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6 Working with Messy Data

Real world data is messy. Very often the data file that you start out with doesn’t have the variables stored in the right format for the analysis you want to do. This can range from the way they are named to how they are coded. At times there might be some missing values in your dataset. We learned how to remove missing values or NAs from our data in session 3. In addition, there maybe a time where you only want to analyze a subset of your data. We learned how to subset a dataset in session 3. In summary, when working with real world data you’ll have to do some data manipulation to get it in the format that you need it. In this session we’ll work with some messy data to learn how to format and properly code variables, create simple frequency tables, and review a few graphing techniques to help us better analyze our data.

Session Outline

1. Messy Data
2. Renaming Columns (Variable Names)
3. Attaching / Detaching
4. Tabulating Data: Constructing Simple Frequency Tables
5. Ordering Factor Variables
6. Summary

Watch introductory video

1. Messy Data

Let’s look at a survey questionnaire administered to undergraduate students aimed to understand their preferences for learning specific technologies as part of their education.

Surveys with close-ended questions typically yield nice and neat datasets of results. However, this short survey produced a pretty complex and messy dataset that makes it difficult to work with.

Let’s take a closer look.

Then, import the undergrad.csv dataset into RStudio. As described in session 3, simply select Import Dataset > From CSV File … from the workspace pane.

Then be sure to convert the data to a data frame

undergrad <-data.frame(undergrad)

Next, view the dataset.

View(undergrad)

Carefully review the data. You’ll notice that is has 39 observations and 11 variables. Take note of what you see as messy with this data. Lots of text? The long column names? The missing data? Also, review the columns that correspond to the responses for questions 1 through 4. The questions require a response on a 7-point Likert scale. However, for question 1 the responses are in text form whereas in questions 2, 3, and 4 are numeric. As we saw in the previous lessons, we need our data in a numeric format of sorts to perform any calculations.

2. Renaming columns (variable names)

As you view the data, you probably noticed that the column names are very long, see Figure 6.1.

Imagine having to type The.following.tool.are.important.to.my.future.career… as a variable name!

It’s difficult to work with a dataset that has long names for variables. Let’s begin with recoding the 11 variable names. We can use the names( ) function to rename each of the columns in undergrad. Just be sure the variable name you type in corresponds to the correct variable. The names need to be written in the order as the columns appear in the data frame.

Create an RScript. Begin by creating an RScript for this lesson. This will help you keep track of your work and repeat commands as necessary. Note: The examples in this lesson will still include the > to indicate an R prompt.

1. Go to File > New > RScript.
3. Rename the undergrad data frame using the names( ) function as described above.
#renaming columns in the undergrad data frame
names(undergrad) <- c("timestamp","excel","access", "statistics", "programming", "iscourse", "cscourse", "topics", "istopics", "onlinecourse", "concentration")
2. Now, view the undergrad data frame to check to see if the column names were changed.
#viewing the undergrad data frame
View(undergrad)

You should see your columns renamed (see Figure 6.2).

3. Attaching / Detaching

You can make your data manipulation and analysis of the undergrad data frame a little more efficient by using the attach( ) function. The attach( ) function enables you to work with a data frame variables directly without explicitly referencing the name of the data frame. It saves the reference to the data frame in your R path. For example, to view the data in the excel variable of the undergrad data frame you would normally have to enter:

undergrad$excel ## [1] "Agree" "Strongly Agree" "Strongly Agree" "Strongly Agree" ## [5] "Agree" "Agree" "Somewhat agree" "Strongly Agree" ## [9] "Strongly Agree" "Strongly Agree" "Strongly Agree" "Strongly Agree" ## [13] "Strongly Agree" "Somewhat agree" "Strongly Agree" "Strongly Agree" ## [17] "Somewhat agree" "Agree" "Strongly Agree" "Strongly Agree" ## [21] "Agree" "Strongly Agree" "Strongly Agree" "Strongly Agree" ## [25] "Agree" "Agree" "Strongly Agree" "Strongly Agree" ## [29] "Strongly Agree" "Strongly Agree" "Agree" "Strongly Agree" ## [33] "Strongly Agree" "Strongly Agree" "Agree" "Strongly Agree" ## [37] "Strongly Agree" "Strongly Agree" "Strongly Agree" In the above example to see all the values for the variable excel you would have to type data frame$variable (e.g. undergrad$excel). Instead you can use the attach( ) function and reference any variable in the undergrad data frame directly. Try using the attach( ) function. #adding undergrad to my R path for easy access and reference of variables attach(undergrad) #referencing the excel variable directly excel ## [1] "Agree" "Strongly Agree" "Strongly Agree" "Strongly Agree" ## [5] "Agree" "Agree" "Somewhat agree" "Strongly Agree" ## [9] "Strongly Agree" "Strongly Agree" "Strongly Agree" "Strongly Agree" ## [13] "Strongly Agree" "Somewhat agree" "Strongly Agree" "Strongly Agree" ## [17] "Somewhat agree" "Agree" "Strongly Agree" "Strongly Agree" ## [21] "Agree" "Strongly Agree" "Strongly Agree" "Strongly Agree" ## [25] "Agree" "Agree" "Strongly Agree" "Strongly Agree" ## [29] "Strongly Agree" "Strongly Agree" "Agree" "Strongly Agree" ## [33] "Strongly Agree" "Strongly Agree" "Agree" "Strongly Agree" ## [37] "Strongly Agree" "Strongly Agree" "Strongly Agree" Note: It’s important to use the detach( ) function when you finished working with a particular data frame. For example, detach(undergrad) # don’t enter this in yet because we are still working with undergrad. 4. Tabulating data: Constructing simple frequency tables and histograms A typical data analysis task is to construct a frequency table or cross-tabulation of one variable to another. Let’s begin by looking at two variables: iscourse and cscourse. These variables refer to the likeliness that the respondent will take another course in the field of Information Systems or Computer Science. We are selecting these two variables because, as we saw before, the responses to this question were coded as numbers. To be sure we can check the data type of each variable using the class( ) function. class(iscourse) ## [1] "integer" class(cscourse) ## [1] "integer" These variables are of type “integer”. This is a numeric data type that only contains positive or negative whole numbers. Integers can never contain decimals. Frequency counts. A simple frequency count of the number of respondents per response category will help us analyze the results. We can use the table( ) function to tabulate the results of each variable: table(iscourse) ## iscourse ## 1 2 3 4 5 6 7 ## 6 3 1 8 5 10 6 table(cscourse) ## cscourse ## 1 2 3 4 5 6 7 ## 3 5 2 6 5 7 11 To interpret the results, look at the first row for each response type. This is a scale that includes the integers 1 through 7 which represents likelihood that respondents will take a course in either computer science or information systems. A response of 1 is extremely unlikely, whereas a response of 7 is extremely likely. Visualizing the results. In chapter 5 we learned how to create a visualization of a frequency table using the hist( ) function. Let’s try it. hist(iscourse, breaks=7, main = "Frequencies students interested in another IS course", xlab = "IS course", col="#4cbea3", labels=TRUE, border="#FFFFFF") hist(cscourse, breaks=7, main = "Frequencies students interested in another CS course", xlab = "CS course", col="#4cbea3", labels=TRUE, border="#FFFFFF") Upon further analysis of the mean, we can see that the respondents report a slightly higher tendency to take a computer science course over an information systems course, but the differences are very small. mean(iscourse) ## [1] 4.461538 mean(cscourse) ## [1] 4.794872 median(cscourse) ## [1] 5 median(iscourse) ## [1] 5 Here we can see that it’s relatively trivial to do this type of analysis on numeric data. Let’s return to our survey and look at the first four questions from the survey questionnaire as displayed in Figure 6.5. Suppose we want to understand if students’ attitude toward specific technologies was important to their career. Specifically we want to look at the variables excel, access, statistics, and programming in the undergrad dataset. Let’s examine the second column of data, excel. The data represents the responses from 39 participants to the question: The following tool is important to my career: Excel Here you may want to simply know the frequency count of the number of respondents per response category: Strongly disagree Disagree Somewhat disagree Neither agree or disagree Somewhat agree Agree Strongly agree We can use the table( ) function to construct a simple frequency table as we did earlier. table(excel) ## excel ## Agree Somewhat agree Strongly Agree ## 9 3 27 Did you notice that the output of the responses only fall within the categories of Agree, Somewhat agree, or Strongly agree? Also, note that the ordering of the table is in alphabetical order. However, it would be easier to interpret the results if we could see them in an order from Strongly Agree to Strongly Disagree. The data is presented unordered. Check the data type using the class( ) function. class(excel) ## [1] "character" You’ll notice that excel is of type character. One way to order the data values correctly and have the ability to plot non-numeric data is to use the factor data type. Refer back to session 2 for more details on factor variables and ordered factor variables. Look above at the output of the table(excel) command. Did you notice that there were only three response types given, even though respondent of the survey had seven to select from? This is because there were no values for Strongly Disagree, Disagree, Somewhat Disagree, and Neutral. This means that every student who completed only selected Agree, Somewhat Agree, or Strongly Agree. Let’s examine the remaining three variables (access, statistics, and programming) and construct frequency tables each of them. table(access) ## access ## Agree Disagree ## 5 5 ## Neither agree or disagree Somewhat agree ## 9 10 ## Somewhat disagree Strongly Agree ## 2 8 table(statistics) ## statistics ## Agree Disagree ## 13 1 ## Neither agree or disagree Somewhat agree ## 2 7 ## Strongly Agree ## 15 table(programming) ## programming ## Agree Disagree ## 12 1 ## Neither agree or disagree Somewhat agree ## 5 6 ## Somewhat disagree Strongly Agree ## 2 13 Next, let’s visualize the access frequency table by creating a histogram. hist(access) Error in hist.default(access) : 'x' must be numeric Oops. It looks like we cannot create a histogram with the data when it is a chaacter. To solve this, we can simply “cast” that variable to a different type. In this case, we want to change the data type of access to a factor and cast access using the as.numeric( ) function to plot our histogram. #reassign access as a factor variable access <-as.factor(access) # cast access as a numeric for plotting. figure06<- hist(as.numeric(access), main = "Responses to the level of importance of learning Microsoft Access", xlab = "Bins by reponse category", col="#4cbea3", labels=TRUE, border="#FFFFFF") figure06 ##$breaks
## [1] 1 2 3 4 5 6
##
## $counts ## [1] 10 9 10 2 8 ## ##$density
## [1] 0.25641026 0.23076923 0.25641026 0.05128205 0.20512821
##
## $mids ## [1] 1.5 2.5 3.5 4.5 5.5 ## ##$xname
## [1] "as.numeric(access)"
##
## $equidist ## [1] TRUE ## ## attr(,"class") ## [1] "histogram" However, we see in Figure 6.6, that the histogram is not very useful in helping us interpret our data. We can clearly see that the values on the x-axis (our ordinal data) do not necessary match what we’d like to see based on our frequency table. Specifically, it’s unclear what the numbers 1-6 mean on the x-axis. Furthermore, it doesn’t seem as though they are in any specific order. For example, if you look at the first bin in the histogram there are 10 observations. If you recall from earlier, the only value that has 10 observations in the access frequency table is “Somewhat Agree”. Let’s return back to the frequency table we created for access. table(access) ## access ## Agree Disagree ## 5 5 ## Neither agree or disagree Somewhat agree ## 9 10 ## Somewhat disagree Strongly Agree ## 2 8 It’s difficult to see which values correspond to which categories (e.g. agree, disagree, etc.). The next section on ordering factor variables will help us work with our ordinal data in a more organized way. 5. Ordering factor variables Let’s quickly look at the access variable: access ## [1] Neither agree or disagree Disagree ## [3] Somewhat agree Strongly Agree ## [5] Somewhat agree Neither agree or disagree ## [7] Disagree Somewhat agree ## [9] Neither agree or disagree Neither agree or disagree ## [11] Agree Disagree ## [13] Neither agree or disagree Neither agree or disagree ## [15] Strongly Agree Neither agree or disagree ## [17] Somewhat agree Somewhat agree ## [19] Strongly Agree Agree ## [21] Somewhat agree Strongly Agree ## [23] Somewhat agree Somewhat agree ## [25] Somewhat agree Agree ## [27] Somewhat disagree Strongly Agree ## [29] Somewhat disagree Disagree ## [31] Neither agree or disagree Strongly Agree ## [33] Strongly Agree Agree ## [35] Neither agree or disagree Somewhat agree ## [37] Agree Disagree ## [39] Strongly Agree ## 6 Levels: Agree Disagree Neither agree or disagree ... Strongly Agree Note the levels. They are unordered. However, the access factor variable is an ordinal variable. This means that there is a relationship between the levels. For example, agree indicates less agreement than strongly agree. This is a pretty common data problem in R when importing a dataset. To order a factor variable we can use the ordered( ) function and set the levels in the order we want them. access_ordered <- ordered(x=access, levels=c("Strongly disagree","Disagree","Somewhat disagree","Neither agree or disagree","Somewhat agree","Agree","Strongly Agree")) View the data and levels for the access variable. access_ordered ## [1] Neither agree or disagree Disagree ## [3] Somewhat agree Strongly Agree ## [5] Somewhat agree Neither agree or disagree ## [7] Disagree Somewhat agree ## [9] Neither agree or disagree Neither agree or disagree ## [11] Agree Disagree ## [13] Neither agree or disagree Neither agree or disagree ## [15] Strongly Agree Neither agree or disagree ## [17] Somewhat agree Somewhat agree ## [19] Strongly Agree Agree ## [21] Somewhat agree Strongly Agree ## [23] Somewhat agree Somewhat agree ## [25] Somewhat agree Agree ## [27] Somewhat disagree Strongly Agree ## [29] Somewhat disagree Disagree ## [31] Neither agree or disagree Strongly Agree ## [33] Strongly Agree Agree ## [35] Neither agree or disagree Somewhat agree ## [37] Agree Disagree ## [39] Strongly Agree ## 7 Levels: Strongly disagree < Disagree < ... < Strongly Agree We can see that the levels represent degrees. For example, the value of Disagree is > (greater than) than the value for Strongly Disagree. Another way to see the attributes of a variable is to use the attributes( ) function. attributes(access_ordered) ##$levels
## [1] "Strongly disagree"         "Disagree"
## [3] "Somewhat disagree"         "Neither agree or disagree"
## [5] "Somewhat agree"            "Agree"
## [7] "Strongly Agree"
##
## $class ## [1] "ordered" "factor" Now that access_ordered is in a logical order, let’s recreate our frequency table. table(access_ordered) ## access_ordered ## Strongly disagree Disagree ## 0 5 ## Somewhat disagree Neither agree or disagree ## 2 9 ## Somewhat agree Agree ## 10 5 ## Strongly Agree ## 8 Note: All response options now appear in the frequency table (i.e. Strongly disagree). Next, create a simple histogram of our access_ordered variable, see Figure 6.7. figure07 <- hist(as.numeric(access_ordered),breaks=7, main = "Responses to the level of importance of learning Microsoft Access", xlab = "Bins by reponse category", col="#4cbea3", labels=TRUE, border="#FFFFFF") figure07 ##$breaks
## [1] 2 3 4 5 6 7
##
## $counts ## [1] 7 9 10 5 8 ## ##$density
## [1] 0.1794872 0.2307692 0.2564103 0.1282051 0.2051282
##
## $mids ## [1] 2.5 3.5 4.5 5.5 6.5 ## ##$xname
## [1] "as.numeric(access_ordered)"
##
## \$equidist
## [1] TRUE
##
## attr(,"class")
## [1] "histogram"

