6.3: Using the Normal Distribution
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The shaded area in the following graph indicates the area to the right of \(x_1\). This area is represented by the probability \(P(X > x_1)\). Some normal tables provide the probability between the mean, 0 for the standard normal distribution, and a specific value such as \(x_1\). This is the unshaded part of the graph from the mean to \(x_1\).
Because the normal distribution is symmetrical , if \(x_1\) were the same distance to the left of the mean the area, probability, in the left tail, would be the same as the shaded area in the right tail. Also, bear in mind that because of the symmetry of this distribution, 0.5 of the probability is to the right of the mean and 0.5 is to the left of the mean.
Calculations of Probabilities
To find the probability for probability density functions with a continuous random variable we need to calculate the area under the function across the values of \(X\) we are interested in. For the normal distribution this seems a difficult task given the complexity of the formula. There is, however, a simply way to get what we want. Here again is the formula for the normal distribution:
\[f(x)=\frac{1}{\sigma \cdot \sqrt{2 \cdot \pi}} \cdot \mathrm{e}^{-\frac{1}{2} \cdot\left(\frac{x-\mu}{\sigma}\right)^{2}}\nonumber\]
Looking at the formula for the normal distribution it is not clear just how we are going to solve for the probability doing it the same way we did it with the previous probability functions. There we put the data into the formula and did the math.
To solve this puzzle we start knowing that the area under a probability density function is the probability.
This shows that the area between \(x_1\) and \(x_2\) is the probability as stated in the formula: \(P (x_1 \leq X \leq x_2)\)
The mathematical tool needed to find the area under a curve is integral calculus. The integral of the normal probability density function between the two points x 1 and x 2 is the area under the curve between these two points and is the probability between these two points.
Doing these integrals is no fun and can be very time consuming. But now, remembering that there are an infinite number of normal distributions out there, we can consider the one with a mean of 0 and a standard deviation of 1. This particular normal distribution is given the name Standard Normal Distribution. Putting these values into the formula it reduces to a very simple equation. We can now quite easily calculate all probabilities for any value of x, for this particular normal distribution, that has a mean of 0 and a standard deviation of 1. These have been produced and are available here in the appendix to the text or everywhere on the web. They are presented in various ways. The table used in this text is the most common presentation and is set up with probabilities for half of the distribution beginning with 0, the mean, and moving outward. The shaded area in the graph at the top of the table in Statistical Tables represents the probability from zero to the specific \(z\) value noted on the horizontal axis.
The only problem is that even with this table, it would be a ridiculous coincidence that our data had a mean of 0 and a standard deviation of 1. The solution is to convert the distribution we have with its mean and standard deviation to this new Standard Normal Distribution. The Standard Normal has a random variable called \(Z\).
Using the standard normal table, typically called the normal table, to find the probability of one standard deviation, go to the \(z\) column, reading down to 1.0 and then read at column 0. That number, \(0.3413\) is the probability from 0 to 1 standard deviation. At the top of the table is the shaded area in the distribution which is the probability for one standard deviation. The table has solved our integral calculus problem. But only if our data has a mean of 0 and a standard deviation of 1.
However, the essential point here is, the probability for one standard deviation on one normal distribution is the same on every normal distribution. If the population data set has a mean of 10 and a standard deviation of 5 then the probability from 10 to 15, one standard deviation, is the same as from 0 to 1, one standard deviation on the standard normal distribution. To compute probabilities, areas, for any normal distribution, we need only to convert the particular normal distribution to the standard normal distribution and look up the answer in the tables. As review, here again is the standardizing formula :
\[z=\frac{x-\mu}{\sigma}\nonumber\]
where \(z\) is the value on the standard normal distribution, \(x\) is the value from a normal distribution one wishes to convert to the standard normal, \(\mu\) and \(\sigma\) are, respectively, the mean and standard deviation of that population. Note that the equation uses \(\mu\) and \(\sigma\) which denotes population parameters. This is still dealing with probability so we always are dealing with the population, with known parameter values and a known distribution. It is also important to note that because the normal distribution is symmetrical it does not matter if the z-score is positive or negative when calculating a probability. One standard deviation to the left (negative z-score) covers the same area as one standard deviation to the right (positive z-score). This fact is why the Standard Normal tables do not provide areas for the left side of the distribution. Because of this symmetry, the z-score formula is sometimes written as:
\[z=\frac{|x-\mu|}{\sigma}\nonumber\]
Where the vertical lines in the equation means the absolute value of the number.
What the standardizing formula is really doing is computing the number of standard deviations \(x\) is from the mean of its own distribution. The standardizing formula and the concept of counting standard deviations from the mean is the secret of all that we will do in this statistics class. The reason this is true is that all of statistics boils down to variation, and the counting of standard deviations is a measure of variation.
This formula, in many disguises, will reappear over and over throughout this course.
Example \(\PageIndex{1}\)
The final exam scores in a statistics class were normally distributed with a mean of 63 and a standard deviation of 5.
a. Find the probability that a randomly selected student scored more than 65 on the exam.
b. Find the probability that a randomly selected student scored less than 85.
Answer a
Let \(X\) = a score on the final exam. \(X \sim N(63, 5)\), where \(\mu = 63\) and \(\sigma = 5\).
Draw a graph.
Then, find \(P(X > 65)\).
\(P(X > 65) = 0.3446\)
\[z_{1}=\frac{x_{1}-\mu}{\sigma}=\frac{65-63}{5}=0.4\nonumber\]
\(P\left(X \geq x_{1}\right)=P\left(Z \geq z_{1}\right)=0.3446\)
The probability that any student selected at random scores more than 65 is 0.3446. Here is how we found this answer.
Answer b
The normal table provides probabilities from zero to the value \(z_1\). For this problem the question can be written as: \(P(X \geq 65) = P(Z \geq z_1)\), which is the area in the tail. To find this area the formula would be \(0.5 – P(X \leq 65)\). One half of the probability is above the mean value because this is a symmetrical distribution. The graph shows how to find the area in the tail by subtracting that portion from the mean, zero, to the \(z_1\) value. The final answer is: \(P(X \geq 63) = P(Z \geq 0.4) = 0.3446\)
\(z_1=\frac{65-63}{5}=0.4\)
Area to the left of \(z_1\) to the mean of zero is \(0.1554\)
\(P(X > 65) = P(Z > 0.4) = 0.5 – 0.1554 = 0.3446\)
\(z=\frac{x-\mu}{\sigma}=\frac{85-63}{5}=4.4\) which is larger than the maximum value on the Standard Normal Table. Therefore, the probability that one student scores less than 85 is approximately one or 100%.
A score of 85 is 4.4 standard deviations from the mean of 63 which is beyond the range of the standard normal table. Therefore, the probability that one student scores less than 85 is approximately one (or 100%).
Exercise \(\PageIndex{1}\)
The golf scores for a school team were normally distributed with a mean of 68 and a standard deviation of three. Find the probability that a randomly selected golfer scored less than 65.
Example \(\PageIndex{2A}\)
A personal computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking, and a myriad of other things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distributed and the standard deviation for the times is half an hour.
a. Find the probability that a household personal computer is used for entertainment between 1.8 and 2.75 hours per day.
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a. Let \(X\) = the amount of time (in hours) a household personal computer is used for entertainment. \(X \sim N(2, 0.5)\) where \(\mu= 2\) and \(\sigma = 0.5\).
Find \(P(1.8 < X < 2.75)\).
The probability for which you are looking is the area between \(X = 1.8\) and \(X = 2.75\). \(P(1.8 < X < 2.75) = 0.5886\)
\(P(1.8 \leq X \leq 2.75) = P(Z_1 \leq Z \leq Z_2)\)
The probability that a household personal computer is used between 1.8 and 2.75 hours per day for entertainment is 0.5886.
Example \(\PageIndex{2B}\)
b. Find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment.
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b. To find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment, find the 25 th percentile, \(k\), where \(P(X < k) = 0.25\).
\(f(z)=0.5-0.25=0.25, \text { therefore } z \approx-0.675(\text { or just } 0.67 \text { using the table) } z=\frac{x-\mu}{\sigma}=\frac{x-2}{0.5}=-0.675 , \text {therefore } x=-0.675 * 0.5+2=1.66\)
The maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment is 1.66 hours.
Exercise \(\PageIndex{2}\)
The golf scores for a school team were normally distributed with a mean of 68 and a standard deviation of three. Find the probability that a golfer scored between 66 and 70.
Example \(\PageIndex{3}\)
In the United States the ages 13 to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of 36.9 years and 13.9 years, respectively.
a. Determine the probability that a random smartphone user in the age range 13 to 55+ is between 23 and 64.7 years old.
Answer
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a. 0.8186
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b. 0.8413
Example \(\PageIndex{4}\)
A citrus farmer who grows mandarin oranges finds that the diameters of mandarin oranges harvested on his farm follow a normal distribution with a mean diameter of 5.85 cm and a standard deviation of 0.24 cm.
a. Find the probability that a randomly selected mandarin orange from this farm has a diameter larger than 6.0 cm. Sketch the graph.
- Answer
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\[z_{1}=\frac{6-5.85}{.24}=.625\nonumber\]
\(P(X \geq 6) = P(Z \geq 0.625) = 0.2670\)
b. The middle 20% of mandarin oranges from this farm have diameters between ______ and ______.
\(f(z)=\frac{0.20}{2}=0.10, \text { therefore } z \approx \pm 0.25\)
\(z=\frac{x-\mu}{\sigma}=\frac{x-5.85}{0.24}=\pm 0.25 \rightarrow \pm 0.25 \cdot 0.24+5.85=(5.79,5.91)\)


4.4: Summary
4.1 Foundations for a Complete Electronic Health Record: Accurate Health History
In every healthcare setting, a health history is taken from each patient and updated regularly. The nurse establishes a trusting relationship with the patient to elicit exact and detailed information to build the foundation of an accurate health record. Often it is the nurse’s responsibility to determine whether the information is valid and reliable and then use professional clinical judgment on the observations and data collected. Application of legal and ethical considerations will guide clinical decision-making. The information gathered and documented is also used by the provider to diagnose and develop a plan of care based on current industry standards and best practice guidelines.
4.2 Data Collection and Documentation
The act of recording patient information discovered in a complete health history is called documenting. Once the nurse understands that the purpose of accurate documenting is to facilitate patient care among the healthcare team, then the nurse can identify what information to document. Patient information can be categorized into subjective and objective data. It is important to remember each healthcare facility will have policies and procedures related to documentation expectations. Additionally, nurses need to master communicating assessment findings in a clear and precise manner to improve patient care outcomes.
4.3 Informatics
The incorporation of informatics with an EHR has drastically improved our healthcare system. As you incorporate the capabilities of an EHR into your daily clinical practice, remember regulations such as meaningful use and privacy laws. Nurses are at the forefront of patient care and often the eyes and ears of when something is not being done correctly. Using knowledge about how and when to release or discuss patient health information will ensure positive patient outcomes.


