5.3.9: Emerging Trends in Ethics, CSR, and Compliance
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- Jamie Hammond and Western Technical College
- OpenStax
Learning Objectives
- Identify forecasts about contemporary ethical and corporate social responsibility issues.
Predicted trends in ethics, compliance, and corporate social responsibility for Fortune 500 companies, governments, groups, and professionals by Navex Global include:
- “A shift in the ‘power of voice in the story of harassment.’” The #MeToo movement has changed everything. “Bill Cosby, Harvey Weinstein, Charlie Rose, Kevin Spacey, Al Franken, Matt Lauer, and Garrison Keillor” and other high-profile public figures have pressured organizations through public forum social media forums to take action. “Victims of harassment have the power of choice. They can make an internal report and hope that their organization responds properly, or they can choose to take their story public.” A clear message for corporations and ethics and compliance officers is, “Create a corporate culture in which employees feel comfortable raising their voices about anything from sexual harassment to feelings of being insulted. This will allow your compliance program to resolve issues before they turn into scandals, and preserve the integrity of your organization’s culture internally and its reputation externally. And don’t ever tolerate retaliation.”
- The “Glassdoor” effect (when people trust online reviews of their companies more than what companies communicate) and the effect of trust when employee messages go viral on social media. Companies need to create “listen-up” cultures by creating internal reporting systems in which leadership and managers listen to and support employees when they raise their voices for the betterment of the company. “This ensures employees know that their report will be heard, taken seriously, and things will change if necessary.”
- Assisting national disasters that suddenly occur causes havoc not only for vulnerable populations but also for unprepared organizations. Ethics and compliance professionals learned from 2017’s natural disasters (hurricanes in particular) to update preparation plans and test emergency hotlines, communications systems, and employee readiness.
- The acceleration of the need for compliance and ethics programs as economies begin again to grow; “growth without ethics and governance does nobody any favors. Growth with ethics and governance won’t simply be a feel-good mantra in 2018, it will be a business imperative.”
- Creating a “culture of compliance” in corporations (a culture of integrity and ethics) over one of “vicious compliance” (an overreliance on laws and regulations). “Finally, and most importantly, leadership accountability is what every employee is watching. In the end, what happens to the top performers who violate the rules will send the loudest message of all to the organization.”
- An increasing need for compliance’s role in prevention and mitigation as cybersecurity evolves. “Compliance must play an integral part in any organization’s cross-functional cyber security program to make sure such efforts are enterprise-wide.”
- Giving new voice to whistle-blowers is predicted as “regulatory scrutiny is increasing, and the voice of the whistleblower in the [Silicon] Valley is growing louder as well.” Corporations need to listen and resolve whistle-blowers’ issues internally before they decide to go outside.
- Managing culture and free speech in the workplace during “polarizing times” continues about “race, gender, sex, sexual orientation, gender identity, national origin, and religion—and people’s right to fair treatment, protection, and the rights and benefits enjoyed by others.”
- Data privacy is becoming a larger concern for chief compliance officers in companies as “privacy laws and the environments they regulate, have evolved.” Creating a safe and respectful workplace is needed.
- The role of the compliance professional evolves and innovates as “old networking models are giving way to online networks that provide new and unprecedented opportunities to share ideas and collaborate.”
Ethics and compliance go hand in hand as stated earlier. With a strong ethical culture, compliance is more effective in preventing risks, and without a compliance program, those who deliberately choose to break laws and codes of conduct would create disorder. “Strong cultures have two elements: A high level of agreement about what is valued and a high level of intensity with regard to those values. “In the long run, a positive culture of integrity is the foundation for an effective ethics and compliance program, which, when properly embedded into an organization, can create a competitive advantage and serve as a valuable organizational asset,” quoted from Keith Darcy, an independent senior advisor to Deloitte & Touche LLP’s Regulatory and Operational Risk practice.7 9
Moral Entrepreneur: A New Component of Ethical Leadership?
An innovative development in ethics and business is the concept of “moral entrepreneur.” 80 Brown and Trevino found that people who have been exposed to a moral entrepreneur are more likely to become a moral entrepreneur themselves because they have experienced its potential and experienced how it is done. Inspiration certainly takes place, and the person’s well-being is improved. 81
Related to moral entrepreneurship is ethical leadership. Brown and Trevino’s ethical leadership is defined as “the demonstration of normatively appropriate conduct through personal actions and interpersonal relationships, and the promotion of such conduct to followers through two-way communication, reinforcement, and decision-making.” Examples of such conduct include openness, honesty, and treating employees fairly and thoughtfully. Social learning theory was used to gain an understanding as to why ethical leadership is important to employees and how it is perceived to work.
Ethical leaders are models of ethical conduct who become the targets of identification and emulation for followers. For leaders to be perceived as ethical leaders and to influence ethics-related outcomes, they must be perceived as attractive, credible, and legitimate. They do this by engaging in behavior that is seen as normatively appropriate (e.g., openness and honesty) and motivated by altruism (e.g., treating employees fairly and considerately). Ethical leaders must also gain followers’ attention to the ethics message by engaging in explicit ethics-related communication and by using reinforcement to support the ethics message. 82
In addition to social learning theory, which focuses on why and how supporters follow a leader, a social development approach to the concept of ethical leadership is also needed because it focuses on the direction that leadership should take. Studies on corporate social responsibility are concerned with how companies can contribute to societal development, not only in the sense of solving social problems but also in the sense of improving social welfare, promoting social progress, and creating new social value.
Muel Kaptein argues that there is a third component to ethical leadership—moral entrepreneurship, in addition to the already defined components of the moral person and the moral manager. 83 His belief is that moral entrepreneurship opens avenues for studying various antecedents and outcomes of ethical leadership that hasn’t been acknowledged adequately to date.
Studies of the antecedents of ethical leadership, at both the situational and personal levels, have found that leaders who have had ethical role models are more likely to become ethical leaders. 84 These studies have also found that the personality traits of agreeableness and conscientiousness are positively related to ethical leadership. And studies on corporate social responsibility are concerned with how companies can contribute to societal development, not only in the sense of solving social problems, but also in the sense of improving social welfare, promoting social progress, and creating new social value.
According to Kaptein, someone who creates a new ethical norm is called a moral entrepreneur.  He differentiates between two kinds of moral entrepreneurs: those who create new norms and those who enforce new norms. The moral entrepreneur experiences something horrific that prompts him to want to do something to solve the issue, as Kaptein describes, a want to correct by translating a preferred norm into legal prohibitions: however, he risks becoming an outsider themselves if they are not unable to congregate support for the new norm.
Kaptein states that the component of moral entrepreneurship complements the two other components of ethical leadership (moral person and moral manager), because it highlights the creation of new norms instead of only following and implementing current ethical norms. Becker suggests that helping others is important and having the altruistic trait is important for a moral entrepreneur. Yurtsever, in developing a scale for moral entrepreneurship personality, suggests that moral entrepreneurs demonstrate high moral virtues, such as justice and honesty. 85 Moreover, being a moral manager is important to be able to get the support of others to follow the new ethical norm. In order to be successful as a norm creator, one needs the support of others. Even though the three components of ethical leadership complement each other, it is still possible for someone to only exhibit one or two of the components, making ethical leadership a multidimensional construct.
Concluding Comments
So, does it pay to be ethical and moral, whether it be entrepreneurs and individuals or corporations and organizations? We have discussed this question in this chapter and presented different arguments with different views. Scholars and ethicists who have debated whether or not corporations can be ethical differ on their responses. One such conference in the INSEAD Business School in France closed with the following statement: “Theoretically, a strong case can be made for the moral responsibility of firms. However, this does not preclude individual moral responsibility for acts as a corporate member. Moreover, it was also evident that considerable concern exists about corporate misconduct going unsanctioned and the possibility that both good and bad corporate behavior is profoundly influenced by the extent to which individuals and corporate entities are held morally responsible.” 86
This statement implies that both corporations and individuals bear the possibility and responsibility for unethical—as well as illegal—acts. While corporations are not individuals, people work and relate in corporate and work settings. That is why organizational leaders and cultures play such important roles in setting the tone and boundaries for what is acceptable (ethically and legally) and what is not. Ethical values and legal, compliance codes of conduct together work to prevent and if necessary seek correction and justice for unlawful actions. As noted earlier, promoting and rewarding ethical actions are more desirable and in the long-term more profitable.
Prooijen and Ellemers’s study using social identity analysis found that individuals are attracted to teams and organizations with positive features such as organizational “competence and achievements” and “moral values and ethical conduct.” Since these two features do not always cohere working environments, the authors’ study had students choose in three different studies which they would prefer in seeking employment, “perceived competence vs morality of a team or organization.” They found that “[r]esults of all three studies converge to demonstrate that the perceived morality of the team or organization has a greater impact on its attractiveness to individuals than its perceived competence.” 87
Concept Check
- What are some emerging national and global issues and trends in ethics and business ethics?
References
79. The Wall Street Journal/CFO Journal Deloitte, 2018, deloitte.wsj.com/riskandcompliance/2018/06/ 04/whistleblower-programs-who-should-be-in-charge-3/
80. Kaptein, M. (2017). The Moral Entrepreneur: A New Component of Ethical Leadership, Journal of Business Ethics. May, Pp. 1-16.
81. Brown, M and Trevino L., “Ethical leadership: A review and future directions”, The Leadership Quarterly , December 2006, Pages 595-616.
82. Ibid.
83. Kaptein, M. (2017). The Moral Entrepreneur: A New Component of Ethical Leadership, Journal of Business Ethics. May, Pp. 1-16.
84. Brown, M and Trevino L., “Ethical leadership: A review and future directions”, The Leadership Quarterly , December 2006, Pages 595-616.
85. Gregory, T., “Small Catholic University, Big Muslim Enrollment Try To Build Bridges”, The Chicago Tribune , January 14, 2017, http://www.chicagotribune.com/news/c...115-story.html .
86. Smith, “Could Your Leadership Style be Influencing Bad Behavior?”, INSEAD Knowledge , October 14, 2014, knowledge.insead.edu/leaders...behaviour-3638.
87. van Prooijen, A.M. and Ellemers, N, “Does it pay to be moral? How indicators of morality and competence enhance organizational and work team attractiveness”, British Journal of Management 26 (2), 225-236


16.4: Transience and Recurrence for Discrete-Time Chains
The study of discrete-time Markov chains, particularly the limiting behavior, depends critically on the random times between visits to a given state. The nature of these random times leads to a fundamental dichotomy of the states.