Let’s compare Figures 6.6 and 6.7 to one another:

par(mfrow = c(1, 2))

hist(as.numeric(access), main = "Figure 6", xlab = "Bins by reponse category", col="#4cbea3", labels=TRUE, border="#FFFFFF")

hist(as.numeric(access_ordered),breaks=7, main = "Figure 7", xlab = "Bins by reponse category", col="#4cbea3", labels=TRUE, border="#FFFFFF")

Finally, you can compute the mean of the access_ordered factor variable by using the mean( ) and as.numeric functions.

mean(as.numeric(access_ordered))
## [1] 4.820513

6. Summary

R Commands and Syntax

• names( ) sets the name of an object. We used to rename columns in a data frame
• attach( ) enables a data frame (or any object) to be added to R’s search path for the easy reference of variables without referencing the data frame.
• detach( ) removes the data frame from R’s search path.
• table( ) function provides a way to create a frequency table
• attributes( ) function returns the features of a variable and type.
• factor( ) a variable type used for categorical data.
• ordered( ) a function for factors to order factor levels.
• histogram( ) draws a histogram, numeric data required.
• as.numeric( ) casts a non-numeric variable into numeric data.

6.1 Exercise 6.1

Create an RMarkdown document to complete the following:

1. Getting to know the data
1. Import the data (http://becomingvisual.com/rfundamentals/summer_winter_olympics.csv)
2. View the data
3. Look at column names
4. Look at dimension of data (rows and columns)
1. Dealing with Data
1. Look at the column names and change names to more meaningful names.
2. The data represent, in order:
1. country
2. number of summer games played, gold, silver, bronze, total,
3. number of winter games played, gold, silver, bronze and total, total
4. total (Winter + Summer) games, gold, silver, bronze, total
1. Summary
1. use table() to find frequency of total summer games played
2. explore the data with other variables
1. Graphs
1. do histogram of summer games (total)
2. do histogram of winter games (total)
3. put above two histograms on one page
4. do two histograms on one page: total summer, total winter medals won
5. is there a correlation between number of medals given out in winter and summer? (do plot)
6. how about number of games each country competes in. Is there correlation between winter and summer?
7. look at distribution of each of the types of medals, by season (6 histograms on one page)
8. redo g with different number of bins (10 instead of 20)
9. explore data on your own

6.2 Exercise 6.2

Create an R Markdown document to complete the Task below:

Merge the columns for the year 2016 for GDP, Life Expectancy, and Employment into a new data frame and clean-up the new table.

You can find the data here:

1. Rename the appropriate columns to “country”, “gdp”, “life_expectancy”, and “employment”.

2. Convert the employment number to percentages by dividing by 100

3. Then round life expectancy to zero decimals and employment to two decimals

4. Create a frequency table for each variable

5. Draw histograms for each variable

6.3 Assignment 6

Use the undersgraduate survey data from http://becomingvisual.com/rfundamentals/undergrad.csv to create ordered factor variables for the excel, statistics and programming variables. In a R Markdown draw histograms for your new ordered factor variables.


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# Consider $f(x)=2\varphi (x)\phi(x)$, then which of the following will be true?

by RAHUl JHa   Last Updated January 16, 2018 18:19 PM

Consider a random variable with the probability density function as $f(x)=2\varphi (x)\phi(x)$, $x$ is a real number, and $\varphi(x)$ is the density function of Standard Normal distribution, $N(0,1)$ while $\phi(x)$ is the cumulative distribution function of Standard Normal distribution, then which of the following will be true:

1. $E(x)>0$
2. $E(x)<0$
3. $P(X\leq0)>0.5$
4. $P(X\geq0)>0.25$

I have no idea how to proceed with the problem. Anything which can make me begin this problem will be helpful. I am totally stuck at this. Thanks in advance.

Tags :

Based on the relationship $\varphi(x) = \frac{d}{dx} \, \phi(x)$, we can obtain the CDF of this density through integration (chain rule). The CDF will be able to tell us how distorted this new distribution is relative to the normal CDF.

If you get stuck on the integration or anything, just let me know.

xkl
January 16, 2018 17:39 PM

Hint for the last two: use integration by parts to show that $P(X \le 0) = .5 - P(X \le 0)$.

