integral part of Andean life for centuries (see Alberti and Mayer 1974; Murra 1975). Annually, traders come through El Angosto from the altiplano of Bolivia, a high plains region at an average altitude of 12,000 feet above sea level. Because of the harsh climate at that altitude, the Bolivian communities rely almost exclusively on herding camelids (llamas and alpacas), having little to no ability to raise needed crops. In the springtime each year, Bolivian traders pass through El Angosto with pack animals (usually llamas) loaded with wool ropes, bags, and dried camelid meat that they produce during the winter months, seeking fresh vegetables for trade.
Although traders negotiate each transaction based on their particular family’s needs, all parties are well aware of the current market value of their animal and vegetable products, as families listen daily to radio broadcasts on trade. I tried my hand at negotiating with Gumercindo, a young trader from San Antonio de Lipes, Bolivia, for a small, handwoven rope. When I asked him the cost of the rope, he looked at me with kind amusement and asked me what I offered to exchange. “Pesos!” I said (Bolivian money). He told me the rope was worth 10 pesos (approximately $10 at that time) but that he needed corn and wheat and that one arroba (approximately 25 pounds) of grain was worth around $12. In other words, I would have to pay the higher cost because he would need to take the money and try to buy an arroba of grain. Most highlanders are more aware of current trade values than even those living in cities.
Other forms of migration affect life in El Angosto. In order to earn cash for manufactured items, many highland families periodically send a family member to work away from the community. The zafra , the annual lowland sugarcane harvest, can usually temporarily absorb anyone willing to work, and young people occasionally seek out urban employment opportunities, such as domestic service in private households. Migration is an enduring part of the fabric of Andean life, binding communities to each other and, ultimately, to each of us.
Mini-Fieldwork Activity
Migration Interviews
For this fieldwork activity, you will compile three ethnographic accounts of migration. Choose three diverse research participants/key informants to interview about their personal histories of moving, as a child and/or an adult, from one home location to another. Some may have moved from one country to another, from one city to another, or even from one house or family to another. Log each of their movements separately, giving the years and duration of the period spent living there, why they moved, how things shifted in their lives as a result of the migration, and their feelings and/or emotions about moving. You may choose to add your own account to this study as well. Once you compile each of the accounts, summarize your findings and compare the accounts to each other, making conclusions about the impact of migration on your participants’ lives.


3.12: Test Yourself- List in the Correct Order
List the steps below in the correct order for each of the following techniques. Write your list on a piece of paper.
Radial Pulse Technique
- Note the rate, rhythm, force, and equality when measuring the radial pulse
- Use the pads of your first three fingers to gently palpate the radial pulse along the radius bone close to the flexor aspect of the wrist
- Press down with your fingers until you can best feel the pulsation
Carotid Pulse Technique
- Gently palpate the carotid artery one at a time
- Note the rate, rhythm, force, and equality when measuring the carotid pulse
- Locate the carotid artery medial to the sternomastoid muscle in the middle third of the neck
- Ask the client to sit upright.
Apical Pulse Technique
- Physically palpate the intercostal spaces to locate the landmark of the apical pulse
- Ask the client to lay flat in a supine position
- Note the rate and rhythm
- Auscultate the apical pulse
Brachial Pulse Technique
- Move your fingers medial from the tendon and about one inch above the antecubital fossa to locate the brachial pulse
- Palpate the bicep tendon in the area of the antecubital fossa
- Note the rate and rhythm
Respiration Technique
- Count for 30 seconds if the rhythm is regular or for a full minute if it is irregular
- Observe the rise and fall of the chest or abdomen
- Leave your fingers in place when you are done counting the pulse, and then begin assessing respiration
- Report the respirations as breaths per minute, as well as whether breathing is relaxed, silent, and has a regular rhythm
Go to the next page to see the correct order of steps for these techniques.


What you’ll learn to do: describe the humanistic, contextual, and evolutionary perspectives of development
Each perspective that we have seen so far emphasizes different aspects of development. We first looked at the psychodynamic approach and how it emphasizes unconscious determinants of behavior. We then turned to the behavioral perspective which emphasizes overt behavior. Now, we’ll turn our attention to the humanistic perspective, which emphasizes empathy and stresses the good in human behavior; it is similar to the cognitive perspective in that it looks more at what people think than at what they do. In this section, we will also look at the contextual perspective, which considers the relationship between individuals and their physical world, cognitive processes, personality, and social worlds. It also examines social and cultural influences on development. And finally, we will briefly examine the evolutionary perspective which focuses on how inherited biological factors underlie development.
Contributors and Attributions
CC licensed content, Original
-
Introduction to the Humanistic, Contextual, and Evolutionary Perspectives of Development.
Authored by
: Sonja Ann Miller for Lumen Learning.
Provided by
: Lumen Learning.
License
:
CC BY: Attribution
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28.5: Advanced topics
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This section introduces a few more advanced topics that allow you to use computer programming to simplifying many tasks. In this section, we will show you how you can write your own program to numerically estimate the value of an integral of any function.
Defining your own functions
Although Python provides many modules and functions, it is often useful to be able to define your own functions. For example, suppose that you would like to define a function that calculates \(\frac{1}{2}x^{2} +\frac{1}{4}x^{3} + \cos(2x)\), for a given value of \(x\). This is done easily using the
def
keyword in Python:
python code \(\PageIndex{1}\)
Defining a function
#import the math module in order to use cos import math as m #define our function and call it myfunction: def myfunction(x): return x**2 / 3 + x**3 / 4 + m.cos(2*x) #Test our function by printing out the result of evaluating it as x = 3 print ( myfunction(3) )
Output
10.710170286650365
A few things to note about the code above:
-
Functions are defined using the
def
keyword followed by the name that we choose for the function (in our case,myfunction
) -
If functions take arguments, those are specified in parenthesis after the name of the function (in our case, we have one argument that we chose to call
x
) - After the name of the function and the arguments, we place a colon
- The code that belongs to the function, after the colon, must be indented (this allows Python to know where the code for the function ends)
-
The function can “return” a value; this is done by using the
return
keyword. -
We used the “operator”
**
to take the power of a number (x**2
), and the operator*
, to multiply numbers. Python would not understand something like2x
; you need to use the multiplication operator, i.e.2*x
.
In the example above, we wrote a Python function to represent a mathematical function. However, one can write a function to execute any set of tasks, not just to apply a mathematical function. Python functions are very useful in order to avoid having to repeatedly type the same code.
Recall that the
numpy
module allows us to apply functions to arrays of numbers, instead of a single number. We can modify the code above slightly so that, if the argument to the function,
x
, is an array, the function will gracefully return an array of numbers to which the function has been applied. This is done by simply replacing the call to the
math
version of the
cos
function by using the
numpy
version:
python code \(\PageIndex{2}\)
Defining a function that works on an array
#import the numpy module in order to use cos to an array import numpy as np #define our function and call it myfunction: def myfunction(x): return x**2 / 3 +x**3 / 4 + np.cos(2*x) #Test our function by printing out the result and evaluating it at x = 3 (same as before) print( myfunction(3) ) #Test it with an array xvals = np.array([1, 2, 3]) print ( myfunction(xvals) )
Output
10.710170286650365 [ 0.1671865 2.67968971 10.71017029]
where we created the array
xvals
using the
numpy
module.
Using a loop to calculate an integral
The ability to define our own functions in Python allows us to easily simplify complex tasks. Using “loops” is another way that computer programming can greatly simplify calculations that would otherwise be very tedious. In a loop, one is able to repeat the same task many times. The example below simply prints out a statement five times:
Python code \(\PageIndex{3}\)
A simple loop
#A loop to print out a statement 5 times: for i in range(5): print("The value of i is ", i)
Output
The value of i is 0 The value of i is 1 The value of i is 2 The value of i is 3 The value of i is 4
A few notes on the code above:
-
The loop is defined by using the keywords
for ... in
-
The value after the keyword
for
is the “iterator” variable and will have a different value each time that the code inside of the loop is run (in our case, we called the variablei
) -
The value after the keyword
in
is an array of values that the iterator will take -
The
range(N)
function returns an array ofN
integer values between0
andN-1
(in our case, this returns the five values \(0,1,2,3,4\)) - The code to be executed at each “iteration” of the loop is preceded by a colon and indented (in the same way as the code for a function also follows a colon and is indented)
We now have all of the tools to evaluate an integral numerically. Recall that the integral of the function \(f(x)\) between \(x_{a}\) and \(x_{b}\) is simply a sum:
\[\begin{aligned} \int_{x_{a}}^{x_{b}}f(x)dx &= \lim_{\Delta x \to 0}\sum _{i=0}^{i=N-1}f(x_{i})\Delta x \\[4pt] \Delta x&=\frac{x_{b}-x_{a}}{N} \\[4pt] x_{i}&=x_{a}+i\Delta x \end{aligned}\]
The limit of \(∆x → 0\) is equivalent to the limit \(N → ∞\). Our strategy for evaluating the integral is:
- Define a Python function for \(f(x)\).