Basic Theory
As usual, our starting point is a probability space \( (\Omega, \mathscr{F}, \P) \), so that \( \Omega \) is the sample space, \( \mathscr{F} \) the \( \sigma \)-algebra of events, and \( \P \) the probability measure on \( (\Omega, \mathscr{F}) \). Suppose now that \( \bs{X} = (X_0, X_1, X_2, \ldots) \) is a (homogeneous) discrete-time Markov chain with (countable) state space \( S \) and transition probability matrix \( P \). So by definition, \[ P(x, y) = \P(X_{n+1} = y \mid X_n = x) \] for \( x, \, y \in S \) and \( n \in \N \). Let \( \mathscr{F}_n = \sigma\{X_0, X_1, \ldots, X_n\} \), the \( \sigma \)-algebra of events defined by the chain up to time \( n \in \N \), so that \( \mathfrak{F} = (\mathscr{F}_0, \mathscr{F}_1, \ldots) \) is the natural filtration associated with \( \bs{X} \).
 Recall that the hitting time to \( A \) is the random variable that gives the first positive time that the chain is in \( A \): \[ \tau_A = \min\{n \in \N_+: X_n \in A\} \] Since the chain may never enter \( A \), the random variable \( \tau_A \) takes values in \( \N_+ \cup \{\infty\} \) (recall our convention that the minimum of the empty set is \( \infty \)). Recall also that \( \tau_A \) is a stopping time for \( \bs{X} \). That is, \( \{\tau_A = n\} \in \mathscr{F}_n \) for \( n \in \N_+ \). Intuitively, this means that we can tell if \( \tau_A = n \) by observing the chain up to time \( n \). This is clearly the case, since explicitly \[ \{\tau_A = n\} = \{X_1 \notin A, \ldots, X_{n-1} \notin A, X_n \in A\} \] When \( A = \{x\} \) for \( x \in S \), we will simplify the notation to \( \tau_x \). This random variable gives the first positive time that the chain is in state \( x \). When the chain enters a set of states \( A \) for the first time, the chain must visit some state in \( A \) for the first time, so it's clear that \[ \tau_A = \min\{\tau_x: x \in A\}, \quad A \subseteq S \] Next we define two functions on \( S \) that are related to the hitting times.
For \( x \in S \), \( A \subseteq S \) (nonempty), and \( n \in \N_+ \) define
- \( H_n(x, A) = \P(\tau_A = n \mid X_0 = x)\)
- \(H(x, A) = \P(\tau_A \lt \infty \mid X_0 = x) \)
So \( H(x, A) = \sum_{n=1}^\infty H_n(x, A) \).
Note that \( n \mapsto H_n(x, A) \) is the probability density function of \( \tau_A \), given \( X_0 = x \), except that the density function may be defective in the sense that the sum \( H(x, A) \) may be less than 1, in which case of course, \( 1 - H(x, A) = \P(\tau_A = \infty \mid X_0 = x) \). Again, when \( A = \{y\} \), we will simplify the notation to \( H_n(x, y) \) and \( H(x, y) \), respectively. In particular, \( H(x, x) \) is the probability, starting at \( x \), that the chain eventually returns to \( x \). If \( x \ne y \), \( H(x, y) \) is the probability, starting at \( x \), that the chain eventually reaches \( y \). Just knowing when \( H(x, y) \) is 0, positive, and 1 will turn out to be of considerable importance in the overall structure and limiting behavior of the chain. As a function on \( S^2 \), we will refer to \( H \) as the hitting matrix of \( \bs{X} \). Note however, that unlike the transition matrix \( P \), we do not have the structure of a kernel. That is, \( A \mapsto H(x, A) \) is not a measure, so in particular, it is generally not true that \( H(x, A) = \sum_{y \in A} H(x, y) \). The same remarks apply to \( H_n \) for \( n \in \N_+ \). However, there are interesting relationships between the transition matrix and the hitting matrix.
\( H(x, y) \gt 0 \) if and only if \( P^n(x, y) \gt 0 \) for some \( n \in \N_+ \).
Proof
Note that \( \{X_n = y\} \subseteq \{\tau_y \lt \infty\} \) for all \( n \in \N_+ \), and \( \{\tau_y \lt \infty\} = \{X_k = y \text{ for some } k \in \N_+\} \). From the increasing property of probability and Boole's inequality it follows that for each \( n \in \N_+ \), \[ P^n(x, y) \le H(x, y) \le \sum_{k=1}^\infty P^k(x, y) \]
The following result gives a basic relationship between the sequence of hitting probabilities and the sequence of transition probabilities.
Suppose that \( (x, y) \in S^2 \). Then \[ P^n(x, y) = \sum_{k=1}^n H_k(x, y) P^{n-k}(y, y), \quad n \in \N_+ \]
Proof
This result follows from conditioning on \( \tau_y \). Starting in state \( x \), the chain is in state \( y \) at time \( n \) if and only if the chain hits \( y \) for the first time at some previous time \( k \), and then returns to \( y \) in the remaining \( n - k \) steps. More formally, \[ P^n(x, y) = \P(X_n = y \mid X_0 = x) = \sum_{k=0}^\infty \P(X_n = y \mid \tau_y = k, X_0 = x) \P(\tau_y = k \mid X_0 = x) \] But the event \( \tau_y = k \) implies \( X_k = y \) and is in \( \mathscr{F}_k \). Hence by the Markov property, \[ \P(X_n = y \mid \tau_y = k, X_0 = x) = \P(X_n = y \mid X_k = y, \tau_y = k, X_0 = x) = \P(X_n = y \mid X_k = y) = P^k(x, y) \] Of course, by definition, \( \P(\tau_y = k \mid X_0 = x) = H_k(x, y) \), so the result follows by substitution.
Suppose that \( x \in S \) and \( A \subseteq S \). Then
- \( H_{n+1}(x, A) = \sum_{y \notin A} P(x, y) H_n(y, A) \) for \( n \in \N_+ \)
- \( H(x, A) = P(x, A) + \sum_{y \notin A} P(x, y) H(y, A) \)
Proof
These results follow form conditioning on \( X_1 \).
- Starting in state \( x \), the chain first enters \( A \) at time \( n + 1 \) if and only if the chain goes to some state \( y \notin A \) at time 1, and then from state \( y \), first enters \( A \) in \( n \) steps. \[ H_{n+1}(x, A) = \P(\tau_A = n + 1 \mid X_0 = x) = \sum_{y \in S} \P(\tau_A = n + 1 \mid X_0 = x, X_1 = y) \P(X_1 = y \mid X_0 = x) \] But \( \P(\tau_A = n + 1 \mid X_0 = x, X_1 = y) = 0 \) for \( y \in A \). By the Markov and time homogeneous properties, \( \P(\tau_A = n + 1 \mid X_0 = x, X_1 = y) = \P(\tau_A = n \mid X_0 = y) = H_n(x, A) \) for \( y \notin A \). Of course \( \P(X_1 = y \mid X_0 = x) = P(x, y) \). So the result follows by substitution.
- Starting in state \( x \), the chain eventually enters \( A \) if and only if it either enters \( A \) at the first step, or moves to some other state \( y \notin A \) at the first step, and then eventually enters \( A \) from \( y \). \[ H(x, A) = \P(\tau_A \lt \infty \mid X_0 = x) = \sum_{y \in S} \P(\tau_A \lt \infty \mid X_1 =y, X_0 = x) \P(X_1 = y \mid X_0 = x)\] But \( \P(\tau_A \lt \infty \mid X_1 = y, X_0 = x) = 1\) for \( y \in A \). By the Markov and homogeneous properties, \(\P(\tau_A \lt \infty \mid X_1 = y, X_0 = x) = \P(\tau_A \lt \infty \mid X_0 = y) = H(y, A) \) for \( y \notin A \). Substituting we have \[ H(x, A) = \sum_{y \in A} P(x, y) + \sum_{y \notin A} P(x, y) H(y, A) = P(x, A) + \sum_{y \notin A} P(x, y) H(y, A) \]
The following definition is fundamental for the study of Markov chains.
Let \( x \in S \).
- State \( x \) is recurrent if \( H(x, x) = 1 \).
- State \( x \) is transient if \( H(x, x) \lt 1 \).
Thus, starting in a recurrent state, the chain will, with probability 1, eventually return to the state. As we will see, the chain will return to the state infinitely often with probability 1, and the times of the visits will form the arrival times of a renewal process. This will turn out to be the critical observation in the study of the limiting behavior of the chain. By contrast, if the chain starts in a transient state, then there is a positive probability that the chain will never return to the state.
Counting Variables and Potentials
Again, suppose that \( A \) is a nonempty set of states. A natural complement to the hitting time to \( A \) is the counting variable that gives the number of visits to \( A \) (at positive times). Thus, let \[ N_A = \sum_{n=1}^\infty \bs{1}(X_n \in A) \] Note that \( N_A \) takes value in \( \N \cup \{\infty\} \). We will mostly be interested in the special case \( A = \{x\} \) for \( x \in S \), and in this case, we will simplify the notation to \( N_x \).
Let \( G(x, A) = \E(N_A \mid X_0 = x) \) for \( x \in S \) and \( A \subseteq S \). Then \( G \) is a kernel on \( S \) and \[ G(x, A) = \sum_{n=1}^\infty P^n(x, A) \]
Proof
Note that \[ G(x, A) = \E \left(\sum_{n=1}^\infty \bs{1}(X_n \in A) \biggm| X_0 = x\right) = \sum_{n=1}^\infty \P(X_n \in A \mid X_0 = x) = \sum_{n=1}^\infty P^n(x, A) \] The interchange of sum and expected value is justified since the terms are nonnegative. For fixed \( x \in S \), \( A \mapsto G(x, A) \) is a positive measure on \( S \) since \( A \mapsto P^n(x, A) \) is a probability measure on \( S \) for each \( n \in \N_+ \). Note also that \( A \mapsto N_A \) is a random, counting measure on \( S \) and hence \( A \mapsto G(x, A) \) is a (deterministic) positive measure on \( S \).
Thus \( G(x, A) \) is the expected number of visits to \( A \) at positive times. As usual, when \( A = \{y\} \) for \( y \in S \) we simplify the notation to \( G(x, y) \), and then more generally we have \( G(x, A) = \sum_{y \in A} G(x, y) \) for \( A \subseteq S \). So, as a matrix on \( S \), \( G = \sum_{n=1}^\infty P^n \). The matrix \( G \) is closely related to the potential matrix \( R \) of \( \bs{X} \), given by \( R = \sum_{n=0}^\infty P^n \). So \( R = I + G \), and \( R(x, y) \) gives the expected number of visits to \( y \in S \) at all times (not just positive times), starting at \( x \in S \). The matrix \( G \) is more useful for our purposes in this section.
The distribution of \( N_y \) has a simple representation in terms of the hitting probabilities. Note that because of the Markov property and time homogeneous property, whenever the chain reaches state \( y \), the future behavior is independent of the past and is stochastically the same as the chain starting in state \( y \) at time 0. This is the critical observation in the proof of the following theorem.
