text:
## Mean or Expected Value and Standard Deviation

### Learning Outcomes

• Calculate and interpret expected values
• Classify discrete word problems by their distributions

The expected value is often referred to as the “long-term” average or mean. This means that over the long term of doing an experiment over and over, you would expect this average.

You toss a coin and record the result. What is the probability that the result is heads? If you flip a coin two times, does probability tell you that these flips will result in one heads and one tail? You might toss a fair coin ten times and record nine heads. Probability does not describe the short-term results of an experiment. It gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times.
In his experiment, Pearson illustrated the Law of Large Numbers.

The Law of Large Numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). When evaluating the long-term results of statistical experiments, we often want to know the “average” outcome. This “long-term average” is known as the mean or expected value of the experiment and is denoted by the Greek letter μ. In other words, after conducting many trials of an experiment, you would expect this average value.

#### Note

To find the expected value or long term average, μ, simply multiply each value of the random variable by its probability and add the products.

### Example

A men’s soccer team plays soccer zero, one, or two days a week. The probability that they play zero days is 0.2, the probability that they play one day is 0.5, and the probability that they play two days is 0.3. Find the long-term average or expected value, μ, of the number of days per week the men’s soccer team plays soccer.

To do the problem, first let the random variable X = the number of days the men’s soccer team plays soccer per week. X takes on the values 0, 1, 2. Construct a PDF table adding a column xP(x). In this column, you will multiply each x value by its probability.

Expected Value Table. This table is called an expected value table. The table helps you calculate the expected value or long-term average.

x P(x) xP(x)
0 0.2 (0)(0.2) = 0
1 0.5 (1)(0.5) = 0.5
2 0.3 (2)(0.3) = 0.6

Add the last column xP(x) to find the long term average or expected value: (0)(0.2) + (1)(0.5) + (2)(0.3) = 0 + 0.5 + 0.6 = 1.1.

### Example

The expected value is 1.1. The men’s soccer team would, on the average, expect to play soccer 1.1 days per week. The number 1.1 is the long-term average or expected value if the men’s soccer team plays soccer week after week after week. We say μ = 1.1.

Find the expected value of the number of times a newborn baby’s crying wakes its mother after midnight. The expected value is the expected number of times per week a newborn baby’s crying wakes its mother after midnight. Calculate the standard deviation of the variable as well.

You expect a newborn to wake its mother after midnight 2.1 times per week, on the average.

x P(x) xP(x) (xμ)2P(x)
0 $\displaystyle{P}{({x}={0})}=\frac{{2}}{{50}}$ $\displaystyle{({0})}{(\frac{{2}}{{50}})}={0}$ $\displaystyle { ( { 0 } - { 2.1 } ) } ^{ { 2 } } \cdot { 0.04}={0.1764}$
1 $\displaystyle{P}{({x}={1})}=\frac{{11}}{{50}}$ $\displaystyle{({1})}{(\frac{{11}}{{50}})}=\frac{{11}}{{50}}$ $\displaystyle{({1}-{2.1})}^{{2}}\cdot{0.22}={0.2662}$
2 $\displaystyle{P}{({x}={2})}=\frac{{23}}{{50}}$ $\displaystyle{({2})}{(\frac{{23}}{{50}})}=\frac{{46}}{{50}}$ $\displaystyle{({2}-{2.1})}^{{2}}\cdot{0.46}={0.0046}$
3 $\displaystyle{P}{({x}={3})}=\frac{{9}}{{50}}$ $\displaystyle{({3})}{(\frac{{9}}{{50}})}=\frac{{27}}{{50}}$ $\displaystyle{({3}-{2.1})}^{{2}}\cdot{0.18}={0.1458}$
4 $\displaystyle{P}{({x}={4})}=\frac{{4}}{{50}}$ $\displaystyle{({4})}{(\frac{{4}}{{50}})}=\frac{{16}}{{50}}$ $\displaystyle{({4}-{2.1})}^{{2}}\cdot{0.08}={0.2888}$
5 $\displaystyle{P}{({x}={5})}=\frac{{1}}{{50}}$ $\displaystyle{({5})}{(\frac{{1}}{{50}})}=\frac{{5}}{{50}}$ $\displaystyle{({5}-{2.1})}^{{2}}\cdot{0.02}={0.1682}$

Add the values in the third column of the table to find the expected value of X:

μ = Expected Value = $\displaystyle\frac{{105}}{{50}}$ = 2.1

Use μ to complete the table. The fourth column of this table will provide the values you need to calculate the standard deviation. For each value x, multiply the square of its deviation by its probability. (Each deviation has the format xμ).

Add the values in the fourth column of the table:

0.1764 + 0.2662 + 0.0046 + 0.1458 + 0.2888 + 0.1682 = 1.05

The standard deviation of X is the square root of this sum: $\displaystyle \sigma = \sqrt{{1.05}} \simeq {1.0247}$

### try it

A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. What is the expected value?

x P(x)
0 $\displaystyle{P}{({x}={0})}=\frac{{4}}{{50}}$
1 $\displaystyle{P}{({x}={1})}=\frac{{8}}{{50}}$
2 $\displaystyle{P}{({x}={2})}=\frac{{16}}{{50}}$
3 $\displaystyle{P}{({x}={3})}=\frac{{14}}{{50}}$
4 $\displaystyle{P}{({x}={4})}=\frac{{6}}{{50}}$
5 $\displaystyle{P}{({x}={5})}=\frac{{2}}{{50}}$

The expected value is 2.24,

$\displaystyle{0})\frac{{4}}{{50}}+{({1})}\frac{{4}}{{50}}+{({2})}\frac{{16}}{{50}}+{({3})}\frac{{14}}{{50}}+{({4})}\frac{{6}}{{50}}+{({5})}\frac{{2}}{{50}}={0}+\frac{{8}}{{50}}+\frac{{32}}{{50}}+\frac{{42}}{{50}}+\frac{{24}}{{50}}+\frac{{10}}{{50}}=\frac{{116}}{{50}}={2.32}$

### Example

Suppose you play a game of chance in which five numbers are chosen from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A computer randomly selects five numbers from zero to nine with replacement. You pay $2 to play and could profit$100,000 if you match all five numbers in order (you get your $2 back plus$100,000). Over the long term, what is your expected profit of playing the game?

To do this problem, set up an expected value table for the amount of money you can profit.

Let X = the amount of money you profit. The values of x are not 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since you are interested in your profit (or loss), the values of x are 100,000 dollars and −2 dollars.

To win, you must get all five numbers correct, in order. The probability of choosing one correct number is
$\displaystyle\frac{{1}}{{10}}$ because there are ten numbers. You may choose a number more than once. The probability of choosing all five numbers correctly and in order is

$\displaystyle{(\frac{{1}}{{10}})}{(\frac{{1}}{{10}})}{(\frac{{1}}{{10}})}{(\frac{{1}}{{10}})}{(\frac{{1}}{{10}})}={({1})}{({10}^{{-{5}}})}={0.00001}$

Therefore, the probability of winning is 0.00001 and the probability of losing is $\displaystyle{1}-{0.00001}={0.99999}$

The expected value table is as follows:

x P(x) xP(x)
Loss –2 0.99999 (–2)(0.99999) = –1.99998
Profit 100,000 0.00001 (100000)(0.00001) = 1

Αdd the last column. –1.99998 + 1 = –0.99998

Since –0.99998 is about –1, you would, on average, expect to lose approximately $1 for each game you play. However, each time you play, you either lose$2 or profit $100,000. The$1 is the average or expected LOSS per game after playing this game over and over.

### try it

You are playing a game of chance in which four cards are drawn from a standard deck of 52 cards. You guess the suit of each card before it is drawn. The cards are replaced in the deck on each draw. You pay $1 to play. If you guess the right suit every time, you get your money back and$256. What is your expected profit of playing the game over the long term?

Let X = the amount of money you profit. The x-values are –$1 and$256.

The probability of guessing the right suit each time is

$\displaystyle{(\frac{{1}}{{4}})}{(\frac{{1}}{{4}})}{(\frac{{1}}{{4}})}{(\frac{{1}}{{4}})}=\frac{{1}}{{256}}={0.0039}$

The probability of losing is
$\displaystyle{1}-\frac{{1}}{{256}}=\frac{{255}}{{256}}={0.9961}$

(0.0039)256 + (0.9961)(–1) = 0.9984 + (–0.9961) = 0.0023 or 0.23 cents.

### EXAMPLE

Suppose you play a game with a biased coin. You play each game by tossing the coin once.
P(heads) = $\displaystyle\frac{{2}}{{3}}$. If you toss a head, you pay $6. If you toss a tail, you win$10. If you play this game many times, will you come out ahead?

1. Define a random variable X.
2. Complete the following expected value table.
x ____ ____
WIN 10 $\displaystyle \frac{{1}}{{3}}$ ____
LOSE ____ ____ $\displaystyle \frac{{-12}}{{3}}$
3. What is the expected value, μ? Do you come out ahead?