7: Consumer Behavior Last updated Save as PDF Page ID 47858 Lumen Learning Lumen Learning 7.1: Reading- Influences on Consumer Decisions 7.2: Reading- Situational Factors 7.3: Reading- Personal Factors 7.4: Reading- Psychological Factors 7.5: Video- Consumer Attitudes and Heinz Baked Beans 7.6: Reading- Social Factors 7.7: Self Check- Factors Influencing Consumer Decisions 7.8: Outcome- B2B Purchasing Decisions 7.9: Reading- Organizational Buyer Behavior 7.10: Video- Complexities of a B2B Solution Sale 7.11: Why It Matters- Consumer Behavior 7.12: Reading- The Organizational Buying Process 7.13: Self Check- B2B Purchasing Decisions 7.14: Putting It Together- Consumer Behavior 7.15: Discussion- Customer Profile 7.16: Outcome- Buying-Process Stages 7.17: Reading- The “Black Box” of Consumer Behavior 7.18: Reading- Buying-Process Stages 7.19: Self Check- Buying Process Stages 7.20: Outcome- Low-Involvement vs. High-Involvement Decisions 7.21: Reading- Low-Involvement vs. High-Involvement Decisions 7.22: Self Check- Low-Involvement vs. High-Involvement Decisions 7.23: Outcome- Factors Influencing Consumer Decisions


6.3: Anxiety Disorders
Occasional anxiety is a normal part of life. You might feel anxious when faced with a problem at work, before taking a test, or making an important decision. But anxiety disorders involve more than temporary worry or fear. For a person with an anxiety disorder, the anxiety does not go away and can get worse over time. The feelings can interfere with daily activities such as job performance, school work, and relationships. There are several different types of anxiety disorders. Examples include generalized anxiety disorder, panic disorder, and social anxiety disorder.
Signs and Symptoms
Generalized Anxiety Disorder
People with generalized anxiety disorder (GAD) display excessive anxiety or worry about everyday problems that lingers for months—even when there is little or no reason to worry about them. People with GAD find it difficult to control their anxiety and stay focused on daily tasks.
Generalized anxiety disorder symptoms include:
- Restlessness or feeling wound-up or on edge
- Being easily fatigued
- Difficulty concentrating or having their minds go blank
- Irritability
- Muscle tension
- Difficulty controlling the worry
- Sleep problems (difficulty falling or staying asleep or restless, unsatisfying sleep)
Excessive worry or anxiety about everyday issues that lasts for 6 months or more may indicate generalized anxiety disorder.
Panic Disorder
People with panic disorder have recurrent unexpected panic attacks, which are sudden periods of intense fear that may include palpitations, pounding heart, or accelerated heart rate; sweating; trembling or shaking; sensations of shortness of breath, smothering, or choking; and feeling of impending doom.
Panic disorder symptoms include:
- Sudden and repeated attacks of intense fear
- Feelings of being out of control during a panic attack
- Intense worries about when the next attack will happen
- Fear or avoidance of places where panic attacks have occurred in the past
Social Anxiety Disorder
People with social anxiety disorder (sometimes called “social phobia”) have a marked fear of social or performance situations in which they expect to feel embarrassed, judged, rejected, or fearful of offending others.
Social anxiety disorder symptoms include:
- Feeling highly anxious about being with other people and having a hard time talking to them
- Feeling very self-conscious in front of other people and worried about feeling humiliated, embarrassed, or rejected, or fearful of offending others
- Being very afraid that other people will judge them
- Worrying for days or weeks before an event where other people will be
- Staying away from places where there are other people
- Having a hard time making friends and keeping friends
- Blushing, sweating, or trembling around other people
- Feeling nauseous or sick to your stomach when other people are around
Evaluation for an anxiety disorder often begins with a visit to a primary care provider. Some physical health conditions, such as an overactive thyroid or low blood sugar, as well as taking certain medications, can imitate or worsen an anxiety disorder. A thorough mental health evaluation is also helpful, because anxiety disorders often co-exist with other related conditions, such as depression or obsessive-compulsive disorder.
Risk Factors
Researchers are finding that genetic and environmental factors, frequently in interaction with one another, are risk factors for anxiety disorders. Specific factors include:
- Shyness, or behavioral inhibition, in childhood
- Being female
- Having few economic resources
- Being divorced or widowed
- Exposure to stressful life events in childhood and adulthood
- Anxiety disorders in close biological relatives
- Parental history of mental disorders
- Elevated afternoon cortisol levels in the saliva (specifically for social anxiety disorder)
Treatments and Therapies
Anxiety disorders are generally treated with psychotherapy, medication, or both.
Psychotherapy
Psychotherapy or “talk therapy” can help people with anxiety disorders. To be effective, psychotherapy must be directed at the person’s specific anxieties and tailored to his or her needs. A typical “side effect” of psychotherapy is temporary discomfort involved with thinking about confronting feared situations.
Cognitive Behavioral Therapy (CBT)
CBT is a type of psychotherapy that can help people with anxiety disorders. It teaches a person different ways of thinking, behaving, and reacting to anxiety-producing and fearful situations. CBT can also help people learn and practice social skills, which is vital for treating social anxiety disorder.
Two specific stand-alone components of CBT used to treat social anxiety disorder are cognitive therapy and exposure therapy. Cognitive therapy focuses on identifying, challenging, and then neutralizing unhelpful thoughts underlying anxiety disorders.
Exposure therapy focuses on confronting the fears underlying an anxiety disorder in order to help people engage in activities they have been avoiding. Exposure therapy is used along with relaxation exercises and/or imagery. One study, called a meta-analysis because it pulls together all of the previous studies and calculates the statistical magnitude of the combined effects, found that cognitive therapy was superior to exposure therapy for treating social anxiety disorder.
CBT may be conducted individually or with a group of people who have similar problems. Group therapy is particularly effective for social anxiety disorder. Often “homework” is assigned for participants to complete between sessions.
Stress-Management Techniques
Stress management techniques and meditation can help people with anxiety disorders calm themselves and may enhance the effects of therapy. While there is evidence that aerobic exercise has a calming effect, the quality of the studies is not strong enough to support its use as treatment. Since caffeine, certain illicit drugs, and even some over-the-counter cold medications can aggravate the symptoms of anxiety disorders, avoiding them should be considered. Check with your physician or pharmacist before taking any additional medications.
The family can be important in the recovery of a person with an anxiety disorder. Ideally, the family should be supportive but not help perpetuate their loved one’s symptoms. Talking with a trusted friend or member of the clergy can also provide support, but it is not necessarily a sufficient alternative to care from an expert clinician.
Medication
Medication does not cure anxiety disorders but often relieves symptoms. Medication can only be prescribed by a medical doctor (such as a psychiatrist or a primary care provider), but a few states allow psychologists to prescribe psychiatric medications.
Medications are sometimes used as the initial treatment of an anxiety disorder, or are used only if there is insufficient response to a course of psychotherapy. In research studies, it is common for patients treated with a combination of psychotherapy and medication to have better outcomes than those treated with only one or the other.
The most common classes of medications used to combat anxiety disorders are antidepressants, anti-anxiety drugs, and beta-blockers (visit Mental Health Medications ). Be aware that some medications are effective only if they are taken regularly and that symptoms may recur if the medication is stopped.
Contributors and Attributions
Public Domain Content
- Anxiety Disorders. Authored by: National Institute of Mental Health. Provided by: National Institutes of Health. Located at: https://www.nimh.nih.gov/health/topics/anxiety-disorders/index.shtml . License: Public Domain: No Known Copyright


6.1: Chronology
410 CE Roman army abandons Britain
476 CE The general Odavacar deposes last Western Roman Emperor
496 CE The Frankish king Clovis converts to Christianity
500s CE Anglo-Saxons gradually take over Britain
533 CE Byzantine Empire conquers the Vandal kingdom in North Africa
535 – 554 CE Byzantine Empire conquers the Ostrogothic kingdom in Italy
560s CE Lombard invasions of Italy begin
580s CE The Franks cease keeping tax registers
597 CE Christian missionaries dispatched from Rome arrive in Britain
610 – 641 CE Heraclius is Byzantine emperor
636 CE Arab Muslims defeat the Byzantine army at the Battle of Yarmouk
670s CE Byzantine Empire begins to lose control of the Balkans to Avars, Bulgars, and Slavs
674 – 678 CE Arabs lay siege to Constantinople but are unsuccessful
711 CE Muslims from North Africa conquer Spain, end of the Visigothic kingdom
717 – 718 CE Arabs lay siege to Constantinople but are unsuccessful
717 CE Leo III becomes Byzantine emperor. Under his rule, the Iconoclast Controversy begins.
732 CE King Charles Martel of the Franks defeats a Muslim invasion of the kingdom at the Battle of Tours
751 CE The Byzantine city of Ravenna falls to the Lombards; Pepin the Short of the Franks deposes the last Merovingian king and becomes king of the Franks; King Pepin will later conquer Central Italy and donate it to the pope
750s CE Duke of Naples ceases to acknowledge the authority of the Byzantine emperor
770s CE Effective control of the city of Rome passes from Byzantium to the papacy
c. 780 – 840 CE The Carolingian Renaissance
782 CE Charlemagne crushes a Saxon rebellion
787 CE Second Council of Nicaea authorizes the use of icons in worship
793 CE Viking raids begin
800 CE Charlemagne crowned Roman emperor by Pope Leo III
830 CE Abbasid caliph Al-Mamun founds the House of Wisdom in Baghdad
843 CE In the Treaty of Verdun, Charlemagne’s three sons, Lothar, Louis, and Charles the Bald, divide his empire among themselves
843 CE Final resolution of the Iconoclast Controversy under Empress Theodora
846 CE Muslim raiders from Aghlabid North Africa sack the city of Rome
c. 843 – 900 CE Macedonian Renaissance
Mid-800s: CE Cyril and Methodius preach Christianity to the Slavic peoples
864 CE Conversion of the Bulgars to Christianity
867 CE Basil I murders the reigning Byzantine emperor and seizes control of the Empire
871 – 899 CE Alfred the Great is king of England. He defeats Norse raiders and creates a consolidated kingdom.
899 CE Defeated by the Pechenegs, the Magyars begin moving into Central Europe
955 CE Otto the Great, king of East Francia, defeats the Magyars in battle
976 – 1025 CE Basil II is Byzantine Emperor
988 CE Vladimir, Grand Prince of Kiev, converts to Christianity


4.1: Two-by-Two and Three-by-Three Determinants
View Two-by-Two and Three-by-Three Determinants on YouTube
Our first introduction to determinants was the definition for the general two-by-two matrix
\[\text{A}=\left(\begin{array}{cc}a&b\\c&d\end{array}\right):\quad\det\text{A}=ad-bc.\nonumber \]
Other widely used notations for the determinant include
\[\det\text{A}=\det\left(\begin{array}{cc}a&b\\c&d\end{array}\right)=|\text{A}|=\left|\begin{array}{cc}a&b\\c&d\end{array}\right|.\nonumber \]
By explicit construction, we have seen that a two-by-two matrix \(\text{A}\) is invertible if and only if \(\det\text{A}\neq= 0\). If a square matrix \(\text{A}\) is invertible, then the equation \(\text{Ax} = \text{b}\) has the unique solution \(\text{x} = \text{A}^{−1}\text{b}\). But if \(\text{A}\) is not invertible, then \(\text{Ax} = \text{b}\) may have no solution or an infinite number of solutions. When \(\det \text{A} = 0\), we say that \(\text{A}\) is a singular matrix.
Here, we would like to extend the definition of the determinant to an \(n\)-by-\(n\) matrix. Before we do so, let us display the determinant for a three-by-three matrix. We consider the system of equations \(\text{Ax} = 0\) and find the condition for which \(\text{x} = 0\) is the only solution. This condition must be equivalent to \(\det \text{A}\neq 0\). With
\[\left(\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right)\left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right)=0,\nonumber \]
one can do the messy algebra of elimination to solve for \(\text{x}_1\), \(\text{x}_2\), and \(\text{x}_3\). One finds that \(\text{x}_1 = \text{x}_2 = \text{x}_3 = 0\) is the only solution when \(\det \text{A}\neq 0\), where the definition is given by
\[\det\text{A}=aei+bfg+cdh-ceg-bdi-afh.\label{eq:1} \]
A way to remember this result for the three-by-three matrix is by the following picture:
The matrix \(\text{A}\) is periodically extended two columns to the right, drawn explicitly here but usually only imagined. Then the six terms comprising the determinant are made evident, with the lines slanting down towards the right getting the plus signs and the lines slanting down towards the left getting the minus signs. Unfortunately, this mnemonic is only valid for three-by-three matrices.


