10.5H: Synaptic Transmission
Synaptic transmission is a chemical event which is involved in the transmission of the impulse via release, diffusion, receptor binding of neurotransmitter molecules and unidirectional communication between neurons.
- Describe the process of synaptic transmission
Key Points
- In a chemical synapse, the pre and post synaptic membranes are separated by a synaptic cleft, a fluid filled space.
- The chemical event is involved in the transmission of the impulse via release, diffusion, receptor binding of neurotransmitter molecules and unidirectional communication between neurons.
- The neurotransmitter termination can occur in three ways – reuptake, enzymatic degradation in the cleft and diffusion.
In a chemical synapse, the pre and post synaptic membranes are separated by a synaptic cleft, a fluid filled space. The chemical event is involved in the transmission of the impulse via release, diffusion, receptor binding of neurotransmitter molecules and unidirectional communication between neurons.
Chemical Synapse
Synaptic vesicles inside a neuron : This pseudocolored image taken with a scanning electron microscope shows an axon terminal that was broken open to reveal synaptic vesicles (blue and orange) inside the neuron.
Neurotransmission at a chemical synapse begins with the arrival of an action potential at the presynaptic axon terminal. When an action potential reaches the axon terminal, it depolarizes the membrane and opens voltage-gated Na + channels. Na + ions enter the cell, further depolarizing the presynaptic membrane. This depolarization causes voltage-gated Ca 2+ channels to open. Calcium ions entering the cell initiate a signaling cascade. A calcium sensing protein binds calcium and interacts with SNARE proteins. These SNARE proteins are involved in the membrane fusion. The synaptic vesicles fuse with the presynaptic axon terminal membrane and empty their contents by exocytosis into the synaptic cleft. Calcium is quickly removed from the terminal.
Fusion of a vesicle with the presynaptic membrane causes neurotransmitters to be released into the synaptic cleft. The neurotransmitter diffuses across the synaptic cleft, binding to receptor proteins on the postsynaptic membrane.
Communication at a chemical synapse : Communication at chemical synapses requires release of neurotransmitters. When the presynaptic membrane is depolarized, voltage-gated Ca2+ channels open and allow Ca2+ to enter the cell. The calcium entry causes synaptic vesicles to fuse with the membrane and release neurotransmitter molecules into the synaptic cleft. The neurotransmitter diffuses across the synaptic cleft and binds to ligand-gated ion channels in the postsynaptic membrane, resulting in a localized depolarization or hyperpolarization of the postsynaptic neuron.
The binding of a specific neurotransmitter causes particular ion channels, in this case ligand-gated channels, on the postsynaptic membrane to open. The binding of a neurotransmitter to its receptor is reversible. As long as it is bound to a post synaptic receptor, a neurotransmitter continues to affect membrane potential. The effects of the neurotransmitter generally lasts few milliseconds before being terminated. The neurotransmitter termination can occur in three ways. First, reuptake by astrocytes or presynaptic terminal where the neurotransmitter is stored or destroyed by enzymes. Second, degradation by enzymes in the synaptic cleft such as acetylcholinesterase. Third, diffusion of the neurotransmitter as it moves away from the synapse.


7.5: Using Singular Value Decompositions
\(\newcommand{\twovec}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix} } \)
\(\newcommand{\threevec}[3]{\begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix} } \)
We've now seen what singular value decompositions are, how to construct them, and how they provide important information about a matrix such as orthonormal bases for the four fundamental subspaces. This puts us in a good position to begin using singular value decompositions to solve a wide variety of problems.
Given the fact that singular value decompositions so immediately convey fundamental data about a matrix, it seems natural that some of our previous work can be reinterpreted in terms of singular value decompositions. Therefore, we'll take some time in this section to revisit some familiar issues, such as least squares problems and principal component analysis, while also looking at some new applications.
Preview Activity 7.5.1.
Suppose that \(A = U\Sigma V^T\) where
vectors \(\mathbf u_j\) form the columns of \(U\text{,}\) and vectors \(\mathbf v_j\) form the columns of \(V\text{.}\)
- What are the dimensions of the matrices \(A\text{,}\) \(U\text{,}\) and \(V\text{?}\)
- What is the rank of \(A\text{?}\)
- Describe how to find an orthonormal basis for \(Col(A)\text{.}\)
- Describe how to find an orthonormal basis for \(Nul(A)\text{.}\)
- If the columns of \(Q\) form an orthonormal basis for \(Col(A)\text{,}\) what is \(Q^TQ\text{?}\)
- How would you form a matrix that projects vectors orthogonally onto \(Col(A)\text{?}\)
Least squares problems
Least squares problems, which we explored in Section 6.5 , arise when we are confronted with an inconsistent linear system \(A\mathbf x=\mathbf b\text{.}\) Since there is no solution to the system, we instead find the vector \(\mathbf x\) minimizing the distance between \(\mathbf b\) and \(A\mathbf x\text{.}\) That is, we find the vector \(\widehat{\mathbf{x}}\text{,}\) the least squares approximate solution, by solving \(A\widehat{\mathbf{x}}=\widehat{\mathbf{b}}\) where \(\widehat{\mathbf{b}}\) is the orthogonal projection of \(\mathbf b\) onto the column space of \(A\text{.}\)
If we have a singular value decomposition \(A=U\Sigma V^T\text{,}\) then the number of nonzero singular values \(r\) tells us the rank of \(A\text{,}\) and the first \(r\) columns of \(U\) form an orthonormal basis for \(Col(A)\text{.}\) This basis may be used to project vectors onto \(Col(A)\) and hence to solve least squares problems.
Before exploring this connection further, we will introduce Sage as a tool for automating the construction of singular value decompositions. One new feature is that we need to declare our matrix to consist of floating point entries. We do this by including
RDF
inside the matrix definition, as illustrated in the following cell.
Activity 7.5.2.
Consider the equation \(A\mathbf x=\mathbf b\) where
-
Find a singular value decomposition for \(A\) using the Sage cell below. What are singular values of \(A\text{?}\)
- What is \(r\text{,}\) the rank of \(A\text{?}\) How can we identify an orthonormal basis for \(Col(A)\text{?}\)
-
Form the reduced singular value decomposition \(U_r\Sigma_rV_r^T\) by constructing the matrix \(U_r\text{,}\) consisting of the first \(r\) columns of \(U\text{,}\) the matrix \(V_r\text{,}\) consisting of the first \(r\) columns of \(V\text{,}\) and \(\Sigma_r\text{,}\) an \(r\times r\) diagonal matrix. Verify that \(A=U_r\Sigma_r V_r^T\text{.}\)
You may find it convenient to remember that, if
B
is a matrix defined in Sage, thenB.matrix_from_columns( list )
andB.matrix_from_rows( list )
can be used to extract columns or rows fromB
. For instance,B.matrix_from_rows([0,1,2])
provides a matrix formed from the first three rows ofB
. - How does the reduced singular value decomposition provide a matrix whose columns are an orthonormal basis for \(Col(A)\text{?}\)
-
Explain why a least squares approximate solution \(\widehat{\mathbf{x}}\) satisfies
\begin{equation*} A\widehat{\mathbf{x}} = U_rU_r^T\mathbf b. \end{equation*}
- What is the product \(V_r^TV_r\) and why does it have this form?
-
Explain why
\begin{equation*} \widehat{\mathbf{x}} = V_r\Sigma_r^{-1}U_r^T\mathbf b \end{equation*}
is a least squares approximate solution by simplifying \(A\widehat{\mathbf{x}} = (U_r\Sigma_r V_r^T)(V_r\Sigma_r^{-1}U_r^T \mathbf b)\text{.}\)
Now use this expression to find \(\widehat{\mathbf{x}}\text{.}\)
This activity demonstrates the power of a singular value decomposition to find a least squares approximate solution for an equation \(A\mathbf x = \mathbf b\text{.}\) Because it immediately provides an orthonormal basis for \(Col(A)\text{,}\) something that we've had to construct by the Gram-Schmidt process in the past, we can easily project \(\mathbf b\) onto \(Col(A)\text{,}\) which results in a simple expression for \(\widehat{\mathbf{x}}\text{.}\)
If \(A=U_r\Sigma_r V_r^T\) is a reduced singular value decomposition of \(A\text{,}\) then a least squares approximate solution to \(A\mathbf x=\mathbf b\) is given by
If the columns of \(A\) are linearly independent, then the equation \(A\widehat{\mathbf{x}} = \widehat{\mathbf{b}}\) has only one solution so there is a unique least squares approximate solution \(\widehat{\mathbf{x}}\text{.}\) Otherwise, the expression in Proposition 7.5.1 produces the solution to \(A\widehat{\mathbf{x}}=\widehat{\mathbf{b}}\) having the shortest length.
The matrix \(A^+ = V_r\Sigma_r^{-1}U_r^T\) is known as the Moore-Penrose psuedoinverse of \(A\text{.}\) When \(A\) is invertible, \(A^{-1} = A^+\text{.}\)
Rank \(k\) approximations
If we have a singular value decomposition for a matrix \(A\text{,}\) we can form a sequence of matrices \(A_k\) that approximate \(A\) with increasing accuracy. This may feel familiar to calculus students who have seen the way in which a function \(f(x)\) can be approximated by a linear function, a quadratic function, and so forth with increasing accuracy.
We'll begin with a singular value decomposition of a rank \(r\) matrix \(A\) so that \(A=U\Sigma V^T\text{.}\) To create the approximating matrix \(A_k\text{,}\) we keep the first \(k\) singular values and set the others to zero. For instance, if \(\Sigma = \begin{bmatrix} 22 & 0 & 0 & 0 & 0 \\ 0 & 14 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix} \text{,}\) we can form matrices
and define \(A_1 = U\Sigma^{(1)}V^T\) and \(A_2 = U\Sigma^{(2)}V^T\text{.}\) Because \(A_k\) has \(k\) nonzero singular values, we know that \(Rank(A_k) = k\text{.}\) In fact, there is a sense in which \(A_k\) is the closest matrix to \(A\) among all rank \(k\) matrices.
Activity 7.5.3.
Let's consider a matrix \(A=U\Sigma V^T\) where
Evaluating the following cell will create the matrices
U
,
V
, and
Sigma
. Notice how the
diagonal_matrix
command provides a convenient way to form the diagonal matrix \(\Sigma\text{.}\)
-
Form the matrix \(A=U\Sigma V^T\text{.}\) What is \(Rank(A)\text{?}\)
-
Now form the approximating matrix \(A_1=U\Sigma^{(1)} V^T\text{.}\) What is \(Rank(A_1)\text{?}\)
- Find the error in the approximation \(A\approx A_1\) by finding \(A-A_1\text{.}\)
-
Now find \(A_2 = U\Sigma^{(2)} V^T\) and the error \(A-A_2\text{.}\) What is \(Rank(A_2)\text{?}\)
- Find \(A_3 = U\Sigma^{(3)} V^T\) and the error \(A-A_3\text{.}\) What is \(Rank(A_3)\text{?}\)
- What would happen if we were to compute \(A_4\text{?}\)
- What do you notice about the error \(A-A_k\) as \(k\) increases?