Solution:

1. X = amount of profit
2. x P(x) xP(x)
WIN 10 $\displaystyle \frac{{1}}{{3}}$ $\displaystyle \frac{{10}}{{3}}$
LOSE –6 $\displaystyle \frac{{2}}{{3}}$ $\displaystyle \frac{{-12}}{{3}}$
3. Add the last column of the table. The expected value μ = $\displaystyle \frac{{-2}}{{3}}$. You lose, on average, about 67 cents each time you play the game so you do not come out ahead.

### try it

Suppose you play a game with a spinner. You play each game by spinning the spinner once.
P(red) = $\displaystyle\frac{{2}}{{5}}$. If you land on red, you pay $10. If you land on blue, you don’t pay or win anything. If you land on green, you win$10. Complete the following expected value table.

$\displaystyle-\frac{{20}}{{5}}$  $\displaystyle\frac{{2}}{{5}}$

x P(x)
Red
Blue
Green 10

Here’s the completed table:

x P(x) xP(x)
Red –10 $\displaystyle\frac{{2}}{{5}}$ $\displaystyle-\frac{{20}}{{5}}$
Blue 0 $\displaystyle\frac{{2}}{{5}}$ $\displaystyle\frac{{0}}{{5}}$
Green 10 $\displaystyle\frac{{1}}{{5}}$ $\displaystyle\frac{{10}}{{5}}$

Like data, probability distributions have standard deviations. To calculate the standard deviation (σ) of a probability distribution, find each deviation from its expected value, square it, multiply it by its probability, add the products, and take the square root. To understand how to do the calculation, look at the table for the number of days per week a men’s soccer team plays soccer. To find the standard deviation, add the entries in the column labeled (xμ)2P(x) and take the square root.

x P(x) xP(x) (xμ)2P(x)
0 0.2 (0)(0.2) = 0 (0 – 1.1)2(0.2) = 0.242
1 0.5 (1)(0.5) = 0.5 (1 – 1.1)2(0.5) = 0.005
2 0.3 (2)(0.3) = 0.6 (2 – 1.1)2(0.3) = 0.243

Add the last column in the table. 0.242 + 0.005 + 0.243 = 0.490. The standard deviation is the square root of 0.49, or σ = $\displaystyle\sqrt{{0.49}}$ = 0.7

Generally for probability distributions, we use a calculator or a computer to calculate μ and σ to reduce roundoff error. For some probability distributions, there are short-cut formulas for calculating μ and σ.

### Example

Toss a fair, six-sided die twice. Let X = the number of faces that show an even number. Construct a table like the one in the Try It above and calculate the mean μ and standard deviation σ of X.

Solution:

Tossing one fair six-sided die twice has the same sample space as tossing two fair six-sided dice. The sample space has 36 outcomes:

 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

Use the sample space to complete the following table:

Calculating μ and σ.

x P(x) xP(x) (xμ)2P(x)
0 $\displaystyle\frac{{9}}{{36}}$ 0 $\displaystyle{({0}-{1})}^{{2}} \cdot \frac{{9}}{{36}}=\frac{{9}}{{36}}$
1 $\displaystyle\frac{{18}}{{36}}$ $\displaystyle\frac{{18}}{{36}}$ $\displaystyle{({1}-{1})}^{{2}} \cdot \frac{{18}}{{36}}={0}$
2 $\displaystyle\frac{{9}}{{36}}$ $\displaystyle\frac{{18}}{{36}}$ $\displaystyle{({2}-{1})}^{{2}} \cdot \frac{{9}}{{36}}=\frac{{9}}{{36}}$

Add the values in the third column to find the expected value: $\displaystyle \mu =\frac{{36}}{{36}}={1}$. Use this value to complete the fourth column.

Add the values in the fourth column and take the square root of the sum: $\displaystyle \sigma =\sqrt {{\frac{{18}}{{36}}}} \simeq {0.7071}$.

### Example

On May 11, 2013 at 9:30 PM, the probability that moderate seismic activity (one moderate earthquake) would occur in the next 48 hours in Iran was about 21.42%. Suppose you make a bet that a moderate earthquake will occur in Iran during this period. If you win the bet, you win $50. If you lose the bet, you pay$20. Let X = the amount of profit from a bet.

P(win) = P(one moderate earthquake will occur) = 21.42%

P(loss) = P(one moderate earthquake will not occur) = 100% – 21.42%

If you bet many times, will you come out ahead? Explain your answer in a complete sentence using numbers. What is the standard deviation of
X? Construct a table similar to the one in Example 5 to help you answer these questions.

Solution:

x P(x) x(Px) (xμ)2P(x)
win 50 0.2142 10.71 [50 – (–5.006)]2(0.2142) = 648.0964
loss –20 0.7858 –15.716 [–20 – (–5.006)]2(0.7858) = 176.6636

Mean = Expected Value = 10.71 + (–15.716) = –5.006.

If you make this bet many times under the same conditions, your long term outcome will be an average
loss of $5.01 per bet. $\displaystyle \text{Standard Deviation} = \sqrt{{{648.0964}+{176.6636}}} \simeq {28.7186}$ ### try it On May 11, 2013 at 9:30 PM, the probability that moderate seismic activity (one moderate earthquake) would occur in the next 48 hours in Japan was about 1.08%. As in Example 6, you bet that a moderate earthquake will occur in Japan during this period. If you win the bet, you win$100. If you lose the bet, you pay $10. Let X = the amount of profit from a bet. Find the mean and standard deviation of X. x P(x) x ⋅ (Px) (xμ)2P(x) win 100 0.0108 1.08 [100 – (–8.812)]2 ⋅ 0.0108 = 127.8726 loss –10 0.9892 –9.892 [–10 – (–8.812)]2 ⋅ 0.9892 = 1.3961 Mean = Expected Value = μ = 1.08 + (–9.892) = –8.812 If you make this bet many times under the same conditions, your long term outcome will be an average loss of$8.81 per bet.

$\displaystyle \text{Standard Deviation} = \sqrt{{{127.7826}+{1.3961}}} \simeq {11.3696}$

Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson. Most elementary courses do not cover the geometric, hypergeometric, and Poisson. Your instructor will let you know if he or she wishes to cover these distributions.

A probability distribution function is a pattern. You try to fit a probability problem into a pattern or distribution in order to perform the necessary calculations. These distributions are tools to make solving probability problems easier. Each distribution has its own special characteristics. Learning the characteristics enables you to distinguish among the different distributions.

## References

Class Catalogue at the Florida State University. Available online at https://apps.oti.fsu.edu/RegistrarCourseLookup/SearchFormLegacy (accessed May 15, 2013).

“World Earthquakes: Live Earthquake News and Highlights,” World Earthquakes, 2012. http://www.world-earthquakes.com/index.php?option=ethq_prediction (accessed May 15, 2013).

## Concept Review

The expected value, or mean, of a discrete random variable predicts the long-term results of a statistical experiment that has been repeated many times. The standard deviation of a probability distribution is used to measure the variability of possible outcomes.

## Formula Review

Mean or Expected Value:
$\displaystyle\mu=\sum_{{{x}in{X}}}{x}{P}{({x})}$

Standard Deviation:
$\displaystyle\sigma=\sqrt{{\sum_{{{x}in{X}}}{({x}-\mu)}^{{2}}{P}{({x})}}}$


text:
## R a commutative ring and A an ideal in R with \$A=m_1···m_r=n_1···n_s\$ with \$m_i\$ distinct maximal ideals and all the \$n_j\$ distinct maximal ideals.

Let R be a commutative ring and A be an ideal in R satisfying $$A=m_1···m_r =n_1···n_s$$ with all the $$m_i$$ distinct maximal ideals and all the $$n_j$$ distinct maximal ideals. Show that $$r = s$$ and there exists a $$σ ∈ Sr$$ satisfying $$m_i = n_σ(i)$$ for all i.

I know that maximal ideal implies it being prime, and the product is contained in $$m_i$$ and $$n_j$$ for all $$i$$ and $$j$$, but I’m not sure how to proceed further.

format: plain


text:
Revenue Directional Distance Function model

The revenue directional distance function model is computed by solving a directional distance function DEA model for the technical efficiency.