7: Multidimensional Data - Analysis of Structure
“Data Mining”, “Big Data”, “Machine Learning”, “Pattern Recognition” phrases often mean all statistical methods, analytical and visual which help to understand the structure of data. Data might be of any kind, but it is usually multidimensional , which is best represented with the table of multiple columns a.k.a. variables (which might be of different types: measurement, ranked or categorical) and rows a.k.a objects. So more traditional name for these methods is “multivariate data analysis” or “multivariate statistics”.
Data mining is often based on the idea of classification , arrange objects into non-intersecting, frequently hierarchical groups. We use classification all the time (but sometimes do not realize it). We open the door and enter the room, the first thing is to recognize (classify) what is inside. Our brain has the outstanding power of classification, but computers and software are speedily advancing and becoming more brain-like. This is why data mining is related with artificial intelligence. There are even methods calling “neural networks”!
In this chapter, along with the other data, we will frequently use the embedded iris data taken from works of Ronald Fisher\(^{[1]}\). There are four characters measured on three species of irises (Figure \(\PageIndex{1}\)), and fifth column is the species name.
References
1. Fisher R.A. 1936. The use of multiple measurements in taxonomic problems. Annals of Eugenics. 7(2): 179–188.


1.3: Love, Some Popular Views
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Learning Objectives
- Identify what the popular views of love typically are and the problems associated with those views.
Watch this
video
or scan the QR code to understand more about why we love.
It has been said that love is that wonderful feeling you will know the instant you have it; that when you fall in love, you will know it. It has been said that love is the solution to the problem of man’s alienation from himself, from others, and from the world. That love is aim-inhibited sex. That love is the result of an act of will. That love is the spirit that draws man’s soul to the heights of truth, beauty, and goodness, and makes him be like the gods. That God is love. That love is holding hands. That love is the power that illumines men’s actions, but so often also plunges them into darkest despair.
All the above sound not unlike the self-styled descriptions of concoctions sold by nineteenth-century medicine men as they hawked their wares to the multitudes. This wonderful elixir picks you up from dropsy, perks you up on those dull, lifeless days, gets you rolling again when you can otherwise hardly stand; it lets insomniacs find blissful sleep, but miraculously also shakes off drowsiness for those who need to be alert. It is unlike anything else ever invented. Its taste is distinctive. The moment you try it, you know it works. It will let you charm your enemies and love your friends more steadfastly, and it will even help you enter the gates of heaven if you should die, which is unlikely if you drink it as a daily tonic.
Love has been described as involving care, responsibility, respect, and knowledge by Erich Fromm in The Art of Loving . Rollo May says there have been four types of love in the Western tradition (page 37 of Love and Will )— sex or lust; eros , the creative drive; philia or friendship; and agape , the love devoted toward the welfare of others. And far too often, I am afraid, love has been thought of as one (or, as above, four) static kind(s) of thing(s), the same or similar for all people and for all time. Love seems to be regarded as something that you either have or feel or that you do not. You are either in love or you are not. There may be slight variations on this theme, but there is only one theme, and it is supposedly basic.
But I think all this is wrong. Love is not described by telling what it can do, any more than water or Dr. John’s soothing snake oil is described by telling what it does—cures thirst or melancholy. Telling what something does is not to explain what it i s . To say, for example, that the heart pumps blood and that blood transports oxygen and carbon dioxide is not to tell someone, who does not already know, what the heart or blood actually are. And I will contend later that there is no one set of ingredients of love — that it is different at different times, for different people, and often for the same person at different times, in different circumstances. What it feels like to love is not a similar kind of question to what it feels like to have an itch; it is more like the question of what it feels like to breathe. But the question of what it feels like to breathe does not just have one answer. It feels different to breathe when you have a knot in your chest than when you don’t. It feels different when you have run further than you are in the condition to, or when you have a mouthful of crackers, or when you are laughing so hard your ribs ache, or when you feel terror or pride or after you have been underwater a bit too long. And, most of the time, and when your mind is on other things, it does not feel like anything at all, because you do not literally feel it.
Likewise, how it feels to love when you are doing dishes or scrubbing floors or running a mile or having intercourse or reading a book or taking an exam or kissing for the first time someone you have silently, secretly, and shyly worshiped a long time, or saying “I do” at the alter, or attending a funeral or feeling guilt or terror or contentedly watching your loved one sleep by your side, or feeling pride, performing surgery, or drowning are very different kinds of things. And this is true whether you are talking about love for a spouse of fifty years, love for a first girlfriend (boyfriend), a spouse on a honeymoon, a son or brother, or clergyman who has been kind in a time of need, your favorite aunt, favorite elementary school teacher, the newest Hollywood (or office) sex symbol, and maybe even your love for pizza. And this is only about how it feels to love, yet I will argue later that love is more than just a (kind of) feeling anyway.
But all this is not to say that love is so unique for different people, or at different times, that nothing of general importance and description can be said about it. Though love is a variety of things and involves a variety of things, the varieties themselves can be meaningfully explained and described, and they can be explained and described simply in terms of everyday experience rather than described away in scientific (or pseudo-scientific) jargon or theory. And though they can be described in specific, accurate, logical, non-mystical and non-mythical prose which will make a reflection, decision, and discussion of love easier and clearer, this will not thereby make love seem prosaic. And it may even heighten both the value of love itself and the meaning, poignancy, and perception poetry about it provides.
Let me first explain, however, that I am not so interested in talking about how the word love is used as in talking about how it should be used, since it is used so differently by so many different people that it is virtually impossible to convey a particular idea to someone else just by using the word. In this regard, it might almost even be better to abandon the word altogether, except that it has such a rich heritage of usage, can serve a useful purpose, is as convenient as any that might be coined in its place, and in many previous contexts can be understood in terms of the analysis I will give. What I am going to do is to try to capture and combine the essences of what people mean, try to mean, or seem to mean by the word “love” in a way that will be representative and significant, yet be more specific, accurate, and helpful. In many cases, however, I think people will, in fact, find it clearer, more accurate, and more useful to think and talk in the terms of the specific components (such as amounts and kinds of feelings, satisfactions, and goodness) that I use to analyze love and other relationships than to use the more encompassing, but more general and vague, word “love” by itself.
In a survey of college students reported in J. Richard Udry’s The Social Context of Marriage (Udry, 1966, p. 177), 40% believed love was a feeling or kind of attraction and said things like: “Love represents a magnetic attraction between two persons.” “Love is a feeling of high emotional affiliation…which sends a person’s ego to dizzying heights.” “Love is the emotional feeling two people receive when they both have sexual and Platonic love in the proper proportions.” Still, another 20% thought love had more to do with companionship and compatibility, and they said things like: “Love is the physical and mental compatibility of two people.” “Love is the end result of a mature union of two compatible personalities.” “Love is helping the other person whenever he needs it…being his companion. It’s having common goals, dreams, and ambitions.” “Love is doing things together and liking it.” Still another 20% thought of love in terms of “giving”: “Love is giving—time, understanding, yourself.” “Love is to give of oneself to another.” “Love is giving trust.” “Love is a give and take relationship— and mostly give.” And 17% responded they thought of love in terms of security: “Love is having security in being wanted and knowing you have someone to rely on.” “When a person is in love, the world is right and a person has security.” Finally, 3% looked at love in terms of efficiency, practicality, or roles: “Love for the girl is cooking for him, washing his clothes and keeping the home in order. For the man, it is providing security, safety, and helping his wife.” “Love to me is faithfulness to my mate and caring for our children.”
I list the results of this survey to show people do use the word differently, though it is easy to prove this yourself simply by asking a few friends how they use the word “love” or what it means to them; you will quickly see a wide difference. Or tell your parents you love someone you know they disapprove of and see how quickly they try to show you what you have is not love for that person but hero-worship, infatuation, sexual longings (being in lust or in heat, not in love), rebellious disrespect for your parents, or whatever.
I would like to take the opportunity to show, rather briefly for now, what is wrong with thinking of love as any of the categories in the above survey, and thereby to show some of the kinds of things a correct or useful theory of love must take into account and thus explain or consider.
If love were the kind of feeling mentioned, then how long should it last, how intense should it be, and how frequently should it occur? If the feeling someday goes away, never or rarely to return, was it really love? If infatuation is also that dizzying kind of feeling, how can one tell the difference between love and infatuation? If love is a feeling and if we have little control over what feelings we have, then what sense could there be in promising eternal love, long-lasting love, or even love through tomorrow? That would seem more a prediction than a promise. If you have to wait to see how long and/or under what conditions the feeling of love lasts in order to tell whether it is truly love or not, then don’t you have to wait for that time or those conditions before you can honestly tell someone you love them?
If love is the kind of compatibility mentioned, then it would seem that all friends were lovers, that people at work who got along well together and helped each other pursue common ends, etc., were lovers, and that, in general, there is little difference between good friendship and love. Further, it would mean two people could not be in love if they had different goals or joys, even though they might get along perfectly well together and have a great fondness for each other. I am not sure what is even meant by physical and mental compatibility. Cannot big people love smaller people; bright people, less bright people; intellectuals love athletes; and vice versa. And if by physical compatibility is meant sexual compatibility, then, aren’t there millions of potential loves for any normal person, since I doubt there is that much sexual incompatibility in the world. “Love is giving” is a popular theme—many sermons in church find this a fond message, usually coupled with some prescription like each person should give 110% or that in most marriages, it is 60/40 (then everyone believes they are the ones giving 60 and getting 40). If giving, though, means being considerate, nice, ethical, doing the right thing, etc., then since we should all be that way anyhow; so does that mean we should all be lovers to each other! It is not clear that giving applies only or specifically to love situations. And if it means always or mostly being altruistic or self- sacrificing, then, as I will argue in the ethics chapter, it is a bad principle. Ethics does not demand self- denial in all, or even most, cases.
I am also not certain what it means to “give” trust, though I assume it means “trusting.” However, we certainly do not always trust children we love to stay out of danger or trouble. And certainly we may not really trust our teenager with driving for the first time—though we may believe showing confidence in him or her is better to do than not to, even if that is to risk a minor accident. And at the adult level, one may not always trust one’s loved one (or even one’s self) to say or do the right thing in various situations, yet one goes forward anyway and simply does not fret over any bad result. For example, minor though it is, one may not trust one’s spouse to make a crucial put-away tennis shot, but it is often better to let the spouse try than to hog the court, because giving him or her the chance or allowing him or her to try is more important than winning some particular point or match. That is not giving trust, however. And it may have more to do with ethical behavior in general anyway. Further, there are certainly people that we trust, that we do not love — some baby sitters, housekeepers, doctors, businessmen, teachers, etc.
Security: being in love certainly does not make the world all right or make all your troubles disappear; just think of loving when one is incurably ill, or think of loving under war conditions or bad economic conditions where it is difficult even to get food or safe shelter. There are many situations in which loved ones are powerless to help one another, and the inability to help a loved one in trouble often causes more distress or agony than does the inability to help a stranger. Love certainly does not always bring peace of mind or security.
Faithfulness seems to be a question related to ethics more than only to love, particularly the ethics of sex (generally), about which I will have more to say later; and childcare, cooking, etc., seems to have more to do simply with having domestic help of some sort (maid, nanny, butler, valet, whatever) than anything specifically related to love.
Some of the things a successful definition or analysis of love must do then is to allow us some way to distinguish between love and friendship, between love and infatuation, between love and unwarranted sacrifice, between love and every day ethical concern for others, between love and “just” sexual or physical attraction, between love and comfort, and between love and an efficient household. I think such an analysis is possible.
Key Takeaways
Realizing that the concept of love is more complex than most people think and that it needs to be analyzed and understood at a deeper level.
Key Terms
Faithfulness seems to be a question related to ethics more than only to love.
Review Questions
- Question: What do you think love is, particularly what is often referred to as romantic love — love of the sort people seek for marriage or intimate relationships — as opposed to brotherly love, family love, or love of humanity? Explain and justify your answer.
- Question : What are the specific components you should consider when defining love?
- Question: What are examples of people that we trust but may not love?


