4.3: Nonmaterial Contributions
Nonmaterial contributions from nature, also called cultural services, include the subjective and psychological aspects of nature that influence our perceptions about quality of life. These contributions can be divided into three subcategories: inspiration and learning support, supporting psychological and physical experiences, and supporting individual and group identities.
Inspiration and learning support
Nature has inspired artists and writers throughout history. Consequently, many books, television programmes, movies, and websites produced for entertainment purposes are based on natural themes. This infusion of nature into popular culture is worth billions of dollars per year. To take one example, the 1994 Disney blockbuster The Lion King , based on the lives of a variety of African savannah animals, generated revenues estimated at just under US $1 billion from theatre attendances alone. It was so successful that three movie sequels, an animated television series, and several video games and books followed. A musical based on The Lion King movie plot continues to be a top-earning title in box-office history for both stage productions and films.
Movies featuring stunning natural landscapes and charismatic wildlife often increase the desire of moviegoers to visit natural areas where they can see these landscapes and animals first-hand. But it can also raise awareness of environmental issues in new audiences. While many documentaries are created with this purpose in mind, such benefits can also extend to blockbuster movies meant for broader audiences (Silk et al., 2018). For example, Disney’s Happy Feet (2006) highlighted the threat of overfishing and plastic pollution to penguins; Avatar (2009) raised awareness of habitat loss and overharvesting; and The Jungle Book (2016) exposed audiences to the plight of pangolins. Such exposure can even lead to environmentally conscious behavioural changes. For example, moviegoers were willing to donate 50% more money to climate mitigation after watching the apocalyptic movie The Day After Tomorrow (2004) (Balmford et al., 2004). Perhaps, in part, due to the influence of environmentally-orientated movies, an increasing number of movie stars (and other celebrities) have started using their stardom as a platform from where they promote biodiversity conservation efforts in Africa (Duthie et al. 2017; see also https://wildfor.life ).
Scientists and engineers often seek inspiration from nature for new technologies or to solve innovation challenges.
Scientists and engineers also sometimes turn to nature to seek inspiration for new technologies or to solve innovation challenges. For example, the water-vapor collecting capacity of the racing stripe darkling beetle ( Stenocara gracilipes ) from the Namib Desert in Namibia (Figure 4.7) inspired engineers who developed self-filling water bottles (Clark, 2012), irrigation systems to overcome drought conditions (Scott, 2011), fog-free windows and mirrors (Parker and Lawrence, 2011), and methods for controlling condensation and frost on aircraft surfaces (Boreyko et al., 2016). While these and other scientific endeavours, collectively known as biomimicry , provide many social and economic benefits, their primary value comes from new knowledge, improved education, and enriched human experiences.
Supporting psychological and physical experiences
While the economic benefits gained from nature incentivises biodiversity protection, many people believe that the aesthetic values of nature provide an even greater incentive for conservation. This principle rests on the fact that nearly everyone enjoys wildlife and landscapes aesthetically. Even city-dwellers who are superficially removed from nature find a sense of relief and well-being when they have opportunities to come in close contact with the natural world. But what if dragonflies and butterflies disappeared? What if our favourite sports team’s mascot ceased to exist in nature? What if there were no more forests filled with bird flocks or monkey troops?
The intangible but desirable aesthetic values people attach to certain aspects of nature are known as amenity values . Amenity values are becoming increasingly important in many local and national economies throughout Africa, in the form ecotourism . At any one time, there are millions of tourists traveling and spending money across Africa to see particular species or to experience unique ecosystems. This includes scuba divers approaching a coral reef (Figure 4.8), birdwatchers visiting a rare species’ stakeout, and people on a safari to view the many flagship species for which Africa’s savannahs are so well known. Ecotourism has long been a major industry in southern and East Africa. For example, ecotourism generated over US $1 billion in annual revenue in the Cape Floristic Region more than a decade ago (Turpie et al., 2003), and has accounted for over 15% of Kenya’s gross domestic product (GDP) at times (WWF and BSI, 2006). Ecotourism is also becoming increasingly important in other parts of Africa. For example, since overcoming periods of social unrest, Burundi, Rwanda, and Uganda have created profitable local industries charging tourists high fees to visit habituated populations of mountain gorillas ( Gorilla beringei beringei , EN). Also, in South Africa, some bird guides earn an average of US $362 per month by showing tourists the unique birds their local area has to offer (Biggs et al., 2011).
In recent years, volunteer-based ecotourism has emerged as a lucrative industry that combines ecotourism with learning opportunities. These organisations offer aspiring conservationists and citizen scientists hands-on experience while bringing financial and other logistical support to rural and protected areas. Many wildlife sanctuaries and conservation NGOs also offer volunteer opportunities and field courses that combine conservation action with local community outreach and education programmes. The research done by professional scientists and citizen science volunteers can be used in locally-relevant educational materials. Biological field stations (Section 13.1.5) often host these activities; the stations can also provide training and jobs for local community members.
The revenue and jobs generated by ecotourism provides a strong and immediate justification to protect biodiversity and restore areas that have been degraded.
The revenue and jobs generated by ecotourism provides a strong and immediate justification to protect areas rich with biodiversity or to restore areas that have been degraded. Ecotourism can even be integrated directly in plans for future development, protection, and restoration. One such example is integrated conservation and development projects (ICDPs, Section 14.3), which provide models for how empowered rural communities can successfully establish accommodation, develop expertise in nature guiding, and sell local handicrafts at curio stores to obtain multiple stable income streams. The revenue obtained from ecotourism also allows local people to move away from unsustainable hunting, fishing, or grazing practices towards lifestyles that can be maintained in the long term.
Still, many of Africa’s ecotourism resources remain under-utilised. To use one example, only a few locations in Africa cater to people who enjoy the thrill of swimming with sharks in their natural habitat. Beyond removing fear and instilling a healthy respect for sharks, this industry also plays an important role in conservation by showing how living sharks bring greater economic benefit than a once-off catch. For example, shark diving at just one location in South Africa is estimated at US $4.4 million annually (Hara et al., 2003); similar industries in the Maldives (Cagua et al., 2014) and Palau (Vianna et al., 2012) generate even more revenue. Presenting unique recreational experiences and a growing global ecotourism sector, more and more African countries will hopefully explore these and other opportunities soon. It is worth noting that the long-term effects of shark diving operations are largely unknown, particularly as it relates to possible behavioural changes from using bait to attract sharks to people, and an active area of current research (Gallagher and Huveneers, 2018).
Although ecotourism can provide many valuable conservation and economic benefits (Thiel et al., 2014), care must always be taken that these activities abide by accepted ethical standards (Hayward et al., 2012). It is also worth remembering that wildlife ecotourism is often geared towards wealthy western markets, making it prohibitively costly to the people who live near facilities, and are most vulnerable to factors such as human-wildlife conflict . As such, it is important to consider what portion of the generated funds are invested locally versus reserved to enrich well-compensated shareholders in the far-away capital. Are local people given opportunities to further their training and education, and to advance their careers within ecotourism organisations? Unfortunately, in many areas of Africa, local people continue to receive only the smallest percentage of money spent by tourists. Similarly, even though national parks themselves may receive large numbers of foreign visitors, governments continue to use only a small percentage of the generated funds on park management (Lindsey et al., 2014; Balmford et al., 2015).
Supporting individual and group identities
Many people care deeply about biodiversity. The thought of a charismatic animal or a special landscape (Figure 4.9) may elicit a strong emotional response, which leads to a desire to protect plants, animals, and natural places. For some people, this desire is associated with a hope to someday see those unique species or landscapes in person. Others do not expect or even desire to see these species and landscape themselves, yet they value their existence. In either case, these individuals recognize the existence values of wildlife and nature—the benefit people receive from simply knowing that an ecosystem or species exists. Bequest values (also known as beneficiary values) is a component of existence values, defined as the perceived benefit people receive from preserving a natural resource or species for future generations.
The desire to ensure the protection of biodiversity has prompted a wide range of people to establish, join, or otherwise contribute to conservation organisations. For many people involved in these organisations, their participation stems from the ethical premise that wildlife are equal to human life, and that biodiversity conservation offers genuine and long-lasting well-being. This environmental philosophy is often described as deep ecology , the ethical premise that species and biodiversity have a right to exist independent of their possible benefits to humans, and that humans have an inherent responsibility to protect species and biodiversity (see also Section 1.4). Deep ecology holds that social structures (including politics, economics, technology, and ideology) must change radically to reduce the destruction of Earth’s biodiversity and to enhance people’s quality of life. It emphasises and prioritises the natural environment, aesthetics, religion, and culture, rather than material consumption. Although the ethical appreciation of biodiversity is similar in deep ecology and conservation biology, deep ecology includes broader goals for personal, social, and political change.
