4.3: Developing Ideas, Innovations, and Inventions
By the end of this section, you will be able to:
- Describe and apply the five stages of creativity
- Discuss innovation as a system for problem solving and much more
- Outline the sequence of steps in developing an invention
The previous section defined creativity, innovation, and invention, and provided examples. You might think of creativity as raw; innovation as transforming creativity into a functional purpose, often meant to eradicate a pain point or to fulfill a need; and invention as a creation that leaves a lasting impact. In this section, you will learn about processes designed to help you apply knowledge from the previous section.
The Creative Process: The Five Stages of Creativity
Raw creativity and an affinity for lateral thinking may be innate, but creative people must refine these skills in order to become masters in their respective fields. They practice in order to apply their skills readily and consistently, and to integrate them with other thought processes and emotions. Anyone can improve in creative efforts with practice. For our purposes, practice is a model for applied creativity that is derived from an entrepreneurial approach ( Figure 4.11 ). 35 It requires:
- Preparation
- Incubation
- Insight
- Evaluation
- Elaboration
Preparation
Preparation involves investigating a chosen field of interest, opening your mind, and becoming immersed in materials, mindset, and meaning. If you have ever tried to produce something creative without first absorbing relevant information and observing skilled practitioners at work, then you understand how difficult it is. This base of knowledge and experience mixed with an ability to integrate new thoughts and practices can help you sift through the ideas quicker. However, relying too heavily on prior knowledge can restrict the creative process. When you immerse yourself in a creative practice, you make use of the products or the materials of others’ creativity. For example, a video-game designer plays different types of video games on different consoles, computers, and online in networks. She or he may play alone, with friends in collaboration, or in competition. Consuming the products in a field gives you a sense of what is possible and indicates boundaries that you may attempt to push with your own creative work. Preparation broadens your mind and lets you study the products, practice, and culture in a field. It is also a time for goal setting. Whether your chosen field is directly related to art and design, such as publishing, or involves human-centric design, which includes all sorts of software and product design efforts, you need a period of open-minded reception to ideas. Repetitive practice is also part of the preparation stage, so that you can understand the current field of production and become aware of best practices, whether or not you are currently capable of matching them. During the preparation stage, you can begin to see how other creative people put meaning into their products, and you can establish benchmarks against which to measure your own creative work.
Incubation
Incubation refers to giving yourself, and your subconscious mind in particular, time to incorporate what you learned and practiced in the preparation stage. Incubation involves the absence of practice. It may look to an outsider as though you are at rest, but your mind is at work. A change of environment is key to incubating ideas. 37 A new environment allows you to receive stimuli other than those directly associated with the creative problem you are working on. It could be as simple as taking a walk or going to a new coffee shop to allow your mind to wander and take in the information you gathered in the previous stage. Mozart stated, “When I am, as it were, completely myself, entirely alone, and of good cheer—say, traveling in a carriage, or walking after a good meal, or during the night when I cannot sleep; it is on such occasions that my ideas flow best and most abundantly.” 38 Incubation allows your mind to integrate your creative problem with your stored memories and with other thoughts or emotions you might have. This simply is not possible to do when you are consciously fixated on the creative problem and related tasks and practice.
Incubation can take a short or a long time, and you can perform other activities while allowing this process to take place. One theory about incubation is that it takes language out of the thought process. If you are not working to apply words to your creative problems and interests, you can free your mind to make associations that go deeper, so to speak, than language. 39 Patiently waiting for incubation to work is quite difficult. Many creative and innovative people develop hobbies involving physical activity to keep their minds busy while they allow ideas to incubate.
Insight
Insight or “illumination” is a term for the “aha!” moment—when the solution to a creative problem suddenly becomes readily accessible to your conscious mind. The “aha!” moment has been observed in literature, in history, and in cognitive studies of creativity. 40 Insights may come all at once or in increments. They are not easily understood because, by their very nature, they are difficult to isolate in research and experimental settings. For the creative entrepreneur, however, insights are a delight. An insight is the fleeting time when your preparation, practice, and period of incubation coalesce into a stroke of genius. Whether the illumination is the solution to a seemingly impossible problem or the creation of a particularly clever melody or turn of phrase, creative people often consider it a highlight in their lives. For an entrepreneur, an insight holds the promise of success and the potential to help massive numbers of people overcome a pain point or problem. Not every insight will have a global impact, but coming up with a solution that your subconscious mind has been working on for some time is a real joy.
Evaluation
Evaluation is the purposeful examination of ideas. You will want to compare your insights with the products and ideas you encountered during preparation. You also will want to compare your ideas and product prototypes to the goals you set out for yourself during the preparation phase. Creative professionals will often invite others to critique their work at this stage. Because evaluation is specific to the expectations, best practices, and existing product leaders in each field, evaluation can take on many forms. You are looking for assurance that your standards for evaluation are appropriate. Judge yourself fairly, even as you apply strict criteria and the well-developed sense of taste you acquired during the preparation phase. For example, you might choose to interview a few customers in your target demographics for your product or service. The primary objective is to understand the customer perspective and the extent to which your idea aligns with their position.
Elaboration
The last stage in the creative process is elaboration , that is, actual production. Elaboration can involve the release of a minimum viable product (MVP) . This version of your invention may not be polished or complete, but it should function well enough that you can begin to market it while still elaborating on it in an iterative development process. Elaboration also can involve the development and launch of a prototype, the release of a software beta, or the production of some piece of artistic work for sale. Many consumer-product companies, such as Johnson & Johnson or Procter & Gamble, will establish a small test market to garner feedback and evaluations of new products from actual customers. These insights can give the company valuable information that can help make the product or service as successful as possible.
At this stage what matters most in the entrepreneurial creative process is that the work becomes available to the public so that they have a chance to adopt it.
Test marketing can reveal much information about the potential users of a product. Visit the Drive Research site on test markets for more information.
Innovation as More than Problem Solving
Innovative entrepreneurs are essentially problem solvers, but this level of innovation—identifying a pain point and working to overcome it—is only one in a series of innovative steps. In the influential business publication Forbes , the entrepreneur Larry Myler notes that problem solving is inherently reactive. 41 That is, you have to wait for a problem to happen in order to recognize the need to solve the problem. Solving problems is an important part of the practice of innovation, but to elevate the practice and the field, innovators should anticipate problems and strive to prevent them. In many cases, they create systems for continuous improvement, which Myler notes may involve “breaking” previous systems that seem to function perfectly well. Striving for continuous improvement helps innovators stay ahead of market changes. Thus, they have products ready for emerging markets, rather than developing projects that chase change, which can occur constantly in some tech-driven fields. One issue with building a system for constant improvement is that you are in essence creating problems in order to solve them, which goes against established culture in many firms. Innovators look for organizations that can handle purposeful innovation, or they attempt to start them. Some innovators even have the goal of innovating far ahead into the future, beyond current capacities. In order to do this, Myler suggests bringing people of disparate experiential backgrounds with different expertise together. These relationships are not guarantees of successful innovation, but such groups can generate ideas independent of institutional inertia. Thus, innovators are problem solvers but also can work with forms of problem creation and problem imagination. They tackle problems that have yet to exist in order to solve them ahead of time.
Let’s examine one multilevel approach to innovation ( Figure 4.12 ). The base is problem solving. The next level up in the pyramid, so to speak, is prevention. The next level is working toward continuous improvement, and at the top of such efforts is creating the capacity to direct the future of your industry or multiple industries so that you can weather disruption in your career or even to create it.
Even if you are not interested in shaping the future of whole industry sectors, developing future-focused innovation practices still is a good idea. It will help you prepare for disruption. The pace of technological change is such that workers at all levels need to be prepared to innovate. Innovation leaders, such as the marketing guru Guy Kawasaki, have built on psychological principles to suggest new ways to approach innovation. According to Kawasaki, innovative products include five key qualities: deep, indulgent, complete, elegant, and emotive—DICEE ( Figure 4.13 ). 42 You can strive to infuse individual innovations with these qualities in practical ways.
Deep
Deep products are based on the logic of innovation that we’ve just established and anticipate users’ needs before they have them. These types of innovations often have masterful designs that are intuitive for new users while still being capable of completing complex tasks. Adobe is an innovative corporation working in several fields, such as software, marketing, and artificial intelligence. Adobe often creates software applications with basic functions that are easily accessible to new users but that also enable experienced users to innovate on their own. 43 Creating a platform for innovation is a hallmark of deep, forward-thinking innovation.
Indulgent
Innovations with lasting power engage users in ways that make them feel special for having purchased the product or for having found the service. Indulgence refers to a depth of quality that does not come from being the fastest solution to a problem. Indulgence may even sound like a negative trait. In humans, it certainly can be, but for someone using an innovative product, feeling indulgent can relate to a richness of experience with the user interface (UI). The UI of a product, particularly a software product, is what the user sees and interacts with. A feeling of indulgence imbues your product with a sense of value and durability that reassures users and encourages them to use your product confidently.
Complete
Kawasaki’s vision of a complete product includes the services wrapped around it and underlying it such that users understand the product well enough to be comfortable using it. Information about how it works and how it is meant to work is readily available. Thus, product innovation must include marketing and other communication efforts. For Kawasaki, this builds the “total user experience.” 44 If you truly have solved a problem in the marketplace, users will understand what that problem is and how your product and related services deliver.
Elegant
Elegance also is part of a product’s UI. It refers to intuitive design that immediately makes sense to consumers. Elegance conveys more information with fewer words. Elegant design is not afraid of negative space or of the occasional pause. Elegant innovations solve problems without creating new ones. For Kawasaki, elegance is the difference between a pragmatic, good innovation and something great.