If \( x, \, y \in S \) then
- \(\P(N_y = 0 \mid X_0 = x) = 1 - H(x, y)\)
- \( \P(N_y = n \mid X_0 = x) = H(x, y) [H(y, y)]^{n-1}[1 - H(y, y)] \) for \( n \in \N_+ \)
Proof
The essence of the proof is illustrated in the graphic above. The thick lines are intended as reminders that these are not one step transitions, but rather represent all paths between the given vertices. Note that in the special case that \( x = y \) we have \[ \P(N_x = n \mid X_0 = x) = [H(x, x)]^n [1 - H(x, x)], \quad n \in \N \] In all cases, the counting variable \( N_y \) has essentially a geometric distribution, but the distribution may well be defective , with some of the probability mass at \( \infty \). The behavior is quite different depending on whether \( y \) is transient or recurrent.
If \( x, \, y \in S \) and \( y \) is transient then
- \( \P(N_y \lt \infty \mid X_0 = x) = 1 \)
- \( G(x, y) = H(x, y) \big/ [1 - H(y, y)] \)
- \( H(x, y) = G(x, y) \big/ [1 + G(y, y)] \)
Proof
- If \( y \) is transient then \( H(y, y) \lt 1 \). Hence using the result above and geometric series, \[\P(N_y \in \N_+ \mid X_0 = x) = \sum_{n=1}^\infty \P(N_y = n \mid X_0 = x) = H(x, y) [1 - H(y, y)] \sum_{n=1}^\infty [H(y, y)]^{n-1} = H(x, y)\] Hence \begin{align*} \P(N_y \lt \infty \mid X_0 = x) & = \P(N_y \in \N \mid X_0 = x) = \P(N_y = 0 \mid X_0 = x) + \P(N_y \in \N_+ \mid X_0 = x) \\ & = [1 - H(x, y)] + H(x, y) = 1 \end{align*}
- Using the derivative of the geometric series, \begin{align*} G(x, y) & = \E(N_y \mid X_0 = x) = \sum_{n=1}^\infty n \P(N_y = n \mid X_0 = x) \\ & = H(x, y) [1 - H(y, y)]\sum_{n=1}^\infty n [H(y, y)]^{n-1} = \frac{H(x, y)}{1 - H(y, y)} \end{align*}
- From (b), \( G(y, y) = H(y, y) \big/[1 - H(y, y)] \) so solving for \( H(y, y) \) gives \( H(y, y) = G(y, y) \big/ [1 + G(y, y)] \). Substituting this back into (b) gives \( G(x, y) = H(x, y)[1 + G(y, y)] \).
if \( x, \, y \in S \) and \( y \) is recurrent then
- \( \P(N_y = 0 \mid X_0 = x) = 1 - H(x, y) \) and \( \P(N_y = \infty \mid X_0 = x) = H(x, y) \)
- \( G(x, y) = 0 \) if \( H(x, y) = 0 \) and \( G(x, y) = \infty \) if \( H(x, y) \gt 0 \)
- \( \P(N_y = \infty \mid X_0 = y) = 1 \) and \( G(y, y) = \infty \)
Proof
- If \( y \) is recurrent, \( H(y, y) = 1 \) and so from the result above , \( \P(N_y = n \mid X_0 = x) = 0 \) for all \( n \in \N_+ \). Hence \( \P(N_y = \infty \mid X_0 = x) = 1 - \P(N_y = 0 \mid X_0 = x) = 1 - H(x, y) \).
- If \( H(x, y) = 0 \) then \( \P(N_y = 0 \mid X_0 = x) = 1 \), so \( \E(N_y \mid X_0 = x) = 0 \). If \( H(x, y) \gt 0 \) then \( \P(N_y = \infty \mid X_0 = x) \gt 0 \) so \( \E(N_y \mid X_0 = x) = \infty \).
- From the result above, \( \P(N_y = n \mid X_0 = y) = 0 \) for all \( n \in \N \), so \( \P(N_y = \infty \mid X_0 = y) = 1 \).
Note that there is an invertible relationship between the matrix \( H \) and the matrix \( G \); if we know one we can compute the other. In particular, we can characterize the transience or recurrence of a state in terms of \( G \). Here is our summary so far:
Let \( x \in S \).
- State \( x \) is transient if and only if \( H(x, x) \lt 1 \) if and only if \( G(x, x) \lt \infty \).
- State \( x \) is recurrent if and only if \( H(x, x) = 1 \) if and only if \( G(x, x) = \infty \).
Of course, the classification also holds for the potential matrix \( R = I + G \). That is, state \( x \in S \) is transient if and only if \( R(x , x) \lt \infty \) and state \( x \) is recurrent if and only if \( R(x, x) = \infty \).
Relations
The hitting probabilities suggest an important relation on the state space \( S \).
For \( (x, y) \in S^2 \), we say that \( x \) leads to \( y \) and we write \( x \to y \) if either \( x = y \) or \( H(x, y) \gt 0 \).
It follows immediately from the result above that \( x \to y \) if and only if \( P^n(x, y) \gt 0 \) for some \( n \in \N \). In terms of the state graph of the chain, \( x \to y \) if and only if \( x = y \) or there is a directed path from \( x \) to \( y \). Note that the leads to relation is reflexive by definition: \( x \to x \) for every \( x \in S \). The relation has another important property as well.
The leads to relation is transitive : For \( x, \, y, \, z \in S \), if \( x \to y \) and \( y \to z \) then \( x \to z \).
Proof
If \( x \to y \) and \( y \to z \), then there exist \( j, \, k \in N \) such that \( P^j(x, y) \gt 0 \) and \( P^k(y, z) \gt 0 \). But then \( P^{j+k}(x, z) \ge P^j(x, y) P^k(y, z) \gt 0 \) so \( x \to z \).
The leads to relation naturally suggests a couple of other definitions that are important.
Suppose that \( A \subseteq S \) is nonempty.
- \( A \) is closed if \( x \in A \) and \( x \to y \) implies \( y \in A \).
- \( A \) is irreducible if \( A \) is closed and has no proper closed subsets.
Suppose that \( A \subseteq S \) is closed. Then
- \( P_A \), the restriction of \( P \) to \( A \times A\), is a transition probability matrix on \( A \).
- \( \bs{X} \) restricted to \( A \) is a Markov chain with transition probability matrix \( P_A \).
- \( (P^n)_A = (P_A)^n \) for \( n \in \N \).
Proof
- If \( x \in A \) and \( y \notin A \), then \( x \) does not lead to \( y \) so in particular \( P(x, y) = 0 \). It follows that \( \sum_{y \in A} P(x, y) = 1 \) for \( x \in A \) so \( P_A \) is a transition probability matrix.
- This follows from (a). If the chain starts in \( A \), then the chain remains in \( A \) for all time, and of course, the Markov property still holds.
- Again, this follows from (a).
Of course, the entire state space \( S \) is closed by definition. If it is also irreducible, we say the Markov chain \( \bs{X} \) itself is irreducible . Recall that for a nonempty subset \( A \) of \( S \) and for \( n \in \N \), the notation \( P_A^n \) refers to \( (P_A)^n \) and not \( (P^n)_A \). In general, these are not the same, and in fact for \( x, \, y \in A \), \[ P_A^n(x, y) = \P(X_1 \in A, \ldots, X_{n-1} \in A, X_n = y \mid X_0 = x) \] the probability of going from \( x \) to \( y \) in \( n \) steps, remaining in \( A \) all the while. But if \( A \) is closed, then as noted in part (c), this is just \( P^n(x, y) \).
Suppose that \( A \) is a nonempty subset of \( S \). Then \( \cl(A) = \{y \in S: x \to y \text{ for some } x \in A\} \) is the smallest closed set containing \( A \), and is called the closure of \( A \). That is,
- \( \cl(A) \) is closed.
- \( A \subseteq \cl(A) \).
- If \( B \) is closed and \( A \subseteq B \) then \( \cl(A) \subseteq B \)
Proof
- Suppose that \( x \in \cl(A) \) and that \( x \to y \). Then there exists \( a \in A \) such that \( a \to x \). By the transitive property, \( a \to y \) and hence \( y \in \cl(A) \).
- If \( x \in A \) then \( x \to x \) so \( x \in \cl(A) \).
- Suppose that \( B \) is closed and that \( A \subseteq B \). If \( x \in \cl(A) \), then there exists \( a \in A \) such that \( a \to x \). Hence \( a \in B \) and \( a \to x \). Since \( B \) is closed, it follows that \( x \in B \). Hence \( \cl(A) \subseteq B \).
Recall that for a fixed positive integer \( k \), \( P^k \) is also a transition probability matrix, and in fact governs the \( k \)-step Markov chain \( (X_0, X_k, X_{2 k}, \ldots) \). It follows that we could consider the leads to relation for this chain, and all of the results above would still hold (relative, of course, to the \( k \)-step chain). Occasionally we will need to consider this relation, which we will denote by \( \underset{k} \to \), particularly in our study of periodicity.
Suppose that \( j, \, k \in \N_+ \). If \( x \, \underset{k}\to \, y \) and \( j \mid k \) then \( x \, \underset{j}\to \, y \).
Proof
If \( x \, \underset{k}\to \, y \) then there exists \( n \in \N \) such that \( P^{n k}(x, y) \gt 0 \). If \( j \mid k \), there exists \( m \in \N_+ \) such that \( k = m j \). Hence \( P^{n m j}(x, y) \gt 0 \) so \( x \, \underset{j}\to \, y \).
By combining the leads to relation \( \to \) with its inverse, the comes from relation \( \leftarrow \), we can obtain another very useful relation.
For \( (x, y) \in S^2 \), we say that \( x \) to and from \( y \) and we write \( x \leftrightarrow y \) if \( x \to y \) and \( y \to x \).
By definition, this relation is symmetric : if \( x \leftrightarrow y \) then \( y \leftrightarrow x \). From our work above, it is also reflexive and transitive. Thus, the to and from relation is an equivalence relation. Like all equivalence relations, it partitions the space into mutually disjoint equivalence classes . We will denote the equivalence class of a state \( x \in S \) by \[ [x] = \{y \in S: x \leftrightarrow y\} \] Thus, for any two states \( x, \, y \in S\), either \( [x] = [y] \) or \( [x] \cap [y] = \emptyset \), and moreover, \( \bigcup_{x \in S} [x] = S \).
Two negative results:
- A closed set is not necessarily an equivalence class.
- An equivalence class is not necessarily closed.