Biodiversity forms the basis of spiritual, celebratory, and other social-cohesion experiences for many people.
Biodiversity also forms the basis of spiritual, celebratory, and other social-cohesion experiences for many people. It ensures people experience a sense of place and belonging, reminds them of childhood experiences, and gives a sense of connection when they experience natural sights, sounds and smells. This is especially true for Africans, many of whom attach deeply-held spiritual, cultural, and symbolic values to the environment. Even the money of most, perhaps all, African countries features aspects of nature, as if to add a little extra (if only symbolic) value to those coins and bills. All these factors play a major role in people’s sense of who we are—our identity.


6: Finite Difference Approximation
We introduce here numerical differentiation, also called finite difference approximation. This technique is commonly used to discretize and solve partial differential equations.
Finite difference formulas
Consider the Taylor series approximation for \(y(x+h)\) and \(y(x-h)\) , given by
\[\begin{aligned} &y(x+h)=y(x)+h y^{\prime}(x)+\frac{1}{2} h^{2} y^{\prime \prime}(x)+\frac{1}{6} h^{3} y^{\prime \prime \prime}(x)+\frac{1}{24} h^{4} y^{\prime \prime \prime \prime}(x)+\ldots, \\ &y(x-h)=y(x)-h y^{\prime}(x)+\frac{1}{2} h^{2} y^{\prime \prime}(x)-\frac{1}{6} h^{3} y^{\prime \prime \prime}(x)+\frac{1}{24} h^{4} y^{\prime \prime \prime \prime}(x)+\ldots . \end{aligned} \nonumber \]
The standard definitions of the derivatives give the first-order approximations
\[\begin{aligned} &y^{\prime}(x)=\frac{y(x+h)-y(x)}{h}+\mathrm{O}(h), \\ &y^{\prime}(x)=\frac{y(x)-y(x-h)}{h}+\mathrm{O}(h) . \end{aligned} \nonumber \]
The more widely-used second-order approximation is called the central-difference approximation and is given by
\[y^{\prime}(x)=\frac{y(x+h)-y(x-h)}{2 h}+\mathrm{O}\left(h^{2}\right) . \nonumber \]
The finite difference approximation to the second derivative can be found from considering
\[y(x+h)+y(x-h)=2 y(x)+h^{2} y^{\prime \prime}(x)+\frac{1}{12} h^{4} y^{\prime \prime \prime \prime}(x)+\ldots, \nonumber \]
from which we find
\[y^{\prime \prime}(x)=\frac{y(x+h)-2 y(x)+y(x-h)}{h^{2}}+\mathrm{O}\left(h^{2}\right) . \nonumber \]
Often a second-order method is required for \(x\) on the boundaries of the domain. For a boundary point on the left, a second-order forward difference method requires the additional Taylor series
\[y(x+2 h)=y(x)+2 h y^{\prime}(x)+2 h^{2} y^{\prime \prime}(x)+\frac{4}{3} h^{3} y^{\prime \prime \prime}(x)+\ldots \nonumber \]
We combine the Taylor series for \(y(x+h)\) and \(y(x+2 h)\) to eliminate the term proportional to \(h^{2}\) :
\[y(x+2 h)-4 y(x+h)=-3 y(x)-2 h y^{\prime}(x)+\mathrm{O}\left(\mathrm{h}^{3}\right) \text {. } \nonumber \]
Therefore,
\[y^{\prime}(x)=\frac{-3 y(x)+4 y(x+h)-y(x+2 h)}{2 h}+\mathrm{O}\left(h^{2}\right) . \nonumber \]
For a boundary point on the right, we send \(h \rightarrow-h\) to find
\[y^{\prime}(x)=\frac{3 y(x)-4 y(x-h)+y(x-2 h)}{2 h}+\mathrm{O}\left(h^{2}\right) . \nonumber \]
Example: the Laplace equation
As an example of the finite difference technique, let us consider how to discretize the two dimensional Laplace equation
\[\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) \Phi=0 \nonumber \]
on the rectangular domain \([0,1] \times[0,1]\) . We assume here that the values of \(\Phi\) are known on the boundaries of the domain. We form a two-dimensional grid with \(N\) intervals in each direction together with end points that lie on the boundaries by writing
\[\begin{array}{ll} x_{i}=i h, & i=0,1, \ldots, N, \\ y_{j}=j h, & j=0,1, \ldots, N, \end{array} \nonumber \]
where \(h=1 / N\) . We will also denote the value of \(\Phi\left(x_{i}, y_{j}\right)\) by \(\Phi_{i, j}\) . The finite difference approximation for the second derivatives at the interior point \(\left(x_{i}, y_{j}\right)\) then results in an equation that we write in the form
\[4 \Phi_{i, j}-\Phi_{i+1, j}-\Phi_{i-1, j}-\Phi_{i, j+1}-\Phi_{i, j-1}=0, \nonumber \]
valid for \(i=1,2, \ldots, N-1\) and \(j=1,2, \ldots, N-1\) . The boundary values \(\Phi_{0, j}, \Phi_{N, j}, \Phi_{i, 0}\) , and \(\Phi_{i, N}\) are assumed to be given.
One can observe that (6.10) represents a system of \((N-1)^{2}\) linear equations for \((N-1)^{2}\) unknowns. We can write this as a matrix equation if we decide how to order the unknowns \(\Phi_{i, j}\) into a vector. The standard ordering is called natural ordering, and proceeds from the bottom of the grid along rows moving towards the top. This orders the unknowns as
\[\Phi=\left[\Phi_{1,1}, \Phi_{2,1}, \ldots, \Phi_{(N-1), 1}, \ldots, \Phi_{1,(N-1)}, \Phi_{2(N-1)}, \ldots, \Phi_{(N-1),(N-1)}\right]^{T} . \nonumber \]
To illustrate the construction of the matrix equation, we consider the case \(N=3\) , with two interior points in each direction. The four resulting linear equations with the boundary terms written on the right-hand-side are
\[\begin{aligned} &4 \Phi_{1,1}-\Phi_{2,1}-\Phi_{1,2}=\Phi_{0,1}+\Phi_{1,0} \\ &4 \Phi_{2,1}-\Phi_{1,1}-\Phi_{2,2}=\Phi_{3,1}+\Phi_{2,0} \\ &4 \Phi_{1,2}-\Phi_{2,2}-\Phi_{1,1}=\Phi_{0,2}+\Phi_{1,3} \\ &4 \Phi_{2,2}-\Phi_{1,2}-\Phi_{2,1}=\Phi_{3,2}+\Phi_{2,3} \end{aligned} \nonumber \]
and the corresponding matrix equation is
\[\left(\begin{array}{rrrr} 4 & -1 & -1 & 0 \\ -1 & 4 & 0 & -1 \\ -1 & 0 & 4 & -1 \\ 0 & -1 & -1 & 4 \end{array}\right)\left(\begin{array}{l} \Phi_{1,1} \\ \Phi_{2,1} \\ \Phi_{1,2} \\ \Phi_{2,2} \end{array}\right)=\left(\begin{array}{l} \Phi_{0,1}+\Phi_{1,0} \\ \Phi_{3,1}+\Phi_{2,0} \\ \Phi_{0,2}+\Phi_{1,3} \\ \Phi_{3,2}+\Phi_{2,3} \end{array}\right) . \nonumber \]
The pattern here may not be obvious, but the Laplacian matrix decomposes into 2 -by- 2 block matrices.
With some thought, a general result can be obtained for an \(N_{x}\) -by- \(N_{y}\) grid of internal points (with \(N_{x}+1\) and \(N_{y}+1\) intervals in the \(x\) - and y-directions, respectively). The Laplacian matrix then decomposes into \(N_{x}\) -by- \(N_{x}\) block matrices. The diagonal contains \(N_{y}\) of these \(N_{x}\) -by- \(N_{x}\) block matrices, each of which are tridiagonal with a 4 on the diagonal and a \(-1\) on the off-diagonals. Immediately above and below these block matrices on the diagonal are \(N_{y}-1\) block matrices also of size \(N_{x}\) -by- \(N_{x}\) , each of which are diagonal with \(-1\) along the diagonal.
MATLAB code for the Laplacian matrix can be found on the web in the function sp_laplace.m. This code was written by Bordner and Saied in 1995, and I have written a more modern and faster version of this code in sp_laplace_new.m.