In this activity, the approximating matrix \(A_k\) has rank \(k\) because its singular value decomposition has \(k\) nonzero singular values. We then saw how the difference between \(A\) and the approximations \(A_k\) decreases as \(k\) increases, which means that the sequence \(A_k\) forms better approximations as \(k\) increases.
Another way to represent \(A_k\) is with a reduced singular value decomposition so that \(A_k = U_k\Sigma_kV_k^T\) where
Notice that the rank \(1\) matrix \(A_1\) then has the form \(A_1 = \mathbf u_1\begin{bmatrix}\sigma_1\end{bmatrix} \mathbf v_1^T = \sigma_1\mathbf u_1\mathbf v_1^T\) and that we can similarly write:
Given two vectors \(\mathbf u\) and \(\mathbf v\text{,}\) the matrix \(\mathbf u~\mathbf v^T\) is called the outer product of \(\mathbf u\) and \(\mathbf v\text{.}\) (The dot product \(\mathbf u\cdot\mathbf v=\mathbf u^T\mathbf v\) is sometimes called the inner product .) An outer product will always be a rank \(1\) matrix so we see above how \(A_k\) is obtained by adding together \(k\) rank \(1\) matrices, each of which gets us one step closer to the original matrix \(A\text{.}\)
Principal component analysis
In Section 7.3 , we explored principal component analysis as a technique to reduce the dimension of a dataset. In particular, we constructed the covariance matrix \(C\) from a demeaned data matrix and saw that the eigenvalues and eigenvectors of \(C\) tell us about the variance of the dataset in different directions. We referred to the eigenvectors of \(C\) as principal components and found that projecting the data onto a subspace defined by the first few principal components frequently gave us a way to visualize the dataset. As we added more principal components, we retained more information about the original dataset. This feels similar to the rank \(k\) approximations we have just seen so let's explore the connection.
Suppose that we have a dataset with \(N\) points, that \(A\) represents the demeaned data matrix, that \(A = U\Sigma V^T\) is a singular value decomposition, and that the singular values are \(A\) are denoted as \(\sigma_i\text{.}\) It follows that the covariance matrix
Notice that \(\frac1N \Sigma\Sigma^T\) is a diagonal matrix whose diagonal entries are \(\frac1N\sigma_i^2\text{.}\) Therefore, it follows that
is an orthgonal diagonalization of \(C\) showing that
- the principal components of the dataset, which are the eigenvectors of \(C\text{,}\) are given by the columns of \(U\text{.}\) In other words, the left singular vectors of \(A\) are the principal components of the dataset.
-
the variance in the direction of a principal component is the associated eigenvalue of \(C\) and therefore
\begin{equation*} V_{\mathbf u_i} = \frac1N\sigma_i^2. \end{equation*}
Activity 7.5.4.
Let's revisit the iris data set that we studied in Section 7.3 . Remember that there are four measurements given for each of 150 irises and that each iris belongs to one of three species.
Evaluating the following cell will load the dataset and define the demeaned data matrix \(A\) whose shape is \(4\times150\text{.}\)
-
Find the singular values of \(A\) using the command
A.singular_values()
and use them to determine the variance \(V_{\mathbf u_j}\) in the direction of each of the four principal components. What is the fraction of variance retained by the first two principal components? -
We will now write the matrix \(\Gamma = \Sigma V^T\) so that \(A = U\Gamma\text{.}\) Suppose that a demeaned data point, say, the 100th column of \(A\text{,}\) is written as a linear combination of principal components:
\begin{equation*} \mathbf x = c_1\mathbf u_1+c_2\mathbf u_2+c_3\mathbf u_3+c_4\mathbf u_4. \end{equation*}
Explain why \(\fourvec{c_1}{c_2}{c_3}{c_4}\text{,}\) the vector of coordinates of \(\mathbf x\) in the basis of principal components, appears as 100th column of \(\Gamma\text{.}\)
- Suppose that we now project this demeaned data point \(\mathbf x\) orthogonally onto the subspace spanned by the first two principal components \(\mathbf u_1\) and \(\mathbf u_2\text{.}\) What are the coordinates of the projected point in this basis and how can we find them in the matrix \(\Gamma\text{?}\)
- Alternatively, consider the approximation \(A_2=U_2\Sigma_2V_2^T\) of the demeaned data matrix \(A\text{.}\) Explain why the 100th column of \(A_2\) represents the projection of \(\mathbf x\) onto the two-dimensional subspace spanned by the first two principal components, \(\mathbf u_1\) and \(\mathbf u_2\text{.}\) Then explain why the coefficients in that projection, \(c_1\mathbf u_1 + c_2\mathbf u_2\text{,}\) form the two-dimensional vector \(\twovec{c_1}{c_2}\) that is the 100th column of \(\Gamma_2=\Sigma_2 V_2^T\text{.}\)
-
Now we've seen that the columns of \(\Gamma_2 = \Sigma_2 V_2^T\) form the coordinates of the demeaned data points projected on to the two-dimensional subspace spanned by \(\mathbf u_1\) and \(\mathbf u_2\text{.}\) In the cell below, find a singular value decomposition of \(A\) and use it to form the matrix
Gamma2
. When you evaluate this cell, you will see a plot of the projected demeaned data plots, similar to the one we created in Section 7.3 .
In our first encounter with principal component analysis, we began with a demeaned data matrix \(A\text{,}\) formed the covariance matrix \(C\text{,}\) and used the eigenvalues and eigenvectors of \(C\) to project the demeaned data onto a smaller dimensional subspace. In this section, we have seen that a singular value decomposition of \(A\) provides a more direct route: the left singular vectors of \(A\) form the principal components and the approximating matrix \(A_k\) represents the data points projected onto the subspace spanned by the first \(k\) principal components. The coordinates of a projected demeaned data point are given by the columns of \(\Gamma_k = \Sigma_kV_k^T\text{.}\)
Image compressing and denoising
In addition to principal component analysis, the approximations \(A_k\) of a matrix \(A\) obtained from a singular value decomposition can be used in image processing. Remember that we studied the JPEG compression algorithm, whose foundation is the change of basis defined by the Discrete Cosine Transform, in Section 3.3 . We will now see how a singular value decomposition provides another tool for both compressing images and removing noise in them.
Activity 7.5.5.
Evaluating the following cell loads some data that we'll use in this activity. To begin, it defines and displays a \(25\times15\) matrix \(A\text{.}\)
-
If we interpret 0 as black and 1 as white, this matrix represents an image as shown below.
We will explore how the singular value decomposition helps us to compress this image.
- By inspecting the image represented by \(A\text{,}\) identify a basis for \(Col(A)\) and determine \(Rank(A)\text{.}\)
-
The following cell plots the singular values of \(A\text{.}\) Explain how this plot verifies that the rank is what you found in the previous part.
-
There is a command
approximate(A, k)
that creates the approximation \(A_k\text{.}\) Use the cell below to define \(k\) and look at the images represented by the first few approximations. What is the smallest value of \(k\) for which \(A=A_k\text{?}\) -
Now we can see how the singular value decomposition allows us to compress images. Since this is a \(25\times15\) matrix, we need \(25\cdot15=375\) numbers to represent the image. However, we can also reconstruct the image using a small number of singular values and vectors:
\begin{equation*} A = A_k = \sigma_1\mathbf u_1\mathbf v_1^T + \sigma_2\mathbf u_2\mathbf v_2^T + \ldots + \sigma_k\mathbf u_k\mathbf v_k^T. \end{equation*}
What are the dimensions of the singular vectors \(\mathbf u_i\) and \(\mathbf v_i\text{?}\) Between the singular vectors and singular values, how many numbers do we need to reconstruct \(A_k\) for the smallest \(k\) for which \(A=A_k\text{?}\) This is the compressed size of the image.
- The compression ratio is the ratio of the uncompressed size to the compressed size. What compression ratio does this represent?
-
Next we'll explore an example based on a photograph.
-
Consider the following image consisting of an array of \(316\times310\) pixels stored in the matrix \(A\text{.}\)
Plot the singular values of \(A\text{.}\)
-
Use the cell below to study the approximations \(A_k\) for \(k=1, 10, 20, 50, 100\text{.}\)
Notice how the approximating image \(A_k\) more closely approximates the original image \(A\) as \(k\) increases.
What is the compression ratio when \(k=50\text{?}\) What is the compression ratio when \(k=100\text{?}\) Notice how a higher compression ratio leads to a lower quality reconstruction of the image.
-
Consider the following image consisting of an array of \(316\times310\) pixels stored in the matrix \(A\text{.}\)
-
A second, related application of the singular value decomposition to image processing is called
denoising
. For example, consider the image represented by the matrix \(A\) below.
This image is similar to the image of the letter "O" we first studied in this activity, but there are splotchy regions in the background that result, perhaps, from scanning the image. We think of the splotchy regions as noise, and our goal is to improve the quality of the image by reducing the noise.
-
Plot the singular values below. How are the singular values of this matrix similar to those represented by the clean image that we considered earlier and how are they different?
-
There is a natural point where the singular values dramatically decrease so it makes sense to think of the noise as being formed by the small singular values. To denoise the image, we will therefore replace \(A\) by its approximation \(A_k\text{,}\) where \(k\) is the point at which the singular values drop off. This has the effect of setting the small singular values to zero and hence eliminating the noise. Choose an appropriate value of \(k\) below and notice that the new image appears to be somewhat cleaned up as a result of removing the noise.
-
Plot the singular values below. How are the singular values of this matrix similar to those represented by the clean image that we considered earlier and how are they different?
Several examples illustrating how the singular value decomposition compresses images are available at this page from Tim Baumann.
Analyzing Supreme Court cases
As we've seen, a singular value decomposition concentrates the most important features of a matrix into the first singular values and singular vectors. We will now use this observation to extract meaning from a large dataset giving the voting records of Supreme Court justices. A similar analysis appears in the paper A pattern analysis of the second Rehnquist U.S. Supreme Court by Lawrence Sirovich.