Emotive
Emotive innovations evoke the intended emotion and demand to be admired and shared. In other words, truly great innovations create fandoms, not just consumer bases. You can’t force people to love your product, but you can give them experiences that create a sense of excitement and anticipation of what you might come up with next.
Developing an Invention
The general process of inventing involves systematic and practical steps that might include linear and nonlinear thinking. You might think that only people with innate artistic skills are creative and that only geniuses become innovators and inventors, but much of creativity is driven by being immersed in a practice. You can build and foster your own creativity. Your idea of an inventor might be someone like Johannes Gutenberg, who developed the printing press. The spread of printing ultimately redrew the map of Europe and resulted in the foundation of new centers of learning. Gutenberg’s supposed spark actually was more of a slow burn. He was creative and innovative—one of history’s most famous inventors—but his printing press, like all other inventions, was a synthesis of existing technologies. Gutenberg’s most important innovation was his use of moveable, interchangeable metal type instead of entire hand-carved wooden blocks of text ( Figure 4.14 ). Perfecting his printing process took decades and left him all but broke. 45 The notion of the inventor’s single stroke of genius is mostly myth. The people that history remembers usually worked very hard to develop their creativity, to become familiar with the processes and tools that were ripe for innovation in their time, and ultimately to make something so unique that society recognizes it as an invention.
The old adage claims that “necessity is the mother of invention,” but an innovator needs experience in a field, creative effort, and knowledge to be a successful inventor. Entrepreneurship means taking your efforts and knowledge, and finding a market where your invention can first survive, then thrive.
One model for developing an invention is the first five steps of a plan adapted from Sourcify.com, which specializes in connecting product developers with manufacturers. 46 This process is succinct and includes suggestions for building a team along the way ( Figure 4.15 ).
Step 1: Educate Yourself
Before your inventive product can do battle with other inventions, you will need to educate yourself. To prepare yourself to weather the competition, you need to learn as much as you can about the current investing climate, current product development opportunities, and current leadership approaches. Even if you are not deeply interested in leadership dogma, it helps to know what the current trends are in leadership and product development. To succeed as an inventor in a vast marketplace, you need to understand the rules, written and unwritten, of the industry and competitive landscape. The product development process can be quite involved. The process can vary by industry and by availability of resources.
Part of educating yourself is also gaining an understanding of your own strengths and weaknesses, and how those relate to your leadership style. A leadership style inventory can help you better understand your approach to leading others. This is just one example of many that exist to give you a starting point. As the inventor of your product or service, you will manage/lead others as you attempt to make your idea a reality. Also, the environment can constantly change. For this reason, it is important to understand the basic tenets of leadership and management in a dynamic work atmosphere. Many sources will give you insights into the challenges of management.
A leadership style inventory can help you understand your leadership style and how to adapt your own style to other situations and people.
Another key step in educating yourself is to find out which kinds of contributors you are going to need to build a successful entrepreneurial team. Building a team is essential to making your invention a reality. Even those who invent alone—and they are quite rare—must have a development team, a manufacturing and/or service team, a marketing team, and other members with specific skill sets such as coders, graphic designers, test marketers, and more.
Step 2: Stay Organized
Most tip sheets for inventors suggest that you find a method for organizing your creativity so that you don’t spend time trying to remember previous ideas, plans, and decisions. You must organize information related to your business idea, your business plan, and your potential teammates in the process.
Contact management software has been popular for decades. Nowadays, you can investigate many other productivity and team-chat tools. Research ways to organize information about the people you plan to work with and hope to work for. The team-chat program Slack (www.slack.com) enables you to create specific topics for team members to discuss and collaborate on. Slack offers several features to help keep employees connected. Insightly (www.insightly.com) is a customer relationship management tool to stay better connected to your customers. Ryver and Glip incorporate task management. Flock and Microsoft Teams offer a host of features, with Microsoft leveraging its corporate position to bring about deployment in more than 200,000 organizations. Select the tool set that works best for you and consider paying for the software that offers the precise team communication functionality and utility you need. 47
Step 3: Conduct Market Research
Market research is an obvious must, but many entrepreneurs fail to go as deeply as they should in researching their competition. You must be aware of current and future competitors so that you are prepared to compete in the marketplace when you are actually ready. Being the best on paper now won’t be much use when you enter the marketplace with an MVP in six to eight months in competition with competitors’ new products and updates.
What should you consider with regard to team development when you’re looking at the competition? Within the legal limits of any noncompete clauses, you should be shopping the competition for potential team members. The best leaders are always seeking talented people. If you sense that someone would be a good fit for your team, that they have not only the skill set but also the temperament that would help put your invention in the market, do not be afraid to reach out to them. How you reach out is something you must research for each industry. In some industries, you will have to be highly secretive. Part of market research is understanding the market well enough to understand the soft skills you need to find contributors who are already working in the industry or in an adjacent one
Step 4: Conduct Patent Research
If you expect to apply for a patent, take the time to read up on policies and procedures. Officials in the US Patent Office, or in similar bureaus in other countries, decide whether an invention is worthy of receiving a patent. A patentable invention must meet the criteria of being novel, useful, and nonobvious; it must be proven to be workable. 48 Those three standards—novel, useful, and nonobvious—are subjective. So is the concept of invention, but conceptualizing invention this way sets a high bar for entrepreneurs who truly wish to make a social impact. Developing an invention that is patentable also creates a barrier against competition, which can make the difference between business success and failure. There are two types of patents.  If a patent is granted, the inventor has a window of time in which to secure further funding, work to produce the product, and try to gain mass-market adoption. 49 After all is said and done, you can apply your creativity to social innovations, product innovations, or service innovations. If you can combine enough innovations, add your unique creativity, and create something that survives the diffusion chasm, you can truly invent something new.
The patent basics page of the US Patent and Trademark Office’s website is roughly forty pages long. 50 The utility patent process includes a thirteen-step flow chart 51 that outlines the process. The patent office encourages you to use a registered patent attorney or agent. If you are skilled and diligent enough to secure a patent, you should expect to pay fees and file paperwork to maintain it for years after it is granted. We’ve already discussed the keys to securing a patent, but to reiterate, here is how an invention is defined in US patent law: “In the language of the statute, any person who ‘invents or discovers any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof, may obtain a patent,’ subject to the conditions and requirements of the law.” 52
When building a team to make your invention a reality, finding a patent lawyer or agent is key. Even those who have advocated for hiring patent attorneys in the past now suggest that hiring a patent agent might work. What’s the difference? Patent attorneys often bill by the hour, but they offer a full suite of legal advice. As the author Stephen Key indicates, patent agents are narrowly focused on helping you get and defend your patent. 53 The other limitation that Key mentions is that patent agents may write a patent application in such a way that you are less prepared to protect your invention against future legal challenges. Key quotes Gene Quinn, a leading attorney on intellectual property and patent law: “By the time you realize that you are sitting on a million-dollar invention it will be too late to do anything about it.…Patent agents as a general rule tend to be very good at describing what it is that you as an inventor show up with.” What they tend to be much less good at is describing what your invention could be. They also frequently will use terms that are more concrete and limiting than would a patent attorney. Attorneys are taught the art of being hyper-specific, which is necessary at times, but also the art of being anything but specific.” 54 Patents cannot be vague, but they can be written with just the right amount of specificity to protect against similar products that may arise and threaten your market share.
Step 5: Develop a Prototype
Developing a prototype can be the most fun or the most tedious part of inventing. Much of your attitude toward developing a prototype depends on available resources, technology, and expertise. In this text, we reference the concept of the lean startup from time to time. In the lean startup model, the prototype is most often an MVP. As we saw earlier, an MVP is a version of your invention that may not be polished or complete in terms of how you envisioned it, but it functions well enough and looks good enough that you can begin to market it with reasonable hopes that it will be adopted. For other inventions, you may need to build a more advanced prototype. This requires serious investment capital, but the payoff is that users will interact with a version of the product that looks and functions more like what you had in mind during your ideation phase. As an inventor, you are responsible for establishing quality control minimums for your product. You may have to compromise on your vision, but you should not compromise on basic functionality or basic levels of quality in materials.
You have many options at the prototype development stage. You can build the prototype yourself or with a small team. You can partner with design/invention firms that specialize in helping inventors create, but you must be very careful and involve your legal representation when working with such firms to be sure that you maintain the patents and other rights to your invention. Many inventors have partnered with such firms only to see their intellectual property stolen. Another option is to get funding for your invention on Kickstarter or some other crowdfunding site, but again you must beware that establishing such a campaign puts your idea in the public sphere. “Copycatters are monitoring crowdfunding platforms like Kickstarter and watching for trendy products to go viral,” according to Amedeo Ferraro, an intellectual property attorney. 55 Competing companies, particularly in foreign markets, actively scout Kickstarter and similar platforms for new ideas that they can manufacture and bring to market before your crowdfunding project has run its course. Perhaps most chilling is this comment regarding legal protections that do not function, even when inventors take precautions to protect their intellectual property when working with some Chinese firms: “But even with these protections, there’s no guarantee that you can stop someone from copycatting your product. [One U.S. intellectual property lawyer] said that the problem lies not in China’s courts but enforcing rulings. Winning a case against one factory is relatively easy. But suing every factory and winning is expensive and time consuming.” 56 For this reason, some inventors prefer to start small and local, if possible. It can be better for them to start with a trusted team striving for a small profit and a good market position than to see the market flooded with copycat products.


6.5: Functions and Computability
In Section \(1.3\) we made the off-hand comment that most functions are not defined by rules (by which we meant instructions for computing the function). We consider a rule to be an instruction (in some language) of finite length. Functions that are unambiguously defined by a rule of finite length are called computable, or recursive functions. Naturally there is a complicated mathematical definition of recursive functions, but we shall dispense with the formalities and say that a function is recursive, or computable, if there is an instruction (of finite length) for finding the image of any element in the domain. How many computable functions are there?