An alternative solution method, which we will later make use of in \(\$ 17.2 .4\) , includes the boundary values in the solution vector. If one of the boundary conditions is at position \(n\) in this vector, then row \(n\) of the left-hand-side matrix will have just a one on the diagonal with all other elements equal to zero (i.e, a row of the identity matrix), and the corresponding element in the right-hand-side vector will have the value at the boundary. Here, with the total number of grid points (including the boundary points) in the \(x\) - and \(y\) directions given by \(N_{x}\) and \(N_{y}\) , the left-hand-side matrix is then generated using \(A=\) sp_laplace_new (N_X, N_Y), and all the rows corresponding to the boundary values are replaced with the corresponding rows of the identity matrix. This formulation may be slightly easier to code, and also easier to incorporate other more general boundary conditions.


38.6: Bone - Cell Types in Bones
- Distinguish among the four cell types in bone
Bone consists of four types of cells: osteoblasts, osteoclasts, osteocytes, and osteoprogenitor (or osteogenic) cells. Each cell type has a unique function and is found in different locations in bones. The osteoblast, the bone cell responsible for forming new bone, is found in the growing portions of bone, including the periosteum and endosteum. Osteoblasts, which do not divide, synthesize and secrete the collagen matrix and calcium salts. As the secreted matrix surrounding the osteoblast calcifies, the osteoblast becomes trapped within it. As a result, it changes in structure, becoming an osteocyte, the primary cell of mature bone and the most common type of bone cell. Each osteocyte is located in a space (lacuna) surrounded by bone tissue. Osteocytes maintain the mineral concentration of the matrix via the secretion of enzymes. As is the case with osteoblasts, osteocytes lack mitotic activity. They are able to communicate with each other and receive nutrients via long cytoplasmic processes that extend through canaliculi (singular = canaliculus), channels within the bone matrix.
If osteoblasts and osteocytes are incapable of mitosis, then how are they replenished when old ones die? The answer lies in the properties of a third category of bone cells: the osteogenic cell. These osteogenic cells are undifferentiated with high mitotic activity; they are the only bone cells that divide. Immature osteogenic cells are found in the deep layers of the periosteum and the marrow. When they differentiate, they develop into osteoblasts. The dynamic nature of bone means that new tissue is constantly formed, while old, injured, or unnecessary bone is dissolved for repair or for calcium release. The cell responsible for bone resorption, or breakdown, is the osteoclast, which is found on bone surfaces, is multinucleated, and originates from monocytes and macrophages (two types of white blood cells) rather than from osteogenic cells. Osteoclasts continually break down old bone while osteoblasts continually form new bone. The ongoing balance between osteoblasts and osteoclasts is responsible for the constant, but subtle, reshaping of bone.
Key Points
- Osteogenic cells are the only bone cells that divide.
- Osteogenic cells differentiate and develop into osteoblasts which, in turn, are responsible for forming new bones.
- Osteoblasts synthesize and secrete a collagen matrix and calcium salts.
- When the area surrounding an osteoblast calcifies, the osteoblast becomes trapped and transforms into an osteocyte, the most common and mature type of bone cell.
- Osteoclasts, the cells that break down and reabsorb bone, stem from monocytes and macrophages rather than osteogenic cells..
- There is a continual balance between osteoblasts generating new bone and osteoclasts breaking down bone.
Key Terms
- osteoclast : a large multinuclear cell associated with the resorption of bone
- osteocyte : a mature bone cell involved with the maintenance of bone
- osteoprogenitor : a stem cell that is the precursor of an osteoblast
- canaliculus : any of many small canals or ducts in bone or in some plants
- periosteum : a membrane surrounding a bone
- endosteum : a membranous vascular layer of cells which line the medullary cavity of a bone
- lacuna : a small opening; a small pit or depression; a small blank space; a gap or vacancy; a hiatus
- osteoblast : a mononucleate cell from which bone develops


7.2: The Golden Ratio and Fibonacci Sequence
In this section, we will discuss a very special number called the Golden Ratio. It is an irrational number, slightly bigger than 1.6, and it has (somewhat surprisingly) had huge significance in the world of science, art and music. It was also discovered that this number has an amazing connection with what is called the Fibonacci Sequence, originally studied in the context of biology centuries ago. This unexpected link among algebra, biology, and the arts suggests the mathematical unity of the world and is sometimes discussed in philosophy as well.
Golden Ratio
With one number \(a\) and another smaller number \(b\), the ratio of the two numbers is found by dividing them. Their ratio is \(a/b\). Another ratio is found by adding the two numbers together \(a+b\) and dividing this by the larger number \(a\). The new ratio is \((a+b)/a\). If these two ratios are equal to the same number, then that number is called the Golden Ratio. The Greek letter \(\varphi\) (phi) is usually used to denote the Golden Ratio.
For example, if \(b = 1\) and \(a / b=\varphi\), then \(a=\varphi\). The second ratio \((a+b)/a\) is then \((\varphi+1) / \varphi\). Because these two ratios are equal, this is true:
\[\varphi=\dfrac{\varphi+1}{\varphi}\nonumber \]
(This equation has two solutions, but only the positive solution is referred to as the Golden Ratio \(\varphi\)).
One way to write this number is
\[\varphi=\dfrac{1+\sqrt{5}}{2} \nonumber \]
\(\sqrt{5}\) is the positive number which, when multiplied by itself, makes \(5: \sqrt{5} \times \sqrt{5}=5\).
The Golden Ratio is an irrational number. If a person tries to write the decimal representation of it, it will never stop and never make a pattern, but it will start this way: 1.6180339887... An interesting thing about this number is that you can subtract 1 from it or divide 1 by it, and the result will be the same.
\[\varphi-1=1.6180339887 \ldots-1=0.6180339887 \nonumber \]
\[1 / \varphi=\frac{1}{1.6180339887}=0.6180339887 \nonumber \]
Golden rectangle
If the length of a rectangle divided by its width is equal to the Golden Ratio, then the rectangle is called a "golden rectangle.” If a square is cut off from one end of a golden rectangle, then the other end is a new golden rectangle. In the picture, the big rectangle (blue and pink together) is a golden rectangle because \(a / b=\varphi\). The blue part (B) is a square. The pink part by itself (A) is another golden rectangle because \(b /(a - b)=\varphi\).
Assume that \(\varphi=\dfrac{a}{b}\) , and \(\varphi\) is the positive solution to \(\varphi^{2}-\varphi-1=0\). Then , \(\dfrac{a^{2}}{b^{2}}-\dfrac{a}{b}-\dfrac{b}{b}=0\). Multiply by \(b^{2}, a^{2}-a b-b^{2}=0\). So, \(a^{2}-a b=b^{2}\). Thus, \(a(a-b)=b^{2}\). We then get \(\dfrac{a}{b}=\dfrac{b}{a-b}\). Both sides are \(\varphi\) .
Fibonacci Sequence
The Fibonacci sequence is a list of numbers. Start with 1, 1, and then you can find the next number in the list by adding the last two numbers together. The resulting (infinite) sequence is called the Fibonacci Sequence. Since we start with 1, 1, the next number is 1+1=2. We now have 1, 1, 2. The next number is 1+2=3. We now have 1, 1, 2, 3. The next number is 2+3=5. The next one is 3+5=8, and so on. Each of these numbers is called a Fibonacci number. Originally, Fibonacci (Leonardo of Pisa, who lived some 800 years ago) came up with this sequence to study rabbit populations! He probably had no idea what would happen when you divide each Fibonacci number by the previous one, as seen below.
|
Fibonacci number |
divided by the one before |
ratio |
|---|---|---|
|
1 |
||
|
1 |
1/1 |
= 1.0000 |
|
2 |
2/1 |
= 2.0000 |
|
3 |
3/2 |
= 1.5000 |
|
5 |
5/3 |
= 1.6667 |
|
8 |
8/5 |
= 1.6000 |
|
13 |
13/8 |
= 1.6250 |
|
21 |
21/13 |
= 1.6154... |
|
34 |
34/21 |
= 1.6190... |
|
55 |
55/34 |
= 1.6177... |
|
89 |
89/55 |
= 1.6182... |
|
... |
... |
... |
|
|
= 1.6180... |
Here is a very surprising fact:
Note
The ratio of two consecutive Fibonacci numbers approaches the Golden Ratio.
It turns out that Fibonacci numbers show up quite often in nature. Some examples are the pattern of leaves on a stem, the parts of a pineapple, the flowering of artichoke, the uncurling of a fern and the arrangement of a pine cone. The Fibonacci numbers are also found in the family tree of honeybees.
Meanwhile, many artists and music researchers have studied artistic works in which the Golden Ratio plays an integral role. These include the works of Michelangelo, Da Vinci, and Mozart. Interested readers can find many resources and videos online. Perhaps it is not surprising that numbers like 3, 5, 8, and 13 are rather important in music theory; just take a quick look at the piano keys!