Example
Consider the trivial Markov chain with state space \( S = \{0, 1\} \) and transition matrix \( P = \left[\begin{matrix} 0 & 1 \\ 0 & 1\end{matrix}\right] \). So state 0 leads deterministically to 1 in one step, while state 1 is absorbing. For the leads to relation, the only relationships are \( 0 \to 0 \), \( 0 \to 1 \), and \( 1 \to 1 \). Thus, the equivalence classes are \( \{0\} \) and \( \{1\} \).
- The entire state space \( S \) is closed, but is not an equivalence class.
- \( \{0\} \) is an equivalence class but is not closed.
On the other hand, we have the following result:
If \( A \subseteq S \) is irreducible, then \( A \) is an equivalence class.
Proof
Fix \( x \in A \) (recall that closed sets are nonempty by definition). Since \( A \) is closed it follows that \( [x] \subseteq A \). Since \( A \) is irreducible, \( \cl(y) = A \) for each \( y \in A \) and in particular, \( \cl(x) = A \). It follows that \( x \leftrightarrow y \) for each \( y \in A \). Hence \( A \subseteq [x] \).
The to and from equivalence relation is very important because many interesting state properties turn out in fact to be class properties , shared by all states in a given equivalence class. In particular, the recurrence and transience properties are class properties.
Transient and Recurrent Classes
Our next result is of fundamental importance: a recurrent state can only lead to other recurrent states.
If \( x \) is a recurrent state and \( x \to y \) then \( y \) is recurrent and \( H(x, y) = H(y, x) = 1 \).
Proof
The result trivially holds if \( x = y \), so we assume \( x \ne y \). Let \( \alpha(x, y) \) denote the probability, starting at \( x \), that the chain reaches \( y \) without an intermediate return to \( x \). It must be the case that \( \alpha(x, y) \gt 0 \) since \( x \to y \). In terms of the graph of \( \bs{X} \), if there is a path from \( x \) to \( y \), then there is a path from \( x \) to \( y \) without cycles. Starting at \( x \), the chain could fail to return to \( x \) by first reaching \( y \) without an intermediate return to \( x \), and then from \( y \) never reaching \( x \). From the Markov and time homogeneous properties, it follows that \( 1 - H(x, x) \ge \alpha(x, y)[1 - H(y, x)] \ge 0 \). But \( H(x, x) = 1 \) so it follows that \( H(y, x) = 1 \). So we now know that there exist positive integers \( j, \, k \) such that \( P^j(x, y) \gt 0 \) and \( P^k(y, x) \gt 0 \). Hence for every \( n \in \N \), \[ P^{j + k + n}(y, y) \ge P^k(y, x) P^n(x, x) P^j(x, y) \] Recall that \( G(x, x) = \infty \) since \( x \) is recurrent. Thus, summing over \( n \) in the displayed equation gives \( G(y, y) = \infty \). Hence \( y \) is recurrent. Finally, reversing the roles of \( x \) and \( y \), if follows that \( H(x, y) = 1 \)
From the last theorem, note that if \( x \) is recurrent, then all states in \( [x] \) are also recurrent. Thus, for each equivalence class, either all states are transient or all states are recurrent. We can therefore refer to transient or recurrent classes as well as states.
If \( A \) is a recurrent equivalence class then \( A \) is irreducible.
Proof
Suppose that \( x \in A \) and that \( x \to y \). Since \( x \) is recurrent, \( y \) is also recurrent and \( y \to x \). Hence \( x \leftrightarrow y \) and so \( y \in A \) since \( A \) is an equivalence class. Suppose that \( B \subseteq A \) is closed. Since \( B \) is nonempty by definition, there exists \( x \in B \) and so \( x \in A \) also. For every \( y \in A \), \( x \leftrightarrow y \) so \( y \in B \) since \( B \) is closed. Thus \( A = B \) so \( A \) is irreducible.
If \( A \) is finite and closed then \( A \) has a recurrent state.
Proof
Fix \( x \in A \). Since \( A \) is closed, it follows that \( \P(N_A = \infty \mid X_0 = x) = 1 \). Since \( A \) is finite, it follows that \( \P(N_y = \infty \mid X_0 = x) \gt 0 \) for some \( y \in A \). But then \( y \) is recurrent.
If \( A \) is finite and irreducible then \( A \) is a recurrent equivalence class.
Proof
Note that \( A \) is an equivalence class by a result above , and \( A \) has a recurrent state by previous result . It follows that all states in \( A \) are recurrent.
Thus, the Markov chain \( \bs{X} \) will have a collection (possibly empty) of recurrent equivalence classes \( \{A_j: j \in J\} \) where \( J \) is a countable index set. Each \( A_j \) is irreducible. Let \( B \) denote the set of all transient states. The set \( B \) may be empty or may consist of a number of equivalence classes, but the class structure of \( B \) is usually not important to us. If the chain starts in \( A_j \) for some \( j \in J \) then the chain remains in \( A_j \) forever, visiting each state infinitely often with probability 1. If the chain starts in \( B \), then the chain may stay in \( B \) forever (but only if \( B \) is infinite) or may enter one of the recurrent classes \( A_j \), never to escape. However, in either case, the chain will visit a given transient state only finitely many time with probability 1. This basic structure is known as the canonical decomposition of the chain, and is shown in graphical form below. The edges from \( B \) are in gray to indicate that these transitions may not exist.
Staying Probabilities and a Classification Test
Suppose that \( A \) is a proper subset of \( S \). Then
- \(P_A^n(x, A) = \P(X_1 \in A, X_2 \in A, \ldots, X_n \in A \mid X_0 = x) \) for \( x \in A \)
- \( \lim_{n \to \infty} P_A^n(x, A) = \P(X_1 \in A, X_2 \in A \ldots \mid X_0 = x) \) for \( x \in A \)
Proof
Recall that \( P_A^n \) means \( (P_A)^n \) where \( P_A \) is the restriction of \( P \) to \( A \times A \).
- This is a consequence of the Markov property, and is the probability that the chain stays in \( A \) at least through time \( n \), starting in \( x \in A \).
- This follows from (a) and the continuity theorem for decreasing events. This is the probability that the chain stays in \( A \) forever, starting in \( x \in A \).
Let \( g_A \) denote the function defined by part (b), so that \[ g_A(x) = \P(X_1 \in A, X_2 \in A, \ldots \mid X_0 = x), \quad x \in A \] The staying probability function \( g_A \) is an interesting complement to the hitting matrix studied above. The following result characterizes this function and provides a method that can be used to compute it, at least in some cases.
For \( A \subset S \), \( g_A \) is the largest function on \( A \) that takes values in \( [0, 1] \) and satisfies \( g = P_A g \). Moreover, either \( g_A = \bs{0}_A \) or \( \sup\{g_A(x): x \in A\} = 1 \).
Proof
Note that \( P_A^{n+1} \bs{1}_A = P_A P_A^n \bs{1}_A\) for \( n \in \N \). Taking the limit as \( n \to \infty \) and using the bounded convergence theorem gives \( g_A = P_A g_A \). Suppose now that \( g \) is a function on \( A \) that takes values in \( [0, 1] \) and satisfies \( g = P_A g \). Then \( g \le \bs{1}_A \) and hence \( g \le P_A^n \bs{1}_A \) for all \( n \in \N \). Letting \( n \to \infty \) it follows that \( g \le g_A \). Next, let \( c = \sup\{g_A(x): x \in A\} \). Then \( g_A \le c \, \bs{1}_A \) and hence \( g_A \le c \, P_A^n \bs{1}_A \) for each \( n \in \N \). Letting \( n \to \infty \) gives \( g_A \le c \, g_A \). It follows that either \( g_A = \bs{0}_A \) or \( c = 1 \).
Note that the characterization in the last result includes a zero-one law of sorts: either the probability that the chain stays in \( A \) forever is 0 for every initial state \( x \in A \), or we can find states in \( A \) for which the probability is arbitrarily close to 1. The next two results explore the relationship between the staying function and recurrence.
Suppose that \( \bs{X} \) is an irreducible, recurrent chain with state space \( S \). Then \( g_A = \bs{0}_A \) for every proper subset \( A \) of \( S \).
Proof
Fix \( y \notin A \) and note that \( 0 \le g_A(x) \le 1 - H(x, y) \) for every \( x \in A \). But \( H(x, y) = 1 \) since the chain is irreducible and recurrent. Hence \( g_A(x) = 0 \) for \( x \in A \).
Suppose that \( \bs{X} \) is an irreducible Markov chain with state space \( S \) and transition probability matrix \( P \). If there exists a state \( x \) such that \( g_A = \bs{0}_A \) where \( A = S \setminus \{x\} \), then \( \bs{X} \) is recurrent.
Proof
With \( A \) as defined above, note that \(1 - H(x, x) = \sum_{y \in A} P(x, y) g_A(y) \). Hence \( H(x, x) = 1 \), so \( x \) is recurent. Since the \( \bs{X} \) is irreducible, it follows that \( \bs{X} \) is recurrent.
More generally, suppose that \( \bs{X} \) is a Markov chain with state space \( S \) and transition probability matrix \( P \). The last two theorems can be used to test whether an irreducible equivalence class \( C \) is recurrent or transient. We fix a state \( x \in C \) and set \( A = C \setminus \{x\} \). We then try to solve the equation \( g = P_A g \) on \( A \). If the only solution taking values in \( [0, 1] \) is \( \bs{0}_A \), then the class \( C \) is recurrent by the previous result . If there are nontrivial solutions, then \( C \) is transient. Often we try to choose \( x \) to make the computations easy.
Computing Hitting Probabilities and Potentials
We now know quite a bit about Markov chains, and we can often classify the states and compute quantities of interest. However, we do not yet know how to compute:
- \( G(x, y) \) when \( x \) and \( y \) are transient
- \( H(x, y) \) when \( x \) is transient and \( y \) is transient or recurrent.
These problems are related, because of the general inverse relationship between the matrix \( H \) and the matrix \( G \) noted in our discussion above. As usual, suppose that \( \bs{X} \) is a Markov chain with state space \( S \), and let \( B \) denote the set of transient states. The next result shows how to compute \( G_B \), the matrix \( G \) restricted to the transient states. Recall that the values of this matrix are finite.
\( G_B \) satisfies the equation \( G_B = P_B + P_B G_B \) and is the smallest nonnegative solution. If \( B \) is finite then \( G_B = (I_B - P_B)^{-1} P_B \).
Proof
First note the \( (P^n)_B = (P_B)^n \) since a path between two transient states can only pass through other transient states. Thus \( G_B = \sum_{n=1}^\infty P_B^n \). From the monotone convergence theorem it follows that \( P_B G_B = G_B - P_B \).  Then \( U = \sum_{k=1}^n P_B^k + P_B^{n+1} U \) for each \( n \in \N_+\). Hence \( U \ge \sum_{k=1}^n P_B^k \) for every \( n \in \N_+ \) and therefore \( U \ge G_B \). It follows that \( (I_B - P_B)(I_B + G_B) = I_B \). If \( B \) is finite, the matrix \( I_B - P_B \) is invertible.