Taylor
January 16, 2018 17:59 PM

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Quantum routing with fast reversals
We present methods for implementing arbitrary permutations of qubits under interaction constraints. Our protocols make use of previous methods for rapidly reversing the order of qubits along a path. Given nearest-neighbor interactions on a path of length n , we show that there exists a constant ϵ ≈ 0.034 such that the quantum routing time is at most ( 1 − ϵ ) n , whereas any swap-based protocol needs at least time n − 1 . This represents the first known quantum advantage over swap-based routing methods and also gives improved quantum routing times for realistic architectures such as grids. Furthermore, we show that our algorithm approaches a quantum routing time of 2 n / 3 in expectation for uniformly random permutations, whereas swap-based protocols require time n asymptotically. Additionally, we consider sparse permutations that route k ≤ n qubits and give algorithms with quantum routing time at most n / 3 + O ( k 2 ) on paths and at most 2 r / 3 + O ( k 2 ) on general graphs with radius r .
Authors:
; ; ; ; ;
Award ID(s):
Publication Date:
NSF-PAR ID:
10296764
Journal Name:
Quantum
Volume:
5
Page Range or eLocation-ID:
533
ISSN:
2521-327X
1. The Swap gate is a ubiquitous tool for moving information on quantum hardware, yet it can be considered a classical operation because it does not entangle product states. Genuinely quantum operations could outperform Swap for the task of permuting qubits within an architecture, which we call routing. We consider quantum routing in two models: (1) allowing arbitrary two-qubit unitaries, or (2) allowing Hamiltonians with norm-bounded interactions. We lower bound the circuit depth or time of quantum routing in terms of spectral properties of graphs representing the architecture interaction constraints, and give a generalized upper bound for all simple connected $n$-vertex graphs. In particular, we give conditions for a superpolynomial classical-quantum routing separation, which exclude graphs with a small spectral gap and graphs of bounded degree. Finally, we provide examples of a quadratic separation between gate-based and Hamiltonian routing models with a constant number of local ancillas per qubit and of an $\Omega(n)$ speedup if we also allow fast local interactions.


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## Chemsitry

Imagine that the gas shown in the simulation is an ideal gas such as helium. Notice that the final number of moles of gas is 1.00 mol for each experimental run. You can find the final volume of the gas using the y axis of the graph shown. Consider an experimental run at 273 K where the initial number of moles (n1) is actually 1.00 mol, and the final number of moles (n2) is 2.00 mol. Use the simulation to find the volume (V1) of 1.00 mol of helium at 273 K, and calculate the final volume (V2).


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## Classical, exceptional, and exotic holonomies: A status report.(English)Zbl 0882.53014

Besse, Arthur L. (ed.), Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, Luminy, France, 12–18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 93-165 (1996).
This is a lucid report, written by a leading expert of the subject, dealing with “the status of the problem of determining the groups that can occur as the irreducible holonomy of a torsion-free affine connection on some manifold”.
The problem of prescribed holonomy, i.e., which subgroups $$H$$ of $$GL(V)$$, where $$V$$ is an $$n$$-dimensional vector space, can be considered as the holonomy (groups) of some torsion-free connection on some $$n$$-dimensional manifold $$M$$ was systematically treated by M. Berger. In his fundamental paper [Bull. Soc. Math. France 83, 279-330 (1955; Zbl 0068.36002)], M. Berger gave a list of the possible holonomies, claiming also that it was complete up to a finite number of missing terms. This work triggered a lot of research since then. However, the classification of non-metric holonomies turned out to be incomplete and it was proved that exotic holonomies, representing missing entries from Berger’s original list, do exist.
The report starts with a paragraph devoted to the holonomy and $$G$$-structures along with the necessary algebraic machinery needed in the formulation of Berger’s criteria, under which $$H$$ can occur as the holonomy of a torsion-free connection. It is also explained why the general problem can be handled by a classification of torsion-free $$H$$-structures, which in turn can be treated by methods of Cartan-Kähler theory.
Paragraph 2 contains Berger’s list related with the metric cases, mainly the possible irreducible holonomies for pseudo-Riemannian metrics which are not locally symmetric. Each entry of the list is reviewed individually and there is an analysis of the particular problem involved. There are also included references of recent contributions related with certain structures and the question whether they do or do not satisfy Berger’s criteria (some of these structures belong to the so-called exceptional cases).
The non-metric case, i.e., the possible irreducible holonomies for affine connections which are not locally symmetric and do not preserve any nonzero quadratic form, are treated in Paragraph 3. Again there is an extensive and detailed analysis of the entries of Berger’s corresponding list (with some modifications), grouped in certain categories (affine, conformal, symplectic, etc.), as well as information about the relevant connections and their space.
Paragraph 4 is a short account of exotic holonomies, discovered in the late ‘80s. As mentioned earlier, they correspond to irreducibly acting groups which can occur as holonomy groups but are not included in Berger’s general list. According to the author, the full classification of the possible exotic examples is far from complete.
Summarizing, the report is a very informative presentation of the development and mathematics involved in the subject of prescribed holonomy. The reader, especially the non-expert, will learn a lot from it.
For the entire collection see [Zbl 0859.00016].

### MSC:

 53B05 Linear and affine connections 53C10 $$G$$-structures 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 58A15 Exterior differential systems (Cartan theory)

Zbl 0068.36002


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CBSE (Arts) Class 11CBSE
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# A Point R With X-coordinate 4 Lies on the Line Segment Joining the Pointsp (2, –3, 4) and Q (8, 0, 10). Find the Coordinates of the Point R. - CBSE (Arts) Class 11 - Mathematics

ConceptThree - Dimensional Geometry Coordinates of a Point in Space

#### Question

A point R with x-coordinate 4 lies on the line segment joining the pointsP (2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R.

[Hint suppose R divides PQ in the ratio k: 1. The coordinates of the point R are given by ((8k + 2)/(k+1), (-3)/(k+1), (10k + 4)/(k+1))

#### Solution

The coordinates of points P and Q are given as P (2, –3, 4) and Q (8, 0, 10).

Let R divide line segment PQ in the ratio k:1.

Hence, by section formula, the coordinates of point R are given by

Is there an error in this question or solution?

#### APPEARS IN

NCERT Solution for Mathematics Textbook for Class 11 (2018 to Current)
Chapter 12: Introduction to Three Dimensional Geometry
Q: 5 | Page no. 279

#### Video TutorialsVIEW ALL [1]

Solution A Point R With X-coordinate 4 Lies on the Line Segment Joining the Pointsp (2, –3, 4) and Q (8, 0, 10). Find the Coordinates of the Point R. Concept: Three - Dimensional Geometry - Coordinates of a Point in Space.
S


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### Stability of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws

Xiao-Biao Lin and Stephen Schecter

Abstract: In contrast to a viscous regularization of a system of $n$ conservation laws, a Dafermos regularization admits many self-similar solutions of the form $$u=u(X/T)$$. In particular, it is known in many cases that Riemann solutions of a system of conservation laws have nearby self-similar smooth solutions of an associated Dafermos regularization. We refer to these smooth solutions as {\em Riemann-Dafermos solutions}. After a change of coordinates, Riemann-Dafermos solutions become stationary, and their time-asymptotic stability as solutions of the Dafermos regularization can be studied by linearization. We study the stability of Riemann-Dafermos solutions near Riemann solutions consisting of $n$ Lax shock waves. We show, by studying the essential spectrum of the linearized system in a weighted function space, that stability is determined by eigenvalues only. We then use asymptotic methods to study the eigenvalues and eigenfunctions. We find there are fast eigenvalues of order $$\frac{1}{\epsilon}$$ and slow eigenvalues of order one. The fast eigenvalues correspond to fast convergence of initial data to traveling wave solutions in singular layers, while the slow eigenvalues correspond to convection in regular layers connected by traveling waves in singular layers. For an example from gas dynamics, we show that all the slow eigenvalues are stable.

Paper:
Available as PostScript (457 Kbytes) or gzipped PostScript (164 Kbytes; uncompress using gunzip).
Author(s):
Xiao-Biao Lin, <xblin@math.ncsu.edu>
Stephen Schecter, <schecter@math.ncsu.edu>
Publishing information:
Comments:
Submitted by:
<schecter@math.ncsu.edu> April 4 2002.

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# Sensory Systems/NonPrimates

Primates are animals belonging to the class of mammals. Primates include humans and the nonhuman primates, the apes, monkeys, lemurs, tree-shrews, lorises, bush babies and tarsiers. They are characterized by a voluminous and complicated forebrain. Most have excellent sight and are highly adapted to an arboreal existence, including in some species the possession of a prehensile tail. Non primates on the other hand often posses smaller brains. But as we learn more about the rest of the animal world, it’s becoming clear that non-primates are pretty intelligent too. Some examples include pigs, octopus, and crows.[1]

In many branches of mythology, the crow plays a shrewd trickster, and in the real world, crows are proving to be quite a clever species. Crows have been found to engage in feats such as tool use, the ability to hide and store food from season to season, episodic-like memory, and the ability to use personal experience to predict future conditions.

As it turns out, being piggy is actually a pretty smart tactic. Pigs are probably the most intelligent domesticated animal on the planet. Although their raw intelligence is most likely commensurate with a dog or cat, their problem-solving abilities top those of felines and canine pals.

If pigs are the most intelligent of the domesticated species, octopuses take the cake for invertebrates. Experiments in maze and problem-solving have shown that they have both short-term and long-term memory. Octopuses can open jars, squeeze through tiny openings, and hop from cage to cage for a snack. They can also be trained to distinguish between different shapes and patterns. In a kind of play-like activity (one of the hallmarks of higher intelligence species) octopuses have been observed repeatedly releasing bottles or toys into a circular current in their aquariums and then catching them.

## Neural Mechanism for Song Learning in Zebra Finches

### Introduction

Over the past four decades songbirds have become a widely used model organism for neuroscientists studying complex sequential behaviours and sensory-guided motor learning. Like human babies, young songbirds learn many of the sounds they use for communication by imitating adults. One songbird in particular, the zebra finch (Taeniopygia guttata), has been the focus of much research because of its proclivity to sing and breed in captivity and its rapid maturation. The song of an adult male zebra finch is a stereotyped series of acoustic signals with structure and modulation over a wide range of time scales, from milliseconds to several seconds. The adult zebra finch song comprises a repeated sequence of sounds, called a motif, which lasts about a second. The motif is composed of shorter bursts of sound called syllables, which often contain sequences of simpler acoustic elements called notes as shown in Fig.1. The songbirds learning system is a very good model to study the sensory-motor integration because the juvenile bird actively listens to the tutor and modulates its own song by correcting for errors in the pitch and offset. The neural mechanism and the architecture of the song bird brain which plays a crucial role in learning is similar to the language processing region in frontal cortex of humans. Detailed study of the hierarchical neural network involved in the learning process could provide significant insights into the neural mechanism of speech learning in humans.