Reference
- References (17)
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0


6.5: Exercises
Suppose that \(G\) is a finite group with an element \(g\) of order \(5\) and an element \(h\) of order \(7\text{.}\) Why must \(|G| \geq 35\text{?}\)
Suppose that \(G\) is a finite group with \(60\) elements. What are the orders of possible subgroups of \(G\text{?}\)
Prove or disprove: Every subgroup of the integers has finite index.
Prove or disprove: Every subgroup of the integers has finite order.
List the left and right cosets of the subgroups in each of the following.
- \(\langle 8 \rangle\) in \({\mathbb Z}_{24}\)
- \(\langle 3 \rangle\) in \(U(8)\)
- \(3 {\mathbb Z}\) in \({\mathbb Z}\)
- \(A_4\) in \(S_4\)
- \(A_n\) in \(S_n\)
- \(D_4\) in \(S_4\)
- \({\mathbb T}\) in \({\mathbb C}^\ast\)
- \(H = \{ (1), (123), (132) \}\) in \(S_4\)
Describe the left cosets of \(SL_2( {\mathbb R} )\) in \(GL_2( {\mathbb R})\text{.}\) What is the index of \(SL_2( {\mathbb R} )\) in \(GL_2( {\mathbb R})\text{?}\)
Verify Euler's Theorem for \(n = 15\) and \(a = 4\text{.}\)
Use Fermat's Little Theorem to show that if \(p = 4n + 3\) is prime, there is no solution to the equation \(x^2 \equiv -1 \pmod{p}\text{.}\)
Show that the integers have infinite index in the additive group of rational numbers.
Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.
Let \(H\) be a subgroup of a group \(G\) and suppose that \(g_1, g_2 \in G\text{.}\) Prove that the following conditions are equivalent.
- \(\displaystyle g_1 H = g_2 H\)
- \(\displaystyle H g_1^{-1} = H g_2^{-1}\)
- \(\displaystyle g_1 H \subset g_2 H\)
- \(\displaystyle g_2 \in g_1 H\)
- \(\displaystyle g_1^{-1} g_2 \in H\)
If \(ghg^{-1} \in H\) for all \(g \in G\) and \(h \in H\text{,}\) show that right cosets are identical to left cosets. That is, show that \(gH = Hg\) for all \(g \in G\text{.}\)
What fails in the proof of Theorem 6.8 if \(\phi : {\mathcal L}_H \rightarrow {\mathcal R}_H\) is defined by \(\phi( gH ) = Hg\text{?}\)
Suppose that \(g^n = e\text{.}\) Show that the order of \(g\) divides \(n\text{.}\)
The cycle structure of a permutation \(\sigma\) is defined as the unordered list of the sizes of the cycles in the cycle decomposition \(\sigma\text{.}\) For example, the permutation \(\sigma = (12)(345)(78)(9)\) has cycle structure \((2,3,2,1)\) which can also be written as \((1, 2, 2, 3)\text{.}\)
Show that any two permutations \(\alpha, \beta \in S_n\) have the same cycle structure if and only if there exists a permutation \(\gamma\) such that \(\beta = \gamma \alpha \gamma^{-1}\text{.}\) If \(\beta = \gamma \alpha \gamma^{-1}\) for some \(\gamma \in S_n\text{,}\) then \(\alpha\) and \(\beta\) are conjugate .
If \(|G| = 2n\text{,}\) prove that the number of elements of order \(2\) is odd. Use this result to show that \(G\) must contain a subgroup of order 2.
Suppose that \([G : H] = 2\text{.}\) If \(a\) and \(b\) are not in \(H\text{,}\) show that \(ab \in H\text{.}\)
If \([G : H] = 2\text{,}\) prove that \(gH = Hg\text{.}\)
Let \(H\) and \(K\) be subgroups of a group \(G\text{.}\) Prove that \(gH \cap gK\) is a coset of \(H \cap K\) in \(G\text{.}\)
Let \(H\) and \(K\) be subgroups of a group \(G\text{.}\) Define a relation \(\sim\) on \(G\) by \(a \sim b\) if there exists an \(h \in H\) and a \(k \in K\) such that \(hak = b\text{.}\) Show that this relation is an equivalence relation. The corresponding equivalence classes are called double cosets . Compute the double cosets of \(H = \{ (1),(123), (132) \}\) in \(A_4\text{.}\)
Let \(G\) be a cyclic group of order \(n\text{.}\) Show that there are exactly \(\phi(n)\) generators for \(G\text{.}\)
Let \(n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}\text{,}\) where \(p_1, p_2, \ldots, p_k\) are distinct primes. Prove that
\[ \phi(n) = n \left( 1 - \frac{1}{p_1} \right) \left( 1 - \frac{1}{p_2} \right)\cdots \left( 1 - \frac{1}{p_k} \right)\text{.} \nonumber \]
Show that
\[ n = \sum_{d \mid n} \phi(d) \nonumber \]
for all positive integers \(n\text{.}\)


We can not
dedicate
We can not
consecrate
We can not
hallow — this ground
Abraham Lincoln, Gettysburg Address
It means to
try to tell your kids everything you thought you’d have the next 10 years to tell them in just a few months.
It means to
make sure everything is buttoned up so that it will be as easy as possible for your family.
It means to
say your goodbyes.
Steve Jobs, Stanford Commencement Speech
These men
are our fathers, grandfathers and even great-grandfathers.
These men
have been and will be apart of our lives in ways that go far beyond the Tokyo Raid.
These men
are the crews that have made sure our families’ lives were on the right path.
Speech on the Doolittle reunion
This was the moment
when the rise of the oceans began to slow and our planet began to heal.
This was the moment
when we ended a war, and secured our nation, and restored our image as the last, best hope on Earth.
This was the moment
, this was the time when we came together to remake this great nation so that it may always reflect our very best selves and our highest ideals.
Barack Obama Democratic Nomination Victory Speech – Change We Can Believe In
REFERENCES
Anderson, C. (2016).
TED Talks: The official TED guide to public speaking.
Mariner.
Anderson, C. (2013). How to Give a killer presentation. Harvard Business Review.
https://hbr.org/2013/06/how-to-give-a-killer-presentation
Ausubel, D. P. (1968).
Educational psychology
. Holt, Rinehart, & Winston.
Baker, E. E. (1965). The immediate effects of perceived speaker disorganization on speaker credibility and audience attitude change in persuasive speaking.
Western Speech, 29
, 148–161.
Belch, G. E., & Belch, M. A.
(2001). Advertising and promotion: An integrated marketing communications perspective.
McGraw-Hill.
Bush, G.W. (2015). President George W. Bush addresses a joint Congress about the War on Terror. [Video]. YouTube.
https://www.youtube.com/watch?v=TYnx-c8pF34
Standard YouTube License.
Bostrom, R. N., & Waldhart, E. S. (1988). Memory models and the measurement of listening.
Communication Education, 37
, 1–13.
https://doi.org/10.1080/03634528809378699
Carnegie, D. (1998).
How to win friends and influence people.
Pocket Books.
Churchhill Central: Life and words of Sir Winston Churchill.
https://www.churchillcentral.com/
Dlugan, A. (2009). How to use the rule of three in your speeches. Six Minutes.
How to Use the Rule of Three in Your Speeches (dlugan.com)
Duarte, N. & Sanchez, P. (2016).
Illuminate: Ignite change through speeches, stories, ceremonies, and symbols.
Portfolio and Penguin.
Ferrera, A. (2019). My identity is a superpower–not an obstacle. [Video]. YouTube.
https://www.ted.com/talks/america_ferrera_my_identity_is_a_superpower_not_an_obstacle/transcript?language=en#t-191931
Standard YouTube License.
Gamache, J.A. (2007). Being Mr. G, JA Gamache, Toastmasters World Championships. [Video]. YouTube.
https://www.youtube.com/watch?v=YoW-T2_6OJo
Standard YouTuve License
Hogan, J. (2018). The tale of two robes. Julie Hogan. [Video]. YouTube.
https://youtu.be/w3Z3jdubP4Y
Standard YouTube License.
Jobs, S. (2008). Steve Jobs’ 2005 Stanford commencement address. [Video]. YouTube.
https://www.youtube.com/watch?v=Hd_ptbiPoXM
Standard YouTube License.
Kalb, I. (2013). Marketers must understand the power of three.
Business Insider.
https://www.businessinsider.com/using-the-power-of-three-to-your-marketing-advantage-2013-5
Knols, B. (2012).  [Video]. YouTube.
https://www.ted.com/talks/bart_knols_3_new_ways_to_kill_mosquitoes?utm_campaign=tedspread&utm_medium=referral&utm_source=tedcomshare
Standard YouTube License.
LeFrancois, G. R. (1999).
Psychology for teaching
(10th ed.) Wadsworth.
Lineham, D. R. (n.d). Rule of three in speechwriting.
https://davelinehan.com/rule-of-three-speechwriting/
McCroskey, J.C. & Teven, J.J. (1999). Goodwill: A re-examination of the construct and its measurement.
