6.8: Reading- Software Development Process
Introduction
In software engineering, a software development methodology (also known as a system development methodology , software development life cycle , software development process , software process ) is a division of software development work into distinct phases (or stages) containing activities with the intent of better planning and management. It is often considered a subset of the systems development life cycle. The methodology may include the pre-definition of specific deliverables and artifacts that are created and completed by a project team to develop or maintain an application.
Common methodologies include waterfall, prototyping, iterative and incremental development, spiral development, rapid application development, extreme programming and various types of agile methodology. Some people consider a life-cycle “model” a more general term for a category of methodologies and a software development “process” a more specific term to refer to a specific process chosen by a specific organization. For example, there are many specific software development processes that fit the spiral life-cycle model.
In practice
A variety of such frameworks have evolved over the years, each with its own recognized strengths and weaknesses. One software development methodology framework is not necessarily suitable for use by all projects. Each of the available methodology frameworks are best suited to specific kinds of projects, based on various technical, organizational, project and team considerations.
Software development organizations implement process methodologies to ease the process of development. Sometimes, contractors may require methodologies employed, an example is the U.S. defense industry, which requires a rating based on process models to obtain contracts. The international standard for describing the method of selecting, implementing and monitoring the life cycle for software is ISO/IEC 12207.
A decades-long goal has been to find repeatable, predictable processes that improve productivity and quality. Some try to systematize or formalize the seemingly unruly task of designing software. Others apply project management techniques to designing software. Without effective project management, software projects can easily be delivered late or over budget. With large numbers of software projects not meeting their expectations in terms of functionality, cost, or delivery schedule, it is effective project management that appears to be lacking.
Organizations may create a Software Engineering Process Group (SEPG), which is the focal point for process improvement. Composed of line practitioners who have varied skills, the group is at the center of the collaborative effort of everyone in the organization who is involved with software engineering process improvement.
A particular development team may also agree to programming environment details, such as which integrated development environment is used, and one or more dominant programming paradigms, programming style rules, or choice of specific software libraries or software frameworks. These details are generally not dictated by the choice of model or general methodology.
History
The software development methodology (also known as SDM) framework didn’t emerge until the 1960s. According to Elliott (2004) the systems development life cycle (SDLC) can be considered to be the oldest formalized methodology framework for building information systems. The main idea of the SDLC has been “to pursue the development of information systems in a very deliberate, structured and methodical way, requiring each stage of the life cycle from inception of the idea to delivery of the final system, to be carried out rigidly and sequentially” within the context of the framework being applied. The main target of this methodology framework in the 1960s was “to develop large scale functional business systems in an age of large scale business conglomerates. Information systems activities revolved around heavy data processing and number crunching routines.”
Methodologies, processes, and frameworks range from specific proscriptive steps that can be used directly by an organization in day-to-day work, to flexible frameworks that an organization uses to generate a custom set of steps tailored to the needs of a specific project or group. In some cases a “sponsor” or “maintenance” organization distributes an official set of documents that describe the process. Specific examples include:
- 1970s
- Structured programming since 1969
- Cap Gemini SDM, originally from PANDATA, the first English translation was published in 1974. SDM stands for System Development Methodology
- 1980s
- Structured systems analysis and design method (SSADM) from 1980 onwards
- Information Requirement Analysis/Soft systems methodology
- 1990s
- Object-oriented programming (OOP) developed in the early 1960s, and became a dominant programming approach during the mid-1990s
- Rapid application development (RAD), since 1991
- Dynamic systems development method (DSDM), since 1994
- Scrum, since 1995
- Team software process, since 1998
- Rational Unified Process (RUP), maintained by IBM since 1998
- Extreme programming, since 1999
- 2000s
- Agile Unified Process (AUP) maintained since 2005 by Scott Ambler
Approaches
Several software development approaches have been used since the origin of information technology, in two main categories. Typically an approach or a combination of approaches is chosen by management or a development team.
“Traditional” methodologies such as waterfall that have distinct phases are sometimes known as software development life cycle (SDLC) methodologies, though this term could also be used more generally to refer to any methodology. A “life cycle” approach with distinct phases is in contrast to Agile approaches which define a process of iteration, but where design, construction, and deployment of different pieces can occur simultaneously.
Waterfall development
The waterfall model is a sequential development approach, in which development is seen as flowing steadily downwards (like a waterfall) through several phases, typically:
- Requirements analysis resulting in a software requirements specification
- Software design
- Implementation
- Testing
- Integration, if there are multiple subsystems
- Deployment (or Installation)
- Maintenance
The first formal description of the method is often cited as an article published by Winston W. Royce in 1970 although Royce did not use the term “waterfall” in this article. The basic principles are:
- Project is divided into sequential phases, with some overlap and splashback acceptable between phases.
- Emphasis is on planning, time schedules, target dates, budgets and implementation of an entire system at one time.
- Tight control is maintained over the life of the project via extensive written documentation, formal reviews, and approval/signoff by the user and information technology management occurring at the end of most phases before beginning the next phase.
The waterfall model is a traditional engineering approach applied to software engineering. A strict waterfall approach discourages revisiting and revising any prior phase once it is complete. This “inflexibility” in a pure waterfall model has been a source of criticism by supporters of other more “flexible” models. It has been widely blamed for several large-scale government projects running over budget, over time and sometimes failing to deliver on requirements due to the Big Design Up Front approach. Except when contractually required, the waterfall model has been largely superseded by more flexible and versatile methodologies developed specifically for software development. See Criticism of Waterfall model.
The waterfall model is also commonly taught with the mnemonic A Dance in the Dark Every Monday, representing Analysis, Design, Implementation, Testing, Documentation and Execution, and Maintenance.
Prototyping
Software prototyping, is the development approach of activities during software development, the creation of prototypes, i.e., incomplete versions of the software program being developed.
The basic principles are:
- Not a standalone, complete development methodology, but rather an approach to handle selected parts of a larger, more traditional development methodology (i.e. incremental, spiral, or rapid application development (RAD)).
- Attempts to reduce inherent project risk by breaking a project into smaller segments and providing more ease-of-change during the development process.
- User is involved throughout the development process, which increases the likelihood of user acceptance of the final implementation.
- Small-scale mock-ups of the system are developed following an iterative modification process until the prototype evolves to meet the users’ requirements.
- While most prototypes are developed with the expectation that they will be discarded, it is possible in some cases to evolve from prototype to working system.
- A basic understanding of the fundamental business problem is necessary to avoid solving the wrong problems.
Incremental development
Various methods are acceptable for combining linear and iterative systems development methodologies, with the primary objective of each being to reduce inherent project risk by breaking a project into smaller segments and providing more ease-of-change during the development process.
The basic principles are:
- A series of mini-Waterfalls are performed, where all phases of the Waterfall are completed for a small part of a system, before proceeding to the next increment, or
- Overall requirements are defined before proceeding to evolutionary, mini-Waterfall development of individual increments of a system, or
- The initial software concept, requirements analysis, and design of architecture and system core are defined via Waterfall, followed by iterative Prototyping, which culminates in installing the final prototype, a working system.
Iterative and incremental development
Iterative development prescribes the construction of initially small but ever-larger portions of a software project to help all those involved to uncover important issues early before problems or faulty assumptions can lead to disaster.
Spiral development
The basic principles are:
- Focus is on risk assessment and on minimizing project risk by breaking a project into smaller segments and providing more ease-of-change during the development process, as well as providing the opportunity to evaluate risks and weigh consideration of project continuation throughout the life cycle.
- “Each cycle involves a progression through the same sequence of steps, for each part of the product and for each of its levels of elaboration, from an overall concept-of-operation document down to the coding of each individual program.”
- Each trip around the spiral traverses four basic quadrants: (1) determine objectives, alternatives, and constraints of the iteration; (2) evaluate alternatives; Identify and resolve risks; (3) develop and verify deliverables from the iteration; and (4) plan the next iteration.
- Begin each cycle with an identification of stakeholders and their “win conditions”, and end each cycle with review and commitment.
Rapid application development
Rapid application development (RAD) is a software development methodology, which favors iterative development and the rapid construction of prototypes instead of large amounts of up-front planning. The “planning” of software developed using RAD is interleaved with writing the software itself. The lack of extensive pre-planning generally allows software to be written much faster, and makes it easier to change requirements.
The rapid development process starts with the development of preliminary data models and business process models using structured techniques. In the next stage, requirements are verified using prototyping, eventually to refine the data and process models. These stages are repeated iteratively; further development results in “a combined business requirements and technical design statement to be used for constructing new systems”.
The term was first used to describe a software development process introduced by James Martin in 1991. According to Whitten (2003), it is a merger of various structured techniques, especially data-driven Information Engineering, with prototyping techniques to accelerate software systems development.
The basic principles of rapid application development are:
- Key objective is for fast development and delivery of a high quality system at a relatively low investment cost.
- Attempts to reduce inherent project risk by breaking a project into smaller segments and providing more ease-of-change during the development process.
- Aims to produce high quality systems quickly, primarily via iterative Prototyping (at any stage of development), active user involvement, and computerized development tools. These tools may include Graphical User Interface (GUI) builders, Computer Aided Software Engineering (CASE) tools, Database Management Systems (DBMS), fourth-generation programming languages, code generators, and object-oriented techniques.
- Key emphasis is on fulfilling the business need, while technological or engineering excellence is of lesser importance.
- Project control involves prioritizing development and defining delivery deadlines or “timeboxes.” If the project starts to slip, emphasis is on reducing requirements to fit the timebox, not in increasing the deadline.
- Generally includes joint application design (JAD), where users are intensely involved in system design, via consensus building in either structured workshops, or electronically facilitated interaction.
- Active user involvement is imperative.
- Iteratively produces production software, as opposed to a throwaway prototype.
- Produces documentation necessary to facilitate future development and maintenance.
- Standard systems analysis and design methods can be fitted into this framework.
Agile development
“Agile software development” refers to a group of software development methodologies based on iterative development, where requirements and solutions evolve via collaboration between self-organizing cross-functional teams. The term was coined in the year 2001 when the Agile Manifesto was formulated.
Agile software development uses iterative development as a basis but advocates a lighter and more people-centric viewpoint than traditional approaches. Agile processes fundamentally incorporate iteration and the continuous feedback that it provides to successively refine and deliver a software system.