The makeup of the Supreme Court was unusually stable during a period from 1994-2005 when it was led by Chief Justice William Rehnquist. This is sometimes called the second Rehnquist court . The justices during this period were:
- William Rehnquist
- Antonin Scalia
- Clarence Thomas
- Anthony Kennedy
- Sandra Day O'Connor
- John Paul Stevens
- David Souter
- Ruth Bader Ginsburg
- Stephen Breyer
During this time, there were 911 cases in which all nine judges voted. We would like to understand patterns in their voting.
Activity 7.5.6.
Evaluating the following cell loads and displays a dataset describing the votes of each justice in these 911 cases. More specifically, an entry of +1 means that the justice represented by the row voted with the majority in the case represented by the column. An entry of -1 means that justice was in the minority. This information is also stored in the \(9\times911\) matrix \(A\text{.}\)
The justices are listed, very roughly, in order from more conservative to more progressive.
In this activity, it will be helpful to visualize the entries in various matrices and vectors. The next cell displays the first 50 columns of the matrix \(A\) with white representing an entry of +1, red representing -1, and black representing 0.
-
Plot the singular values of \(A\) below. Describe the significance of this plot, including the relative contributions from the singular values \(\sigma_k\) as \(k\) increases.
-
Form the singular value decomposition \(A=U\Sigma V^T\) and the matrix of coefficients \(\Gamma\) so that \(A=U\Gamma\text{.}\)
-
We will now study a particular case, the second case appearing as the column of \(A\) indexed by
1
. There is a commanddisplay_column(A, k)
that provides a visual display of the \(k^{th}\) column of a matrix \(A\text{.}\) Describe the justices' votes in the second case. -
Also, display the first left singular vector \(\mathbf u_1\text{,}\) the column of \(U\) indexed by \(0\text{,}\) and the column of \(\Gamma\) holding the coefficients that express the second case as a linear combination of left singular vectors.
What does this tell us about how the second case is constructed as a linear combination of left singular vectors? What is the significance of the first left singular vector \(\mathbf u_1\text{?}\)
-
Let's now study the \(48^{th}\) case, which is represented by the column of \(A\) indexed by
47
. Describe the voting pattern in this case. -
Display the second left singular vector \(\mathbf u_2\) and the vector of coefficients that express the \(48^{th}\) case as a linear combination of left singular vectors.
Describe how this case is constructed as a linear combination of singular vectors. What is the significance of the second left singular vector \(\mathbf u_2\text{?}\)
-
The data in Table 7.5.2 describes the number of cases decided by each possible vote count.
How do the singular vectors \(\mathbf u_1\) and \(\mathbf u_2\) reflect this data? Would you characterize the court as leaning toward the conservatives or progressives? Use these singular vectors to explain your response.
Table 7.5.2. Number of cases by vote count Vote count # of cases 9-0 405 8-1 89 7-2 111 6-3 118 5-4 188 -
Cases decided by a 5-4 vote are often the most impactful as they represent a sharp divide among the justices and, often, society at large. For that reason, we will now focus on the 5-4 decisions. Evaluating the next cell forms the \(9\times188\) matrix \(B\) consisting of 5-4 decisions.
Form the singular value decomposition of \(B=U\Sigma V^T\) along with the matrix \(\Gamma\) of coefficients so that \(B=U\Gamma\) and display the first left singular vector \(\mathbf u_1\text{.}\) Study how the \(7^{th}\) case, indexed by
6
, is constructed as a linear combination of left singular vectors.What does this singular vector tell us about the make up of the court and whether it leans towards the conservatives or progressives? -
Display the second left singular vector \(\mathbf u_2\) and study how the \(6^{th}\) case, indexed by
5
, is constructed as a linear combination of left singular vectors.What does \(\mathbf u_2\) tell us about the relative importance of the justices' voting records? - By a swing vote , we mean a justice who is less inclined to vote with a particular bloc of justices but instead swings from one bloc to another with the potential to sway close decisions. What do the singular vectors \(\mathbf u_1\) and \(\mathbf u_2\) tell us about the presence of voting blocs on the court and the presence of a swing vote? Which justice represents the swing vote?
Summary
This section has demonstrated some uses of the singular value decomposition. Because the singular values appear in decreasing order, the decomposition has the effect of concentrating the most important features of the matrix into the first singular values and singular vectors.
- Because the first left singular vectors form an orthonormal basis for \(Col(A)\text{,}\) a singular value decomposition provides a convenient way to project vectors onto \(Col(A)\) and therefore to solve least squares problems.
-
A singular value decomposition of a rank \(r\) matrix \(A\) leads to a series of approximations \(A_k\) of \(A\) where
\begin{align*} A \approx A_1 & = \sigma_1\mathbf u_1\mathbf v_1^T\\ A \approx A_2 & = \sigma_1\mathbf u_1\mathbf v_1^T + \sigma_2\mathbf u_2\mathbf v_2^T\\ A \approx A_3 & = \sigma_1\mathbf u_1\mathbf v_1^T + \sigma_2\mathbf u_2\mathbf v_2^T + \sigma_3\mathbf u_3\mathbf v_3^T\\ \vdots & \\ A = A_r & = \sigma_1\mathbf u_1\mathbf v_1^T + \sigma_2\mathbf u_2\mathbf v_2^T + \sigma_3\mathbf u_3\mathbf v_3^T + \ldots + \sigma_r\mathbf u_r\mathbf v_r^T \end{align*}
In each case, \(A_k\) is the rank \(k\) matrix that is closest to \(A\text{.}\)
- If \(A\) is a demeaned data matrix, the left singular vectors give the principal components of \(A\) and the variance in the direction of a principal component can be easily expressed in terms of the corresponding singular value.
- The singular value decomposition has many applications. In this section, we looked at how the decomposition is used in image processing through the techniques of compression and denoising.
- Because the first few left singular vectors contain the most important features of a matrix, we can use a singular value decomposition to extract meaning from a large dataset as we did when analyzing the voting patterns of the second Rehnquist court.
Exercises 7.5.7Exercises
Suppose that
- Find the singular values of \(A\text{.}\) What is \(Rank(A)\text{?}\)
- Find the sequence of matrices \(A_1\text{,}\) \(A_2\text{,}\) \(A_3\text{,}\) and \(A_4\) where \(A_k\) is the rank \(k\) approximation of \(A\text{.}\)
Suppose we would like to find the best quadratic function
fitting the points
- Set up a linear system \(A\mathbf x = \mathbf b\) describing the coefficients \(\mathbf x = \threevec{\beta_0}{\beta_1}{\beta_2}\text{.}\)
- Find the singular value decomposition of \(A\text{.}\)
- Use the singular value decomposition to find the least squares approximate solution \(\widehat{\mathbf{x}}\text{.}\)
Remember that the outer product of two vector \(\mathbf u\) and \(\mathbf v\) is the matrix \(\mathbf u~\mathbf v^T\text{.}\)
-
Suppose that \(\mathbf u = \twovec{2}{-3}\) and \(\mathbf v=\threevec201\text{.}\) Evaluate the outer product \(\mathbf u~\mathbf v^T\text{.}\) To get a clearer sense of how this works, perform this operation without using technology.
How is each of the columns of \(\mathbf u~\mathbf v^T\) related to \(\mathbf u\text{?}\)
- Suppose \(\mathbf u\) and \(\mathbf v\) are general vectors. What is \(Rank(\mathbf u~\mathbf v^T)\) and what is a basis for its column space \(Col(\mathbf u~\mathbf v^T)\text{?}\)
- Suppose that \(\mathbf u\) is a unit vector. What is the effect of multiplying a vector by the matrix \(\mathbf u~\mathbf u^T\text{?}\)
Evaluating the following cell loads in a dataset recording some features of 1057 houses. Notice how the lot size varies over a relatively small range compared to the other features. For this reason, in addition to demeaning the data, we'll scale each feature by dividing by its standard deviation so that the range of values is similar for each feature. The matrix \(A\) holds the result.
-
Find the singular values of \(A\) and use them to determine the variance in the direction of the principal components.
- For what fraction of the variance do the first two principal components account?
-
Find a singular value decomposition of \(A\) and construct the matrix \(2\times1057\) matrix \(B\) whose entries are the coordinates of the demeaned data points projected on to the two-dimensional subspace spanned by the first two principal components. You can plot the projected data points using
list_plot(B.columns())
. - Study the entries in the first two principal components \(\mathbf u_1\) and \(\mathbf u_2\text{.}\) Would a more expensive house lie on the left, right, top, or bottom of the plot you constructed?
- In what ways does a house that lies on the far left of the plot you constructed differ from an average house? In what ways does a house that lies near the top of the plot you constructed differ from an average house?
Let's revisit the voting records of justices on the second Rehnquist court. Evaluating the following cell will load the voting records of the justices in the 188 cases decided by a 5-4 vote and store them in the matrix \(A\text{.}\)
-
The cell above also defined the 188-dimensional vector \(\mathbf v\) whose entries are all 1. What does the product \(A\mathbf v\) represent? Use the following cell to evaluate this product.
- How does the product \(A\mathbf v\) tell us which justice voted in the majority most frequently? What does this say about the presence of a swing vote on the court?
- How does this product tell us whether we should characterize this court as leaning conservative or progressive?
- How does this product tell us about the presence of a second swing vote on the court?
- Study the left singular vector \(\mathbf u_3\) and describe how it reinforces the fact that there was a second swing vote. Who was this second swing vote?
The following cell loads a dataset that describes the percentages with which justices on the second Rehnquist court agreed with one another. For instance, the entry in the first row and second column is 72.78, which means that Justices Rehnquist and Scalia agreed with each other in 72.78% of the cases.
- Examine the matrix \(A\text{.}\) What special structure does this matrix have and why should we expect it to have this structure?
-
Plot the singular values of \(A\) below. For what value of \(k\) would the approximation \(A_k\) be a reasonable approximation of \(A\text{?}\)
-
Find a singular value decomposition \(A=U\Sigma V^T\) and examine the matrices \(U\) and \(V\) using, for instance,
n(U, 3)
. What do you notice about the relationship between \(U\) and \(V\) and why should we expect this relationship to hold? -
The command
approximate(A, k)
will form the approximating matrix \(A_k\text{.}\) Study the matrix \(A_1\) using thedisplay_matrix
command. Which justice or justices seem to be most agreeable, that is, most likely to agree with other justices? Which justice is least agreeable? -
Examine the difference \(A_2-A_1\) and describe how this tells us about the presence of voting blocs and swing votes on the court.
Suppose that \(A=U_r\Sigma_rV_r^T\) is a reduced singular value decomposition of the \(m\times n\) matrix \(A\text{.}\) The matrix \(A^+ = V_r\Sigma_r^{-1}U_r^T\) is called the Moore-Penrose inverse of \(A\text{.}\)
- Explain why \(A^+\) is an \(n\times m\) matrix.