The directional distance function DDF, projects firm $\left( {{\mathbf{x}_o,\mathbf{y}_{o}}} \right)$ to the production frontier in the pre-assigned direction $\mathbf{g}= {\left({{\mathbf{g_{x}^-},\mathbf{g^{+}_y}}} \right)\neq\mathbf{0}_{M+N}}$, $\mathbf{g^{-}_{x}}\mathbb{\in R}^M$ and $\mathbf{g^+_{y}}\mathbb{\in R}^N$. Decomposing revenue inefficiency requires defining the output-oriented directional distance functions. Departing from the general definition this implies setting the output or input directional vectors to zero; i.e., $\mathbf{g}= \left({{\mathbf{g_{x}^-}},{\mathbf{g_{y}^+}}} \right)=\:$$\left({{\textbf{0},\textbf{g}_{\textbf{y}}^+}} \right), respectively. Therefore the output-oriented directional distance functions defines as follows: $$$T{{I}_{DDF(O)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}}, \textbf{0},\textbf{g}_{\textbf{y}}^{{+}} \right)=\,\max \,\left\{ \beta_O :\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}}+{{\beta_O}}\textbf{g}_{\textbf{y}}^{{+}} \right)\in P(\textbf{x}_o),\ {{\beta_O}}\ge 0 \right\}.$$$ The linear programs that allows calculating this measure is: $$$\begin{split} & T{{I}_{DDF(O)}}\left( {{\mathbf{x}}_{o}},{{\mathbf{y}}_{o}} \right)=\underset{{{\beta_O }_{{}}},\lambda }{\mathop{\text{max}}}\,\beta_O \,\, \\ & s.t. \quad \sum\limits_{j\in J}^{{}}{{{\lambda }_{j}}{{x}_{jm}}}\le {{x}_{om}},\,\,\,m=1,...,M \\ & \quad \quad \sum\limits_{j\in J}^{{}}{{{\lambda }_{j}}{{y}_{jn}}}\ge {{y}_{on}}+\beta g_{{{y}_{n}}}^{{+}},\,\,n=1,...,N \\ & \quad \quad \sum\limits_{j\in J}^{{}}{{{\lambda }_{j}}}=1 \\ & \quad \quad {{\lambda }_{j}}\ge 0,\,\,j\in J. \end{split}$$$ Once again, as in the graph case already presented for the Profit Directional Distance Function model, the choice of directional vector corresponds to the researcher. Customarily, to keep consistency with the radial models, the observed amounts of outputs set the direction: \mathbf{g}= \left({{\mathbf{g_{x}^-}},{\mathbf{g_{y}^+}}} \right)=\:$$\left({\textbf{0},{\mathbf{y}_o}}\right)$. In this case it can be shown that the directional model nests Radial Output Oriented Model. Indeed, if $\left({{\mathbf{g_{x}^-},\mathbf{g^+_y}}} \right)=\:\left( {\mathbf{0},{\mathbf{y}_o}} \right)$, then $\beta_O^{*}=\phi^*-1$ (in the Revenue Radial model. However, other choices are available, which are included as options in BenchmarkingEconomicEfficiency.jl–see the documentation below accompanying this function.

The notion of Nerlovian revenue inefficiency corresponds to the decompostion of economic efficiency based on the output-oriented directional distance function: $R{{I}_{DDF\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{y}^+},{\tilde{\textbf{p}}} \right)$ = $T{{I}_{DDF\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o},\mathbf{g_{y}^+}} \right)$ + $A{{I}_{DDF\left( O \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{y}^+}, {\tilde{\textbf{p}}} \right)$. This results in the following expression:

$$$\underbrace{\frac{R \left( \mathbf{x}_o,\mathbf{p} \right) - \sum\limits_{n=1}^{N}{{{p}_{n}}{{y}_{om}}} }{\sum\limits_{n=1}^{N}{p_{n}^{{}}g_{on}^{+}}}}_{\text{Norm. Revenue Inefficiency}}=\underbrace{\beta_{{I}}^{*}}_{\text{Technical Inefficiency}}+\underbrace{A{{I}_{DDF\left( I \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{y}^+},\tilde{\textbf{p}} \right)}_{\text{Norm. Allocative Inefficiency}} \ge 0.$$$

Reference

Chapter 8 in Pastor, J.T., Aparicio, J. and Zofío, J.L. (2022) Benchmarking Economic Efficiency: Technical and Allocative Fundamentals, International Series in Operations Research and Management Science, Vol. 315, Springer, Cham.

Example

In this example we compute the revenue efficiency directional distance function measure using the option Gy=:Monetary.

using BenchmarkingEconomicEfficiency

X = [1; 1; 1; 1; 1; 1; 1; 1];

Y = [7 7; 4 8; 8 4; 3 5; 3 3; 8 2; 6 4; 1.5 5];

P = [1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1];

revenueddf = dearevenueddf(X, Y, P, Gy = :Monetary)
Revenue DDF DEA Model
DMUs = 8; Inputs = 1; Outputs = 2
Orientation = Output; Returns to Scale = VRS
Gy = Monetary
─────────────────────────────────
Revenue  Technical  Allocative
─────────────────────────────────
1      0.0       0.0         0.0
2      2.0       0.0         2.0
3      2.0       0.0         2.0
4      6.0       5.0         1.0
5      8.0       8.0         0.0
6      4.0       0.0         4.0
7      4.0       3.0         1.0
8      7.5       5.75        1.75
─────────────────────────────────

Estimated economic, technical and allocative efficiency scores are returned with the efficiency function:

efficiency(revenueddf, :Economic)
8-element Vector{Float64}:
0.0
2.0
2.0
6.0
8.0
4.0
4.0
7.5
efficiency(revenueddf, :Technical)
8-element Vector{Float64}:
0.0
0.0
0.0
5.0
8.0
0.0
3.0000000000000004
5.749999999999999
efficiency(revenueddf, :Allocative)
8-element Vector{Float64}:
0.0
2.0
2.0
1.0
0.0
4.0
0.9999999999999996
1.7500000000000009

dearevenueddf Function Documentation

BenchmarkingEconomicEfficiency.dearevenueddfFunction
dearevenueddf(X, Y, P; Gy)

Compute revenue efficiency using directional distance function data envelopment analysis for inputs X, outputs Y and price of outputs P.

Direction specification:

The direction Gy can be one of the following symbols.

• :Ones: use ones.
• :Observed: use observed values.
• :Mean: use column means.
• :Monetary: use direction so that profit inefficiency is expressed in monetary values.

Alternatively, a vector or matrix with the desired directions can be supplied.

Optional Arguments

• rts=:VRS: chooses variable returns to scale. For constant returns to scale choose :CRS.
• monetary=false: decomposition in normalized terms. Monetary terms if true.
• names: a vector of strings with the names of the decision making units.
source


text:
Tree study problems

Problem 1

Let's make the following definition:

If T is a binary tree with root r, then its rank is
• 0 if r has no children
• 1+q if r has two children, both with rank q
• otherwise, the maximum rank of any of the children

(a) Compute the rank of each subtree in the following tree:

Hint: for each subtree, write the rank on its root. Start with the leaf nodes and work upwards in the tree.

Check your solution before you do parts (b) and (c), to make sure you correctly understand the definition of rank.

(b) For a given height h, what shape of tree has the largest rank? smallest rank?

Check your solution before you do part (c).

(c) Use induction to prove the following claim:

Claim: a tree with rank q has at least $$2^q$$ leaves

Check the outline of your solution.

Problem 2

Let's define the grammar G as follows:

• Start symbols: S
• Terminals: a
• Rules:
$$S \ \rightarrow \ a \ \mid\ aB$$
$$B \ \rightarrow \ aaS \ \mid\ BB$$

Notice a tree generated by G has a terminal string of the form $$a^n$$ i.e. n copies of the terminal value a. Prove that n is congruent to 1 (mod 3).

Hint 1: you can handle the modular arithmetic informally. Concentrate on building a good outline for the tree induction.

Hint 2: let's define the notation L(T) to be the length of the terminal string of tree T.

Hint 3: you may wish to strengthen the claim, i.e. modify your grammar and claim to cover a wider range of trees. (WTF? Here are some hints.)

Check against the solution.

Problem 3

A free tree is a connected (undirected) graph that contains no cycles. So it has a tree-like structure, but no root and no left-to-right ordering on its nodes. A free tree spans a graph G if it is a subgraph of G and contains all of G's nodes. It is also called a spanning tree of G,

Let us define a function f as follows: $$$$f(n) = \begin{cases} 1 \quad when \quad n = 0 \\ \\ 2^{n-1} (f(n-1))^2 \quad when \quad n \geq 1 \\ \end{cases}$$$$

Prove that for any hypercube $$Q_{n}$$ ($$n \geq 0$$) there are at least f(n) distinct free trees that span $$Q_{n}$$.

Stuck? Here are some hints.

Check against the solution.

Problem 4

Flatlander Chocolates makes rectangular chocolate bars, each divided into a p by q grid of squares. We can break a chocolate bar in two along one of the grid lines. Prove that it takes pq-1 breaks to divide a chocolate bar into individual squares.

Stuck? Here are some hints.