There are many agile methodologies, including:
- Dynamic systems development method (DSDM)
- Kanban
- Scrum
Code and fix
“Code and fix” development is not so much a deliberate strategy as a result of schedule pressure on software developers. Without much of a design in the way, programmers immediately begin producing code. At some point, testing begins (often late in the development cycle), and the unavoidable bugsmust then be fixed before the product can be shipped. Programming without a planned-out design is also known as cowboy coding.
Lightweight methodologies
A lightweight methodology has a small number of rules. Some of these methodologies are also considered “agile”.
- Adaptive Software Development by Jim Highsmith, described in his 1999 book Adaptive Software Development
- Crystal Clear family of methodologies with Alistair Cockburn,
- Extreme Programming (XP), promoted by people such as Kent Beck and Martin Fowler. In extreme programming, the phases are carried out in extremely small (or “continuous”) steps compared to the older, “batch” processes. The (intentionally incomplete) first pass through the steps might take a day or a week, rather than the months or years of each complete step in the Waterfall model. First, one writes automated tests, to provide concrete goals for development. Next is coding (by programmers working in pairs, a technique known as “pair programming”), which is complete when all the tests pass, and the programmers can’t think of any more tests that are needed. Design and architecture emerge from refactoring, and come after coding. The same people who do the coding do design. (Only the last feature — merging design and code — is common to all the other agile processes.) The incomplete but functional system is deployed or demonstrated for (some subset of) the users (at least one of which is on the development team). At this point, the practitioners start again on writing tests for the next most important part of the system. [10]
- Feature Driven Development (FDD) developed (1999) by Jeff De Luca and Peter Coad
- ICONIX – UML-based object modeling with use cases, a lightweight precursor to the Rational Unified Process
Other
Other high-level software project methodologies include:
- Chaos model – The main rule is always resolve the most important issue first.
- Incremental funding methodology – an iterative approach
- Structured systems analysis and design method – a specific version of waterfall
- Slow programming, as part of the larger Slow Movement, emphasizes careful and gradual work without (or minimal) time pressures. Slow programming aims to avoid bugs and overly quick release schedules.
- V-Model (software development) – an extension of the waterfall model
- Unified Process (UP) is an iterative software development methodology framework, based on Unified Modeling Language (UML). UP organizes the development of software into four phases, each consisting of one or more executable iterations of the software at that stage of development: inception, elaboration, construction, and guidelines. Many tools and products exist to facilitate UP implementation. One of the more popular versions of UP is theRational Unified Process (RUP).
Process meta-models
Some “process models” are abstract descriptions for evaluating, comparing, and improving the specific process adopted by an organization.
- ISO/IEC 12207 is the international standard describing the method to select, implement, and monitor the life cycle for software.
- The Capability Maturity Model Integration (CMMI) is one of the leading models and based on best practice. Independent assessments grade organizations on how well they follow their defined processes, not on the quality of those processes or the software produced. CMMI has replaced CMM.
- ISO 9000 describes standards for a formally organized process to manufacture a product and the methods of managing and monitoring progress. Although the standard was originally created for the manufacturing sector, ISO 9000 standards have been applied to software development as well. Like CMMI, certification with ISO 9000 does not guarantee the quality of the end result, only that formalized business processes have been followed.
- ISO/IEC 15504 Information technology — Process assessment also known as Software Process Improvement Capability Determination (SPICE), is a “framework for the assessment of software processes”. This standard is aimed at setting out a clear model for process comparison. SPICE is used much like CMMI. It models processes to manage, control, guide and monitor software development. This model is then used to measure what a development organization or project team actually does during software development. This information is analyzed to identify weaknesses and drive improvement. It also identifies strengths that can be continued or integrated into common practice for that organization or team.
- Soft systems methodology – a general method for improving management processes
- Method engineering – a general method for improving information system processes
Formal methods
Formal methods are mathematical approaches to solving software (and hardware) problems at the requirements, specification, and design levels. Formal methods are most likely to be applied to safety-critical or security-critical software and systems, such as avionics software. Software safety assurance standards, such as DO-178B, DO-178C, and Common Criteria demand formal methods at the highest levels of categorization.
For sequential software, examples of formal methods include the B-Method, the specification languages used in automated theorem proving, RAISE, and the Z notation.
Formalization of software development is creeping in, in other places, with the application of Object Constraint Language (and specializations such as Java Modeling Language) and especially with model-driven architecture allowing execution of designs, if not specifications.
For concurrent software and systems, Petri nets, process algebra, and finite state machines (which are based on automata theory – see also virtual finite state machine or event driven finite state machine) allow executable software specification and can be used to build up and validate application behavior.
Another emerging trend in software development is to write a specification in some form of logic—usually a variation of first-order logic (FOL)—and then to directly execute the logic as though it were a program. The OWL language, based on Description Logic (DL), is an example. There is also work on mapping some version of English (or another natural language) automatically to and from logic, and executing the logic directly. Examples are Attempto Controlled English, and Internet Business Logic, which do not seek to control the vocabulary or syntax. A feature of systems that support bidirectional English-logic mapping and direct execution of the logic is that they can be made to explain their results, in English, at the business or scientific level.
- Software Development Process. Provided by : Wikipedia. Located at : https://en.wikipedia.org/wiki/Software_development_process . License : CC BY: Attribution
- Waterfall-Kayak-Drop, Action-Sequence. Authored by : Leigh Anderson. Located at : https://www.flickr.com/photos/leighanderson/8514593748/ . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives


<IP_ADDRESS>: Project Managers
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Information-systems projects are notorious for going over budget and being delivered late. In many cases, a failed IT project can spell doom for a company. A project manager is responsible for keeping projects on time and on budget. This person works with the stakeholders of the project to keep the team organized and communicates the status of the project to management. A project manager does not have authority over the project team; instead, the project manager coordinates schedules and resources in order to maximize the project outcomes. A project manager must be a good communicator and an extremely organized person. A project manager should also have good people skills. Many organizations require each of their project managers to become certified as a project management professional (PMP) .


3.7: Selecting and Moving Text
Learning Objectives
- Reorganize text within a paragraph.
To select text, click and drag your cursor across the text.
The text that is highlighted in gray is selected text. Selected text can be copied, cut, or pasted. Many features and settings in Microsoft Office are applied only to text that is selected.
Moving Text
Like files and folders, there are many ways to move text once it is selected:
- Cut using Ctrl+X (Cmnd+X), move your cursor to the new location of text, then paste to the new location with Ctrl+V (Cmnd+V)
- Copy using Ctrl+C (Cmnd+C), move your cursor to the new location of text, then paste to the new location with Ctrl+V (Cmnd+V)
- Right-click selected text, select either Copy or Cut, then right-click the new location of text and select Paste
You can move text between pages, between paragraphs, or even between programs or windows. For example, you can cut text you typed in a Word document and paste it into an email, or you can copy text from the internet and paste it into a PowerPoint presentation.
In many programs, you can also drag text to a new location within the same program. For example, in Microsoft Word, you can select text, then click and drag it to a new location within the Word document.
Practice Question
Review
Contributors and Attributions
- Selecting and Moving Text. Authored by : Lumen Learning. License : CC BY: Attribution
- Word 2016 - Move, Copy, and Delete Text. Authored by : Professor Adam Morgan. Located at : https://youtu.be/f8jBx5_dz7w . License : All Rights Reserved


16.17: Potential Matrices
Prelimnaries
This is the third of the introductory sections on continuous-time Markov chains . So our starting point is a time-homogeneous Markov chain \( \bs{X} = \{X_t: t \in [0, \infty)\} \) defined on an underlying probability space \( (\Omega, \mathscr{F}, \P) \) and with discrete state space \( (S, \mathscr{S}) \). Thus \( S \) is countable and \( \mathscr{S} \) is the power set of \( S \), so every subset of \( S \) is measurable, as is every function from \( S \) into another measurable space. In addition, \( S \) is given the discret topology so that \( \mathscr{S} \) can also be thought of as the Borel \( \sigma \)-algebra. Every function from \( S \) to another topological space is continuous. Counting measure \( \# \) is the natural measure on \( (S, \mathscr{S}) \), so in the context of the general introduction, integrals over \( S \) are simply sums. Also, kernels on \( S \) can be thought of as matrices, with rows and sums indexed by \( S \), so the left and right kernel operations are generalizations of matrix multiplication. As before, let \( \mathscr{B} \) denote the collection of bounded functions \( f: S \to \R \). With the usual pointwise definitions of addition and scalar multiplication, \( \mathscr{B} \) is a vector space. The supremum norm on \( \mathscr{B} \) is given by \[ \|f\| = \sup\{\left|f(x)\right|: x \in S\}, \quad f \in \mathscr{B} \] Of course, if \( S \) is finite, \( \mathscr{B} \) is the set of all real-valued functions on \( S \), and \( \|f\| = \max\{\left|f(x)\right|: x \in S\}\) for \(f \in \mathscr{B} \). The time space is \( ([0, \infty), \mathscr{T}) \) where as usual, \( \mathscr{T} \) is the Borel \( \sigma \)-algebra on \( [0, \infty) \) corresponding to the standard Euclidean topology. Lebesgue measure is the natural measure on \( ([0, \infty), \mathscr{T}) \).
In our first point of view, we studied \( \bs{X} \) in terms of when and how the state changes. To review briefly, let \( \tau = \inf\{t \in (0, \infty): X_t \ne X_0\} \). Assuming that \( \bs{X} \) is right continuous, the Markov property of \( \bs{X} \) implies the memoryless property of \( \tau \), and hence the distribution of \( \tau \) given \( X_0 = x \) is exponential with parameter \( \lambda(x) \in [0, \infty) \) for each \( x \in S \). The assumption of right continuity rules out the pathological possibility that \( \lambda(x) = \infty \), which would mean that \( x \) is an instantaneous state so that \( \P(\tau = 0 \mid X_0 = x) = 1 \). On the other hand, if \( \lambda(x) \in (0, \infty) \) then \( x \) is a stable state , so that \( \tau \) has a proper exponential distribution given \( X_0 = x \) with \( \P(0 \lt \tau \lt \infty \mid X_0 = x) = 1 \). Finally, if \( \lambda(x) = 0 \) then \( x \) is an absorbing state , so that \( \P(\tau = \infty \mid X_0 = x) = 1 \). Next we define a sequence of stopping times: First \( \tau_0 = 0 \) and \( \tau_1 = \tau\). Recursively, if \( \tau_n \lt \infty \) then \( \tau_n = \inf\left\{t \gt \tau_n: X_t \ne X_{\tau_n}\right\} \), while if \( \tau_n = \infty \) then \( \tau_{n+1} = \infty \). With \( M = \sup\{n \in \N: \tau_n \lt \infty\} \) we define \( Y_n = X_{\tau_n} \) if \( n \in \N \) with \( n \le M \) and \( Y_n = Y_M \) if \( n \in \N \) with \( n \gt M \). The sequence \( \bs{Y} = (Y_0, Y_1, \ldots) \) is a discrete-time Markov chain on \( S \) with one-step transition matrix \( Q \) given by \(Q(x, y) = \P(X_\tau = y \mid X_0 = x)\) if \( x, \, y \in S \) with \( x \) stable, and \( Q(x, x) = 1\) if \( x \in S \) is absorbing. Assuming that \( \bs{X} \) is regular , which means that \( \tau_n \to \infty \) as \( n \to \infty \) with probability 1 (ruling out the explosion event of infinitely many transitions in finite time), the structure of \( \bs{X} \) is completely determined by the sequence of stopping times \( \bs{\tau} = (\tau_0, \tau_1, \ldots) \) and the embedded discrete-time jump chain \( \bs{Y} = (Y_0, Y_1, \ldots) \). Analytically, the distribution \( \bs{X} \) is determined by the exponential parameter function \( \lambda \) and the one-step transition matrix \( Q \) of the jump chain.
