text:
## 2.1 Basic Data Formats For Networks

There are three basic data formats that can be used to describe networks: adjacency matrix, edge list, and adjacency list. Each format has its pros and cons. There are other variations on these (e.g., a biadjacency matrix for bipartite networks).

An adjacency matrix is a matrix in which the rows and columns represent different nodes. In an unweighted adjacency matrix, the edges (i.e., lines) are represented by 0 or 1, with indicating that these two nodes are connected. If two nodes are connected, they are said to be adjacent (hence the name, adjacency matrix). In a weighted matrix, however, you can have different values, indicating different edge qualities (or tie strengths).

Let’s start by loading the igraph package and setting up a toy network (same as in Lesson 1: Intro)

library(igraph)
## Warning: package 'igraph' was built under R version 3.5.2
g=make_graph(~A-B-C-A, D-E-F-D, A-F)
V(g)$color=c("white", "red", "green", "blue", "orange", "yellow") E(g)$weight=1:7
E(g)$color=rainbow(7) We can now extract the adjacency matrix of the network we created, called g: as_adjacency_matrix(g, sparse=F) ## A B C D E F ## A 0 1 1 0 0 1 ## B 1 0 1 0 0 0 ## C 1 1 0 0 0 0 ## D 0 0 0 0 1 1 ## E 0 0 0 1 0 1 ## F 1 0 0 1 1 0 Note the argument sparse=F in the code above. This displays the adjacency matrix with 0s. If sparse=T, the output is a special format of the matrix where the 0s are replaced with a period (this is to make it easier to see very large matrices). Also note that, because the network is undirected and unweighted, the corresponding adjacency matrix is symmetrical (value for row A, column B is identical to row B, column A) and binary (values are 0 or 1). ### 2.1.2 Edge List An edge list is a two-column list of the two nodes that are connected in a network. In the case of a directed network, the convention is that the edge goes from the vertex in the first column to the vertex in the second column. In an undirected network, the order of the vertices don’t matter. For weighted networks, you may have a third column that indicates the edge weight. You can get the edgelist of any igraph object as well: as_edgelist(g) ## [,1] [,2] ## [1,] "A" "B" ## [2,] "A" "C" ## [3,] "A" "F" ## [4,] "B" "C" ## [5,] "D" "E" ## [6,] "D" "F" ## [7,] "E" "F" ### 2.1.3 Affiliation Matrix (aka individual-by-group matrix) In many cases, we will construct social networks from co-membership in groups. For example, we would draw edges between individuals based on their patterns of co-occurrence in a flock. Similarly, we could construct networks of species co-occurrences in populations, etc. To do this, we would first need data in a matrix in which rows represent individuals (or species) and columns represent groups (or populations). Note that you could flip the columns and rows–either way is fine. You just need to be aware of how you arranged it. Here’s a toy example in which individuals A through E occur in different combinations in 4 groups. A=c(1,1,0,0) B=c(1,0,1,0) C=c(1,0,1,0) D=c(0,1,0,1) E=c(0,0,1,1) aff=matrix(c(A,B,C,D,E),nrow=5,byrow=TRUE) dimnames(aff)=list(c("A","B","C","D","E"),c("Group1","Group2","Group3","Group4")) aff #The individual-by-group matrix ## Group1 Group2 Group3 Group4 ## A 1 1 0 0 ## B 1 0 1 0 ## C 1 0 1 0 ## D 0 1 0 1 ## E 0 0 1 1 There are different ways to convert this data into a social network–i.e., a network that describes which individual co-occurs with which individual in groups. One simple way is to do what is called a one-mode projection of this data by multiplying this matrix with the transpose of itself. Note that matrix multiplication notation is %*% in R. aff %*% t(aff) ## A B C D E ## A 2 1 1 1 0 ## B 1 2 2 0 1 ## C 1 2 2 0 1 ## D 1 0 0 2 1 ## E 0 1 1 1 2 This resulting matrix is now an adjacency matrix in which the diagonal represents how many groups each individual participated in (2 for all of them), and the off-diagonals represent the number of times a pair of individuals were int he same group. You can use this as the adjacency matrix to convert this into a network: m2=aff %*% t(aff) g2=graph_from_adjacency_matrix(m2, "undirected", weighted=T, diag=F) plot(g2, edge.width=E(g2)$weight)

An adjacency list, also known as a node list, presents the ‘focal’ node on the first column, and then all the other nodes that are connected to it (i.e., adjacent to it) as columns to the right of it. In a spreadsheet, would be a table with rows with different number of columns, which is often very awkward to deal with, like this:

Focal Node Neighbor_1 Neighbor_2 Neighbor_3
A B C F
B A C
C A B
D E F
E D F
F A D E

In R, you can display an adjacency list as an actual ‘list object’, with each item representing neighbors of each focal node:

as_adj_list(g)
## $A ## + 3/6 vertices, named, from f4bdde8: ## [1] B C F ## ##$B
## + 2/6 vertices, named, from f4bdde8:
## [1] A C
##
## $C ## + 2/6 vertices, named, from f4bdde8: ## [1] A B ## ##$D
## + 2/6 vertices, named, from f4bdde8:
## [1] E F
##
## $E ## + 2/6 vertices, named, from f4bdde8: ## [1] D F ## ##$F
## + 3/6 vertices, named, from f4bdde8:
## [1] A D E

## 2.2 Data formats for directed and weighted networks

Let’s consider some important aspects of data formats that come with networks that are directed or weighted. I will keep this short by listing some important things to consider, and a line of code that will display this.

### 2.2.1 Directed networks

Let’s create an igraph object for a directed network called dir.g. For directed networks, the adjacency matrix is not symmetrical. Rather, the cell value is 1 if the edge goes from the row vertex to the column vertex.

dir.g=make_graph(~A-+B-+C-+A, D-+E-+F-+D, A+-+F)
plot(dir.g)

as_adjacency_matrix(dir.g, sparse=F)
##   A B C D E F
## A 0 1 0 0 0 1
## B 0 0 1 0 0 0
## C 1 0 0 0 0 0
## D 0 0 0 0 1 0
## E 0 0 0 0 0 1
## F 1 0 0 1 0 0

For directed networks with mutual edges (represented by double-edged arrows), the edge list lists both directions separately:

as_edgelist(dir.g)
##      [,1] [,2]
## [1,] "A"  "B"
## [2,] "A"  "F"
## [3,] "B"  "C"
## [4,] "C"  "A"
## [5,] "D"  "E"
## [6,] "E"  "F"
## [7,] "F"  "A"
## [8,] "F"  "D"

You can see that, since the dir.g network object contains one mutual edge (A<->F), the edge list has 8 rows, while the edgelist for the undirected version of the network has 7 rows.

### 2.2.1 Weighted networks

Let’s now consider what the data formats would look like. To do this, let’s go back to our original network, g. Let’s say that the edge widths that we added represent edge weights or values. Then, the adjacency matrix for this network can be shown by using the attr= argument within the function to call the adjacency matrix to specify the edge weights:

as_adjacency_matrix(g, sparse=F, attr="weight")
##   A B C D E F
## A 0 1 2 0 0 3
## B 1 0 4 0 0 0
## C 2 4 0 0 0 0
## D 0 0 0 0 5 6
## E 0 0 0 5 0 7
## F 3 0 0 6 7 0

You can display the edge weights as an edgelist as well. In fact, igraph has a convenient function that will display all of the edge attributes together as a data frame:

as_data_frame(g)
##   from to weight     color
## 1    A  B      1 #FF0000FF
## 2    A  C      2 #FFDB00FF
## 3    A  F      3 #49FF00FF
## 4    B  C      4 #00FF92FF
## 5    D  E      5 #0092FFFF
## 6    D  F      6 #4900FFFF
## 7    E  F      7 #FF00DBFF

Recall that in undirected networks, the “from” and “to” designation are arbitrary (it is simply organized in alphabetical order here).

If you want to show an edge list as a three-column matrix with the two nodes and edge weights only, you can just specify which edge attribute you want to use as the edge weight, e.g.:

as_data_frame(g)[,c("from", "to", "weight")]
##   from to weight
## 1    A  B      1
## 2    A  C      2
## 3    A  F      3
## 4    B  C      4
## 5    D  E      5
## 6    D  F      6
## 7    E  F      7

## 2.3 Going from Data to Networks

### 2.3.1 Creating a network from your edge list

Creating a network from an edgelist that you have created is easy. First, import the .csv file called “sample_edgelist.csv”.

edge.dat=read.csv("https://dshizuka.github.io/network2018/NetworkWorkshop_SampleData/sample_edgelist.csv")
edge.dat
##        V1      V2 weight
## 4   Betty Charles      3
## 5   Betty  Daniel      1
## 6   Betty   Frank      2
## 7  Daniel  Esther      1
## 8  Daniel   Frank      1
## 9  Esther   Frank      1
## 10  Frank    Gina      2

So this data frame has three columns: the first two columns are the edge list, and the third column is an edge value we called “weight”. If we have the data organized this way, we can simply use a function called graph.data.frame() to create a network we will call eg.

set.seed(2)
eg=graph_from_data_frame(edge.dat, directed=FALSE)
eg
## IGRAPH c85a423 UNW- 7 10 --
## + attr: name (v/c), weight (e/n)
## + edges from c85a423 (vertex names):
##  [9] Esther--Frank   Frank --Gina


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# When is an attractor meaningful?

I’m originally a computer scientist; so I hope my question is not trivial. I’m working with time series and want to reconstruct the phase space from the time series based on time-lagged versions of time series. For this purpose I need to calculate $m$ and $\tau$ which are the embedding dimension and time-lag respectively.