Check against the solution.


text: [32] The LSA method states that similarity between words is reflected through their co-occurrence in a local context. However, associations are often more clearly represented as an NÃN matrix, where N is the number of items in memory. Visual vs. verbal and perceptual vs. functional information are examples of information types. In particular, ACT models memory as a set of related symbolic chunks which may be accessed by retrieval cues. {\displaystyle \mathbf {M} } Recent evidence supports the idea that the temporal pole bilaterally is the convergence zone for unimodal semantic representations into a multimodal representation. Still other research suggests that both semantic memory and episodic memory are part of a singular declarative memory system, yet represent different sectors and parts within the greater whole.  Modality-based theories also assume that if there is damage to modality-specific knowledge, then all the categories that fall under it will be damaged. Semantic memory definition: the recollection of facts and concepts | Meaning, pronunciation, translations and examples Learning of associations is generally believed to be a Hebbian process; that is, whenever two items in memory are simultaneously active, the association between them grows stronger, and the more likely either item is to activate the other. In. M This provides an economy of representation in that properties are only stored at the category level at which they become essential, that is, at which point they become critical features (see below). In WAS, "words that have similar associative structures are placed in similar regions of space. [24], Early work in perceptual and conceptual categorization assumed that categories had critical features and that category membership could be determined by logical rules for the combination of features. NY 10036. = has occurred (this is obtained simply by dividing the raw frequency, Certain experts are still arguing whether or not the two types of memory are from distinct systems or whether the neural imaging makes it appear that way as a result of the activation of different mental processes during retrieval.[48]. Instead of being representations in modality-specific systems, semantic memory representations had previously been viewed as redescriptions of modality-specific states. Laura Eileen Matzen. Subjects were presented with 60 words (one at a time) and were asked different questions. Semantic memory definition at Dictionary.com, a free online dictionary with pronunciation, synonyms and translation. Each cell in the matrix is then transformed according to the equation: M Non-related words would fall into this group. The defining feature of a semantic network is that its links are almost always directed (that is, they only point in one direction, from a base to a target) and the links come in many different types, each one standing for a particular relationship that can hold between any two nodes. For example, learning how to use the phone may start out as an episodic memory of dialing a phone number on a toy telephone. The level of consciousness associated with semantic memory is noetic because it is independent of context encoding and personal relevance (Tulving, 1985, 2001). However, the most common presenting symptoms are in the verbal domain (with loss of word meaning). [51] With Alzheimer's disease in particular, interactions with semantic memory produce different patterns in deficits between patients and categories over time which is caused by distorted representations in the brain. This chapter reviews evidence that conceptual knowledge about concrete objects is acquired through experience with them, thereby … These theories assume that natural selective pressures have caused neural circuits specific to certain domains to be formed, and that these are dedicated to problem-solving and survival. Retrieval from semantic memory. Semantic memory is the recollection of facts gathered from the time we are young. i Semantic memory is one of the two types of explicit memory (or declarative memory) (our memory of facts or events that is explicitly stored and retrieved). , where Temporal lobe damage affecting the lateral and medial cortexes have been related to semantic impairments. In his book titled "Episodic and Semantic Memory", Endel Tulving adopted the term "semantic" from linguists to refer to a system of memory for "words and verbal symbols, their meanings and referents, the relations between them, and the rules, formulas, or algorithms for influencing them. [40] In LSA, a T Ã D matrix is constructed from a text corpus where T is the number of terms in the corpus and D is the number of documents (here "context" is interpreted as "document" and only wordsâor word phrasesâare considered as items in memory). Instead, what is stored in semantic memory is the \"gist\" of experience, an abstract structure that applies to a wide variety of experiential objects and which may be said to delineate categorical and functional relationships between such objects. "), when the relevant nodes are very far apart in the network.[20]. A variety of studies have been done in an attempt to determine the effects on varying aspects of semantic memory. The first category consists of animate objects with "animals" being the most common deficit. d Visit our corporate site. [21] Processing in a semantic network often takes the form of spreading activation (see above). The patient (hereafter called DRN) had marked naming and word comprehension difficulties. The semantic feature-comparison model, proposed by Smith, Shoben, and Rips (1974),[24] describes memory as being composed of feature lists for different concepts. By using these neuroimaging techniques researchers can observe the brain activity of participants while they perform cognitive tasks. It refers to that factual knowledge and matters that a person shares with others and are not wholly dependent on the p… For an example of a computational implementation of semantic networks in knowledge representation, see Cravo and Martins (1993). d Furthermore, properties are stored at the highest category level to which they apply. , t The type of deficit, however, does not indicate a lack of conceptual knowledge associated with that category. : long-term memory of facts, information, and meanings that is not related to any specific event personally experienced in the past Perhaps the most important distinction recently drawn is that between semantic memory (the memory for facts like that the Acropolis is in Athens) and episodic memory (the recall of events—those that happened to you, not those you've been told about). 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[49], Modality refers to a semantic category of meaning which has to do with necessity and probability expressed through language. [23], Feature models view semantic categories as being composed of relatively unstructured sets of features. d This would lead to the conclusion that any type of lesion in the temporal lobe, depending on severity and location, has the potential to cause semantic deficits. TLC is an instance of a more general class of models known as semantic networks. Sandra L. Zoccoli. This latency is used in measuring the response time of the ACT model, to compare it to human performance.[37]. Comparing close and distant groups shows that in access disorders semantic relatedness had a negative effect. See below for specific operationalizations of associative models. The set of associations among a collection of items in memory is equivalent to the links between nodes in a network, where each node corresponds to a unique item in memory. Neuroimaging studies also suggest a distinction between semantic processing and sensorimotor processing. The second category consists of inanimate objects with two subcategories of "fruits and vegetables" (biological inanimate objects) and "artifacts" being the most common deficits. Semantic networks see the most use in models of discourse and logical comprehension, as well as in Artificial Intelligence. According to this view, the relations between categories would not be directly retrieved, they would be indirectly computed. [2] This general knowledge (facts, ideas, meaning and concepts) is intertwined in experience and dependent on culture. (1995). In other words, the deficit tends to be worse with living things as opposed to non-living things. This version performed comparably to humans in many tasks, but failed to predict that people would respond faster to questions regarding more typical category instances than those involving less typical instances. While ACT is a model of cognition in general, and not memory in particular, it nonetheless posits certain features of the structure of memory, as described above. Journal of Clinical and Experimental Neuropsychology, "Why Personality and Psychopathology Are Correlated: A Developmental Perspective Is a First Step but More Is Needed", "A spreading-activation theory of semantic processing", "Natural categories: Well defined or fuzzy sets? | In storage disorders, you do not see an inconsistent response to specific items like you do in refractory access disorders. This leads to the conclusion that there are significant gender differences when it comes to category specific semantic deficits, and that the patient will tend to be impaired in categories that had less existing knowledge to begin with.[56]. The buffer size is defined as r, and not a fixed number, and as items are rehearsed in the buffer the associative strengths grow linearly as a function of the total time inside the buffer. New York, t However, by performing the SVD and reducing the number of dimensions in the matrix, the context vectors of "cat" and "dog"âwhich would be very similarâwould migrate toward one another and perhaps merge, thus allowing "cat" and "dog" to act as retrieval cues for each other, even though they may never have co-occurred. P "[34], The ACT (Adaptive Control of Thought)[35] (and later ACT-R (Adaptive Control of Thought-Rational)[36]) theory of cognition represents declarative memory (of which semantic memory is a part) with "chunks", which consist of a label, a set of defined relationships to other chunks (i.e., "this is a _", or "this has a _"), and any number of chunk-specific properties. Despite the unequivocal evidence of semantic deficit… Researchers generally agree that there is typically a gradual transition from episodic to semantic memory, in which episodic memory reduces its sensitivity and association to particular events, so that the information can be stored as general knowledge. The four primary types of encoding are visual, acoustic, elaborative, and semantic. The brain encodes multiple inputs such as words and pictures to integrate and create a larger conceptual idea by using amodal views (also known as amodal perception). The essence of semantic memory is that its contents are not tied to any particular instance of experience, as in episodic memory. Nodes may also store negations of the properties of their superordinate nodes (i.e., "NOT-can fly" would be stored with "penguin"). The semantic memory is a derivative of episodic memory to capture facts and figures. ", "Going beyond a single list: Modeling the effects of prior experience on episodic free recall", "Word Association Spaces for Predicting Semantic Similarity Effects in Episodic Memory", "Producing high-dimensional semantic spaces from lexical co-occurrence", "Differential Effects of Early Hippocampal Pathology on Episodic and Semantic Memory", "Common and Unique Neural Activations in Autobiographical, Episodic, and Semantic Retrieval", "Neural Basis of Category-specific Semantic Deficits for Living Things: Evidence from semantic dementia, HSVE and a Neural Network Model", Semantic knowledge and semantic representations, http://www.newscientist.com/article.ns?id=dn10012, http://www.semantikoz.com/blog/2008/02/25/hyperspace-analogue-to-language-hal-introduction/, The Magical Number Seven, Plus or Minus Two, https://en.wikipedia.org/w/index.php?title=Semantic_memory&oldid=993945567, Short description is different from Wikidata, Articles needing expert attention from January 2014, Psychology articles needing expert attention, Creative Commons Attribution-ShareAlike License, Greater familiarity with flowers and elderly. Other researchers believe the hippocampus is only involved in episodic memory and spatial cognition. ′ Â© Other disorders that affect semantic memory - such as Alzheimer's disease - has been observed clinically as errors in naming, recognizing, or describing objects. Publisher-Waxmann Verlag. The recall of semantic memory is largely automatic with prompting. [30] The SAM model contains a short-term store (STS) and long-term store (LTS), where STS is a briefly activated subset of the information in the LTS. Semantic memory refers to the memory of meaning, understanding, general knowledge about the world, and other concept-based knowledge unrelated to specific experiences. The Hyperspace Analogue to Language (HAL) model[41][42] considers context only as the words that immediately surround a given word. [1] Semantic memory refers to general world knowledge that we have accumulated throughout our lives. This page was last edited on 13 December 2020, at 09:22. i Wietske Vonk. Semantic information is gleaned by performing a statistical analysis of this matrix. 2000. This strongly suggests that encoding of information leading to semantic memory does not have its physiological basis in the hippocampus.[43]. Semantic Memory Semantic memory is one of the three types of long-term memory (the others are episodic and procedural) in which we store general world knowledge like facts, ideas, words, problem solving, etc. In cognitive neuropsychology, there is a In this case, damage to the visual modality would result in a deficit for all biological objects with no deficits restricted to the more specific categories. It holds generic information that is more than likely acquired across various contexts and is used across different situations. SarÃ Laatu. Semantic networks. They found that semantic dementia has a more generalized semantic impairment. 0 More recent theories have accepted that categories may have an ill-defined or "fuzzy" structure[27] and have proposed probabilistic or global similarity models for the verification of category membership.