In our second point of view, we studied \( \bs{X} \) in terms of the collection of transition matrices \( \bs{P} = \{P_t: t \in [0, \infty)\} \), where for \( t \in [0, \infty) \), \[ P_t(x, y) = \P(X_t = y \mid X_0 = x), \quad (x, y) \in S^2 \] The Markov and time-homogeneous properties imply the Chapman-Kolmogorov equations \( P_s P_t = P_{s+t} \) for \( s, \, t \in [0, \infty) \), so that \( \bs{P} \) is a semigroup of transition matrices. The semigroup \( \bs{P} \), along with the initial distribution of \( X_0 \), completely determines the distribution of \( \bs{X} \). For a regular Markov chain \( \bs{X} \), the fundamental integral equation connecting the two points of view is \[ P_t(x, y) = I(x, y) e^{-\lambda(x) t} + \int_0^t \lambda(x) e^{-\lambda(x) s} Q P_{t - s} (x, y) \, ds, \quad (x, y) \in S^2 \] which is obtained by conditioning on \( \tau \) and \( X_\tau \). It then follows that the matrix function \( t \mapsto P_t \) is differentiable, with the derivative satisfying the Kolmogorov backward equation \( P_t^\prime = G P_t \) where the generator matrix \( G \) is given by \[ G(x, y) = -\lambda(x) I(x, y) + \lambda(x) Q(x, y), \quad (x, y) \in S^2 \] If the exponential parameter function \( \lambda \) is bounded, then the transition semigroup \( \bs{P} \) is uniform , which leads to stronger results. The generator \( G \) is a bounded operator on \( \mathscr{B} \), the backward equation holds as well as a companion forward equation \( P_t^\prime = P_t G \), as operators on \( \mathscr{B} \) (so with respect to the supremum norm rather than just pointwise). Finally, we can represent the transition matrix as an exponential: \( P_t = e^{t G} \) for \( t \in [0, \infty) \).
In this section, we study the Markov chain \( \bs{X} \) in terms of a family of matrices known as potential matrices . This is the least intuitive of the three points of view, but analytically one of the best approaches. Essentially, the potential matrices are transforms of the transition matrices.
Basic Theory
We assume again that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a regular Markov chain on \( S \) with transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \). Our first discussion closely parallels the general theory, except for simplifications caused by the discrete state space.
Definitions and Properties
For \( \alpha \in [0, \infty) \), the \( \alpha \)-potential matrix \( U_\alpha \) of \( \bs{X} \) is defined as follows: \[ U_\alpha(x, y) = \int_0^\infty e^{-\alpha t} P_t(x, y) \, dt, \quad (x, y) \in S^2 \]
- The special case \( U = U_0 \) is simply the potential matrix of \( \bs{X} \).
- For \( (x. y) \in S^2 \), \( U(x, y) \) is the expected amount of time that \( \bs{X} \) spends in \( y \), starting at \( x \).
- The family of matrices \( \bs{U} = \{U_\alpha: \alpha \in (0, \infty)\} \) is known as the reolvent of \( \bs{X} \).
Proof
Since \( t \mapsto P_t(x, y) \) is continuous, \( U_\alpha(x, y) \) makes sense for \( (x, y) \in S^2 \). The interpretation of \( U(x, y) \) involves an interchange of integrals: \[ U(x, y) = \int_0^\infty P_t(x, y) \, dt = \int_0^\infty \E[\bs{1}(X_t = y) \mid X_0 = x] \, dt = \E\left( \int_0^\infty \bs{1}(X_t = y) \, dt \biggm| X_0 = x\right) \] The inside integral is the Lebesgue measure of \( \{t \in [0, \infty): X_t = y\} \).
It's quite possible that \( U(x, y) = \infty \) for some \( (x, y) \in S^2 \), and knowing when this is the case is of considerable interest. If \( f: S \to \R \) and \( \alpha \ge 0 \), then giving the right operation in its many forms, \begin{align*} U_\alpha f(x) & = \sum_{y \in S} U_\alpha(x, y) f(y) = \int_0^\infty e^{-\alpha t} P_t f(x) \, dt \\ & = \int_0^\infty e^{-\alpha t} \sum_{y \in S} P_t(x, y) f(y) = \int_0^\infty e^{-\alpha t} \E[f(X_t) \mid X_0 = x] \, dt, \quad x \in S \end{align*} assuming, as always, that the sums and integrals make sense. This will be the case in particular if \( f \) is nonnegative (although \( \infty \) is a possible value), or as we will now see, if \( f \in \mathscr{B} \) and \( \alpha \gt 0 \).
If \( \alpha \gt 0 \), then \( U_\alpha(x, S) = \frac{1}{\alpha} \) for all \( x \in S \).
Proof
For \( x \in S \), \[ U_\alpha(x, S) = \int_0^\infty e^{-\alpha t} P_t(x, S) \, dt = \int_0^\infty e^{-\alpha t} dt = \frac{1}{\alpha} \]
It follows that for \( \alpha \in (0, \infty) \), the right potential operator \( U_\alpha \) is a bounded, linear operator on \( \mathscr{B} \) with \( \|U_\alpha\| = \frac{1}{\alpha} \). It also follows that \( \alpha U_\alpha \) is a probability matrix. This matrix has a nice interpretation.
If \( \alpha \gt 0 \) then \( \alpha U_\alpha (x, \cdot) \) is the conditional probability density function of \( X_T \) given \( X_0 = x \), where \( T \) is independent of \( \bs{X} \) and has the exponential distribution on \( [0, \infty) \) with parameter \( \alpha \).
Proof
Suppose that \( (x, y) \in S^2 \). The random time \( T \) has PDF \( f(t) = \alpha e^{-\alpha t} \) for \( t \in [0, \infty) \). Hence, conditioning on \( T \) gives \[ \P(X_T = y \mid X_0 = x) = \int_0^\infty \alpha e^{-\alpha t} \P(X_T = y \mid T = t, X_0 = x) \, dt \] But by the substitution rule and the assumption of independence, \[ \P(X_T = y \mid T = t, X_0 = x) = \P(X_t = y \mid T = t, X_0 = x) = \P(X_t = y \mid X_0 = x) = P_t(x, y) \] Substituting gives \[ \P(X_T = y \mid X_0 = x) = \int_0^\infty \alpha e^{-\alpha t} P_t(x, y) \, dt = \alpha U_\alpha(x, y)\]
So \( \alpha U_\alpha \) is a transition probability matrix, just as \( P_t \) is a transition probability matrix, but corresponding to the random time \( T \) (with \( \alpha \in (0, \infty) \) as a parameter), rather than the deterministic time \( t \in [0, \infty) \). The potential matrix can also be interpreted in economic terms. Suppose that we receive money at a rate of one unit per unit time whenever the process \( \bs{X} \) is in a particular state \( y \in S \). Then \( U(x, y) \) is the expected total amount of money that we receive, starting in state \( x \in S \). But money that we receive later is of less value to us now than money that we will receive sooner. Specifically, suppose that one monetary unit at time \( t \in [0, \infty) \) has a present value of \( e^{-\alpha t} \) where \( \alpha \in (0, \infty) \) is the inflation factor or discount factor . Then \( U_\alpha(x, y) \) is the total, expected, discounted amount that we receive, starting in \( x \in S \). A bit more generally, suppose that \( f \in \mathscr{B} \) and that \( f(y) \) is the reward (or cost, depending on the sign) per unit time that we receive when the process is in state \( y \in S \). Then \( U_\alpha f(x) \) is the expected, total, discounted reward, starting in state \( x \in S \).
\( \alpha U_\alpha \to I \) as \( \alpha \to \infty \).
Proof
Note first that with a change of variables \( s = \alpha t \), \[ \alpha U_\alpha = \int_0^\infty \alpha e^{-\alpha t} P_t \, dt = \int_0^\infty e^{-s} P_{s/\alpha} \, ds \] But for \( s \in [0, \infty) \), \( s / \alpha \to 0 \) and hence \( P_{s/\alpha} \to I \) as \( \alpha \to \infty \). The result then follows from the dominated convergence theorem.
If \( f: S \to [0, \infty) \), then giving the left potential operation in its various forms, \begin{align*}f U_\alpha(y) & = \sum_{x \in S} f(x) U_\alpha(x, y) = \int_0^\infty e^{-\alpha t} f P_t (y) \, dt\\ & = \int_0^\infty e^{-\alpha t} \left[\sum_{x \in S} f(x) P_t(x, y)\right] dt = \int_0^\infty e^{-\alpha t} \left[\sum_{x \in S} f(x) \P(X_t = y) \right] dt, \quad y \in S \end{align*} In particular, suppose that \( \alpha \gt 0 \) and that \( f \) is the probability density function of \( X_0 \). Then \( f P_t \) is the probability density function of \( X_t \) for \( t \in [0, \infty) \), and hence from the last result, \( \alpha f U_\alpha \) is the probability density function of \( X_T \), where again, \( T \) is independent of \( \bs{X} \) and has the exponential distribution on \( [0, \infty) \) with parameter \( \alpha \). The family of potential kernels gives the same information as the family of transition kernels.
The resolvent \(\bs{U} = \{U_\alpha: \alpha \in (0, \infty)\} \) completely determines the family of transition kernels \( \bs{P} = \{P_t: t \in (0, \infty)\} \).