Communication Monographs. 66
(1), 90-103.
https://doi.org/10.1080/03637759909376464
McCroskey, J.C. & Mehrley, R.S. (2009). The effects of disorganization and nonfluency on attitude change and source credibility.
Speech Monographs
.
https://doi.org/10.1080/03637756909375604
McCroskey, J.C. & Young, T. J. (1981). Ethos and credibility: The construct and its measurement after three decades.
Central States Speech Journal, 32,
24-34.
https://doi.org/10.1080/10510978109368075
Obama, B. (2004). Keynote speech to Democratic National Convention. [Video]. YouTube.
https://youtu.be/eWynt87PaJ0
Standard YouTube License.
Obama, B. (2008). Obama victory speech-Change we can believe in. [Video]. YouTube.
https://www.youtube.com/watch?v=7wJ-2Zu_Iic
Standard YouTube License.
Obama, M. (2011). Address at West Point Graduation Banquet. [Video]. YouTube.
https://youtu.be/hS01mK7xMDs
Standard YouTube License.
Raa Rams. (2018). Raa 2018 Keynote Speaker. [Video]. YouTube.
https://www.youtube.com/watch?v=g6OchXV-Mo8
Standard YouTube License.
Smith, R. G. (1951).
Speech Monographs, 18
, 292–301.
https://doi.org/10.1080/03637755109375045
Shu, S. B. & Carlson, K.A. (2014).
When three charms but four alarms: Identifying the optimal number of claims in persuasion settings.
Journal of Marketing, 78(1).
https://doi.org/10.1509/jm.11.0504
Thompson, E. C. (1960). An experimental investigation of the relative effectiveness of organizational structure in oral communication.
Southern Speech Journal, 26
, 59–69.
https://doi.org/10.1080/10417946009371589
Torgovnick May, K. (2012). How to give more persuasive presentations: A Q & A with Nancy Duarte. TED Blog.
https://blog.ted.com/how-to-give-more-persuasive-presentations-a-qa-with-nancy-duarte/
Tucker, B. & LeHew, M. (n.d)
Exploring public speaking
(4th edition).
https://alg.manifoldapp.org/projects/exploring-public-speaking


10.5: Sample Documentation
selected template will load here
This action is not available.
Patient reports no previous history for bone trauma, disease, infection, injury, or deformity. No symptoms of joint stiffness, pain, swelling, limited function, or muscle weakness. Patient is able to perform and manage regular daily activities without limitations and reports consistent exercise consisting of walking 2 miles for 5 days a week. Joints and muscles are symmetrical bilaterally. No swelling, deformity, masses, or redness upon inspection. Nontender palpation of joints without crepitus. Full ROM of the arms and legs with smooth movement. Upper and lower extremity strength is rated at 5 out of 5. Patient is able to maintain full resistance of muscle without tenderness or discomfort.
Patient reports “I felt a pop in my right ankle while playing basketball this afternoon” and “My right ankle hurts when trying to walk on it.” Pain is constant and worsens with weight-bearing. Patient rates pain at 4/10 at rest and 9/10 with walking and describes pain as an “aching, burning feeling.” Ibuprofen and ice decrease pain. Right ankle is moderately swollen laterally and anteriorly with tenderness to palpation but no erythema, warmth, or obvious deformity. Color, motion, and sensation are intact distal to the ankle. ROM of the right ankle is limited and produces moderate pain. Minimal eversion and inversion demonstrated. Patient is unable to bear weight on the right ankle. Dr. Smith notified and an order for an ankle X-ray received. The right ankle was elevated and ice applied while the patient waits for the X-ray.


4.1.2: Radiation as Particles
Radiation as particles
It's hard to imagine radiation moving as waves through empty space without a medium to transfer the wave form. For instance, the waves created when you drop a rock into a pool of water require molecules of H 2 O to propagate them. Though we describe electromagnetic radiation as invisible waves of energy, at the smallest scale it behaves as a particle, like when light is emitted by a single atom or molecule. When energy is given off there is a change in the orbital pattern of the electrons that surround the nucleus of an atom. As the orbit changes, a bundle of energy called a " photon " is released. However, particles of light differ from particles of matter: they have no mass, occupy no space, and travel at the speed of light, 2.9998 X 10 8 m s -1 . The amount of energy carried by a photon varies inversely with wavelength, the shorter the wavelength, the more energetic the photon.
Learn more about the nature of radiation by "Digging Deeper into the Nature of Radiation: Radiation Laws" or skip and continue reading.
In order to understand the processes and explain patterns of weather and climate, it is helpful to dig a bit deeper into a few laws related to energy and radiation.
Stefan-Boltzmann's Law: The total energy emitted by a black body is proportional the 4th power of its absolute temperature. Substances that emit the maximum amount of radiation for their temperature in all wavelengths are called black bodies. Also known as Stefan's law, it can be expressed as
\[E\,=\sigma T^4\]
where Ε is energy emitted per second from a unit are of a black body with a temperature of T(in Kelvin) and \(\sigma\) is the Stefan-Bloltzmann constant of 5.670 X 10 -8 J K -4 m -2 s -1
Wein's Law: The hotter the substance, the shorter the wavelength of emission. Wein's Law, introduced earlier in this chapter can be expressed as
\[\lambda_{(max)}\,=\,\frac{a}{T}\]
where a is the constant = 2897 μm K and T is temperature (in Kelvin). Using Wein's law reveals why the Sun emits most of its radiation as shortwave and the earth as longwave. Using average surface temperatures for the Sun and Earth
Sun: \(\lambda_{(max)}\,=\,\frac{2897\mu m\,K}{5778\,K}\,=\,0.50\,\mu m\)
Earth: \(\lambda_{(max)}\,=\,\frac{2897\mu m\,K}{287\,K}\,=\,10\,\mu m\)
Hence, the Sun emits most of its energy in the shortwave end of the electromagnetic spectrum while the Earth emits most of its energy in the longwave end.
Inverse Square Law : The intensity of light emanating from a point source is inversely proportional to the square of the distance from the source. The inverse square law explains why the intensity of radiation received diminishes as one moves away from the source. Notice how light (radiation) spreads as you move away from point S. Imagine a light sensor placed in the path of the light at distance d and measuring the intensity of light on one unit of area (A). As we move further away from the source, the light continues to spread, though our sensor still measures the light hitting one unit of area. When we reach a distance of 2d, the light now has spread to 4 units of area, but only 1/4 of the light falls on the sensor. If we move to 3 times the distance (3d) the light spreads further and now 1/9 of the light falls on the sensor.
The diminishing intensity of light as one moves away from the source is expressed mathematically as,
\[S_d\,=\,S_0\,(\frac{d_0}{d})^2\]
where S is the solar intensity at some distance d, and S 0 is the intensity at some reference distance d 0 .
Kirchoff's Law: Good absorbers are good emitters at a particular wavelength, and poor absorbers are poor emitters at the same wavelengths. In other words, some things, like the gases of Earth's atmosphere, are selective absorbers and emitters. Kirchoff's law helps explain phenomena like the greenhouse effect and why the warmest evening occur when it is cloudy. Carbon dioxide is a good absorber and emitter of of infrared radiation and thus plays an important role in the greenhouse effect. Clouds are also good absorbers and emitters of infrared radiation, creating warmer conditions than cloudless conditions during the night.