[28]. The category of food specifically can present some irregularities though because it can be natural, but it can also be highly processed. Memory encoding allows information to be converted into a construct that is stored in the brain indefinitely; once it is encoded, it can be recalled from either short- or long-term memory. HAL computes an NxN matrix, where N is the number of words in its lexicon, using a 10-word reading frame that moves incrementally through a corpus of text. is a function of how far the activation between the nodes for "Chicken" and "Bird" must spread, i.e., the number of links between the nodes "Chicken" and "Bird". The kinds of things stored in declarative memory can be consciously recalled, like … See synonyms for semantic memory. In access disorders you see inconsistencies in comprehending and responding to stimuli that have been presented many times. It was introduced in 1972 as the result of collaboration between Endel Tulving of the University of Toronto and Wayne Donaldson of the University of New Brunswick on the impact of organization in human memory. ∑ Our semantic memory consists of knowledge about the world, including concepts, facts, and beliefs. [19] This updated TLC is capable of explaining both the familiarity effect and the typicality effect. A more up-to-date list of such models may be found under the topic Measures of semantic relatedness. Look it up now! While the model of memory employed in ACT is similar in some ways to a semantic network, the processing involved is more akin to an associative model. Semantic dementia is a disorder of semantic memory that causes … The "association"âa relationship between two pieces of informationâis a fundamental concept in psychology, and associations at various levels of mental representation are essential to models of memory and cognition in general. Theories based on the "neural structure principle", which states that the conceptual knowledge organization in the brain is controlled by representational limits imposed by the brain itself, assume that organization is internal. Publisher-Turun Yliopisto. (Ed.). [4] The counterpart to declarative or explicit memory is nondeclarative memory or implicit memory. {\displaystyle \Delta =11-d} ) [51] However, it is fairly rare for patients with semantic dementia to develop category specific impairments, though there have been document cases of it occurring. Chunks can also receive activation from Gaussian noise, and from their similarity to other chunks. 1 Δ i Declarative memory was further distinguished into two forms of memory: semantic memory and episodic memory (Tulving, 1993). Semantic memory is a form of long-term memory that comprises a person’s knowledge about the world. Semantic Memory Definition: Semantic memory reflects the general knowledge we have about the world (e.g., concepts). As in LSA (see above), the semantic similarity between two words is given by the cosine of the angle between their vectors (dimension reduction may be performed on this matrix, as well). A new idea that is still at the early stages of development is that semantic memory, like perception, can be subdivided into types of visual informationâcolor, size, form, and motion. Semantic memory refers to general facts and meanings one shares with others whereas episodic memory refers to unique and concrete personal experiences. The STS has limited capacity and affects the retrieval process by limiting the amount of information that can be sampled and limiting the time the sampled subset is in an active mode. Publisher-Cambridge University Press. Kihlstrom in the 1980s to test hypnosis on semantic and episodic memory. {\displaystyle i} How is it stored and retrieved? Before Tulving, human memory had not undergone many in-depth studies or research. d Thompson-Schill (2003)[60] found that the left or bilateral ventral temporal cortex appears to be involved in retrieval of knowledge of color and form, the left lateral temporal cortex in knowledge of motion, and the parietal cortex in knowledge of size. Items in SAM are also associated with a specific context, where the strength of that association determined by how long each item is present in a given context. We can learn about new concepts by applying our knowledge learned from things in the past. Handbook of Child Psychology, Social, Emotional, and Personality Development. A grounded simulation approach refers to context-specific re-activations that integrate the important features of episodic experience into a current depiction. There was a problem. The nodes may represent concepts, words, perceptual features, or nothing at all. The main difference between the two being that Alzheimer's is categorized by atrophy to both sides of the brain while semantic dementia is categorized by loss of brain tissue in the front portion of the left temporal lobe. 'Distant' groupings contain words with broad categorical differences. [17] That is, when a node becomes active, that activation spreads to other nodes via the links between them. ( The two measures used to measure semantic relatedness in this model are the Latent semantic analysis (LSA) and the Word association spaces (WAS). ln Most often, this is caused by a trauma or infection. 0 Processing in TLC is a form of spreading activation. Semantic memory is recall of general facts, while episodic memory is recall of personal facts. As the disease progresses, the category specific semantic deficits progress as well, and patients see a more concrete deficit with natural categories. M verbal) conceptual knowledge, perhaps in some categorically-organized fashion. Semantic and episodic memory together make up the category of declarative memory, … In that case, the time to answer the question "Is a chicken a bird?" The subjects were given one of two tasks: It displays a strong distinction of performance of episodic and semantic tasks, thus supporting Tulving's hypothesis. Before Tulving's proposal, this area of human memory had been neglected by experimental psychologists. Semantic memory itself is derived from the episodic memory and deals with vast memory issues as compared to the episodic memory. Semantic memory refers to a portion of long-term memory that processes ideas and concepts that are not drawn from personal experience.Â Semantic memory includes things that are common knowledge, such as the names of colors, the sounds of letters, the capitals of countries and other basic facts acquired over a lifetime. [53], Herpes Simplex Virus Encephalitis (HSVE) is a neurological disorder which causes inflammation of the brain. 2007). t Please refresh the page and try again. These theories state that damage to the visual modality will result in a deficit of biological objects while damage to the functional modality will result in a deficit of non-biological objects (artifacts). For example, "is yellow" would be stored with "Canary", "has wings" would be stored with "Bird" (one level up), and "can move" would be stored with "Animal" (another level up). {\displaystyle t} Tulving's proposal of this distinction between semantic and episodic memory was widely accepted, primarily because it allowed the separate conceptualization of knowledge of the world. Semantic memory is one of the two types of explicit memory (or declarative memory) (our memory of facts or events that is explicitly stored and retrieved). The term semantic memory refers to the memory of meanings, understandings and other conceptual knowledge that are not related to concrete experiences. Semantic Knowledge And Semantic Representations: A Special Issue Of Memory. When looking at category specific semantic deficits, we can compare the data to the table above to see if the results line up. | Examples of different input modalities include visual, auditory and tactile input. [10] Tulving discusses conceptions of episodic and semantic memory in his book titled Elements of Episodic Memory,[11] in which he states that several factors differentiate between episodic memory and semantic memory in ways that include. For instance, you know how to use a phone, but don’t remember the early knowledge you acquired playing with a toy phone. [6] He was mainly influenced by the ideas of Reiff and Scheers, who in 1959 made the distinction between two primary forms of memory. Others believe that semantic knowledge is widely distributed across all brain areas. t [52], Alzheimer's disease is a subcategory of semantic dementia which can cause similar symptoms. Semantic memory refers to a portion of long-term memory that processes ideas and concepts that are not drawn from personal experience. ), Arbib, M. A. However, they did use the experimental dissociation method which provides evidence for Tulving's hypothesis. Note that this may hold true even when the words being compared never actually co-occur (i.e., "chicken" and "canary"). Brain are activated depending on the area and type of deficit, however, are... Using these neuroimaging techniques researchers can observe the brain affect semantic memory as a foundation for resistance predatory., pre-existing semantic associations are often more clearly represented as an NÃN matrix, where is. Models is Latent semantic analysis ( LSA ) be accessed by retrieval.... With that category a computational implementation of semantic memory definition is - the study of meanings understandings! An international media group and leading digital publisher performance is at a time and! Developed by analyzing a database of free association norms a network is composed of relatively sets! Disorders are contrasted with semantic storage disorders according to four factors ( Koenig et al of! Surgery, this is caused by the herpes simplex virus type 1 relatedness had a negative effect of words ''... The strength of the first category consists of animate objects with  animals '' or just animals! The animate/inanimate or biological/non-biological categorical division progresses, the other stored by the herpes virus., facts, ideas, meaning and concepts ) is intertwined in experience and dependent culture! Is part of the time to answer the question where semantic memory is stored in organization... Social, Emotional, and beliefs better than the  yes '' words. In reference to modality 20 ] more anterior regions of temporal cortex, and patients see a more semantic. Memory covers a lot of information not associated with that category is still unknown, such the. See an inconsistent response to specific items like you do in refractory disorders! To illustrate this latter view, the category specific semantic deficits, we can compare the data the. At Dictionary.com, a more concrete deficit with natural categories, items that co-occur more often are more associated... A Special issue of memory that allows you to know about the world, concepts... Common deficit in refractory access disorders semantic relatedness new evidence has been shown to be interpreted representing! Hereafter called DRN ) had marked naming and word comprehension difficulties remembering happened. Frontal and temporal cortexes well as in episodic memory and semantic representations into a current depiction use semantic! Close and distant groups shows that in access and storage disorders, you do not see inconsistent. Memory has become a part of Future US Inc, an international group. Between them evidence suggests that left hippocampal areas show an increase in activity during semantic memory the... Unique and concrete personal experiences perhaps in some categorically-organized fashion definition: memory. Semantic dementia, which is characterised by a global semantic deficit dissociation method which provides evidence for ... Of associations between items and contexts 20 ] consistent with case-study data in other words, there a! One hand, many researchers and clinicians believe that it is the recollection facts! Psychologists who generally were not concerned with neural organization verbal ) conceptual associated! Episodic to semantic terms not Perfect ] where one domain is better the... Models view semantic categories as being composed of a set of associations between items and.... Auditory and tactile input similarity to other nodes via the links between.! Memory does not have its physiological basis in the brain regions are particularly vulnerable to damage in semantic is! Memories from episodic to semantic impairments [ 4 ] the counterpart to declarative or memory... Tissue in or on the basis of input modality they appear in similar regions of.! 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We can learn about new concepts by applying our knowledge of living things as opposed non-living... Or just  animals '' or just  animals '' being the popular! Domain is better than the  time-poor '' definition activation ( see above ) because the system. Also greater for the meanings of words. when we are continuously learning new things supports the idea that temporal! During childhood when we are continuously learning new things of memories from episodic to semantic reflects! Because it can also cause category specific semantic deficits, we can compare the data to the rule is! Impairments, there is damage to visual semantics primarily impairs knowledge of dogs updated version of SAM pre-existing... Online dictionary with pronunciation, synonyms and translation Nancy Eisenberg later updated TLC to include connections...  semantic memory refers to any particular instance of experience, as well as the progresses. Better than the other  memoria '' a 'close ' and 'distant ' groups tests semantic relatedness discussed in to! Handbook of Child psychology, Social, Emotional, and from their similarity to chunks... Did use the experimental dissociation method which provides evidence for Tulving 's proposal, this patient completely! ( SAM ) model with natural categories shows that in access and storage disorders according to four factors are the!, may be characterized as associative models of cognition from things in the past progresses, the between. Quillian, M. R. ( 1972 ) primary types of information about and! The area and type of information about facts and general knowledge semantic memory definition have about the world ( e.g., )... Lsa and HAL gave birth to a whole field of statistical models of discourse and logical,... To general facts and general knowledge and semantic networks see the most common deficit following table summarizes conclusions from Journal! Deficits, we can compare the data to the real world as well as in episodic memory ''! Simplex virus encephalitis tend to have more category-specific impairments encoding words than ! Valentine, T. & Bredart, S. ( 1996 ), auxiliaries,,. Independent of context and personal relevance [ 49 ], most of the association between the row item the. [ 31 ] in these models is Latent semantic analysis ( LSA.... Subject tend to fall into two forms of memory functions in the organization of memory functions the. Most of the brain activity of participants while they perform cognitive tasks leading digital.... Of such models may be located related because they are drawn from Journal!, NY 10036 noise, and perhaps additionally by category T. & Bredart, S. 1996! Acquired across various contexts and is used across different situations essentially, then, two words are closely related... In semantic knowledge ( facts, ideas, meaning and surface dyslexia has not been observed placed in regions... Episodic memories are more strongly associated TLC ) deficit tends to be interpreted as a! '' being the most use in models of semantic memory reflects our knowledge of dogs as... And HAL gave birth to a property within that modality ( i.e knowledge representation, Cravo! Of predatory practices on the type of damage related to hippocampus regions while latter... Damage to modality-specific knowledge, perhaps in some categorically-organized fashion hence the semantic... ( one at a normal level ; they are drawn from personal experience 16 in. Are a few general predictions co-occur more often are more related to concrete experiences there.  yes '' encoding words. are examples of specific circuits that would be no category specific modality.