7.3.1: General Characteristics of Viruses
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- Ying Liu
- City College of San Francisco
Learning Objectives
- State 2 living and 2 nonliving characteristics of viruses.
- List 3 criteria used to define a virus.
- Discuss why bacteria can be cultivated on synthetic media such as nutrient broth whereas viruses cannot.
- Define bacteriophage.
Viruses are infectious agents with both living and nonliving characteristics. They can infect animals, plants, and even other microorganisms. Viruses that infect only bacteria are called bacteriophages and those that infect only fungi are termed mycophages . There are even some viruses called virophages that infect other viruses.
| Living Characteristics of Viruses | Nonliving Characteristics of Viruses |
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Recently, viruses have been declared as living entities based on the large number of protein folds encoded by viral genomes that are shared with the genomes of cells. This indicates that viruses likely arose from multiple ancient cells.
The vast majority of viruses contain only one type of nucleic acid: DNA or RNA, but not both. Virus are totally dependent on a host cell for replication (i.e., they are strict intracellular parasites.) Furthermore, v iral components must assemble into complete viruses (virions) to go from one host cell to another. Since viruses lack metabolic machinery of their own and are totally dependent on their host cell for replication, they cannot be grown in synthetic culture media. Animal viruses are normally grown in animals, embryonated eggs, or in cell cultures where in animal host cells are grown in a synthetic medium and the viruses are then grown in these cells.
Summary
- Viruses are infectious agents with both living and nonliving characteristics.
- Living characteristics of viruses include the ability to reproduce – but only in living host cells – and the ability to mutate.
- Nonliving characteristics include the fact that they are not cells, have no cytoplasm or cellular organelles, and carry out no metabolism on their own and therefore must replicate using the host cell's metabolic machinery.
- Viruses can infect animals, plants, and even other microorganisms.
- Since viruses lack metabolic machinery of their own and are totally dependent on their host cell for replication, they cannot be grown in synthetic culture media.


1.2.2: Applied Microbiology
The information gained by microbiologists can be applied to many medicinal and commercial endeavors.
Learning Objectives
- Explain applied microbiology
Related Chapter Learning Outcomes:
- Describe the field of microbiology
- Appreciate the variety of impacts of microbes and microbiology on humans and the environment
Key Points
- Using knowledge gained by microbiologists studying microbes, several fields of applied microbiology have formed.
- While food and medicinal applications are a big portion of applied microbiology, the study of microbes has lead to entire commercial industries which affect almost all aspects of human life.
- There are a myriad of practical applications that microbiology contributes to, including several parts of food production and medicinal applications.
Microbiology is the study of microbes, which affect almost every aspect of life on the earth. In addition, there are huge commercial and medicinal benefits in understanding microbes. The application of this understanding is known as applied microbiology. There are many different types of applied microbiology which can be briefly defined as follows:
Medical Microbiology
Medical microbiology is the study of the pathogenic microbes and the role of microbes in human illness. This includes the study of microbial pathogenesis and epidemiology and is related to the study of disease pathology and immunology.
Pharmaceutical Microbiology
The study of microorganisms that are related to the production of antibiotics, enzymes, vitamins, vaccines, and other pharmaceutical products. Pharmaceutical microbiology also studies the causes of pharmaceutical contamination and spoil.
Industrial Microbiology
The exploitation of microbes for use in industrial processes. Examples include industrial fermentation and waste-water treatment. Closely linked to the biotechnology industry. This field also includes brewing, an important application of microbiology.
Microbial Biotechnology
The manipulation of microorganisms at the genetic and molecular level to generate useful products.
Food Microbiology and Dairy Microbiology
The study of microorganisms causing food spoilage and food-borne illness. Microorganisms can also produce foods, for example by fermentation.
Agricultural Microbiology
The study of agriculturally relevant microorganisms. This field can be further classified into the following subfields:
- Plant microbiology and plant pathology – The study of the interactions between microorganisms and plants and plant pathogens.
- Soil microbiology – The study of those microorganisms that are found in soil.
- Veterinary microbiology – The study of the role in microbes in veterinary medicine or animal taxonomy.
Environmental microbiology
The study of the function and diversity of microbes in their natural environments. This involves the characterization of key bacterial habitats such as the rhizosphere and phyllosphere, soil and groundwater ecosystems, open oceans or extreme environments (extremophiles). This field includes other branches of microbiology such as: microbial ecology (the role of microbes in ecosystems), geomicrobiology (interactions between microbes and their physical and chemical surroundings), microbial diversity, water microbiology (the study of those microorganisms that are found in water), aeromicrobiology (the study of airborne microorganisms) and epidemiology (the study of the incidence, spread, and control of disease). Bioremediation involves the application of microbial processes to remove pollutants from the environment.
This is by no means an exhaustive list of the different types of applied microbiology, but gives an indication of the expansive variety of the field and some of the benefits these studies entail.


8: Advanced Theories of Covalent Bonding
We have examined the basic ideas of bonding, showing that atoms share electrons to form molecules with stable Lewis structures and that we can predict the shapes of those molecules by valence shell electron pair repulsion (VSEPR) theory. These ideas provide an important starting point for understanding chemical bonding. But these models sometimes fall short in their abilities to predict the behavior of real substances. How can we reconcile the geometries of s, p, and d atomic orbitals with molecular shapes that show angles like 120° and 109.5°? Furthermore, we know that electrons and magnetic behavior are related through electromagnetic fields.
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- 8.0: Prelude to Covalent Bonding
- Yet oxygen demonstrates very different magnetic behavior than nitrogen. We can pour liquid nitrogen through a magnetic field with no visible interactions, while liquid oxygen is attracted to the magnet and floats in the magnetic field. We need to understand the additional concepts of valence bond theory, orbital hybridization, and molecular orbital theory to understand these observations.
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- 8.1: Valence Bond Theory
- Valence bond theory describes bonding as a consequence of the overlap of two separate atomic orbitals on different atoms that creates a region with one pair of electrons shared between the two atoms. When the orbitals overlap along an axis containing the nuclei, they form a σ bond. When they overlap in a fashion that creates a node along this axis, they form a π bond.
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- 8.2: Hybrid Atomic Orbitals
- We can use hybrid orbitals, which are mathematical combinations of some or all of the valence atomic orbitals, to describe the electron density around covalently bonded atoms. These hybrid orbitals either form sigma (σ) bonds directed toward other atoms of the molecule or contain lone pairs of electrons. We can determine the type of hybridization around a central atom from the geometry of the regions of electron density about it.
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- 8.3: Multiple Bonds
- Multiple bonds consist of a σ bond located along the axis between two atoms and one or two π bonds. The σ bonds are usually formed by the overlap of hybridized atomic orbitals, while the π bonds are formed by the side-by-side overlap of unhybridized orbitals. Resonance occurs when there are multiple unhybridized orbitals with the appropriate alignment to overlap, so the placement of π bonds can vary.
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- 8.4: Molecular Orbital Theory
- Molecular orbital (MO) theory describes the behavior of electrons in a molecule in terms of combinations of the atomic wavefunctions. The resulting molecular orbitals may extend over all the atoms in the molecule. Bonding molecular orbitals are formed by in-phase combinations of atomic wavefunctions, and electrons in these orbitals stabilize a molecule. Antibonding molecular orbitals result from out-of-phase combinations and electrons in these orbitals make a molecule less stable.
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- 8.E: Advanced Theories of Covalent Bonding (Exercises)
- These are homework exercises to accompany the Textmap created for "Chemistry" by OpenStax. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here. In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.
Contributors
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