- If \(A\) is an invertible, square matrix, explain why \(A^+=A^{-1}\text{.}\)
- Explain why \(AA^+\mathbf b=\widehat{\mathbf{b}}\text{,}\) the orthogonal projection of \(\mathbf b\) onto \(Col(A)\text{.}\)
- Explain why \(A^+A\mathbf x=\widehat{\mathbf{x}}\text{,}\) the orthogonal projection of \(\mathbf x\) onto \(Col(A^T)\text{.}\)
In Subsection 5.1 , we saw how some linear algebraic computations are sensitive to round off error made by a computer. A singular value decomposition can help us understand when this situation can occur.
For instance, consider the matrices
The entries in these matrices are quite close to one another, but \(A\) is invertible while \(B\) is not. It seems like \(A\) is almost singular. In fact, we can measure how close a matrix is to being singular by forming the condition number , \(\sigma_1/\sigma_n\text{,}\) the ratio of the largest to smallest singular value. If \(A\) were singular, the condition number would be undefined because the singular value \(\sigma_n=0\text{.}\) Therefore, we will think of matrices with large condition numbers as being close to singular.
-
Define the matrix \(A\) and find a singular value decomposition. What is the condition number of \(A\text{?}\)
-
Define the left singular vectors \(\mathbf u_1\) and \(\mathbf u_2\text{.}\) Compare the results \(A^{-1}\mathbf b\) when
- \(\mathbf b=\mathbf u_1+\mathbf u_2\text{.}\)
- \(\mathbf b=2\mathbf u_1+\mathbf u_2\text{.}\)
Notice how a small change in the vector \(\mathbf b\) leads to a small change in \(A^{-1}\mathbf b\text{.}\)
-
Now compare the results \(A^{-1}\mathbf b\) when
- \(\mathbf b=\mathbf u_1+\mathbf u_2\text{.}\)
- \(\mathbf b=\mathbf u_1+2\mathbf u_2\text{.}\)
Notice now how a small change in \(\mathbf b\) leads to a large change in \(A^{-1}\mathbf b\text{.}\)
-
Previously, we saw that, if we write \(\mathbf x\) in terms of left singular vectors \(\mathbf x=c_1\mathbf v_1+c_2\mathbf v_2\text{,}\) then we have
\begin{equation*} \mathbf b=A\mathbf x = c_1\sigma_1\mathbf u_1 + c_2\sigma_2\mathbf u_2. \end{equation*}
If we write \(\mathbf b=d_1\mathbf u_1+d_2\mathbf u_2\text{,}\) explain why \(A^{-1}\mathbf b\) is sensitive to small changes in \(d_2\text{.}\)
Generally speaking, a square matrix \(A\) with a large condition number will demonstrate this type of behavior so that the computation of \(A^{-1}\) is likely to be affected by round off error. We call such a matrix ill-conditioned .


5.4: Classical School
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- Alison S. Burke, David Carter, Brian Fedorek, Tiffany Morey, Lore Rutz-Burri, & Shanell Sanchez
- Southern Oregon University via OpenOregon
During the Enlightenment, citizens and social thinkers began to question how they were ruled. In the Leviathan (1651), Hobbes made a few assumptions about human beings. [1] He assumed humans were at conflict with one another, pursued their self-interests, and were rational. Moreover, people would create authority figures out of fear of others, and people should democratically create rules that all citizens must follow. Hobbes wanted a new type of government, one that was ruled by the people and not by monarchs. He believed people had natural rights such as life, liberty, and the pursuit of happiness. If we grant the assumption that people are rational, we would assume people have the ability to consider the possible consequences of their actions. Hobbes was one of the first social contract thinkers. Social contract thinkers believed people would invest in the laws of their society if, and only if, they know government protected them from those who break the law. People will give up a little of their self-interests as long as everyone reciprocates.
Building on Hobbes and other social contract thinkers at the time, humans were assumed to have free will and were rational beings. We can choose one action over another based on perceived benefits and possible consequences. Moreover, human beings are hedonistic. Hedonism is the assumption that people will see maximum pleasure and avoid pain (punishment). Consequently, if we grant the assumptions of classical theory, we can hold people 100% responsible for their actions because it was a choice. These assumptions have been the basis for the American criminal justice system since its inception. Although theories may have changed the landscape of understanding criminal behavior and may have changed the philosophies of punishments over time, the criminal justice system has maintained the assumption that crime is a choice. Hence, we can hold offenders 100% responsible for their actions.
Cesare Beccaria (1738-1794) was an Italian mathematician and economist. He was shocked by the unfair treatment of the accused. In protest, he anonymously wrote An Essay on Crimes and Punishment (1764), which attacked Europe’s use of harsh treatment. [2] Ideally, he wanted to change the excessive and cruel punishment by applying rationalistic, social contract ideas. At the time, judges had tremendous power to determine guilt and create laws based on their decisions. Intellectuals well received his essay at the time, but the Catholic Church banned it. His ideas were exceptionally radical at the time, mainly because his writing questioned the power structures at the time.
Beccaria laid out his ideas about legal reform including how and why to create effective punishments. His Essay was highly influential during the Enlightenment, and it may have served as a model during the creation of the American Criminal Justice system. For example, Beccaria advocated that punishments should fit the crime and be proportional to the harm done, laws should only be determined by the legislature, judges should only determine guilt, and every person should be treated equally under the law. He claimed the sole purpose of the law was to deter people from committing the crime. Deterrence can be accomplished if the punishment is certain, swift, and severe. These may seem like common sense today, but they were considered radical ideas at the time.
Jeremy Bentham (1748-1832) was an English philosopher and regarded as a founder of utilitarianism, which is the belief that decisions are considered right or wrong depending on their effect. He believed a person’s expectation of the future was most predictive for deterrence. Therefore, the utility of punishment should be severe enough to deter people from crime. Punishment would promote happiness throughout society by maximizing social benefits. He helped popularize classical theory throughout Europe. [3]
-
Hobbes, T. (1651/1968).
Leviathan
. Baltimore, MD: Penguin Books.
↵
-
Beccaria, C. (1963).
On crimes and punishments
(H. Paolucci, Trans.). Indianapolis, IN: Bobbs-Merril. (Original work published in 1764)
↵
-
Bentham, J. (1823).
Introduction to the principles of morals and legislation
. Oxford, UK: Oxford University Press.
↵


2.11: Words and Numbers (An Aside)
In business and perhaps in Finance in particular, memorization is less important than active, creative thinking. Some people find math intimidating, but it does not have to be. By and large, Finance does not require the memorization of formulas. However, one must understand the mathematics.
This is an important distinction. If you regard formulas as memorization, it may become an unpleasant and, worse, a difficult task. Instead, think of formulas as shorthand explanations of the relationships between important financial concepts, between prices and interest rates, for example. If this is done, the relationships will seem clear and logical, and memorization should prove unnecessary, or at least, less so.
Some people consider themselves word- rather than number-persons. Think of numbers as words! Indeed, they are linguistically meaningful. Mathematical symbols are a shorthand way of saying something that can also be said – at greater length – in words.
Over the course of the semester, the student will be exposed to numerous mathematical formulae.
Do not worry about your difficulties in mathematics; I can assure you mine are still even greater.
-Albert Einstein


Electron Affinity
Skills to Develop
- Define electron affinity
- Describe the periodic trend of electron affinity
Electron affinity is a measure of how much an atom wants to gain an electron, becoming an anion. Unfortunately 2 different definitions are used: intro textbooks use 1 definition and everyone else uses the other! I think you should use the standard advanced definition, according to which electron affinity EA = IE 0 , the energy of this reaction:
\[A^{−}(g) \rightarrow A(g) + e^{−}(g)\]
The other definition just has the opposite sign, because it's the energy of this process:
\[A(g) + e^{−}(g) \rightarrow A^{−}(g)\]
Why does Electron Affinity matter?
Like ionization energy, electron affinity tells us how likely an atom is to steal electrons from other atoms, or just convince them to transfer their loyalty partially.
Predicting Relative Electron Affinities
It pretty much follows the same pattern you would expect based on ionization energy. If the new electron goes into an orbital that feels high effective nuclear charge, gaining an electron is good. If it goes into an orbital that is in a higher shell or subshell, or that feels low effective nuclear charge, gaining an electron is bad. The only other thing is that it's easier to add electrons to bigger atoms, because then they can spread out and not have so much electron-electron repulsion. For this reason, chlorine has more favorable EA than fluorine (in the gas phase, fluorine is still much more reactive).
Look at EA for yourself!
Go to Ptable's electron affinity page. See the general trend (bigger middle and right) with big effects at the end and middle of each block (noble gases, mercury family, alkaline earths, nitrogen, manganese).


20.3: The Writers Guild of America in the National Context
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- Miranda Banks & David Hesmondhalgh
- University of California Press
In early November 2007, certain quarters of Los Angeles transformed overnight into walking districts. For the next five months, five days a week, dozens of writers, often spectacled, wearing jeans and T-shirts and always with picket signs, walked for hours in front of various gates of the major Hollywood studios. Across the country, dozens more in New York bundled up and braved the cold to protest their rights of labor and rates of compensation. These professional film and television writers walked en masse to protest stalled negotiations with the American trade organization the Alliance of Motion Picture and Television Producers (AMPTP). For the first time in nineteen years, the Writers Guild of America (WGA) was on strike. Nationally, a poll conducted two weeks after negotiations broke off showed that 63 percent of Americans sided with the striking workers (with 4 percent favoring the studios, 33 percent unsure). 19
It is rare in the United States to see striking workers marching in a number of areas across the two largest cities in the country. Even more notable was the fact that these employees were neither blue-collar laborers nor white-collar workers. They were no-collar workers. 20 Unlike earlier strikes, this time writer-producers and showrunners also walked the picket lines, arguing that they could not separate their work as producers from their role as writers. The guild leadership specifically targeted showrunners early in the negotiations to get their support, not just for labor action but to read the letter of the law in such a way that their role as producers could not be separated from their role as writers. While as producers they were part of management, as writers they were employees of the studio. While some faces were familiar—Tina Fey, Rob Reiner—others had names that were familiar to audiences: Norman Lear and James L. Brooks. Still others were attached to beloved products that suddenly disappeared from homes across the globe. Writers were now positioned—in their role marching around the outside of studio buildings—as industry workers fighting for their rights.