We shall restrict our investigation to functions from \(\mathbb{N}\) to \(\mathbb{N}\) . We consider functions as graphs of functions. That is, every subset of \(P(\mathbb{N} \times \mathbb{N})\) that satisfies the definition of a function is a function in \(\mathbb{N}^{\mathbb{N}}\) . Are all such functions computable? It is obvious that \[2^{\aleph_{0}} \leq\left|\mathbb{N}^{\mathbb{N}}\right| \text {. }\] (Why?) In fact you can show that the sets are bijective. So there are uncountably many functions in \(\mathbb{N}^{\mathbb{N}}\) . How many instructions for computing functions are there? An instruction is a finite string, or sequence, of symbols. For instance, an instruction for the function that squares natural numbers is \[f(x)=x^{2} .\] This is a finite sequence of seven symbols. The instruction gives enough information to compute the image of any natural number. There are many other rules for computing this function. For instance the rule \[f(x)=x \cdot x\] obviously defines the same function, but the instruction is different - it contains one more symbol. Consider the set of all possible instructions for computing functions of natural numbers. How are the instructions formulated? One produces a finite sequence of symbols that forms an explicit guide for computing the image of any natural number.
Let \(X\) be the set of all symbols appearing in instructions for computing functions of natural numbers. The set \(X\) will include letters, digits, symbols for operations, symbols for relations and potentially any other symbol that you might see in a book on mathematics. How large is \(X\) ? If you require that every symbol appear in some actual dictionary, it would clearly be finite. You will probably wish to allow any natural number to appear in the instruction. However, although there are infinitely many natural numbers, we need only ten symbols to name them all. It seems that we can reasonably require that \(X\) is finite, but as it turns out, we can allow for \(X\) to be countably infinite without changing our conclusion.
If there is any language with countably many symbols in which the set of all instructions for computing functions could be written, then we may assume that \(X\) is countable. If \(F\) is an instruction or rule (and hence a finite sequence of symbols from \(X\) ), then there is \(N \in \mathbb{N}\) such that \[F \in X^{N} \text {. }\] So it is easily seen that the set of all possible instructions for elements of \(\mathbb{N}^{\mathbb{N}}, I\) , satisfies \[I \preceq \bigcup_{N \in \mathbb{N}} X^{N} .\] For \(N \in \mathbb{N}, X^{N}\) is the direct product of \(N\) factors of \(X\) , and by Theorem \(6.20\) , \[\left|X^{N}\right| \leq \aleph_{0} .\] The set \(\bigcup_{N \in \mathbb{N}} X^{N}\) is the countable union of countable sets, and by Theorem \(6.16\) is countable. Therefore there are uncountably many functions of natural numbers that are not defined by rules.
For a more thorough treatment of set theory, see the book [5] by Yiannis Moschovakis.


4.10: Learning Activities
Learning Activities
(Answers to “Learning Activities” can be found in the “Answer Key” at the end of the book. Answers to interactive activity elements will be provided within the element as immediate feedback.)
Instructions: Apply what you’ve learned in this chapter by creating a nursing care plan using the following scenario. Use the template in Appendix B as a guide.
The client, Mark S., is a 57-year-old male who was admitted to the hospital with “severe” abdominal pain that was unable to be managed in the Emergency Department. The physician has informed Mark that he will need to undergo some diagnostic tests. The tests are scheduled for the morning.
After receiving the news about his condition and the need for diagnostic tests, Mark begins to pace the floor. He continues to pace constantly. He keeps asking the nurse the same question (“How long will the tests take?”) about his tests over and over again. The patient also remarked, “I’m so uptight I will never be able to sleep tonight.” The nurse observes that the client avoids eye contact during their interactions and that he continually fidgets with the call light. His eyes keep darting around the room. He appears tense and has a strained expression on his face. He states, “My mouth is so dry.” The nurse observes his vital signs to be: T 98, P 104, R 30, BP 180/96. The nurse notes that his skin feels sweaty (diaphoretic) and cool to the touch.
Critical Thinking Activity:
-
- Group (cluster) the subjective and objective data.
- Create a problem-focused nursing diagnosis (hypothesis).
- Develop a broad goal and then identify an expected outcome in “SMART” format.
- Outline three interventions for the nursing diagnosis to meet the goal. Cite an evidence-based source.
- Imagine that you implemented the interventions that you identified. Evaluate the degree to which the expected outcome was achieved: Met – Partially Met – Not Met.
Query \(\PageIndex{1}\)
Query \(\PageIndex{2}\)
Query \(\PageIndex{3}\)


22.7: Magnetic Force on a Current-Carrying Conductor
Learning Objectives
By the end of this section, you will be able to:
- Describe the effects of a magnetic force on a current-carrying conductor.
- Calculate the magnetic force on a current-carrying conductor.
Because charges ordinarily cannot escape a conductor, the magnetic force on charges moving in a conductor is transmitted to the conductor itself.
We can derive an expression for the magnetic force on a current by taking a sum of the magnetic forces on individual charges. (The forces add because they are in the same direction.) The force on an individual charge moving at the drift velocity \(v_{d}\) is given by \(F = qv_{d}B\sin \theta \). Taking \(B\) to be uniform over a length of wire \(l\) and zero elsewhere, the total magnetic force on the wire is then \(F = \left(qv_{d}B \sin \theta \right) \left(N\right)\), where \(N\) is the number of charge carriers in the section of wire of length \(l\). Now, \(N = nV\), where \(n\) is the number of charge carriers per unit volume and \(V\) is the volume of wire in the field. Noting that \(V = Al\), where \(A\) is the cross-sectional area of the wire, then the force on the wire is \(F = \left(qv_{d}B\sin\theta\right))\left(nAl\right)\). Gathering terms,
\[F = \left(nqAv_{d}\right)lB\sin\theta . \label{22.8.1}\]
Because \(nqAv_{d} = I\) (see Current ),
\[F = Il B \sin \theta \label{22.8.1b}\]
is the equation for magnetic force on a length \(l\) of wire carrying a current \(I\) in a uniform magnetic field \(B\), as shown in Figure \(\PageIndex{2}\). If we divide both sides of this expression by \(l\), we find that the magnetic force per unit length of wire in a uniform field is \(\frac{}{} = IB \sin \theta \). The direction of this force is given by RHR-1, with the thumb in the direction of the current \(I\). Then, with the fingers in the direction of \(B\), a perpendicular to the palm points in the direction of \(F\), as in Figure 2.
Example \(\PageIndex{1}\): Calculating Magnetic Force on a Current-Carrying Wire: A Strong Magnetic Field
Calculate the force on the wire shown in Figure \(\PageIndex{1}\), given \(B = 1.50 T\), \(l = 5.00 cm\), and \(I = 20.0 A\).
Strategy
The force can be found with the given information by using \(F = \pi B \sin\theta\) and noting that the angle \(\theta\) between \(I\) and \(B\) is \(90^{\circ}\), so that \(sin \theta = 1\).
Solution
Entering the given values into \(F = \pi B \sin \theta\) yields
\[F = \pi B \sin \theta = \left(20.0 A\right) \left(0.0500 m\right) \left(1.50 T\right) \left(1\right). \nonumber\]
The units for tesla are \(1 T = \frac{N}{A \cdot m}\); thus,
\[F = 1.50 \,N. \nonumber\]
Discussion
This large magnetic field creates a significant force on a small length of wire.
Magnetic force on current-carrying conductors is used to convert electric energy to work. (Motors are a prime example—they employ loops of wire and are considered in the next section.) Magnetohydrodynamics (MHD) is the technical name given to a clever application where magnetic force pumps fluids without moving mechanical parts (Figure \(\PageIndex{3}\)).
A strong magnetic field is applied across a tube and a current is passed through the fluid at right angles to the field, resulting in a force on the fluid parallel to the tube axis as shown. The absence of moving parts makes this attractive for moving a hot, chemically active substance, such as the liquid sodium employed in some nuclear reactors. Experimental artificial hearts are testing with this technique for pumping blood, perhaps circumventing the adverse effects of mechanical pumps. (Cell membranes, however, are affected by the large fields needed in MHD, delaying its practical application in humans.) MHD propulsion for nuclear submarines has been proposed, because it could be considerably quieter than conventional propeller drives. The deterrent value of nuclear submarines is based on their ability to hide and survive a first or second nuclear strike. As we slowly disassemble our nuclear weapons arsenals, the submarine branch will be the last to be decommissioned because of this ability (Figure \(\PageIndex{4}\)). Existing MHD drives are heavy and inefficient—much development work is needed.
Summary
- The magnetic force on current-carrying conductors is given by \[F = Il B \sin \theta, \nonumber\] where \(\) is the current, \(l\) is the length of a straight conductor in a uniform magnetic field \(B\), and \(\theta\) is the angle between \(I\) and \(B\). The force follows RHR-1 with the thumb in the direction of \(I\).


Section 01: Basic rules for SL
In designing a proof system, we could just start with disjunctive syllogism and modus ponens. Whenever we discovered a valid argument which could not be proven with rules we already had, we could introduce new rules. Proceeding in this way, we would have an unsystematic grab bag of rules. We might accidently add some strange rules, and we would surely end up with more rules than we need.
Instead, we will develop what is called a natural deduction system. In a natural deduction system, there will be two rules for each logical operator: an introduction rule that allows us to prove a sentence that has it as the main logical operator and an elimination rule that allows us to prove something given a sentence that has it as the main logical operator.