Skip to main content
-
ask
-
specific amount of money or type of assistance requested in a pitch
-
corporate narrative
-
story about how a successful company grew from something small into something broad and powerful, serving thousands or even millions of people; it may have marketing purposes, and it may serve as an inspirational story for leaders and employees
-
elevator pitch
-
abbreviated pitch, a memorized talk that can get you into the door to put your full pitch deck on display
-
entrepreneurial contest
-
any entrepreneurial competition other than a pitch competition; contests may entail submitting business models, long-form prospectuses, or forms tailored to contest specifications
-
fear of missing out (FOMO)
-
need to keep up with peers and the personas they represent in social settings, particularly on social media
-
iceberg principle
-
writing principle useful for preparing contest documents and pitch decks; following this principle, each word or sentence in the submission or presentation should be backed up with a wealth of supporting information and details that go unspoken in the pitch or proposal, but which you have at the ready should an investor take interest in the details and your knowledge of them
-
ideation
-
purposeful process of opening up one’s mind to new trains of thought that branch out in many directions from a stated purpose or problem
-
mission statement
-
statement of an organization’s reason for being that defines its primary long-term goal, explains what the venture will do, what value it will provide to the end customer, and how it will accomplish that action
-
noncompete agreement
-
formal agreement that the person signing the document will not work for another organization that is a direct competitor of their current employer during employment and usually for a specified length of time after employment ends
-
nondisclosure agreement (NDA)
-
contract that carries harsh financial penalties should one party to the agreement disclose what they know of the proprietary information
-
pitch
-
formal presentation in which you ask for something and that is delivered (usually) to potential investors in a startup
-
pitch competition
-
competition that may have cash prizes or mentorship opportunities for winners; you are expected to present your business idea in the form of a presentation, usually with a pitch deck
-
pitch deck
-
slide presentation that you create that gives a quick overview of your product and what you’re asking for
-
problem-solution narrative
-
brief story of a sentence or two that states a problem affecting many people and offering a solution that is innovative, unique, insightful, or all of these
-
user interface (UI)
-
what a customer sees when they use your product; having a variant of a UI is not necessarily grounds for establishing a patent, but having unique design in the broader sense may help you to secure patent protection
-
value proposition
-
summary describing the benefits (value) customers can expect from a particular product or service
-
work-for-hire agreement
-
agreement between an individual employee and a company states that ownership of innovations belongs to the company, even if an individual makes substantial contributions to the product


8.5: Montejo v. Louisiana 556 US 778 (2009)
MONTEJO V. LOUISIANA SUPREME COURT OF THE UNITED STATES
MONTEJO v . LOUISIANA
certiorari to the supreme court of louisiana
No. 07–1529. Argued January 13, 2009—Decided May 26, 2009
At a preliminary hearing required by Louisiana law, petitioner Montejo was charged with first-degree murder, and the court ordered the appointment of counsel. Later that day, the police read Montejo his rights under Miranda v. Arizona , 384 U. S. 436 , and he agreed to go along on a trip to locate the murder weapon. During the excursion, he wrote an inculpatory letter of apology to the victim’s widow.  At trial, his letter was admitted over defense objection, and he was convicted and sentenced to death. Affirming, the State Supreme Court rejected his claim that the letter should have been suppressed under the rule of Michigan v. Jackson , 475 U. S. 625 , which forbids police to initiate interrogation of a criminal defendant once he has invoked his right to counsel at an arraignment or similar proceeding. The court reasoned that Jackson ’s prophylactic protection is not triggered unless the defendant has actually requested a lawyer or has otherwise asserted his Sixth Amendment right to counsel; and that, since Montejo stood mute at his hearing while the judge ordered the appointment of counsel, he had made no such request or assertion.
Held:
1. Michigan v. Jackson should be and now is overruled. Pp. 3–18.
(a) The State Supreme Court’s interpretation of Jackson would lead to practical problems. Requiring an initial “invocation” of the right to counsel in order to trigger the Jackson presumption, as the court below did, might work in States that require an indigent defendant formally to request counsel before an appointment is made, but not in more than half the States, which appoint counsel without request from the defendant. Pp. 3–6.
(b) On the other hand, Montejo’s solution is untenable as a theoretical and doctrinal matter. Eliminating the invocation requirement entirely would depart fundamentally from the rationale of Jackson, whose presumption was created by analogy to a similar prophylactic rule established in Edwards v. Arizona , 451 U. S. 477 , to protect the Fifth Amendment-based Miranda right. Both Edwards and Jackson are meant to prevent police from badgering defendants into changing their minds about the right to counsel once they have invoked it, but a defendant who never asked for counsel has not yet made up his mind in the first instance. Pp. 6–13.
(c) Stare decisis does not require the Court to expand significantly the holding of a prior decision in order to cure its practical deficiencies. To the contrary, the fact that a decision has proved “unworkable” is a traditional ground for overruling it. Payne v. Tennessee , 501 U. S. 808 , 827. Beyond workability, the relevant factors include the precedent’s antiquity, the reliance interests at stake, and whether the decision was well reasoned. Pearson v. Callahan , 555 U. S. ___, ___. The first two cut in favor of jettisoning Jackson: the opinion is only two decades old, and eliminating it would not upset expectations, since any criminal defendant learned enough to order his affairs based on Jackson ’s rule would also be perfectly capable of interacting with the police on his own. As for the strength of Jackson ’s reasoning, when this Court creates a prophylactic rule to protect a constitutional right, the relevant “reasoning” is the weighing of the rule’s benefits against its costs. Jackson’ s marginal benefits are dwarfed by its substantial costs. Even without Jackson , few badgering-induced waivers, if any, would be admitted at trial because the Court has taken substantial other, overlapping measures to exclude them. Under Miranda , any suspect subject to custodial interrogation must be advised of his right to have a lawyer present. 384 U. S., at 474. Under Edwards , once such a defendant “has invoked his [Miranda] right,” interrogation must stop. 451 U. S., at 484. And under Minnick v. Mississippi , 498 U. S. 146 , no subsequent interrogation may take place until counsel is present. Id., at 153. These three layers of prophylaxis are sufficient. On the other side of the equation, the principal cost of applying Jackson ’s rule is that crimes can go unsolved and criminals unpunished when uncoerced confessions are excluded and when officers are deterred from even trying to obtain confessions. The Court concludes that the Jackson rule does not “pay its way,” United States v. Leon , 468 U. S. 897 , 907–908, n. 6, and thus the case should be overruled. Pp. 13–18.
2. Montejo should nonetheless be given an opportunity to contend that his letter of apology should have been suppressed under the Edwards rule . He understandably did not pursue an Edwards objection, because Jackson offered broader protections, but the decision here changes the legal landscape. Pp. 18–19.
06–1807 (La.), 974 So. 2d 1238, vacated and remanded.
Scalia, J., delivered the opinion of the Court, in which Roberts, C. J., and Kennedy, Thomas, and Alito, JJ., joined. Alito, J., filed a concurring opinion, in which Kennedy, J., joined. Stevens, J., filed a dissenting opinion, in which Souter and Ginsburg, JJ., joined, and in which Breyer, J., joined, except for n. 5. Breyer, J., filed a dissenting opinion.
Contributors and Attributions
- Authored by : US Government . Located at : https://supreme.justia.com/cases/federal/us/556/778/ . License : Public Domain: No Known Copyright


3.5: Spelling
selected template will load here
This action is not available.
In the age of spell checkers, it’s hard to imagine that spelling errors would top the most common error list, but they do. First, it’s important to remember to run the spell checker. Even though you have to remember to check your spell checker, spell checkers are a good place to start when it comes to spelling errors.
Still, there are plenty of spelling errors the spell checker won’t catch. In addition to possibly creating wrong-word errors, as discussed earlier, spell checkers also miss misspelled words like names and other proper nouns, as illustrated in the following example:
The spell checker missed these errors. Here is the corrected version of the sentence:
So, be sure to run your spell checker, but, then, don’t forget to double and triple check your writing. Spell checkers do not catch everything.


8.1: Fundamental Equation of Thermodynamics
Let’s summarize some of the results from the first and second law of thermodynamics that we have seen so far. For reversible processes in closed systems:
\[\begin{equation} \begin{aligned} \text{From 1}^{\text{st}} \text{ Law:} \qquad \quad & dU = đQ_{\mathrm{REV}}-PdV \\ \text{From The Definition of Entropy:} \qquad \quad & dS = \dfrac{đQ_{\mathrm{REV}}}{T} \rightarrow đQ_{\mathrm{REV}} = TdS \\ \\ \Rightarrow \quad & dU = TdS - PdV. \end{aligned} \end{equation} \label{8.1.1} \]
Equation \ref{8.1.1} is called the fundamental equation of thermodynamics since it combines the first and the second laws. Even though we started the derivation above by restricting to reversible transformations only, if we look carefully at Equation \ref{8.1.1}, we notice that it exclusively involves state functions. As such, it applies to both reversible and irreversible processes. The fundamental equation, however, remains constrained to closed systems. This fact restricts its utility for chemistry, since when a chemical reaction happens, the mass in the system will change, and the system is no longer closed.
At the end of the 19 th century, Josiah Willard Gibbs (1839–1903) proposed an important addition to the fundamental equation to account for chemical reactions. Gibbs was able to do so by introducing a new quantity that he called the chemical potential:
The chemical potential is the amount of energy absorbed or released due to a change of the particle number of a given chemical species.
The chemical potential of species \(i\) is usually abbreviated as \(\mu_i\), and it enters the fundamental equation of thermodynamics as:
\[ dU = TdS-PdV+\sum_i\mu_i dn_i, \nonumber \]
where \(dn_i\) is the differential change in the number of moles of substance \(i\), and the summation extends over all chemical species in the system.
According to the fundamental equation, the internal energy of a system is a function of the three variables entropy, \(S\), volume, \(V\), and the numbers of moles \(\{n_i\}\). 1 Because of their importance in determining the internal energy, these three variables are crucial in thermodynamics. Under several circumstances, however, they might not be the most convenient variables to use. 2 To emphasize the important connections given by the fundamental equation, we can use the notation \(U(S,V,\{n_i\})\) and we can term \(S\), \(V\), and \(\{n_i\}\) natural variables of the energy.