-
Create an array,
xvals
, of \(N\) values of \(x\) between \(x_{a}\) and \(x_{b}\). -
Evaluate the function for all those values and store those into an array,
fvals
. -
Loop over all of the values in the array
fvals
, multiply them by \(∆x\), and sum them together.
Let’s use Python to evaluate the integral of the function \(f(x) = 4x^{3}+3x^{2}+5\) between \(x = 1\) and \(x = 5\):
Python Code \(\PageIndex{4}\)
Numerical integration of a function
#import numpy to work with arrays: import numpy as np #define our function def f(x): return 4*x**3 + 3*x**2 + 5 #Make N and the range of integration variables: N - 1000 xmin = 1 xmax = 5 #create the array of values of x between xmin and xmax xvals = np.linspace(xmin, xmax, N) #evaluate the function at all those values of x fvals = f(xvals) #calculate delta x deltax = (xmax - xmin) / N #initialize the sum to be zero: sum = 0 #loop over the values fvals and add them to the sum for fi in fvals: sum = sum + fi*deltax #print the result: print("The integral between {} and {} using {} steps is {:.2f}".format(xmin, xmax, N, sum))
Output
The integral between 1 and 5 using 1000 steps is 768.42
One can easily integrate the above function analytically and obtain the exact result of \(768\). The numerical answer will approach the exact answer as we make \(N\) bigger. Of course, the power of numerical integration is to use it when the function cannot be integrated analytically.
Exercise \(\PageIndex{1}\)
What value of \(N\) should you use above in order to get within \(0.01\) of the exact analytic answer?
- Answer
-


3.1: Introduction
As discussed in Chapter 2 Asset Management, asset component inventory and condition assessment are important steps in any infrastructure management process. They provide essential information for maintenance and rehabilitation decision making.
For immobile and long-lasting physical assets with explicit geophysical locations, inventory can be relatively simple. Data records for asset location, size, age and other pertinent information can be created. As new assets are added or retired, the data records need to be updated, either as transactions occur or on a periodic basis. Computerized asset facilities management (CAFM) systems and computerized maintenance management systems (CMMS) systems with or without mapping capability are often useful for these inventories.
For linear and network physical assets (e.g., pipelines, electrical distribution systems, etc.) inventory is increasingly complex. Like fixed location systems, data records can be created and updated; however, accurate inventories for the systems are specifically dependent on management systems with geospatial capabilities (e.g., geographic information systems (GIS)) that include a mapping capability so that these assets can be displayed on a two, or even three-dimensional map.
Mobile assets may be more difficult to inventory since their locations might not be known at any given time. However, records of acquiring and retiring mobile assets provide a means of keeping an inventory listing. Manual inspection or bar code readers provide a means of identifying the locations of the various assets. Figure 3.1 illustrates a typical mobile asset in the form of a postal truck used for carrying mail and parcels. Replacement parts for infrastructure such as elevators represent another class of potentially mobile assets that might be inventoried by an infrastructure manager.


3.3: Power Sets
We have already seen that using union, intersection, set difference, and complement we can create new sets (in the same universe) from existing sets. In this section, we will describe another way to generate new sets; however, the new sets will not “live" in the same universe this time. The following set is always a set of subsets. That is, its elements are themselves sets.
Definition 3.27. If \(S\) is a set, then the power set of \(S\) is the set of subsets of \(S\) . The power set of \(S\) is denoted \(\tcboxmath{\mathcal{P}(S)}\) .
You can see that a power set of \(S\) is not composed of elements of \(S\) , but rather it is composed of subsets of \(S\) , and none of these subsets are elements of \(S\) .
For example, if \(S=\{a,b\}\) , then \(\mathcal{P}(S)=\{\emptyset, \{a\}, \{b\}, S\}\) . It follows immediately from the definition that \(A\subseteq S\) if and only if \(A\in\mathcal{P}(S)\) .
Problem 3.28. For each of the following sets, find the power set.
- \(A=\{\circ, \triangle, \square\}\)
- \(B=\{a,\{a\}\}\)
- \(C=\emptyset\)
- \(D=\{\emptyset\}\)
Problem 3.29. How many subsets do you think that a set with \(n\) elements has? What if \(n=0\) ? You do not need to prove your conjecture at this time. We will prove this later using mathematical induction.
It is important to realize that the concepts of element and subset need to be carefully delineated. For example, consider the set \(A=\{x,y\}\) . The object \(x\) is an element of \(A\) , but the object \(\{x\}\) is both a subset of \(A\) and an element of \(\mathcal{P}(A)\) . This can get confusing rather quickly. Consider the set \(B\) from Problem 3.28. The set \(\{a\}\) happens to be an element of \(B\) , a subset of \(B\) , and an element of \(\mathcal{P}(B)\) . The upshot is that it is important to pay close attention to whether “ \(\subseteq\) " or “ \(\in\) " is the proper symbol to use.
Since the next theorem is a biconditional proposition, you need to write two distinct subproofs, one for “ \(S\subseteq T\) implies \(\mathcal{P}(S)\subseteq \mathcal{P}(T)\) ", and another for “ \(\mathcal{P}(S)\subseteq \mathcal{P}(T)\) implies \(S\subseteq T\) ".
Theorem 3.30. Let \(S\) and \(T\) be sets. Then \(S\subseteq T\) if and only if \(\mathcal{P}(S)\subseteq \mathcal{P}(T)\) .
Problem 3.31. Let \(S\) and \(T\) be sets. Determine whether each of the following statements is true or false. If the statement is true, prove it. If the statement is false, provide a counterexample.
- \(\mathcal{P}(S\cap T) \subseteq \mathcal{P}(S)\cap\mathcal{P}(T)\)
- \(\mathcal{P}(S)\cap\mathcal{P}(T)\subseteq \mathcal{P}(S\cap T)\)
- \(\mathcal{P}(S\cup T)\subseteq \mathcal{P}(S)\cup\mathcal{P}(T)\)
- \(\mathcal{P}(S)\cup\mathcal{P}(T)\subseteq \mathcal{P}(S\cup T)\)
While power sets provide a useful way of generating new sets, they also play a key role in Georg Cantor ’s (1845–1918) investigation into the “size" of sets. Cantor’s Theorem (see Theorem 9.64) states that the power set of a set—even if the set is infinite—is always “larger" than the original set. One consequence of this is that there are different sizes of infinity and no largest infinity. Mathematics is awesome.


4.1: Statically Equivalent Systems
Two sets of forces are considered statically equivalent if they cause the same set of reaction forces on a body. Because this is true, the two statically equivalent sets of forces are interchangeable in statics analysis.
Determining if Forces are Statically Equivalent:
To determine if two sets of forces are statically equivalent, you must solve for the reaction forces in both cases. If the reaction forces are the same then the two sets of forces must be statically equivalent. For any one set of forces, there are an infinite number of sets of forces that are statically equivalent to original set of forces.
Finding a Single Equivalent Point Force:
In statics analysis, we are usually looking to simplify a problem by turning multiple forces into a single, statically equivalent force. To find a single point force that is equivalent to multiple point forces you can use the following procedure.
- Solve for the reaction forces in the original scenario.
- Draw a new free body diagram with these reaction forces. You will also add one force with an unknown magnitude, direction, and point of application to your diagram. This is the single point load that will be equivalent to your original set of forces.
- Write out the equations of equilibrium for this scenario, including the known values for the reaction forces.
- First, solve the force equations to find the \(x\) and \(y\) components of this unknown force (or \(x\), \(y\) and \(z\) components for a 3D problem). This can be used to find the magnitude and direction of the statically equivalent point force.
- Next, use the moment equation (or equations, for 3D problems) to determine the location of the statically equivalent point force.
Example \(\PageIndex{1}\)
Determine if the two sets of forces shown below are statically equivalent.
- Solution


2.9: Computer Simulation of Electric Circuits
These same programs can be fantastic aids to the beginning student of electronics, allowing the exploration of ideas quickly and easily with no assembly of real circuits required. Of course, there is no substitute for actually building and testing real circuits, but computer simulations certainly assist in the learning process by allowing the student to experiment with changes and see the effects they have on circuits. Throughout this book, I’ll be incorporating computer printouts from circuit simulation frequently in order to illustrate important concepts. By observing the results of a computer simulation, a student can gain an intuitive grasp of circuit behavior without the intimidation of abstract mathematical analysis.