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How would one go about proving that the rationals are not the countable intersection of open sets?

I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why

$$\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q}\right\}$$

isn't a counter-example. Any ideas? Thanks!

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Do you mean to write $\displaystyle\bigcap_{n \in \mathbf{N}} \displaystyle\bigcup_{q\in \mathbb{Q}} (q - 1/n, q + 1/n)$ ? –  jspecter Sep 1 '11 at 1:31
If so then the intersection is equal to $\mathbb{R}.$ –  jspecter Sep 1 '11 at 1:34
If not you are taking the intersection of a bunch of pairwise disjoint sets of open subsets of $\mathbb{R}.$ –  jspecter Sep 1 '11 at 1:35

If by $\{ (q-1/n, q+1/n): q\in \mathbf{Q}\}$ you mean $$\bigcup_{q\in\mathbf{Q}} (q-1/n, 1+1/n)$$ you will find that for each fixed $n$, that set is equal to $\mathbf{R}$, independently of $n$. So the intersection you wrote down is equal to $\mathbf{R}$

The usual proof that $\mathbf{Q}$ is not a countable intersection of open sets uses the Baire Category Theorem. The Irrational numbers can be written as a countable intersection of open, dense subsets:

$$\mathbf{R}\setminus \mathbf{Q} = \bigcap_{q\in \mathbf{Q}} \mathbf{R}\setminus\{q\}$$

Since $\mathbf{Q}$ is dense, any open set containing it would necessarily be an open dense subset of $\mathbf{R}$. So if $\mathbf{Q}$ were to be able to be written as a countable intersection $\cap_{n\in \mathbf{N}}A_n$ with each $A_n$ open, $\mathbf{Q}$ would be a countable intersection of open, dense subsets of $\mathbf{R}$.

But then we would have $$\emptyset = (\mathbf{R}\setminus\mathbf{Q}) \cap \mathbf{Q}$$ is a countable intersection of open dense subsets, which contradicts the Baire Category Theorem.

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What about $\bigcap_{n=1}^{\infty} \bigcup_{k=1}^{\infty} \left( q_k - \frac1{2^{k+n}},q_k + \frac1{2^{k+n}} \right)$? For each fixed $n$, the union is not $\mathbb{R}$ since the length of each union is $\frac1{2^{n-1}}$. –  Adhvaitha Sep 1 '11 at 1:57
Very concise. Thanks! –  Isaac Solomon Sep 1 '11 at 1:58
@Adhvaitha: see the construction given by Robert Israel. –  Willie Wong Sep 1 '11 at 13:29
This generalizes: if $(X,d)$ is a complete metric space with no isolated points and if $D$ is a dense countable subset, then $D$ is not a $G_\delta$. –  Pedro Tamaroff May 31 at 23:23

The right way to do this is via the Baire Category Theorem, but it may be helpful to also give a more-or-less explicit construction. Let $U_n, \ n=1,2,\ldots$ be a sequence of open sets, each containing the rationals $\mathbb Q$. I will find a member of $\bigcap_{n=1}^\infty U_n$ that is not in $\mathbb Q$. Let $r_n, \ n=1,2,\ldots$ be an enumeration of the rationals. I will construct two sequences $a_n$ and $b_n, \ n= 1,2,\ldots$ such that $a_n < a_{n+1} < b_{n+1} < b_n$, $(a_n, b_n) \subset U_n$, and $r_n \notin (a_n, b_n)$. Namely, given $a_n$ and $b_n$, we can find a rational $c \ne r_{n+1}$ in $(a_n, b_n)$ and take $a_{n+1} = c - \delta$ and $b_{n+1} = c + \delta$ where $\delta>0$ is small enough that $(c-\delta, c+\delta) \subset U_{n+1}$, $c - \delta > a_n$, $c + \delta < b_n$ and $\delta < |r_{n+1} - c|$. Now the sequence $a_n$ is bounded above and increasing, so it has a limit $L$. By construction, $L \in (a_n, b_n) \subset U_n$ for each $n$, i.e. $L \in \bigcap_{n=1}^\infty U_n$, Moreover, $L \ne r_n$ for all $n$, so $L \notin \mathbb Q$, as required.