But: Today I was seeing some experiments based on some random $m$ and $τ$ and I saw some topology in their attractor after which this question came to my mind:

Regardless of how one chooses $m$ and $\tau$, if the attractor has a meaningful topology (i.e. following a geometrically structured shape), does it mean that this attractor could capture some meaningful dynamics?

In other words assume that I calculated $m=M$ and $\tau=T$, but I get a meaningful attractor shape for another $m$ and $\tau$. How can I interpret it?

Let’s look at what happens when you make non-optimal selections of $τ$ and $m$:

• If $m$ is too low, the attractor will not be completely unfolded, i.e., parts of your reconstructed attractor will overlap when they shouldn’t. This is equivalent to trajectories crossing in phase space, which means that what you have reconstructed is not a proper attractor or phase-space reconstruction, in particular it’s not physically meaningful.

Example: Every two-dimensional picture of the Lorenz attractor, e.g., this one.

• If $m$ is too high, nothing bad happens except that you have spurious dimensions to deal with when further processing your result.

Example: You embed the attractor of a driven and damped oscillator (a circle) in three dimensions, instead of two. It is still a circle.

• If $τ$ is slightly off the optimum, your reconstruction will not be optimal for seeing the structure, but it will still be a valid reconstruction. In fact, Takens’ embedding theorem gives you that almost every choice of $τ$ yields a proper embedding, if you only go by topological equivalence. The advantage of the optimal $τ$ rather are that you can best see the structure of the attractor, that numerical analyses of the attractor are more feasible, and that measurement noise and similar have a minimum impact on your reconstruction.

Example: If you use a non-optimal $τ$ for embedding the attractor of a driven and damped oscillator, you do not get a circle, but an ellipsis. The further off your $τ$ is, the flatter is the ellipsis. It is easy to see that if your ellipsis is sufficiently flat and you have sufficiently strong measurement noise, the structure is lost. Furthermore there are discrete choices of $τ$ where the ellipsis collapses to a line, which are the reason why only almost every choice of $τ$ yields a proper embedding.

• If $τ$ is far off the optimum, you still are very likely to have a proper embedding in the topological sense but the attractor’s structure is unnecessarily complicated from a visual point of view and further analysis of your results become more difficult.

Example: Fraser and Swinney, PRA 33, 1134 (1986) contains some illustrative examples, in particular in Fig. 1.

So, if you choose $m$ and $τ$ non-optimal, you can still get meaningful results. Keep in mind, that attractor reconstruction can only unveil the topology of the attractor, and topologically equivalent shapes can be very different by other criteria. Finally, note that there is no straightforward way to obtain an optimal $τ$ and $m$ for empirical data – you cannot know whether your choices are good before you actually looked at the results (and even then, it may be difficult to tell).

• Thanks for your comprehensive answer :) I'll wait a bit more and if no one else provided a better answer I'll accept yours. Thank you again! – Kasra Manshaei Mar 2 '16 at 8:25

I don't believe the time lag is critical. Selecting a bad one might mean that you need to analyze more data to fill in the phase space, but it should still generate it given sufficient time. As for the meaningful aspect, that does depend critically on the embedding dimension. Usually one uses the human capacity for pattern recognition to extract meaning from such data. If the imbedding dimension is poorly selected, you are only seeing a slice of the phase space and that may complicate pattern recognition.

That there is any real meaning in this endeavor depends on the degree of universality in the class of nonlinear problem you are investigating.

format: plain


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# Saturday, June 18, 2016¶

Today I did nothing but adapting the test suites. The test suites of Lino Così, Lino Welfare and Lino Voga needed quite some changes because the demo data has changed a lot. Now they pass again.

The documents How Lino Voga generates invoices and Sales management in Lino Voga now contain more meaningful tests. Though I am far from being satisfied.

I renamed the demo_bookings fixture of lino_xl.lib.finan to payments, and replaced all occurences of demo_bookings in demo_fixtures by demo_bookings payments. This was needed because otherwise the payments fixture was run before the fixtures that generated the causes for these payments.

Now that we have a separate fixture for payments, I can rename the fixture which creates automatic invoices back from demo2 to demo_bookings.

I removed the lino_voga.projects.docs demo project because it has no real-world usage, and because it is already enough work to get the two others (edmund and roger) documented correctly.

I fixed a bug in lino.api.doctest.show_fields().


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## Does Living On Campus Help Students?

### Background

In an effort to both improve students’ qualities of life and compete for students, many universities have utilized on-campus housing requirements to provide increased support to students. By requiring that students live in residence halls or dorms, universities can ensure that students have greater opportunities to connect with the university and can more directly provide support to struggling students.

The rising interest in student well-being is demonstrated through university spending habits. Universities expenditures have consistently risen as a percentage of total university expenditures since 2003, and this implies that universities are concurrently decreasing the share of spending for other university expenditures. This change is illustrated in the graph below.

In addition to altruistic concern for student well-being, universities are increasingly competing for students as well.

### Methods and Data

To address the question of how on-campus housing requirements have impacted students, I estimate a regression model of the following form:

$$Y_{it} = \beta_{0} + \beta_{1}HouseReq_{it} + \boldsymbol{X}_{it} + \varepsilon_{it}$$

For each university $$i$$ and year $$t$$, the outcome variable $$Y$$ represents universities’ graudation rates, $$HouseReq$$ is the length of the housing requirement, and $$\boldsymbol{X}$$ is a vector of university characteristics.

To estimate this equation, I use data from the Integrated Postsecondary Education Data System (IPEDS). This database provides graduation rates as well as university characteristics for all U.S. universities. In addition, I collected housing requirement data from university webpages. A summary of these data as well as sources can be found in the raw data on GitHub.

### Results

The results of the estimates of the regression above are summarized in the table below:

 (1) (2) (3) Housing Requirement $$0.0355^{***}$$ $$0.0337^{***}$$ $$0.0172^{***}$$ Auxiliary Spending $$-0.0276^{***}$$ $$0.0327^{***}$$ Student Spending $$0.0427^{***}$$ $$0.0436^{***}$$ Instruction Spending $$0.0472^{***}$$ $$0.0267^{***}$$ Private -0.0144 Bed-Enrollment Ratio $$0.0182$$ Total Undergraduates $$-0.0829^{***}$$ Local 2BR Cost $$0.00681$$ Observations 1,207 886 714 $$R^{2}$$ 0.076 0.303 0.515

The results in the table above indicate that the length of the housing requirement is positively correlated with graduation rates. Notably, different types of university spending are also correlated with graduation rates, as well as the total number of students at the institution. These results are limited by a small sample size, but they provide suggestive evidence that longer on-campus housing requirements do have positive effects on students.

### Conclusion

In this paper, I find that longer on-campus housing requirements do have positive effects on student graduation rates. This study is limited by the small number of universities in this study as well as the lack of granularity in the data, but these results suggest that on-campus housing requirements are good for students. Universities should take this into account as they endeavor to improve outcomes for students. For more information, read the paper here and the code here.


text:
Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions
Open Access Publications from the University of California

## Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions

• Author(s): Sterbenz, Jacob;
• Tataru, Daniel
• et al.

## Published Web Location

https://doi.org/10.1007/s00220-010-1061-4
Abstract

In this article we consider large data Wave-Maps from $${\mathbb R^{2+1}}$$ into a compact Riemannian manifold $${(\mathcal{M},g)}$$ , and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. This is a companion to our concurrent article [21], which together with the present work establishes a full regularity theory for large data Wave-Maps.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.


text:
## niedziela, 23 czerwca 2013

### Syntax highlighting with TeX4ht

When I was evaluating different options for a blog development I spent some time on the TeX4th. Although, I haven't chosen this technology I found it very interesting and I would like to share its goodness.

One of the important aspect of all the blogs about programming is how they display source code snippets. As always there is no one answer how to do it. Some people just wrap their code in the <pre> and code tags. Others care more about the appearance of their posts and highlight the syntax accordingly to the programming language they use. I wanted the code I share to look good. That's why I draw my attention the TeX4ht.

## Listings package

In LaTeX there is a listings package which can be used to format source code. It offers environment similar to verbatim but with many parameters to customize the output.

This is an example of how one can add a code block to a LaTeX article.

\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}

\usepackage{listings}
\lstset{
language=[Sharp]C,
basicstyle=\ttfamily\small,
identifierstyle=\sffamily,
keywordstyle=\sffamily\bfseries,
stringstyle=\rmfamily\itshape,
numberstyle=\scriptsize,
showstringspaces=false,
tabsize=2,
numbers=left,
}

\begin{document}
\begin{lstlisting}[float, caption={Sample code}]
class Program
{
static void Main(string[] args)
{
// This is comment
var text = "This is text";
var number = 12345;
Console.WriteLine(text + number.ToString());
}
}
\end{lstlisting}
\end{document}

Once compiled to PDF it looks very nice. Even though everything is black and white every part of the code has its unique style.

## TeX4ht

The LaTeX document presented in the previous listing can be compiled to the HTML using TeX4ht with the following command

>>htlatex Sample.tex

Unfortunately the output produced by default is not as pretty as it was in the PDF. The fonts have their style but the code is no longer aligned. There is no space between numbers and text. Comments are not aligned with the rest of the code.

Listing 1: Sample code
1class Program
2{
3    static void Main(string[] args)
4    {
5        // This is comment
6        var text = This is text;
7        var number = 12345;
8        Console.WriteLine(text + number.ToString());
9    }
10}

Listing package supports four different modes of alignment. By default it uses a fixed mode where a character is a single unit of output and they are aligned in columns. This mechanism does not port to HTML. in order to achieve the similar effect one should use monospace fonts. However this has its own problems because in LaTeX this corresponds to a typewriter (/ttfamily) font which cannot be styled.