5.1: Nerve Fibers and the Dynamic Strang Quartet
Introduction
Up to this point we have largely been concerned with
- Deriving linear systems of algebraic equations (from considerations of static equilibrium) and
- The solution of such systems via Gaussian elimination.
In this module we hope to begin to persuade the reader that our tools extend in a natural fashion to the class of dynamic processes. More precisely, we shall argue that
- Matrix Algebra plays a central role in the derivation of mathematical models of dynamical systems and that,
- With the aid of the Laplace transform in an analytical setting or the Backward Euler method in the numerical setting, Gaussian elimination indeed produces the solution.
Nerve Fibers and the Dynamic Strang Quartet
Gathering Information
A nerve fiber's natural electrical stimulus is not direct current but rather a short burst of current, the so-called nervous impulse .
The typical value of a cell's membrane capacitance is
\[c = 1\frac{\mu F}{cm^{2}} \nonumber\]
where \(\mu F\) denotes micro-Farad. Recalling our variable conventions, the capacitance of a single compartment is
\[C_{m} = 2 \pi a \frac{l}{N} c \nonumber\]
and runs parallel to each \(R_{m}\), see Figure 1. This figure also differs from the simpler circuit from the introductory electrical modeling module in that it possesses two edges to the left of the stimuli. These edges serve to mimic that portion of the stimulus current that is shunted by the cell body. If \(A_{cb}\) denotes the surface area of the cell body, then it has
\(C_{cb} = A_{cb} c\)
\(R_{cb} = A_{cb} \rho_{m}\)
Updating the Strang Quartet
We ask now how the static Strang Quartet of the introductory electrical module should be augmented.
Updating (S1')
Regarding (S1') we proceed as before. The voltage drops are
\[e_{1} = x_{1} \nonumber\]
\[e_{2} = x_{1}-E_{m} \nonumber\]
\[e_{3} = x_{1}-x_{2} \nonumber\]
\[e_{4} = x_{2} \nonumber\]
\[e_{5} = x_{2}-E_{m} \nonumber\]
\[e_{6} = x_{2}-x_{3} \nonumber\]
\[e_{7} = x_{3} \nonumber\]
\[e_{8} = x_{3}-E_{m} \nonumber\]
and so
\[\begin{array}{ccccc} {\textbf{e} = \textbf{b}-A \textbf{x}}&{\text{where}}&{\textbf{b} = (-E_{m}) \begin{pmatrix} {0}\\ {1}\\ {0}\\ {0}\\ {1}\\ {0}\\ {0}\\ {1} \end{pmatrix}}&{\text{and}}&{A = \begin{pmatrix} {-1}&{0}&{0}\\ {-1}&{0}&{0}\\ {-1}&{1}&{0}\\ {0}&{-1}&{0}\\ {0}&{-1}&{0}\\ {0}&{-1}&{1}\\ {0}&{0}&{-1}\\ {0}&{0}&{-1} \end{pmatrix}} \end{array} \nonumber\]
Updating (S2)
To update (S2) we must now augment Ohm's law with
The current through a capacitor is proportional to the time rate of change of the potential across it.
This yields, (denoting derivative by '),
\[y_{1} = C_{cb}e_{1}′ \nonumber\]
\[y_{2} = \frac{e_{2}}{R_{cb}} \nonumber\]
\[y_{3} = \frac{e_{3}}{R_{i}} \nonumber\]
\[y_{4} = C_{m}e_{4}′ \nonumber\]
\[y_{5} = \frac{e_{5}}{R_{m}} \nonumber\]
\[y_{6} = \frac{e_{6}}{R_{i}} \nonumber\]
\[y_{7} = C_{m} e_{7}′ \nonumber\]
\[y_{8} = \frac{e_{8}}{R_{m}} \nonumber\]
or, in matrix terms,
\[\textbf{y} = G \textbf{e}+C \textbf{e}' \nonumber\]
where
\[G = \begin{pmatrix} {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{\frac{1}{R_{cb}}}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{\frac{1}{R_{i}}}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{\frac{1}{R_{m}}}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{\frac{1}{R_{i}}}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{\frac{1}{R_{m}}} \end{pmatrix} \nonumber\]
and
\[C = \begin{pmatrix} {C_{cb}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{C_{m}}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{C_{m}}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0} \end{pmatrix} \nonumber\]
are the conductance and capacitance matrices.
Updating (S3)
As Kirchhoff's Current law is insensitive to the type of device occupying an edge, step (S3) proceeds exactly as before.
\[i_{0}-y_{1}-y_{2}-y_{3} = 0 \nonumber\]
\[y_{3}-y_{4}-y_{5}-y_{6} = 0 \nonumber\]
\[y_{6}-y_{7}-y_{8} = 0 \nonumber\]
or, in matrix terms,
\[\begin{array}{ccc} {A^{T} \textbf{y} = -\textbf{f}}&{where}&{\textbf{f} = \begin{pmatrix} {i_{0}}\\ {0}\\ {0} \end{pmatrix}^{T}} \end{array} \nonumber\]
Step (S4): Assembling
Step (S4) remains one of assembling,
\[(A^{T} \textbf{y} = -\textbf{f}) \Rightarrow (A^{T} (G \textbf{e}+C \textbf{e}') = -\textbf{f}) \Rightarrow (A^{T} (G(\textbf{b}-A\textbf{x})+C (\textbf{b}'-A\textbf{x}')) = -\textbf{f}) \nonumber\]
becomes
\[A^{T}CA \textbf{x}'+A^{T}GA \textbf{x} = A^{T}G \textbf{b}+\textbf{f}+A^{T}C \textbf{b}' \nonumber\]
This is the general form of the potential equations for an RC circuit. It presumes of the user knowledge of the initial value of each of the potentials,
\[\textbf{x}(0) = X \nonumber\]
Regarding the circuit of Figure 1, and letting \(G = \frac{1}{R}\), we find
\[\begin{array}{cc} {A^{T}CA = \begin{pmatrix} {C_{cb}}&{0}&{0}\\ {0}&{C}&{0}\\ {0}&{0}&{C} \end{pmatrix}}&{A^{T}CA = \begin{pmatrix} {G_{cb}+G_{i}}&{-G_{i}}&{0}\\ {-G_{i}}&{2G_{i}+G_{m}}&{-G_{i}}\\ {0}&{-G_{i}}&{G_{i}+G_{m}} \end{pmatrix}} \end{array} \nonumber\]
\[\begin{array}{cc} {A^{T}G \textbf{b} = E_{m} \begin{pmatrix} {G_{cb}}\\ {G_{m}}\\ {G_{m}} \end{pmatrix}}&{A^{T}C \textbf{b}' = \begin{pmatrix} {0}\\ {0}\\ {0} \end{pmatrix}} \end{array} \nonumber\]
and an initial (rest) potential of
\[x_{0} = E_{m} \begin{pmatrix} {1}\\ {1}\\ {1} \end{pmatrix}^{T} \nonumber\]
Modes of Attack
We shall now outline two modes of attack on such problems. The Laplace Transform is an analytical tool that produces exact, closed-form solutions for small tractable systems and therefore offers insight into how larger systems 'should' behave. The Backward-Euler method is a technique for solving a discretized (and therefore approximate) version of Equation. It is highly flexible, easy to code, and works on problems of great size. Both the Backward-Euler and Laplace Transform methods require, at their core, the algebraic solution of a linear system of equations. In deriving these methods we shall find it more convenient to proceed from the generic system
\[\textbf{x}′ = B\textbf{x}+\textbf{g} \nonumber\]
With respect to our fiber problem
\[B = (-(A^{T}CA)^{-1}) A^{T}GA \nonumber\]
\[= \begin{pmatrix} {\frac{-(G_{cb}+G_{i})}{C_{cb}}}&{\frac{G_{i}}{C_{cb}}}&{0}\\ {\frac{G_{i}}{C_{m}}}&{\frac{-(2G_{i}+G_{m})}{C_{m}}}&{\frac{G_{i}}{C_{m}}}\\ {0}&{\frac{G_{i}}{C_{m}}}&{\frac{-(G_{i}+G_{m})}{C_{m}}} \end{pmatrix} \nonumber\]
and
\[\textbf{g} = (A^{T}CA)^{-1}(A^{T}G \textbf{b}+\textbf{f}) = \begin{pmatrix} {\frac{G_{cb}E_{m}+i_{0}}{C_{cb}}}\\ {\frac{E_{m}G_{m}}{C_{m}}}\\ {\frac{E_{m}G_{m}}{C_{m}}} \end{pmatrix} \nonumber\]