In addition to the rules for each logical operator, we will also have a reiteration rule. If you already have shown something in the course of a proof, the reiteration rule allows you to repeat it on a new line. For instance:
When we add a line to a proof, we write the rule that justifies that line. We also write the numbers of the lines to which the rule was applied. The reiteration rule above is justified by one line, the line that you are reiterating. So the ‘R 1’ on line 2 of the proof means that the line is justified by the reiteration rule (R) applied to line 1.
Obviously, the reiteration rule will not allow us to show anything new . For that, we will need more rules. The remainder of this section will give introduction and elimination rules for all of the sentential connectives. This will give us a complete proof system for SL. Later in the chapter, we introduce rules for quantifiers and identity.
Conjunction
Think for a moment: What would you need to show in order to prove \(E\) & \(F\)?
Of course, you could show \(E\) & \(F\) by proving \(E\) and separately proving \(F\). This holds even if the two conjuncts are not atomic sentences. If you can prove [(\(A\)∨J) → \(V\) ] and [(\(V\) → \(L\)) ↔ (\(F\) ∨ \(N\))], then you have effectively proven
[(\(A\)∨J) → \(V\) ]&[(\(V\) → \(L\)) ↔ (\(F\) ∨ \(N\))].
So this will be our conjunction introduction rule, which we abbreviate &I:
A line of proof must be justified by some rule, and here we have ‘&I \(m\), \(n\).’ This means: Conjunction introduction applied to line \(m\) and line \(n\). These are variables, not real line numbers; \(m\) is some line and \(n\) is some other line. In an actual proof, the lines are numbered 1,2,3,... and rules must be applied to specific line numbers. When we define the rule, however, we use variables to underscore the point that the rule may be applied to any two lines that are already in the proof. If you have \(K\) on line 8 and \(L\) on line 15, you can prove (\(K\) & \(L\)) at some later point in the proof with the justification ‘&I 8, 15.’
Now, consider the elimination rule for conjunction. What are you entitled to conclude from a sentence like \(E\) & \(F\)? Surely, you are entitled to conclude \(E\); if \(E\) & \(F\) were true, then \(E\) would be true. Similarly, you are entitled to conclude \(F\). This will be our conjunction elimination rule, which we abbreviate &E:
When you have a conjunction on some line of a proof, you can use &E to derive either of the conjuncts. The &E rule requires only one sentence, so we write one line number as the justification for applying it.
Even with just these two rules, we can provide some proofs. Consider this argument.
The main logical operator in both the premise and conclusion is conjunction. Since conjunction is symmetric, the argument is obviously valid. In order to provide a proof, we begin by writing down the premise. After the premises, we draw a horizontal line— everything below this line must be justified by a rule of proof. So the beginning of the proof looks like this:
From the premise, we can get each of the conjuncts by &E. The proof now looks like this:
The rule &I requires that we have each of the conjuncts available somewhere in the proof. They can be separated from one another, and they can appear in any order. So by applying the &I rule to lines 3 and 2, we arrive at the desired conclusion. The finished proof looks like this:
This proof is trivial, but it shows how we can use rules of proof together to demonstrate the validity of an argument form. Also: Using a truth table to show that this argument is valid would have required a staggering 256 lines, since there are eight sentence letters in the argument.
Disjunction
If \(M\) were true, then \(M\)∨\(N\) would also be true. So the disjunction introduction rule (∨\(I\)) allows us to derive a disjunction if we have one of the two disjuncts:
Notice that \(\mathcal{B}\) can be any sentence whatsoever. So the following is a legitimate proof:
It may seem odd that just by knowing \(M\) we can derive a conclusion that includes sentences like \(A\), \(B\), and the rest— sentences that have nothing to do with \(M\). Yet the conclusion follows immediately by ∨I. This is as it should be: The truth conditions for the disjunction mean that, if \(\mathcal{A}\) is true, then \(\mathcal{A}\)∨\(\mathcal{B}\) is true regardless of what \(\mathcal{B}\) is. So the conclusion could not be false if the premise were true; the argument is valid.
Now consider the disjunction elimination rule. What can you conclude from \(M\) ∨ \(N\)? You cannot conclude \(M\). It might be \(M\)’s truth that makes \(M\) ∨ \(N\) true, as in the example above, but it might not. From \(M\)∨\(N\) alone, you cannot conclude anything about either \(M\) or \(N\) specifically. If you also knew that \(N\) was false, however, then you would be able to conclude \(M\).
This is just disjunctive syllogism, it will be the disjunction elimination rule (∨\(E\)).
Conditional
Consider this argument:
The argument is certainly a valid one. What should the conditional introduction rule be, such that we can draw this conclusion?
We begin the proof by writing down the premise of the argument and drawing a horizontal line, like this:
If we had¬\(R\) as a further premise, we could derive \(F\) by the ∨E rule. We do not have ¬\(R\) as a premise of this argument, nor can we derive it directly from the premise we do have— so we cannot simply prove \(F\). What we will do instead is start a subproof , a proof within the main proof. When we start a subproof, we draw another vertical line to indicate that we are no longer in the main proof. Then we write in an assumption for the subproof. This can be anything we want. Here, it will be helpful to assume ¬\(R\). Our proof now looks like this:
It is important to notice that we are not claiming to have proven ¬\(R\). We do not need to write in any justification for the assumption line of a subproof. You can think of the subproof as posing the question: What could we show if ¬\(R\) were true? For one thing, we can derive \(F\). So we do:
This has shown that if we had ¬\(R\) as a premise, then we could prove \(F\). In effect, we have proven ¬\(R\) → \(F\). So the conditional introduction rule (→I) will allow us to close the subproof and derive ¬\(R\) → \(F\) in the main proof. Our final proof looks like this:
Notice that the justification for applying the →I rule is the entire subproof. Usually that will be more than just two lines.
It may seem as if the ability to assume anything at all in a subproof would lead to chaos: Does it allow you to prove any conclusion from any premises? The answer is no, it does not. Consider this proof:
It may seem as if this is a proof that you can derive any conclusions \(\mathcal{B}\) from any premise \(\mathcal{A}\). When the vertical line for the subproof ends, the subproof is closed . In order to complete a proof, you must close all of the subproofs. And you cannot close the subproof and use the R rule again on line 4 to derive \(\mathcal{B}\) in the main proof. Once you close a subproof, you cannot refer back to individual lines inside it.
Closing a subproof is called discharging the assumptions of that subproof. So we can put the point this way: You cannot complete a proof until you have discharged all of the assumptions besides the original premises of the argument.
Of course, it is legitimate to do this:
This should not seem so strange, though. Since \(\mathcal{B}\)→\(\mathcal{B}\) is a tautology, no particular premises should be required to validly derive it. (Indeed, as we will see, a tautology follows from any premises.)
Put in a general form, the →I rule looks like this:
When we introduce a subproof, we typically write what we want to derive in the column. This is just so that we do not forget why we started the subproof if it goes on for five or ten lines. There is no ‘want’ rule. It is a note to ourselves and not formally part of the proof.
Although it is always permissible to open a subproof with any assumption you please, there is some strategy involved in picking a useful assumption. Starting a subproof with an arbitrary, wacky assumption would just waste lines of the proof. In order to derive a conditional by the →I, for instance, you must assume the antecedent of the conditional in a subproof.
The →I rule also requires that the consequent of the conditional be the last line of the subproof. It is always permissible to close a subproof and discharge its assumptions, but it will not be helpful to do so until you get what you want.
Now consider the conditional elimination rule. Nothing follows from \(M\) → \(N\) alone, but if we have both \(M\) → \(N\) and \(M\), then we can conclude \(N\). This rule, modus ponens, will be the conditional elimination rule (→E).
Now that we have rules for the conditional, consider this argument:
We begin the proof by writing the two premises as assumptions. Since the main logical operator in the conclusion is a conditional, we can expect to use the →I rule. For that, we need a subproof— so we write in the antecedent of the conditional as assumption of a subproof:
We made \(P\) available by assuming it in a subproof, allowing us to use →E on the first premise. This gives us \(Q\), which allows us to use →E on the second premise. Having derived \(R\), we close the subproof. By assuming \(P\) we were able to prove \(R\), so we apply the →I rule and finish the proof.
Biconditional
The rules for the biconditional will be like double-barreled versions of the rules for the conditional.
In order to derive \(W\) ↔ \(X\), for instance, you must be able to prove \(X\) by assuming \(W\) and prove \(W\) by assuming \(X\). The biconditional introduction rule (↔I) requires two subproofs. The subproofs can come in any order, and the second subproof does not need to come immediately after the first— but schematically, the rule works like this:
The biconditional elimination rule (↔E) lets you do a bit more than the conditional rule. If you have the left-hand subsentence of the biconditional, you can derive the right-hand subsentence. If you have the right-hand subsentence, you can derive the left-hand subsentence. This is the rule:
Negation
Here is a simple mathematical argument in English:
Assume there is some greatest natural number. Call it \(A\).
That number plus one is also a natural number.
Obviously, \(A\) + 1 > \(A\).
So there is a natural number greater than \(A\).
This is impossible, since \(A\) is assumed to be the greatest natural number.
.˙. There is no greatest natural number.
This argument form is traditionally called a reductio . Its full Latin name is reductio ad absurdum , which means ‘reduction to absurdity.’ In a reductio, we assume something for the sake of argument— for example, that there is a greatest natural number. Then we show that the assumption leads to two contradictory sentences— for example, that \(A\) is the greatest natural number and that it is not. In this way, we show that the original assumption must have been false.
The basic rules for negation will allow for arguments like this. If we assume something and show that it leads to contradictory sentences, then we have proven the negation of the assumption. This is the negation introduction (¬I) rule:
For the rule to apply, the last two lines of the subproof must be an explicit contradiction: some sentence followed on the next line by its negation. We write ‘for reductio’ as a note to ourselves, a reminder of why we started the subproof. It is not formally part of the proof, and you can leave it out if you find it distracting.