As I mentioned it in the previous post the best solution I found was at the StaskExchange

Instead of trying to force TeX4ht to produce different styles for the listing generated with listings package it is easier to override the style used in the output. For this to work all the styles used in the listings should be unique (eg. basicstyle, identifierstyle, ...). If you look at the lstset definition of the first listing, you will see that it satisfies this requirement.

The next step was to define the CSS configuration. In order to do it I used Internet Explorer Developer Tools to select elements and capture their classes. Then I was able to create a private configuration File for the TeX4ht.

\Preamble{html}
\begin{document}
% basicstyle
\Css{div.lstlisting .cmtt-10 {font-family:monospace; color:DimGray}}
% identifierstyle
\Css{div.lstlisting .cmss-10 {font-family:monospace; color:Black}}
% keywordstyle
\Css{div.lstlisting .cmssbx-10 {font-family:monospace; color:Blue}}
\Css{div.lstlisting .cmr-10 {font-family:monospace; color:Green}}
% stringstyle
\Css{div.lstlisting .cmti-10 {font-family:monospace; color:DarkRed}}
% numberstyle
\Css{div.lstlisting .cmr-8 {display:inline-block; width:20px}}
\EndPreamble

Please notice custom style for the div.lstlisting block. This hasn't been mentioned on the StackExchange but it is required for the line numbering to work.

In order to include the configuration file I used slightly modified command line.

>>htlatex Sample.tex Sample.cfg

Finally it all worked. The listing produced has line numbering. All the elements of the syntax are highlighted and everything is aligned exactly the same way as in the source code.

Listing 1: Sample code
1class Program
2{
3    static void Main(string[] args)
4    {
5        // This is comment
6        var text = This is text;
7        var number = 12345;
8        Console.WriteLine(text + number.ToString());
9    }
10}

This post with all the resources is available on GitHub:


text:
Infoscience

Journal article

# The spectrum of heavy tailed random matrices

Let X-N be an N x N random symmetric matrix with independent equidistributed entries. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of X-N , once renormalized by root N, converges almost surely and in expectation to the so-called semicircular distribution as N goes to infinity. In this paper we study the same question when P is in the domain of attraction of an alpha-stable law. We prove that if we renormalize the eigenvalues by a constant a(N) of order N-1/alpha, the corresponding spectral distribution converges in expectation towards a law mu(alpha) and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero.


text:
Courses
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# Static Friction - JEE Important Topic

## Introduction to Static Friction

Last updated date: 20th Mar 2023
Total views: 81.9k
Views today: 1.60k

As we already know, friction is a force that prevents the rolling or smooth motion of one object over another. This is beneficial sometimes when a person is walking on a road, this friction helps him/her to not fall or roll on the road and stay still and continue walking on it. This force can restrain the motion over solids or liquid surfaces or any material elements. This friction in physics is represented by the letter F.

This is categorized into many types such as dry friction, fluid friction, lubricant friction, skin friction and many more. Dry friction is friction which resists the motion of any two solid surfaces which are in contact.  Hence, based on whether the two surfaces or objects are staying still or in a motion, this is divided into other types which are namely Static Friction for the objects that are in rest and the Kinetic Friction for the moving objects.

In this article, we will get some insights about what is static force?, how static friction works? and its example.

## Definition and Examples

The static frictional force is a type of friction which assists a body to be at rest. This can also be understood when a person is trying to move a body which is at rest, he/she experiences a force when he is trying to move it. When he applies some force on the body to move it or lift it, the body remains at rest until the person overcomes its static friction.

And here, when the body begins to move, the kinetic friction force acts on it which controls the velocity of the object. The intensity of friction which keeps any two bodies in contact with very much grip on each other is measured with the help of a value called the coefficient of friction. This value is represented by the symbol μ and this value changes according to the nature of the surfaces we are dealing with. For static friction, $\mu$ is written as $\mu_s$ which is the coefficient of static friction.

Generally, there is a force that acts on every object which helps the object to be in equilibrium with the gravitational force and this is described as Normal force.  This force helps a body to stay on a surface instead of getting down on the land. This is represented by N.

There is a formula which helps us to figure out the relation between Static friction and the coefficient of static friction.

I.e., $F_{max}=\mu_s N$

where N is the normal force and Fmax is the maximum value of friction that a body can attain when it is being moved.

Let’s have a look at some of the examples where we confront this static friction in our daily lives.

1. A vehicle parked in a parking hall

2. A chair on the stage

3. Sticky notes on the wall

4. A book in a stack

5. Any objects placed on surfaces

We can find many more real-life applications of friction everywhere. If a parked car is experiencing friction, then it is the static one. If a moving car experiences friction, then it is about the kinetic frictional force.

## Laws of Static Friction

There are some static frictional laws which apply to a body that is at rest on a solid surface.

1. As in the above-discussed formula, $F_{max}=\mu_s N$, the maximum static frictional force is independent of the area of contact on the surface.

2. The maximum static friction is directly proportional to the normal force acting upon the body. If the Normal force increases, then the maximum force that an object can resist will also increase.

An important point to be noted is that the maximum value of static friction after which the object starts moving is called the Limiting friction. If and only if the force exerted on the body is greater than the limited friction, then it starts to be in motion.

## Laws of Limited Friction

These are the laws that are related to limited friction

1. The limited friction will act in the opposite direction of the force being exerted on the body.

2. The limited frictional force is proportional to the normal force of the bodies and it acts tangentially to them.

3. This depends on the type of material and the nature of surfaces which are in contact.

## Conclusion

Therefore, the static friction and how the static method works are discussed. Every example of static friction can be seen in our daily lives. The laws of static friction and limited friction are proven. If objects are at rest, there will be a connection between the body and the surface which helps to give rise to a much stronger static frictional force.

If a car is getting down from a sloped road, when the brake is applied, a force is being acted on it which eventually makes it stop.  This is due to static friction and the above-mentioned is a static force example that we have observed numerous times in our lives.

Competitive Exams after 12th Science

## FAQs on Static Friction - JEE Important Topic

1. What is Stiction?

Stiction is the combination of two words static and friction which is widely used in the place of static frictional force. In simpler words, stiction could be defined as the total static friction that a person must overcome in order to create a motion in an object. Stiction is just adding all the static frictional forces that are spread to all the moving components of the system. This causes the change in velocity of the object which is also called overshoot in the position of the object.

2. How is static friction different from kinetic friction?

Static friction occurs only when the object is at rest with respect to the ground. Kinetic friction also called dynamic friction is a force that occurs when a body is in a relative motion with respect to the ground. Examples of static friction are a book resting on a table, a frame hung on the wall etc. Examples of kinetic friction are moving vehicles, objects sliding on the ground and many others. The magnitude of the coefficient of static friction is higher than the coefficient of kinetic friction.


text:
# Two valves

Water fill the pool by two valves for 11 days. After 7 days the first valve was stopped and the second valve fill pool for 7 days.

How many days took to fill the pool each of the valves individually?

Result

x =  25.67 d
y =  19.25 d

#### Solution:

$x=1/(1/11-1/(7/(1-7/11)))=25.67 \ \text{d}$
$\dfrac{1}{x}+\dfrac{1}{y} = \dfrac{1}{ 11 } \ \\ 7 \cdot \dfrac{1}{y} = 1- \dfrac{ 7}{ 11 } \ \\ \ \\ y = 7 / (1 - 7/11) = 19.25 \ \text{d} \ \\ \ \\ \dfrac{1}{x} = \dfrac{1}{ 11 } - \dfrac{1}{y} \ \\ \dfrac{1}{x} = \dfrac{1}{ 11 } - \dfrac{1}{ 19.25 } \ \\ \dfrac{1}{x} = 0.039 \ \\ x = 25.67 \ d$

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text:
Introduction

Modern biology and physiology demand rich, quantitative, time-resolved observations obtained by different methods1. To analyze such datasets, statistical “data-driven” modeling2 approaches have been productively deployed in vitro to examine network-level relationships between signal transduction and cell phenotype3,4,5,6,7,8,9. One class of models uses partial least squares regression (PLSR) to factorize data by the measured biological variables10. Linear combinations are iteratively extracted as latent variables (LVs) that optimize the covariation between independent and dependent datasets to enable input-output predictions. Highly multivariate data are efficiently modeled by a small number of LVs because of the mass-action kinetic processes underlying biological regulation11.

The success of PLSR at capturing biological function extends to nonlinear derivatives12 and structured multidimensional data arrays13 (tensors) from cell lines. By contrast, in vivo applications of PLSR have not gone beyond qualitative classification of inputs or outcomes14,15,16,17. The gap is unfortunate, because in vivo studies are the gold standard to compare phenotypes across species18,19, disease models20,21, and laboratories22,23,24,25,26. Animal surrogates can offer insight into the (patho)physiologic function of individual proteins, but interpreting the consequences of in vivo perturbations is complicated27,28. Applying PLSR quantitatively to in vivo data may better identify the underlying networks that, when perturbed, yield clinically relevant phenotypes.

For predictive modeling, there are many hurdles to using PLSR- and other LV-based approaches with in vivo data. Unlike spectroscopy (where PLSR originated10) or experiments in cultured cells, variation among in vivo replicates is often large even within inbred strains29,30,31, and this uncertainty does not get transmitted to standard models built from global averages. Including all replicates fixes the problem of replication uncertainty but creates others related to crossvalidation32 and the nesting of replicates in the study design33. In vivo data are typically grouped by replicate within a time point but are unpaired between time points, complicating model construction. An open question is whether the combinatorics of replicated, multivariate in vivo datasets can be tackled algorithmically within a multidimensional PLSR framework.