Now that we can compute \( G_B \), we can also compute \( H_B \) using the result above . All that remains is for us to compute the hitting probability \( H(x, y) \) when \( x \) is transient and \( y \) is recurrent. The first thing to notice is that the hitting probability is a class property.
Suppose that \( x \) is transient and that \( A \) is a recurrent class. Then \( H(x, y) = H(x, A) \) for \( y \in A \).
That is, starting in the transient state \( x \in S \), the hitting probability to \( y \) is constant for \( y \in A \), and is just the hitting probability to the class \( A \). As before, let \( B \) denote the set of transient states and suppose that \( A \) is a recurrent equivalence class. Let \( h_A \) denote the function on \( B \) that gives the hitting probability to class \( A \), and let \( p_A \) denote the function on \( B \) that gives the probability of entering \( A \) on the first step: \[ h_A(x) = H(x, A), \; p_A(x) = P(x, A), \quad x \in B \]
\( h_A = p_A + G_B p_A \).
Proof
First note that \( \P(\tau_A = n \mid X_0 = x) = (P_B^{n-1} p_A)(x) \) for \( n \in \N_+ \). The result then follows by summing over \( n \).
This result is adequate if we have already computed \( G_B \) (using the result in above , for example). However, we might just want to compute \( h_A \) directly.
\( h_A \) satisfies the equation \( h_A = p_A + P_B h_A \) and is the smallest nonnegative solution. If \( B \) is finite, \( h_A = (I_B - P_B)^{-1} p_A \).
Proof
First, conditioning on \( X_1 \) gives \( h_A = p_A + P_B h_A \). Next suppose that \( h \) is nonnegative and satisfies \( h = p_A + P_B h \). Then \( h = p_A + \sum_{k=1}^{n-1} P_B^k p_A + P_B^n h \) for each \( n \in \N_+ \). Hence \( h \ge p_A + \sum_{k=1}^{n-1} P_B^k p_A \). Letting \( n \to \infty \) gives \( h \ge h_A \). The representation when \( B \) is finite follows from the result above .
Examples and Applications
Finite Chains
Consider a Markov chain with state space \( S = \{a, b, c, d\} \) and transition matrix \( P \) given below:
\[ P = \left[ \begin{matrix} \frac{1}{2} & \frac{2}{3} & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{matrix} \right] \]- Draw the state graph.
- Find the equivalent classes and classify each as transient or recurrent.
- Compute the matrix \( G \).
- Compute the matrix \( H \).
Answer
- \( \{a, b\} \) recurrent; \( \{c\} \) recurrent; \( \{d\} \) transient.
- \( G = \left[ \begin{matrix} \infty & \infty & 0 & 0 \\ \infty & \infty & 0 & 0 \\ 0 & 0 & \infty & 0 \\ \infty & \infty & \infty & \frac{1}{3} \end{matrix} \right] \)
- \( H = \left[ \begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \frac{2}{3} & \frac{2}{3} & \frac{1}{3} & \frac{1}{4} \end{matrix} \right] \)
Consider a Markov chain with state space \( S = \{1, 2, 3, 4, 5, 6\} \) and transition matrix \( P \) given below:
\[ P = \left[ \begin{matrix} 0 & 0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \frac{1}{4} & 0 & \frac{1}{2} & 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{3} & 0 & \frac{2}{3} & 0 \\ 0 & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} \end{matrix} \right] \]- Sketch the state graph.
- Find the equivalence classes and classify each as recurrent or transient.
- Compute the matrix \( G \).
- Compute the matrix \( H \).
Answer
- \( \{1, 3, 5\} \) recurrent; \( \{2, 6\} \) transient; \( \{4\} \) recurrent.
- \( G = \left[ \begin{matrix} \infty & 0 & \infty & 0 & \infty & 0 \\ \infty & \frac{1}{2} & \infty & \infty & \infty & 2 \\ \infty & 0 & \infty & 0 & \infty & 0 \\ 0 & 0 & 0 & \infty & 0 & 0 \\ \infty & 0 & \infty & 0 & \infty & 0 \\ \infty & \frac{1}{2} & \infty & \infty & \infty & 1 \end{matrix} \right] \)
- \( H = \left[ \begin{matrix} 1 & 0 & 1 & 0 & 1 & 0 \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{matrix} \right] \)
Consider a Markov chain with state space \( S = \{1, 2, 3, 4, 5, 6\} \) and transition matrix \( P \) given below:
\[ P = \left[ \begin{matrix} \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{4} & \frac{3}{4} & 0 & 0 & 0 & 0 \\ \frac{1}{4} & 0 & \frac{1}{2} & \frac{1}{4} & 0 & 0 \\ \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} \\ 0 & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{matrix} \right] \]- Sketch the state graph.
- Find the equivalence classes and classify each as recurrent or transient.
- Compute the matrix \( G \).
- Compute the matrix \( H \).
Answer
- \( \{1, 2\} \) recurrent; \( \{3, 4\} \) transient; \( \{5, 6\} \) recurrent.
- \( G = \left[ \begin{matrix} \infty & \infty & 0 & 0 & 0 & 0 \\ \infty & \infty & 0 & 0 & 0 & 0 \\ \infty & \infty & \frac{7}{5} & \frac{4}{5} & \infty & \infty \\ \infty & \infty & \frac{4}{5} & \frac{3}{5} & \infty & \infty \\ 0 & 0 & 0 & 0 & \infty & \infty \\ 0 & 0 & 0 & 0 & \infty & \infty \end{matrix} \right] \)
- \( H = \left[ \begin{matrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ \frac{4}{5} & \frac{4}{5} & \frac{7}{12} & \frac{1}{2} & \frac{1}{5} & \frac{1}{5} \\ \frac{3}{5} & \frac{3}{5} & \frac{1}{3} & \frac{3}{8} & \frac{2}{5} & \frac{2}{5} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} \right] \)
Special Models
Read again the definitions of the Ehrenfest chains and the Bernoulli-Laplace chains. Note that since these chains are irreducible and have finite state spaces, they are recurrent.
Read the discussion on recurrence in the section on the reliability chains.
Read the discussion on random walks on \( \Z^k \) in the section on the random walks on graphs.
Read the discussion on extinction and explosion in the section on the branching chain.
Read the discussion on recurrence and transience in the section on queuing chains.
Read the discussion on recurrence and transience in the section on birth-death chains.


1.4: Sumerian City-States
Rich soils and the availability of water in the southern area, commonly known as Sumer, drew settlers. Prior to 3,000 BCE, Sumerians, whose origins remain a subject of debate, founded a number of independent cities in Lower Mesopotamia. In these cities, Sumerians had organized religions, centralized governments, social hierarchies, and access to trade networks. As these cities expanded, their leaders claimed control over adjacent territories, forming at least a dozen city-states , which became the basic organizational structure of Sumerian civilization in the third millennium BCE. By incorporating the surrounding territories into city-states, urban centers were able to draw on more resources.
Sumerian cities had certain characteristics in common. First, a temple complex or a ziggurat was usually the visual focus of the urban landscape. Sumerians believed that their entire city belonged to its main deity, and built a massive temple, the most important building in the city, to be the dwelling place of their city’s main god or goddess. Next to these temples were built complexes that housed the priests and priestesses who served the city’s deity. In addition to attending to the religious needs of the community, temples complexes also owned land, managed industries, were involved in trade, and acted as banks. Their wide-ranging roles meant that temples often had additional outbuildings, like granaries and storage sheds, in the surrounding countryside.
Sumerians were polytheistic , meaning they worshipped multiple gods and goddesses. Because Sumerians believed each god had a family, they also built smaller shrines and temples dedicated to these divine family members. Therefore, each city would have a number of temples while many Sumerian homes had small altars dedicated to other gods.
Sometimes, urban temples or ritual spaces were built atop a ziggurat , a solid rectangular tower made of sun-dried mud bricks. By the late third millennium BCE, temples in many of the Sumerian city-states were raised on platforms or else situated on a ziggurat to stress their significance to the surrounding community. The best-preserved ziggurat, the Great Ziggurat of Ur, was constructed with an estimated 720,000 baked bricks and rose to a height of about 100 feet, and was dedicated to their patron deity, the moon goddess Nanna.
Viewing nature as unpredictable, people brought offerings to their city’s temple complexes or ziggurat, hoping to please the gods who controlled the natural forces of their world. Priests and priestesses collected and redistributed the offerings, demonstrating the vital roles they played in Sumerian society. The relatively privileged position of priests and priestesses also shows Sumerian social stratification (the development of a hierarchy) and how agricultural surpluses supported the specialization of labor. Some of the early leaders of Sumerian cities may have been “priest-kings,” who attained elevated positions through their association with the temples. The later rulers of city-states definitely supported the temples, claiming to be acting on behalf of the gods who brought divine favor to their followers.
Sumerian city-states had local rulers, who lived in large palaces. But most of these local rulers were not considered kings. Archeologists have dated the earliest known royal palaces to c. 2600 BCE. Thus, Sumerian city-states had centralized governments with secular rulers by at least that timeframe. While there does seem to have been a sense of inhabiting a shared space in Southern Mesopotamia, referred to as “the Land” in written records, city-states had distinctive identities. In part, their distinctive identities revolved around their main deity. The rulers of city-states alternately supported, competed with, and fought against one another.
The Sumerian King List (Figure \(\PageIndex{2}\)), a manuscript that listed early kings and described their reigns (with some presumably fictive and exaggerated elements), provides evidence of these alliances, competition, and war. For example, it describes En-mebarages as the second to last king of the 1st Dynasty of Kish, “…who carried away as spoil the weapons of the land of Elam, became king, and reigned 900 years…” 1 Local rulers often came to power after proving themselves militarily.
The Sumerian King List recognized only rulers who had established control over multiple city-states as kings (with the title of lugal in the Sumerian language) belonging to distinct dynasties. While it lasted, a dynasty generally passed down the kingship through the male line. The seat of power, held by hereditary kings, shifted from city-state to city-state with the rise and fall of dynasties through the third millennium BCE. Significantly, the Sumerian King List began its recorded history “when kingship came down from heaven,” legitimizing secular kings through their association with gods. Sumerian kings led armies, collected taxes, organized labor for state projects, and meted out justice. At the top of the hierarchy and with control over multiple city-states, kings expected obedience from local rulers and their subjects, and support from the priests and priestesses of the temples.
Kings, local rulers, priests, and priestesses held influential positions in Sumerian societies. However, farmers, taken as a whole, made up an estimated 90% of the population. Other skilled people included animal-breeders, merchants, craftspeople, fishermen, doctors, soldiers, architects, and scribes. Surplus agricultural production collected as tribute as well as wealth generated by trade supported such labor specialization.