## Octopus

### Introduction

One of the most interesting non-primate is the octopus. The most interesting feature about this non-primate is its arm movement. In these invertebrates, the control of the arm is especially complex because the arm can be moved in any direction, with a virtually infinite number of degrees of freedom. In the octopus, the brain only has to send a command to the arm to do the action—the entire recipe of how to do it is embedded in the arm itself. Observations indicate that octopuses reduce the complexity of controlling their arms by keeping their arm movements to set, stereotypical patterns. To find out if octopus arms have minds of their own, the researchers cut off the nerves in an octopus arm from the other nerves in its body, including the brain. They then tickled and stimulated the skin on the arm. The arm behaved in an identical fashion to what it would in a healthy octopus. The implication is that the brain only has to send a single move command to the arm, and the arm will do the rest.

In this chapter we discuss in detail the sensory system of an octopus and focus on the sensory motor system in this non-primate.

#### Octopus - The intelligent non-primate

The Common Octopus, Octopus vulgaris.

Octopuses have two eyes and four pairs of arms, and they are bilaterally symmetric. An octopus has a hard beak, with its mouth at the center point of the arms. Octopuses have no internal or external skeleton (although some species have a vestigial remnant of a shell inside their mantle), allowing them to squeeze through tight places. Octopuses are among the most intelligent and behaviorally flexible of all invertebrates.

The most interesting feature of the octopuses is their arm movements. For goal directed arm movements, the nervous system in octopus generates a sequence of motor commands that brings the arm towards the target. Control of the arm is especially complex because the arm can be moved in any direction, with a virtually infinite number of degrees of freedom. The basic motor program for voluntary movement is embedded within the neural circuitry of the arm itself.[2]

### Arm Movements in Octopus

In the hierarchical organization in octopus, the brain only has to send a command to the arm to do the action. The entire recipe of how to do it is embedded in the arm itself. By the use of the arms octopus walks, seizes its pray, or rejects unwanted objects and also obtains a wide range of mechanical and chemical information about its immediate environment.

Octopus arms, unlike human arms, are not limited in their range of motion by elbow, wrist, and shoulder joints. To accomplish goals such as reaching for a meal or swimming, however, an octopus must be able to control its eight appendages. The octopus arm can move in any direction using virtually infinite degrees of freedom. This ability results from the densely packed flexible muscle fibers along the arm of the octopus.

Observations indicate that octopuses reduce the complexity of controlling their arms by keeping their arm movements to set, stereotypical patterns.[3] For example, the reaching movement always consists of a bend that propagates along the arm toward the tip. Since octopuses always use the same kind of movement to extend their arms, the commands that generate the pattern are stored in the arm itself, not in the central brain. Such a mechanism further reduces the complexity of controlling a flexible arm. These flexible arms are controlled by an elaborate peripheral nervous system containing 5 × 107 neurons distributed along each arm. 4 × 105 of these are motor neurons, which innervate the intrinsic muscles of the arm and locally control muscle action.

Whenever it is required, the nervous system in octopus generates a sequence of motor commands which in turn produces forces and corresponding velocities making the limb reach the target. The movements are simplified by the use of optimal trajectories made through vectorial summation and superposition of basic movements. This requires that the muscles are quite flexible.

### The Nervous System of the Arms

The eight arms of the octopus are elongated, tapering, muscular organs, projecting from the head and regularly arranged around the mouth. The inner surface of each arm bears a double row of suckers, each sucker alternating with that of the opposite row. There are about 300 suckers on each arm.[4]

The arms perform both motor and sensory functions. The nervous system in the arms of the octopus is represented by the nerve ganglia, subserving motor and inter-connecting functions. The peripheral nerve cells represent the sensory systems. There exists a close functional relationship between the nerve ganglia and the peripheral nerve cells.

#### General anatomy of the arm

The muscles of the arm can be divided into three separate groups, each having a certain degree of anatomical and functional independence:

1. Intrinsic muscles of the arm,
2. Intrinsic muscles of the suckers, and
3. Acetabulo-brachial muscles (connects the suckers to the arm muscles).

Each of these three groups of muscles comprises three muscle bundles at right angles to one another. Each bundle is innervated separately from the surrounding units and shows a remarkable autonomy.In spite of the absence of a bony or cartilaginous skeleton, octopus can produce arm movements using the contraction and relaxation of different muscles. Behaviorally, the longitudinal muscles shorten the arm and play major role in seizing objects carrying them to mouth, and the oblique and transverse muscles lengthen the arms and are used by octopus for rejecting unwanted objects.

Cross section of an octopus arm: The lateral roots innervate the intrinsic muscles, the ventral roots the suckers.

Six main nerve centers lie in the arm and are responsible for the performance of these sets of muscles. The axial nerve cord is by far the most important motor and integrative center of the arm. The eight cords one in each arm contains altogether 3.5 × 108 neurons. Each axial cord is linked by means of connective nerve bundles with five sets of more peripheral nerve centers, the four intramuscular nerve cords, lying among the intrinsic muscles of the arm, and the ganglia of the suckers, situated in the peduncle just beneath the acetabular cup of each sucker.

All these small peripheral nerves contain motor neurons and receive sensory fibers from deep muscle receptors which play the role of local reflex centers. The motor innervation of the muscles of the arm is thus provided not only by the motor neurons of the axial nerve cord, which receives pre-ganglionic fibers from the brain, but also by these more peripheral motor centers.

#### Sensory Nervous system

The arms contain a complex and extensive sensory system. Deep receptors in the three main muscle systems of the arms, provide the animal with a widespread sensory apparatus for collecting information from muscles. Many primary receptors lie in the epithelium covering the surface of the arm. The sucker, and particularly its rim, has the greatest number of these sensory cells, while the skin of the arm is rather less sensitive. Several tens of thousands of receptors lie in each sucker.

Three main morphological types of receptors are found in arms of an octopus. These are round cells, irregular multipolar cells, and tapered ciliated cells. All these elements send their processes centripetally towards the ganglia. The functional significance of these three types of receptors is still not very well known and can only be conjectured. It has been suggested that the round and multipolar receptors may record mechanical stimuli, while ciliated receptors are likely to be chemo-receptors.

The ciliated receptors do not send their axons directly to the ganglia but the axons meet encapsulated neurons lying underneath the epithelium and make synaptic contacts with the dendritic processes of these. This linkage helps in reduction of input between primary nerve cells. Round and multipolar receptors on the other hand send their axons directly to the ganglia where the motor neurons lie.

### Functioning of peripheral nervous system in arm movements

Behavioral experiments suggest that information regarding the movement of the muscles does not reach the learning centers of the brain, and morphological observations prove that the deep receptors send their axons to peripheral centers such as the ganglion of the sucker or the intramuscular nerve cords.[5] The information regarding the stretch or movement of the muscles is used in local reflexes only.

When the dorsal part of the axial nerve cord that contains the axonal tracts from the brain is stimulated by electrical signals, movements in entire arm are still noticed. The movements are triggered by the stimulation which is provided and is not directly driven by the stimuli coming from the brain. Thus, arm extensions are evoked by stimulation of the dorsal part of the axial nerve cord. In contrast, the stimulation of the muscles within the same area or the ganglionic part of the cord evokes only local muscular contractions. The implication is that the brain only has to send a single move command to the arm, and the arm will do the rest.

A dorsally oriented bend propagates along the arm causing the suckers to point in the direction of the movement. As the bend propagates, the part of the arm proximal to the bend remains extended. For further conformations that an octopus arm has a mind of its own, the nerves in an octopus arm have been cut off from the other nerves in its body, including the brain. Movements resembling normal arm extensions were initiated in amputated arms by electrical stimulation of the nerve cord or by tactile stimulation of the skin or suckers.

It has been noted that the bend propagations are more readily initiated when a bend is created manually before stimulation. If the fully relaxed arm is stimulated, the initial movement is triggered by the stimuli, which follows the same bend propagation. The nervous system of the arm thus, not only drives local reflexes but controls complex movements involving the entire arm.

These evoked movements are almost kinematically identical to the movements of freely behaving octopus. When stimulated, a severed arm shows an active propagation of the muscle activity as in natural arm extensions. Movements evoked from similar initial arm postures result in similar paths, while different starting postures result in different final paths.

As the extensions evoked in denervated octopus arms are qualitatively and kinematically similar to natural arm extensions, an underlying motor program seems to be controlling the movements which are embedded in the neuromuscular system of the arm, which does not require central control.

## Fish

Fish are aquatic animals with great diversity. There are over 32’000 species of fish, making it the largest group of vertebrates.

The lateral line sensory organ shown on a shark.

Most fish possess highly developed sense organs. The eyes of most daylight dwelling fish are capable of color vision. Some can even see ultra violet light. Fish also have a very good sense of smell. Trout for example have special holes called “nares” in their head that they use to register tiny amounts of chemicals in the water. Migrating salmon coming from the ocean use this sense to find their way back to their home streams, because they remember what they smell like. Especially ground dwelling fish have a very strong tactile sense in their lips and barbels. Their taste buds are also located there. They use these senses to search for food on the ground and in murky waters.

Fish also have a lateral line system, also known as the lateralis system. It is a system of tactile sense organs located in the head and along both sides of the body. It is used to detect movement and vibration in the surrounding water.

### Function

Fish use the lateral line sense organ to sense prey and predators, changes in the current and its orientation and they use it to avoid collision in schooling.

Coombs et al. have shown [1] that the lateral line sensory organ is necessary for fish to detect their prey and orient towards it. The fish detect and orient themselves towards movements created by prey or a vibrating metal sphere even when they are blinded. When signal transduction in the lateral lines is inhibited by cobalt chloride application, the ability to target the prey is greatly diminished.

The dependency of fish on the lateral line organ to avoid collisions in schooling fish was demonstrated by Pitcher et al. in 1976, where they show that optically blinded fish can swim in a school of fish, while those with a disabled lateral line organ cannot [2].