The Writers Guild of America was first established as the Screen Writers Guild in 1933, though it was not granted a contract until 1942. The WGA, which comprises East and West branches, is the bargaining agent for professional writers who craft film, television, news, animation, streaming media, and video game scripts for American signatory companies. The Writers Guild has gone on strike six times, in 1959–1960, 1973, 1981, 1985, 1988, and 2007–2008. Three of these industry-wide walkouts were protracted, lasting many months. As they had in every previous strike, in 2007–2008, these American writers marched in circles and demanded their rights, not as artisans but as workers in a media industry. This time, though, because of the globalization of film and television distribution, as well as the rise of YouTube—where many striking writers went to speak directly to audiences—more people than ever before were aware of a strike among working writers. Not just in the United States, but globally. And not just audiences, but other writers as well.
For the writers under its protection, the WGA as a guild provides union-oriented services: it convenes and mobilizes members, addresses their concerns, negotiates and enforces contracts, lobbies on behalf of its members, and represents the face of screenwriters to the outside world. But it is its final directive—preserving the art and craft of writing—that most clearly illuminates the subtle difference between a union and a guild. The WGA sees its protection, teaching, and preservation of the work of writing as the additive dimension that distinguishes it from a traditional trade union.
Yet during moments of economic crisis or labor negotiations, writers often feel compelled to define themselves as a union first and foremost. Bob Barbash, a writer on Zane Grey Theater , explained how this perception played out during a strike in 1960: “A tremendous amount of people in the Guild . . . resent the word ‘union.’ . . .  Now that is not a Guild. That’s a union, man. When you are walking there and you are trying to stop people from crossing the line. We are an unusual group because we like to think of ourselves as [part of a] super, upper [tier of] intelligence. That we don’t work on a loading dock . . . but if you are going to have a union, you are a union.” 21 In contrast, the term guild implies a focus less on working conditions and more on championing the artistry of the profession. The difference is not merely one of terminology: it has resulted in a recurring tug-of-war across the entertainment industries between different groups of writers and sometimes even within an individual writer’s conception of what they do and how their interests ought to be represented. 22 The internal friction is captured in shifting definitional terms such as artist , worker , creative , laborer .
Writers must join the guild if they have surpassed a certain quantity of work with a company that has signed as a contractual partner on the guild’s collective bargaining agreement. A signatory company can be as vast as a multinational corporation or as limited as a small pro-union production company. An associate writer amasses units to gain full membership, and today writers must belong to either the WGA East (which uses the acronym WGAE) or the Writers Guild West (which prefers WGAw), depending on geography. The guild’s stated objectives are voluminous. It contracts minimum rates for specific types of work, determines writers’ screen credits, ensures payment of residuals, provides pensions and health benefits for members, engages in national policy debates concerning writers’ interests, and provides continuing education for members and the community. Some writers have seen their induction into the guild as a sign of having “made it” in the industry. Others have felt membership to be a weighty burden foisted upon them. And still others have paid little attention to what membership meant. Then there are those who view membership as a life raft. Barbara Corday, creator of Cagney & Lacey , expressed deep gratitude for the benefits afforded to veteran writers: “First of all, having residuals. Lifetime medical insurance as a backup to Medicare, as a secondary insurance. How many people outside of Congress have things like that? It’s just phenomenal .” 23
Corralling this disparate group of workers, however, is an arduous task. The guild brings together thousands of individuals who predominantly perform solitary work. As Hal Kanter, creator of the series Julia , noted in the 1970s, “We writers are, collectively, a strange group of creatures and it’s a frequent source of amazement to me that the Guild is such a well-run zoo!” 24 John Furia Jr., writer for The Singing Nun and president of the WGAw from 1973 to 1975, laughed as he pointed out, “We are the most individualistic group to band together.” 25 Phyllis White, who worked on writing teams for various television series from the 1950s through the 1980s, noted the paradox of singular writers with unique voices aligning for a collective cause: “It’s a Guild of individuals as no other union is. You’ve got the Teamsters and there are a certain number of Teamsters who do the same job. . . . They do the same hours. They do the same thing. We don’t. . . . Trying to amalgamate this group . . . [of] nearly 5,000 into one union now is horrendous. It’s amazing that it works at all.” 26 White’s sweeping claims around the specialness of writers’ work are problematic: many trade unions cover diverse members with distinct job descriptions, and the work of writers is not as rarefied as she proclaims. 27 And yet the notion of collecting a community of workers who usually work alone does pose distinct difficulties.
Another major challenge for the Writers Guild is that it coexists with a number of other guilds and unions in the media industries. The other groups that negotiate with signatory companies include the Directors Guild of America (DGA), which represents directors, assistant directors, unit production managers, and production associates; the Screen Actors Guild–American Federation of Television and Radio Artists (SAG-AFTRA), which represents actors, extras, broadcast journalists, and puppeteers, among others; and the International Alliance of Theatrical Stage Employees (IATSE), which represents a diverse set of industry workers, from electricians to set carpenters, makeup artists, prop masters, cinematographers, editors, and art directors. The other three organizations service vastly larger constituencies than the WGA, and have needs so diverse that a united front proves tricky—especially when it comes time to negotiate with the monolithic Alliance of Motion Picture and Television Producers (AMPTP). The AMPTP is an enormous bargaining unit that digests the concerns of hundreds of production companies, networks, and studios and then delivers a proposal—representing the united group’s interests—to the negotiating table. Whereas in standard bargaining a union tries to garner advantage by playing off one company against another, the AMPTP positions itself so that the three creative guilds must jostle with each other, grabbing for scraps at the table. This tactic, called reverse pattern bargaining, forces each guild into what one member called “a kind of a chess game between the three unions.” 28


4.6: Post-positions
Post-Positions Video
Instructor Video: Post-positions
English has a class of words called “prepositions”. Prepositions are the words that precede a phrase and indicate a relationship between the phrase and another word in the sentence. Hindi has a similar class of words known as “post-positions”. As the name implies, post-positions come before the word they modify. In other respects, however, they are very similar to prepositions in English.
Simple Post-positions
Simple post-positions consist of a single word.
Examples
| Post-position | English | Examples | |
| Hindi | English | ||
| में | in | मेरे शहर में | In my town |
| पर | on, | मेज़ पर | On a table |
| at | घर पर | At home | |
| से | from | भारत से | from India |
| by | ट्रेन से | by train | |
| with | कलम से | with a pen | |
| since | बचपन से | since childhood | |
| का/की/ के | ‘s |
सोने का कमरा
घर की खिड़की घर के लोग |
bed room
House’s window (window of a house) House’s people (people of a house) |
| को | to |
लड़के को
आप को |
to a boy
to you |
Compound Post-positions
Hindi has many compound post-positions as well. Compound post-positions consist of two or more words. Most compound post-positions begin with the post-position के, की, or से.
Examples
| Post-position | English | Examples | |
| Hindi | English | ||
| x के लिये | for x | घर के लिये | For house |
| x के बारे में | about x | शहर के बारे में | About town |
| x की वजह से / के कारण | b ecause of x | आप की वजह से / के कारण | Because of you |
| x से दूर | far from x | घर से दूर | far from home |
| x के नज़दीक / पास | near x | घर के नज़दीक / पास | near home |
| x से / के पहले | before x | स्कूल से / के पहले | before school |
| x के बाहर | outside x | घर के बाहर | outside home |
| x के अंदर | inside the x | घर के अंदर | inside the home |
| x के बाद | after x | स्कूल के बाद | after school |
| x के ऊपर | above/ on/ on top of x | घर के ऊपर | above/ on/ on top of home |
| x के नीचे | under/ beneath/ below x | छत के नीचे | under/ beneath/ below roof |
| x की तरफ़ | towards x | आप की तरफ़ | towards you |
| x के सामने | in front of x | आप के सामने | in front of you |
| x के पीछे | behind x | आप के पीछे | behind you |
| x के आगे | ahead of x | आप के आगे | ahead of you |
| x के साथ | together with/ with/ along with x | आप के साथ | together with/ with/ along with you |
| x और y के बीच | between x and y | स्कूल और घर के बीच | between school and home |
Test yourself
https://openbooks.lib.msu.edu/ll151/?p=855
Fill in the blanks with suitable post-position.
https://openbooks.lib.msu.edu/ll151/?p=855
धन्यवाद !


1.6: Pregnancy and Fetal Development
Chapter Objectives
- Identify maternal and fetal changes that occur during each prenatal trimester
- Discuss possible dysfunction or complications that can occur during each trimester of pregnancy
- Describe a typical birth process along with at least four variations that could occur
- Describe the pros and cons associated with homebirth and hospital birth
- Discuss possible causes and symptoms of miscarriage and stillbirth
Pregnancy and Birthing Options
“Making the decision to have a baby (or adopt a child) is momentous. It is to decide forever to have your heart go walking around outside your body.” ~ Elizabeth Stone
“That she bear children is not a woman’s significance. But that she bear herself, that is her supreme and risky fate.” ~ D.H. Lawrence
To Have or Not to Have Children
One of the bigger choices we make as humans is the choice to have and raise children or to not have children. Sometimes it is not a choice at all but the result of circumstances given to us. Why might a person choose to have kids and what would the arguments be for choosing to remain childless? What circumstances might cause someone who so badly wants a child to be without? How might someone deal with an unexpected pregnancy when having a child was not part of the plan?

Pregnancy
A missed period is often the first clue that a woman might be pregnant though she might suspect pregnancy even sooner. Symptoms such as headache, fatigue, and breast tenderness, can occur even before a missed period. The wait to know can be emotional. These days, many women first use home pregnancy tests (HPT) to learn they are pregnant.
All pregnancy tests work by detecting Human Chorionic Gonadotropic (hCG) in the urine or blood only present a woman is pregnant. hCG is made when a fertilized egg implants in the uterus and rapidly builds up in your body during early pregnancy.
Unplanned Pregnancy
In the event of an unplanned pregnancy many thoughts may come to mind. Will your partner welcome the news? Can you afford to care for the baby? Will choices such as past drug use addiction affect the health of the unborn baby? Will having a baby keep me from finishing school or pursuing a career?
First steps to take when a person fins out they are pregnant:
- Start taking care of yourself right away. Take 400 to 800 micrograms (400 to 800 mcg or 0.4 to 0.8 mg) folic acid every. Stop alcohol, tobacco, and drug use.
- Make a doctor’s visit to confirm your pregnancy. Discuss your health and issues that could affect your pregnancy. Ask for help quitting smoking. Find out what you can do to take care of yourself and your unborn baby.
- Ask your doctor to recommend a counselor who you can talk to about your situation.
- Seek support in someone you trust and respect.