To simulate circuits on computer, I make use of a particular program called SPICE, which works by describing a circuit to the computer by means of a listing of text. In essence, this listing is a kind of computer program in itself, and must adhere to the syntactical rules of the SPICE language. The computer is then used to process, or “run,” the SPICE program, which interprets the text listing describing the circuit and outputs the results of its detailed mathematical analysis, also in text form. Many details of using SPICE are described in volume 5 (“Reference”) of this book series for those wanting more information. Here, I’ll just introduce the basic concepts and then apply SPICE to the analysis of these simple circuits we’ve been reading about.
First, we need to have SPICE installed on our computer. As a free program, it is commonly available on the internet for download, and in formats appropriate for many different operating systems. In this book, I use one of the earlier versions of SPICE: version 2G6, for its simplicity of use.
Next, we need a circuit for SPICE to analyze. Let’s try one of the circuits illustrated earlier in the chapter. Here is its schematic diagram:
This simple circuit consists of a battery and a resistor connected directly together. We know the voltage of the battery (10 volts) and the resistance of the resistor (5 Ω), but nothing else about the circuit. If we describe this circuit to SPICE, it should be able to tell us (at the very least), how much current we have in the circuit by using Ohm’s Law (I=E/R).
SPICE cannot directly understand a schematic diagram or any other form of graphical description. SPICE is a text-based computer program, and demands that a circuit be described in terms of its constituent components and connection points. Each unique connection point in a circuit is described for SPICE by a “node” number. Points that are electrically common to each other in the circuit to be simulated are designated as such by sharing the same number. It might be helpful to think of these numbers as “wire” numbers rather than “node” numbers, following the definition given in the previous section. This is how the computer knows what’s connected to what: by the sharing of common wire, or node, numbers. In our example circuit, we only have two “nodes,” the top wire and the bottom wire. SPICE demands there be a node 0 somewhere in the circuit, so we’ll label our wires 0 and 1:
In the above illustration, I’ve shown multiple “1” and “0” labels around each respective wire to emphasize the concept of common points sharing common node numbers, but still this is a graphic image, not a text description. SPICE needs to have the component values and node numbers given to it in text form before any analysis may proceed.
Creating a text file in a computer involves the use of a program called a text editor . Similar to a word processor, a text editor allows you to type text and record what you’ve typed in the form of a file stored on the computer’s hard disk. Text editors lack the formatting ability of word processors (no italic , bold , or underlined characters), and this is a good thing, since programs such as SPICE wouldn’t know what to do with this extra information. If we want to create a plain-text file, with absolutely nothing recorded except the keyboard characters we select, a text editor is the tool to use.
If using a Microsoft operating system such as DOS or Windows, a couple of text editors are readily available with the system. In DOS, there is the old Edit text editing program, which may be invoked by typing edit at the command prompt. In Windows (3.x/95/98/NT/Me/2k/XP), the Notepad text editor is your stock choice. Many other text editing programs are available, and some are even free. I happen to use a free text editor called Vim , and run it under both Windows 95 and Linux operating systems. It matters little which editor you use, so don’t worry if the screenshots in this section don’t look like yours; the important information here is what you type , not which editor you happen to use.
To describe this simple, two-component circuit to SPICE, I will begin by invoking my text editor program and typing in a “title” line for the circuit:
We can describe the battery to the computer by typing in a line of text starting with the letter “ v ” (for “Voltage source”), identifying which wire each terminal of the battery connects to (the node numbers), and the battery’s voltage, like this:
This line of text tells SPICE that we have a voltage source connected between nodes 1 and 0, direct current (DC), 10 volts. That’s all the computer needs to know regarding the battery. Now we turn to the resistor: SPICE requires that resistors be described with a letter “r,” the numbers of the two nodes (connection points), and the resistance in ohms. Since this is a computer simulation, there is no need to specify a power rating for the resistor. That’s one nice thing about “virtual” components: they can’t be harmed by excessive voltages or currents!
Now, SPICE will know there is a resistor connected between nodes 1 and 0 with a value of 5 Ω. This very brief line of text tells the computer we have a resistor (” r ”) connected between the same two nodes as the battery (1 and 0), with a resistance value of 5 Ω.
If we add an .end statement to this collection of SPICE commands to indicate the end of the circuit description, we will have all the information SPICE needs, collected in one file and ready for processing. This circuit description, comprised of lines of text in a computer file, is technically known as a netlist , or deck :
Once we have finished typing all the necessary SPICE commands, we need to “save” them to a file on the computer’s hard disk so that SPICE has something to reference to when invoked. Since this is my first SPICE netlist, I’ll save it under the filename “ circuit1.cir ” (the actual name being arbitrary). You may elect to name your first SPICE netlist something completely different, just as long as you don’t violate any filename rules for your operating system, such as using no more than 8+3 characters (eight characters in the name, and three characters in the extension: 12345678.123 ) in DOS.
To invoke SPICE (tell it to process the contents of the circuit1.cir netlist file), we have to exit from the text editor and access a command prompt (the “DOS prompt” for Microsoft users) where we can enter text commands for the computer’s operating system to obey. This “primitive” way of invoking a program may seem archaic to computer users accustomed to a “point-and-click” graphical environment, but it is a very powerful and flexible way of doing things. Remember, what you’re doing here by using SPICE is a simple form of computer programming, and the more comfortable you become in giving the computer text-form commands to follow—as opposed to simply clicking on icon images using a mouse—the more mastery you will have over your computer.
Once at a command prompt, type in this command, followed by an [Enter] keystroke (this example uses the filename circuit1.cir ; if you have chosen a different filename for your netlist file, substitute it):
Here is how this looks on my computer (running the Linux operating system), just before I press the [Enter] key:
As soon as you press the [Enter] key to issue this command, text from SPICE’s output should scroll by on the computer screen. Here is a screenshot showing what SPICE outputs on my computer (I’ve lengthened the “terminal” window to show you the full text. With a normal-size terminal, the text easily exceeds one page length):
SPICE begins with a reiteration of the netlist, complete with title line and .end statement. About halfway through the simulation it displays the voltage at all nodes with reference to node 0. In this example, we only have one node other than node 0, so it displays the voltage there: 10.0000 volts. Then it displays the current through each voltage source. Since we only have one voltage source in the entire circuit, it only displays the current through that one. In this case, the source current is 2 amps. Due to a quirk in the way SPICE analyzes current, the value of 2 amps is output as a negative (-) 2 amps.
The last line of text in the computer’s analysis report is “total power dissipation,” which in this case is given as “2.00E+01” watts: 2.00 x 10 1 , or 20 watts. SPICE outputs most figures in scientific notation rather than normal (fixed-point) notation. While this may seem to be more confusing at first, it is actually less confusing when very large or very small numbers are involved. The details of scientific notation will be covered in the next chapter of this book.
One of the benefits of using a “primitive” text-based program such as SPICE is that the text files dealt with are extremely small compared to other file formats, especially graphical formats used in other circuit simulation software. Also, the fact that SPICE’s output is plain text means you can direct SPICE’s output to another text file where it may be further manipulated. To do this, we re-issue a command to the computer’s operating system to invoke SPICE, this time redirecting the output to a file I’ll call “ output.txt ”:
SPICE will run “silently” this time, without the stream of text output to the computer screen as before. A new file, output1.txt , will be created, which you may open and change using a text editor or word processor. For this illustration, I’ll use the same text editor ( Vim ) to open this file:
Now, I may freely edit this file, deleting any extraneous text (such as the “banners” showing date and time), leaving only the text that I feel to be pertinent to my circuit’s analysis:
Once suitably edited and re-saved under the same filename ( output.txt in this example), the text may be pasted into any kind of document, “plain text” being a universal file format for almost all computer systems. I can even include it directly in the text of this book—rather than as a “screenshot” graphic image—like this:
Incidentally, this is the preferred format for text output from SPICE simulations in this book series: as real text, not as graphic screenshot images.
To alter a component value in the simulation, we need to open up the netlist file ( circuit1.cir ) and make the required modifications in the text description of the circuit, then save those changes to the same filename, and re-invoke SPICE at the command prompt. This process of editing and processing a text file is one familiar to every computer programmer. One of the reasons I like to teach SPICE is that it prepares the learner to think and work like a computer programmer, which is good because computer programming is a significant area of advanced electronics work.
Earlier we explored the consequences of changing one of the three variables in an electric circuit (voltage, current, or resistance) using Ohm’s Law to mathematically predict what would happen. Now let’s try the same thing using SPICE to do the math for us.