### Anatomy

The lateral lines are visible as two faint lines that run along either side of the fish body, from its head to its tail. They are made up of a series of mechanoreceptor cells called neuromasts. These are either located on the surface of the skin or are, more frequently, embedded within the lateral line canal. The lateral line canal is a mucus filled structure that lies just beneath the skin and transduces the external water displacement through openings from the outside to the neuromasts on the inside. The neuromasts themselves are made up of sensory cells with fine hair cells that are encapsulated by a cylindrical gelatinous cupula. These reach either directly into the open water (common in deep sea fish) or into the lymph fluid of the lateral line canal. The changing water pressures bend the cupula, and in turn the hair cells inside. Similar to the hair cells in all vertebrate ears, a deflection towards the shorter cilia leads to a hyperpolarization (decrease of firing rate) and a deflection in the opposite direction leads to depolarization (increase of firing rate) of the sensory cells. Therefore the pressure information is transduced to digital information using rate coding that is then passed along the lateral line nerve to the brain. By integrating many neuromasts through their afferent and efferent connections, complex circuits can be formed. This can make them respond to different stimulation frequencies and consequently coding for different parameters, like acceleration or velocity [3].

Some scales of the lateral line (center) of a Rutilus rutilus

In sharks and rays, some neuromasts have undergone an interesting evolution. They have evolved into electroreceptors called ampullae of Lorenzini. They are mostly concentrated around the head of the fish and can detect a change of electrical stimuli as small as 0.01 microvolt [4]. With this sensitive instrument these fish are able to detect tiny electrical potentials generated by muscle contractions and can thus find their prey over large distances, in murky waters or even hidden under the sand. It has been suggested that sharks also use this sense for migration and orientation, since the ampullae of Lorenzini are sensitive enough to detect the earth’s electromagnetic field.

### Convergent Evolution

Cephalopods:

Cephalopods such as squids, octopuses and cuttlefish have lines of ciliated epidermal cells on head and arms that resemble the lateral lines of fish. Electrophysiological recordings from these lines in the common cuttlefish (Sepia officinalis) and the brief squid (Lolliguncula brevis) have identified them as an invertebrate analogue to the mechanoreceptive lateral lines of fish and aquatic amphibians [5].

Crustaceans:

Another convergence to the fish lateral line is found in some crustaceans. Contrary to fish, they don’t have the mechanosensory cells on their body, but have them spaced at regular intervals on long trailing antennae. These are held parallel to the body. This forms two ‘lateral lines’ parallel to the body that have similar properties to those of fish lateral lines and are mechanically independent of the body [6].

Mammals:

In aquatic manatees the postcranial body bears tactile hairs. They resemble the mechanosensory hairs of naked mole rats. This arrangement of hair has been compared to the fish lateral line and complement the poor visual capacities of the manatees. Similarly, the whiskers of harbor seals are known to detect minute water movements and serve as a hydrodynamic receptor system. This system is far less sensitive than the fish equivalent. [7]

## Flies

### Introduction

Halteres of the Crane fly

Halteres are sensory organs present in many flying insects. Widely thought to be an evolutionary modifcation of the rear pair of wings on such insects, halteres provide gyroscopic sensory data, vitally important for flight. Although the fly has other relevant systems to aid in flight, the visual system of the fly is too slow to allow for rapid maneuvers. Additionally, to be able to fly adeptly in low light conditions, a requirement to avoid predation, such a sensory system is necessary. Indeed, without halteres, flies are incapable of sustained, controlled flight. Since the 18th century, scientists have been aware of the role halteres play in flight, but it was only recently that the mechanisms by which they operate have been better explored. [6] [7]

### Anatomy

The haltere evolved from the rearmost of two pairs of wings. While the first has maintained its usage for flight, the posterior pair has lost its flight functions and has adopted a slightly different shape. The haltere is visually comprised of three structural components: a knob-shaped end, a thin shaft, and a slightly wider base. The knob contains approximately 13 innervated hairs, while the base contains two chordotonal organs, each innervated by about 20-30 nerves. Chordotonal organs are sense organs thought to be solely responsive to extension, though they remain relatively unknown. The base is also covered by around 340 campaniform sensilla, which are small fibers which respond preferentially to compression in the direction in which they are elongated. Each of these fibers is also innervated. Relative to the stalk of the haltere, both the chordotonal organs and the campaniform sensilla have an orientation of approximately 45 degrees, which is optimal for measuring bending forces on the haltere. The halteres move contrary (anti-phase) to the wings during flight. The sensory components can be categorized into three groups [8]): those sensitive to vertical oscillations of the haltere, including the dorsal and ventral scapal plates, dorsal and ventral Hicks papillae (both the plates and papillae are subcategories of the aforementioned campaniform sensilla), and the small chordotonal organ. The basal plate (another manifestation of the sensilla) and the large chordotonal organ are sensitive to gyroscopic torque acting on the haltere, and there is also a population of undifferentiated papillae which are responsive to all strains acting on the base of the haltere. This provides an additional method for flies to distinguish between the direction of force being applied to the haltere.

#### Genetics

As Homeobox genes were being discovered and explored for the first time, it was found that the deletion or inactivation of the Hox gene Ultrabithorax (Ubx) causes the halteres to develop into a normal pair of wings. This was a very compelling early result as to the nature of Hox genes. Manipulations to the Antennapedia gene can similarly cause legs to become severely deformed, or can cause a set of legs to develop instead of antennae on the head.

### Function

The halteres function by detecting Coriolis forces, sensing the movement of air across the potentially rotating fly body. Studies have indicated that the angular velocity of the body is encoded by the Coriolis forces measured by the halteres [8]. Active halteres can recruit any neighboring units, influencing nearby muscles and causing dramatic changes in the flight dynamics. Halteres have been shown to have extremely fast response times, allowing these flight changes to be performed much more quickly than if the fly were to rely on its visual system. In order to distinguish between different rotational components, such as pitch and roll, the fly must be able to combine signals from the two halteres, which must not be coincident (coincident signals would diminish the ability of the fly to differentiate the rotational axes). The halteres are capable of contributing to image stabilization, as well as in-flight attitude control, which was established by numerous authors noting a reaction from the head and wings to inputs from the components of the rotation rate vector. contributions from halteres to head and neck movements have been noted, explaining their role in gaze stabilization. The fly therefore uses input from the halteres to establish where to fixate its gaze, an interesting integration of the two senses.

### Mathematics

Recordings have indicated that halteres are capable of responding to stimuli at the same (double-wingbeat) frequency as Coriolis forces, the proof of concept that allows further mathematical analysis of how these measurements can occur. The vector cross-product of the halteres' angular velocity and the rotation of the body provide the Coriolis force vector to the fly. This force is at the same frequency as the wingbeat in both the pitch and roll planes, and is doubly fast in the yaw plane. Halteres are capable of providing a rate damping signal to affect rotations. This is because the Coriolis force is proportional to the fly's own rotation rate. By measuring the Coriolis force, the halteres can send an appropriate signal to their affiliated muscles, allowing the fly to properly control its flight. The large amplitude of haltere motion allows for the calculation of the vertical and horizontal rates of rotation. Because of the large disparity in haltere movement between vertical and horizontal movement, Ω1, the vertical component of the rotation rate, generates a force of double the frequency of the horizontal component. It is widely thought that this twofold frequency difference is what allows the fly to distinguish between the vertical and horizontal components. If we assume that the haltere moves sinusoidally, a reasonably accurate approximation of its real-world behavior, the angular position γ can be modeled as: $\gamma = \frac{\pi}{ 2}\sin(\omega t)$ where ω is the haltere beat frequency, and the amplitude is 180, a close approximation to the real life range of motion. The body rotational velocities can be computed, given the known rates (the roll, pitch, and yaw components are labeled below with 1, 2, and 3, respectively) from the two halteres' (Ωb being the left and Ωc being the right haltere) reference frames, respective to the body of the fly with the following calculations [7]:

$W_{1} = - \frac{\Omega_{b3} + \Omega_{c3} }{2\sin(\alpha)}$
$W_{2} = \frac{\Omega_{b3} - \Omega_{c3} }{2\cos(\alpha)}$

$W_{3} = - \frac{\Omega_{b1} + \Omega_{c1} }{2}$

α represents the haltere angle of rotation from the body plane, and the Ω terms are, as mentioned, the angular velocity of the haltere with respect to the body. Knowing this, one could roughly simulate input to the halteres using the equation for forces on the end knob of a haltere:

$F = mg - ma_{i} - ma_{F} - m\dot{\Omega_{i}}\times r_{i} -m\Omega_{i}\times (\Omega_{i}\times r_{i} ) - 2m\Omega_{i} \times v_{i}$

m is the mass of the knob of the haltere, g is the acceleration due to gravity, ri, vi,} and ai are the position, velocity, and acceleration of the knob relative to the body of the fly in the i direction, aF is the fly's linear acceleration, and Ωi and Ώi are the angular velocity and acceleration components for the direction i, respectively, of the fly in space. The Coriolis force is simulated by the 2mΩ × vi term. Because the sensory signal generated is proportional to the forces exerted on the halteres, this would allow the haltere signal to be simulated. If attempting to reconcile the force equation with the rotational component equations, it is worthwhile to remember that the force equation must be calculated separately for both halteres.

## Butterflies

The sense of balance of butterflies sits at the base of the antennae.

Butterflies and moth keep their balance with Johnston's organ: this is an organ at the base of a butterfly's antennae, and is responsible for maintaining the butterfly's sense of balance and orientation, especially during flight.

## Spider´s Visual System

### Introduction

While the highly developed visual systems of some spider species have been subject to extensive studies since many decades, terms like animal intelligence or cognition were not usually used in the context of spider studies. Instead, spiders were traditionally portrayed as rather simple, instinct driven animals (Bristowe 1958, Savory 1928), processing visual input in pre-programmed patterns rather than actively interpreting the information received from their visual apparatus towards appropriate reactions. While Although this still seems to be the case in a majority of spiders, which primarily interact with the world through tactile sensation rather than by visual cues, some spider species have shown surprisingly intelligent use of their eyes. Considering its limited dimensions within the body, a spider´s optical apparatus and visual processing perform extremely well.[9] Recent research points towards a very sophisticated use of visual cues in a spider´s world when investigating topics such as the complex hunting schemes of the vision-guided jumping spiders (Salticidae) taking huge leaps of up to 30 times their own body length onto prey or a wolf spider´s (Lycosidae) ability to visually recognize asymmetries in potential mates. Even in the case of the night-active Cupiennius salei (Ctenidae), relying primarily on other sensory organs, or the ogre-faced Dinopis hunting at night by spinning small webs and throwing them at approaching prey, the visual system is still highly developed. Findings like these are not only fascinating but are also inspiring other scientific and engineering fields such as robotics and computer-guided image analysis.