To see how the rule works, suppose we want to prove the law of non-contradiction: ¬(\(G\)&¬\(G\)). We can prove this without any premises by immediately starting a subproof. We want to apply ¬I to the subproof, so we assume (\(G\)&¬\(G\)). We then get an explicit contradiction by &E. The proof looks like this:
The ¬E rule will work in much the same way. If we assume ¬\(\mathcal{A}\) and show that it leads to a contradiction, we have effectively proven \(\mathcal{A}\). So the rule looks like this:


20.6: Molecules Talk- Selecting Molecular Communication and Complexity
In our complex human society, we define communication by its specificity . Without a careful choice of words, our speech would be at best, a source of magnificent misunderstanding, or just plain babel! What does this mean for prebiotic chemistries?
In terms of prebiotic chemical evolution, selection by definition would have favored protective accumulation of longer-lived molecular aggregates. Over time, the same selective imperatives would create webs of such aggregates, increasing the range and specificity of molecular interactions in a challenging environment. If this were to have occurred in an enclosed proto-cellular space, it should have resulted in a primitive molecular communication and a growing complexity (another property of life!). In fact, alof the properties of life must have accompanied the achievement of more and more complex intermolecular communication. Simply put, a prebiotic (or for that matter a cellular) genetic change that alters the rate of one catalytic reaction (if not destructive) will drive the selection of changes in components of other, interconnected metabolic chemistries. If molecular communication required the evolution of catalytic specificity, then the final elaboration of complexity and order as a property of life further requires the selection of mechanisms of regulation and coordination .
A. Intermolecular Communication Leads to an Early Establishment of Essential Interconnected Chemistries
Earlier, we suggested that inorganic catalyst precursors to biological enzymes were probably minerals embedded in clay or other substrata, providing surfaces that would naturally aggregate organic molecules and catalyze repetitive reactions. Either the initial objects of prebiotic selection included external stable monomers and polymers, outside or as seems more likely, inside proto-cells. Later, selection would have favored polymers that enhanced growth and reproduction of successful aggregates. These polymers were likely those that catalyzed their own synthesis, perhaps collaborating with inorganic catalytic minerals. The result would be the elaboration of a web of interconnected chemical reactions between molecules with high affinity for each other, thereby increasing the specificity of those reactions. In the context of life origins and evolution, co-catalysis describes the activities of these interconnected metabolic reactions.
As noted, high-affinity interactions are inherently protective . During prebiotic chemical/metabolic evolution, protected stable molecular assemblies would be targets of selection. Continuing co-evolution of catalysts, substrates and co-catalytic reaction sets would lead to more and more sophisticated molecular communication . Once established, efficient biochemical reaction sets would be constrained against significant evolutionary change. Any change (mutation) that threatened this efficiency would mean the end of a prebiotic chemical (or for that matter, cell) lineage! This explains why we find common pathways for energy-generation (e.g., autotrophic and fermentative), reproduction (replication), and information storage and retrieval (DNA, RNA, protein synthesis) in all of LUCA’s descendants.
Sophisticated, effective communication requires coordination. In fact, effective communication is defined by coordination , the capacity to make chemical decisions . Selection of molecular aggregates that sequestered metabolic reactions behind a semipermeable membrane ensure that only certain molecules communicate with each other. This sequestration is likely to have occurred repeatedly during chemical evolution, beginning with the synthesis of larger, polymeric molecules and possibly, an aggregation of primitive lipoidal molecules. We can think of increasingly effective catalysis in an enclosed environment as a conversation mediated by good speakers ! Coordination is a property that likely co-evolved with life itself!
B. Origins of Coordination
Let’s look some possible structures churning around in the prebiotic chemistry set that might have self-assembled, or sequestered compatible chemistries of life. Along with the alkaline vent biofilm compartment , coacervates , proteinoid microspheres and liposomes have been considered as possible progenitors of biological membranes. Each can be made in the laboratory. They are demonstrably semipermeable, and in some cases can even replicate! Micrographs and the production of coacervates, proteinoid microspheres and liposomes are shown below
Oparin had proposed that the action of sunlight in the absence of oxygen could cause ionized, oppositely charged organic molecules (e,g, amino acids, carbohydrates, etc.) to form droplets from organic molecules in his primordial soup. These coacervates were actually produced in 1932, visualized by microscopy and demonstarted to be a semi-permeable compartment. They even behaved as if they were able to grow and reproduce (also as Oparin originally suggested they might).
In the 1950s, Sidney Fox produced proteinoid microspheres from short peptides that formed spontaneously from aqueous amino acid solutions heated to dryness (not unlike what happens in the tidal pool scenario of polymer formation from organic monomers). These can be seen by light and electron microscopy.
While liposomes are easily made in a laboratory, it isn’t clear that they existed on a pre-biotic earth. Nevertheless, cell membranes must have had acquired their phospholipid bilayer structure by the time of LUCA since we all have them! Prior to LUCA, chemical rearrangenments must have occurred to enable incorporation of a phospholipid bilayer into whatever starting boundary life started with.
We have already considered the biofilm proposed for cellular origins in an alkaline vent. The formation of such biofilms would have separated acidic ocean protons from the interior of such protocells, creating a proton gradient. Such a gradient could have driven the early evolution of chemiososis as a means to create chemical energy, complete with the eventual selection of ATP synthases and the enzymes of proton transport, again because all cells descendent from LUCA’s posess these biochemistries. Of course, proteinoid microspheres, coacervates, biofilm-based ‘membranes and liposomes are not alive, and are therefore not cells. But one or another of them must have been where the enhanced coordination of molecular communication required for life began.
354 Protected Molecular Communication: Semipermeable Membranes
An important take-home message here is that whatever the original structure of the first cells, they arose soon after the organic chemical prerequisites of life began to acquire familiar metabolic functions. We need to see chemical and structural progress to cellularity as concurrent metabolic evolutionary events. At some point, selection of sequestered biochemistries led to protocells , then to the first cells, each with all of the properties of life.
Finally, selection of highly specific communication between cellular molecules allowed cells themselves to talk to one another, engage in group activities, and eventually join together to form multicellular organisms. Multicellularity is of course a characteristic of many if not most eukaryotes. But watch a great TED Talk on bacterial intercellular communication by Dr. Bonnie Bassler at Intercellular Communication in Bacteria .
C. Origins of Information Storage and Retrieval in an RNA World
Let us accept for now that molecular communication began concurrently with the packaging of interconnected co-catalytic sets into semipermeable structures. Then the most ‘fit’ of these structures were selected for efficient coordination of meaningful, timely chemical messages. Ultimately, coordination requires information processing , storage and retrieval , something we recognize in Francis Crick’s Central Dogma of information flow from DNA to RNA to protein. Cells and organisms do coordination quite well, but what do its beginnings look like? The answer may lie in the pre-biotic RNA world we discussed earlier. The Central Dogma, modified to account for reverse transcription and the behavior of retroviruses, is shown below.
We do not really know how cells came to rely on DNA to store, pass on and mobilize genetic information, but we have presented reasons to believe that the first replicating nucleic acid was RNA, creating an RNA world . Here is the evidence that leads us to this conclusion.
- Based on the stem-and-loop and other structures that form when RNA molecules undergo internal H-bonding, we know that RNAs can take on varied and intricate shapes.
- Diverse conformations are consistent with the evolution of specificity in the interaction of RNAs with themselves and/or with other molecules in the prebiotic environment.
- RNAs, either alone as autocatalysts (for example, self-splicing mRNAs) or in catalytic ribonucleoprotein complexes (e.g., ribosomes) exist in cells today.
- Some of these RNAs (specifically rRNAs), have a long phylogenetic heritage, shared by cells in all three domains of life.
The propensity of single stranded RNA molecules to fold based on internal H-bonding can lead to those diverse three-dimensional shapes (tertiary structure). These structures could have interacted with other molecules in a environment. Because they could be replicated according to different prebiotic scenarios, the same RNAs could also pass on simple genetic information contained in their base sequences. The combination of informational and catalytic properties in a single molecule is illustrated below.
The capacity of RNAs as catalysts and warehouses of genetic information speaks to an efficient candidate for the first dual or multi-purpose polymer, a property that is not known and cannot be demonstrated for DNA. Read more about the proposed ‘RNA worlds’ in which life may have begun in Cech TR (2012) [ The RNA Worlds in Context . In Cold Spring Harbor Perspectives in Biology (Cold Spring Harbor, NY: Cold Spring Harbor press) 4(7):a006742e].
355 Self-Replication: Information, Communication, and Coordination
What might RNA catalysis beyond self-replication have looked like in simpler times? Consider the interaction between a two hypothetical RNAs and different hypothetical amino acids bound to each, shown below.
The binding of each RNA to its amino acid would be a high affinity, specific interaction based on charge and shape complementarity. Likewise, the two RNAs seen in the illustration must have a high affinity for each other, also based on chemical and physical complementarities. One can even envision some strong H-bonding between bases in the two RNAs that might displace intra-strand H-bonding (not shown here). The result is that the two amino acids are brought together in a way that catalyzes peptide bond formation. This will require an input of free energy (recall that peptide bond is one of the most energy intensive reaction in cells). For now, assume a chemical energy source and let us focus on the specificities required for RNA catalytic activity.
We know now that tRNAs the intermediaries between nucleic acids and polypeptide synthesis. So it’s fair to ask how the kind of activity illustrated above could have led to the tRNA-amino acid interactions we see today. There is no obvious binding chemistry between today’s amino acids and RNAs, but there may be a less obvious legacy of the proposed bindings. This has to do with the fact that the genetic code is universal, which means that any structural relationship between RNA and amino acids must have been selected early (at the start!) of cellular life on earth. Here is the argument.