If we were to triple the voltage in our last example circuit from 10 to 30 volts and keep the circuit resistance unchanged, we would expect the current to triple as well. Let’s try this, re-naming our netlist file so as to not over-write the first file. This way, we will have both versions of the circuit simulation stored on the hard drive of our computer for future use. The following text listing is the output of SPICE for this modified netlist, formatted as plain text rather than as a graphic image of my computer screen:
Just as we expected, the current tripled with the voltage increase. Current used to be 2 amps, but now it has increased to 6 amps (-6.000 x 10 0 ). Note also how the total power dissipation in the circuit has increased. It was 20 watts before, but now is 180 watts (1.8 x 10 2 ). Recalling that power is related to the square of the voltage (Joule’s Law: P=E 2 /R), this makes sense. If we triple the circuit voltage, the power should increase by a factor of nine (3 2 = 9). Nine times 20 is indeed 180, so SPICE’s output does indeed correlate with what we know about power in electric circuits.
If we want to see how this simple circuit would respond over a wide range of battery voltages, we can invoke some of the more advanced options within SPICE. Here, I’ll use the “ .dc ” analysis option to vary the battery voltage from 0 to 100 volts in 5 volt increments, printing out the circuit voltage and current at every step. The lines in the SPICE netlist beginning with a star symbol (” * ”) are comments . That is, they don’t tell the computer to do anything relating to circuit analysis, but merely serve as notes for any human being reading the netlist text.
The .print command in this SPICE netlist instructs SPICE to print columns of numbers corresponding to each step in the analysis:
If I re-edit the netlist file, changing the .print command into a .plot command, SPICE will output a crude graph made up of text characters:
In both output formats, the left-hand column of numbers represents the battery voltage at each interval, as it increases from 0 volts to 100 volts, 5 volts at a time. The numbers in the right-hand column indicate the circuit current for each of those voltages. Look closely at those numbers and you’ll see the proportional relationship between each pair: Ohm’s Law (I=E/R) holds true in each and every case, each current value being 1/5 the respective voltage value, because the circuit resistance is exactly 5 Ω. Again, the negative numbers for current in this SPICE analysis is more of a quirk than anything else. Just pay attention to the absolute value of each number unless otherwise specified.
There are even some computer programs able to interpret and convert the non-graphical data output by SPICE into a graphical plot. One of these programs is called Nutmeg , and its output looks something like this:
Note how Nutmeg plots the resistor voltage v(1) (voltage between node 1 and the implied reference point of node 0) as a line with a positive slope (from lower-left to upper-right).
Whether or not you ever become proficient at using SPICE is not relevant to its application in this book. All that matters is that you develop an understanding for what the numbers mean in a SPICE-generated report. In the examples to come, I’ll do my best to annotate the numerical results of SPICE to eliminate any confusion, and unlock the power of this amazing tool to help you understand the behavior of electric circuits.


16.16: Transition Matrices and Generators of Continuous-Time Chains
16. Transition Matrices and Generators of Continuous-Time Chains
Preliminaries
This is the second of the three introductory sections on continuous-time Markov chains. Thus, suppose that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a continuous-time Markov chain defined on an underlying probability space \( (\Omega, \mathscr{F}, \P) \) and with state space \( (S, \mathscr{S}) \). By the very meaning of Markov chain , the set of states \( S \) is countable and the \( \sigma \)-algebra \( \mathscr{S} \) is the collection of all subsets of \( S \). So every subset of \( S \) is measurable, as is every function from \( S \) to another measurable space. Recall that \( \mathscr{S} \) is also the Borel \( \sigma \) algebra corresponding to the discrete topology on \( S \). With this topology, every function from \( S \) to another topological space is continuous. Counting measure \( \# \) is the natural measure on \( (S, \mathscr{S}) \), so in the context of the general introduction, integrals over \( S \) are simply sums. Also, kernels on \( S \) can be thought of as matrices, with rows and sums indexed by \( S \). The left and right kernel operations are generalizations of matrix multiplication.
A space of functions on \( S \) plays an important role. Let \( \mathscr{B} \) denote the collection of bounded functions \( f: S \to \R \). With the usual pointwise definitions of addition and scalar multiplication, \( \mathscr{B} \) is a vector space. The supremum norm on \( \mathscr{B} \) is given by \[ \|f\| = \sup\{\left|f(x)\right|: x \in S\}, \quad f \in \mathscr{B} \] Of course, if \( S \) is finite, \( \mathscr{B} \) is the set of all real-valued functions on \( S \), and \( \|f\| = \max\{\left|f(x)\right|: x \in S\}\) for \(f \in \mathscr{B} \).
In the last section, we studied \( \bs{X} \) in terms of when and how the state changes. To review briefly, let \( \tau = \inf\{t \in (0, \infty): X_t \ne X_0\} \). Assuming that \( \bs{X} \) is right continuous, the Markov property of \( \bs{X} \) implies the memoryless property of \( \tau \), and hence the distribution of \( \tau \) given \( X_0 = x \) is exponential with parameter \( \lambda(x) \in [0, \infty) \) for each \( x \in S \). The assumption of right continuity rules out the pathological possibility that \( \lambda(x) = \infty \), which would mean that \( x \) is an instantaneous state so that \( \P(\tau = 0 \mid X_0 = x) = 1 \). On the other hand, if \( \lambda(x) \in (0, \infty) \) then \( x \) is a stable state , so that \( \tau \) has a proper exponential distribution given \( X_0 = x \) with \( \P(0 \lt \tau \lt \infty \mid X_0 = x) = 1 \). Finally, if \( \lambda(x) = 0 \) then \( x \) is an absorbing state , so that \( \P(\tau = \infty \mid X_0 = x) = 1 \). Next we define a sequence of stopping times: First \( \tau_0 = 0 \) and \( \tau_1 = \tau\). Recursively, if \( \tau_n \lt \infty \) then \( \tau_n = \inf\left\{t \gt \tau_n: X_t \ne X_{\tau_n}\right\} \), while if \( \tau_n = \infty \) then \( \tau_{n+1} = \infty \). With \( M = \sup\{n \in \N: \tau_n \lt \infty\} \) we define \( Y_n = X_{\tau_n} \) if \( n \in \N \) with \( n \le M \) and \( Y_n = Y_M \) if \( n \in \N \) with \( n \gt M \). The sequence \( \bs{Y} = (Y_0, Y_1, \ldots) \) is a discrete-time Markov chain on \( S \) with one-step transition matrix \( Q \) given by \(Q(x, y) = \P(X_\tau = y \mid X_0 = x)\) if \( x, \, y \in S \) with \( x \) stable, and \( Q(x, x) = 1\) if \( x \in S \) is absorbing. Assuming that \( \bs{X} \) is regular , which means that \( \tau_n \to \infty \) as \( n \to \infty \) with probability 1 (ruling out the explosion event of infinitely many transitions in finite time), the structure of \( \bs{X} \) is completely determined by the sequence of stopping times \( \bs{\tau} = (\tau_0, \tau_1, \ldots) \) and the discrete-time jump chain \( \bs{Y} = (Y_0, Y_1, \ldots) \). Analytically, the distribution \( \bs{X} \) is determined by the exponential parameter function \( \lambda \) and the one-step transition matrix \( Q \) of the jump chain.
In this section, we sill study the Markov chain \( \bs{X} \) in terms of the transition matrices in continuous time and a fundamentally important matrix known as the generator . Naturally, the connections between the two points of view are particularly interesting.
The Transition Semigroup
Definition and basic Properties
The first part of our discussion is very similar to the treatment for a general Markov processes, except for simplifications caused by the discrete state space. We assume that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \).
The transition probability matrix \( P_t \) of \( \bs{X} \) corresponding to \( t \in [0, \infty) \) is \[ P_t(x, y) = \P(X_t = y \mid X_0 = x), \quad (x, y) \in S^2 \] In particular, \( P_0 = I \), the identity matrix on \( S \)
Proof
The mapping \( y \mapsto P_t(x, y) \) is the PDF of \( X_t \) given \( X_0 = x \). Hence \( P_t \) is a probability matrix. That is, \( P_t(x, y) \ge 0 \) for \( (x, y) \in S^2 \) and \( \sum_{y \in S} P_t(x, y) = 1 \) for \( x \in S \). Trivially, \( P_0 = I \) by definition.