10.2.3: Cultural Diversity and Gender Norms
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Figure \(2.3.1\): Mojave group. Photo Source: Wikimedia Commons, the free media repository
There is some variation in cultural gender-role standards both within the United States and across cultures; however, within the United States, for example, standards vary depending on ethnicity, age, education, and occupation. These variables will be discussed more fully in coming chapters. But now that we’ve explored the definitions and constructions of sex and gender, let’s more closely examine some traditional gendered norms from cultures that don’t totally reflect Eurocentric, Westernized ways of doing gender.
Mojave Native Americans
First we’ll talk about the Mojave in North America. This culture taught and reinforced parallel institutional structures for males and females. 15 For example, the Mojave husbands and wives worked together to farm their fields. Men planted and watered the crops, and women harvested them. Both sexes took part in storytelling, music and artwork, and traditional medicine. 16
In Mohave society, pregnant women believed they had dreams predicting the anatomic sex of their children. These dreams also sometimes included hints of their child’s future gender variant status. A boy who “acted strangely” before he participated in the boy’s puberty ceremonies in the Mohave tribe would be considered for the transvestite ceremony. The ceremony itself was meant to surprise the boy. Other nearby settlements would receive word to come and watch. A circle of onlookers would sing special songs. If the boy danced like a woman, it confirmed his status as an alyha. He was then taken to a river to bathe, and was given a skirt to wear. The ceremony would permanently change his gender status within the tribe. He then took up a female name. The alyha would imitate many aspects of female life, including menstruation, puberty observations, pregnancy, and birth. The alyha were considered great healers, especially in curing sexually transmitted diseases like syphilis.
Oaxaca, Mexico
Figure \(2.3.2\) Muxe: Mexican men who take on the traditional dress and gender roles of Mexican women within their communities.
The Juchitán in Oaxaca, Mexico, practice gender norms that do not comply with traditional Western practices. For example, the women traditionally run businesses, wear colorfully bold traditional clothing, and hold their heads firmly high. The women are regarded as being empowered and the tolerance of homosexuality and transgender individuals has been part of their cultural tradition. men who take on the traditional roles of women, referred to as “Muxes,” are not only accepted, but cherished as symbols of good luck. This community exemplifies an alternative gender system unlike the binary gender categories that have been established throughout many parts of the world.
Indian Hijras
Figure \(2.3.4\) Hijra
In Indian Hindu culture, when compared to the native North Americans, the gender system is essentially binary, but the ideas themselves are quite different from Western thoughts. These ideas often come from religious contexts. Some Hindu origin myths feature androgynous or hermaphroditic ancestors. Ancient poets often showed this idea by presenting images with mixed physical attributes between the two sexes. These themes still exist in the culture, and are even still institutionalized. The most prominent group are the hijras. 17 Hijras, today, are not seen either male or female, but rather are typically identified as “hijra.” “Being a hijra means making a commitment that gives social support and some economic security, as well as a cultural meaning, linking them to the larger world.” 18
Brazil
As in Indian culture, Brazilian culture does follow a gender binary, just not the traditional western one. Rather than men and women, certain areas of Brazil have men and not-men. Men are masculine, and anyone who displays feminine qualities falls under the category of “not-man.” This concept is a result of sexual penetration as the deciding factor of gender. Any one who is penetrated is not-male.  Unlike in native North America and India, the existence of the travestí is not from a religious context. It is an individual’s choice to become a travestí. Born as males, they go to extensive measures to try to appear female. The travestí recognize they are not female, and that they cannot ever become female. Instead, their culture is based on this man/not-man premise. 20
Thailand
Figure \(2.3.5\) A Kathoey. Photo Source: https://maytermthailand.org/2015/04/27/the- third-gender-in-thailand-kathoey/
In Thailand the term kathoey is used by both males and females and allows and refers to cross-dressing and adopting masculine and feminine identities opposite their assigned birth gender. Up until the 1970s cross-dressing men and women could all come under the term kathoey, however the term has been dropped for the cross-dressing masculine females who are now referred to as tom . 21 As a result of the shifts, kathoey today is most commonly understood as a male transgender category, and these people are now sometimes referred to as “lady-boys.” Kathoey is derived from the Buddhist myth that describes three original human sex/genders, male, female, and a biological hermaphrodite or kathoey. 22 Kathoey is not defined as merely being a variant between male or female but as an independently existing third sex.
Nigeria
The Nigerian Yoruba social life and gender roles do not duplicate those found in the West. Instead of focusing on gender distinctions, this culture typically focuses on age distinctions. In addition, men who choose to wear women’s clothing, jewelry, and cosmetics will be labeled as “wife of the god,” as only women can be wives of mortal men.
Waria, Indonesia
Waria is a term used for the third gender in Indonesia outside of the masculine and feminine ideals in this Islamic nation. The waria are born male but live along a continuum of gender identity not constrained to the traditional Western interpretation of the masculinity. The term “waria” includes individuals who continue to identify as male but who imitate certain femi nine mannerisms, and can occasionally wear makeup, jewelry, and women’s clothing. Others identify so closely as female that they are able to pass as female in their daily interactions in society.
Sistergirls and Brotherboys, Australia
In Australia, indigenous transgendered people are known as "sistergirls" and "brotherboys". As in some other native cultures, there is evidence that transgender and intersex people were much more accepted in their society before colonization. Therefore, the Eurocentric tendencies that shaped Western gender ideals have also played a huge part in the contemporary view of the sistergirls and brotherboys in Australia today. For example, today there are more stigmas attached to these individuals. But through an increasing number of support groups specifically aimed at sistergirls and brotherboys, perhaps times will change again. If there’s anything we’ve learned from gender so far, it’s that it is constantly being constructed and re -constructed.
Mahu, Hawaii
In traditional Hawaiian culture, creative expression of gender and sexuality was celebrated as an authentic part of the human experience.  A multiple gender tradition existed among the Kanaka Maoli indigenous people. The mahu could be biological males or females inhabiting a gender role somewhere between or encompassing both the masculine and feminine. Their social role is sacred as educators and conservators of ancient traditions and rituals. The arrival of Europeans and the colonization of Hawaii nearly eliminated the native culture, and today mahu face discrimination in a culture dominated by white Eurocentric ideology.
Suggested Articles for Further Reading:
Candace West; Don H. Zimmerman. (1987). Doing Gender . Gender and Society, Vol. 1, No. 2. (Jun., 1987), pp. 125-151.
Patricia A. Adler, Steven J. Kless and Peter Adler. (1992). Socialization to Gender Roles: Popularity among Elementary School Boys and Girls. Sociology of Education. Vol. 65, No. 3 (Jul., 1992), pp. 169-187.
Suggested Films:
Middle Sexes: Refining He and She (2005)
Examines the diversity of human sexual and gender variance around the globe, with commentary by scientific experts and first-handaccounts of people who do not conform to a simple male/female binary.
Two Spirits (2009)
Filmmaker Lydia Nibley explores the cultural context behind a tragic and senseless murder. Fred Martinez was a Navajo youth slain at the age of 16 by a man who bragged to his friends that he 'bug-smashed a fag'. But Fred was part of an honored Navajo tradition -the 'nadleeh', or 'two-spirit', who possesses a balance of masculine and feminine traits. Through telling Fred's story, Nibley reminds us of the values that America's indigenous peoples have long embraced. - Written by Outfest
Half the Sky (2012)
This documentary—filmed in 10 countries with narrations from celebrities such as Olivia Wilde, Eva Mendes and Meg Ryan—tellsuplifting stories of women around the world who are fighting back against systemic oppression. The film presents gender equality as the unfinished business of the our time and highlights women who are working to improve everything from healthcare to education.
15
Hill, W. W. (1935). The status of the hermaphrodite and transvestite in Navaho culture.
American Anthropologist
, 37,
273-279.
16
Nanda, Serena. Gender Diversity: Crosscultural Variations. Waveland Press, 1999. Print. Pages 21-23
17
Nanda, Serena. Gender Diversity: Crosscultural Variations. Waveland Press, 1999. Print.pg. 27,28
18
Ibid
19
Kulick, D. "The Gender of Brazilian Transgendered Prostitutes." American Anthropologist 99.3 (1997): 574-85. Page
578
20
Ibid
21
Nanda, Serena. Gender Diversity: Crosscultural Variations. Waveland Press, 1999. Print. Page 73
22
Ibid


7.1: Curvature
Consider the smooth curve in Figure \(7.1.1\). The curvature of the curve at a point is a measure of how drastically the curve bends away from its tangent line, and this curvature is often studied in a multivariable calculus course. The radius of curvature at a point corresponds to the radius of the circle that best approximates the curve at this point. The radius \(r\) of this circle is the reciprocal of the curvature \(k\) of the curve at the point: \(k = \dfrac{1}{r}\text{.}\)
The curvature of a surface (such as the graph of a function \(z = f(x,y)\)) at a particular point is a measure of how drastically the surface bends away from its tangent plane at the point. There are three fundamental types of curvature. A surface has positive curvature at a point if the surface lives entirely on one side of the tangent plane, at least near the point of interest. The surface has negative curvature at a point if it is saddle-shaped, in the sense that the tangent plane cuts through the surface. Between these two cases is the case of zero curvature. In this case the surface has a line along which the surface agrees with the tangent plane. For instance, a cylinder has zero curvature, as suggested in Figure \(7.1.2(c)\).
This informal description of curvature makes use of how the surface is embedded in space. One of Gauss' great theorems, one he called his Theorem Egregium, states that the curvature of a surface is an intrinsic property of the surface. The curvature doesn't change if the surface is bent without stretching, and our tireless two-dimensional inhabitant living in the space can determine the curvature by taking measurements.
A two-dimensional bug living in the hyperbolic plane, the projective plane, or the Euclidean plane would notice that a small circle's circumference is related to its radius by the Euclidean formula \(c \approx 2\pi r\text{.}\) In Euclidean geometry this formula applies to all circles, but in the non-Euclidean cases, the observant bug might notice in large circles a significant difference between the actual circumference of a circle and the circumference predicted by \(c = 2\pi r\text{.}\) Large circles about a cup-shaped point with positive curvature will have circumference less than that predicted by Euclidean geometry. This fact explains why a large chunk of orange peel fractures if pressed flat onto a table. Large circles drawn around a saddle-shaped point with negative curvature will have circumference greater than that predicted by the Euclidean formula.
Calculus may be used to precisely capture this deviation between the Euclidean-predicted circumference \(2\pi r\) and the actual circumference \(c\) for circles of radius \(r\) in the different geometries.
Recall that in the hyperbolic plane, \(c = 2\pi \sinh(r)\text{;}\) in the Euclidean plane \(c = 2\pi r\text{;}\) and in the elliptic plane \(c = 2\pi \sin(r)\text{.}\) In Figure \(7.1.3\) we have graphed the ratio \(\dfrac{c}{2\pi r}\) where \(c\) is the circumference of a circle with radius \(r\) in (a) the hyperbolic plane; (b) the Euclidean plane; and (c) the elliptic plane.
In all three cases, the ratio \(\dfrac{c}{2\pi r}\) approaches \(1\) as \(r\) shrinks to \(0\). Furthermore, in all three cases, the derivative of the ratio approaches \(0\) as \(r \to 0^+\text{.}\) But with the second derivative of the ratio we may distinguish these geometries. It can be shown that the curvature at a point is proportional to this second derivative evaluated in the limit as \(r \rightarrow 0^+\text{.}\) We will not derive this formula for curvature but will use this working definition as it appears in Thurston's book [11] .
Suppose a circle of radius \(r\) about a point \(p\) is drawn in a space 1 , and its circumference is \(c\text{.}\) The curvature of the space at \(p\) is
\[ k = -3 \lim_{r \to 0^+}\dfrac{d^2}{dr^2}\bigg[\dfrac{c}{2\pi r}\bigg]\text{.} \]
the term `space' is intentionally vague here. Our space needs to have a well-defined metric, so that it makes sense to talk about radius and circumference. The space might be the Euclidean plane, the hyperbolic plane or the sphere. Other spaces are discussed in Section 7.5.
Since we are interested in worlds that are homogeneous and isotropic, we will focus our attention on worlds in which the curvature is the same at all points. That is, we investigate surfaces of constant curvature.
Example 7.1.1: The Curvature of a Sphere
Consider the sphere with radius \(s\) in the following diagram, and note the circle centered at the north pole \(N\) having surface radius \(r\text{.}\) The circle is parallel to the plane \(z = 0\text{,}\) has Euclidean radius \(x\text{,}\) and hence circumference \(2\pi x\text{.}\)
But \(x = s\sin(\theta)\) and \(r = \theta\cdot s\text{,}\) from which we deduce \(x = s\sin(\dfrac{r}{s})\text{,}\) and in terms of the surface radius \(r\) of the circle, its circumference is
\[ c = 2\pi s\sin\bigg(\dfrac{r}{s}\bigg)\text{.} \]
The curvature of the sphere at \(N\) is thus
\[ k = -3 \lim_{r \to 0^+}\dfrac{d^2}{dr^2}\bigg[\dfrac{2\pi s\sin(\dfrac{r}{s})}{2\pi r}\bigg]\text{.} \]
Cancelling the \(2\pi\) terms and replacing \(\sin(\dfrac{r}{s})\) with its power series expansion, we have
\begin{align*} k & = -3 \lim_{r \to 0^+}\dfrac{d^2}{dr^2}\left[\dfrac{s(\dfrac{r}{s}-\dfrac{r^3}{6s^3}+\dfrac{r^5}{120s^5}-\cdots)}{r}\right]\\ & = -3 \lim_{r \to 0^+}\dfrac{d^2}{dr^2}\left[1-\dfrac{r^2}{6s^2}+\dfrac{r^4}{120s^4}-\cdots\right]\\ & = -3\lim_{r \to 0^+}\left[\dfrac{-1}{3s^2} + \dfrac{12r^2}{120s^4} - \cdots \right]\text{.} \end{align*}
Note that all the terms of the second derivative after the first have powers of \(r\) in the numerator, so these terms go to \(0\) as \(r \to 0^+\text{,}\) and the curvature of the sphere at the north pole is \(\dfrac{1}{s^2}\text{.}\) In fact because the sphere is homogeneous, the curvature at any point is
\[ k=\dfrac{1}{s^2}\text{.} \]
Example 7.1.2: Curvature of the Hyperbolic Plane
Because hyperbolic geometry is homogeneous and its transformations preserve circles and lengths, the curvature is the same at all points in the hyperbolic plane. We choose to compute the curvature at the origin.
Recall, the circumference of a circle in \((\mathbb{D},\cal{H})\) is \(c = 2\pi \sinh(r).\) To compute the curvature, use the power series expansion
\[ \sinh(r)=r + \dfrac{r^3}{3!}+\dfrac{r^5}{5!}+\cdots\text{.} \]
\begin{align*} k & = -3 \lim_{r \to 0^+}\dfrac{d^2}{dr^2}\left[\dfrac{2\pi \sinh(r)}{2\pi r}\right]\\ & = -3 \lim_{r \to 0^+}\dfrac{d^2}{dr^2}\left[1+\dfrac{r^2}{3!}+\dfrac{r^4}{5!}+\cdots\right]\\ & = -3 \lim_{r \to 0^+}\left[\dfrac{1}{3}+\dfrac{12r^2}{5!}+\cdots\right]\text{.} \end{align*}
Again, each term of the second derivative after the first has a power of \(r\) in its numerator, so in the limit as \(r \to 0^+\text{,}\) each of these terms vanishes. Thus, the curvature of the hyperbolic plane in (\(\mathbb{D},{\cal H})\) is \(k = -1.\)
Exercises
Use our working definition to show that the curvature of the projective plane in elliptic geometry is \(1\). Recall, \(c = 2\pi\sin(r)\) in this geometry.
Use our working definition to explain why the curvature of the Euclidean plane is \(k = 0\text{.}\)