- In the case of the numbers of moles we include them in curly brackets to indicate that there might be more than one, depending on how many species undergo chemical reactions.
- For example, we don’t always have a simple way to calculate or to measure the entropy.


1.14: Intentions
-
- Last updated
- Save as PDF
Before reading this chapter, it is important to understand something that seems obvious, but that people don’t pay attention to because they do it so automatically. That is that when people do things they don’t always think heavily about what it is they are going to do. When people do things like move their hand to open a door, or just move their hand in a certain fashion, they don’t say to themselves “ok, now I am going to move my hand in this way” – they simply do it. They are conscious that they are moving their hand in that way, but they could be a lot more conscious of it if they thought about it a lot before they did it. It is almost like you are doing these simple actions unconsciously because you don’t really think about them before you do it, but are only minorly aware that you are doing it. There are degrees and types of examples for this, for instance typing is much more automatic than opening a door.
When someone has an intention, or does anything such as thinking something or doing something without thought, what is the exact mental process that lies behind that action? What combination of emotions, feelings and thoughts makes that happen? Here is what is at the bottom of the "Emotion is a Combination of Feeling and Thought" chapter:
“Emotion always precedes thought; thought is always just going to be an explanation of emotion. Everything in the end turns out to be an emotion in your system, so therefore everything is really an emotion. When you say "I want to leave" the feeling of you wanting to leave is always going to precede the thought. Actually first you quickly understand what it is that you are feeling when you realize what it is you are feeling as an unconscious thought process, then you have a more regular feeling about it, and then you are able to verbalize that feeling into a thought. Unless something is said to you instead of you thinking it, in which case the process is reversed. First it is a thought because it is expressed that way, then it is a feeling, and then it is a quick unconscious thought process to think about what was said.”
So there is an unconscious thought process before everything you think/do, however there are also patterns of feelings which are also there. The feelings described are an important part of it, when you do something there isn't an unconscious thought right before you do it. You first have the unconscious thought when you have the original feeling that caused you to want to do that thing - you first have a feeling that you want to do something, then you understand what that feeling means as an unconscious thought, and then that is translated back into a feeling which remains there until you do the action. So the unconscious thought is not right before you do the thing, the feeling is there before you do it because feelings are faster than thoughts, so your mind has the feeling ready at hand to act on the unconscious thought process.  Remember from the instinctual frog example that feelings are faster than thoughts, and feelings are also unconscious thoughts so they can also store information to do. This is the frog example in the chapter “Thoughts”:
“The definition of intellect and thoughts is almost understanding (those concrete things). Emotion is feeling, completely separate from facts or information. All facts and information are going to be about things that cause feeling, however, since all things that happen cause feelings and all facts and information are about things that happen. So facts and information are just feelings organized in a logical manner. Intellect and thought also generates feelings when those thoughts are processed in your mind. Since thought is really only about feelings, it is logical that thought actually has root in feelings. For example, all events are really feelings in the mind, so thoughts are actually just comparing feelings. You take two feelings and can arrive at one thought. Take the feeling of a frog moving and the feeling of a threat of danger. The two feelings combined equal the idea or thought that the frog needs to move when there is danger - the thought is actually just understanding how feelings interact. All thought is is the understanding of how feelings and real events interact with themselves. Feeling is what provides the motivation to arrive at the answer (the thought). If you just had the facts, there is a threat, and the frog can jump, you aren't going to arrive at the conclusion that the frog should jump away. You need to take the feeling that there is a threat and the feeling that the frog can jump and then combine the two sensory images in your head to arrive at the answer.
That shows how all intellect is powered and motivated by emotion. It also shows that frogs have thoughts; the frog has to have the thought to jump away when it sees a threat, as a thought is just the combination of two feelings resulting in the resulting feeling of wanting to move away. That process of feelings is like a thought process. Thoughts are a little different for humans, however, because humans have such a large memory that they are able to compare this experience to all the other experiences in their life while the frog only remembers the current situation and is programmed (brain wiring) to jump away. The frog doesn't have a large enough memory to learn from new information and change its behavior. That shows how humans are very similar to frogs in how they process data (in one way at least), and that one thing that separates a human from a frog is a larger memory which can store lots of useful information and potential behavioral patterns.”
It would be too slow for you to just do something based on an unconscious thought process, you would have to wait to have this unconscious thought right before you do the thing, instead of having the thought at one point in time and storing it, and then doing the thing later on. If it is just an instinctual reaction, however, it is just a feeling that you are responding to because it is too fast to have an unconscious thought process. It is just a manner of the definition of what an unconscious thought is - that it is going to be more like a thought than a feeling - which is also an unconscious thought, so it depends how you view it.
If it is an instinctual, immediate reaction, say if you slam a door on your hand then you are going to say "ouch" - that is a thought that resulted from two feelings, the feeling of pain and the feeling that you need to express that pain. The thought is so fast you might consider it unconscious, that is also like in the frog example.
It gets even more complicated than that - this is in the "Life Occurs in Sharp Spikes" chapter of the book:
“Everything that is processed, not just spikes, follows the sequence of thought to emotion to feeling. That is because thoughts are clearer than emotions and feelings, and emotions are more similar to thoughts than feelings are (discussed previously) so when you see something or hear something or whatnot for the first time, it is clearer in your mind. Then it becomes less clear and you think about it unconsciously. You think about it unconsciously because it takes further processing in order to isolate the feeling that that things gives you. Some things are just too complicated to feel them right away. Other things, however, can be felt right away, say if you are touching something the feeling arises right away. That is because the physical stimulus is more immediate than emotional stimulus.
Emotional things, however, are simply to complicated to "feel" them right away, they need to be processed first. That is logical, just take looking at anything, say a book. In order to feel the feelings that the book causes in you, you are going to have to at least unconsciously think about it first (that is, after you start paying attention to it, which you do by starting to think about it or just see it and notice it more than you usually notice things in the area). Since you don't need to think about physical stimulus since it is just a physical stimulus, (not something like vision) you don't really unconsciously process it.”
That shows that it is really all mixed in - thoughts, emotions and feelings - that there isn't just an unconscious thought process but you could also just say that feelings or thoughts are first - this is because when you process something you might think about it first, and it certainly feels this way because when you are processing something it is a very intellectual experience, it is clear in your mind and it feels like you are thinking about the thing so clearly that you must be using thoughts instead of emotions. I say that things are first clear in your mind when you first see it or whatnot, - that would be the "thought" but then it is an emotion, and you do that (make it into an emotion) to isolate the feeling the thing causes in you, so then you feel it (after you isolate the feeling) - thought to emotion to feeling.
So when you have an intention to do something could it be that first it is an unconscious thought and then you just do it? First you are going to have an unconscious thought about it, then you are going to have a conscious thought about it (because it is an intention) and then you are going to do it. Your conscious thought about it may or may not be verbal, you don’t have to think about everything verbally in order to do it. You do have a conscious thought about it because that is almost the definition of intention, your intent. If you don’t have a conscious thought about it then it is more instinctual, or it could be a mix of the two. Everything someone does is going to be on the spectrum somewhere between complete intention and completely instinctual.
Intentions and instincts (or things you do) aren’t just thoughts, but feelings and emotions are often involved as well, where do they fit in? First an emotion could start an intention, and then it would be an unconscious thought process, and then it might become another emotion because you can feel everything (you are going to feel the thought, or have a feeling about it) and feelings are very fast so this feeling can fit into the time after you think about it and before you do the action, or after the initiating event and before the unconscious or conscious thought process. When you do think it is very fast, in fact your thinking might be slow, but there is one point in time where your thinking leads to a conclusion and that is culmination is considered to be when you had the “thought” because it is a conscious thought that your mind understands, but leading up to that conscious thought (which could be verbal or not verbal) was unconscious thoughts (or thinking) because it is hard to reach difficult conclusions instantly. This thought is then held in your mind until you do the action, it prepares your mind for the action, and during that time that thought might generate a certain feeling – maybe fear or a lack of confidence. This feeling is then used when you do the intention, because when you do something you do it so fast that you don't "think" about it right before you do it, but you use the feeling that is “storing” the thought.
You might not have feelings about it and your action might not be swayed by feeling, but if it is then your thoughts might be under the influence of your feelings. Your feelings might cause you to stop doing the thing if you are too afraid, for example.