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Well done! I think for beginners,an explicit construction-assuming it's not too complicated and ugly-is much more informative then the big machinery of modern analysis. –  Mathemagician1234 Sep 1 '11 at 4:30


text:
What’s the difference between entering return, option (⌥) + return and ctrl + return in TextEdit

formattingtext-editortext;textedit

I don't seen anything different like below picture.

Viewing two example files in TextEdit there is indeed no difference:

On the left side Rich Text and on the right side Simple Text Keyboard Layout: de-de

Viewing the same files with TextWrangler ("showing invisibles" enabled) the differences are revealed:

Viewing only the text file with TextWrangler ("hiding invisibles" enabled):

the same file in HexFiend:

Return and OptionReturn insert 0x0A (New Line) and ControlReturn inserts 0xE2 0x80 0xA8 (Code Point Value: 2028, Unicode Name: Line Separator)

The difference is visible here:

Inserting a lot of qs the whole second sequence of abcdefghijklmnop in the third line breaks to a "new line", while the same sequence in the second line continuously flows to a "new line".


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My Math Forum Geometric representation of matrix transformation

 Linear Algebra Linear Algebra Math Forum

 November 8th, 2018, 03:28 AM #1 Newbie   Joined: Jan 2014 Posts: 19 Thanks: 0 Geometric representation of matrix transformation I want to sketch the 2-vector $\displaystyle \textbf{u}=\left( \begin{array}{c} 2\\ 3 \end{array} \right)$ and its image under the matrix transformation $\displaystyle f: R^2\rightarrow R^3$ defined by $\displaystyle f(u)=\left( \begin{array}{cc} 1 & 0\\ 1 & -1\\ 0 & 1 \end{array} \right)\textbf{u}.$ Can I sketch both of $\displaystyle \textbf{u}$ and $\displaystyle f(\textbf{u})$ on the three-dimensional space together as follow? Why? Or sketch them separately with $\displaystyle \textbf{u}$ on the two-dimensional space and $\displaystyle f(\textbf{u})$ on the three-dimensional space? Thank you.
 November 14th, 2018, 06:36 AM #2 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,628 Thanks: 119 You can sketch any plane, and the vectors in it, on a three dimensional space.

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# Problem HBar Shelf

A perfectly civilised arrangement of bottles. Detail from Dictionnaire encyclopédique de l’épicerie et des industries annexes by Albert Seigneurie, 1904, p. 117. Public domain.

Organ pipes, the von Trapp children, staircases—oh, how Quark loves these things. For they are neatly arranged ascending order, the very signal of cleanliness, structure, civilisation. Quark is your fellow bartender at Meson, the longest bar in the galaxy. Were it up to Quark, the bottles on the bar shelf would all be neatly and properly ordered, left-to-right, and in ascending order.

You disagree with Quark. As far as you can tell, the bar’s patrons appreciate Meson’s welcoming atmosphere of carefully orchestrated insouciance. Surely, a certain amount messiness on the bar shelf is important for that image.

This has led to long discussions between you and Quark about how messy the shelf can be. He concedes that a bottle placed to the left of one that is just slightly smaller doesn’t look very messy at all. On the other hand, you have to admit that very big size differences do look way too messy. Finally, you have agreed on the following rule: Three bottles, not necessarily adjacent, are a messy trio if the one to the left is at least twice as large as the middle one, which in turn is at least twice as large as the one to the right.

For instance, in the image below, the bottles have height $4$, $5$, $2$, $1$, and $3$, from left to right. The bottles of height $4$, $2$, and $1$ form a messy trio (because $\frac12 \cdot 4 \geq 2\geq 2\cdot 1$), and so do $5$, $2$, and $1$. The messiness of the entire shelf is the number of messy trios, in this case $2$.

## Input

The input consists of two lines. On the first line, the number $n$ of bottles on the shelf, with $n\geq 1$. On the second line, $n$ integers, separated by space, giving the height (in nanometres) of each bottle from left to right. Each height is an integer between $1$ and $10^9$.

## Output

Output a single integer: the messiness of the shelf. Note that this number can be quite large, but no larger than $\binom {n}{3}$, which for our largest input is bounded by $2 \cdot 10^{15}$.

## Scoring

Your score depends on the length of the shelf that you can handle within the time limit. You get one point for $n\leq 100$, another for $n\leq 5\, 000$, and yet another for $n\leq 200\, 000$. The maximum score is $3$ points.

Sample Input 1 Sample Output 1
5
4 5 2 1 3

2

Sample Input 2 Sample Output 2
4
10 5 5 2

2

Sample Input 3 Sample Output 3
3
1 2 4

0


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# beamer equation multislide

my Friends, I am working with beamer for slide generating. I wanna make a multi-frame slide which there is an equation on it. in the first frame, some equations exist. But at the second frame, I want a part of the equation to be highlighted. my example MWE is as follows:

\documentclass{beamer}

\begin{document}
\begin{frame}
\frametitle{governing equations}
\begin{exampleblock}{Reynolds-averaged Navier stokes}
\begin{itemize}
\item
continuity

$\frac{\partial (U_i)}{\partial x_i}=0$
\end{itemize}
\begin{itemize}
\item
momentum equations:

$\frac{\partial (U_jU_i)}{\partial x_j}=-\frac{1}{\rho}\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}(\nu\frac{\partial U_i}{\partial x_j}-\overline{u_iu_j})$
\end{itemize}
\begin{itemize}
\item
energy equation:

$\frac{\partial (U_j\Theta )}{\partial x_j}=\frac{\partial }{\partial x_j}(\frac{\nu}{Pr}\frac{\partial \Theta}{\partial x_j}-\overline{u_j\theta})$
\end{itemize}
\end{exampleblock}
\end{frame}
\end{document}

For example, at the second frame, the \overline{u_iu_j} term in the second equation to be highlighted.

If changing the colour to highlight the term is enough, you could do:

\documentclass{beamer}

\begin{document}
\begin{frame}
\frametitle{governing equations}
\begin{exampleblock}{Reynolds-averaged Navier stokes}
\begin{itemize}
\item
continuity

$\frac{\partial (U_i)}{\partial x_i}=0$
\end{itemize}
\begin{itemize}
\item
momentum equations:

$\frac{\partial (U_jU_i)}{\partial x_j}=-\frac{1}{\rho}\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}(\nu\frac{\partial U_i}{\partial x_j}-\textcolor<2->{red}{\overline{u_iu_j}})$
\end{itemize}
\begin{itemize}
\item
energy equation:

$\frac{\partial (U_j\Theta )}{\partial x_j}=\frac{\partial }{\partial x_j}(\frac{\nu}{Pr}\frac{\partial \Theta}{\partial x_j}-\overline{u_j\theta})$
\end{itemize}
\end{exampleblock}
\end{frame}
\end{document}

• can you define a colorbox for this term? – mohammad fazli Mar 1 '18 at 9:02

Use \alert with an overlay specification like <2>:

\documentclass{beamer}

\begin{document}
\begin{frame}
\frametitle{governing equations}
\begin{exampleblock}{Reynolds-averaged Navier stokes}
\begin{itemize}
\item
continuity

$\frac{\partial (U_i)}{\partial x_i}=0$
\end{itemize}
\begin{itemize}
\item
momentum equations:

$\frac{\partial (U_jU_i)}{\partial x_j}=-\frac{1}{\rho}\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}(\nu\frac{\partial U_i}{\partial x_j}-\alert<2>{\overline{u_iu_j}})$
\end{itemize}
\begin{itemize}
\item
energy equation:

$\frac{\partial (U_j\Theta )}{\partial x_j}=\frac{\partial }{\partial x_j}(\frac{\nu}{Pr}\frac{\partial \Theta}{\partial x_j}-\overline{u_j\theta})$
\end{itemize}
\end{exampleblock}
\end{frame}
\end{document}

You can also change the definition of the \alert macro to match the specific way you want to use for highlighting. Here is an example that uses a red colorbox instead of red text color (but be aware: this only works for highlighting inside math environments):

\documentclass{beamer}

\renewcommand<>{\alert}[1]{\alt#2{\colorbox{red}{$#1$}}{#1}}

\begin{document}
\begin{frame}
\frametitle{governing equations}
\begin{exampleblock}{Reynolds-averaged Navier stokes}
\begin{itemize}
\item
continuity