Proof
Note that for \( (x, y) \in S^2 \), the function \( \alpha \mapsto U_\alpha(x, y) \) on \( (0, \infty) \) is the Laplace transform of the function \( t \mapsto P_t(x, y) \) on \( [0, \infty) \). The Laplace transform of a continuous function determines the function uniquely.
Although not as intuitive from a probability view point, the potential matrices are in some ways nicer than the transition matrices because of additional smoothness. In particular, the resolvent \( \{U_\alpha: \alpha \in [0, \infty)\} \), along with the initial distribution, completely determine the finite dimensional distributions of the Markov chain \( \bs{X} \). The potential matrices commute with the transition matrices and with each other.
Suppose that \( \alpha, \, \beta, \, t \in [0, \infty) \). Then
- \( P_t U_\alpha = U_\alpha P_t = \int_0^\infty e^{-\alpha s} P_{s+t} ds\)
- \( U_\alpha U_\beta = U_\beta U_\alpha = \int_0^\infty \int_0^\infty e^{-\alpha s} e^{-\beta t} P_{s+t} ds \, dt \)
Proof
The interchanges of matrix multiplication and integrals below are interchanges of sums and integrals, and are justified since the underlying integrands are nonnegative. The other tool used is the semigroup property of \( \bs{P} = \{P_t: t \in [0, \infty)\} \). You may want to write out the proofs explicitly to convince yourself
- First, \[ U_\alpha P_t = \left(\int_0^\infty e^{-\alpha s} P_s \, ds\right) P_t = \int_0^\infty e^{-\alpha s} P_s P_t \, ds = \int_0^\infty e^{-\alpha s} P_{s+t} \, ds \] Similarly \[ P_t U_\alpha = P_t \int_0^\infty e^{-\alpha s} P_s \, ds = \int_0^\infty e^{-\alpha s} P_t P_s \, ds = \int_0^\infty e^{-\alpha s} P_{s+t} \, ds\]
- First \[U_\alpha U_\beta = \left(\int_0^\infty e^{-\alpha s} P_s \, ds \right) \left(\int_0^\infty e^{-\beta t} P_t \, dt \right) = \int_0^\infty \int_0^\infty e^{-\alpha s} e^{-\beta t} P_s P_t \, ds \, dt = \int_0^\infty \int_0^\infty e^{-\alpha s} e^{-\beta t} P_{s+t} \, ds \, dt\] The other direction is similar.
The equations above are matrix equations, and so hold pointwise. The same identities hold for the right operators on the space \( \mathscr{B} \) under the additional restriction that \( \alpha \gt 0 \) and \( \beta \gt 0 \). The fundamental equation that relates the potential kernels, known as the resolvent equation , is given in the next theorem:
If \( \alpha, \, \beta \in [0, \infty) \) with \( \alpha \le \beta \) then \( U_\alpha = U_\beta + (\beta - \alpha) U_\alpha U_\beta \).
Proof
If \( \alpha = \beta \) the equation is trivial, so assume \( \alpha \lt \beta \). From the previous result, \[ U_\alpha U_\beta = \int_0^\infty \int_0^\infty e^{-\alpha s} e^{-\beta t} P_{s + t} \, dt \, ds \] The transformation \( u = s + t, \, v = s \) maps \( [0, \infty)^2 \) one-to-one onto \( \{(u, v) \in [0, \infty)^2: u \ge v\} \). The inverse transformation is \( s = v, \, t = u - v \) with Jacobian \( -1 \). Hence we have \begin{align*} U_\alpha U_\beta & = \int_0^\infty \int_0^u e^{-\alpha v} e^{-\beta(u - v)} P_u \, dv \, du = \int_0^\infty \left(\int_0^u e^{(\beta - \alpha) v} dv\right) e^{-\beta u} P_u \, du \\ & = \frac{1}{\beta - \alpha} \int_0^\infty \left[e^{(\beta - \alpha) u} - 1\right] e^{-\beta u} P_u du\\ & = \frac{1}{\beta - \alpha}\left(\int_0^\infty e^{-\alpha u} P_u \, du - \int_0^\infty e^{-\beta u} P_u \, du\right) = \frac{1}{\beta - \alpha}\left(U_\alpha - U_\beta \right) \end{align*} Simplifying gives the result. Note that \( U_\beta \) is finite since \( \beta \gt 0 \), so we don't have to worry about the dreaded indeterminate form \( \infty - \infty \).
The equation above is a matrix equation, and so holds pointwise. The same identity holds for the right potential operators on the space \( \mathscr{B} \), under the additional restriction that \( \alpha \gt 0 \).
Connections with the Generator
Once again, assume that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a regular Markov chain on \( S \) with transition semigroup \( \bs{P} = \{P_t: t \in [0, \infty)\} \), infinitesimal generator \( G \), resolvent \( \bs{U} = \{U_\alpha: \alpha \in (0, \infty)\} \), exponential parameter function \( \lambda \), and one-step transition matrix \( Q \) for the jump chain. There are fundamental connections between the potential \( U_\alpha \) and the generator matrix \( G \), and hence between \( U_\alpha \) and the function \( \lambda \) and the matrix \( Q \).
If \( \alpha \in (0, \infty) \) then \( I + G U_\alpha = \alpha U_\alpha \). In terms of \( \lambda \) and \( Q \), \[ U_\alpha(x, y) = \frac{1}{\alpha + \lambda(x)} I(x, y) + \frac{\lambda(x)}{\alpha + \lambda(x)} Q U_\alpha(x, y), \quad (x, y) \in S^2 \]
Proof 1
First, \[ G U_\alpha = G \int_0^\infty e^{-\alpha t} P_t \, dt = \int_0^\infty e^{-\alpha t} G P_t \, dt = \int_0^\infty e^{-\alpha t} P_t^\prime \, dt \] Passing \( G \) through the integrand is justified since \( G P_t(x, y) \) is a sum with just one negative term for \( (x, y) \in S^2 \). The second identity in the displayed equation follows from the backward equation. Integrating by parts then gives \[ G U_\alpha = e^{-\alpha t} P_t \biggm|_0^\infty + \int_0^\infty \alpha e^{-\alpha t} P_t \, dt = -I + \alpha U_\alpha \]
Proof 2
This proof use the fundamental integral equation relating \( \bs{P} \), \( \lambda \), and \( Q \) as well as the definition of \( U_\alpha \) and interchanges of integrals. The interchange is justified since the integrand is nonnegative. So for \( \alpha \in [0, \infty) \) and \( (x, y) \in S^2 \), \begin{align*} U_\alpha(x, y) & = \int_0^\infty e^{-\alpha t} P_t(x, y) \, dt \\ & = \int_0^\infty e^{-\alpha t} \left[e^{-\lambda(x) t} I(x, y) + \lambda(x) e^{-\lambda(x) t} \int_0^t e^{\lambda(x) r} Q P_r(x, y) \, dr \right] dt \\ & = I(x, y) \int_0^\infty e^{-[\alpha + \lambda(x)]t} dt + \lambda(x) \int_0^\infty \int_0^t e^{-[\alpha + \lambda(x)]t} e^{\lambda(x) r} Q P_r(x, y) \, dr \, dt \\ & = \frac{1}{\alpha + \lambda(x)} I(x, y) + \lambda(x) \int_0^\infty \int_r^\infty e^{-[\alpha + \lambda(x)]t} e^{\lambda(x) r} Q P_r(x, y) \, dt \, dr \\ & = \frac{1}{\alpha + \lambda(x)} I(x, y) + \frac{\lambda(x)}{\alpha + \lambda(x)} \int_0^\infty e^{-[\alpha + \lambda(x)]r} e^{\lambda(x) r} QP_r(x, y) \, dr \\ & = \frac{1}{\alpha + \lambda(x)} I(x, y) + \frac{\lambda(x)}{\alpha + \lambda(x)} \int_0^\infty e^{-\alpha r} Q P_r (x, y) \, dr = \frac{1}{\alpha + \lambda(x)} I(x, y) + \frac{\lambda(x)}{\alpha + \lambda(x)} Q U_\alpha(x, y) \end{align*}
Proof 3
Recall that \( \alpha U_\alpha(x, y) = \P(X_T = y \mid X_0 = x) \) where \( T \) is independent of \( \bs{X} \) and has the exponential distribution with parameter \( \alpha \). This proof works by conditioning on whether \( T \lt \tau_1 \) or \( T \ge \tau_1 \): \[\alpha U_\alpha(x, y) = \P(X_T = y \mid X_0 = x, T \lt \tau_1) \P(T \lt \tau_1 \mid X_0 = x) + \P(X_T = y \mid X_0 = x, T \ge \tau_1) \P(T \ge \tau_1 \mid X_0 = x)\] But \( X_0 = x \) and \( T \lt \tau_1 \) imply \( X_T = x \) so \( \P(X_T = y \mid X_0 = x, T \lt \tau_1) = I(x, y) \). And by a basic property of independent exponential variables that we have seen many times before, \[ \P(T \lt \tau_1 \mid X_0 = x) = \frac{\alpha}{\alpha + \lambda(x)} \] Next, for the first factor in the second term of the displayed equation, we condition on \( X_{\tau_1} \): \[ \P(X_T = y \mid X_0 = x, T \ge \tau_1) = \sum_{z \in S} \P(X_T = y \mid X_0 = x, X_{\tau_1} = z, T \ge \tau_1) \P(X_{\tau_1} = z \mid X_0 = x, T \ge \tau_1) \] But by the strong Markov property, given \( X_{\tau_1} = z \), we can restart the clock at time \( \tau_1 \) in state \( z \). Moreover, by the memoryless property and independence, the distribution of \( T - \tau_1 \) given \( T \ge \tau_1 \) is the same as the distribution of \( T \), mainly exponential with parameter \( \alpha \). It follows that \[ \P(X_T = y \mid X_0 = x, X_{\tau_1} = z, T \ge \tau_1) = \P(X_T = y \mid X_0 = z) = \alpha U_\alpha(z, y) \] Also, \( X_{\tau_1} \) is independent of \( \tau_1 \) and \( T \) so \[\P(X_{\tau_1} = z \mid X_0 = x, T \ge \tau_1) = Q(x, z)\] Finally using the basic property of exponential distributions again, \[ \P(T \ge \tau_1 \mid X_0 = x) = \frac{\lambda(x)}{\alpha + \lambda(x)} \] Putting all the pieces together we have \[ \alpha U_\alpha(x, y) = \frac{\alpha}{\alpha + \lambda(x)} I(x, y) = \frac{\lambda(x)}{\alpha + \lambda(x)} \sum_{z \in S} Q(x, z) \alpha U_\alpha(z, y) = \frac{\alpha}{\alpha + \lambda(x)} I(x, y) + \frac{\lambda(x)}{\alpha + \lambda(x)} Q \alpha U_\alpha (x, y) \]
As before, we can get stronger results if we assume that \( \lambda \) is bounded, or equivalently, the transition semigroup \( \bs{P} \) is uniform.