Trying to Get Pregnant
Some women want children but either cannot conceive or miscarry multiple times. Many couples face infertility when trying to start or grow a family. About one-third of the time, it is a female problem. In another one-third of cases, it is the man with the fertility problem. For the remaining one-third, both partners have fertility challenges or no cause is found.
Causes of infertility
Age – Women generally have some decrease in fertility starting in their early 30s. As a woman ages, normal changes that occur in her ovaries and eggs make it harder to become pregnant. Even though menstrual cycles continue to be regular in a woman’s 30s and 40s, the eggs that ovulate each month are of poorer quality than those from her 20s. As a woman nears menopause, the ovaries may not release an egg each month, which also can make it harder to get pregnant.
Health problems – Some women have diseases or conditions that affect their hormone levels, which can cause infertility. Women with polycystic ovary syndrome (PCOS) rarely or never ovulate. Failure to ovulate is the most common cause of infertility in women.
Common problems with a woman’s reproductive organs, like uterine fibroids, endometriosis, and pelvic inflammatory disease can worsen with age and also affect fertility. These conditions might cause a blockage in the fallopian tubes to be blocked preventing the egg from getting to the uterus.
Lifestyle factors – Certain lifestyle factors also can have a negative effect on a woman’s fertility. Examples include smoking, alcohol use, weighing much more or much less than an ideal body weight, a lot of strenuous exercise, and having an eating disorder. Stress also can affect fertility.
When to see your doctor
You should talk to your doctor about your fertility if:
* You are younger than 35 and have not been able to conceive after one year of frequent sex without birth control.
- You are age 35 or older and have not been able to conceive after six months of frequent sex without birth control.
- You believe you or your partner might have fertility problems in the future (even before you begin trying to get pregnant).
- You or your partner has a problem with sexual function or libido.
If you are having fertility issues, your doctor can refer you to a fertility specialist. About 9 in 10 cases of infertility are treated with drugs or surgery. Don’t delay seeing your doctor as age also affects the success rates of these treatments. For some couples, adoption or foster care offers a way to share their love with a child and build a family.
Increasing Likelihood of Pregnancy
Being aware of the menstrual cycle and the changes in the body that happen during mensuration can help a woman know when she is likely to get pregnant.
There are ways you can keep track of your fertile times in the menstrual cycle. They are:
- Basal body temperature method – Basal body temperature is your temperature at rest as soon as you awake in the morning. A woman’s basal body temperature rises slightly with ovulation. So by recording this temperature daily for several months, a woman will be able to predict the most fertile days.
- Calendar method – This involves recording your menstrual cycle on a calendar for eight to 12 months. The first day of a period is Day 1. Circle Day 1 on the calendar and record down the total number of days a cycle it lasts each time. Days 12-14 of the cycle will be a woman’s most fertile days.
- Cervical mucus method (also known as the ovulation method) – This involves being aware of the changes in your cervical mucus throughout the month. The hormones that control the menstrual cycle also change the kind and amount of mucus before and during ovulation. The greatest amount of mucus appears just before ovulation.
Preconception Care: Why Preconception Health Matters
Preconception health is a woman’s health before she becomes pregnant. It means knowing how health conditions and risk factors could affect a woman or her unborn baby if she becomes pregnant. For example, some foods, habits, and medicines can harm a baby — even before conception. Some health problems, such as diabetes, also can affect pregnancy.
It is important for any person thinking about their health, but with half of all pregnancies unplanned it is especially important to consider behaviors that may affect the health of a baby. Unplanned pregnancies are at greater risk of preterm birth and low birth weight babies. Despite important advances in medicine and prenatal care, about 1 in 8 babies is born too early. Researchers are trying to find out why and how to prevent preterm birth. But experts agree that women need to be healthy before becoming pregnant. By taking action on health issues and risks before pregnancy, a person can prevent problems that might affect mom or baby later.
Improving Preconception Health
Ideally, couple should prepare for pregnancy at least three months before getting pregnant. Some actions, such as quitting smoking, reaching a healthy weight, or adjusting medicines may need to start even earlier.
The five most important things a woman can do for preconception health are:
- Take 400 to 800 micrograms (400 to 800 mcg or 0.4 to 0.8 mg) of folic acid every day if you are planning or capable of pregnancy to lower your risk of some birth defects of the brain and spine, including spina bifida.
- Stop smoking and drinking alcohol.
- If you have a medical condition, be sure it is under control. Some conditions that can affect pregnancy or be affected by it include asthma, diabetes, oral health, obesity, or
- Talk to your doctor about any over-the-counter and prescription medicines you are using. These include dietary or herbal supplements. Be sure your vaccinations are up to date.
- Avoid contact with toxic substances or materials that could cause infection at work and at home. Stay away from chemicals and cat or rodent feces.
Anatomical and Physiological Changes that Occur During Pregnancy
A full-term pregnancy lasts approximately 270 days (approximately 38.5 weeks) from conception to birth. Because it is easier to remember the first day of the last menstrual period (LMP) than to estimate the date of conception, obstetricians set the due date as 284 days (approximately 40.5 weeks) from the LMP. This assumes that conception occurred on day 14 of the woman’s cycle, which is usually a good approximation. The 40 weeks of an average pregnancy are usually discussed in terms of three trimesters , each approximately 13 weeks. We will express embryonic and fetal ages in terms of weeks from conception. The period of time required for full development of a fetus in utero is referred to as gestation (gestare = “to carry” or “to bear”).
A developing human is referred to as a zygote for the first 2 weeks after conception, an embryo during weeks 3–8, and a fetus from the ninth week of gestation until birth.
Implantation
Following fertilization, the zygote and its associated membranes, continue their travel through the fallopian tube toward the uterus. During its journey to the uterus, the zygote undergoes five or six rapid mitotic cell divisions. At the end of the first week, the blastocyst comes in contact with the uterine wall and adheres to it, embedding itself in the uterine lining (referred to as implantation). ( Figure 2 ). Implantation can be accompanied by minor bleeding. The zygote typically implants in the fundus of the uterus or on the posterior wall. However, if the endometrium is not fully developed and ready to receive the zygote, the zygote and associated cells will detach and find a better spot. A significant percentage (50–75 percent) of zygotes fail to implant; when this occurs, the cell bundle is shed with the endometrium during menses. The high rate of implantation failure is one reason why successful pregnancy typically requires several ovulation cycles.
Just a few days after implantation, the mass of cells has secreted enough hCG (human growth hormone) for an at-home urine pregnancy test to give a positive result. However, in one to two percent of cases, the embryo implants either outside the uterus or in a region of uterus that can create complications for the pregnancy.
Dysfunction
During Implantation
In the vast majority of
ectopic pregnancies
, the embryo does not complete its journey to the uterus and implants in the uterine tube, referred to as a tubal pregnancy. However, there are also ovarian ectopic pregnancies (in which the egg never left the ovary) and abdominal ectopic pregnancies (in which an egg was “lost” to the abdominal cavity during the transfer from ovary to uterine tube, or in which an embryo from a tubal pregnancy re-implanted in the abdomen).
Tubal pregnancies can be caused by scar tissue within the tube following a sexually transmitted bacterial infection. The scar tissue impedes the progress of the embryo into the uterus—in some cases “snagging” the embryo and, in other cases, blocking the tube completely. Approximately one half of tubal pregnancies resolve spontaneously. Implantation in a uterine tube causes bleeding, which appears to stimulate smooth muscle contractions and expulsion of the embryo. In the remaining cases, medical or surgical intervention is necessary.
Even if the embryo has successfully found its way to the uterus, it does not always implant in an optimal location (the fundus or the posterior wall of the uterus). Placenta previa can result if an embryo implants close to the internal opening of the cervix. As the fetus grows, the placenta can partially or completely cover the opening of the cervix ( Figure 4 ). Although it occurs in only 0.5 percent of pregnancies, placenta previa is the leading cause of antepartum hemorrhage (profuse vaginal bleeding after week 24 of pregnancy but prior to childbirth).
Development of the Placenta
During the first several weeks of development, the cells of the endometrium (uterus) nourish the embryo. During prenatal weeks 4–12, the developing placenta gradually takes over the role of feeding the embryo, and the endometrial cells are no longer needed. The mature placenta is composed of tissues derived from the embryo, as well as maternal tissues of the endometrium. The placenta connects to the embryo via the umbilical cord , which carries deoxygenated blood and wastes from the fetus through two umbilical arteries; nutrients and oxygen are carried from the mother to the fetus through the single umbilical vein.
Embryotic Development
Within the first 8 weeks of gestation, a developing embryo establishes the rudimentary structures of all of its organs and tissues. The heart begins its development in the embryo as a tube-like structure, connected via capillaries. Cells of the primitive tube-shaped heart are capable of electrical conduction and contraction. The heart begins beating in the beginning of the fourth week, although it does not actually pump embryonic blood until a week later, when the oversized liver has begun producing red blood cells. (This is a temporary responsibility of the embryonic liver that the bone marrow will assume during fetal development.) During weeks 4–5, the eye pits form, limb buds become apparent, and the rudiments of the pulmonary system are formed.
During the sixth week, uncontrolled fetal limb movements begin to occur. The gastrointestinal system develops too rapidly for the embryonic abdomen to accommodate it, and the intestines temporarily loop into the umbilical cord. Paddle-shaped hands and feet develop fingers and toes. By week 7, the facial structure is more complex and includes nostrils, outer ears, and lenses ( Figure 12 ). By the eighth week, the head is nearly as large as the rest of the embryo’s body, and all major brain structures are in place. The external genitalia are apparent, but at this point, male and female embryos are indistinguishable. Bone begins to replace cartilage in the embryonic skeleton through the process of ossification. By the end of the embryonic period, the embryo is approximately 3 cm (1.2 in) from crown to rump and weighs approximately 8 g (0.25 oz).
Sexual Differentiation
Sexual differentiation does not begin until the fetal period, during weeks 9–12. Embryonic males and females, though genetically distinguishable, are morphologically identical. Gonads that can develop into male or female sexual organs, are connected to a central cavity called the cloaca via Müllerian ducts (female)and Wolffian ducts (male).
During male fetal development, the gonads become the testes and associated epididymis. The Müllerian ducts degenerate. The Wolffian ducts become the vas deferens.
During female fetal development, the gonads develop into ovaries. The Wolffian ducts degenerate. The Müllerian ducts become the uterine tubes and uterus.
Fetal Development
During weeks 9–12 of fetal development, the brain continues to expand, the body elongates, and ossification of bones continues. Fetal movements are frequent during this period, but are jerky and not well-controlled. The bone marrow begins to take over the process of red blood cell production. The eyes are well-developed by this stage, but the eyelids are fused shut. The fingers and toes begin to develop nails. By the end of week 12, the fetus measures approximately 9 cm (3.5 in) from crown to rump.