One important outcome of labor specialization was innovation in metalworking. In approximately 2900 BCE, metalworkers began producing bronze, which was stronger than copper. Stronger weapons and farming tools gave Sumerians advantages when it came to combat and agricultural production.
Social stratification is further evident as some Sumerians and even institutions, including temples, owned slaves. Slaves performed a variety of tasks like construction, weaving, agricultural and domestic labor, tending animals, and even administrative work as scribes. Some slaves were chattel slaves, meaning that society treated them as property with no rights. Usually, chattel slaves were prisoners of war or slaves bought from outside communities. If they tried to run away, the law required slaves to be returned. The more widespread type of servitude in most Sumerians societies was likely debt slavery, which was generally temporary until a debtor paid off a loan and its interest.
Sumerians developed cuneiform , a written script of wedge-shaped marks, around 3200 BCE. Cuneiform was one of the earliest, if not the very first, written script in the world. Merchants, scribes, administrators, priests, and others kept written records describing financial transactions, court proceedings, administrative decisions, and architectural plans. They also wrote legends, epic poems, chants, and prayers. Most people were not literate, so scribes—who had been specially trained in scribal schools—generated many of the records.
In addition to copying written passages, scribes learned arithmetic. The Sumerians system of arithmetic was based on the number 60, which we still use to divide time into hours, minutes, and seconds. Scribes used styluses made of reed to write on clay tablets that were dried and could be stored. The discovery of cuneiform tablets has greatly aided archaeologists, but only a small percentage of the tablets found to date have been translated.
From archeological finds, scholars suspect that Sumerian cities were divided into neighborhoods by occupation and according to kinship groups, but uncertainties about specifics linger. Legal documents and tax records show that people owned property in both the cities and the countryside. Also, evidence suggests social stratification, as some Sumerians owned fairly large chunks of land, while others had much smaller plots or presumably no land at all. Wills, court proceedings, and temple documents show that land and temple offices were usually bought or else acquired through military or other service to the state.
A man inherited land, property, offices, and their attendant obligations to the state (like reoccurring military service) from his father. The eldest son seems to have frequently inherited a larger share than younger brothers and had been given control over the family home. He was tasked with performing regular rituals to honor dead ancestors, who were usually buried underneath the home. From the written documents, we also get glimpses into other aspects of Sumerian life, like marriage and divorce.
Sumerians viewed marriage as a contract between two families and, as a result, the male heads of the two families arranged a couple’s marriage. Both families contributed resources to seal the union or complete the marriage contract. The man’s family gave gifts or money and hosted a feast, while the woman’s family amassed a dowry. Although a woman did not automatically receive an inheritance upon the death of her father, she could expect (and use the court system to make sure she got) to receive a dowry, even if it came from her father’s estate after his death.
Divorce was possible but sometimes led to social ostracism or even punishment if there were accusations of misconduct, such consequences being especially the case for the woman. Records indicate that polygamy was not common, but wealthier men did keep slave-girls as concubines. Overall, Sumerians considered marriage an essential institution in that it brought families together and ensured the continuation of the family lineage.
Legends, myths, poems, and literary texts tell us about Sumerians, too. For example, we can explore their values and views of the afterlife through reading the Epic of Gilgamesh , a legendary king who may have lived around 2700 BCE. The epic had multiple versions and was told orally before it was first written down in cuneiform in about 2000 BCE. Following the heroic exploits of Gilgamesh and his companion, Enkidu, it emphasize the importance of values such as loyalty and humility. In one section, the epic describes a very gloomy afterlife where “people see no light, they sit in darkness,” reflecting Mesopotamian beliefs that the afterlife was miserable for all, even those who had lived virtuously. Additionally, it portrays the environment as potentially violent and hostile as in its flood story, which in some ways resembles the account of Noah and the flood found in Genesis. Ultimately, Gilgamesh fails at his quest to find eternal life but learns to work for the glory of the gods and for worthwhile human achievements.
Cuneiform tablets with pictures of dancers and singers, as well as instruments found in graves, suggest that Sumerians placed importance on music, using it for entertainment and ritual purposes. Trying to learn more about Sumerian music, scholars and other enthusiasts have replicated these instruments and presented their best estimation of Sumerian scales and tuning. The documentary evidence suggests that hymns from the ancient Sumerian city-states were shared with later Mesopotamian empires and even spread into the Mediterranean world.
1 J.N. Postgate, Early Mesopotamia: Society and Economy at the Dawn of History, (London: Routledge, 1994): 28.
2 Ibid.
3 Excerpt from: The Epic of Gilgamesh, trans. Nancy Sandars (New York: Penguin Books, 1960), http://web.archive.org/ web/20010217041824/www.humanities.ccny.cuny.edu/...r/gilgames.htm


7: Confidence Intervals and Sample Size
In this chapter, you will learn to construct and interpret confidence intervals. You will also learn a new distribution, the Student's-t, and how it is used with these intervals. Throughout the chapter, it is important to keep in mind that the confidence interval is a random variable. It is the population parameter that is fixed.
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- 7.1: Confidence Intervals
- In this chapter, you will learn to construct and interpret confidence intervals. You will also learn a new distribution, the Student's-t, and how it is used with these intervals. Throughout the chapter, it is important to keep in mind that the confidence interval is a random variable. It is the population parameter that is fixed.
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- 7.3: Confidence Intervals for the Mean with Unknown Standard Deviation
- We rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation ss as an estimate for σσ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
Contributors and Attributions
-
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


4.1: Respiratory System Introduction
Learning Objectives
- Apply the rules of medical language to build, analyze, spell, pronounce, abbreviate, and define terms as they relate to the respiratory system
- Identify meanings of key word components of the respiratory system
- Categorize diagnostic, therapeutic, procedural, or anatomic terms related to the respiratory system
- Use terms related to the respiratory system
- Use terms related to the diseases and disorders of the respiratory system
Introduction to the Respiratory System
The primary function of the respiratory system is to provide oxygen to body tissues and to remove carbon dioxide. This chapter will begin by reviewing common word components related to the respiratory system that can be used to build definitions of respiratory terminology. Other respiratory terms, whose definitions cannot be easily built from word components, will be described in context based on the anatomy and physiology of the respiratory system, as well as common respiratory diseases and disorders. Medical specialists, diagnostic tests, procedures, and equipment related to the respiratory system will also be discussed.
View Figure 4.1 [1] for an illustration of major anatomic structures of the respiratory system.
- “ 2301 Major Respiratory Organs.jpg ” by OpenStax College is licensed under CC BY 3.0 ↵


10.5D: Resting Membrane Potentials
- Describe the resting membrane potential of cells
The potential difference in a resting neuron is called the resting membrane potential. This causes the membrane to be polarized. The value of the resting membrane potential varies from −40mV to −90mV in a different types of neurons.
The resting membrane potential exists only across the membrane. Most of the time, the difference in ionic composition of the intracellular and extracellular fluids and difference in ion permeability generates the resting membrane potential difference.
The interactions that generate the resting potential are modeled by the Goldman equation. It is based on the charges of the ions in question, as well as the difference between their inside and outside concentrations and the relative permeability of the plasma membrane to each ion where:
\[Em=RTFln(PK[K+]out+PNa[NA+]out+PCl[Cl−]inPK[K+]in+PNa[Na+]in+PCl[Cl−]out)\]
Goldman equation: R is the universal gas constant, equal to 8.314 joules·K−1·mol −1 T is the absolute temperature, measured in kelvins (= K = degrees Celsius + 273.15) F is the Faraday constant, equal to 96,485 coulombs·mol−1 or J·V−1·mol −1
The three ions that appear in this equation are potassium (K + ), sodium (Na + ), and chloride (Cl − ). The Goldman formula essentially expresses the membrane potential as an average of the reversal potentials for the individual ion types, weighted by permeability. In most animal cells, the permeability to potassium is much higher in the resting state than the permeability to sodium. Consequently, the resting potential is usually close to the potassium reversal potential.
Key Points
- The potential difference in a resting neuron is called the resting membrane potential.
- The value of the resting membrane potential varies from -40mV to -90mV in a different types of neurons.
- Most of the time, the difference in ionic composition of the intracellular and extracellular fluids and difference in ion permeability generates the resting membrane potential difference.
- The Goldman formula essentially expresses the membrane potential as an average of the reversal potentials for the individual ion types weighted by permeability
Key Terms
- resting membrane potential : The potential difference in a resting neuron that causes its membrane to be polarized.
- Goldman equation : Models the interactions that generate resting membrane potential.


29.2B: Gnathostomes - Jawed Fishes
- Differentiate among the types of jawed fishes
Gnathostomes or “jaw-mouths” are vertebrates that possess jaws. One of the most significant developments in early vertebrate evolution was the development of the jaw, which is a hinged structure attached to the cranium that allows an animal to grasp and tear its food. The evolution of jaws allowed early gnathostomes to exploit food resources that were unavailable to the jawless animals. In early evolutionary history, there were gnathostomes (jawed fishes) and agnathans (jawless fishes). Gnathostomes later evolved into all tetrapods (animals with four limbs) including amphibians, birds, and mammals.
Early gnathostomes were jawed fishes that possessed two sets of paired fins, which increased their ability to maneuver accurately. These paired fins were pectoral fins, located on the anterior body, and pelvic fins, on the posterior. The evolution of the jaw combined with paired fins permitted gnathostomes to expand from the sedentary suspension feeding of jawless fishes and become mobile predators. The gnathostomes’ ability to exploit new nutrient sources led to their evolutionary success during the Devonian period. Two early groups of gnathostomes were the acanthodians and placoderms, which arose in the late Silurian period and are now extinct. Most modern gnathostomes belong to the clades Chondrichthyes and Osteichthyes.
Chondrichthyes: Cartilaginous Fishes
The clade Chondrichthyes consists of sharks, rays, and skates, together with sawfishes and a few dozen species of fishes called chimaeras, or “ghost,” sharks. Chondrichthyes are jawed fishes that possess paired fins and a skeleton made of cartilage. This clade arose approximately 370 million years ago in the early or middle Devonian.
Most cartilaginous fishes live in marine habitats, although a few species live in fresh water for part or all of their lives. Most sharks are carnivores that feed on live prey, either swallowing it whole or using their jaws and teeth to tear it into smaller pieces. Shark teeth probably evolved from the jagged scales that cover their skin called placoid scales. Some species of sharks and rays are suspension feeders that feed on plankton.