9.5: Food Safety
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To minimize the risk of serious adverse health consequences or death from consumption of contaminated produce, the Food and Drug Administration (FDA) is establishing science-based minimum standards for the safe growing, harvesting, packing, and holding of produce, meaning fruits and vegetables grown for human consumption.
FDA is establishing these standards as part of implementation of the FDA Food Safety and Modernization Act. These standards do not apply to produce that is rarely consumed raw, produce for personal or on-farm consumption, or produce that is not a raw agricultural commodity. In addition, produce that receives commercial processing that adequately reduces the presence of microorganisms of public health significance is eligible for exemption from the requirements of this rule. The rule sets forth procedures, processes, and practices that minimize the risk of serious adverse health consequences or death, including those reasonably necessary to prevent the introduction of known or reasonably foreseeable biological hazards into or onto produce and to provide reasonable assurances that the produce is not adulterated on account of such hazards. The FDA expects the rule to reduce foodborne illness associated with the consumption of contaminated produce.
Foodborne Illnesses
While the American food supply is among the safest in the world, the Federal government estimates that there are about 48 million cases of foodborne illness annually —the equivalent of sickening 1 in 6 Americans each year. Each year these illnesses result in an estimated 128,000 hospitalizations and 3,000 deaths. These cases have driven authorities to seriously consider methods and regulations that help prevent the spread of foodborne illness.
Food Handling
Most foodborne illnesses can be avoided if you take the following precautionary steps:
CLEAN: Always wash your food, hands, counters, and cooking tools.
- Wash hands in warm soapy water for at least 20 seconds (sing Happy Birthday while washing and the 20 seconds will be over). Do this before and after touching food.
- Wash cutting boards, dishes, forks, spoons, knives and counter tops with hot soapy water. Do this after working with each food item.
- Rinse fruits and veggies well
- Clean the lids on canned goods before opening.
SEPARATE (Keep Apart): Keep raw foods apart from other raw foods. Germs can spread from one food to another.
- Keep raw meat, poultry, seafood, and eggs away from other foods in the shopping cart, bags, and the refrigerator.
- Do not reuse marinades used on raw foods unless first brought to a boil first.
- Use a separate cutting board and utensils for raw meats, and a designated plate and clean utensils for raw foods. This prevents cross-contamination.