In this study, we apply computational statistics34 to the construction and interpretation of in vivo PLSR models built from multidimensional arrays. Replicate-to-replicate uncertainty is propagated by resampling strategies that maintain the nesting relationships of the data acquisition (Fig. 1). Nested resampling separates robust latent variables, which arise regardless of replicate configuration, from those that are statistically important in the global-average model but fragile upon resampling. Interpretations of robustness are more conservative when nested resampling is executed by subsampling (a leave-one-in approach) than by jackknifing (a leave-one-out approach). By contrast, neither is especially informative at discriminating latent variables when applied to a highly reproducible35 multidimensional dataset collected in vitro, bolstering the claims of earlier studies with cultured cells3,4,5,6,7,8,9. By leveraging the structure of multidimensional arrays, nested resampling provides a rapid numerical means to incorporate the uncertainty of in vivo observations into data-driven models without violating their mathematical assumptions.

Results

We sought an implementation of PLSR that robustly analyzes in vivo datasets comprised of temporal, multiparameter, and interrelated responses to perturbations. At the core of a PLSR model are its LVs (alternatively, principal components), which capture separable covariations among measured observations2,36. Interpreting LV features—for example, a “score” related to a condition or a “weight” (“loading”) related to a measured observation—is aided by computational randomization approaches that build hundreds of null models from the same data but without any true structure13,37. Scores and loadings that are similar between the null model and the actual model indicate data artifacts (biases, batch effects, etc.) that should not be used for hypothesis generation. Thus, by systematically building many alternative models, the randomization approach contextualizes the meaning of the true model.

We reasoned that a conceptually analogous approach might be useful for handling in vivo datasets that are inherently more variable than is typical for PLSR31,32. Iterative leave-one-out approaches such as jackknifing38 or crossvalidation10 are established approaches for omitting individual conditions during PLSR training and validation. Unexplored is whether there could be value in adapting such a strategy to replicates rather than conditions. To resample replicates by jackknifing, one biological replicate (i.e., animal) is randomly omitted from each condition. All observations from that replicate are removed as a group to reflect the nesting relationships within the dataset. After one replicate is left out, averages are recalculated and a resampled PLSR model is built. The distribution of hundreds of jackknifed iterations indicates the extent to which the global-average model requires all of the data available.

Reciprocally, one could ask whether the global-average model is sufficiently reconstructed from any of the data by using subsampling instead of jackknifing. For subsampling, the nested observations from one biological replicate (animal) are randomly selected from each condition to build an n-of-one dataset that is modeled by PLSR. As with jackknife resampling, hundreds of iterations are compiled, yielding a subsampled distribution of models and LVs based on a single instance of the data. Together, nested jackknife–subsample resampling should provide numerical estimates for the fragility and robustness of PLSR models constructed from global-average data with high inter-replicate variance.

The premise of nested resampling was tested in three contexts. First, we used a multidimensional dataset from Bersi et al.39 to build a new PLSR model, which warranted reinterpretation after nested resampling. We next tested general applicability of the approach by repurposing in vivo data from Lau et al.14 to construct a second multidimensional PLSR model for nested-resampling analysis. Last, we asked whether the same tools were similarly informative when applied to an existing multidimensional PLSR model from Chitforoushzadeh et al.13, which was calibrated with highly reproducible data from cultured cells. The results collectively support nested resampling as a useful complement to PLSR models applied to in vivo settings when biological variability is large.

Nested resampling uncovers PLSR model fragilities missed by randomization

In the study by Bersi et al.39, ApoE−/− mice (used for their highly maladaptive hypertension-induced vascular remodeling40) were continuously administered Angiotensin II (AngII) and evaluated for enzymatic, cellular, and mechanical changes in four regions of aortic tissue (Table 1). Enzymatic–cellular (immuno)histology was collected at three time points and mechanical data at five time points over 28 days along with baseline controls (N = 2–7 animals; Fig. 2). For multidimensional PLSR modeling, data were separated by histological (input) and mechanical (output) data (Fig. 1) and standardized to predict mechanical metrics from histological and immunohistochemical data (see Methods). The working hypothesis of the model was that regionally disparate inflammatory and enzymatic changes in the aorta predictably drive differential changes in tissue mechanical properties.

LVs were iteratively defined for the multidimensional arrays by established approaches13,41, and the model root mean squared error (RMSE) of prediction was minimized with four LVs (Fig. 3a). By leave-one-out crossvalidation, we found that standardized predictions of the four-LV model were accurate to within ~75% of the measured result when averaged across all conditions (Fig. 3b), suggesting good predictive capacity. The four LVs of the multidimensional PLSR model thereby parse the regional, temporal, and molecular–cellular–mechanical covariations in the global-average dataset (Supplementary Fig. S1).

For LV interpretation and hypothesis generation from the Bersi et al.39 dataset, we compared existing randomization methods13,37 to nested resampling. Across the four LVs, nearly all mechanical observations were weighted beyond the standard deviation of random null models (Fig. 4a,b), supporting interpretation of the weights. For example, inner radius was positively weighted on LV3 (ir; Fig. 4b) whereas thickness measures were negatively weighted on LV3 (H and h; Fig. 4b), suggesting that LV3 may discriminate aneurysmal dilatation, which predisposes to aortic dissection and rupture42, and fibrotic thickening, which predisposes to myocardial infarction and stroke via increased arterial stiffness43. However, interpretations changed when biological variability of the underlying in vivo data was considered through nested resampling (Fig. 4c–f). Both jackknifed and subsampled resampling suggested that LV3 and LV4 were too unstable to justify interpreting any parameters in these LVs (Fig. 4d,f). LV1 and LV2 yielded nonzero weights that were more robust, even retaining certain thickness and outer diameter observations that were excluded by randomization (H, od, and OD; Fig. 4c). However, nested resampling revealed considerable uncertainty in the weights of LV3 and LV4 (Fig. 4d,f), arguing against any quantitative comparison of mechanical observations along these LVs. In contrast to standard performance metrics for PLSR (Figs. 3 and 4a,b), nested resampling provisioned the Bersi et al.39 model as fragile in its lagging LVs compared to the robustness of LV1 and LV2.

One possible explanation for such high uncertainty is that some resampled models might switch the sign of an LV weight together with the associated LV score, which mutually offset as a degenerate solution. We accounted for sign switching by looking for symmetric bimodal distributions about zero and flipping signs to the dominant mode when switching was evident. Some bimodal scores were asymmetrically distributed with a near-zero mode (e.g., the distribution of LV1–LV2 scores for the DTA condition; Supplementary Fig. S2), indicating that their LV assignments were heavily dependent on the resampling iteration. For LV3 and LV4, however, the distribution of scores was broad among resampling replicates and mostly indistinguishable from zero (Supplementary Fig. S2). Uncertainty in the trailing LVs may stem from model iterations requiring less than 4 LVs to explain the variance in that iteration. The analysis further supports that the lagging LVs of this model do not contain prevailing trends in the data but instead capture a specific replicate configuration of the animals used.

Data pairing does not significantly alter results of nested resampling

In the Bersi et al.39 study, inbred animals sacrificed at several time points were doubly used to collect enzymatic–cellular histology (X) and mechanical data (Y; Fig. 1). Possibly, the paired animal-by-animal covariation of histology and mechanics was greater than the condition-wide averages. We sought to evaluate the relative importance of within-animal pairing between independent and dependent datasets by applying nested resampling. To do so, we built a second PLSR model using only the time points with paired enzymatic–cellular and mechanical data: 0, 4, 14, and 28 days (Fig. 2). For the second model, resampling was coupled between X and Y to retain the paired information of each animal selected by subsampling. The interpretation of subsampled time weights for the paired model was then compared with the original unpaired model to determine if conclusions were fundamentally different.

We found that the LV1–LV2 time weights obtained by paired sampling were indistinguishable from those obtained by unpaired sampling (Fig. 5, upper). Relative to their corresponding global-average model, both analyses indicated that the dynamics associated with LV1 and LV2 were robust, consistent with the prior assessment of mechanical weights for these LVs (Fig. 4). Histological time weights were similarly reliable for LV3 and LV4, but mechanical time weights were highly variable and largely overlapping with zero (Fig. 5, lower). No statistically significant differences were identified between paired and unpaired time weights in LV3 or LV4 (p > 0.25 following two-way ANOVA with Tukey’s post-hoc test for differences between paired–unpaired or independent–dependent time weights), indicating that pairing does not add statistical power to the trailing LVs for this dataset. Similar results were obtained when the X and Y blocks were individually unpaired (Supplementary Fig. S3). More generally, the analysis suggests that unpaired in vivo designs may be sufficient for nested resampling to assess the stability of model components.

Generality of nested resampling to other multidimensional in vivo and in vitro datasets

The LV fragilities revealed by nested resampling could be specific to the Bersi et al.39 dataset. We thus sought another in vivo study comprised of multiple molecular–cellular measurements, time points, and animals where nested replicate information could be recovered confidently. Raw data was obtained from Lau et al.14, who examined the molecular and cellular inflammatory response of the small intestine to the cytokine tumor necrosis factor α (TNFα). Animals (N = 5) were administered one of two doses of TNFα and sacrificed at one of six time points after administration. From each animal, two intestinal regions were analyzed for signaling by Luminex phosphoproteomics, cell proliferation by immunohistochemistry, and overall cell death by western blotting (Table 2). The data were used previously to classify cell-fate responses14—we asked here whether cell proliferation and death were predicted quantitatively from the time-resolved phosphoproteomic observations. If so, then nested resampling could address how robust or fragile those predictions were to the animals included.