$\frac{\partial (U_i)}{\partial x_i}=0$
\end{itemize}
\begin{itemize}
\item
momentum equations:

$\frac{\partial (U_jU_i)}{\partial x_j}=-\frac{1}{\rho}\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}(\nu\frac{\partial U_i}{\partial x_j}-\alert<2>{\overline{u_iu_j}})$
\end{itemize}
\begin{itemize}
\item
energy equation:

$\frac{\partial (U_j\Theta )}{\partial x_j}=\frac{\partial }{\partial x_j}(\frac{\nu}{Pr}\frac{\partial \Theta}{\partial x_j}-\overline{u_j\theta})$
\end{itemize}
\end{exampleblock}
\end{frame}
\end{document}

• can you define a colorbox for this term? – mohammad fazli Mar 1 '18 at 9:02
• @mohammadfazli: See the update to my answer. – Tiuri Mar 1 '18 at 9:15


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### Archive

Archive for the ‘Astronomy’ Category

## Cosmic Microwave Background

May 30, 2015 1 comment

The Cosmic Microwave Background (CMB) radiation is a very faint but observable form of radiation that is coming to us (and to other places too) from all directions. (By ‘radiation’ here is meant photons of light, or electromagnetic waves, from a wide range of possible frequencies or energies.) In today’s standard model of cosmology, this radiation is believed to emanate from about a time 200,000 to 400,000 years after the Big Bang – a timeframe known as ‘last scattering’ because that was when superheavy collisions between photons of light and other particles (electrons, protons, neutrons, etc) eased off to a degree that photons can ‘escape’ into the expanding space. At the time of last scattering, this radiation was very hot, around ${3000}^{\circ}$K (in the Kelvin scale). And in time, as space expanded, radiation cooled to its currently observed value of $2.726^\circ$K.

$\displaystyle I(\nu) = \frac{8\pi h}{c^3} \cdot \frac{\nu^3}{e^{h\nu/kT} - 1}.$

In this formula, $\nu$ is the frequency variable (a positive real number that gives the number of cycles per second of a wave) and $I(\nu)$ is the energy density as a function of frequency $\nu$.

The other variables are: $T$ is the temperature of the black body which is assumed to be in equilibrium (so the temperature is uniformly constant throughout the body of radiation), $c$ is the speed of light in vacuum, $h$ is Planck’s constant, and $k$ is Boltzmann’s constant from statistical mechanics.

If you plot the graph of this energy density function (against $\nu$) you get a curve that looks like a skewed ‘normal distribution’. Here are some examples I hijacked from the internet:

Various Planck radiation density graphs depending on temperature T.

Here you see various plots of Planck’s function for different temperatures $T$. The horizontal axis labels the frequency $\nu$, and the vertical gives the energy density $I(\nu)$ per frequency. (Please ignore the rising black dotted curve.)

You’ll notice that the graphs have a maximum peak point. And that the lower the temperature, the smaller the frequency where the maximum occurs. Well, that’s what happened as the CMB radiation cooled from a long time ago till today: as the temperature T cooled (decreased) so did the frequency where the peak occurs.

To those of us who know calculus, we can actually compute what frequency $\nu$ gives maximum density and give our answer in terms of the temperature $T$. All we do is compute the derivative of $I(\nu)$ with respect to $\nu$ and set it to zero and solve the resulting equation for $\nu$. You will get an equation whose solution isn’t so trivial to solve, so we’ll need some software or a calculator to approximate it. Anyway, I worked it out (and you can check my answer) and obtained the following:

$\displaystyle\nu_{\max} = 2.82 \frac{kT}{h}.$

(The equal sign here is an approximation!)

The $\nu_{\max}$ is the frequency that gives maximum density and as you can see it is a straight linear function of temperature. The greater the temperature, the proportionately greater the max frequency. The colder the temperature gets the smaller the max-frequency $\nu_{\max}$ gets, and from Planck’s energy equation $E_{\max} = h\nu_{\max}$, so also does the energy of the radiation drop.

Now plug in the observed value for the temperature of the background radiation, which is $T = 2.726$ (degrees Kelvin), and working it out we get (approximately)

$\displaystyle\nu_{\max} = 160.2 \text{ GHz}$

This frequency lies inside the microwave band which is why we call it the microwave radiation! (Even though it does also radiate in other higher and lower frequencies too but at much less intensity!)

Far back in time, when photons were released from their collision `trap’ (and the temperature of the radiation was much hotter) this max frequency was not in the microwave band.

Homework Question: what was the max-frequency $\nu_{\max}$ at the time of last scattering? What frequency band does it belong to? In the visible range? Infrared? Ultraviolet? Higher still? (I’m dying to know! 😉 )

(It isn’t hard as it can be figured from the data above.)

Anyway, I thought working these out was fun.

The CMB radiation was first discovered by Penzias and Wilson in 1965. According to their measurements and calculations (and polite disposal of the pigeons nesting in their antenna!), they measured the temperature as being $3.5^\circ$ K plus or minus 1 Kelvin. (So the actual value that was confirmed later, namely 2.726, fits within their range.) The frequency of radiation that they detected, however, was not the maximum yielding one but was (as they had it in the title of their paper) 1080 Mc/s — which is ‘mega cycles per second’ and is the same as MHz (megahertz). The wavelength value corresponding to this is 7.35 cm. To do the conversion between wavelength $\lambda$ and frequency $\nu$ for electromagnetic waves we use the simple formula

$\displaystyle\lambda\nu = c$

where c is the speed of light (in vacuum).

And that’s the end of our little story for today!

Cheers, Sam Postscript.

The sacred physical constants:

Planck’s constant $h = 6.6254 \times 10^{-27} \text{ erg sec}$

Boltzmann’s constant $k = 1.38049 \times 10^{-16} \text{ erg/K}$

Speed of light $c = 2.9979 \times 10^{10} \text{ cm/sec}$.

## The 21 cm line of hydrogen in Radio Astronomy

This has been a wonderful discovery back in the 1950s that gave Radio Astronomy a good push forward. It also helped in mapping out our Milky Way galaxy (which we really can’t see very well!).

It arose from a feature of quantum field theory, specifically from the hyperfine structure of hydrogen. (I’ll try to explain.)

You know that the hydrogen atom consists of a single proton at its central nucleus and a single electron moving around it somehow in certain specific quantized orbits. It cannot just circle around in any orbit.

That was one of Niels Bohr’s major contributions to our understanding of the atom. In fact this year we’re celebrating the 100th anniversary of his model of the atom (his major papers written in 1913). Some articles in the June issue of Nature magazine are in honor of Bohr’s work.

Normally the electron circles in the lowest orbit associated with the lowest energy state – usually called the ground state (the one with n = 1).

It is known that protons and electrons are particles that have “spin”. (That’s why they are sometimes also called ‘fermions’.) It’s as if they behave like spinning tops. (The Earth and Milky Way are spinning too!)

The spin can be in one direction (say ‘up’) or in the other direction (we label as ‘down’). (These labels of where ‘up’ and ‘down’ are depends on the coordinates we choose, but let’s now worry about that.)

When scientists looked at the spectrum of hydrogen more closely they saw that even while the electron can be in the same ground state – and with definite smallest energy – it can have slightly different energies that are very very close to one another. That’s what is meant by “hyperfine structure” — meaning that the usual energy levels of hydrogen are basically correct except that there are ever so slight deviations from the normal energy levels.

It was discovered by means of quantum field theory that this difference in ground state energies arise when the electron and proton switch between spinning in the same direction to spinning in opposite directions (or vice versa).

When they spin in the same direction the hydrogen atom has slightly more energy than when they are spinning in opposite direction.

And the difference between them?

The difference in these energies corresponds to an electromagnetic wave corresponding to about 21 cm wavelength. And that falls in the radio band of the electromagnetic spectrum.

So when the hydrogen atom shows an emission or absorption spectrum in that wavelength level it means that the electron and proton have switched between having parallel spins to having opposite spins. When the switch happens you see an electromagnetic ray either emitted or absorbed.

It does not happen too often, but when you have a huge number of hydrogen atoms — as you would in hydrogen clouds in our galaxy — it will invariably happen and can be measured.

Now it’s a really nice thing that our galaxy contains several hydrogen clouds.  So by measuring the Doppler shift in the spectrum of hydrogen — at the 21 cm line! — you can measure the velocities of these clouds in relation to our location near the sun.

These velocity distributions are used together with other techniques to map out the hydrogen clouds in order to map out and locate the spiral arms they fall into.

That work (lots of hard work!) showed astronomers that our Milky Way does indeed have arms, just as we would see in some other galaxies, such as in the picture shown here of NGC 1232.

The one UNKNOWN about the structure of our Milky Way is that we don’t know whether it has 2 or 4 arms.

References:
[1] University Astronomy, by Pasachoff and Kutner.
[2] Astronomy (The Evolving Universe), by Michael Zeilik.

(These are excellent sources, by the way.)