Note that since we are assuming that the Markov chain is homogeneous, \[ P_t(x, y) = \P(X_{s + t} = y \mid X_s = x), \quad (x, y) \in S^2 \] for every \( s, \, t \in [0, \infty) \). The Chapman-Kolmogorov equation given next is essentially yet another restatement of the Markov property. The equation is named for Andrei Kolmogorov and Sydney Chapman,
Suppose that \( \bs{P} = \{P_t: t \in [0, \infty)\} \) is the collection of transition matrices for the chain \( \bs{X} \). Then \( P_s P_t = P_{s+t} \) for \( s, \, t \in [0, \infty) \). Explicitly, \[ P_{s+t}(x, z) = \sum_{y \in S} P_s(x, y) P_t(y, z), \quad x, \, z \in S \]
Proof
We condition on \( X_s \). \[ P_{s+t}(x, z) = \P(X_{s + t} = z \mid X_0 = x) = \sum_{y \in S} \P(X_{s+t} = z \mid X_s = y, X_0 = x) \P(X_s = y \mid X_0 = x) \] But by the Markov and time homogeneous properties, \[ \P(X_{s+t} = z \mid X_s = y, X_0 = x) = \P(X_{s+t} = z \mid X_s = y) = P_t(y, z) \] Of course by definition, \( \P(X_s = y \mid X_0 = x) = P_s(x, y) \). So the first displayed equation above becomes \[ P_{s+t}(x, y) = \sum_{y \in S} P_s(x, y) P_t(y, z) = P_s P_t(x, z) \]
Restated in another form of jargon, the collection \( \bs{P} = \{P_t: t \in [0, \infty)\} \) is a semigroup of probability matrices. The semigroup of transition matrices \( \bs{P}\), along with the initial distribution, determine the finite-dimensional distributions of \( \bs{X} \).
Suppose that \( X_0 \) has probability density function \( f \). If \( (t_1, t_2, \ldots, t_n) \in [0, \infty)^n \) is a time sequence with \( 0 \lt t_1 \lt \cdots \lt t_n \) and \( (x_0, x_1, \ldots, x_n) \in S^{n+1} \) is a state sequence, then \[ \P\left(X_0 = x_0, X_{t_1} = x_1, \ldots X_{t_n} = x_n\right) = f(x_0) P_{t_1}(x_0, x_1) P_{t_2 - t_1}(x_1, x_2) \cdots P_{t_n - t_{n-1}}(x_{n-1}, x_n) \]
Proof
To simplify the notation, we will just give the cases \( n = 1 \) and \( n = 2 \), which capture the essence of the proof. First suppose \( x, \, y \in S \) and \( t \in [0, \infty) \). Then \[ \P(X_0 = x, X_t = y) = \P(X_0 = x) \P(X_t = y \mid X_0 = x) = f(x) P_t(x, y) \] Next suppose that \( x, \, y, \, z \in S \) and \( s, \, t \in [0, \infty) \) with \( s \lt t \). Then \[ \P(X_0 = x, X_s = y, X_t = z) = \P(X_t = z \mid X_0 = x, X_s = y) \P(X_0 = x, X_s = y) \] But by the Markov and time homogeneous properties, \( \P(X_t = z \mid X_0 = x, X_s = y) = P_{t - s}(y, z) \). By the \( n = 1 \) case, \( \P(X_0 = x, X_s = y) = f(x) P_s(x, y) \). Hence \[ \P(X_0 = x, X_s = y, X_t = z) = f(x) P_s(x, y) P_{t-s}(y, z) \]
As with any matrix on \( S \), the transition matrices define left and right operations on functions which are generalizations of matrix multiplication. For a transition matrix, both have natural interpretations.
Suppose that \( f: S \to \R \), and that either \( f \) is nonnegative or \( f \in \mathscr{B} \). Then for \( t \in [0, \infty) \), \[ P_t f(x) = \sum_{y \in S} P_t(x, y) f(y) = \E[f(X_t) \mid X_0 = x], \quad x \in S \] The mapping \( f \mapsto P_t f \) is a bounded, linear operator on \( \mathscr{B} \) and \( \|P_t\| = 1 \).
Proof
Since \( P_t(x, \cdot) \) is the conditional probability density function of \( X_t \) given \( X_0 = x \), it follows that \( P_t f(x) = \E[f(X_t) \mid X_0 = x] \). The statement about \( f \mapsto P_t f \) follows from general results on probability kernels.
If \( f \) is nonnegative and \( S \) is infinte, then it's possible that \( P_t f(x) = \infty \). In general, the left operation of a positive kernel acts on positive measures on the state space. In the setting here, if \( \mu \) is a positive (Borel) measure on \( (S, \mathscr{S}) \), then the function \( f: S \to [0, \infty) \) given by \( f(x) = \mu\{x\} \) for \( x \in S \) is the density function of \( \mu \) with respect to counting measure \( \# \) on \( (S, \mathscr{S}) \). This simply means that \( \mu(A) = \sum_{x \in A} f(x) \) for \( A \subseteq S \). Conversely, given \( f: S \to [0, \infty) \), the set function \( \mu(A) = \sum_{x \in A} f(x) \) for \( A \subseteq S \) defines a positive measure on \( (S, \mathscr{S}) \) with \( f \) as its density function. So for the left operation of \( P_t \), it's natural to consider only nonnegative functions.
If \( f: S \to [0, \infty) \) then \[ f P_t(y) = \sum_{x \in S} f(x) P_t(x, y), \quad y \in S\] If \( X_0 \) has probability density function \( f \) then \( X_t \) has probability density function \( f P_t \).

It follows that if \( X_0 \) has an invariant probability density function \( f \), then \( X_t \) has probability density function \( f \) for every \( t \in [0, \infty) \), so \( \bs{X} \) is identically distributed. Invariant and limiting distributions are fundamentally important for continuous-time Markov chains.
Standard Semigroups
Suppose again that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a Markov chain on \( S \) with transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \). Once again, continuity assumptions need to be imposed on \( \bs{X} \) in order to rule out strange behavior that would otherwise greatly complicate the theory. In terms of the transition semigroup \( \bs{P} \), here is the basic assumption:
The transition semigroup \( \bs{P} \) is standard if \( P_t(x, x) \to 1 \) as \( t \downarrow 0 \) for each \( x \in S \).
Since \( P_0(x, x) = 1 \) for \( x \in S \), the standard assumption is clearly a continuity assumption. It actually implies much stronger smoothness properties that we will build up by stages.
If the transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \) is standard, then the function \( t \mapsto P_t(x, y) \) is right continuous for each \( (x, y) \in S^2 \).
Proof
First note that if \( (x, y) \in S^2 \) with \( x \ne y \) then \( P_h(x, y) \le 1 - P_h(x, x) \to 0 \) as \( h \downarrow 0 \). Hence \( P_h(x, y) \to I(x, y) \) as \( h \downarrow 0 \) for all \( (x, y) \in S^2 \). Suppose next that \( t \in (0, \infty) \) and \( (x, y) \in S^2 \). By the semigroup property, \[ P_{t+h}(x, y) = P_t P_h(x, y) = \sum_{z \in S} P_t(x, z) P_h(z, y) \] But \( P_h(z, y) \to I(z, y) \) as \( h \downarrow 0 \) so by the bounded convergence theorem, \( P_{t+h}(x, y) \to P_t(x, y) \) as \( h \downarrow 0 \).
Our next result connects one of the basic assumptions in the section on transition times and the embedded chain with the standard assumption here.
If the Markov chain \( \bs{X} \) has no instantaneous states then the transition semigroup \( \bs{P}\) is standard.
Proof
Given \( X_0 = x \in S \) note that \( \tau \gt t \) implies \( X_t = x \). Hence \[ P_t(x, x) = \P(X_t = x \mid X_0 = x) \ge \P(\tau \gt t \mid X_0 = x) = e^{-\lambda(x) t} \] Since \( \bs{X} \) has no instantaneous states, \( 0 \le \lambda(x) \lt \infty\) so \( e^{-\lambda(x) t} \to 1 \) as \( t \downarrow 0 \).
Recall that the non-existence of instantaneous states is essentially equivalent to the right continuity of \( \bs{X} \). So we have the nice result that if \( \bs{X} \) is right continuous, then so is \( \bs{P} \). For the remainder of our discussion, we assume that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a regular Markov chain on \( S \) with transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \), exponential function \( \lambda \) and one-step transition matrix \( Q \) for the jump chain. Our next result is the fundamental integral equations relating \( \bs{P} \), \( \lambda \), and \( Q \).