### General structure of a spider´s visual system

A spider´s anatomy primarily consists of two major body segments, the prosoma and the opisthosoma, which are also known as the cephalothorax and abdomen, respectively. All extremities as well as the sensory organs including the eyes are located in the prosoma. Other than the visual system of arthropods featuring compound eyes, modern arachnid eyes are ocelli (simple eyes consisting of a lens covering a vitreous fluid-filled pit with a retina at the bottom), of which spiders have six or eight, characteristically arranged in three or four rows across the prosoma´s carapace. Overall, 99% of all spiders have eight eyes and of the remaining 1% almost all have six. Spiders with only six eyes lack the “principal eyes”, which are described in detail below.

The pairs of eyes are called anterior median eyes (AME), anterior lateral eyes (ALE), posterior median eyes (PME), and posterior lateral eyes (PLE). The large principal eyes facing forward are the anterior median eyes, which provide the highest spatial resolution to a spider, at the cost of a very narrow field of view. The smaller forward-facing eyes are the anterior lateral eyes with a moderate field of view and medium spatial resolution. The two posterior eye pairs are rather peripheral, secondary eyes with wide field of view. They are extremely sensitive and suitable for low-light conditions. Spiders use their secondary eyes for sensing motion, while their principal eyes allow shape and object recognition. In contrast to insect vision, a visually-based spider´s brain is almost completely devoted to vision, as it receives only the optic nerves and consists of only the optic ganglia and some association centers. The brain is apparently able to recognize object motion, but even more to also classify the counterpart into a potential mate, rival or prey by seeing legs (lines) at a particular angle to the body. Such stimulus will result in a spider displaying either courtship or threatening signs respectively.

### A Spider´s eyes

Although spider eyes may be described as “camera eyes”, they are very different in their details from the “camera eyes” of mammals or any other animals. In order to fit a high-resolution eye into such a small body, neither an insect´s compound eyes nor spherical eyes, as we humans have them, would solve the problem. The ocelli found in spiders are the optically better solution, as their resolution is not limited by refractive effects at the lens which would be the case with compound eyes. When replacing the eye of a spider by a compound eye of the same resolving power, it would simply not fit into the spider´s prosoma. By using ocelli, the spatial acuity of some spiders is more similar to that of a mammal than to that of an insect, with a huge size difference and only a few thousand photocells, e.g. in a jumping spider´s eye, as compared to more than 150 million photocells in the human retina.

#### Principal eyes

The anterior median eyes (AME), which are present in most spider species, are also called the principal eyes. Details about the principal eye´s structure and its components are illustrated in the figure below and are explained in the following by going through the AME of the jumping spider Portia (family Salticidae), which is famous for its high-spatial-acuity eyes and vision-guided behavior despite its very small body size of 4.5-9.5 mm.

When a light beam enters the principal eye it firstly passes a large corneal lens. This lens features a long focal length enabling it to magnify even distant objects. The combined field of view of the two principal eyes´ corneal lenses would cover about 90° in front of the salticid spider, however a retina with the desired acuity would be too large to fit inside a spider´s eye. The surprising solution is a small, elongated retina, which lies behind a long, narrow tube and a second lens (a concave pit) at its end. Such combination of a corneal lens (with a long focal length) and a long eye tube (magnifying the image from the corneal lens) resembles a telephoto system, making the pair of principal eyes similar to a pair of binoculars.

The salticid spider captures light beams successively on four retina layers of receptors, which lie behind each other (in contrast, the human retina is arranged in only one plane). This structure allows not only a larger number of photoreceptors in a confined area but also enables color vision, as the light is split into different colours (chromatic aberration) by the lens system. Different wavelengths of light thus come into focus at different distances, which correspond to the positions of the retina´s layers. While salticids discern green (layer 1 – ~580 nm, layer 2 – ~520-540 nm), blue (layer 3 – ~480-500 nm) and ultraviolet (layer 4 – ~360 nm) using their principal eyes, it is only the two rearmost layers (layers 1 and 2) which allow shape and form detection due to their close receptor spacing.

As in human eyes, there is a central region in layer 1 called the “fovea”, where the inter-receptor spacing was measured to about 1 μm. This was found to be optimal, as the telephoto optical system provides images precise enough to be sampled in this resolution, but any closer spacing would reduce the retina´s sampling quality due to quantum-level interference between adjacent receptors. Equipped with such eyes, Portia exceeds any insect by far when it comes to visual acuity: While the dragonfly Sympetrum striolatus has the highest acuity known for insects (0.4°), the acuity of Portia is ten times higher (0.04°) with much smaller eyes. The human eye with 0.007° acuity is only five times better than Portia´s. With such visual precision, Portia would be technically able to discriminate two objects which are 0.12 mm apart from a distance of 200 mm. The spatial acuity of other salticid eyes is usually not far behind that of Portia.[10][11][12]

#### Principal eye retina movements

Such spectacular visual abilities come at a price within small animals as the jumping spiders: The retina in each of Portia´s principal eyes has only 2-5° field of view, while its fovea even captures only 0.6° field of view. This results from the principal retina having elongated boomerang-like shapes which span about 20° vertically and only 1° horizontally, corresponding to about six receptor rows. This severe limitation is compensated by sweeping the eye tube over the whole image of the scene using eye muscles, of which jumping spiders have six. These are attached to the outside of the principal eye tube and allow the same three degrees of freedom – horizontal, vertical, rotation – as in human eyes. Principal retinae can move by as much as 50° horizontally and vertically and rotate about the optical axis (torsion) by a similar amount.

Spiders making sophisticated use of visual cues move their principal eyes´ retinae either spontaneously, in “saccades” fixating the fovea on a moving visual target (“tracking”), or by “scanning”, which serves presumably for pattern recognition. It seems today, that spiders scan a scene sequentially by moving the eye-tube in complex patterns, allowing it to process high amounts of visual information despite their very limited brain capacities.

The spontaneous retinal movements, so-called “microsaccades”, are a mechanism thought to prevent the photoreceptor cells of the anterior-median eyes from adapting to a motionless visual stimulus. Cupiennius spiders, which feature 4 eye muscles - two dorsal and two ventral ones – continuously perform such microsaccades of 2° to 4° in the dorso-median direction, lasting about 80 ms (when fixed to a holder). The 2-4° of microsaccadic movements match closely to Cupiennius´ angle of about 3° between the receptor cells, supporting the idea of its function preventing adaption. In contrast, retinal movements elicited by mechanical stimulation (directing an air puff onto the tarsus of the second walking leg) can be considerably larger than the spontaneous retinal movements, with deflections up to 15°. Such stimulus increases eye muscle activity from being spontaneously active at 12 ± 1 Hz at the resting level to 80 Hz with the air puff stimulation applied. Active retinal movement of the two principal eyes is however never activated simultaneously during such experiments and no correlation exists between the two eyes regarding their direction either. These two mechanisms, spontaneous microsaccades as well as active “peering” by active retinal movement, seemingly allow spiders to follow and analyze stationary visual targets efficiently using only their principal eyes without reinforcing the saccadic movements by body movements.

However, there is another factor influencing visual capacities of a spider´s eye, which is the problem of keeping objects at different distances in focus. In human eyes, this is solved by accommodation, i.e. changing the shape of the lens, but salticids take a different approach: the receptors in layer 1 of their retina are arranged on a “staircase” at different distances from the lens. Thus, the image of any object, whether a few centimeters or some meters in front of the eye, will be in focus on some part of the layer-1 staircase. Additionally, the salticid can swing the eye tubes side to side without moving the corneal lenses and will thus sweep the staircase of each retina across the image of the corneal lense, sequentially obtaining a sharp image of the object.

The resulting visual performance is impressive: Jumping spiders such as Portia focus accurately on an object at distances between 2 centimeters to infinity, being able to see up to about 75 centimeters in practice. The time needed to recognize objects is however relatively long (seemingly in the range of 10-20 s) because of the complex scanning process needed to capture high-quality images from such tiny eyes. Due to this limitation, it is very difficult for spiders such as Portia to identify much larger predators fast enough because of the predator´s size, making the small spider an easy prey for birds, frogs and other predators.[13][14]

#### Blurry vision for distance estimation

An unexpected finding recently surprised researchers, when it was shown that jumping spiders use a technique called blurry vision to estimate their distance to previously recognized prey before taking a jump. Where humans achieve depth perception using binocular vision and other animals do so by moving their heads around or measuring ultrasound responses, jumping spiders perform this task within their principal eyes. As in other jumping spider species, the principal eyes of Hasarius adansoni feature four retinal layers with the two bottom ones featuring photocells responding to green impulses. However, green light will only ever focus sharply on the bottom one, layer 1, due to its distance from the inner lens. Layer 2 would receive focused blue light, however these photoreceptor cells are not sensitive to blue and receive a fuzzy green image instead. Interestingly, the amount of blur depends on the distance of an object from the spider´s eye – the closer it is, the more out of focus it will appear on the second retina layer. At the same time, the first retina layer 1 always receives a sharp image due to its staircase structure. Jumping spiders are thus able to estimate depth using a single unmoving eye by comparing the images of the two bottom retina layers. This was confirmed by letting spiders jump at prey in an arena flooded with green light versus red light of equal brightness. Without the ability to use the green retina layers, jumping spiders would repeatedly fail to judge distance accurately and miss their jump.