Weeks 13–16 are marked by sensory organ development. The eyes move closer together; blinking motions begin, although the eyes remain sealed shut.  The scalp begins to grow hair.
During approximately weeks 16–20, as the fetus grows and limb movements become more powerful, the mother may begin to feel quickening , or fetal movements. However, space restrictions limit these movements and typically force the growing fetus into the “fetal position,” with the arms crossed and the legs bent at the knees. Oil glands coat the skin with a waxy, protective substance called vernix that protects and moisturizes the skin and may provide lubrication during childbirth. A silky hair called lanugo also covers the skin during weeks 17–20, but it is shed as the fetus continues to grow. Extremely premature infants sometimes exhibit residual lanugo.
Developmental weeks 21–30 are characterized by rapid weight gain, which is important for maintaining a stable body temperature after birth. During this period, the fetus grows eyelashes. The eyelids are no longer fused and can be opened and closed. The lungs begin producing surfactant, a substance that reduces surface tension in the lungs and assists proper lung expansion after birth. In male fetuses, the testes descend into the scrotum near the end of this period. The fetus at 30 weeks measures 28 cm (11 in) from crown to rump and exhibits the approximate body proportions of a full-term newborn, but still is much leaner.
The fetus continues to lay down fat under the skin from week 31 until birth which causes and the skin transitions from red and wrinkled to soft and pink. Lanugo is shed, and the nails grow to the tips of the fingers and toes. Immediately before birth, the average crown-to-rump length is 35.5–40.5 cm (14–16 in), and the fetus weighs approximately 2.5–4 kg (5.5–8.8 lbs). Once born, the newborn is no longer confined to the fetal position, so subsequent measurements are made from head-to-toe instead of from crown-to-rump. At birth, the average length is approximately 51 cm (20 in).
Maternal Changes
The 40 weeks of an average pregnancy are usually discussed in terms of three trimesters, each approximately 13 weeks. During the second and third trimesters, the pre-pregnancy uterus—about the size of a fist—grows dramatically to contain the fetus, causing a number of anatomical changes in the mother.
Effects of Hormones
Virtually all of the effects of pregnancy can be attributed in some way to the influence of hormones—particularly estrogens, progesterone, and hCG. Relaxin, another hormone secreted by the placenta, helps prepare the mother’s body for childbirth. It increases the elasticity of the pubic joint and pelvic ligaments, making room for the growing fetus and allowing expansion of the pelvic outlet for childbirth. Relaxin also helps dilate the cervix during labor.
Weight Gain
The second and third trimesters of pregnancy are associated with dramatic changes in maternal anatomy and physiology. The most obvious anatomical sign of pregnancy is the dramatic enlargement of the abdominal region, coupled with maternal weight gain. This weight results from the growing fetus as well as the enlarged uterus, amniotic fluid, and placenta. Additional breast tissue and dramatically increased blood volume also contribute to weight gain ( Table 2 ). Surprisingly, fat storage accounts for only approximately 2.3 kg (5 lbs) in a normal pregnancy and serves as a reserve for the increased metabolic demand of breastfeeding.
During the first trimester, the mother does not need to consume additional calories to maintain a healthy pregnancy. However, a weight gain of approximately 0.45 kg (1 lb) per month is common. During the second and third trimesters, the mother’s appetite increases, but it is only necessary for her to consume an additional 300 calories per day to support the growing fetus. Most women gain approximately 0.45 kg (1 lb) per week.
| Contributors to Weight Gain During Pregnancy (Table 2) | ||
|---|---|---|
| Component | Weight (kg) | Weight (lb) |
| Fetus | 3.2–3.6 | 7–8 |
| Placenta and fetal membranes | 0.9–1.8 | 2–4 |
| Amniotic fluid | 0.9–1.4 | 2–3 |
| Breast tissue | 0.9–1.4 | 2–3 |
| Blood | 1.4 | 4 |
| Fat | 0.9–4.1 | 3–9 |
| Uterus | 0.9–2.3 | 2–5 |
| Total | 10–16.3 | 22–36 |
Changes in Organ Systems During Pregnancy
As the woman’s body adapts to pregnancy, characteristic physiologic changes occur. These changes can sometimes prompt symptoms often referred to collectively as the common discomforts of pregnancy.
Digestive and Urinary System Changes
Nausea and vomiting, sometimes triggered by an increased sensitivity to odors, are common during the first few weeks to months of pregnancy. This phenomenon is often referred to as “ morning sickness ,” although the nausea may persist all day. The source of pregnancy nausea is thought to be the increased circulation of pregnancy-related hormones. By about week 12 of pregnancy, nausea typically subsides.
A common gastrointestinal complaint during the later stages of pregnancy is gastric reflux, or heartburn, which results from the upward, constrictive pressure of the growing uterus on the stomach.
The downward pressure of the uterus also compresses the urinary bladder, leading to frequent urination. In addition, the maternal urinary system processes both maternal and fetal wastes, further increasing the total volume of urine.
Circulatory System Changes
Blood volume increases substantially during pregnancy, so that by childbirth, it exceeds its preconception volume by 30 percent, or approximately 1–2 liters. The greater blood volume helps to manage the demands of fetal nourishment and fetal waste removal. In conjunction with increased blood volume, the pulse and blood pressure also rise moderately during pregnancy. As the fetus grows, the uterus compresses underlying pelvic blood vessels, hampering venous return from the legs and pelvic region. As a result, many pregnant women develop varicose veins or hemorrhoids.
Respiratory System Changes
During the second half of pregnancy, the volume of gas inhaled or exhaled by the lungs per minute increases by 50 percent to compensate for the oxygen demands of the fetus and the increased maternal metabolic rate. The growing uterus exerts upward pressure on the diaphragm, decreasing the volume of each inspiration and potentially causing shortness of breath. During the last several weeks of pregnancy, the pelvis becomes more elastic, and the fetus descends lower in a process called lightening. This typically alleviates shortness of breath.
Skin Changes
The skin stretches extensively to accommodate the growing uterus, breast tissue, and fat deposits on the thighs and hips. Torn connective tissue cause stretch marks on the abdomen, which appear as red or purple marks during pregnancy that fade to a silvery white color in the months after childbirth.
An increase in melanin-stimulating hormone, in conjunction with estrogens, darkens the areolae and creates a line of pigment from the umbilicus to the pubis called the linea nigra ( Figure 2 ). Melanin production during pregnancy may also darken or discolor skin on the face to create a chloasma, or “mask of pregnancy.”
Check for Understanding
- What are some reasons a person may decide to have or not have children?
- How can a person increase the likelihood of becoming pregnant or contributing to a pregnancy?
- What are some common causes of infertility?
- What are some important factors to consider in preconception health?
- What are the various stages of development by trimester for embryo/fetus?
- What are some maternal changes during pregnancy?


1.2.5: Scales without Units
Lesson
Let's explore a different way to express scales.
Exercise \(\PageIndex{1}\): One to One Hundred
A map of a park says its scale is 1 to 100.
- What do you think that means?
- Give an example of how this scale could tell us about measurements in the park.
Exercise \(\PageIndex{2}\): Apollo Lunar Module
Your teacher will give you a drawing of the Apollo Lunar Module. It is drawn at a scale of 1 to 50.
- The “legs” of the spacecraft are its landing gear. Use the drawing to estimate the actual length of each leg on the sides. Write your answer to the nearest 10 centimeters. Explain or show your reasoning.
- Use the drawing to estimate the actual height of the Apollo Lunar Module to the nearest 10 centimeters. Explain or show your reasoning.
- Neil Armstrong was 71 inches tall when he went to the surface of the Moon in the Apollo Lunar Module. How tall would he be in the drawing if he were drawn with his height to scale? Show your reasoning.
- Sketch a stick figure to represent yourself standing next to the Apollo Lunar Module. Make sure the height of your stick figure is to scale. Show how you determined your height on the drawing.
Are you ready for more?
The table shows the distance between the Sun and 8 planets in our solar system.
- If you wanted to create a scale model of the solar system that could fit somewhere in your school, what scale would you use?
- The diameter of Earth is approximately 8,000 miles. What would the diameter of Earth be in your scale model?
| planet | average distance (millions of miles) |
|---|---|
| Mercury | \(35\) |
| Venus | \(67\) |
| Earth | \(93\) |
| Mars | \(142\) |
| Jupiter | \(484\) |
| Saturn | \(887\) |
| Uranus | \(1,784\) |
| Neptune | \(2,795\) |
Exercise \(\PageIndex{3}\): Same Drawing, Different Scales
A rectangular parking lot is 120 feet long and 75 feet wide.
- Lin made a scale drawing of the parking lot at a scale of 1 inch to 15 feet. The drawing she produced is 8 inches by 5 inches.
- Diego made another scale drawing of the parking lot at a scale of 1 to 180. The drawing he produced is also 8 inches by 5 inches.
- Explain or show how each scale would produce an 8 inch by 5 inch drawing.
- Make another scale drawing of the same parking lot at a scale of 1 inch to 20 feet. Be prepared to explain your reasoning.
- Express the scale of 1 inch to 20 feet as a scale without units. Explain your reasoning.
Summary
In some scale drawings, the scale specifies one unit for the distances on the drawing and a different unit for the actual distances represented. For example, a drawing could have a scale of 1 cm to 10 km.
In other scale drawings, the scale does not specify any units at all. For example, a map may simply say the scale is 1 to 1,000. In this case, the units for the scaled measurements and actual measurements can be any unit, so long as the same unit is being used for both. So if a map of a park has a scale 1 to 1,000, then 1 inch on the map represents 1,000 inches in the park, and 12 centimeters on the map represent 12,000 centimeters in the park. In other words, 1,000 is the scale factor that relates distances on the drawing to actual distances, and \(\frac{1}{1000}\) is the scale factor that relates an actual distance to its corresponding distance on the drawing.
A scale with units can be expressed as a scale without units by converting one measurement in the scale into the same unit as the other (usually the unit used in the drawing). For example, these scales are equivalent:
- 1 inch to 200 feet
- 1 inch to 2,400 inches (because there are 12 inches in 1 foot, and \(200\cdot 12=2,400\))
- 1 to 2,400
This scale tells us that all actual distances are 2,400 times their corresponding distances on the drawing, and distances on the drawing are \(\frac{1}{2400}\) times the actual distances they represent.
Glossary Entries
Definition: Scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.
Definition: Scale Drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.
Practice
Exercise \(\PageIndex{4}\)
A scale drawing of a car is presented in the following three scales. Order the scale drawings from smallest to largest. Explain your reasoning. (There are about 1.1 yards in a meter, and 2.54 cm in an inch.)