For \( t \in [0, \infty) \), \[ P_t(x, y) = I(x, y) e^{-\lambda(x) t} + \int_0^t \lambda(x) e^{-\lambda(x) s} Q P_{t - s} (x, y) \, ds, \quad (x, y) \in S^2 \]
Proof
If \( x \) is an absorbing state, then the equation trivially holds, since \( \lambda(x) = 0 \) and \( P_t(x, y) = I(x, y) \). So suppose that \( x \) is a stable state, and as above, let \( \tau = \inf\{t \in [0, \infty): X_t \ne X_0\} \). Given \( X_0 = x \), \( \tau \) has a proper exponential distribution with parameter \( \lambda(x) \in (0, \infty) \). Taking cases, \[ P_t(x, y) = \P(X_t = y \mid X_0 = x) = \P(X_t = y, \tau \gt t \mid X_0 = x) + \P(X_t = y, \tau \le t \mid X_0 = x) \] The first term on the right is 0 if \( y \ne x \) and is \( \P(\tau \gt t \mid X_0 = x) = e^{-\lambda(x) t} \) if \( y = x \). In short, \[ \P(X_t = y, \tau \gt t \mid X_0 = x) = I(x, y) e^{-\lambda(x)s} \] For the second term on the right in the displayed equation, we condition on \( \tau \) and \( Y_1 = X_\tau \). By a result in the last section on transition times and the embedded chain, the joint PDF of \( (\tau, Y_1) \) at \( s \in [0, \infty) \) and \( z \in S \), given \( X_0 = x \), is \( \lambda(x) e^{-\lambda(x) s} Q(x, z) \) (continuous in time, discrete in space). Also, given \(\tau = s \in [0, t] \) and \( Y_1 = z \in S \), we can use the strong Markov property to
restart the clock
at \( s \) giving \[ \P(X_t = y \mid X_0 = x, \tau = s, Y_1 = z) = \P(X_{t-s} = y \mid X_0 = z) = P_{t-s}(z, y) \] Putting the pieces together we have \[ \P(X_t = y, \tau \le t \mid X_0 = x) = \int_0^t \lambda(x) e^{-\lambda(x) s} \sum_{z \in S} Q(x, z) P_{t-s}(z, y) \, ds = \int_0^t \lambda(x) e^{-\lambda(x) s} QP_{t - s} (x, y) \, ds\]
We can now improve on the continuity result that we got earlier. First recall the leads to relation for the jump chain \( \bs{Y} \): For \( (x, y) \in S^2 \), \( x \) leads to \( y \) if \( Q^n(x, y) \gt 0 \) for some \( n \in \N \). So by definition, \( x \) leads to \( x \) for each \( x \in S \), and for \( (x, y) \in S^2 \) with \( x \ne y \), \( x \) leads to \( y \) if and only if the discrete-time chain starting in \( x \) eventually reaches \( y \) with positive probability.
For \( (x, y) \in S^2 \),
- \( t \mapsto P_t(x, y) \) is continuous.
- If \( x \) leads to \( y \) then \( P_t(x, y) \gt 0 \) for every \( t \in (0, \infty) \).
- If \( x \) does not lead to \( y \) then \( P_t(x, y) = 0 \) for every \( t \in (0, \infty) \).
For \( t \in [0, \infty) \), we can use the change of variables \( r = t - s \) in the fundamental integral equation to get \[ P_t(x, y) = I(x, y) e^{-\lambda(x) t} + \lambda(x) e^{-\lambda(x) t} \int_0^t e^{\lambda(x) r} Q P_r (x, y) \, dr, \quad (x, y) \in S^2 \]
Proof
- In the displayed equation, \( r \mapsto P_r(x, y) \) is right continuous for every \( (x, y) \in S^2 \), and hence by the bounded convergence theorem again, so is \( r \mapsto QP_r(x, y) \). Since the integrand in the displayed equation is bounded and right continuous, the integral is a continuous function of \( t \). Hence \( t \mapsto P_t(x, y) \) is continuous for \( (x, y) \in S^2 \).
- For \( x \in S \), note that \( P_t(x, x) \ge e^{-\lambda(x) t} \gt 0 \) for \( t \in [0, \infty) \). If \( x \) leads to \( y \) and \( x \ne y \) then there exists \( n \in \N_+ \) and \( (x_1, x_2, \ldots, x_{n-1}) \in S^{n-1} \) such that \( Q(x, x_1) \gt 0, \, \ldots Q(x_{n-1}, y) \gt 0\). Then \[ P_t(x, y) = \P(X_t = y \mid X_0 = x) \ge \P(Y_1 = x_1, \ldots, Y_{n-1} = x_{n-1}, Y_n = y, \tau_n \le t \lt \tau_{n+1}) \gt 0 \]
- This is clear from the definition of the embedded chain \( \bs{Y} \).
Parts (b) and (c) are known as the Lévy dichotomy , named for Paul Lévy. It's possible to prove the Lévy dichotomy just from the semigroup property of \( \bs{P} \), but this proof is considerably more complicated. In light of the dichotomy, the leads to relation clearly makes sense for the continuous-time chain \( \bs{X} \) as well as the discrete-time embedded chain \( \bs{Y} \).
The Generator Matrix
Definition and Basic Properties
In this discussion, we assume again that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a regular Markov chain on \( S \) with transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \), exponential parameter function \( \bs{\lambda} \) and one-step transition matrix \( Q \) for the embedded jump chain. The fundamental integral equation above now implies that the transition probability matrix \( P_t \) is differentiable in \( t \). The derivative at \( 0 \) is particularly important.
The matrix function \( t \mapsto P_t \) has a (right) derivative at 0: \[ \frac{P_t - I}{t} \to G \text { as } t \downarrow 0 \] where the infinitesimal generator matrix \( G \) is given by \( G(x, y) = -\lambda(x) I(x, y) + \lambda(x) Q(x, y) \) for \( (x, y) \in S^2 \).
Proof
As before the change of variables \( r = t - s \) in the fundamental integral equation gives \[ P_t(x, y) = I(x, y) e^{-\lambda(x) t} + \lambda(x) e^{-\lambda(x) t} \int_0^t e^{\lambda(x) r} Q P_r (x, y) \, dr \] The first term is clearly differentiable in \( t \), and the second term is also differentiable in \( t \) since we now know that the integrand is a continuous function of \( r \). The result then follows from standard calculus.
Note that \( \lambda(x) Q(x, x) = 0 \) for every \( x \in S \), since \( \lambda(x) = 0 \) is \( x \) is absorbing, while \( Q(x, x) = 0 \) if \( x \) is stable. So \( G(x, x) = -\lambda(x) \) for \( x \in S \), and \( G(x, y) = \lambda(x) Q(x, y) \) for \( (x, y) \in S^2 \) with \( y \ne x \). Thus, the generator matrix \( G \) determines the exponential parameter function \( \lambda \) and the jump transition matrix \( Q \), and thus determines the distribution of the Markov chain \( \bs{X} \).
Given the generator matrix \( G \) of \( \bs{X} \),
- \( \lambda(x) = -G(x, x) \) for \( x \in S \)
- \( Q(x, y) = - G(x, y) \big/ G(x, x)\) if \( x \in S \) is stable and \( y \in S - \{x\} \)
The infinitesimal generator has a nice interpretation in terms of our discussion in the last section. Recall that when the chain first enters a stable state \( x \), we set independent, exponentially distributed
timers
on (x, y), for each \( y \in S - \{x\} \). Note that \( G(x, y) \) is the exponential parameter for the timer on \( (x, y) \). As soon as an alarm sounds for a particular \( (x, y) \), the chain moves to state \( y \) and the process continues.
The generator matrix \( G \) satisfies the following properties for every \( x \in S \):
- \( G(x, x) \le 0 \)
- \( \sum_{y \in S} G(x, y) = 0 \)
The matrix function \( t \mapsto P_t \) is differentiable on \( [0, \infty) \), and satisfies the Kolmogorov backward equation : \( P^\prime_t = G P_t \). Explicitly, \[ P^\prime_t(x, y) = -\lambda(x) P_t(x, y) + \sum_{z \in S} \lambda(x) Q(x, z) P_t(z, y), \quad (x, y) \in S^2 \]
Proof
The proof is just like before, and follows from standard calculus and the integral equation \[ P_t(x, y) = I(x, y) e^{-\lambda(x) t} + \lambda(x) e^{-\lambda(x) t} \int_0^t e^{\lambda(x) r} Q P_r (x, y) \, dr \]
The backward equation is named for Andrei Kolmogorov. In continuous time, the transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \) can be obtained from the single, generator matrix \( G \) in a way that is reminiscent of the fact that in discrete time, the transition semigroup \( \bs{P} = \{P^n: n \in \N\} \) can be obtained from the single, one-step matrix \( P \). From a modeling point of view, we often start with the generator matrix \( G \) and then solve the the backward equation, subject to the initial condition \( P_0 = I \), to obtain the semigroup of transition matrices \( \bs{P} \).
As with any matrix on \( S \), the generator matrix \( G \) defines left and right operations on functions that are analogous to ordinary matrix multiplication. The right operation is defined for functions in \( \mathscr{B} \).