Suppose that \( \lambda \) is bounded and \( \alpha \in (0, \infty) \). Then as operators on \( \mathscr{B} \) (and hence also as matrices),
- \( I + G U_\alpha = \alpha U_\alpha \)
- \( I + U_\alpha G = \alpha U_\alpha \)
Proof
Since \( \lambda \) is bounded, \( G \) is a bounded operator on \( \mathscr{B} \). The proof of (a) then proceeds as before. For (b) we know from the forward and backward equations that \( G P_t = P_t G \) for \( t \in [0, \infty) \) and hence \( G U_\alpha = U_\alpha G \) for \( \alpha \in (0, \infty) \).
As matrices, the equation in (a) holds with more generality than the equation in (b), much as the Kolmogorov backward equation holds with more generality than the forward equation. Note that \[ U_\alpha G(x, y) = \sum_{z \in S} U_\alpha(x, z) G(z, y) = -\lambda(y) U_\alpha(x, y) + \sum_{z \in S} U_\alpha(x, z) \lambda(z) Q(z, y), \quad (x, y) \in S^2 \] If \( \lambda \) is unbounded, it's not clear that the second sum is finite.
Suppose that \( \lambda \) is bounded and \( \alpha \in (0, \infty) \). Then as operators on \( \mathscr{B} \) (and hence also as matrices),
- \( U_\alpha = (\alpha I - G)^{-1}\)
- \( G = \alpha I - U_\alpha^{-1}\)
Proof
- This follows immediately from the previous result, since \( U_\alpha (\alpha I - G) = I \) and \( (\alpha I - G) U_\alpha = I \)
- This follows from (a): \(\alpha I - G = U_\alpha^{-1} \) so \( G = \alpha I - U_\alpha^{-1} \)
So the potential operator \( U_\alpha \) and the generator \( G \) have a simple, elegant inverse relationship. Of course, these results hold in particular if \( S \) is finite, so that all of the various matrices really are matrices in the elementary sense.
Examples and Exercises
The Two-State Chain
Let \( \bs{X} = \{X_t: t \in [0, \infty)\} \) be the Markov chain on the set of states \( S = \{0, 1\} \), with transition rate \( a \in [0, \infty) \) from 0 to 1 and transition rate \( b \in [0, \infty) \) from 1 to 0. To avoid the trivial case with both states absorbing, we will assume that \( a + b \gt 0 \). The first two results below are a review from the previous two sections.
The generator matrix \( G \) is \[ G = \left[\begin{matrix} -a & a \\ b & -b\end{matrix}\right] \]
The transition matrix at time \( t \in [0, \infty) \) is \[ P_t = \frac{1}{a + b} \left[\begin{matrix} b & a \\ b & a \end{matrix} \right] - \frac{1}{a + b} e^{-(a + b)t} \left[\begin{matrix} -a & a \\ b & -b\end{matrix}\right], \quad t \in [0, \infty) \]
Now we can find the potential matrix in two ways.
For \( \alpha \in (0, \infty) \), show that the potential matrix \( U_\alpha \) is \[ U_\alpha = \frac{1}{\alpha (a + b)} \left[\begin{matrix} b & a \\ b & a \end{matrix}\right] - \frac{1}{(\alpha + a + b)(a + b)} \left[\begin{matrix} -a & a \\ b & -b\end{matrix}\right] \]
- From the definition.
- From the relation \( U_\alpha = (\alpha I - G)^{-1} \).
Computational Exercises
Consider the Markov chain \( \bs{X} = \{X_t: t \in [0, \infty)\} \) on \( S = \{0, 1, 2\} \) with exponential parameter function \( \lambda = (4, 1, 3) \) and jump transition matrix \[ Q = \left[\begin{matrix} 0 & \frac{1}{2} & \frac{1}{2} \\ 1 & 0 & 0 \\ \frac{1}{3} & \frac{2}{3} & 0\end{matrix}\right] \]
- Draw the state graph and classify the states.
- Find the generator matrix \( G \).
- Find the potential matrix \( U_\alpha \) for \( \alpha \in (0, \infty) \).
Answer
- The edge set is \( E = \{(0, 1), (0, 2), (1, 0), (2, 0), (2, 1)\} \). All states are stable.
- The generator matrix is \[ G = \left[\begin{matrix} -4 & 2 & 2 \\ 1 & -1 & 0 \\ 1 & 2 & -3 \end{matrix}\right] \]
- For \( \alpha \in (0, \infty) \), \[ U_\alpha = (\alpha I - G)^{-1} = \frac{1}{15 \alpha + 8 \alpha^2 + \alpha^3} \left[\begin{matrix} 3 + 4 \alpha + \alpha^2 & 10 + 2 \alpha & 2 + 2 \alpha \\ 3 + \alpha & 10 + 7 \alpha + \alpha^2 & 2 \\ 3 + \alpha & 10 + 2 \alpha & 2 + 5 \alpha + \alpha^2\end{matrix}\right] \]
Special Models
Read the discussion of potential matrices for chains subordinate to the Poisson process.


8.11.4: Regulatory Elements in Evolution
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- Ying Liu
- City College of San Francisco
Mutations can occur in both cis -elements and trans -factors; both can result in altered patterns of gene expression. If an altered pattern of gene expression results in a selective advantage (or at least do not produce a major disadvantage), they may be selected and maintained in future populations. They may even contribute to the evolution of new species. An example of a sequence change in an enhancer is found in the Pitx gene.
Example: Pitx expression in Stickleback
The three-spined stickleback (Figure \(\PageIndex{11}\)) provides an example of natural selection of a mutation in a cis -regulatory element. This fish occurs in two forms: (1) populations that inhabit deep, open water and have a spiny pelvic fin that deters larger predator fish from feeding on them; (2) populations from shallow water environments and lack this spiny pelvic fin. In shallow water, it appears that a long, spiny pelvic fin would be a disadvantage because it frequently contacts the sediment at the bottom of the pond and allows parasitic insects in the sediment to invade the stickleback. Researchers compared gene sequences of individuals from both deep and shallow water environments as shown in Figure \(\PageIndex{11}\). They observed that in embryos from the deep-water population, a gene called Pitx was expressed in several groups of cells, including those that developed into the pelvic fin. Embryos from the shallow-water population expressed Pitx in the same groups of cells as the other population, with an important exception: Pitx was not expressed in the pelvic fin primordium in the shallow-water population. Further genetic analysis showed that the absence of Pitx gene expression from the developing pelvic fin of shallow-water stickleback was due to the absence (mutation) of a particular enhancer element upstream of Pitx .
Example: Hemoglobin expression in placental mammals.
Hemoglobin is the oxygen-carrying component of red blood cells (erythrocytes). Hemoglobin usually exists as tetramers of four non-covalently bound hemoglobin molecules (Fig 12.12). Each hemoglobin molecule consists of a globin polypeptide with a covalently attached heme molecule. Heme is made through a specialized metabolic pathway and is then bound to globin polypeptide through post-translational modification .
The composition of the tetramers changes during development (Figure \(\PageIndex{13}\)). From early childhood onward, most tetramers are of the type \(\mathbf{\alpha}\) 2 \(\mathbf{\beta}\) 2 , which means they contain of two copies of each of two slightly different globin proteins named \(\alpha\) and \(\beta\). A small amount of adult hemoglobin is \(\alpha\) 2 \(\delta\) 2 , which has \(\delta\) globin instead of the more common \(\beta\) globin. Other tetrameric combinations predominate before birth: \(\zeta\) 2 \(\varepsilon\) 2 is most abundant in embryos, and \(\alpha\) 2 \(\gamma\) 2 is most abundant in fetuses. Although the six globin proteins (\(\alpha\) = alpha, \(\beta\) = beta , \(\gamma\) = gamma, \(\delta\) =delta, \(\varepsilon\) =epsilon , \(\zeta\) = zeta) are very similar to each other, they do have slightly different functional properties. For example, fetal hemoglobin has a higher oxygen affinity than adult hemoglobin, allowing the fetus to more effectively extract oxygen from maternal blood. The specialized \(\gamma\) globin genes that are characteristic of fetal hemoglobin are found only in placental mammals.
Each of these globin polypeptides is encoded by a different gene. In humans, globin genes are located in clusters on two chromosomes (Figure \(\PageIndex{14}\)). We can infer that these clusters arose through a series of duplications of an ancestral globin gene. Gene duplication events can occur through rare errors in processes such as DNA replication, meiosis, or transposition. The duplicated genes can accumulate mutations independently of each other. Mutations can occur in either the regulatory regions (e.g. promoter regions), or in the coding regions, or both. In this way, the promoters of globin genes have evolved to be expressed at different phases of development, and to produce proteins optimized for the prenatal environment.
Of course, not all mutations are beneficial: some mutations can lead to inactivation of one or more of the products of a gene duplication. This can produce what is called a pseudogene . Examples of pseudogenes (\(\psi\)) are also found in the globin clusters. Pseudogenes have mutations that prevent them from being expressed at all. The globin genes provide an example of how gene duplication and mutation, followed by selection, allows genes to evolve specialized expression patterns and functions. Many genes have evolved as gene families in this way, although they are not always clustered together as are the globins.


40.1.1: Plant Growth
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Growth in plants occurs chiefly at meristems where rapid mitosis provides new cells. As these cells differentiate, they provide new plant tissue.
Stem Growth
In stems , mitosis in the apical meristem of the shoot apex (also called the terminal bud ) produces cells that enable the stem to grow longer and, periodically, cells that will give rise to leaves. The point on the stem where leaves develop is called a node . The region between a pair of adjacent nodes is called the internode . The internodes in the terminal bud are very short so that the developing leaves grow above the apical meristem that produced them and thus protect it. New meristems, the lateral buds , develop at the nodes, each just above the point where a leaf is attached. When the lateral buds develop, they produces new stem tissue, and thus branches are formed.
Under special circumstances (such as changes in photoperiod), the apical meristem is converted into a flower bud . This develops into a flower. The conversion of the apical meristem to a flower bud "uses up" the meristem so that no further growth of the stem can occur at that point. However, lateral buds behind the flower can develop into branches.