#### Secondary eyes

In contrast to the principal eyes responsible for object analysis and discrimination, a spider´s secondary eyes act as motion detectors and therefore do not feature eye muscles to analyze a scene more extensively. Depending on their arrangement on the spider´s carapace, secondary eyes enable the animal to have panoramic vision detecting moving objects almost 360° around its body. The anterior and posterior lateral eyes (i.e. secondary eyes) only feature a single type of visual cells with a maximum spectral sensitivity for green colored light of ~535-540 nm wavelength. The number and arrangement of secondary eyes differs significantly between or even within different spider families, as does their structure: Large secondary eyes can contain several thousand rhabdomeres (the light-sensitive parts of the retina) and support hunters or nocturnal spiders with their high sensitivity to light, while small secondary eyes contain at most a few hundred rhabdomeres and only providing basic movement detection. Differently from the principal eyes which are everted (the rhabdomeres point towards the light), the secondary eyes of a spider are inverted, i.e. their rhabdomeres point away from the light, as is the case for vertebrates like the human eye. Spatial resolution of the secondary eyes e.g. in the extensively studied Cupiennius salei is greatest in horizontal direction, enabling the spider to analyse horizontal movements well even with the secondary eyes, while vertical movement may not be especially important when living in a “flat world”.

The reaction time of jumping spiders´ lateral eyes is comparably slow and amounts to 80-120 ms, measured with a 3°-sized (inter-receptor angle) square stimulus travelling past the animal´s eyes. The minimum stimulus travel distances, until the spider reacts, are 0.1° at a stimulus velocity of 1°/s, 1° at 9°/s and 2.5° at 27°/s. This means that a jumping spider´s visual system detects motion even if an object is travelling only a tenth of the secondary eyes´ inter-receptor angle at slow speed. If the stimulus gets even smaller to a size of only 0.5°, responds occur only after long delays, indicating that they lie at the spiders´ limit of perceivable motion.

Secondary eyes of (night-active) spiders usually feature a tapetum behind the rhabdomeres, which is a layer of crystals reflecting light back to the receptors to increase visual sensitivity. This allows night-hunting spiders to have eyes with an aperture as large as f/0.58 enabling them to capture visual information even in ultra-low-light conditions. Secondary eyes containing a tapetum thus easily reveal a spider´s location at night when illuminated e.g. by a flashlight.[15][16]

### Central nervous system and visual processing in the brain

As anywhere in neuroscience, we still know very little about a spider´s central nervous system (CNS), especially regarding its functioning in visually controlled behavior. Of all the spiders, the CNS of Cupiennius has been studied most extensively, focusing mainly on the CNS structure. As of today, only little is known about electrophysiological properties of central neurons in Cupiennius, and even less about other spiders in this regard.

The structure of a spider´s nervous system is closely related to its body´s subdivisions, but instead of being spread all over the body, the nervous tissue is enormously concentrated and centralized. The CNS is made up of two paired, rather simple nerve cell clusters (ganglia), which are connected to the spider´s muscles and sensory systems by nerves. The brain is formed by fusion of these ganglia in the head segments ahead of and behind the mouth and fills the prosoma largely with nervous tissue, while no ganglia exist in the abdomen. Looking at the spider´s brain, it receives direct inputs from only one sensory system, the eyes - unlike any insects and crustaceans. The eight optic nerves enter the brain from the front and their signals are processed in two optic lobes in the anterior region of the brain. When a spider´s behavior is especially dependent on vision, as in the case of the jumping spider, the optic ganglia contribute up to 31% of the brain´s volume, indicating the brain to be almost completely devoted to vision. This score still amounts to 20% for Cupiennius, whereas other spiders like Nephila and Ephebopus come in at only 2%.

The distinction between principal and secondary eyes persists in the brain. Both types of eyes have their own visual pathway with two separate neuropil regions fulfilling distinct tasks. Thus spiders evidently process the visual information provided by their two eye types in parallel, with the secondary eyes being specialized for detecting horizontal movement of objects and the principal eyes being used for the detection of shape and texture.

### Two visual systems in one brain

While principal and secondary eyesight seems to be distinct in spiders´ brains, surprising inter-relations between both visual systems in the brain are known as well. In visual experiments principal eye muscle activity of Cupiennius was measured while covering either its principal or secondary eyes. When stimulating the animals in a white arena with short sequences of moving black bars, the principal eyes moved involuntarily whenever a secondary eye detected motion within its visual field. This activity increase of the principal eye muscles, compared to no stimulation presented, would not change when covering the principal eyes with black paint, but would stop with the secondary eyes masked. Thus it is now clear, that only the input received from secondary eyes controls principal eye muscle activity. Also, a spider´s principal eyes do not seem to be involved in motion detection, which is only the secondary eyes´ responsibility.

Other experiments using dual-channel telemetric registration of the eye muscle activities of Cupiennius have shown that the spider actively peers into the walking direction: The ipsilateral retina of the principal eyes was measured to shift with respect to the walking direction before, during and after a turn, while the contralateral retina remained in its resting position. This happened independently from the actual light conditions, suggesting a “voluntary” peering initiated by the spider´s brain.

### Pattern recognition using principal eyes

Recognition of shape and form by jumping spiders is believed to be accomplished through a scanning process of the visual field, which consists of a complex set of rotations (torsional movements) and translations of the anterior-median eyes´ retinae. As described in the section “Principal eye retina movements”, a spider´s retinae are narrow and shaped like boomerangs, which can be matched with straight features by sweeping over the visual scene. When investigating a novel target, the eyes scan it in a stereotyped way: By moving slowly from side to side at speeds of 3-10° per second and rotating through ± 25°, horizontal and torsional retina movement allows the detection of differently positioned and rotated lines. This method can be understood as template matching where the template has elongated shape and produces a strong neural response whenever the retina matches a straight feature in the scene. This identifies a straight line with little or no further processing necessary.

A computer vision algorithm for straight line detection as an optimization problem (da Costa, da F. Costa) was inspired by the jumping spider´s visual system and uses the same approach of scanning a scene sequentially using template matching. While the well-known Hough Transform allows robust detection of straight visual features in an image, its efficiency is limited due to the necessity to calculate a good part or even the whole parameter space while searching for lines. In contrast the alternative approach used in salticid visual systems suggests searching the visual space by using a linear window, which allows adaptive searching schemes during the straight line search process without the need to systematically calculate the parameter space. Also, solving the straight line detection in such a way allows to understand it as an optimization problem, which makes efficient processing by computers possible. While it is necessary to find appropriate parameters controlling the annealing-based scanning experimentally, the approach taking a jumping spider´s path of straight line detection was proven to be very effective, especially with properly set parameters.[17]

### Visually-guided behavior

#### Discernment of visual targets

The ability of discerning between slightly different visual targets has been shown for Cupiennius salei, although this species relies mainly on its mechanosensory systems during prey catching or mating behavior. When presenting two targets at a distance of 2 m to the spider, its walking path depends on their visual appearance: Having to choose between two identical targets such as vertical bars, Cupiennius shows no preference. However the animal strongly prefers a vertical bar to a sloping bar or a V-shaped target.

The discrimination of different targets has been shown to be only possible with the principal eyes uncovered, while the spider is able to detect the targets using any of the eyes. This suggests that many spiders´ anterior-lateral (secondary) eyes are capable of much more than simply object movement detection. With all eyes covered, the spider exhibits totally undirected walking paths.

Placing Cupiennius in total darkness however results not only in undirected walks but also elicits a change of gait: Instead of using all eight legs the spider will only walk with six and employ the first legs as antennae, comparable to a blind person´s cane. In order to feel the surroundings the extended forelegs are moved up and down as well as sideways. This is specific to the first leg pair only, influenced solely by the visual input when the normal room light is switched to the invisible infrared light.

#### Vision-based decision making in jumping spiders

The behavior of jumping spiders after having detected movement with the eyes depends on three factors: the target´s size, speed and distance. If it has more than twice the spider´s size, the object is not approached and the spider tries to escape if it comes towards her. If the target has adequate size, its speed is visually analyzed using the secondary eyes. Fast moving targets with a speed of more than 4°/s are chased by jumping spiders, guided by her anterior-lateral eyes. Slower objects are carefully approached and analyzed with the anterior-median (i.e. principal) eyes to determine whether it is prey or another spider of the same species. This is seemingly achieved by applying the above described straight line detection, to find out whether a visual target features legs or not. While jumping spiders have shown to approach potential prey of appropriate characteristics as long as it moves, males are pickier in deciding whether their current counterpart might be a potential mate.

#### Potential mate detection

Experiments have shown that drawings of a central dot with leg-like appendages on the sides will result in courtship displays, suggesting that visual feature extraction is used by jumping spiders to detect the presence and orientation of linear structures in the target. Additionally, a spider´s behavior towards a considered conspecific spider depends on different factors such as sex and maturity of both involved spiders and whether it is mating time. Female wolf spiders, Schizocosa ocreata, even discern asymmetries in male secondary sexual characters when choosing their mate, possibly to avoid developmental instability in their offspring. Conspicuous tufts of bristles on a male´s forelegs, which are used for visual courtship signaling, appear to influence female mate choice and asymmetry of these body parts in consequence of leg loss and regeneration apparently reduces female receptivity to such male spiders.[18]

#### Secondary eye-guided hunting

A jumping spider´s stalking behavior when hunting insect prey is comparable to a cat stalking birds. If something moves within the visual field of the secondary eyes, they initiate a turn to bring the larger, forward-facing pair of principal eyes into position for classifying the object´s shape into mate, rival or prey. Even very small, low contrast dot stimuli moving at slow or fast speeds elicit such orientation behavior. Like Cupiennius, jumping spiders are also able to use their secondary eyes for more sophisticated tasks than just motion detection: Presenting visual prey cues to salticids with only visual information from the secondary eyes available and both primary eyes covered, results in the animal exhibiting complete hunting sequences. This suggests that the anterior lateral eyes of jumping spiders may be the most versatile components of their visual system. Besides detecting motion, the secondary eyes obviously also feature a spatial acuity which is good enough to direct complete visually-guided hunting sequences.