If \( f \in \mathscr{B} \) then \( Gf \) is given by \[ G f(x) = -\lambda(x) f(x) + \sum_{y \in S} \lambda(x) Q(x, y) f(y), \quad x \in S \]
Proof
By definition, \[ G f(x) = \sum_{y \in S} G(x, y) f(y) = -\lambda(x) f(x) + \sum_{y \in S - \{x\}} \lambda(x) Q(x, y) f(y) \] In the second term, we can sum over all \( y \in S \) since \( \lambda(x) = 0 \) if \( x \) is absorbing and \( Q(x, x) = 0 \) if \( x \) is stable. Note that \( G f \) is well defined since \[ \sum_{y \in S-\{x\}} \lambda(x) Q(x, y) \left|f(x)\right| \le \sum_{y \in S-\{x\}} \lambda(x) Q(x, y) \|f\| = \lambda(x) \|f\| \]
But note that \( G f \) is not in \( \mathscr{B} \) unless \( \lambda \in \mathscr{B} \). Without this additional assumption, \( G \) is a linear operator from the vector space \( \mathscr{B} \) of bounded functions from \( S \) to \( \R \) into the vector space of all functions from \( S \) to \( \R \). We will return to this point in our next discussion.
Uniform Transition Semigroups
We can obtain stronger results for the generator matrix if we impose stronger continuity assumptions on \( \bs{P} \).
The transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \) is uniform if \( P_t(x, x) \to 1 \) as \( t \downarrow 0 \) uniformly in \( x \in S \).
If \( \bs{P} \) is uniform, then the operator function \( t \mapsto P_t \) is continuous on the vector space \( \mathscr{B} \).
Proof
The statement means that for \( f \in \mathscr{B} \), the function \( t \mapsto P_t f \) is continuous with respect to the supremum norm on \( \mathscr{B} \).
As usual, we want to look at this new assumption from different points of view.
The following are equivalent:
- The transition semigroup \( \bs{P} \) is uniform.
- The exponential parameter function \( \lambda \) is bounded.
- The generator matrix \( G \) defiens a bounded linear operator on \( \mathscr{B} \).
Proof
From our remarks above we know that \( \lambda \in \mathscr{B} \) if and only if the generator matrix \( G \) defines a bounded linear operator on \( \mathscr{B} \). So we just need to show the equivalence of (a) and (b). If \( \lambda \in \mathscr{B} \) then \[ P_t(x, x) = \P(X_t = x \mid X_0 = x) \ge \P(\tau \gt t \mid X_0 = x) = \exp[-\lambda(x) t] \ge \exp(-\|\lambda\|t) \] The last term converges to 1 as \( t \downarrow 0 \) uniformly in \( x \).
So when the equivalent conditions are satisfied, the Markov chain \( \bs X = \{X_t: t \in [0, \infty)\} \) is also said to be uniform. As we will see in a later section, a uniform, continuous-time Markov chain can be constructed from a discrete-time Markov chain and an independent Poisson process. For a uniform transition semigroup, we have a companion to the backward equation.
Suppose that \( \bs{P} \) is a uniform transition semigroup. Then \( t \mapsto P_t \) satisfies the Kolmogorov forward equation \( P^\prime_t = P_t G \). Explicitly, \[ P^\prime_t(x,y) = -\lambda(y) P_t(x, y) + \sum_{z \in S} P_t(x, z) \lambda(z) Q(z, y), \quad (x, y) \in S^2 \]
The backward equation holds with more generality than the forward equation, since we only need the transition semigroup \( \bs{P} \) to be standard rather than uniform . It would seem that we need stronger conditions on \( \lambda \) for the forward equation to hold, for otherwise it's not even obvious that \( \sum_{z \in S} P_t(x, z) \lambda(z) Q(z, y) \) is finite for \( (x, y) \in S \). On the other hand, the forward equation is sometimes easier to solve than the backward equation, and the assumption that \( \lambda \) is bounded is met in many applications (and of course holds automatically if \( S \) is finite).
As a simple corollary, the transition matrices and the generator matrix commute for a uniform semigroup: \( P_t G = G P_t \) for \( t \in [0, \infty) \). The forward and backward equations formally look like the differential equations for the exponential function. This actually holds with the operator exponential.
Suppose again that \( \bs{P} = \{P_t: t \in [0, \infty)\} \) is a uniform transition semigroup with generator \( G \). Then \[ P_t = e^{t G} = \sum_{n=0}^\infty \frac{t^n}{n!} G^n, \quad t \in [0, \infty) \]
Proof
First \( e^{t G} \) is well defined as a bounded linear operator on \( \mathscr{B} \) for \( t \in [0, \infty) \) (and hence also simply as a matrix), since \( G \) is a bounded linear operator on \( \mathscr{B} \). Trivially \( e^{0 G} = I\), and by basic properties of the matrix exponential, \[ \frac{d}{dt} e^{t G} = G e^{t G}, \quad t \in (0, \infty) \] It follows that \( P_t = e^{t G} \) for \( t \in [0, \infty) \).
We can characterize the generators of uniform transition semigroups. We just need the minimal conditions that the diagonal entries are nonpositive and the row sums are 0.
Suppose that \( G \) a matrix on \( S \) with \( \|G\| \lt \infty \). Then \( G \) is the generator of a uniform transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \) if and only if for every \( x \in S \),
- \( G(x, x) \le 0 \)
- \(\sum_{y \in S} G(x, y) = 0\)
Proof
We know of course that if \( G \) is the generator of a transition semigroup, then conditions (a) and (b) hold. For the converse, we can use the previous result. Let \[ P_t = e^{t G} = \sum_{n=0}^\infty \frac{t^n}{n!} G^n, \quad t \in [0, \infty) \] which makes sense since \( G \) is bounded in norm. Then \( P_t(x, y) \ge 0 \) for \( (x, y) \in S^2 \). By part (b), \( \sum_{y \in S} G^n(x, y) = 0 \) for every \( x \in S \) and \( n \in \N_+ \), and hence \( \sum_{y \in S} P_t(x, y) = \sum_{y \in S} I(x, y) = 1 \) for \( x \in S \). Finally, the semigroup property is a consequence of the law of exponents, which holds for the exponential of a matrix. \[ P_s P_t = e^{s G} e^{t G} = e^{(s+t) G} = P_{s+t} \]
Examples and Exercises
The Two-State Chain
Let \( \bs{X} = \{X_t: t \in [0, \infty)\} \) be the Markov chain on the set of states \( S = \{0, 1\} \), with transition rate \( a \in [0, \infty) \) from 0 to 1 and transition rate \( b \in [0, \infty) \) from 1 to 0. This two-state Markov chain was studied in the previous section. To avoid the trivial case with both states absorbing, we will assume that \( a + b \gt 0 \).
The generator matrix is \[ G = \left[\begin{matrix} -a & a \\ b & -b\end{matrix}\right] \]
Show that for \( t \in [0, \infty) \), \[ P_t = \frac{1}{a + b} \left[\begin{matrix} b & a \\ b & a \end{matrix} \right] - \frac{1}{a + b} e^{-(a + b)t} \left[\begin{matrix} -a & a \\ b & -b\end{matrix}\right] \]
- By solving the Kolmogorov backward equation.
- By solving the Kolmogorov forward equation.
- By computing \( P_t = e^{t G} \).
You probably noticed that the forward equation is easier to solve because there is less coupling of terms than in the backward equation.
Define the probability density function \( f \) on \( S \) by \( f(0) = \frac{b}{a + b} \), \( f(1) = \frac{a}{a + b} \). Show that
- \( P_t \to \frac{1}{a + b} \left[\begin{matrix} b & a \\ b & a \end{matrix} \right] \) as \(t \to \infty \), the matrix with \( f \) in both rows.
- \( f P_t = f \) for all \( t \in [0, \infty) \), so that \( f \) is invariant for \( \bs{P} \).
- \( f G = 0 \).
Computational Exercises
Consider the Markov chain \( \bs{X} = \{X_t: t \in [0, \infty)\} \) on \( S = \{0, 1, 2\} \) with exponential parameter function \( \lambda = (4, 1, 3) \) and embedded transition matrix \[ Q = \left[\begin{matrix} 0 & \frac{1}{2} & \frac{1}{2} \\ 1 & 0 & 0 \\ \frac{1}{3} & \frac{2}{3} & 0\end{matrix}\right] \]
- Draw the state graph and classify the states.
- Find the generator matrix \( G \).
- Find the transition matrix \( P_t \) for \( t \in [0, \infty) \).
- Find \( \lim_{t \to \infty} P_t \).
Answer
- The edge set is \( E = \{(0, 1), (0, 2), (1, 0), (2, 0), (2, 1)\} \). All states are stable.