The drawing is of a typical woody dicot, the horse chestnut, as seen during the dormant season. The leaves have dropped off, leaving a leaf scar ; the dots inside each leaf scar show where the vascular bundles (xylem and phloem) had entered the petiole of the leaf. A flower had been produced the season before, so that during the season just ended two branches had grown out on either side of the flower bud scar . Lenticels are openings that allow oxygen and carbon dioxide to diffuse between the living cells of the stem and the air.
The growth of roots in described in a separate page,


12.23: War
Famine and disease were not the only disasters to strike late medieval Europe. The fourteenth century also saw an increase in both civil wars and wars between states. The Holy Roman Empire saw nearly a decade of civil war (1314 – 1326) between rival emperors and, because of the close relations of their kings, Sweden, Denmark, and Norway experienced frequent combinations of civil and interstate war until the 1397 Union of Kalmar brought the three together under one crown.
The longest-running of these wars was between England and France, the so-called Hundred Years’ War (1337 – 1453). In 1328, the French king Charles IV died without a direct heir. England’s king, Edward III (r. 1327 – 1377), related to the French royal family, claimed to be rightful heir to the crown of France. The resulting war would last over a century, although it was broken by frequent, lengthy truces. Although France had many more people than England, the kingdom of England was often able to defeat it. The main reason was that the English kings made increasing use of trained, disciplined infantry armies. Horses are effective in battle against raiders or other horsemen. A horse, however, is less effective when an infantry formation is able to present a solid front against the horses and use missile weapons on those horses before they can close with their enemy. Using a combination of archers and infantry, the English were able to inflict severe defeats on the French at both Crécy (26 August 1346) and Poitiers (19 September 1356).
The war was particularly hard on the civilians of the French countryside: the method of waging war of a pre-modern army often involved invading enemy territory and burning crops, looting villages, and murdering civilians. French peasants, who had suffered first from the Plague and then from war, rose in rebellion in 1358, but this rebellion was ruthlessly crushed, with the peasants slaughtered and leaders brutally executed.
The Hundred Years’ War would spill over into Spain, which itself was suffering from a vicious war between Castile and Aragon that eventually caused a Castilian civil war, with both French and English intervening.
The wars of the fourteenth and especially fifteenth century saw not only an increasing use of trained, professional armies, but also the employment of gunpowder weapons, invented in Song China and first seen in Europe in the early 1300s. At first, firearms were limited to heavy, cumbersome artillery pieces that were deployed from fixed points. Their use on the battlefield and in sieges was limited, although by the fifteenth century, cannons could blast open the gates of most existing fortifications. By the mid-1400s, the harquebus, a man-transportable firearm, appeared on the battlefield in Spain, bringing gunpowder to the individual infantryman.


3.3: Chapter Review
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Solve the following linear programming problems by the graphical method.
1) Mr. Shoemacher has $20,000 to invest in two types of mutual funds: a High-Yield Fund and an Equity Fund. The High-Yield fund has an annual yield of 12%, while the Equity fund earns 8%. He would like to invest at least $3000 in the High-Yield fund and at least $4000 in the Equity fund. How much should he invest in each to maximize his annual yield, and what is the maximum yield?
2) Dr. Lum teaches part-time at two community colleges, Hilltop College and Serra College. Dr. Lum can teach up to 5 classes per semester. For every class he teaches at Hilltop College, he needs to spend 3 hours per week preparing lessons and grading papers. For each class at Serra College, he must do 4 hours of work per week. He has determined that he cannot spend more than 18 hours per week preparing lessons and grading papers. If he earns $6,000 per class at Hilltop College and $7,500 per class at Serra College, how many classes should he teach at each college to maximize his income, and what will be his income?
3) Mr. Shamir employs two part-time typists, Inna and Jim, for his typing needs. Inna charges $15 an hour and can type 6 pages an hour, while Jim charges $18 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each typist to minimize his typing costs, and what will be the total cost?
4) Mr. Boutros wants to invest up to $20,000 in two stocks, Cal Computers and Texas Tools. The Cal Computers stock is expected to yield a 16% annual return, while the Texas Tools stock promises a 12% yield. Mr. Boutros would like to earn at least $2,880 this year. According to Value Line Magazine's safety index (1 highest to 5 lowest), Cal Computers has a safety number of 3 and Texas Tools has a safety number of 2. How much money should he invest in each to minimize the safety number? Note: A lower safety number means less risk.
5) A store sells two types of copy machines: compact (low capacity) and standard (which takes more space). The store can sell up to 90 copiers a month. A maximum of 1080 cubic feet of storage space is available. A compact copier requires 6 cu. ft. of storage space, and a standard copier requires 18 cu. ft.. The compact and standard copy machines take, respectively, 1 and 1.5 sales hours of labor.
A maximum of 99 hours of labor is available. The profit from each of these copiers is $60 and $80, respectively, how many of each type should be sold to maximize profit, and what is the maximum profit?
6) A company manufactures two types of cell phones, a Basic model and a Pro model. The Basic model generates a profit of $100 per phone and the Pro model has a profit of $150 per phone. On the assembly line the Basic phone requires 7 hours, while the Pro model takes 11 hours. The Basic phone requires one hour and the Pro phone needs 3 hours for finishing, which includes loading software. Both phones require one hour for testing. On a particular production run the company has available 1,540 work hours on the assembly line, 360 work hours for finishing, and 200 work hours in the testing department. How many cell phones of each type should be produced to maximize profit, and what is that maximum profit?
7) John wishes to choose a combination of two types of cereals for breakfast - Cereal A and Cereal B. A small box (one serving) of Cereal A costs $0.50 and contains 10 units of vitamins, 5 units of minerals, and 15 calories. A small box(one serving) of Cereal B costs $0.40 and contains 5 units of vitamins, 10 units of minerals, and 15 calories. John wants to buy enough boxes to have at least 500 units of vitamins, 600 units of minerals, and 1200 calories. How many boxes of each food should he buy to minimize his cost, and what is the minimum cost?
8) Jessica needs at least 60 units of vitamin A, 40 units of vitamin B, and 140 units of vitamin C each week. She can choose between Costless brand or Savemore brand tablets. A Costless tablet costs 5 cents and contains 3 units of vitamin A, 1 unit of vitamin B, and 2 units of vitamin C. A Savemore tablet costs 7 cents and contains 1 unit of A, 1 of B, and 5 of C. How many tablets of each kind should she buy to minimize cost, and what is the minimum cost?
9) A small company manufactures two products: A and B. Each product requires three operations: Assembly, Finishing and Testing. Product A requires 1 hour of Assembly, 3 hours of Finishing, and 1 hour of Testing. Product B requires 3 hours of Assembly, 1 hour of Finishing, and 1 hour of Testing. The total work-hours available per week in the Assembly division is 60, in Finishing is 60, and in Testing is 24. Each item of product A has a profit of $50, and each item of Product B has a profit of $75. How many of each should be made to maximize profit? What is the maximum profit?
10) A factory manufactures two products, A and B. Each product requires the use of three machines, Machine I, Machine II, and Machine III. The time requirements and total hours available on each machine are listed below.
| Machine I | Machine II | Machine III | |
|---|---|---|---|
| Product A | 1 | 2 | 4 |
| Product B | 2 | 2 | 2 |
| Total hours | 70 | 90 | 160 |
If product A generates a profit of $60 per unit and product B a profit of $50 per unit, how many units of each product should be manufactured to maximize profit, and what is the maximum profit?
11) A company produces three types of shoes, formal, casual, and athletic, at its two factories, Factory I and Factory II. The company must produce at least 6000 pairs of formal shoes, 8000 pairs of casual shoes, and 9000 pairs of athletic shoes. Daily production of each factory for each type of shoe is:
| Factory I | Factory II | |
|---|---|---|
| Formal | 100 | 100 |
| Casual | 100 | 200 |
| Athletic | 300 | 100 |
Operating Factory I costs $1500 per day and it costs $2000 per day to operate Factory II. How many days should each factory operate to complete the order at a minimum cost, and what is the minimum cost?
12) A professor gives two types of quizzes, objective and recall. He plans to give at least 15 quizzes this quarter. The student preparation time for an objective quiz is 15 minutes and for a recall quiz 30 minutes. The professor would like a student to spend at least 5 hours (300 minutes) preparing for these quizzes above and beyond the normal study time. The average score on an objective quiz is 7, and on a recall type 5, and the professor would like the students to score at least 85 points on all quizzes. It takes the professor one minute to grade an objective quiz, and 1.5 minutes to grade a recall type quiz. How many of each type should he give in order to minimize his grading time?
13) A company makes two mixtures of nuts: Mixture A and Mixture B. Mixture A contains 30% peanuts, 30% almonds and 40% cashews and sells for $5 per pound. Mixture B contains 30% peanuts, 60% almonds and 10% cashews and sells for $3 a pound. The company has 540 pounds of peanuts, 900 pounds of almonds, 480 pounds of cashews. How many pounds of each of mixtures A and B should the company make to maximize profit, and what is the maximum profit?


6.2: The Nature of Light
In this chapter, we study the basic properties of light. In the next few chapters, we investigate the behavior of light when it interacts with optical devices such as mirrors, lenses, and apertures.
-
- 6.2.1: The Propagation of Light
- The index of refraction of a material is \(n = \frac{c}{v}\), where v is the speed of light in a material and c is the speed of light in a vacuum. The ray model of light describes the path of light as straight lines. The part of optics dealing with the ray aspect of light is called geometric optics. Light can travel in three ways from a source to another location: (1) directly from the source through empty space; (2) through various media; and (3) after being reflected from a mirror.
-
- 6.2.7: Polarization
- Polarization is the attribute that wave oscillations have a definite direction relative to the direction of propagation of the wave. The direction of polarization is defined to be the direction parallel to the electric field of the EM wave. Unpolarized light is composed of many rays having random polarization directions. Unpolarized light can be polarized by passing it through a polarizing filter or other polarizing material. The process of polarizing light decreases its intensity by a factor of
Thumbnail: An EM wave, such as light, is a transverse wave. The electric \(\overrightarrow{E}\) and magnetic \(\overrightarrow{B}\) fields are perpendicular to the direction of propagation. The direction of polarization of the wave is the direction of the electric field.


2.2E: Use a General Strategy to Solve Linear Equations (Exercises)
Practice Makes Perfect
Solve Equations Using the General Strategy
In the following exercises, determine whether the given values are solutions to the equation.