- 1 in to 1 ft
- 1 in to 1 m
- 1 in to 1 yd
Exercise \(\PageIndex{5}\)
Which scales are equivalent to 1 inch to 1 foot? Select all that apply.
- 1 to 12
- \(\frac{1}{12}\) to 1
- 100 to 0.12
- 5 to 60
- 36 to 3
- 9 to 108
Exercise \(\PageIndex{6}\)
A model airplane is built at a scale of 1 to 72. If the model plane is 8 inches long, how many feet long is the actual airplane?
Exercise \(\PageIndex{7}\)
Quadrilateral A has side lengths 3, 6, 6, and 9. Quadrilateral B is a scaled copy of A with a shortest side length equal to 2. Jada says, “Since the side lengths go down by 1 in this scaling, the perimeter goes down by 4 in total.” Do you agree with Jada? Explain your reasoning.
(From Unit 1.1.3)
Exercise \(\PageIndex{8}\)
Polygon B is a scaled copy of Polygon A using a scale factor of 5. Polygon A’s area is what fraction of Polygon B’s area?
(From Unit 1.1.6)
Exercise \(\PageIndex{9}\)
Figures R, S, and T are all scaled copies of one another. Figure S is a scaled copy of R using a scale factor of 3. Figure T is a scaled copy of S using a scale factor of 2. Find the scale factors for each of the following:
- From T to S
- From S to R
- From R to T
- From T to R
(From Unit 1.1.5)


2.1.7: Some Final Thoughts
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Jean-Paul Sartre’s Being and Nothingness (1943) is considered the defining text of modern existentialism. Sartre was an atheist, so the brief introduction to existentialism in this chapter went in the direction of atheism. However, Frankl and May were not atheists, and one of May’s most influential mentors, as well as a close personal friend, was Paul Tillich. Tillich remains a well-known and respected existential philosopher in the spiritual tradition. May went so far as to say that Tillich’ book The Courage to Be might be the best and most understandable presentation of existentialism as an approach to life that has been written in English (May, 1983).
It is also interesting to note that both Frankl and May were significantly influenced by Alfred Adler. Frankl worked closely with Adler for a time, and May took a summer course with Adler. Both cite Adler regularly in their writings. Adler’s focus on the childhood struggle against one’s own inferiority, his emphasis on social interest as a responsible means to superiority, and his recognition of the dangers inherent in seeking superiority at the expense of others, all fit well with the existential perspective on making responsible choices in living one’s life. This point emphasizes, once again, the profound influence that Adler has had on psychology, and that he is in all probability the most under-recognized figure in the history of psychology.
In 1897, William James published an essay entitled Is Life Worth Living? (James, 1897/1992).  However, for most people this is not the case, and there is no magic way to give everyone such an optimistic point of view. So, James presents a series of arguments that one might use with suicidal people (that is the term he uses) in order to convince them that life is worth living. He relies heavily on religious faith, though not on any particular religion, but also leads into a discussion of existential thought. Approximately a decade before Frankl and May were even born, James wrote the following words:
…Suppose, however thickly evils crowd upon you, that your unconquerable subjectivity proves to be their match, and that you find a more wonderful joy than any passive pleasure can bring in trusting ever in the larger whole. Have you not now made life worth living on these terms?…This life is worth living, we can say, since it is what we make it, from the moral point of view , and we are determined to make it from that point of view, so far as we have anything to do with it, a success…These, then, are my last words to you: Be not afraid of life. Believe that life is worth living, and your belief will help create the fact. (pp. 501-503; James, 1897/1992)
The challenges that we all face in trying to live authentic lives, the challenges of making responsible and ethical choices that are true to who we ourselves are, can be difficult. In a fascinating book entitled Not a Genuine Black Man , Brian Copeland (2006) talks about his family’s racial struggles during the civil rights movement and the difficulties he faces today as a Black man who has adopted many so-called “White” cultural values. Copeland insists, however, that we cannot so easily claim that any given value or personal interest belongs only to one group of people:
…When all is said and done, I AM indeed a Genuine Black Man - because I am resilient. That’s what being black in America is truly about: resilience…I stayed on my feet through taunts and harassment, through police intimidation and bigoted nuns, through schoolyard bullies and Sylvester, through my mother’s death and bouts of sometimes crippling depression. I am still standing.
I am black because, as my friend Mr. Wilkins once told me, people should be called what they want to be called. I have the right and the ability to determine my identity regardless of what other blacks or whites say. I am not an “oreo,” nor am I “still a nigger.” I am a man. I am a black man.
No one person or group of individuals holds the monopoly on what in this society is the “true” black experience. My world is as “black” as that of Malcolm X, Colin Powell, Snoop Dogg, Jesse Jackson, Usher, Bill Cosby, or Diddy. As their experiences in America are unique, mine is unique - yet it is the same. It is as valid as that of the poor African American living in “the ‘hood,” the rich black rapper balancing a lifestyle of fame and violence, and the black scholar working to better this world through academic dissertation. It is as authentic as the experiences of those who marched with Dr. King for civil rights and those who defy the black community by arguing the conservative point of view.
It is the “true” black experience because it is my experience… (pp. 243-244; Copeland, 2006)
discussion question \(\PageIndex{1}\)
Brian Copeland talked about how difficult it can be to live an authentic life when you don’t meet the expectations of others. Have you ever gone against the advice of family or friends? Did it prove to be the right decision, or did it at least help you to feel better about your own confidence in yourself?


5.7: Discussion Quetions
Explain the difference between an idea and an entrepreneurial opportunity. Why is it important to recognize if your idea is truly an entrepreneurial opportunity or only an idea?
2 .Is now a good time to start a business? Why or why not?
3 .If you were thinking about starting a clothing store for teens and young women in your hometown, what census information do you think would be useful to you?
4 .What new patterns or behaviors have you noticed within your society or the country’s society?
5 .What unused resources are present in your own life that could be translated into a shared economy idea for a new venture?
6 .What spin-off ideas can you think of that support a gig economy? For example, what services or products might gig employees value or gig employers value?
7 .What are some of the reasons for business failures? If you were starting a business, what are some actions you could take to increase your potential for success?
8 .Many entrepreneurs like Chris Johnson, who invented the ramen noodle cooker and Palms Barber had ordinary problems they were trying to solve. What are some examples of ordinary problems you could solve if you could invent a product?
9 .How do social enterprises like Bee Love help the community? What problem was Palms Barber trying to solve? What problems did the company end up solving?


5.6: Warranties
What you’ll learn to do: describe warranties
In this section you will learn about the different types of warranties that are attached to the goods and services that businesses provide.
Learning Objectives
- Explain express warranties
- Explain the warranties provided by the Uniform Commercial Code
Warranties
A warranty is a guarantee or promise that provides assurance by one party to another party that specific facts or conditions are true or will happen. Some warranties run with a product, such that a manufacturer makes the warranty to a consumer with whom the manufacturer has no direct relationship. For example, your iPhone came with a warranty, but the people at Apple don’t know who you are. This fact doesn’t negate the warranty that came with your phone, though. A warranty may be express or implied, depending on whether the warranty is explicitly provided (typically written).
Types of Warranties
Express Warranties
Express warranties are just that—expressed in writing. When you purchased a new television set, for example, somewhere among all of the instructions on how to connect it to your Blue-Ray player and Wi-Fi and XBox was a small card that expressly stated what warranty the manufacturer was providing to you as the purchaser.
Implied Warranties
Implied warranties are unwritten promises that arise from the nature of the transaction and the inherent understanding by the buyer rather than from the express representations of the seller. In the United States, the Uniform Commercial Code provides that the following two warranties are implied unless they are explicitly disclaimed (such as an “as is” statement):
- The warranty of merchantability is implied unless expressly disclaimed or the sale is identified with the phrase “as is.” To be “merchantable,” the goods must reasonably conform to an ordinary buyer’s expectations. For example, a fruit that looks and smells good but has hidden defects may violate the warranty if its quality does not meet the standards for such fruit “as passes ordinarily in the trade.”
- The warranty of fitness for a particular purpose is implied unless disclaimed when a buyer relies upon the seller to select the goods to fit a specific request. For example, this warranty is violated when a buyer asks a mechanic to provide tires for use on snowy roads and receives tires that are unsafe to use in snow.
The following video explains one’s legal rights under implied warranties.


11.10: Avoidable Costs
Learning Outcomes
- Describe avoidable costs
It is Thursday night and you are deciding whether you should go out for dinner or stay in and make a pizza. If you decide on the pizza, a quick run to the grocery store would be needed. But dinner at your local Mexican restaurant is sure sounding nice. What costs could be avoided if you chose to eat the pizza at home? What if you chose to go out for dinner?
By choosing to go out to eat, you can avoid the cost of groceries, so the cost of groceries is an avoidable cost. Conversely, by choosing to eat at home, you can avoid the cost of the restaurant meal, so the cost of the meal is an avoidable cost.  The mortgage on your house is not an avoidable cost, as whether you choose to eat at a restaurant, or eat at home, the mortgage payment is still due.
So we would all like to avoid costs, right? But what is an avoidable cost?
An avoidable cost is one that can be eliminated completely depending on the alternative we pick.
An avoidable cost is a relevant cost, while unavoidable costs are irrelevant costs. Since we have to pay the mortgage no matter what, we can disregard that cost when we make decisions, right?
Let’s look at another example. Let’s revisit our friends at Simply Yoga.
Simply Yoga is looking at streamlining its class schedule. Currently, they have 50 classes running, with 10 students each. They would like to streamline to 20 classes with 25 students each to save money. What would be an avoidable cost in this situation?
| Classes taken | 500 | 500 |
|---|---|---|
| Number of classes offered | 20 | 50 |
| Revenue ($14/class) | $7,000 | $7,000 |
| Expenses | ||
| Wages and salaries ($7/person or $84/class) | $3,500 | $4,200 |
| Yoga supplies | $250 | $300 |
| Utilities (300 + $10/hour) | $500 | $800 |
| Rent | $500 | $500 |
| Insurance | $100 | $100 |
| Other expenses | $250 | $300 |
| Total expense | $5,100 | $6,200 |
| Net operating income | $1,900 | $800 |
The additional wages, supplies, utilities and other expenses could be avoided by reducing the number of classes. Rent and insurance are unavoidable costs, as they will happen regardless of how many classes happen or how many students attend.
So we can disregard rent and insurance as we figure out how to make Simply Yoga more profitable, right? Those costs will make no difference in the decision we ultimately make or how many classes we hold.
Practice Questions