- The code is indeed universal (or nearly so)
-
There is a correlation between the chemical properties of amino acids and their codons, for example
- Charged (polar) amino acids are encoded by triplet codons with more G (guanine) bases.
- Codons for uncharged amino acids more often contain a middle U (uracil) than any other base.
These correlations would mean that an early binding of amino acids to specifically folded RNAs was replaced in evolution by enzyme-catalyzed covalent attachment of an amino acid to a ‘correct’ tRNA, such as we see today.
What forces might have selected separation of the combined template/informational functions from most of the catalytic activities of RNAs? Perhaps it was the selection of the greater diversity of structure (i.e., shape) that folded polypeptides can achieve, compared to folded RNAs. After all, polypeptides are strings of 20 different amino acids compared to the four bases that make up nucleic acids. This potential for molecular diversity would in turn accelerate the pace of chemical (and ultimately cellular) evolution. A scenario for the transition from earlier self-replicating RNA events to the translation of proteins from mRNAs is illustrated here.
Adaptor RNAs in the illustration will become tRNAs. The novel, relatively unfolded RNA depicts a presumptive mRNA. Thus, even before the entry of DNA into our RNA world, it is possible to imagine the selection of the defining features of the genetic code and mechanism of translation (protein synthesis) that characterizes all life on the planet. Next, we consider “best-speculations” of how RNA-based information storage and catalytic chemistries might have made the evolutionary transition to DNA-based information storage and predominantly protein based enzyme catalysis.
D. From Ribosomes to Enzymes; From RNA to DNA
The term co-catalysis could very well describe biochemical reactions in which a catalyst accelerates a chemical reaction whose product feeds back in some way on its own synthesis. We saw this in action when we discussed allosteric enzyme regulation and the control of biochemical pathways. Catalytic feedback loops must have been significant events in the evolution of the intermolecular communication and metabolic coordination required for life.
Here we will consider some scenarios for the transition from an RNA world to something more recognizable as today’s nucleic acid information storage and proteinbased catalytic metabolism.
1. Ribozymes Branch Out: Replication, Transcription, and Translation
If RNAs catalyzed their own replication, it may have resembled the autocatalytic replication of AATE. At the same time, some RNAs may also have attracted amino acids to their surfaces and catalyzed peptide bond formation, as already described. Shapely prebiotic RNAs may therefore have catalyzed synthesized peptides, some of which would eventually take over catalysis of RNA synthesis! The scenario is summarized below.
356 Information Storage and Retrieval in an RNA World
Selection favoring the synthesis of short oligopeptides and polypeptides is consistent with a catalytic diversification that led to the dominance of protein catalysts, i.e., enzymes. The primitive enzyme shown here must have been selected because at first, it assisted the autocatalytic replication of the RNA itself! Over time, the enzyme would evolve along with the RNA. This co-evolution then eventually replaced autocatalytic RNA replication with the enzyme-catalyzed RNA synthesis we recognize as transcription today. In this scenario, self-splicing premRNAs and ribozymes are surviving remnants of an RNA world!
Let’s turn now to some ideas about how an RNA world could make the transition to the DNA-RNA-protein world we have today
2. Transfer of Information Storage from RNA to DNA
The transfer of function from RNA to DNA is by no means a settled issue among students of life origins and early evolution. A best guess is that the elaboration of protein enzymes begun in the RNA world would lead to reverse transcriptase-like enzymes that copied RNA information into DNA molecules. DNA information may have been selected because DNA is chemically more stable than RNA. The basic transfer scenario is illustrated below.
All cells alive today store information in DNA (only some viruses have an RNA genome). Therefore, transition to the use of DNA as an information molecule must have preceded the origin of life. At least, it must have occurred in the cells from which the LUCA arose. Details of this key change involve evolutionary steps yet to be worked out to everyone’s satisfaction!
E. The Evolution of Biochemical Pathways
The tale of the evolution of enzymes from ribozymes and of informational DNA from RNA, and the other metabolic chemistries behind prebiotic semipermeable boundaries is ongoing in cells today. Undoubtedly, early cellular metabolism involved only reactions crucial to life, catalyzed by a limited number of enzymes. But, if evolution inexorably trends towards greater complexity of molecular communication and coordination, in other words, towards increasingly refined regulation of metabolism, how did the repertoire of enzymes get larger, and how did biochemical pathways become more elaborate? We answered the first question elsewhere, when we discussed gene duplication (e.g., by unequal crossing over). The duplicate genes encoding the same enzyme provided the raw material for new enzymes and new enzymatic functions.
Whether in cells or in prebiotic structures, we can hypothesize how a new chemical reaction could evolve. For example, assume that a cell acquires molecule D required for an essential function, from an external, environmental source. What happens if levels of D in the environment become limiting ?  That is, unless cells that already have a duplicated, redundant gene that has mutated and now encodes an enzyme with the ability to make D in the cell. Such a cell might have existed with other cells without the mutation, but a D-limited environment would select the mutant cell for survival and reproduction. Imagine the scenario illustrated below.
358 Origins and Evolution of Biochemical Pathways
In a similar scenario, a mutation in a duplicated gene could result in a novel enzyme activity that can convert some molecule (e.g., C or D) in the cell into a new molecular product. If the new enzyme and molecular product do not kill or debilitate the cell, the cell might survive to be selected by some future exigency.


5.2: What is Curriculum?
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In early care and education, we value how children process information, we recognize how important their feelings are, and we place great significance on how children learn to socialize with others. As intentional teachers, our primary goal is to incorporate curriculum that supports the “whole-child” in all domains of development including social, emotional, physical, and intellectual. The word “curriculum” can mean different things to different people. For some, curriculum can provide the framework for learning, for others it can be a variety of planned activities, still for others curriculum is a way to drive learning outcomes, goals and objectives. Curriculum can be developed by teachers, or it can be purchased as part of a prefabbed program. Curriculum can be child-directed, and it can be teacher-directed. According to a joint position paper by the National Association for the Education of Young Child (NAEYC) and the National Association of Early Childhood Specialists in State Department of Education (NAECS-SDE), “Curriculum is an organized framework that delineates the content children are to learn, the processes through which children achieve identified curricular goals, what teachers do to help achieve these goals, and the context in which teaching and learning occur” (2009). In other words, curriculum helps teachers organize the day to day activities, it outlines the learning goals and outcomes that teachers need to assess, and lastly, curriculum provides teachers with the guidance and structure that they need to teach. [52]


3.1: – in vitro Neuronal Models
A variety of in vitro models can be used to examine circuits as well as to test many of the techniques mentioned in this unit and to help understand development of the nervous system. Often these in vitro models have been developed to provide a simplified understanding of the much more complex in vivo condition. These reduced complexity models allow for a simplified approach to studying key neuronal processes on both the cellular and molecular level. At the same time, some tissue preparations suffer several limitations due to their simplicity, reducing direct in vivo comparisons.
Expression systems and immortalized cell lines
The simplest in vitro electrophysiological models include heterologous and recombinant expression systems which are cells/cell lines that can be maintained in culture for an extended period of time. The cells/cell lines typically used as heterologous ( e.g. , Xenopus oocytes; or recombinant expression systems ( e.g. , human embryonic kidney 293 (HEK-293) cells, Chinese hamster ovary (CHO) cells)) that are easily maintained, allow for manual and automated electrophysiological techniques and express high levels of desired protein within a short period of time that can be consistently observed. As such, these systems have been used extensively to evaluate the pharmacological properties and structure-function relationships of multiple neuron ion-channels. However, despite their simplicity and ubiquitous use, these cells lack many of the complexities associated with neuronal function within the intact brain (e.g. network associations, glial interactions, and developmental regulation) — a disadvantage when modeling the brain. Furthermore, these cells are typically of a non-neuronal origin and thus lack the same sophisticated level of cellular architecture, sub-cellular organization or biochemistry associated with native neuronal preparations.
Early efforts to address these non-neuronal concerns focused on neuronal cells derived from mouse neuroblastoma C-1300 tumor (e.g. N1E-115)) or the human SH-SY5Y neuroblastoma cell line. However, subsequent advances in molecular biology enable the use of neural stem cells (NSCs). NSCs are uncommitted cells with self-renewal potential and the ability to differentiate into cells of all neural lineages. These cells can be derived from several sources such as pluripotent embryonic stem cells isolated from the blastocyst, human umbilical cord blood, induced pluripotent stem cells and multipotent somatic progenitors derived from several tissues including the CNS. Electrophysiologically, these cells possess Na + , K + and Ca 2+ currents that resemble the known patterns described for their in vivo neuronal counterparts, even at early stages of differentiation. Furthermore, these cells are also capable of forming rudimentary, yet functional, glutamatergic and GABAergic synapses in culture. Limitations in the use of cells obtained from adults offer limited neural lineage potential and senesce after only a few passages ( Jakel, Schneider, & Svendsen, 2004 ). Moreover, NSC cultures may possess mixtures of both undifferentiated and differentiated neurons, for which some neurons are developmentally immature, and thus hinder extrapolation of data to the adult in vivo condition.
Dissociated neuronal primary cultures
Increasing in complexity, dissociated neuronal primary cultures represent another common tissue preparation. These cultures are mechanically and enzymatically dissociated from various brain regions ( e.g. , hippocampus, cortex, cerebellum, striatum, midbrain, superior cervical ganglion, etc.) and consist of either one predominant neuronal cell type, a co-mixture of different neuronal populations or mixed neuronal–glial cultures. Dissociated neurons and astrocytes retain much of their functional capacity in vitro enabling these preparations to address many important processes observed in the in vivo condition such as network dynamics and neuronal-glial interactions but dissociated neurons cannot be maintained in culture for extended periods of time and thus are required to be freshly isolated and grown on a regular basis.