- The generator matrix is \[ G = \left[\begin{matrix} -4 & 2 & 2 \\ 1 & -1 & 0 \\ 1 & 2 & -3 \end{matrix}\right] \]
- For \( t \in [0, \infty) \), \[ P_t = \frac{1}{15} \left[\begin{matrix} 3 + 12 e^{-5 t} & 10 - 10 e^{-3 t} & 2 - 12 e^{-5 t} + 10 e^{-3 t} \\ 3 - 3 e^{-5 t} & 10 + 5 e^{-3 t} & 2 + 3 e^{-5t} - 5 e^{-3 t} \\ 3 - 3 e^{-5 t} & 10 - 10 e^{-3 t} & 2 + 3 e^{-5 t} + 10 e^{-3 t} \end{matrix}\right] \]
- \[ P_t \to \frac{1}{15} \left[\begin{matrix} 3 & 10 & 2 \\ 3 & 10 & 2 \\ 3 & 10 & 2 \end{matrix}\right] \]
Special Models
Read the discussion of generator and transition matrices for chains subordinate to the Poisson process.
Read the discussion of the infinitesimal generator for continuous-time birth-death chains.
Read the discussion of the infinitesimal generator for continuous-time queuing chains.
Read the discussion of the infinitesimal generator for continuous-time branching chains.


7.4: Pants
Pants Pattern Simplicity
Place patterns on fabric along straight of grain.Cut out pants pattern pieces.Transfer marking symbols in pants pattern pieces. |
With right sides together, place back of one pant leg to front of one pant leg and stitch inseam.Then stitch side seam of pant same pant leg.  Serge Pants. |
Serge around the top of the pants.Fold under 1.25” and press in place.Topstitch at 1” around the top of the pants leaving a 1 inch opening. |
Serge around the top of the pants.Fold under 1.25” and press in place.Topstitch at 1” around the top of the pants leaving a 1” opening for elasticcasing.Cut 3/4” elastic to desired waist measurement. Place a safety pin in the end of the elastic. Thread through the opening at the top of the pants.Once the elastic is complete threaded and ends are showing through theopening, stitch elastic together at ends.Topstitch the opening closed. |
Mark hem.Trim if necessary and serge hemline.Turn under hem at desired length and pin.Machine stitch hem at 1” mark above hem line. Press. |


28.5A: Phylum Echinodermata
- Describe the characteristics of echinodermata
Phylum Echinodermata
Echinodermata are so named owing to their spiny skin (from the Greek “echinos” meaning “spiny” and “dermos” meaning “skin”). This phylum is a collection of about 7,000 described living species. Echinodermata are exclusively marine organisms. Sea stars, sea cucumbers, sea urchins, sand dollars, and brittle stars are all examples of echinoderms. To date, no freshwater or terrestrial echinoderms are known.
Morphology and Anatomy
Adult echinoderms exhibit pentaradial symmetry and have a calcareous endoskeleton made of ossicles, although the early larval stages of all echinoderms have bilateral symmetry. The endoskeleton is developed by epidermal cells and may possess pigment cells that give vivid colors to these animals, as well as cells laden with toxins. Echinoderms possess a simple digestive system which varies according to the animal’s diet. Starfish are mostly carnivorous and have a mouth, oesophagus, two-part pyloric stomach with a pyloric duct leading to the intestine and rectum, with the anus located in the center of the aboral body surface. In many species, the large cardiac stomach can be everted and digest food outside the body. Gonads are present in each arm. In echinoderms such as sea stars, every arm bears two rows of tube feet on the oral side which help in attachment to the substratum. These animals possess a true coelom that is modified into a unique circulatory system called a water vascular system. The more notably distinct trait, which most echinoderms have, is their remarkable powers of regeneration of tissue, organs, limbs, and, in some cases, complete regeneration from a single limb.
Water Vascular System
Echinoderms possess a unique ambulacral or water vascular system, consisting of a central ring canal and radial canals that extend along each arm. Water circulates through these structures and facilitates gaseous exchange as well as nutrition, predation, and locomotion. The water vascular system also projects from holes in the skeleton in the form of tube feet. These tube feet can expand or contract based on the volume of water (hydrostatic pressure) present in the system of that arm.
The madreporite is a light-colored, calcerous opening used to filter water into the water vascular system of echinoderms. Acting as a pressure-equalizing valve, it is visible as a small red or yellow button-like structure (similar to a small wart) on the aboral surface of the central disk of a sea star. Close up, it is visibly structured, resembling a “madrepore” colony. From this, it derives its name. Water enters the madreporite on the aboral side of the echinoderm. From there, it passes into the stone canal, which moves water into the ring canal. The ring canal connects the radial canals (there are five in a pentaradial animal), and the radial canals move water into the ampullae, which have tube feet through which the water moves. By moving water through the unique water vascular system, the echinoderm can move and force open mollusk shells during feeding.
Other Body Systems
The nervous system in these animals is a relatively simple structure with a nerve ring at the center and five radial nerves extending outward along the arms. Structures analogous to a brain or derived from fusion of ganglia are not present in these animals.
Podocytes, cells specialized for ultrafiltration of bodily fluids, are present near the center of echinoderms. These podocytes are connected by an internal system of canals to the madreporite.
Echinoderms are sexually dimorphic and release their eggs and sperm cells into water; fertilization is external. In some species, the larvae divide asexually and multiply before they reach sexual maturity. Echinoderms may also reproduce asexually, as well as regenerate body parts lost in trauma.
Key Points
- Echinoderms live exclusively in marine systems; they are widely divergent, with over 7,000 known species in the phylum.
- Echinoderms have pentaradial symmetry and a calcareous endoskeleton that may possess pigment cells that give them a wide range of colors, as well as cells that possess toxins.
- Echinoderms have a water vascular system composed of a central ring of canals that extend along each arm, through which water circulates for gaseous exchange and nutrition.
- Echinoderms have a very simple nervous system, comprised of a nerve ring at the center and five radial nerves extending outward along the arms; there is no structure resembling a brain.
- There are two sexes in echinoderms, which each release their eggs and sperm into the water; here, the sperm will fertilize the eggs.
- Echinoderms can reproduce asexually by regeneration.
Key Terms
- madreporite : a lightcolored calcerous opening used to filter water into the water vascular system of echinoderms
- podocyte : cells that filter the bodily fluids in echinoderms
- pentaradial symmetry : a variant of radial symmetry that arranges roughly equal parts around a central axis at orientations of 72° apart
- water vascular system : a hydraulic system used by echinoderms, such as sea stars and sea urchins, for locomotion, food and waste transportation, and respiration
- ampulla : the dilated end of a duct


16.5: Childhood Obesity and the Bad Mother
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Concerns over food safety can also be extended to include foods high in salt, sugar and fat. These do not cause immediate harm in the same way that melamine-tainted milk does, but their cumulative effects can still be dangerous. The World Health Organization has warned that unhealthy diets are a leading global risk to health because of their link to illnesses like heart disease and stroke. In fact, these are the two biggest killers in the world, each causing more deaths every year than HIV/AIDS, lung cancer and road accidents combined. This aspect of food malnutrition – ‘mal’ meaning bad rather than insufficient – should be just as worrying as the existence of food shortages. In the United Kingdom, the public debate about malnutrition has paid particular attention to children’s diets. Some of the debate has focused on problems experienced during childhood itself. For example, in 2014 it was reported that the consumption of sugary foods and drinks had contributed to 25,000 children aged five to nine being admitted to hospital to get rotten teeth pulled out. But mostly it has focused on childhood obesity and the risk this poses for children later in life. Under pressure from campaigners, including doctors and other health professionals, successive British governments have introduced policies to promote dietary change. Restrictions have been placed on junk food adverts, minimum nutritional standards have been applied to school meals, families have been targeted with healthy lifestyle campaigns, and food manufacturers have been asked to lower the salt, sugar and fat content of their products. To cap this off, a ‘sugar tax’ on high-sugar soft drinks was announced in 2016.
Despite first impressions, these internal debates have actually had an international dimension. In this respect it is important to remember that the United Kingdom is a nation-state made up of four countries (England, Scotland, Wales and Northern Ireland), with the latter three each having some devolved political powers of their own. As such, policy debates about diet have often become proxy wars over the further devolution of power away from the central state. This happened in 2014 when the first minister of Scotland declared that the Scottish policy to offer more free school meals to pupils showed that Scotland would be better off as an independent country. International data has also been used to defend or discredit domestic policy proposals. The successful campaign to tax sugary drinks, spearheaded by the celebrity chef Jamie Oliver, constantly referred to a similar policy introduced in Mexico to show that what worked there could work in the United Kingdom.
International comparison has also been used in depictions of national identity. British newspapers have run countless stories saying it has become a nation beset by increasing obesity. For some people, especially those with right-wing political views, this has been taken as evidence that the British are becoming lazy and that standards of parenting have worsened. Since childhood obesity is positively correlated with poverty, meaning that children from poorer backgrounds are more likely to be overweight, this interpretation also produced a divisive image of the nation. Put simply, it implied that poor parents were to blame for the country’s moral failings. Moreover, since it is women that tend to be the primary caregivers, the figure of the bad parent inevitably assumed a female face.