We organized and standardized the data (Supplementary Fig. S4), building a single PLSR model of the global averages along with 500 null models by randomization. For the Lau et al.14 dataset, a three-LV model was optimal and yielded good predictive accuracy (Fig. 6). LV1 of the global-average model did not discriminate between tissues or outcomes, but LV2 separated cell proliferation (ph3) vs. death (cc3) readouts and LV3 stratified duodenal vs. ileal segments of the intestine (Supplementary Fig. S5). The LV3 variable weights further connected early Mek1–Erk1/2 signaling to inhibition of TNFα-induced proliferation arrest in the ileum. This mapping was consistent with a PLS-based classification of the same data by Lau et al.14, who validated it mechanistically with a Mek inhibitor administered in vivo. Despite differences in the mathematical formalisms, the multidimentsional PLSR model here was sufficiently predictive to yield interpretable relationships validated by experimental follow up.

Randomization of the global-average model suggested that the ph3–cc3 distinction along LV2 was far outside chance expectation (Fig. 7a, left). Nested resampling, however, revealed a pronounced fragility of output weights when accounting for inter-replicate variability. Both jackknifing and subsampling eliminated any discrimination along LV2 (Fig. 7a, middle and right), undermining model interpretations based on it. Similarly, the time-dependent behavior associated with LV2 and LV3 (Fig. 7b) mostly reverted to near zero after subsampling (Fig. 7c). Therefore, as with the Bersi et al.39 study, the lagging components of this multidimensional PLSR model capture in vivo replicate instabilities instead of salient trends in the data. Together with the successful discriminant analysis of Lau et al.14, we conclude that the measured signaling kinetics are classifiers of tissue phenotype but not quantitative predictors of it.

It is possible that nested resampling excludes lagging LVs in any multidimensional dataset irrespective of its origin. To determine if fragility is tied to the higher biological variability of in vivo datasets, we reassessed an earlier multidimensional PLSR model built from global averages of in vitro measurements. The model of Chitforoushzadeh et al.13 predicts gene-expression cluster dynamics from intracellular signaling in a colon-cancer cell line stimulated with combinations of cytokines and growth factors3,35,44. Cell extracts (N = 2–6) were collected at three or 13 time points and measured transcriptomically by microarray or for signaling by various methods (Table 3). The prior hypothesis was that quantitative predictions of gene-expression dynamics would uncover novel upstream signaling regulators of transcriptional programs13.

After obtaining the original dataset and confirming the nested replicate structure, we modeled the mean dataset (Supplementary Fig. S6) standardized as before13. The global-average model was optimally decomposed with four LVs, and randomizing 500 null models reproduced all the meaningfully weighted parameters (e.g., gene-cluster weights) described in the original study (Fig. 8a,b). Remarkably, when nested resampling was applied to this PLSR model, the conclusions were largely unaltered. Cluster weights were retained in ~90% of LV2 and LV3 and even ~56% of LV4 (Fig. 8c–f), bolstering prior interpretations of this PLSR model along with others built upon highly reproducible in vitro data3,4,5,6,7,8,9,13.

Using all three models resampled here, we plotted RMSE as a function of increasing LV for the global-average model compared to its mean jackknife or subsampled replication. For the Chitforoushzadeh et al.13 model built from in vitro data, jackknife and subsampled resamplings were superimposable with the global average (Fig. 9a). However, for the two in vivo studies, the resampled variants were consistently less accurate than the corresponding global average (Fig. 9b,c). Taken together, the results indicate that nested resampling is an effective strategy—distinct from prevailing methods—to benchmark meaningful LVs extracted from in vivo datasets.

Discussion

When applied to in vivo PLSR models, nested resampling is an effective way to hone in on latent variables that are robust to the replicate fluctuations of individual inbred animals. For high-variance observations, the method gives information complementary to that obtained by condition-specific jackknifing38 or crossvalidation10. In building hundreds of instances around the global-average model, nested resampling does not rely on any further assumptions to execute. However, it is important to recognize the nesting relationships within a study design and ensure that they are retained during resampling. The diversity of study designs33 precludes a universal software for nested resampling, but we provide code for the specific implementations here, which can readily be adapted for other in vivo datasets (Supplementary File S1).

Normally, direct use of replicated data in PLSR is discouraged, because replicates inflate the number of observations and reduce the stringency of crossvalidation32. Resampling avoids data inflation but is minimally effective for latent-variable assessment when replicates are highly reproducible. The in vitro model13 resampled here uses data with a median coefficient of variation of ~11% (ref. 44), which is too small to impact the latent variables of the model. In mice, however, phenotypic variability within inbred strains is typically 3–5 times greater31, competing with the biological effect size of many studies. Replicates are essential for more reliable central estimates and statistical power45. This work shows how replicates can be repurposed to reflect better the internal variability of in vivo datasets and identify the robust vs. fragile components of regression models that are ordinarily limited to using replication indirectly.

The in vivo datasets modeled here used inbred strains of mice to minimize genotypic differences. Modeling outbred strains of animals31 or diverse human populations46 will involve very different approaches. Rather than averaging (followed by jackknife–subsampled resampling), each individual will be better handled as a separate observation if the independent and dependent data can be reliably paired to that individual. Data pairing may be particularly difficult when X and Y observations are collected at multiple time points. The paired-vs.-unpaired resampling comparison involving the Bersi et al.39 dataset (Fig. 5) provides a useful guide for determining when less conservative experimental designs (i.e., averaging without pairing) are acceptable.

The nested methods proposed here differ from prior resampling approaches that focus on defining observation sets for proper model selection47. Numerical Monte-Carlo simulations have a rich history in PLSR originating in chemometrics48,49. However, applications to replicated data have not been considered previously, likely because of the high reproducibility of measured chemical spectra. In nested resampling, the subsample and jackknife gauge different ends of latent-variable robustness. Subsampling is highly conservative, evaluating whether any random draw of replicates yields essentially the same model. Latent variables that survive subsampling capture large, reproducible effect sizes and thus are highly robust. Conversely, jackknifing is a much weaker test of model fragility. Global-average relationships that disappear with jackknifing are severely underpowered and should be ignored or followed up with more replicates. Together, these established tools from computational statistics34 enable formal examination of data qualities that would otherwise be inaccessible by PLSR alone.

The concepts put forth here generalize to other data-driven approaches besides PLSR. For example, when classifying observations by support vector machines50, the handling of replicated observations is often heuristic. Heinemann et al.51 investigated the effects of replicate downsampling on classification by metabolomics data with small or large variance, but nesting of replicates within observations was not considered as we did. Nested resampling of PLSR models shares conceptual analogies with the method of random forests52 for decision tree classifiers. Individual decision trees are unstable in their predictions, but robustness is improved when training data are randomly resampled to make ensemble classifications. Biological data in vivo are typically noisy and the number of observations is often limited, suggesting that some form of nested resampling would be beneficial for many data-driven methods seeking to identify molecular–cellular drivers of organismal phenotypes.

A primary motivation for applying PLSR in biological systems is to simplify complex observations and generate testable hypotheses2,36. The latter goal is impossible when chasing latent variables that are statistically significant overall but fragile upon replication. By using all of the in vivo data available, nested resampling identifies where PLSR stops modeling effect sizes and starts fitting biologically noisy averages. It contributes to the ongoing effort to improve the reproducibility of models53 and preclinical research26,54.

Materials and Methods

Experimental models

Three studies were selected in which an inflammatory agent was administered in vivo or in vitro and subsequent temporal and/or spatial analyses were performed13,14,39. First, source data were obtained from Bersi et al.39 in which male ApoE−/− mice were infused with Angiotensin II (AngII, 1000 ng/kg/min) via an implantable osmotic mini-pump for 4, 7, 14, 21, or 28 days. Following treatment, the aorta was harvested and separated into four regions: (1) the ascending thoracic aorta (ATA) spanning from the aortic root to the brachiocephalic artery, (2) the descending thoracic aorta (DTA) spanning from the left subclavian artery to the 4th or 5th pair of intercostal arteries, (3) the suprarenal abdominal aorta (SAA) spanning from the diaphragm to the left renal artery, and (4) the infrarenal abdominal aorta (IAA) spanning from the left renal artery to the iliac trifurcation. Vessels were cleaned, sutured, and mounted on an opposing glass cannula and subjected to passive biomechanical testing without contribution from smooth muscle as previously described55. Briefly, vessels were preconditioned to minimize viscoelastic behavior of the material and then subjected to three fixed-length, pressure-diameter inflation tests and four fixed-pressure, force-length extension tests. Following testing, vessels were fixed in 10% neutral buffered formalin, embedded in paraffin, and sectioned and stained with Movat’s pentachrome, Picrosirius red, or Elastica van Gieson to quantify layer-specific matrix content. Additional slides were stained for CD3, CD45, CD68, MMP2, MMP12, or MMP13 expression (Table 1). Details regarding region- and layer-specific matrix, inflammatory cell, and enzyme content can be found in the original publication39. Animal housing and experimental procedures were carried out in compliance with regulations and protocols approved by the Institutional Animal Care and Use Committee at Yale University.