So there is an unconscious thought before every intention, that is what thought is, it is figuring out what you are going to do, and you are going to have to figure out what it is that you are going to do first before you do it. Unless it is like the frog example where you just feel it at the same time that you do it, but in that case the feelings are mixed in with the thoughts, so then it is a matter of how you define "thought". Thought is really a conclusion (not a partial thought, which could be an emotion), so you take two feelings and come at a conclusion, which is the thought, then you do the thing, and that means that you do have an unconscious thought right before the intention, the feeling really is a thought, it is just so fast that it is a feeling and a thought. So right before you do something there can be a feeling - which is also a thought, that causes you to do it finally. So is it a thought or is it a feeling? The feeling is the drive behind the thought (or thinking), which builds up along with the feeling. The feeling is powering the thought (or thinking) because it is so instinctual. So things that are more instinctual are going to be faster and involve more feelings, feelings can speed up thoughts (this is obvious with the instinctual example, where instinct then is really just powerful feelings causing you to think very fast).
So if you do anything there is going to be unconscious thoughts before you do it, because thoughts are just understanding real things. That includes if you have intentions, only intentions (since they are more conscious) are going to involve conscious thoughts as well as unconscious ones, unless it is an intention you intended to do unconsciously. The reason intentions involve unconscious thoughts as well is because you need to think to arrive at the conclusion, and most thinking isn’t completely consciously understood. How many people can think without using words, yet understand what it is that they are thinking? You can understand that you are going to do a certain thing without using words, but you can’t think for a long period of time without using words and still follow your thought process. Complicated non-verbal thought processes are unconscious. And almost all thoughts and everything you do is going to be complicated - and therefore they are going to involve long unconscious thinking about them (by long I just mean longer than instantaneous, which would be what you would do if it was instinctual).
So right before you do something there is going to be something in your mind that understands what it is you are going to do, this is a thought because it is real (versus feelings which are things which you just feel). You might even "feel" the thought really. That is what happens right before you do something. However, leading up to that final thought/feeling it is going to be like described before; first you might have a feeling. If humans were computers I would say that first it starts with its programming and then it has the thought, but for humans feelings are their programming – so humans first have feelings and then we have thoughts. Feelings can originate from thoughts however, so it is then a which came first, the chicken or the egg debate. But if the original feeling started because of a thought, the thought was more further away in time from the feeling -by a few seconds at least – that is because conscious thoughts (verbal ones) have space of time around them, if you think, “I am going to shoot” you don’t shoot as quickly as you would if you just understood that you were going to shoot, the conscious verbal thought slows you down. So when you have an intention or when you are going to think something (which is what thoughts are - they can be verbal because you can express anything verbally almost, including all intentions) then that follows the process of feeling to unconscious thought to feeling again to store it. I said before “a feeling, then an unconscious thought process, then a more general feeling”.
I said that because the first feeling is just the real feeling of the intention you are going to have - which you could say is an unconscious thought because as discussed previously all feelings are unconscious thoughts - and it is clear they are when you realize it is an intention, which is going to be doing something real, and intellect is understanding things that are real. So the first feelings/thoughts are when you first feel that you want to do something, then you need to unconsciously think about it to realize what it is you want to do exactly (this is not a conscious non-verbal thought, but an unconscious one), and then you have a more specific or general feeling about it (by general there I really mean larger or more clear) to store that clear thought, the general feeling then is going to be more clear because you now unconsciously understand what it is that you are going to do, and then it is a real conscious thought and then you could translate that conscious thought to a verbal thought or an action.
So to explain the statement, "first it is a feeling, then it is an unconscous thought process, and then it is a more general feeling and then you are able to make that feeling into a conscious thought (or do an action which would stem from that clear thought)" - that was originally said in the book at the end of the "Emotion is a Combination of Feeling and Thought" chapter in this form - "actually first you quickly understand what it is that you are feeling when you realize what it is you are feeling as an unconscious thought process, then you have a more regular feeling about it, and then you are able to verbalize that feeling into a thought". Whether someone’s state before they have that thought is one that started with an emotion or without an emotion, that state must have originated from a previous state, or from some other previous stimulus. In terms of someone’s first feelings, their first feelings probably came from physical feelings before the brain was developed in the womb. First people would have just physical feelings, not deep emotional ones because all there is in the beginning is sensory stimulation - mostly feeling your own body and your surroundings.
So the first thoughts/feelings originated from physical stimulus, like, "ouch that hurts". Or "that looks cool". After the human develops they can have thoughts and feelings that can originate from sensory stimulation, physical stimulation, or other thoughts and feelings. But that doesn't explain what happens right before someone thinks something or does something. It explains that originally there are those things which would cause the intention, but not how the intention is formed. Since humans have strong emotions, many intentions are going to be formed from emotion. Intentions are also going to be formed from conscious / unconscious thinking. Feelings are also going to have elements of thoughts, however (so it isn’t either feeling or thought that originated the intention, it might be both at the same time). Say if you want to switch a switch - it is going to be a progression of feeling/thought. That is, it is going to take time for you to realize what it is you want to do, so it could be feeling and thinking all along, and at some point in that feeling/thinking you are going to realize fully what you want to do, and then you could call it a thought because it is completely formed (this thought might be conscious or it might remain unconscious and only later become conscious). When you realize you want to switch a switch it isn't instantaneous, but it takes time. But when you do switch the switch instantaneously, are you acting off of the thought or the feeling? You are probably acting off of the feeling, the thought was a period in time a while ago, but that thought started the feeling of you wanting to do it, which lead to you switching the switch off of the feeling instead of the thought. Unless you happen to do the thing right after you finally figure out what it is you want to do, then you could say that the thought made you do it.
That reveals that you are always going to have some feeling about what it is you are going to do right before you do it, because then you “think” or “feel” what it is you are going to do. It isn’t going to be as strong in terms of thought as when you first thought of what it was you were going to do, because you don’t need to think as much to realize what it is you are going to do. You are probably going to be feeling more than thinking right before you do it because you are going to be excited about doing something, you already realized what you were going to do which was the thought part, now it is time for the feeling part. The thought is still there of course otherwise you wouldn’t know what to do, however right before you do it feeling is probably going to dominate.
Right before you do something your mind needs to get ready to do it, and you need to remind yourself what it is you need to do and that you need to do it. So that means your mind probably feels something based on what it is you are going to do. This feeling can be simulated if you read a book and then later reflect on how you feel about the book. Reading the book in this instance would be the original thought process, and reflecting on it later would be simulating the feeling right before you do something. You don’t need to think about everything in the book to understand the feeling that the book causes you. You don’t need to think as hard to understand the same things because it was already understood at one point. The second time it is easier. That is like when you first have an unconscious thought process to understand what you are going to, when you are going to do it later you already understand what you are going to do, you simply then “feel” what it is you are going to do because it is more clearly understood, it is understood emotionally now (more instinctual) so you don’t need to “think” as much as you did before. Emotion replaces thought because emotion is easier than thought. Someone isn’t going to think unless they have to, you basically have already done the hard part, so the second time you bring it up the thought would be reduced and the emotion would remain. The further excitement of being about to do the thing would raise the emotion even more. But here learned is another thing, if you think about something once the next times you bring it up (especially if you bring it up right after you figure it out) it is going to be much easier to understand so thought is going to be reduced and feeling raised relatively.
So in other words, before the thought or your understanding of what it is you are going to do is complete, you are going or are not going to be having emotions that are encouraging this thought process or affecting this thought process. Emotion and intelligence are intertwined. That is why first comes the emotion, then the complete thought, and then you might have an emotion about that thought itself as well, - in other words the state of the emotion you are feeling is probably going to evolve as the thought does. This reveals that while emotion is unconscious thought, not all unconscious thought is emotion.
Humans don't just say things without thinking about them first, so everything is going to be unconscious first. Speech is much much slower than your thoughts are, and unless you start saying something and don't know the complete sentence before you say it, you are going to have the entire thing thought out first. So technically everything starts with an unconscious thought. However this thought has levels of understanding, there are levels to which you understand the thought, that is why you can't just say everything all at once, you usually have to think about it for a bit first. When people think, it takes time to think, and they don't think unconsciously in sentences. They think unconsciously with emotions, thoughts, visualizations, anything your mind can simulate. When they think unconsciously with emotions you could be taking large emotional experiences and trying to analyze them, or little ones, you could be combining different experiences, or combining emotion with thought or emotion with visualization (etc.). Your mind doesn't just use sentences to figure out what it wants to do, that would take too long. Sentences are actually just sounds that represent things, you don't need to simulate a sound in your head in order to think.  It isn't the firing of one neuron once that makes a complete sentence. There is a progression of thought. This is obvious because when you are doing a problem, say a math problem, you often can reach the answer without having to say anything. What is happening is that you are thinking about things unconsciously, maybe you are visualizing the number of things you need to visualize to find the answer (say adding 1 to 1 you have to visualize the separate objects, and then visualize the two objects together).