Three-dimensional (3D) neuronal organoid models
The 3D neuronal model represents the next level of complexity for CNS in vitro models. Like the two-dimensional (2D) preparations discussed above, 3D brain cell cultures can consist of a co-mixture of different neuronal and non-neuronal populations. Interestingly, instead of being cultured in a traditional planar monolayer, 3D brain cultures are created up to 10 cell diameters thick within reaggregate or spherical cultures (i.e. spheroids), hydrogel/scaffold cultures or rotary bioreactor cultures with cell aggregates or microcarriers . When grown in a 3D environment, neural cells demonstrate better survivability and behave differently when compared to traditional 2D-models. As such, these models promote better development of native voltage-gated ion-channel functionality, resting membrane potentials, intracellular Ca2 + dynamics, Na+/H+ exchange, enhanced neurogenesis and differentiation, synapse formation, neuronal mobility and axon myelination ( Lancaster & Knoblich, 2014; Lancaster et al., 2013; LaPlaca et al., 2010; van Vliet et al., 2007 ).


3.6: Researching the Topic
Many written assignments in your nursing program will require you to research your topic before drafting and writing your paper. Although researching a topic is not the main focus of this book, here are a few ideas related to this stage of your journey to becoming a writer:
- Consult library resources —there are many workshops you can participate in and online resources such as research guides, information sheets, and tutorials that you can access on the library website. As an example, check out the Ryerson University library resources: Research help and Research help guide: Find sources or check out your own library research resources.
- Talk to librarians —they have a wealth of information about how to research topics. Librarians can teach you how to best use literature search tools such as databases and search engines that organize the information that you will be looking for throughout your nursing program. As part of your post-secondary education, you will have free access to these search engines and many articles and resources.
- Take a research course during your post-secondary nursing education that will further expose you to researching your topic.
Think about your research journey as an ongoing and iterative process in which you move back and forth between the stages (see Figure 3.4 ). You begin by choosing a topic, then brainstorm relevant keywords to search in the databases, and then evaluate your resources. Now, you need to reflect on and develop skills in how to identify keywords and search databases.
Tips to consider as you research
Here are a few tips for you to consider as you are researching a topic, choosing keywords, and searching the various databases:
- Search specific peer-reviewed journals and review the issues and volumes of that journal.
- Become familiar with common databases to search for information related to a topic. In nursing, common databases include CINAHL, Medline, Proquest Nursing, and Evidence Based Medicine (EBM) Reviews. More information can be found at: Top Databases for Nursing .
- Decide on the keywords to search. For example, if you are writing a paper about nursing interventions related to helping patients manage stress, you might search keywords such as “stress management” with a related word such as “coping.” Check out this link and scroll to the bottom of the page about Medical Subject Headings (MeSH) to help you narrow your search to a specific concept. You can also look at the keywords used by the authors in the journal articles that you are reviewing.


7.3: Adiabatic Temperature Change and Stability
In "The Atmosphere" we discovered that air temperature usually decreases with an increase in elevation through the troposphere. The decrease in temperature with elevation is called the environmental lapse rate of temperature or normal lapse rate of temperature . Recall that the normal lapse rate of temperature is the average lapse rate of temperature of .65 o C / 100 meters. The environmental lapse rate of temperature is the actual vertical change in temperature on any given day and can be greater or less than .65 o C / 100 meters. Also recall that the decrease in temperature with height is caused by increasing distance from the source of energy that heats the air, the Earth's surface. Air is warmer near the surface because it's closer to its source of heat. The further away from the surface, the cooler the air will be. It's like standing next to a fire, the closer you are the warmer you'll feel. Temperature change caused by an exchange of heat between two bodies is called diabatic temperature change . There is another very important way to change the temperature of air called adiabatic temperature change.
Adiabatic temperature change of air occurs without the addition or removal of energy. That is, there is no exchange of heat with the surrounding environment to cause the cooling or heating of the air. The temperature change is due to work done on a parcel of air by the external environment, or work done by a parcel of air on the air that surrounds it. What kind of work can be done? The work that is done is the expansion or compression of air.
Imagine an isolated parcel of air that is moving vertically through the troposphere. We know that air pressure decreases with increasing elevation. As the parcel of air moves upward the pressure exerted on the parcel decreases and the parcel expands in volume as a result. In order to expand (i.e.. do work), the parcel must use its internal energy to do so. As the air expands, the molecules spread out and ultimately collide less with one another. The work of expansion causes the air's temperature to decrease. You might have had personal experience with this kind of cooling if you've let the air out of an automobile or bicycle tire. Air inside the tire is under a great deal of pressure, and as it rushes outside it moves into a lower pressure environment. In so doing, the parcel quickly expands against the outside environment air. By placing your hand in front of the valve stem, you can feel the air cool as it expands. This is called adiabatic cooling .
As air descends through the troposphere it experiences increasing atmospheric pressure. This causes the parcel volume to decrease in size, squeezing the air molecules closer together. In this case, work is being done on the parcel. As the volume shrinks, air molecules bounce off one another more often ricocheting with greater speed. The increase in molecular movement causes an increase in the temperature of the parcel. This process is referred to as adiabatic warming .
The rate at which air cools or warms depends on the moisture status of the air. If the air is dry, the rate of temperature change is 1 o C/100 meters and is called the dry adiabatic rate (DAR). If the air is saturated, the rate of temperature change is .6 o C/100 meters and is called the saturated adiabatic rate (SAR). The DAR is a constant value, that is, it's always 1 o C/100 meters. The SAR varies somewhat with how much moisture is in the air, but we'll assume it to be a constant value here. The reason for the difference in the two rates is due to the liberation of latent heat as a result of condensation. As saturated air rises and cools, condensation takes place. Recall that as water vapor condenses, latent heat is released. This heat is transferred into the other molecules of air inside the parcel causing a reduction in the rate of cooling.
Stability of Air
Adiabatic temperature change is an important factor in determining the stability of the air. We can think of air stability as the tendency for air to rise or fall through the atmosphere under its own "power". Stable air has a tendency to resist movement. On the other hand, unstable air will easily rise. What gives air "power" to rise? The tendency for air to rise or fall depends on the adiabatic and environmental lapse rates.
Stable air
Stable atmospheric conditions prevail when the environmental lapse rate is less than the saturated adiabatic rate. An example of this condition is shown in Figure \(\PageIndex{1}\). At the surface (0 meters) both the parcel of air (red line) and the air of the surrounding environment (blue line) have the same temperature. The surrounding air is changing its temperature at a rate of .65 o C/100 meters. The parcel on this day is "dry" and will rise and cool at a temperature of 1 o C/100 meters. After giving the parcel a slight upward push, it rises to a level of 1000 meters where it cools to a temperature of 20 o C. A measurement of the air surrounding the parcel shows a temperature of 23.5 o C. In other words, the parcel is colder (and more dense) than the surrounding air at 1000 meters. If the uplift mechanism ceased, the parcel of air would return to the surface.
Unstable air
Air is unstable when the environmental lapse rate is greater than the dry adiabatic rate. Under these conditions, a rising parcel of air is warmer and less dense than the air surrounding it at any given elevation. Figure \(\PageIndex{2}\) depicts unstable conditions. Follow up the graph for the rising parcel of air. Note that at any elevation above the surface the parcel temperature is higher than the air that surrounds it. Even as it reaches the dew point temperature at 2000 meters, the air remains warmer than the surrounding air. As a result it continues to rise and cool at the saturated adiabatic rate. Vertically developed clouds are likely to develop under unstable conditions such as this.
According to the graph above, the air is absolutely stable at letter A, B, C, or D?
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C


35.13: The Peripheral Nervous System - Autonomic Nervous System
- Explain the function of the autonomic nervous system
Autonomic Nervous System
The autonomic nervous system (ANS) serves as the relay between the central nervous system (CNS) and the internal organs. It controls the lungs, the heart, smooth muscle, and exocrine and endocrine glands, largely without conscious control. It can continuously monitor the conditions of these different systems and implement changes as needed. Signaling to the target tissue usually involves two synapses. A preganglionic neuron (originating in the CNS) synapses to a neuron in a ganglion that, in turn, synapses on the target organ. There are two divisions of the autonomic nervous system that often have opposing effects: the sympathetic nervous system and the parasympathetic nervous system.
Sympathetic Nervous System
The sympathetic nervous system is responsible for the “fight or flight” response that occurs when an animal encounters a dangerous situation. One way to remember this is to think of the surprise a person feels when encountering a snake (“snake” and “sympathetic” both begin with “s”). Examples of functions controlled by the sympathetic nervous system include an accelerated heart rate and inhibited digestion, both of which help prepare an organism’s body for the physical strain required to escape a potentially dangerous situation or to fend off a predator.
Most preganglionic neurons in the sympathetic nervous system originate in the spinal cord. The axons of these neurons release acetylcholine on postganglionic neurons within sympathetic ganglia (the sympathetic ganglia form a chain that extends alongside the spinal cord). The acetylcholine activates the postganglionic neurons. Postganglionic neurons then release norepinephrine onto target organs. As anyone who has ever felt a rush before a big test, speech, or athletic event can attest, the effects of the sympathetic nervous system are quite pervasive. This is both because one preganglionic neuron synapses on multiple postganglionic neurons, amplifying the effect of the original synapse, and because the adrenal gland also releases norepinephrine (and the closely-related hormone epinephrine) into the blood stream. The physiological effects of this norepinephrine release include dilating the trachea and bronchi (making it easier for the animal to breathe), increasing heart rate, and moving blood from the skin to the heart, muscles, and brain (so the animal can think and run). The strength and speed of the sympathetic response helps an organism avoid danger. Scientists have found evidence that it may also increase long term potentiation in neurons, allowing the animal to remember the dangerous situation and avoid it in the future.