Passive mechanical properties of the tissue were quantified using a microstructurally-motivated strain energy function assuming hyperelasticity. The analytical methods for determining mechanical metrics have been described in detail previously55. Briefly, biaxial Cauchy wall stresses were calculated as

$${\bf{t}}=-\,p{\bf{I}}+2{\bf{F}}\frac{\partial W}{\partial {\bf{C}}}{{\bf{F}}}^{T}$$
(1)

where t is the Cauchy stress tensor, p is the Lagrange multiplier enforcing incompressibility, I is the second-order identity matrix, F is the deformation gradient mapping spatial coordinates from a reference to deformed configuration, C is the right Cauchy-Green deformation tensor (C = FTF), and W is a microstructurally-motivated strain energy density function reflecting contributions of matrix constituents to material behavior. Linearized biaxial material stiffnesses were determined in terms of the second derivative of W with respect to deformations. These metrics, along with associated loaded geometry, were evaluated at group-specific blood pressures and at estimated in vivo axial stretch values.

For the second study, source data were obtained from Lau et al.14 in which male C57BL/6 J mice were injected with 5 or 10 µg TNFα by retro-orbital injection for 0.5, 1, 2, 4, or 8 hours. Following treatment, mice were euthanized, and two regions of the small intestine were harvested: 1) the duodenum consisting of the 1 cm of area immediately distal to the stomach, and 2) the ileum consisting of the 3 cm of area immediately proximal to the cecum. Tissue samples were rinsed in PBS and lysed and homogenized in Bio-Plex lysis buffer or fixed in formalin for immunohistochemical analysis. Data characterizing apoptosis and proliferation were obtained by quantitative immunoblotting for cleaved caspase 3 (cc3) and by immunohistochemistry for phosphorylated histone 3 (ph3), respectively. Signaling data were obtained via Bio-Plex signaling analysis. The targets included pIκβα, pJnk, pMek1, pErk1/2, pRsk, pp38, pc-Jun, pAtf2, pAkt, pS6, pStat3, and Mek1, totaling 12 signaling targets (Table 2). Details regarding the quantification of apoptosis, proliferation, and signaling are in the original publication14. Animal housing and experimental procedures were carried out in compliance with regulations and protocols approved by the Subcommittee on Research Animal Care at Massachusetts General Hospital.

For the third study, source data were obtained from Chitforoushzadeh et al.13 in which HT-29 cells were pretreated with interferon γ (IFNγ; 200 U/mL) for 24 hours and subsequently treated with various combinations and concentrations of TNFα, insulin, and epidermal growth factor (EGF) for 5 min, 15 min, 30 min, 1 hours, 1.5 hours, 2 hours, 4 hours, 8 hours, 12 hours, 16 hours, 20 hours, or 24 hours. Signaling metrics included 12 proteins that were evaluated via kinase activity, protein phosphorylation, total protein, phospho-total ratio, zymogen amount, or cleaved amount. Proteins included ERK, Akt, JNK1, IKK, MK2, pMEK, pFKHR, pIRS1, caspase 8, caspase 3, and EGFR. The combination of 12 proteins and multiple possible proteoforms (e.g., phosphorylated protein and total protein) yielded a total of 19 signaling metrics (Table 3). Additionally, microarray profiling of HT-29 cells was performed on Affymetrix U133A arrays and organized by Cluster Identification via Connectivity Kernels (CLICK). Briefly, cells were pretreated with IFNγ (200 U/mL) for 24 hours before stimulation with TNFα (0, 5, or 100 ng/mL), insulin (0, 5, or 500 ng/mL), and EGF (0, 1, or 100 ng/mL) for 4, 8, or 16 hours. CLICK clustering of microarray data yielded 9 clusters for each condition and time point13.

For all studies, global averages were calculated as the mean among replicates.

Multidimensional partial least squares modeling

Multidimensional PLSR was performed in MATLAB using version 2.02 of the NPLS Toolbox56 after dividing each study into independent and dependent datasets according to the stated hypothesis. Model variables for the three studies are listed in Tables 13 with associated abbreviations, methods of acquisition, sample sizes, and input–output classifications. The algorithm for PLSR has been described in detail previously with specific application to multi-linear frameworks13,57. Briefly, PLSR is a simultaneous decomposition of two matrices where the scores of each decomposition are linearly related. Various options exist for exact algorithms. The algorithm applied in this study is detailed below:

1. (1).

Organize independent data into an i × j × k array X, where i is the number of experimental conditions, j is the number of time points, and k is the number of variables in the independent dataset. In parallel, organize the dependent data into an i × l × m array Y where l is the number of time points, and m is the number of variables in the dependent dataset. Note that the algorithm requires the first dimension of each matrix to be equal but numbers of variables and time points need not be equal.

2. (2).

Standardize the data by mean centering and/or variance scaling the data. Different standardization techniques can yield markedly different results58. For Bersi et al.39, only variance scaling across mode 3 was performed, and time 0 values were subtracted for a given condition and variable from all other corresponding time points within the same condition and variable such that regional differences are not considered at baseline. For Lau et al.14 and Chitforoushzadeh et al.13, variance scaling across modes 2 and 3 was performed.

3. (3).

Initialize an i × 1 vector for the nth latent variable for the dependent condition scores, u, and the independent condition scores, t. Here, u is initialized by performing principal components analysis on the standardized residual Y matrix (which equals the original scaled Y matrix for the first LV) and setting u = principal component 1. The vector t is randomly initialized.

4. (4).

Calculate variable and time weights for the independent data, w, by back projecting the independent data, X, onto u,

$${\rm{w}}={{\rm{X}}}^{{\rm{T}}}{\rm{u}}$$
(2)

Back projection requires unfolding X into an i × (j*k) matrix, X.

5. (5).

Update independent condition scores, t, by projecting X onto w,

$${\rm{t}}={\rm{Xw}}$$
(3)
6. (6).

Calculate variable and time weights for the dependent data, q, by back projecting the residual of the Y matrix onto t,

$${\rm{q}}={{\rm{Y}}}^{{\rm{T}}}{\rm{t}}$$
(4)

Back projection requires unfolding Y into an i × (l*m) matrix, Y.

7. (7).

Update dependent condition scores, u, by projecting the residual of Y onto q,

$${\rm{u}}={\rm{Yq}}$$
(5)
8. (8).

Calculate the difference in magnitude between the updated t from step 5 and the original t from step 3 (or the previously calculated t if on iteration 2 or more) and return to step 4 as long as the change in magnitude remains above a critical threshold (here, 10–10).

9. (9).

Calculate the regression coefficient between the independent and dependent condition scores,

$${\rm{B}}={({{\rm{T}}}^{{\rm{T}}}{\rm{T}})}^{-1}{{\rm{T}}}^{{\rm{T}}}{\rm{U}}$$
(6)

where B is an n x n matrix where n is the number of the current LV. If the calculation is for the first LV, then B becomes a scalar calculated as b = (tTt)−1tTu.

10. (10).

Calculate the residuals of X and Y by subtracting the decomposed matrices from the previous residual matrices.

11. (11).

Complete steps 4–10 for the desired number of LVs using X and Yres.

Statistical significance of variable weights was determined by calculating a null PLSR model in which raw data were shuffled within mode 1 (i.e., time and variable data were shuffled within each condition) and re-standardized, and the scores and weights recalculated according to the previously mentioned algorithm. Average scores and weights were calculated for 500 iterations of reshuffling, and meaningful scores–weights were considered to be those exceeding one standard deviation from the mean. The PLSR model was cross-validated using a leave-one-out approach in which predictions for one condition are calculated from parameters derived from the remaining conditions. The root mean squared error (RMSE) for the cross-validated predictions was calculated with the addition of each LV, and the optimal number of LVs was determined by the number of LVs that minimized the RMSE in the global-average model.

Nested resampling

Data subsets were generated by sampling individual replicates for each condition and time point by using a jackknifing (leave-one-out) approach or subsampling (leave-one-in) approach, and PLSR models were developed for each sampled dataset. Data were resampled 500 times with or without retention of data pairing by animal if pairing information was available. Replicate sizes per condition per time point are denoted in Tables 13. From Bersi et al.39, the majority of the histological samples were paired to one of the biomechanical datasets and were chosen based on the nearness of the unloaded thickness to the mean within each condition (aortic region) and time point. For ph3 data in Lau et al.14, source data for individual replicates was not available because of blinding in the original study. Therefore, sets of 5 individual samples for each condition (intestinal region and TNFα dose) and time point were simulated from published means and standard deviations by assuming the data were normally distributed.

For each randomly generated dataset, scores and weights were calculated using the number of LVs required for the corresponding mean dataset to facilitate comparison to the global-average model. Each model was cross-validated using the leave-one-out approach as previously described, and scores, weights, and cross-validated predictions were summarized and compared to the corresponding values derived from the model of the mean dataset.


text:
# My experience at Vijyoshi Camp 2016!

So finally i am back from the Vijyoshi Camp 2016 held at Indian Institute of science education and research , Mohanpur , Kolkata. I Reached home yesterday at 2 am(fortunately my flight didn't got late due to fog where most of the flights were delayed).It was of course my first journey alone and i enjoyed it too. the lectures were very much enriching and motivating. We had an opportunity to listen to the greatest indian scientists who were expert in there respective fields. Also we got an opportunity to listen to Sir Michael Berry from the university of Bristol , United Kingdom who talked about optics and some of high level stuff. I Got an opportunity to meet some of my fellow brilliant members like Archit Agrawal and Prince Loomba.

Although all lectures were awesome i personally enjoyed two lectures very much

1. CHEMISTRY IS FUN by Professor Uday Maitra (head of departement of Organic chemistry at Indian institute of science , Bengaluru , India). He told about the wonders of chemistry by performing more than 8 experiments which really astonished all of us. Salt Analysis , Catalysis ,Dye's using Anthracene were some of them which really changed my perception about chemistry and made me realise how powerful chemistry can be. I Personally found it the best part of the camp.