#### Prey “face recognition”

Visual cues also play an important role for jumping spiders (salticids) when discriminating between salticid and non-salticid prey using principal eyesight. To this end a salticid prey´s large principal eyes provide critical cues, to which the jumping spider Portia fimbriata reacts by exhibiting cryptic stalking tactics before attacking (walking very slowly with palps retracted and freezing when faced). This behavior is only used when identifying a prey as salticid. This was exploited in experiments presenting computer-rendered, realistic three-dimensional lures with modified principal eyes to Portia fimbriata. While intact virtual lures resulted in cryptic stalking, lures without or with smaller principal eyes than usual (as sketched in the figure on the right) elicited different behavior. Presenting virtual salticid prey with only one anterior-median eye or a regular lure with two enlarged secondary eyes elicited cryptic stalking behavior suggesting successful recognition of a salticid, while P. fimbriata froze less often when faced by a Cyclops-like lure (a single principal eye centered between the two secondary eyes). Lures with square-edged principal eyes were usually not classified as a salticid, indicating that the shape of the principal eyes´ edges are an important cue to identify fellow salticids.[19]

#### Jumping decisions from visual features

Spiders in the genus Phidippus have been tested within a study for their willingness to cross inhospitable open space by placing visual targets on the other side of a gap. It was found that whether the spider takes the risk of crossing open ground or not is mainly dependent on factors like distance to target, relative target size compared to distance and the target´s color and shape. In independent test runs, the spider moved to tall, distant targets equally often as to short, close targets, with both objects appearing equally sized on the spider´s retina. When giving the choice of moving to either white or green grass-like targets, the spiders consistently chose the green target irrespective of its contrast with the background, thus proving their ability to use color discernment in hunting situations.[20]

#### Identifying microhabitat traits by visual cues

Presented with manipulated real plants and photos of plants, Psecas chapoda (a bromeliad-dwelling salticid spider) is able to detect a favorable microhabitat by visually analyzing architectural features of the host plant´s leaves and rosette. By using black-and-white photos, any potential influence of other cues, such as color and smell, on host plant selection by the spider could be excluded during a study, leaving only shape and form as discerning characteristics. Even when having to decide solely from photographs, Psecas chapoda consistently preferred rosette-shaped plants (Agavaceae) with narrow and long leaves over differently looking plants, which proves that some spider species are able to evaluate and distinguish physical structure of microhabitats only on the basis of shape from visual cues of plant traits.[21]

## References

1. K. Gammon, Life’s Little Mysteries (http://www.lifeslittlemysteries.com/1647- smartest-non-primates.html) . TechMediaNetwork.
2. G. S. et al., Control of Octopus Arm Extension by a Peripheral Motor Program . Science 293, 1845, 2001.
3. Y. Gutfreund, Organization of octopus arm movements: a model system for study- ing the control of flexible arms. Journal of Neuroscience 16, 7297, 1996.
4. P. Graziadei, The anatomy of the nervous system of Octopus vulgaris, J. Z. Young. Clarendon, Oxford, 1971.
5. M. J. Wells, The orientation of octopus. Ergeb. Biol. 26, 40-54, 1963.
6. J. L. Fox and T. L. Daniel (2008), "A neural basis for gyroscopic force measurement in the halteres of Holorusia.", J Comp Physiol 194: 887-897
7. a b Rhoe A. Thompson (2009), "Haltere Mediated Flight Stabilization in Diptera: Rate Decoupling, Sensory Encoding, and Control Realization.", PhD thesis (University of Florida)
8. a b J. W. S. Pringle (1948), "The gyroscopic mechanism of the halteres of diptera.", Phil Trans R Soc Lond B 233 (602): 347-384
9. F. G. Barth: A Spider´s World: Senses and Behavior. ISBN 978-3-642-07557-5, Springer-Verlag Berlin, Heidelberg. (2002)
10. D. P. Harland, R. R. Jackson: 'Eight-legged cats' and how they see - a review of recent research on jumping spiders (Araneae: Salticidae). Department of Zoology, University of Canterbury (2000)
11. A. Schmid: Different functions of different eye types in the spider Cupiennius salei. The Journal of Experimental Biology 201, 221–225 (1998)
12. S. Yamashita, H. Tateda: Spectral Sensitivities of Jumping Spider Eyes. J. comp. Physiol. 105, 29-41 (1976)
13. D. P. Harland, R. R. Jackson: Inﬂuence of cues from the anterior medial eyes of virtual prey on Portia ﬁmbriata, an araneophagic jumping spider. The Journal of Experimental Biology 205, 1861–1868 (2002)
14. A. Schmid, C. Trischler: Active sensing in a freely walking spider: Look where to go. Journal of Insect Physiology 57 p.494–500 (2011)
15. D. B. Zurek, X. J. Nelson: Hyperacute motion detection by the lateral eyes of jumping spiders. Vision Research 66 p.26–30 (2012)
16. D. B. Zurek, A. J. Taylor, C. S. Evans, X. J. Nelson: The role of the anterior lateral eyes in the vision-based behaviour of jumping spiders. The Journal of Experimental Biology 213, 2372-2378 (2010)
17. F. M. G. da Costa, L. da F. Costa: Straight Line Detection as an Optimization Problem: An Approach Motivated by the Jumping Spider Visual System. In: Biologically Motivated Computer Vision, First IEEE International Workshop, BMVC 2000, Seoul, Korea (2000)
18. G.W. Uetz, E. I. Smith: Asymmetry in a visual signaling character and sexual selection in a wolf spider. Behav Ecol Sociobiol (1999) 45: 87–93
19. D. P. Harland, R. R. Jackson: Inﬂuence of cues from the anterior medial eyes of virtual prey on Portia ﬁmbriata, an araneophagic jumping spider. The Journal of Experimental Biology 205, 1861–1868 (2002)
20. R. R. Jackson, D. P. Harland: One small leap for the jumping spider but a giant step for vision science. THE JOURNAL OF EXPERIMENTAL BIOLOGY, JEB Classics p.2129-2132
21. P. M. de Omena, and G. Q. Romero: Using visual cues of microhabitat traits to ﬁnd home: the case study of a bromeliad-living jumping spider (Salticidae). Behavioral Ecology 21:690–695 (2010)


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# Difficult Percentages Solved QuestionAptitude Discussion

 Q. $B$ as a percentage of $A$ is equal to $A$ as a percentage of $(A + B)$. Find $B$ as a percentage of $A$.
 ✔ A. $62\%$ ✖ B. $73\%$ ✖ C. $41\%$ ✖ D. $57\%$

Solution:
Option(A) is correct

From the question stem, we know

$\dfrac{B}{A}=\dfrac{A}{A+B}$

As $B$ is a percent of $A$, let us assume $B=Ax$

Then, equation (1) can be re-written as

$x=\dfrac{1}{1+x}$

$\Rightarrow x(1+x)=1$

After this step, trial and error works best. The solution is $62\%$, answer choice ($A$) is the correct one.

Alternatively, solving, we get

$x^2+x+1=0$

$\Rightarrow x=\dfrac{-1+\sqrt{5}}{2}$

or

$\Rightarrow x=\dfrac{-1-\sqrt{5}}{2}$

So,

$\Rightarrow x=\dfrac{-1+\sqrt{5}}{2}$

$=0.62=62\%$

## (1) Comment(s)

Kotresh
()

If the percentage are different then how to solve such type of questions?


text:
# Math Help - dense subsets

1. ## dense subsets

in $C[a,b]$, any dense subset is possible other then set of polynomials.

2. ## Re: dense subsets

Originally Posted by saravananbs
in $C[a,b]$, any dense subset is possible other then set of polynomials.
Yes, the algebra generated by $\mathcal{F}=\{e^{nx}:n\in\mathbb{Z}\}$ contains the constant functions and separate points. Now, use the Stone-Weierstrass theorem.


text:
# Comparing mean, median ,mode

Compare the mean, median and mode for the set of numbers

{$$3, 4, 6, 7, 7, 7, 8, 9, 9, 10$$}

Which of the following is true?

Try this: All of my problems

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## Linear Recurrence Relations

Take your recursion skills to the next level. If you've got a recurrence relation but no computer, how can you find a closed form? What about asymptotic behavior? How fast do rabbits reproduce?

# Repeated Roots

The sequence $$\{x_n\}$$ satisfies the relation $\begin{array} &x_1 = 2, &x_2 = 9, &x_{n+2} = 6 x_{n+1} - 9 x_n &(n = 1, 2, 3, \ldots). \end{array}$ What is the value of $$x_5?$$

The sequence $$\{x_n\}$$ satisfies the relation $\begin{array} &x_1 = 3, &x_2 = 9, &x_3 = 29, &x_{n+3} = 9 x_{n+2} - 26 x_{n+1} + 24 x_n, \end{array}$ where $$n$$ is a positive integer. Find the closed-form expression of $$x_n.$$

The sequence $$\{x_n\}$$ satisfies the relation $\begin{array} &x_1 = -3, &x_2 = -9, &x_3 = 9, &x_{n+3} = 9 x_{n+2} - 27 x_{n+1} + 27 x_n, \end{array}$ where $$n$$ is a positive integer. Find the closed-form expression of $$x_n.$$

The sequence $$\{x_n\}$$ satisfies the relation $\begin{array} &x_1 = -7, &x_2 = 0, &x_3 = 6, &x_{n+3} = 2 x_{n+2} + 5 x_{n+1} - 6 x_n, \end{array}$ where $$n$$ is a positive integer. What is the value of $$x_6?$$

The sequence $$\{x_n\}$$ satisfies the relation $\begin{array} &x_1 = 0, &x_2 = 1, &x_{n+2} + 2 x_{n+1} + x_n = 0 &(n = 1, 2, 3, \ldots). \end{array}$ If $$x_n$$ can be expressed as $a r^n + b n r^n,$ what is the value of $$\frac{b}{a}?$$

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Get our free extension to see links to code for papers anywhere online!

# Signal propagation in continuous approximations of binary neural networks

The training of stochastic neural network models with binary ($\pm1$) weights and activations via a deterministic and continuous surrogate network is investigated. We derive, using mean field theory, a set of scalar equations describing how input signals propagate through the surrogate network. The equations reveal that these continuous models exhibit an order to chaos transition, and the presence of depth scales that limit the maximum trainable depth. Moreover, we predict theoretically and confirm numerically, that common weight initialization schemes used in standard continuous networks, when applied to the mean values of the stochastic binary weights, yield poor training performance. This study shows that, contrary to common intuition, the means of the stochastic binary weights should be initialised close to $\pm 1$ for deeper networks to be trainable.