Sharks have well-developed sense organs that aid them in locating prey, including a keen sense of smell and electroreception. Organs called ampullae of Lorenzini enable sharks to detect the electromagnetic fields that are produced by all living things, including their prey. Only aquatic or amphibious animals possess electroreception. Sharks, together with most fishes and aquatic and larval amphibians, also have a sense organ called the lateral line, which is used to detect movement and vibration in the surrounding water. It is often considered homologous to “hearing” in terrestrial vertebrates. The lateral line is visible as a darker stripe that runs along the length of a fish’s body.
Rays and skates comprise more than 500 species and are closely related to sharks. They can be distinguished from sharks by their flattened bodies, pectoral fins that are enlarged and fused to the head, and gill slits on their ventral surface. Like sharks, rays and skates have a cartilaginous skeleton. Most species are marine and live on the sea floor, with nearly a worldwide distribution.
Osteichthyes: Bony Fishes
Members of the clade Osteichthyes, also called bony fish, are characterized by a bony skeleton. The vast majority of present-day fish belong to this group, which consists of approximately 30,000 species, making it the largest class of vertebrates in existence today.
Nearly all bony fish have an ossified skeleton with specialized bone cells (osteocytes) that produce and maintain a calcium phosphate matrix. A few groups of Osteichthyes, such as sturgeons and paddlefish, have primarily cartilaginous skeletons, but retain some bony elements. The skin of bony fish is often covered by overlapping scales. Skin glands secrete mucus that reduces drag when swimming and aids the fish in osmoregulation. Like sharks, bony fish have a lateral line system that detects vibrations in water. All bony fish use gills for gas exchange. Water is drawn over gills that are located in chambers covered and ventilated by a protective, muscular flap called the operculum. Many bony fish also have a swim bladder, a gas-filled organ that helps to control the buoyancy of the fish.
Bony fish are further divided into two extant clades: Actinopterygii (ray-finned fish) and Sarcopterygii (lobe-finned fish). Actinopterygii, the ray-finned fish include many familiar fish, such as tuna, bass, trout, and salmon, among others. Ray-finned fish are named for their fins that are webs of skin supported by bony spines called rays. In contrast, the fins of Sarcopterygii are fleshy and lobed, supported by bone. Although most members of this clade are extinct, living members include the less-familiar lungfishes and coelacanths. Early Sarcopterygii evolved into modern tetrapods, including reptiles, amphibians, birds, and mammals.
Contributions and Attributions
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest/?collection=col11448/latest . License : CC BY: Attribution
- hagfish. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/hagfish . License : CC BY-SA: Attribution-ShareAlike
- agnathan. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/agnathan . License : CC BY-SA: Attribution-ShareAlike
- lamprey. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/lamprey . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Fishes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest...e_29_02_01.jpg . License : CC BY: Attribution
- Ostracoderm digital recreation.. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Ostracoderm_digital_recreation..jpg . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Fishes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest...e_29_02_02.jpg . License : CC BY: Attribution
- Osteichthyes. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/Osteichthyes . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest/?collection=col11448/latest . License : CC BY: Attribution
- ossified. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/ossified . License : CC BY-SA: Attribution-ShareAlike
- Chondrichthyes. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/Chondrichthyes . License : CC BY-SA: Attribution-ShareAlike
- operculum. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/operculum . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Fishes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest/Figure_29_02_01.jpg . License : CC BY: Attribution
- Ostracoderm digital recreation.. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Ostracoderm_digital_recreation..jpg . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Fishes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest/Figure_29_02_02.jpg . License : CC BY: Attribution
- OpenStax College, Fishes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest...29_02_07ab.jpg . License : CC BY: Attribution
- OpenStax College, Fishes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest...e_29_02_03.jpg . License : CC BY: Attribution
- OpenStax College, Fishes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44687/latest...e_29_02_04.jpg . License : CC BY: Attribution
Key Points
- Early jawed fish (gnathostomes) were able to exploit new nutrient sources because of their jaws and paired fins.
- Chondrichthyes includes all jawed fish with cartilagenous skeletons, such as sharks, rays, skates, and chimaeras.
- Osteichthyes includes all jawed fish with ossified (bony) skeletons; this includes the majority of modern fish.
- Osteichthyes can be further separated into Actinopterygii (the ray-finned fishes) and Sarcopterygii (lobe-finned fishes).
- The majority of modern fish species are actinopterygii, from trout to clownfish.
- Early Sarcopterygii (lobe-finned fishes) evolved into modern tetrapods, including reptiles, amphibians, birds, and mammals.
Key Terms
- ossified : composed of bone, which is a calcium phosphate matrix created by special cells called osteoblasts
- operculum : a covering flap or lidlike structure in plants and animals, such as a gill cover
- Chondrichthyes : a taxonomic class within the subphylum Vertebrata: the cartilaginous fish
- Osteichthyes : a taxonomic class within the subphylum vertebrata: the bony fish


12.4: Sample Documentation
Sample Documentation of Expected Findings
The patient denies abdominal pain, nausea, vomiting, bloating, constipation, diarrhea, urinary pain, urgency or frequency, change in appetite, food intolerance, dysphagia, or personal or family history. Abdominal contour is flat and symmetric. No visible lesions, pulsations, or peristalsis noted. Bowel sounds present and normoactive. Patient denies pain with palpation; no masses noted.
Sample Documentation of Unexpected Findings
The patient reports generalized abdominal pain, along with nausea and vomiting for the last two days. Abdomen is slightly distended. Bowel sounds hypoactive in all four quadrants. Pain reported at 7/10 and guarding noted with palpation of the RLQ. Dr. Smith notified at 0730.


6.10: Rearranging Tables
Learning Objectives
- Add, delete, and move columns and rows.
At times, you may find it necessary to change the number of rows or columns within your table, move columns or rows, or even alter the standard width of rows and columns. All these tasks are easy to accomplish within Excel, often with only a few clicks of your mouse.
Add Columns and Rows
Note for this section, the word column will be used throughout but the same steps can be used to add a row.
Method 1
After you have entered several columns of data into a worksheet, you may realize you need to add another column in the middle of what you have already typed. The easiest way to add a column within an existing set of columns involves only two clicks of your mouse.
- Identify the area where you want to insert a new column.
- Right-click on the column to the immediate right of the area where you wish to add a column.
- From the menu select Insert.
- A new column will appear to the left of the column that you right-clicked.
Method 2
The same result can be achieved using the Insert button in the Cells group of the ribbon. The new column will be inserted to the left of the column you clicked on. In this screenshot, a column has been inserted.
Note that when you insert a new row, it will appear above the row you click on.
Practice Question
Delete Columns and Rows
Note for this section, the word column will be used throughout, but the same steps can be used to delete a row.
Method 1
After you have entered several columns of data into a worksheet, you may realize you need to remove one column. The easiest way to delete a column uses two clicks of your mouse.
- Right-click on the column you wish to delete.
- From the menu select Delete.
Method 2
The same result can be achieved using the Delete button in the Cells group of the ribbon. The column currently selected will be the one deleted. In this screenshot the column has been deleted.
Move Columns and Rows
Note for this section, the word column will be used throughout, but the same steps can be used to move a row.
Method 1
After you have entered several columns of data into a worksheet, you may realize you need to move a column to a different location within the set. Once again, Excel makes it quite easy to move data around a worksheet.
- Right-click on the column you wish to move.
- From the pop-up window select Cut. The column you have selected should become outlined in a dashed border.
- Next, right-click on the column you wish to move your column to the LEFT of.
- A new dialog window will appear. Select Insert Cut Cells. The column you selected to move should then appear in this new location.
Method 2
The same result can be achieved using a combination of keyboard shortcuts and dialog windows.
- Right-click on the column you wish to move.
- Press Ctrl+X. The column you have selected should become outlined in a dashed border.
- Next, right-click on the column you wish to move your column to the LEFT of.
- A dialog window will appear. From it select Insert Cut Cells. The column you wish to move should then appear in the new location.
Practice Question
Contributors and Attributions
- Rearranging Tables. Authored by : Shelli Carter. Provided by : Lumen Learning. License : CC BY: Attribution


3: BrdU assay
Summary
BrdU is a synthetic nucleoside analog, like the nucleoside thymidine, except the methyl group is replaced with a bromine atom. BrdU can be incorporated into DNA whenever thymidine normally would be, including during DNA replication and repair. Therefore, BrdU is often used to label cells that are undergoing or have recently undergone DNA replication.
Also known as:
Bromodeoxyuridine assay, 5-bromo-2′-deoxyuridine assay
Samples needed
Live cells are treated with BrdU, which incorporates into DNA being replicated. Subsequent steps depend on how sample will be visualized, for instance by immunofluorescence microscopy or immunohistochemistry .
Method
Cells are treated with BrdU for a defined length of time. Any cells undergoing S-phase within the window when BrdU was present will have BrdU incorporated into their DNA. BrdU can be recognized by an anti-BrdU antibody, then visualized by the desired method. In order for BrdU to be accessible for the antibody to bind, the DNA has to be denatured.
Interpretation
If a cell is positive for BrdU, it was going through S phase when BrdU was present.
Special cases
To assess formation of single-strand DNA (ssDNA)
Normally, DNA has to be denatured for the BrdU antibody to detect any BrdU incorporated into the DNA. However, if a) BrdU labeling occurs over a long period so that all DNA is labeled and b) BrdU staining is performed under non-denaturing conditions, the only BrdU that will be accessible to the antibody will be in areas where DNA is single-stranded for some other reason (besides denaturing), like replication fork collapse. So, in this case, staining correlates with the amount of ssDNA in a sample.
Thumbnail
"BrdU staining (green).jpg" by Jason Snyder is licensed under CC BY 2.0 .
Description: BrdU (green) and arc (red) staining of neurons.
Author
Katherine Mattaini, Tufts University


10.8: Getting Tough - Initiatives for Punishment and Accountability
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- Last updated
- Save as PDF
- Alison S. Burke, David Carter, Brian Fedorek, Tiffany Morey, Lore Rutz-Burri, & Shanell Sanchez
- Southern Oregon University via OpenOregon
The 1980s saw a huge shift in the way states and federal laws were addressing juvenile law. Gangs, gun violence, and drugs drew attention to the identification, punishment, and prevention of violent and chronic youth offenders. The Office of Juvenile Justice and Delinquency Prevention (OJJDP) focused research on youth violence and state and local programming. Attention focused on the identification and control of serious, violent, and chronic offenders. [1]
At the state level, lawmakers enacted policies to crack down on youth crime. In the mid-1990s the idea of the juvenile superpredator – youth so impulsively violent, remorseless, and have no respect for human life- led to widespread reform and more punitive approaches to juvenile crime and delinquency. This included more punitive sentences, lowering the age at which a juvenile could be tried as an adult, and loosening the provisions for trying juveniles in adult court. The motto “adult time for adult crime” drove accountability initiatives and get-tough campaigns. A youth was no longer seen as vulnerable minors in need of protection and treatment. Instead, the narrative changed and they were seen as violent monsters acting “with no conscience and no empathy”, a statement Hillary Clinton has publicly regretted saying.