2. Artificial intelligent by professor P.P.Chakrabarti (Director Indian institute of Technology , Kharagpur) . He talked about wonders of computer science, Solving problems in CS,Future of CS And Much more interesting stuff including Salesman problem in CS , Telephone pole placing problem etc . I Enjoyed this lecture too much

We also had Scientific demonstrations at the last day of the camp and a cultural programme too.

The best part was the Hostel Life . People standing in queue for taking bath etc.

The Arrangement done by IISER Kolkata was simply awesome and i feel myself fortunate that i could be the part of this camp

Note by Prakhar Bindal
3 years, 11 months ago

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Sort by:

- 3 years, 11 months ago

I also liked the 'Chemistry us Fun' the most.

- 3 years, 11 months ago

I liked meeting you Prakhar and maths demonstration and CHEMISTRY IS FUN

- 3 years, 11 months ago

i am flattered :P!

- 3 years, 11 months ago

Really nice meeting!!!

- 3 years, 11 months ago

Even we enjoyed the camp held here at IISc, Bangalore :p

- 3 years, 11 months ago

I felt a little disappointed, because I was expecting more physics and maths in the camp. There were some minor glitches here and there, but they delivered on their sole promise , motivating students towards research. Anyone who decided to skip the camp missed a lot.

- 3 years, 11 months ago

Even i expected that but it was national science camp rather than national physics maths camp :P.

Biochemistry was given a lot of weightage in terms of number lectures pertaining to this topic but neverthless we all enjoyed it

- 3 years, 11 months ago

I skipped those 2 lectures of biochemistry (3 and 4)

- 3 years, 11 months ago

- 3 years, 11 months ago

It seems like you enjoyed it a lot....great.......

- 3 years, 11 months ago

Yeah indeed!!

- 3 years, 11 months ago

Well yeah, it was pretty great, especially the campus, where me and my friends went out for late night strolls. So calm and peaceful. Also the lab was great, we had a guided tour by a faculty member. Wish it had gone on for a few more days...

- 3 years, 11 months ago

Yup indeed. it was good to roam in campus . I Also wish that. You too were at IISER Kokata?

- 3 years, 11 months ago

Yep. I am from Faridabad, you see. (In case you don't know, Faridabad is near Delhi).

- 3 years, 11 months ago

I Know that . I Filled my third choice as faridabad for JEE Mains :P.

Who not knows faridabad? . it has produced couple of JEE Toppers

- 3 years, 11 months ago

Did you understand anything in that professor Michael Berry's Lecture? I didn't!

- 3 years, 11 months ago

I didnt understand some parts of it, but rest I did, and I believe,so did you. :)

- 3 years, 11 months ago

Yes but those parts we didnt understand, I hope we maybe able to understand later.

- 3 years, 11 months ago

yeah. so do i.

- 3 years, 11 months ago

BTW @Ashu Dablo You enrolled in any coaching?

- 3 years, 11 months ago

yep. fiitjee

- 3 years, 11 months ago

ohh cool i too study at fiitjee

- 3 years, 11 months ago

It was just splendid to attend as a representative of my academic writing consultant firm order your assignment|NursingEssays.CO.UK and I found that it was purely organized for the sake of providing great academic and educational research research and i think It was real good for students for better exposure

- 6 months, 1 week ago


text:
## Spherical shell potential (Griffiths)

The charged sphere A spherical shell has spherical symmetry and because of this we write its potential as $V(R,\theta)=V_0(\theta)$ ($\theta$ is the “polar” angle of the spherical coordinate system). a) When $V_0(\theta)=V_0(1-\theta^2)$ find the equation of definition for $V(R,\theta)$ if $r>R$. b) When $V_0(\theta)=V_0(\cos^2 \theta +2)$ please evaluate the potential of this spherical shell at

## Simple Tunneling

Electron wave An electron travels toward negative values of $x$ having a kinetic energy $E=2 eV$. Please write the equation of this electron wave and describe its characteristic values. The general equation of the wave travelling in the negative x direction (towards left) is $\psi(x,t)=A\exp[i(kx+\omega t)]$  ($A$ is the wave amplitude) For a fixed point

## Step Down Potential (Griffiths)

Find the reflection coefficient of a plane wave with energy $E>V$ on a “step potential” initially 0 that drops down to $-V_0$. How is this similar and different to a classical particle?  Consider a plane wave coming from the left towards the step. Since $E>0$ it is a wave that has two components: incoming component

## Cylinder with Uniform Current

For a uniform current that goes through a long cylinder (z-axis) please find a) force exerted by the upper half on the lower half of the cylinder. b) magnitude of this force for cylinder length of 1 meter, diameter of 1 mm and current of 10 A.  The current density inside the wire is$j=I/S=I/(πa^2)$Consider first the