1. \(6y+10=12y\)
a. \(y=\frac{5}{3}\)
b. \(y=−\frac{1}{2}\)
- Answer
-
a. yes
b. no
2. \(4x+9=8x\)
a. \(x=−\frac{7}{8}\)
b. \(x=\frac{9}{4}\)
3. \(8u−1=6u\)
a. \(u=−\frac{1}{2}\)
b. \(u=\frac{1}{2}\)
- Answer
-
a. no
b. yes
4. \(9v−2=3v\)
a. \(v=−\frac{1}{3}\)
b. \(v=\frac{1}{3}\)
In the following exercises, solve each linear equation.
5. \(15(y−9)=−60\)
- Answer
-
\(y=5\)
6. \(−16(3n+4)=32\)
7. \(−(w−12)=30\)
- Answer
-
\(w=−18\)
8. \(−(t−19)=28\)
9. \(51+5(4−q)=56\)
- Answer
-
\(q=3\)
10. \(−6+6(5−k)=15\)
11. \(3(10−2x)+54=0\)
- Answer
-
\(x=14\)
12. \(−2(11−7x)+54=4\)
13 . \(\frac{2}{3}(9c−3)=22\)
- Answer
-
\(c=4\)
14. \(\frac{3}{5}(10x−5)=27\)
15. \(\frac{1}{5}(15c+10)=c+7\)
- Answer
-
\(c=\frac{5}{2}\)
16. \(\frac{1}{4}(20d+12)=d+7\)
17. \(3(4n−1)−2=8n+3\)
- Answer
-
\(n=2\)
18. \(9(2m−3)−8=4m+7\)
19. \(12+2(5−3y)=−9(y−1)−2\)
- Answer
-
\(y=−5\)
20. \(−15+4(2−5y)=−7(y−4)+4\)
21. \(5+6(3s−5)=−3+2(8s−1)\)
- Answer
-
\(s=10\)
22. \(−12+8(x−5)=−4+3(5x−2)\)
23 . \(4(p−4)−(p+7)=5(p−3)\)
- Answer
-
\(p=−4\)
24. \(3(a−2)−(a+6)=4(a−1)\)
25. \(4[5−8(4c−3)]=12(1−13c)−8\)
- Answer
-
\(c=−4\)
26. \(5[9−2(6d−1)]=11(4−10d)−139\)
27. \(3[−9+8(4h−3)]=2(5−12h)−19\)
- Answer
-
\(h=\frac{3}{4}\)
28. \(3[−14+2(15k−6)]=8(3−5k)−24\)
29. \(5[2(m+4)+8(m−7)]=2[3(5+m)−(21−3m)]\)
- Answer
-
\(m=6\)
30. \(10[5(n+1)+4(n−1)]=11[7(5+n)−(25−3n)]\)
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.
31. \(23z+19=3(5z−9)+8z+46\)
- Answer
-
identity; all real numbers
32. \(15y+32=2(10y−7)−5y+46\)
33. \(18(5j−1)+29=47\)
- Answer
-
conditional equation; \(j=\frac{2}{5}\)
34. \(24(3d−4)+100=52\)
35. \(22(3m−4)=8(2m+9)\)
- Answer
-
conditional equation; \(m=165\)
36. \(30(2n−1)=5(10n+8)\)
37. \(7v+42=11(3v+8)−2(13v−1)\)
- Answer
-
contradiction; no solution
38. \(18u−51=9(4u+5)−6(3u−10)\)
39. \(45(3y−2)=9(15y−6)\)
- Answer
-
contradiction; no solution
40. \(60(2x−1)=15(8x+5)\)
41. \(9(14d+9)+4d=13(10d+6)+3\)
- Answer
-
identity; all real numbers
42. \(11(8c+5)−8c=2(40c+25)+5\)
Solve Equations with Fraction or Decimal Coefficients
In the following exercises, solve each equation with fraction coefficients.
43. \(\frac{1}{4}x−\frac{1}{2}=−\frac{3}{4}\)
- Answer
-
\(x=−1\)
44. \(\frac{3}{4}x−\frac{1}{2}=\frac{1}{4}\)
45. \(\frac{5}{6}y−\frac{2}{3}=−\frac{3}{2}\)
- Answer
-
\(y=−1\)
46. \(\frac{5}{6}y−\frac{1}{3}=−\frac{7}{6}\)
47. \(\frac{1}{2}a+\frac{3}{8}=\frac{3}{4}\)
- Answer
-
\(a=\frac{3}{4}\)
48. \(\frac{5}{8}b+\frac{1}{2}=−\frac{3}{4}\)
49. \(2=\frac{1}{3}x−\frac{1}{2}x+\frac{2}{3}x\)
- Answer
-
\(x=4\)
50. \(2=\frac{3}{5}x−\frac{1}{3}x+\frac{2}{5}x\)
51. \(\frac{1}{3}w+\frac{5}{4}=w−\frac{1}{4}\)
- Answer
-
\(w=\frac{9}{4}\)
52. \(\frac{1}{2}a−\frac{1}{4}=\frac{1}{6}a+\frac{1}{12}\)
53. \(\frac{1}{3}b+\frac{1}{5}=\frac{2}{5}b−\frac{3}{5}\)
- Answer
-
\(b=12\)
54. \(\frac{1}{3}x+\frac{2}{5}=\frac{1}{5}x−\frac{2}{5}\)
55. \(\frac{1}{4}(p−7)=\frac{1}{3}(p+5)\)
- Answer
-
\(p=−41\)
56. \(\frac{1}{5}(q+3)=\frac{1}{2}(q−3)\)
57. \(\frac{1}{2}(x+4)=\frac{3}{4}\)
- Answer
-
\(x=−\frac{5}{2}\)
58. \(\frac{1}{3}(x+5)=\frac{5}{6}\)
59. \(\dfrac{4n+8}{4}=\dfrac{n}{3}\)
- Answer
-
\(n=−3\)
60. \(\dfrac{3p+6}{3}=\dfrac{p}{2}\)
61. \(\dfrac{3x+4}{2}+1=\dfrac{5x+10}{8}\)
- Answer
-
\(x=−2\)
62. \(\dfrac{10y−2}{3}+3=\dfrac{10y+1}{9}\)
63. \(\dfrac{7u−1}{4}−1=\dfrac{4u+8}{5}\)
- Answer
-
\(u=3\)
64. \(\dfrac{3v−6}{2}+5=\dfrac{11v−4}{5}\)
In the following exercises, solve each equation with decimal coefficients.
65. \(0.4x+0.6=0.5x−1.2\)
- Answer
-
\(x=18\)
66. \(0.7x+0.4=0.6x+2.4\)
67. \(0.9x−1.25=0.75x+1.75\)
- Answer
-
\(x=20\)
68. \(1.2x−0.91=0.8x+2.29\)
69. \(0.05n+0.10(n+8)=2.15\)
- Answer
-
\(n=9\)
70. \(0.05n+0.10(n+7)=3.55\)
71. \(0.10d+0.25(d+5)=4.05\)
- Answer
-
\(d=8\)
72. \(0.10d+0.25(d+7)=5.25\)
Everyday Math
73 . Fencing Micah has 74 feet of fencing to make a dog run in his yard. He wants the length to be 2.5 feet more than the width. Find the length, L , by solving the equation \(2L+2(L−2.5)=74\).
- Answer
-
\(L=19.75\) feet
74 . Stamps Paula bought $22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was eight less than the number of 49-cent stamps. Solve the equation \(0.49s+0.21 (s−8) =22.82\) for s , to find the number of 49-cent stamps Paula bought.
Writing Exercises
75. Using your own words, list the steps in the general strategy for solving linear equations.
- Answer
-
Answers will vary.
76. Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.
77. What is the first step you take when solving the equation \(3−7(y−4)=38?\) Why is this your first step?
- Answer
-
Answers will vary.
78. If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?
79. If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?
- Answer
-
Answers will vary.
80. For the equation \(0.35x+2.1=3.85\), how do you clear the decimal?
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


6.1: Choosing a Topic
Questions to Consider
When it comes to academic research, choosing a topic is often the hardest part of the process! In many instances, college instructors will provide guidelines or a broad topic for the assignment, but it’s your job to choose a narrower topic within those guidelines. Here are some questions to ask yourself before you decide on a topic to research:
- Have I been assigned a specific topic or can I choose my own?
- Is there a particular current event that interests me or an issue I feel strongly about? Can I select a topic that is closely related to my college major?
- Did the instructor specify what types of information sources I need to use?
Narrowing Your Topic
It is important to select a topic that not only interests you or sparks your curiosity, but also one that is neither too broad or too narrow in scope. For example, if you are interested in climate change and wanted to choose that as a topic for your research paper, you would quickly find yourself overwhelmed with search results and information. You may find millions of search results about all different aspects of climate change, like policy, history, and the environment, and feel lost about where to begin.
An important step in the research process is narrowing down your larger topic into a smaller, more researchable topic. What is it about climate change that is most interesting to you? Is there something in particular that you’ve learned in class that relates to this topic that you could explore with your research? Watch the following video [3:10] to see how to recognize if your topic is too broad or too specific, and to learn some tips on what to do:
Note: Turn on closed captions with the “CC” button or use the text transcript if you prefer to read.
More Strategies
If you are having trouble selecting a topic or narrowing down your topic, here are some strategies that can help:
- Scan your course textbook for ideas
- Browse some magazines or newspapers for current events
- Dive into Wikipedia, or another encyclopedia, and see what sparks your interest
- Talk to your instructor, a librarian, or classmates, for ideas and guidance
- Browse the “issues” and “topic” pages in library databases that include reference sources, like CQ Researcher or Opposing Viewpoints. These topic pages are similar to Wikipedia and give general background information on a topic and link to more specific information, which is helpful when narrowing down your topic. Some library databases have great features for getting your research started, like pro/con viewpoints, background information, timelines, and bibliographies for further reading. If you are unfamiliar with what library databases your college has a subscription to, ask a librarian for help!
Sources
Image: “ Narrowing a Topic ” by Teaching & Learning, Ohio State University Libraries is licensed under CC BY 4.0
“ Picking Your Topic IS Research ” by NC State University Libraries is licensed under CC BY-NC-SA 3.0 US


6.5: Chapter 9- Of their voyage, and how they passed the sea; and of their safe arrival at Cape Cod
The 9. Chap.
Of their vioage, & how they passed y e sea, and of their safe arrivall at Cape Codd.