Parasympathetic Nervous System
While the sympathetic nervous system is activated in stressful situations, the parasympathetic nervous system allows an animal to “rest and digest.” One way to remember this is to think that during a restful situation like a picnic, the parasympathetic nervous system is in control (“picnic” and “parasympathetic” both start with “p”). Parasympathetic preganglionic neurons have cell bodies located in the brainstem and in the sacral (toward the bottom) spinal cord. The axons of the preganglionic neurons release acetylcholine on the postganglionic neurons, which are generally located very near the target organs.
The parasympathetic nervous system resets organ function after the sympathetic nervous system is activated (the common adrenaline dump you feel after a ‘fight-or-flight’ event). Effects of acetylcholine release on target organs include slowing of heart rate, lowered blood pressure, and stimulation of digestion.
Key Points
- The autonomic nervous system controls the workings of internal organs such as the heart, lungs, digestive system, and endocrine systems; it does so without conscious effort.
- The sympathetic nervous system controls the body’s automatic response to danger, increasing the heart rate, dilating the blood vessels, slowing digestion, and moving blood flow to the heart, muscles, and brain.
- The parasympathetic nervous system works in opposition to the sympathetic; during periods of rest it slows the heart rate, lowers the blood pressure, stimulates digestion, and moves blood flow back to the skin.
Key Terms
- preganglionic : describing the nerve fibres that supply a ganglion
- sympathetic nervous system : the part of the autonomic nervous system that under stress raises blood pressure and heart rate, constricts blood vessels and dilates the pupils
- parasympathetic nervous system : one of the divisions of the autonomic nervous system, based between the brain and the spinal cord, that slows the heart and relaxes muscles
- acetylcholine : a neurotransmitter in humans and other animals, which is an ester of acetic acid and choline


2.1: Introduction
In the United States, law and government are interdependent. The US Constitution establishes the basic framework of the federal government and imposes certain limitations on the powers of government. In turn, the various branches of government are intimately involved in making, enforcing, and interpreting the law. Most law comes from Congress and the state legislatures. Courts interpret the laws and apply them to cases. We will learn more about the US Constitution's framework in Chapter 6 , but we will begin learning about the American court system by introducing Article III of the US Constitution.
Laws are meaningless if they are not enforced. Companies have to make many decisions daily, from product development to marketing to maintaining growth. These decisions are based on financial considerations and legal requirements. If a company violates a law, it is often held accountable through litigation in courts.
Because federal judges are appointed for life, businesses cannot directly influence the actions of the judicial branch. However, they can do so indirectly by lobbying Congress on laws that it considers and lobbying the president concerning enforcement priorities. While all states have a comparable three-branch system, in some states [such as Wisconsin], judges obtain office through partisan elections. In such states, businesses can seek to influence the judicial branch by supporting judges whose philosophy favors business generally or a particular industry. For these reasons, in choosing whether to litigate in state or federal court, businesses should consider that federal judges may be more likely to take politically unpopular actions. ~ John W., Judge


2.6: Neurophysiological Background
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Presenting Neurophysiology in its entirety would be enough to fill several books. Fortunately we do not have to concern ourselves with most of these facts. Instead, let's just focus on the aspects that are really relevant to problem solving. Nevertheless this topic is quite complex and problem solving cannot be attributed to one single brain area. Rather there are systems of several brain areas working together to perform a specific task. This is best shown by an example:
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In 1994 Paolo Nichelli and coworkers used the method of PET (Positron Emission Tomography), to localise certain brain areas, which are involved in solving various chess problems. In the following table you can see which brain area was active during a specific task:
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Task
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Location of Brain activity
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determining location of pieces
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Thinking about making a move
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Remembering a pieces move
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Planning and executing strategies
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Pathway from Occipital to Temporal Lobe
(also called the "what"-pathway of visual processing)
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Pathway from Occipital to parietal Lobe
(also called the "where"-pathway of visual processing)
(forming new memories)
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Lobes of the Brain
One of the key tasks, namely
planning and executing strategies
, is performed by a brain area which also plays an important role for several other tasks correlated with problem solving –
the prefrontal cortex (PFC)
. This can be made clear if you take a look at several examples of damages to the PFC and their effects on the ability to solve problems.
Patients with a
lesion
in this brain area have difficulty switching from one behaviouristic pattern to another. A well known example is the
wisconsin card-sorting task
. A patient with a PFC lesion who is told to separate all blue cards from a deck, would continue sorting out the blue ones, even if the experimenter told him to sort out all brown cards. Transferred to a more complex problem, this person would most likely fail, because he is not flexible enough to change his strategy after running into a
dead end
.
Another example is the one of a young homemaker, who had a tumour in the frontal lobe. Even though she was able to cook individual dishes, preparing a whole family meal was an infeasible task for her.
As the examples above illustrate, the structure of our brain seems to be of great importance regarding problem solving, i.e. cognitive life. But how was our cognitive apparatus designed? How did perception-action integration as a central species specific property come about?


15.3: Two-Group Comparison
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One of the simplest inferential analyses is comparing the posttest outcomes of treatment and control group subjects in a randomised posttest-only control group design—such as whether students enrolled to a special program in mathematics perform better than those in a traditional math curriculum. In this case, the predictor variable is a dummy variable (1 = treatment group, 0 = control group), and the outcome variable, performance, is ratio-scaled—e.g., score on a math test following the special program. The analytic technique for this simple design is a one-way ANOVA (one-way because it involves only one predictor variable), and the statistical test used is called a Student’s -test , or -test).
The -test was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness Brewery in Dublin, Ireland to monitor the quality of stout—a dark beer popular with ninteenth century porters in London. Because his employer did not want to reveal the fact that it was using statistics for quality control, Gosset published the test in Biometrika using his pen-name ‘Student’—he was a student of Sir Ronald Fisher—and the test involved calculating the value of t, which was a letter used frequently by Fisher to denote the difference between two groups. Hence, the name Student’s -test, although Student’s identity was known to fellow statisticians.
The -test examines whether the means of two groups are statistically different from one another (non-directional or two-tailed test), or whether one group has a statistically larger (or smaller) mean than the other (directional or one-tailed test). In our example, if we wish to examine whether students in the special math curriculum perform better than those in the traditional curriculum, we have a one-tailed test. This hypothesis can be stated as:
\mu_{2} \qquad \mbox{(alternative hypothesis)} \end{eqnarray*}" decoding="async" height="44" loading="lazy" src=" https://usq.pressbooks.pub/app/uploa...69ee5fc_l3.svg " title="Rendered by QuickLaTeX.com" width="344">
where represents the mean population performance of students exposed to the special curriculum (treatment group) and is the mean population performance of students with traditional curriculum (control group). Note that the null hypothesis is always the one with the ‘equal’ sign, and the goal of all statistical significance tests is to reject the null hypothesis.
How can we infer about the difference in population means using data from samples drawn from each population? From the hypothetical frequency distributions of the treatment and control group scores in Figure 15.2, the control group appears to have a bell-shaped (normal) distribution with a mean score of 45 (on a 0–100 scale), while the treatment group appear to have a mean score of 65. These means look different, but they are really sample means ( ), which may differ from their corresponding population means ( ) due to sampling error. Sample means are probabilistic estimates of population means within a certain confidence interval—95% CI is the sample mean two standard errors, where standard error is the standard deviation of the distribution in sample means as taken from infinite samples of the population. Hence, statistical significance of population means depends not only on sample mean scores, but also on the standard error or the degree of spread in the frequency distribution of the sample means. If the spread is large (i.e., the two bell-shaped curves have a lot of overlap), then the 95% CI of the two means may also be overlapping, and we cannot conclude with high probability ( ) that that their corresponding population means are significantly different. However, if the curves have narrower spreads—i.e., they are less overlapping), then the CI of each mean may not overlap, and we reject the null hypothesis and say that the population means of the two groups are significantly different at .
To conduct the -test, we must first compute a -statistic of the difference in sample means between the two groups. This statistic is the ratio of the difference in sample means relative to the difference in their variability of scores (standard error):
where the numerator is the difference in sample means between the treatment group (Group 1) and the control group (Group 2) and the denominator is the standard error of the difference between the two groups, which in turn, can be estimated as:
is the variance and is the sample size of each group. The -statistic will be positive if the treatment mean is greater than the control mean. To examine if this -statistic is larger than that possible by chance, we must look up the probability or -value associated with our computed -statistic in statistical tables available in standard statistics textbooks or on the Internet or as computed by statistical software programs such as SAS and SPSS. This value is a function of the -statistic, whether the -test is one-tailed or two-tailed, and the degrees of freedom ( ) or the number of values that can vary freely in the calculation of the statistic—usually a function of the sample size and the type of test being performed. The degree of freedom of the t-statistic is computed as:
which often approximates to ( ). If this -value is smaller than a desired significance level (say ) or the highest level of risk (probability) we are willing to take to conclude that there is a treatment effect when in fact there is none (Type I error), then we can reject the null hypotheses.
After demonstrating whether the treatment group has a significantly higher mean than the control group, the next question usually is what is the effect size (ES) or the magnitude of the treatment effect relative to the control group? We can estimate the ES by conducting regression analysis with performance scores as the outcome variable ( ) and a dummy coded treatment variable as the predictor variable ( ) in a two-variable GLM. The regression coefficient of the treatment variable ( ), which is also the slope of the regression line ( ), is an estimate of the effect size. In the above example, since is a dummy variable with two values (0 and 1), , and hence the effect size or is simply the difference between treatment and control means .