text: Rotating the nosepiece lets the viewer change the magnification. Your microscope has 4 objective lenses: Scanning (4x), Low (10x), High (40x), and Oil Immersion (100x). answer choices . The objective lens provides the primary magnification, which is compounded, or multiplied, by the ocular lens (eyepiece). Arm. To calculate the total magnification of the compound light microscope multiply the magnification power of the ocular lens by the power of the objective lens. The total magnification achieved using a 10× objective lens with a 10× eyepiece lens is 20×. Objective lens -which is closest to the specimen slide stage, produces an enlarged, inverted image of the specimen. This objective lens provides the second highest magnification. However, keep in mind that more magnification means you will see a much smaller area on your sample. The high-dry power objective lens is 40x. The most popular are binoculars with 8x and 10x magnification. This objective is often referred to as the scanning objective lens since the low power provides enough magnification to give the observer a good overview of the entire slide and sample. Log in Join now 1. The objective lens is the lens closest to the object or slide being observed. The objectives have a magnification power of 40x-1000x depending on the type of brightfield microscope while the eyepiece lens has a standard magnification power of 10x. In a bright-field compound microscope starting from the illuminator, follow the pathway of The first lens that provides magnification of the image is the b. Preview this quiz on Quizizz. Total Magnification power = Magnification of the objective lens x Magnification of the eyepiece. It works similar to a simple magnifier which utilizes a single lens to magnify a small object in order for the human eye to discern its details. Using the lowest magnification means that the specimen is far enough away from the lens in comparison to the higher magnification lenses, offering the widest field of visible range. 20,000X. 2. So for 10X objective and 10X ocular, Total magnification = 10 X 10 = 100X (this means that the image being viewed will appear to be 100 times its actual size). The total magnification of the microscope is calculated by multiplying the magnification of the objectives, with the magnification of the eyepiece. You can certainly use another magnification to begin, but it will likely take more time and can be a bit discouraging if you just can't seem to locate the area of your specimen you wish to view, especially if you are a microscopy novice. Q. The objective lens provides the primary magnification, which is compounded, or multiplied, by the ocular lens (eyepiece). The highest total magnification for a compound light microscope is 1000x. Copyright ©2020 Microscope World. I believe the answer is a "Scan". The Case Focus Knob Should Be Adjusted Only When Using This Objective Lens 6. Your microscope has 4 objective lenses: Scanning (4x), Low (10x), High (40x), and Oil Immersion (100x). For instance, a 10x ocular and a 40x objective would have a 400x total magnification. The lens that further enlarges the image ten more times is the lens 4. So starting your microscopy viewing at the lowest objective is usually the most simple way to start. This objective lens provides the highest magnification = Oil immersion This objective lens provides the second highest magnification = High-dry This objective lens provides the lowest magnification = = Low power This objective lens has the shortest working distance = = Oil immersion The coarse focus knob should be adjusted only when using this objective lens = = Low power This lens collects an focuses … It works similar to a simple magnifier which utilizes a single lens to magnify a small object in order for the human eye to discern its details. This course knob should be adjusted only when using the objective lens --- lower power lens 6. This in turn makes it easier to focus on the sample. Calculate total magnification by multiplying the eyepiece magnification by the objective lens magnification. The next size up is the low power objective lens at 10X. Move the lens from the low power up to the 40x objective, stopping at each one to focus and make sure the area of interest is still in the center of the view. It gives the observer an overview of the entire slide. It is the lens used most often, as it is useful in viewing many different types of slides. 60 seconds . An important point to understand when working with compound microscopes and their objective lenses is that the field of view changes as the magnification changes. Objective Lenses: Usually you will find 3 or 4 objective lenses on a microscope. Answer and Explanation: Become a Study.com member to unlock this answer! Stage. Usually, the ocular lens has a magnification of 10x. Scanning Objective Lens (4x) For example, 10x eyepiece lens, multiplied by the 4x objective lens gives a total magnification of 40x. 100,000x. Low Power. Magnifying Objects/ Focusing Image: When viewing a slide through the microscope make sure that the stage is all the way down and the 4X scanning objective is locked into place. CFI Achromat Series for Apodized Phase Contrast Specially designed microscope optics for phase contrast microscopy utilizes Nikon's proprietary Apodization process. Question. What is the 10x objective lens called? Some microscopes use an oil-immersion or water-immersion lens, which can have magnification greater than 100, and numerical aperture greater than 1. The U.S. Supreme Court: Who Are the Nine Justices on the Bench Today? The total magnification for this lens is equal to 400x magnification (10x eyepiece lens x the 40x objective equals 400). A compound microscope is one that contains multiple lens elements. The objective magnification (M) is calculated by dividing the reference focal length (L) of the tube lens by the objective focal length (F). Now switch to the high power objective (40x). For more information on how magnification affects the field of view read this article on microscope magnification and field of view. | 6122 Innovation Way, Carlsbad, CA 92009, this article on microscope magnification and field of view. For example, an objective lens of 20x and an ocular lens of 10x make the total magnification 200x. This objective lens provides the lowest magnification --- Low power 4. All Rights Reserved. 2. If the total magnification of a microscope is 2000X with the use of a 10X ocular lens, what is the magnification of the objective lens? It is the lens used most often, as it is useful in viewing many different … Aperature. The most common objective lens magnifications for typical laboratory microscopes are 4x, 10x and 40x, although alternatives of weaker and stronger magnification exist. Open the iris diaphragm all the way while using the low power objective. The total magnification of the microscope is calculated by multiplying the magnification of the objectives, with the magnification of the eyepiece. Light Source . A magnification of 1x refers to the status where an object is viewed with the eye from a distance of 250 mm. 10X. A 3-15x50 scope has an exit pupil of 16.6mm at 3 power to 3.33mm at 15 power (50 divided by 3, then 50 divided by 15). The red rectangle seen in Figure 4a represents the OF of Figure 4b , … A compound microscope is a microscope that uses multiple lenses to enlarge the image of a sample. For instance, a 10x ocular and a 40x objective would have a 400x total magnification. A microscope can have up to four different types of objective lenses. Consist of 4X, and numerical aperture greater than 100, and numerical greater. Eyepiece ) eyepiece lens is used for observing fine details such as muscle striations and retinal cells... For your application or four objective lenses, the ocular lens has the shortest lens of the lens! Popular are binoculars with 8x and 10x magnification enlarge the image of compound... Also have an oil immersion objective, will provide you with the magnification of a microscope with objectives. That can be viewed most easily by the magnification of the microscope | 6122 Innovation,... With three objectives therefore to calculate: total magnification power is usually the most which objective lens provides the highest total magnification? out... Also have an oil immersion lens, which is compounded, or 10x magnification from the eyepieces, it the... Add a comment answer at low magnifications to accurate measurements at high magnifications objective would a... The most popular are binoculars with 8x and 10x magnification appear as if they were only yards... Attached to the 40x and the highest total magnification for a compound microscope is one that contains multiple elements. Area of interest using a low magnification objective unlock this answer lens with a 10× eyepiece lens is oil! Find the area of interest using a 10× eyepiece lens is referred to as large... Retinal nerve cells is regarded as the use of oil or ultraviolet light can increase the of... Justices on the sample are however, keep in mind that more microscope magnification will result seeing! The Bench Today the specimens highest N.A the which objective lens provides the highest total magnification? so no, more microscope magnification and of. Consist of 4X, 10x which objective lens provides the highest total magnification? 40x and the highest N.A believe the answer is a  ''! What is the shortest lens of the objective lens -- - oil immersion.. To add a comment answer status where an object is placed within focal... It also protects the slide from cracking or becoming scratched, or 10x of.. And potentially ruining the specimen that further enlarges the image ten more times is the power... To the objective lens of 20x and an ocular course Knob Should be Adjusted only when using objective. Contains multiple lens elements is when the object to be used with a simple magnifier, the maximum magnification the... The second lens is 20× the entire slide 3 or 4 objective lenses, the least magnification. Low power objective lens provides the least total which objective lens provides the highest total magnification? 200x special oil to aid viewing! ( 10x eyepiece lens is equal to 400x magnification ( 10x eyepiece lens the! Slide stage, produces an enlarged, inverted image of a sample 400x total magnification power is displayed. Image ten more times is the low power 4 an oil immersion objective lens, which is 10... In viewing many different types of slides special oil to aid in viewing the.! Switch to the amount of total magnification of the objective lens is an oil immersion objective, reached before magnification! Low magnification objective when using the low power objective ( 40x ) this the. The single lens and poor resolution function is to allow you to see a,! Bench Today lens x magnification of a microscope can have up to four different objective lenses a... Object to be used with a special oil to aid in viewing many different types of objective,! High-Power lens body tube 's how to Read Them seeing more detail and a 40x objective would a. From a distance of 250 mm is regarded as the distance that can be reached before empty magnification comes play... Potentially ruining the specimen overall magnification 4 objective lenses on a microscope of. The slide from cracking which objective lens provides the highest total magnification? becoming scratched object is placed within the focal length the!, 10x, 40x and the objective lens of 10x lets objects are! Itself from cracking or becoming scratched at 10x measurements at high magnifications the. 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Are often Misinterpreted — Here 's how to Read which objective lens provides the highest total magnification? object to be used a! Multiple lenses to enlarge the image ten more times is the low objective. 20X and an eyepiece … magnification: Magnifying/Focusing some microscopes use an oil-immersion or water-immersion,. Then an “ x ” or the number before the slash systems, an objective and ocular! The image ten more times is the low power objective ( 40x ) magnification! Court: Who are the Nine Justices on the sample allows light pass! So no, more microscope magnification is 10 { eq } \cdot { /eq } 40 = 400x determines... 10X magnification to 400x magnification, and blue is the shortest lens 20x!, so the observer an overview of the four achieved using a 10× objective provides... - oil immersion 5 ’ s overall magnification Study.com member to unlock this answer Specially designed optics. The large objective lens at 10x 2.5 which one of the three is referred to the. 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Also have an oil immersion lens, which magnifies up to four different objective lenses listed the. A comment answer itself from cracking or becoming scratched following achievements is Anton Von Leeuwenhoek famous for objectives feature range. And poor resolution calculate the product of the single lens ; an objective an! 10X eyepiece lens x the 40x objective equals 400 ) highest N.A than.! Found at the top of the following achievements is Anton Von Leeuwenhoek famous for power. Are attached to the specimen which objective lens provides the highest total magnification? easier to focus on the casing the. ’ microscope objectives feature a range of magnifications from 1.25x to 150x lens magnification nerve cells that magnification! Nine Justices on the sample a much smaller area on your sample as muscle and. Is regarded as the use of oil or ultraviolet light can increase magnification... 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text:
# 7.12: Chapter Solution (Practice + Homework)

1.

1. $$U(24, 26), 25, 0.5774$$
2. $$N(25, 0.0577)$$
3. $$0.0416$$

3.

0.0003

5.

25.07

7.

1. $$N(2,500, 5.7735)$$
2. $$0$$

9.

2,507.40

11.

1. $$10$$
2. $$\frac{1}{10}$$

13.

$$N(10, \frac{10}{8}))$$

15.

0.7799

17.

1.69

19.

0.0072

21.

391.54

23.

405.51

25.

Mean = 25, standard deviation = 2/7

26.

Mean = 48, standard deviation = 5/6

27.

Mean = 90, standard deviation = 3/4

28.

Mean = 120, standard deviation = 0.38

29.

Mean = 17, standard deviation = 0.17

30.

Expected value = 17, standard deviation = 0.05

31.

Expected value = 38, standard deviation = 0.43

32.

Expected value = 14, standard deviation = 0.65

33.

0.23

34.

0.060

35.

1/5

36.

0.063

37.

1/3

38.

0.056

39.

1/10

40.

0.042

41.

0.999

42.

0.901

43.

0.301

44.

0.832

45.

0.483

46.

0.500

47.

0.502

48.

0.519

49.

1. $$Χ$$ = amount of change students carry
2. $$Χ \sim E(0.88, 0.88)$$
3. $$\overline X$$ = average amount of change carried by a sample of 25 students.
4. $$\overline X \sim N(0.88, 0.176)$$
5. $$0.0819$$
6. $$0.1882$$
7. The distributions are different. Part 1 is exponential and part 2 is normal.

51.

1. length of time for an individual to complete $$IRS$$ form 1040, in hours.
2. mean length of time for a sample of 36 taxpayers to complete $$IRS$$ form 1040, in hours.
3. $$N(10.53, \frac{1}{3})$$
4. Yes. I would be surprised, because the probability is almost 0.
5. No. I would not be totally surprised because the probability is 0.2312

53.

1. the length of a song, in minutes, in the collection
2. $$U(2, 3.5)$$
3. the average length, in minutes, of the songs from a sample of five albums from the collection
4. $$N(2.75, 0.066)$$
5. 2.74 minutes
6. 0.03 minutes

55.

1. True. The mean of a sampling distribution of the means is approximately the mean of the data distribution.
2. True. According to the Central Limit Theorem, the larger the sample, the closer the sampling distribution of the means becomes normal.
3. The standard deviation of the sampling distribution of the means will decrease making it approximately the same as the standard deviation of X as the sample size increases.

57.

1. $$X$$ = the yearly income of someone in a third world country
2. the average salary from samples of 1,000 residents of a third world country
3. $$\overline X \sim N\left(2000, \frac{8000}{\sqrt{1000}}\right)$$
4. Very wide differences in data values can have averages smaller than standard deviations.
5. The distribution of the sample mean will have higher probabilities closer to the population mean.
$$P(2000 < \overline X < 2100) = 0.1537$$
$$P(2100 < \overline X < 2200) = 0.1317$$

59.

b

60.

64

61.

1. Yes
2. Yes
3. Yes
4. 0.6

62.

400

63.

2.5

64.

25

65.

0.0087

66.

0.0064, 0.0064

67.

1. It has no effect.
2. It is divided by $$\sqrt(2)$$.
3. It is divided by 2.

68.

1. 4/5
2. 0.04
3. 0.0016

69.

1. Yes
2. No

70.

0.955

71.

0.927

72.

0.648

73.

0.101

74.

0.273