Sept r : 6. These troubls being blowne over, and now all being compacte togeather in one shipe, Home [AE] they put to sea againe with a prosperus winde, which continued diverce days togeather, which was some incouragmente unto them; yet according to y e usuall maner many were afflicted with sea-sicknes. And I may not omite hear a spetiall worke of Gods providence. Ther was a proud & very profane yonge man, one of y e Home [91] sea-men, of a lustie, able body, which made him the more hauty; he would allway be contemning y e poore people in their sicknes, & cursing them dayly with greēous execrations, and did not let to tell them, that he hoped to help to cast halfe of them over board before they came to their jurneys end, and to make mery with what they had; and if he were by any gently reproved, he would curse and swear most bitterly. But it plased God before they came halfe seas over, to smite this yong man with a greeveous disease, of which he dyed in a desperate maner, and so was him selfe y e first y t was throwne overbord. Thus his curses light on his owne head; and it was an astonishmente to all his fellows, for they noted it to be y e just hand of God upon him.
After they had injoyed faire winds and weather for a season, they were incountred many times with crosse winds, and mette with many feirce stormes, with which y e shipe was shroudly shaken, and her upper works made very leakie; and one of the maine beames in y e midd ships was bowed & craked, which put them in some fear that y e shipe could not be able to performe y e vioage. So some of y e cheefe of y e company, perceiveing y e mariners to feare y e suffisiencie of y e shipe, as appeared by their mutterings, they entred into serious consulltation with y e m r . & other officers of y e ship, to consider in time of y e danger; and rather to returne then to cast them selves into a Home [92] desperate & inevitable perill. And truly ther was great distraction & differance of opinion amongst y e mariners them selves; faine would they doe what could be done for their wages sake, (being now halfe the seas over,) and on y e other hand they were loath to hazard their lives too desperatly. But in examening of all opinions, the m r . & others affirmed they knew y e ship to be stronge & firme under water; and for the buckling of y e maine beame, ther was a great iron scrue y e passengers brought out of Holland, which would raise y e beame into his place; y e which being done, the carpenter & m r . affirmed that with a post put under it, set firme in y e lower deck, & otherways bounde, he would make it sufficiente. And as for y e decks & uper workes they would calke them as well as they could, and though with y e workeing of y e ship they [46] would not longe keepe stanch, yet ther would otherwise be no great danger, if they did not overpress her with sails. So they com̅ited them selves to y e will of God, & resolved to proseede. In sundrie of these stormes the winds were so feirce, & y e seas so high, as they could not beare a knote of saile, but were forced to hull, for diverce days togither. And in one of them, as they thus lay at hull, in a mighty storme, a lustie yonge man (called John Howland) coming upon some occasion above y e grattings, was, with a seele of the shipe throwne into [y e ] sea; but it pleased God y t he caught hould of y e top-saile Home [93] halliards, which hunge over board, & rane out at length; yet he held his hould (though he was sundrie fadomes under water) till he was hald up by y e same rope to y e brime of y e water, and then with a boat hooke & other means got into y e shipe againe, & his life saved; and though he was something ill with it, yet he lived many years after, and became a profitable member both in church & com̅one wealthe. In all this viage ther died but one of y e passengers, which was William Butten, a youth, servant to Samuell Fuller, when they drew near y e coast. But to omite other things, (that I may be breefe,) after longe beating at sea they fell with that land which is called Cape Cod; the which being made & certainly knowne to be it, they were not a litle joyfull. After some deliberation had amongst them selves & with y e m r . of y e ship, they tacked aboute and resolved to stande for y e southward (y e wind & weather being faire) to finde some place aboute Hudsons river for their habitation. But after they had sailed y t course aboute halfe y e day, they fell amongst deangerous shoulds and roring breakers, and they were so farr intangled ther with as they conceived them selves in great danger; & y e wind shrinking upon them withall, they resolved to bear up againe for the Cape, and thought them selves hapy to gett out of those dangers before night overtooke them, as by Gods providence they did. And y e next day they gott into y e Cape-harbor wher they ridd in Home [94] saftie. A word or too by y e way of this cape; it was thus first named by Capten Gosnole & his company, Home [AF] Anno: 1602, and after by Capten Smith was caled Cape James; but it retains y e former name amongst seamen. Also y t pointe which first shewed those dangerous shoulds unto them, they called Pointe Care, & Tuckers Terrour; but y e French & Dutch to this day call it Malabarr, by reason of those perilous shoulds, and y e losses they have suffered their.
Being thus arived in a good harbor and brought safe to land, they fell upon their knees & blessed y e God of heaven, who had brought them over y e vast & furious ocean, and delivered them from all y e periles & miseries therof, againe to set their feete on y e firme and stable earth, their proper elemente. And no marvell if they were thus joyefull, seeing wise Seneca was so affected with sailing a few miles on y e coast of his owne Italy; as he affirmed, Home [AG] that he had rather remaine twentie years on his way by land, then pass by sea to any place in a short time; so tedious & dreadfull was y e same unto him.
But hear I cannot but stay and make a pause, and stand half amased at this poore peoples presente condition; and so I thinke will the reader too, when he well considers [47] y e same. Being thus passed y e vast ocean, and a sea of troubles before in their preparation (as may be remembred by y t which wente Home [95] before), they had now no freinds to wellcome them, nor inns to entertaine or refresh their weatherbeaten bodys, no houses or much less townes to repaire too, to seeke for succoure. It is recorded in scripture Home [AH] as a mercie to y e apostle & his shipwraked company, y t the barbarians shewed them no smale kindnes in refreshing them, but these savage barbarians, when they mette with them (as after will appeare) were readier to fill their sids full of arrows then otherwise. And for y e season it was winter, and they that know y e winters of y t cuntrie know them to be sharp & violent, & subjecte to cruell & feirce stormes, deangerous to travill to known places, much more to serch an unknown coast. Besids, what could they see but a hidious & desolate wildernes, full of wild beasts & willd men? and what multituds ther might be of them they knew not. Nether could they, as it were, goe up to y e tope of Pisgah, to vew from this willdernes a more goodly cuntrie to feed their hops; for which way soever they turnd their eys (save upward to y e heavens) they could have litle solace or content in respecte of any outward objects. For sum̅er being done, all things stand upon them with a wetherbeaten face; and y e whole countrie, full of woods & thickets, represented a wild & savage heiw. If they looked behind them, ther was y e mighty ocean which they had passed, and was now as a Home [96] maine barr & goulfe to seperate them from all y e civill parts of y e world. If it be said they had a ship to sucour them, it is trew; but what heard they daly from y e m r . & company? but y t with speede they should looke out a place with their shallop, wher they would be at some near distance; for y e season was shuch as he would not stirr from thence till a safe harbor was discovered by them wher they would be, and he might goe without danger; and that victells consumed apace, but he must & would keepe sufficient for them selves & their returne. Yea, it was muttered by some, that if they gott not a place in time, they would turne them & their goods ashore & leave them. Let it also be considred what weake hopes of supply & succoure they left behinde them, y t might bear up their minds in this sade condition and trialls they were under; and they could not but be very smale. It is true, indeed, y e affections & love of their brethren at Leyden was cordiall & entire towards them, but they had litle power to help them, or them selves; and how y e case stode betweene them & y e marchants at their coming away, hath allready been declared. What could now sustaine them but the spirite of God & his grace? May not & ought not the children of these fathers rightly say: Our faithers were Englishmen which came over this great Home [97] ocean, and were ready to perish in this willdernes; Home [AI] but they cried unto y e Lord, and he heard their voyce, and looked on their adversitie, &c. Let them therfore praise y e Lord, because he is good, & his mercies endure for ever. Home [AJ] Yea, let them which have been redeemed of y e Lord, shew how he hath delivered them from y e hand of y e oppressour. When they wandered in y e deserte willdernes out of y e way, and found no citie to dwell in, both hungrie, & thirstie, their sowle was overwhelmed in them. Let them confess before y e Lord his loving kindnes, and his wonderfull works before y e sons of men.


8.1: Why It Matters- Entrepreneurship
Why discuss the role of entrepreneurship in small business?
Figure \(\PageIndex{1}\). What do all of these items have in common? They all represent the efforts of entrepreneurs.
The American psyche often equates size with success, applying a “get big or get out” standard to industries ranging from agriculture and energy to social enterprise. Indeed, that phrase (decades before a variation became associated with living large) was the mantra of Secretary of Agriculture Earl Butz, whose policy changes supported the growth of major agribusiness corporations at the expense of the small family farm. And yet if American history has taught us anything, it’s that we are in charge of our destiny and our definition of success. And that success can be achieved at any size and from any number of incubators (kitchen, garage, dorm room) and ad hoc office spaces.
Large financial institutions and corporations are often considered to be the engines of our economy, but big business is only—literally—half of the story. While the majority of small businesses don’t operate at scale (roughly 70 percent have less than 100 employees) their cumulative economic impact is significant. Through inventions and innovations, entrepreneurs also shape how and how well we communicate, learn, live and experience life. Entrepreneurship matters not only because of the employment and productivity impact, but because small businesses contribute to the resilience of their communities and the nation and, by extension, to stability globally.
In this module, we’ll define small businesses and entrepreneurship, provide perspective on the opportunities and the risks, discuss common motivations and traits and finally the steps involved in developing a business plan.


2.1: Herstory The Emergence of Women Artists in European Art (500 CE CE)
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- 2.1.1: Introduction (1400 - 1600)
- Rinascita, Italian for "rebirth," was a period marking the end of Europe's feudal systems and entering a new form of a cultural and political society built on commerce. The shift from the Middle Ages in Europe to the Renaissance was a revival act of the classical styles of Greek and Roman art, highlighting humanists' progression.
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- 2.1.2: European Medieval Art (500 CE - 1400 CE)
- The Medieval, also known as the Middle Ages period in Europe, was bounded by the fall of the Roman Empire. It leads to the beginning of the Renaissance art period in Europe. At this time, Christianity was established as the primary religion bringing specific ideals and ecclesiastical standards. The fragments from the Roman Empire defined the boundaries of many European countries.
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- 2.1.3: Renaissance Art (1400-1600)
- Economic wealth through trade routes around the world brought money into Europe and the coffers of the church and wealthy families. Artists flourished, workshops and educational opportunities grew, and artists were in demand. Many male artists during the Renaissance received monetary support from families like the Medici and the Catholic Church.
Thumbnail: Diva 45 (Courtesy of the artist, Jylian Gustlin)


6.10: Bibliography
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