21.16: Attenuation in Coaxial Cable
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In this section, we consider the issue of attenuation in coaxial transmission line. Recall that attenuation can be interpreted in the context of the “lumped element” equivalent circuit transmission line model as the contributions of the resistance per unit length \(R'\) and conductance per unit length \(G'\). In this model, \(R'\) represents the physical resistance in the inner and outer conductors, whereas \(G'\) represents loss due to current flowing directly between the conductors through the spacer material.
The parameters used to describe the relevant features of coaxial cable are shown in Figure \(\PageIndex{1}\). In this figure, \(a\) and \(b\) are the radii of the inner and outer conductors, respectively. \(\sigma_{ic}\) and \(\sigma_{oc}\) are the conductivities (SI base units of S/m) of the inner and outer conductors, respectively. Conductors are assumed to be non-magnetic; i.e., having permeability \(\mu\) equal to the free space value \(\mu_0\). The spacer material is assumed to be a lossy dielectric having relative permittivity \(\epsilon_r\) and conductivity \(\sigma_s\).
Figure \(\PageIndex{1}\): Parameters defining the design of a coaxial cable.Resistance per unit length
The resistance per unit length is the sum of the resistances of the inner and outer conductor per unit length. The resistance per unit length of the inner conductor is determined by \(\sigma_{ic}\) and the effective cross-sectional area through which the current flows. The latter is equal to the circumference \(2\pi a\) times the skin depth \(\delta_{ic}\) of the inner conductor, so:
\[R_{ic}' \approx \frac{1}{\left(2\pi a \cdot \delta_{ic} \right) \sigma_{ic} } ~~~ \mbox{for} ~ \delta_{ic} \ll a \nonumber \]
This expression is only valid for \(\delta_{ic} \ll a\) because otherwise the cross-sectional area through which the current flows is not well-modeled as a thin ring near the surface of the conductor. Similarly, we find the resistance per unit length of the outer conductor is
\[R_{oc}' \approx \frac{1}{\left(2\pi b \cdot \delta_{oc} \right) \sigma_{oc} } ~~~ \mbox{for} ~ \delta_{oc} \ll t \nonumber \]
where \(\delta_{oc}\) is the skin depth of the outer conductor and \(t\) is the thickness of the outer conductor. Therefore, the total resistance per unit length is
\begin{align} R' &= R_{ic}' + R_{oc}' \nonumber \\ & \approx \frac{1}{\left(2\pi a \cdot \delta_{ic} \right) \sigma_{ic} } + \frac{1}{\left(2\pi b \cdot \delta_{oc} \right) \sigma_{oc} } \label{m0189_eRp1} \end{align}
Recall that skin depth depends on conductivity. Specifically:
\begin{align} \delta_{ic} &= \sqrt{2/\omega\mu\sigma_{ic}} \\ \delta_{oc} &= \sqrt{2/\omega\mu\sigma_{oc}} \end{align}
Expanding Equation \ref{m0189_eRp1} to show explicitly the dependence on conductivity, we find:
\[R' \approx \frac{1}{2\pi \sqrt{2/\omega\mu_0} } \left[ \frac{1}{a \sqrt{\sigma_{ic}} } + \frac{1}{b \sqrt{\sigma_{oc}} } \right] \nonumber \]
At this point it is convenient to identify two particular cases for the design of the cable. In the first case, “Case I,” we assume \(\sigma_{oc} \gg \sigma_{ic}\). Since \(b > a\), we have in this case
\begin{align} R' &\approx \frac{1}{2\pi \sqrt{2/\omega\mu_0} } \left[ \frac{1}{a \sqrt{\sigma_{ic}} } \right] \nonumber \\ &= \frac{1}{2\pi \delta_{ic} \sigma_{ic} }~\frac{1}{a} ~~~\mbox{(Case~I)}\end{align}
In the second case, “Case II,” we assume \(\sigma_{oc} = \sigma_{ic}\). In this case, we have
\begin{align} R' &\approx \frac{1}{2\pi \sqrt{2/\omega\mu_0} } \left[ \frac{1}{a \sqrt{\sigma_{ic}} } + \frac{1}{b \sqrt{\sigma_{ic}} } \right] \nonumber \\ &= \frac{1}{2\pi \delta_{ic} \sigma_{ic} }~\left[ \frac{1}{a} + \frac{1}{b} \right] ~~~\mbox{(Case~II)}\end{align}
A simpler way to deal with these two cases is to represent them both using the single expression
\[R' \approx \frac{1}{2\pi \delta_{ic} \sigma_{ic} }~\left[ \frac{1}{a} + \frac{C}{b} \right] \nonumber \]
where \(C=0\) in Case I and \(C=1\) in Case II.
Conductance per unit length
The conductance per unit length of coaxial cable is simply that of the associated coaxial structure at DC; i.e.,
\[G' = \frac{2\pi\sigma_s}{\ln\left(b/a\right)} \nonumber \]
Unlike resistance, the conductance is independent of frequency, at least to the extent that \(\sigma_s\) is independent of frequency.
Attenuation
The attenuation of voltage and current waves as they propagate along the cable is represented by the factor \(e^{-\alpha z}\), where \(z\) is distance traversed along the cable. It is possible to find an expression for \(\alpha\) in terms of the material and geometry parameters using:
\[\gamma \triangleq \sqrt{ \left( R' + j\omega L' \right) \left( G' + j\omega C' \right)} = \alpha + j\beta \label{m0189_fGamma} \]
where \(L'\) and \(C'\) are the inductance per unit length and capacitance per unit length, respectively. These are given by
\[L' = \frac{\mu}{2\pi}\ln{\left(b/a\right)} \nonumber \]
and
\[C' = \frac{2\pi\epsilon_0\epsilon_r}{\ln{\left(b/a\right)}} \nonumber \]
In principle we could solve Equation \ref{m0189_fGamma} for \(\alpha\). However, this course of action is quite tedious, and a simpler approximate approach facilitates some additional insights. In this approach, we define parameters \(\alpha_R\) associated with \(R'\) and \(\alpha_G\) associated with \(G'\) such that
\[e^{-\alpha_R z} e^{-\alpha_G z} = e^{-\left(\alpha_R+\alpha_G\right) z} = e^{-\alpha z} \nonumber \]
which indicates
\[\alpha = \alpha_R + \alpha_G \nonumber \]
Next we postulate
\[\alpha_R \approx K_R \frac{R'}{Z_0} \label{m0189_eAlphaR} \]
where \(Z_0\) is the characteristic impedance
\[Z_0 \approx \frac{\eta_0}{2\pi}\frac{1}{\sqrt{\epsilon_r}}\ln{\frac{b}{a}}~~~\mbox{(low loss)} \label{m0189_eZ0cc} \]
and where \(K_R\) is a unitless constant to be determined. The justification for Equation \ref{m0189_eAlphaR} is as follows: First, \(\alpha_R\) must increase monotonically with increasing \(R'\). Second, \(R'\) must be divided by an impedance in order to obtain the correct units of 1/m. Using similar reasoning, we postulate
\[\alpha_G \approx K_G G' Z_0 \label{m0189_eAlphaG} \]
where \(K_G\) is a unitless constant to be determined. The following example demonstrates the validity of Equations \ref{m0189_eAlphaR} and \ref{m0189_eAlphaG}, and will reveal the values of \(K_R\) and \(K_G\).
Example \(\PageIndex{1}\): Attenuation constant for RG-59
RG-59 is a popular form of coaxial cable having the parameters \(a \cong 0.292~\)mm, \(b \cong 1.855\) mm, \(\sigma_{ic} \cong 2.28 \times 10^7\) S/m, \(\sigma_s \cong 5.9 \times 10^{-5}\) S/m, and \(\epsilon_r \cong 2.25\). The conductivity \(\sigma_{oc}\) of the outer conductor is difficult to quantify because it consists of a braid of thin metal strands. However, \(\sigma_{oc}\gg\sigma_{ic}\), so we may assume Case I; i.e., \(\sigma_{oc}\gg\sigma_{ic}\), and subsequently \(C=0\).
Figure \(\PageIndex{2}\): Comparison of \(\alpha=\operatorname{Re}\{\gamma\}\) to \(\alpha_R\), \(\alpha_G\), and \(\alpha_R + \alpha_G\) for \(K_R = K_G = 1/2\). The result for \(\alpha\) has been multiplied by 1.01; otherwise the curves would be too close to tell apart.Figure \(\PageIndex{2}\) shows the components \(\alpha_G\) and \(\alpha_R\) computed for the particular choice \(K_R=K_G=1/2\). The figure also shows \(\alpha_G + \alpha_R\), along with \(\alpha\) computed using Equation \ref{m0189_fGamma}. We find that the agreement between these values is very good, which is compelling evidence that the ansatz is valid and \(K_R=K_G=1/2\).
Note that there is nothing to indicate that the results demonstrated in the example are not generally true. Thus, we come to the following conclusion:
The attenuation constant \(\alpha\approx\alpha_G+\alpha_R\) where \(\alpha_G\triangleq R'/2Z_0\) and \(\alpha_R\triangleq G'Z_0/2\).
Minimizing attenuation
Let us now consider if there are design choices which minimize the attenuation of coaxial cable. Since \(\alpha=\alpha_R+\alpha_G\), we may consider \(\alpha_R\) and \(\alpha_G\) independently. Let us first consider \(\alpha_G\):
\begin{align} \alpha_G &\triangleq \frac{1}{2}G'Z_0 \nonumber \\ &\approx \frac{1}{2} \cdot \frac{2\pi\sigma_s}{\ln\left(b/a\right)} \cdot \frac{1}{2\pi} \frac{\eta_0}{\sqrt{\epsilon_r}} \ln\left(b/a\right) \nonumber \\ &= \frac{\eta_0}{2}~\frac{\sigma_s}{\sqrt{\epsilon_r}}\end{align}
It is clear from this result that \(\alpha_G\) is minimized by minimizing \(\sigma_s/\sqrt{\epsilon_r}\). Interestingly the physical dimensions \(a\) and \(b\) have no discernible effect on \(\alpha_G\). Now we consider \(\alpha_R\):
\begin{align} \alpha_R &\triangleq \frac{R'}{2Z_0} \nonumber \\ & = \frac{1}{2} \frac{ \left(1/ 2\pi \delta_{ic} \sigma_{ic}\right)\left[ 1/a + C/b \right] }{ \left(1/2\pi\right) \left( \eta_0/\sqrt{\epsilon_r} \right) \ln\left(b/a\right) } \nonumber \\ &= \frac{\sqrt{\epsilon_r}}{2\eta_0\delta_{ic} \sigma_{ic}} \cdot \frac{ \left[ 1/a + C/b \right] }{ \ln\left(b/a\right) }\end{align}
Now making the substitution \(\delta_{ic} = \sqrt{2/\omega\mu_0\sigma_{ic}}\) in order to make the dependences on the constitutive parameters explicit, we find:
\[\alpha_R = \frac{1}{2\sqrt{2}\cdot\eta_0} \sqrt{\frac{\omega\mu_0 \epsilon_r}{\sigma_{ic}}} \cdot \frac{ \left[ 1/a + C/b \right] }{ \ln\left(b/a\right) } \nonumber \]
Here we see that \(\alpha_R\) is minimized by minimizing \(\epsilon_r/\sigma_{ic}\). It’s not surprising to see that we should maximize \(\sigma_{ic}\). However, it’s a little surprising that we should minimize \(\epsilon_r\). Furthermore, this is in contrast to \(\alpha_G\), which is minimized by maximizing \(\epsilon_r\). Clearly there is a tradeoff to be made here. To determine the parameters of this tradeoff, first note that the result depends on frequency: Since \(\alpha_R\) dominates over \(\alpha_G\) at sufficiently high frequency (as demonstrated in Figure \(\PageIndex{2}\)), it seems we should minimize \(\epsilon_r\) if the intended frequency of operation is sufficiently high; otherwise the optimum value is frequency-dependent. However, \(\sigma_s\) may vary as a function of \(\epsilon_r\), so a general conclusion about optimum values of \(\sigma_s\) and \(\epsilon_r\) is not appropriate.
However, we also see that \(\alpha_R\) – unlike \(\alpha_G\) – depends on \(a\) and \(b\). This implies the existence of a generally-optimum geometry. To find this geometry, we minimize \(\alpha_R\) by taking the derivative with respect to \(a\), setting the result equal to zero, and solving for \(a\) and/or \(b\). Here we go:
\[\frac{\partial}{\partial a}\alpha_R = \frac{1}{2\sqrt{2}\cdot\eta_0} \sqrt{\frac{\omega\mu_0 \epsilon_r}{\sigma_{ic}}} \cdot \frac{\partial}{\partial a} \frac{\left[ 1/a + C/b \right] }{ \ln\left(b/a\right) } \label{m0189_eDAlpha} \]
This derivative is worked out in an addendum at the end of this section. Using the result from the addendum, the right side of Equation \ref{m0189_eDAlpha} can be written as follows:
\[\frac{1}{2\sqrt{2}\cdot\eta_0} \sqrt{\frac{\omega\mu_0 \epsilon_r}{\sigma_{ic}}} \cdot \left[ \frac{-1}{a^2 \ln\left(b/a\right) } + \frac{ 1/a + C/b }{a \ln^2\left(b/a\right) } \right] \label{m0189_eDAlpha2} \]
In order for \(\partial\alpha_R/\partial a=0\), the factor in the square brackets above must be equal to zero. After a few steps of algebra, we find:
\[\ln\left(b/a\right) = 1+\frac{C}{b/a} \nonumber \]
In Case I (\(\sigma_{oc} \gg \sigma_{ic}\)), \(C=0\) so:
\[b/a=e\cong 2.72 ~~~ \mbox{(Case I)} \nonumber \]
In Case II (\(\sigma_{oc} = \sigma_{ic}\)), \(C=1\). The resulting equation can be solved by plotting the function, or by a few iterations of trial and error; either way one quickly finds
\[b/a\cong 3.59 ~~~ \mbox{(Case II)} \nonumber \]
Summarizing, we have found that \(\alpha\) is minimized by choosing the ratio of the outer and inner radii to be somewhere between \(2.72\) and \(3.59\), with the precise value depending on the relative conductivity of the inner and outer conductors.
Substituting these values of \(b/a\) into Equation \ref{m0189_eZ0cc}, we obtain:
\[Z_0 \approx \frac{59.9~\Omega}{\sqrt{\epsilon_r}} ~~\mbox{to}~~ \frac{76.6~\Omega}{\sqrt{\epsilon_r}} \label{m0189_eZ0Opt} \]
as the range of impedances of coaxial cable corresponding to physical designs that minimize attenuation.
Equation \ref{m0189_eZ0Opt} gives the range of characteristic impedances that minimize attenuation for coaxial transmission lines. The precise value within this range depends on the ratio of the conductivity of the outer conductor to that of the inner conductor.
Since \(\epsilon_r\ge 1\), the impedance that minimizes attenuation is less for dielectric-filled cables than it is for air-filled cables. For example, let us once again consider the RG-59 from Example \(\PageIndex{1}\). In that case, \(\epsilon_r\cong 2.25\) and \(C=0\), indicating \(Z_0\approx 39.9~\Omega\) is optimum for attenuation. The actual characteristic impedance of \(Z_0\) is about \(75~\Omega\), so clearly RG-59 is not optimized for attenuation. This is simply because other considerations apply, including power handling capability (addressed in Section 7.4) and the convenience of standard values (addressed in Section 7.5).
Addendum: Derivative of \(a^2\ln(b/a)\)
Evaluation of Equation \ref{m0189_eDAlpha} requires finding the derivative of \(a^2\ln(b/a)\) with respect to \(a\). Using the chain rule, we find:
\begin{align} \frac{\partial}{\partial a}\left[ a^2\ln\left(\frac{b}{a}\right) \right] = &\left[ \frac{\partial}{\partial a} a^2 \right] \ln\left(\frac{b}{a}\right) \nonumber \\ + &a^2 \left[ \frac{\partial}{\partial a} \ln\left(\frac{b}{a}\right) \right]\end{align}
Note
\[\frac{\partial}{\partial a} a^2 = 2a \nonumber \]
and
\begin{align} \frac{\partial}{\partial a} \ln\left(\frac{b}{a}\right) &= \frac{\partial}{\partial a} \left[ \ln\left(b\right) - \ln\left(a\right) \right] \nonumber \\ &=-\frac{\partial}{\partial a}\ln\left(a\right) \nonumber \\ &=-\frac{1}{a}\end{align}
So:
\begin{align} \frac{\partial}{\partial a}\left[ a^2\ln\left(\frac{b}{a}\right) \right] &= \left[ 2a \right] \ln\left(\frac{b}{a}\right) + a^2 \left[ -\frac{1}{a} \right] \nonumber \\ &= \boxed{ 2a \ln\left(\frac{b}{a}\right) - a } \end{align}
This result is substituted for \(a^2\ln(b/a)\) in Equation \ref{m0189_eDAlpha} to obtain Equation \ref{m0189_eDAlpha2}.


7.6: Introduction to Romantic Opera
This section contains information on major trends and composers in Romantic opera. We will be focusing our study on developments in the Italian and German operatic traditions. There was also a good deal of innovation taking place in Paris in the 19th century, but Verdi (Italian) and Wagner (German) were the most influential composers of the era so we’ll limiting our reading to those two national styles.
This section includes the following pages:
- Slide Show: Romantic Opera
- Early Romantic Opera
- Later Romantic Opera
- Trends in German and Italian Opera
- Giuseppe Verdi
- Rigoletto
- La donna e mobile
- Verdi’s Requiem
- Richard Wagner
- Tristan und Isolde
- Liebestod
- Verismo
- Giacomo Pucinni
- La Boheme
- Che gelida manina
- Si, mi chiamano Mimi
Contributors and Attributions
CC licensed content, Original
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Provided by : Lumen Learning. License : CC BY: Attribution


12.5: Creating an Access Table
Learning Outcomes
- Create a new table
You can add new tables to an existing database by using the commands in the Tables group on the Create tab. Access will automatically create a new table with a table structure built in. Access will also set the field types based on the data entered. The new table will be automatically displayed in Datasheet view, allowing you to take immediate action, i.e., enter change field names, set properties, and enter data directly in to the table.
An alternate way to create a new table is to select Table Design in the Tables group. Access will create a new table and display it in Design View. This approach provides a more traditional albeit manual method of setting the details of the table.
Learn More
For detailed, step-by-step instructions Microsoft Access Help is an excellent resource and can be found by following this link: Creating Tables Access Help
Contributors and Attributions
- Creating an Access Table. Authored by : Robert Danielson. Provided by : Lumen Learning. License : CC BY: Attribution