Waiver and Adult Time
All states have enacted laws that allow juveniles to be tried in adult criminal courts. There are several mechanisms by which a juvenile can be transferred to adult criminal court: prosecutorial, legislative, and judicial waiver . The prosecutorial waiver also is referred to as “Direct File” and “Concurrent Jurisdiction.” With this waiver mechanism, the legislature grants a prosecutor the discretion to determine in which court to file charges against the juvenile. [2] The prosecutor, or district attorney, can choose to file charges in juvenile court or adult criminal court. This procedure does not require a transfer hearing, so the defense is not accorded the opportunity to present evidence in an attempt to avoid the transfer [3]
Legislative waiver , or statutory waiver, identifies certain offenses which have been mandated by state law to be excluded from juvenile court jurisdiction. It is utilized as a method to decrease or eliminate the discretionary powers of judges and prosecutors. For example, the number of state statutes specifies that violent felony offenses such as homicide, rape, and robbery, when committed by older adolescents, are automatically sent to adult criminal court.
In the News: Raising the Age and Raising the Bar
As part of the “Raise the Age” legislation passed in 2017, all minors on Rikers Island awaiting trial or otherwise, have to be moved out of the notorious New York City jail in October 2018. Rikers Island is famed for abuse, corruption, and violence and has begun the 10 years shut down a plan to close the scandal-ridden jail complex. The jail houses some 9,000 inmates, more than 2,000 who are juveniles. The plan is to reduce the jail population while moving the inmates to other facilities throughout New York’s boroughs.
Part of the reduction in the number of inmates stems from the recent law which mandates that 16 and 17 year-olds in New York State will no longer automatically be charged as adults in criminal courts. And the age raises even more, to 18, on October 18, 2 019.
Rikers Island has a sordid history of brutality and inhumane treatment of prisoners. Perhaps the most well-known case in recent history is the story of Kalief Browder, a 16-year-old kid from the Bronx, who was charged with stealing a backpack. Although he claimed he was innocent, he ended up spending three years at Rikers Island, and more than two years were spent in solitary confinement. The charges were eventually dismissed and Browder was released, but the time spent in solitary caused significant and detrimental mental health issues. Tragically, he committed suicide in 2015, just two years after his release. His case garnered national attention prompting New York to ban the use of solitary confinement for inmates under the age of 18.
Research shows that solitary confinement is linked to mental health problems like depression, anxiety, psychosis, and even suicidal ideation. For these reasons, all federal prisons ban solitary confinement for juveniles and most states don’t allow the use of solitary in juvenile facilities. However, solitary is still used in adult prisons. Each year around 200,000 youth are tried as adults and many are sentenced to time in regular, adult prisons. Many of these state jails and prisons still use solitary confinement for the “safety” and “protection” of juveniles housed with adults (Resitvo, 2019).
Raising the age legislation is a step in the right direction and will prevent more juveniles from beginning sent to adult facilities. New York and North Carolina were the last two states in the nation to charge 16 and 17 year-olds as adults up until last year when both amended their laws. The legislation will have a profound impact on New York’s criminal justice system and is seen as a massive win for reformers who have been pushing for better treatment of children at Rikers Island for years.
Listen to the story and read more at: www.wnycstudios.org/story/raise-age-new-york-minors-rikers [4]
Judicial waiver affords the juvenile court judge the authority to transfer a case to adult criminal court. [5] There are three types of judicial waiver: discretionary, presumptive, and mandatory .
The discretionary (regular) transfer allows a judge to transfer a juvenile from juvenile court to adult criminal court. [6] With this type of transfer, the burden of proof rests with the state and the prosecutor must confirm that the juvenile is not amenable to treatment. As discussed previously, in Kent v. United States (383 U.S. 541, 566-67 [1966]), the Supreme Court outlined threshold criteria that must be met before a court can consider waiving a case. These waiver statutes typically include a minimum age, the specified type of offense, a sufficiently serious prior record, or a combination of the three.
Presumptive waiver shifts the burden of proof from the State to the defendant. It is presumptive because it is presumed that it will occur unless the youth can meet the burden of proof and provide a justifiable reason to remain in juvenile court. If the youth is unable to show just cause or sufficient reason why the case should be tried in juvenile court, the case will be transferred and tried in adult court.
The third type of judicial waiver is a mandatory waiver . Mandatory waiver means that a juvenile judge must automatically transfer to adult court juvenile offenders who meet certain criteria, such as age and current offense. In these cases, the role of the judge is simply to confirm that the waiver criteria are met and then to transfer the case to adult court. Mandatory waiver attempts to remove all discretionary powers from the juvenile court judge in transfer proceedings. [7]
In addition to increasing transfer mechanisms, at least 13 states lowered the age of majority to 15, 16, and 17, which allowed the youth of these ages to be automatically tried in adult criminal courts. These were supposed to provide procedures that curbed only the worst of the worst offenders, however, these provisions increased the prosecution of all juvenile offenders and youth of color in particular.
Ted Talks: Alice Goffman In the United States, two institutions guide teenagers on the journey to adulthood: college and prison. Sociologist Alice Goffman spent six years in a troubled Philadelphia neighborhood and saw first-hand how teenagers of African-American and Latino backgrounds are funneled down the path to prison — sometimes starting with relatively minor infractions. In an impassioned talk she asks, “Why are we offering only handcuffs and jail time?” https://www.ted.com/talks/alice_goffman_college_or_prison_two_destinies_one_blatant_injustice?language=en
- Krisberg, B., & Austin,J. (1978). History of the Control and Prevention of Juvenile Delinquency in America. In B. Krisberg & J. Austin (Eds.), The Children of Ishmael: Critical Perspective on Juvenile Justice (pp. 7-50). Palo Alto, CA: Mayfield. ↵
- Feld, B.C. (2001). Race, youth violence, and the changing jurisprudence of waiver. Behavioral Sciences & the Law, 19 (1), 3-22. ↵
- Steiner, B., Hemmens, C., & Bell, V. (2006). Legislative waiver reconsidered: General deterrent effects of statutory exclusion laws enacted post-1979. Justice Quarterly, 23 (1), 34-50 ↵
- Restivo, E. (2019, Feb 14). Stop putting juveniles in solitary confinement. Daily News. www.greensburgdailynews.com/opinion/columns/stop-putting-juveniles-in-solitary-confinement/article_d438d7bc-4e3d-5a9e-97da-22706d6037c8.html ↵
- Hemmens, S., & Bell, C. (2006). Legislative waiver reconsidered: General deterrent effects of statutory exclusion laws enacted Post 1990. Justice Quarterly, 23 (1), p34-59. ↵
- Sanborn, J. (2004). The adultification of youth. In P. Benekos & A. Merlo (Eds.), Controversies in juvenile justice and delinquency (pp. 143-164). Anderson Publishing. ↵
- Burke, A. (2016). Trends of the time. ↵


3.2.1: Introduction
It might seem obvious that not every function, even every arithmetical function, can be computable. There are just too many, whose behavior is too complicated. Functions defined from the decay of radioactive particles, for instance, or other chaotic or random behavior. Suppose we start counting 1-second intervals from a given time, and define the function \(f(n)\) as the number of particles in the universe that decay in the \(n\) -th 1-second interval after that initial moment. This seems like a candidate for a function we cannot ever hope to compute.
But it is one thing to not be able to imagine how one would compute such functions, and quite another to actually prove that they are uncomputable. In fact, even functions that seem hopelessly complicated may, in an abstract sense, be computable. For instance, suppose the universe is finite in time—some day, in the very distant future the universe will contract into a single point, as some cosmological theories predict. Then there is only a finite (but incredibly large) number of seconds from that initial moment for which \(f(n)\) is defined. And any function which is defined for only finitely many inputs is computable: we could list the outputs in one big table, or code it in one very big Turing machine state transition diagram.
We are often interested in special cases of functions whose values give the answers to yes/no questions. For instance, the question “is \(n\) a prime number?” is associated with the function \[\fn{isprime}(n) = \begin{cases} 1 & \text{if $n$ is prime}\\ 0 & \text{otherwise.} \end{cases}\nonumber\] We say that a yes/no question can be effectively decided , if the associated \(1/0\) -valued function is effectively computable.
To prove mathematically that there are functions which cannot be effectively computed, or problems that cannot effectively decided, it is essential to fix a specific model of computation, and show about it that there are functions it cannot compute or problems it cannot decide. We can show, for instance, that not every function can be computed by Turing machines, and not every problem can be decided by Turing machines. We can then appeal to the Church-Turing thesis to conclude that not only are Turing machines not powerful enough to compute every function, but no effective procedure can.
The key to proving such negative results is the fact that we can assign numbers to Turing machines themselves. The easiest way to do this is to enumerate them, perhaps by fixing a specific way to write down Turing machines and their programs, and then listing them in a systematic fashion. Once we see that this can be done, then the existence of Turing-uncomputable functions follows by simple cardinality considerations: the set of functions from \(\Nat\) to \(\Nat\) (in fact, even just from \(\Nat\) to \(\{0, 1\}\) ) are uncountable, but since we can enumerate all the Turing machines, the set of Turing-computable functions is only countably infinite.
We can also define specific functions and problems which we can prove to be uncomputable and undecidable, respectively. One such problem is the so-called Halting Problem. Turing machines can be finitely described by listing their instructions. Such a description of a Turing machine, i.e., a Turing machine program, can of course be used as input to another Turing machine. So we can consider Turing machines that decide questions about other Turing machines. One particularly interesting question is this: “Does the given Turing machine eventually halt when started on input \(n\) ?” It would be nice if there were a Turing machine that could decide this question: think of it as a quality-control Turing machine which ensures that Turing machines don’t get caught in infinite loops and such. The interesting fact, which Turing proved, is that there cannot be such a Turing machine. There cannot be a single Turing machine which, when started on input consisting of a description of a Turing machine \(M\) and some number \(n\) , will always halt with either output \(1\) or \(0\) according to whether \(M\) machine would have halted when started on input \(n\) or not.
Once we have examples of specific undecidable problems we can use them to show that other problems are undecidable, too. For instance, one celebrated undecidable problem is the question, “Is the first-order formula \(A\) valid?”. There is no Turing machine which, given as input a first-order formula \(A\) , is guaranteed to halt with output \(1\) or \(0\) according to whether \(A\) is valid or not. Historically, the question of finding a procedure to effectively solve this problem was called simply “the” decision problem; and so we say that the decision problem is unsolvable. Turing and Church proved this result independently at around the same time, so it is also called the Church-Turing Theorem.