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We’ve sketched out a rough architecture at this point, time to get an estimate of expected cost, and think through how well we can expect it to meet our original design goals. Besides overall user experience, I find it’s important to carefully think through the limitations of any project - what a thing can’t do is important as what it can. (Avoid scope creep!) Lastly, it’s always important to include non-functional design requirements like security, performance, etc, as these can make or break the project.

# Expected Cost

Given that this is a project intended primary for personal use, I’ve been quite careful in making selections that minimize cost wherever possible. As an example, a default installation of some of the software can make use of Virtual Private Clouds with private networks, which require a special interface (“NAT”) for computers on that network to reach the outside. That’s all great for security, but that default NAT device from Amazon can cost as much as a single compute VM, and needs to be running 24/7.

The solution I’ve come up with consists of the following:

• Base infrastructure

• Master node: t2.micro - $0.0116/hr, approx$8/mo

The master is the node that is really managing K8s and helping to determine which compute node gets which work. It needs to always be running, but doesn’t require significant memory or cpu so can be a very low powered instance.

• Admin” node: 2 x t2.small - $0.023/hr, approx$16/mo * 2, $32/mo I designate 2 nodes to be “administration” nodes - nodes that are on 24/7 to run basic connectively functionality like the LDAP user authentication database, the SSH point of entry, and network ingresses as we launch server applications. • Storage • Elastic File System (EFS) -$ 0.30/GB/mo

I use EFS as main storage for home directories. It’s not cheap for large data ($30/mo for 100 GB, for comparison, Dropbox is$9/mo for 1 TB), but it’s extremely convenient and reliable, a true file system (unlike Dropbox), and for configurations and environments is a great no-hassle way to get started. It’s pay for what you use, so if you keep this to just configurations it can be as low as $5/mo. The other great thing about EFS is that it’s really a managed NFS service, so all of your containers can connect to it simultaneously. • Elastic Block Store (EBS) SSD -$ 0.10/GB/mo

SSD storage is required for the root file systems of all nodes. I generally set these to fairly small sizes (10-20 GB), so total cost for my 3 permanent nodes is only ~$5/mo • Elastic Block Store (EBS) Magnetic -$ 0.045/GB/mo

Magnetic block storage is considerably cheaper than EFS, but also is much less convenient as only a single container can mount an EBS volume at any given time. I do use these in the intial tutorials for storing databases (e.g. LDAP data) as only a single container needs access. In future tutorials I plan to show how to launch an NFS server hosted in K8s (based on Ganesha), which will offer a low-cost alternative to EFS for larger data. I’ll also demonstrate use of rook.io for k8s-native distributed storage, backed by EBS.

• Simple Storage Service (S3) - $0.023/GB/mo This is going to be the bread-and-butter long term storage for my environment, as it’s most appropriate for large datasets. While traditional tools like Python are only just starting to work with object storage, Spark can use it quite effectively. • Advanced Compute • Jupyter Hub My initial tutorials will deploy a single-instance juptyerhub that will be shared by any users of the system. The instance can be sized according to administrator preference, and depending on workload can make use of spot priced instances which can often be 80% discounted compared to reserved instances. • Spark Spark is a great candidate for using spot instances - you can have several instances designated as spark administrators that are smaller and always on, then a much larger pool of large workers that are spot instances - they will go in and out of the cluster, but will work hard when they join and will be extremely cost effective. The whole cluster only needs to be launched as needed, and can be destroyed when not in use. • GPU Nodes Like the other compute nodes, these only need to be provisioned when needed. A great advantage with this infrastructure is that all cuda/cudnn drivers are in the docker container, so setup time is on the order of minutes - compared to hours if doing everything from scratch. All told, base cost of the system to leave running 24/7 is ~$50/month, which will allow users to login, edit code, and launch simple jobs in the jupyterhub container. Heavier computation will raise the cost, but in proportion to the size of the job, and since these are generally able to use spot instances the prices can be quite reasonable.

As an example, a t2.xlarge instance that is normally $0.16/hr on the reserved market can generally be had for approx$ 0.05/hr in spot - this has 4 cores and 16 GB of ram, which is plenty for many investigations. For larger work you can go to a m5.12xlarge, a monster with 48 cores and 192 GB ram - normally $2.3/hr on reserved market, but only$ 0.75 on spot.

To give an estimate for personal side projects… let’s say a user is able to really buckle down on a problem every ~3 months and spend a week straight on preliminary investigations, with a weekend of heavy duty compute to finish training, hyperparameter searches, etc. The rest of the time the administration nodes are sufficient for general coding, writing paper abstracts/posters, etc.

In this scenario our cost for the year would be something like:

• 12 x $50 for administration ($600)

• 4 x 7 x 24 x $0.05/hr for a t2.xlarge for exploratory work ($33)

• 4 x 2 x 24 x $0.75/hr for a m5.12xlarge for heavy duty analysis ($144)

• 1 TB data in S3 for the year = 1000 GB * .023/GB/mo * 12 = ($276) Total annual cost:$1053

Even including the purchase of that Chromebook to access these resources, and adding cost to expand for spark and other things, it would still take 2-3 years to match the price of a new MacBook, and you have the advantage of being able to scale to 2-100 nodes and access GPU and other resources whenever you need.

# Security

Security of these kinds of systems are always foremost on my mind. While it’s tempting to focus on just the data science aspects, it’s no fun when your account gets shut down because someone’s gotten in and been routing spam mail through your servers, mining bitcoin, or other uninvited activities.

My typical approach is to have a VPC with public and private subnets, described here. This used to be time consuming to put together, but with Amazon’s wizards it’s become quite easy. Following their wizard will result in a private cloud environment with 2 subnets - one that has direct public access where you would launch any internet-facing servers, and another that accesses websites through a proxy of some sort (e.g. a Network Address Translation (NAT) unit, or other), and that has no direct connect for outside users to access.

This type of setup is very nice from a security perspective as you dramatically reduce your “attack surface area” - the number of computers, ports, and services that are directly exposed to attack. For my purposes I generally just launch a single computer, known as a Bastion Host, that only exposes a single port for tunneling, and that I use to connect to all other internal hosts. I’ll write further posts on how I use ssh tunneling to do this, but it can be surprisingly easy to configure and work with, and it allows you to launch services like Jupyter, Spark, and other web applications inside your private cloud with little to fear as they’re not directly connected to the internet, and as all of your activity is encrypted via your SSH tunnel.

Security is always an engineering challenge, in which you try to find the optimum point that reduces risks of relevant attack vectors to an acceptible level, is feasible to implement and maintain, and feasible for end users to use. For this project I decided that this VPC private-public configuration didn’t meet the requirements, primarily due to ease of implementation for data scientists, and also due to cost with the default configurations. For example, the NAT device that is required for the computers in the private network to get outside connectivity (e.g. to download debian package) costs $0.05/hr, or$ 36/mo! I generally reduce costs by configuring my bastion node to also serve as a NAT device, but that gets into fairly low level linux configuration that is out of the scope of this blog.

The security approach I take here is to leverage Kubernetes itself, along with Amazon security groups, to limit access to relevant nodes and ports. Each of the kubernetes nodes has both a public and private interface, which eliminates the need for the NAT device and dramatically reduces the overhead cost. The default configuration will only expose the SSH port of the computers themselves, the API port for the master (protected by K8s authentication and https), and an SSH port for a container running inside the K8s cluster itself that will serve as our Bastion entry point. We will still be able to launch Jupyter, Spark, and other services on our cluster, and because they are not exposed with ingresses or with security groups they will be inaccessible by unauthorized users.

# Limitations

Like any system, ours will not be all things to all people, and that’s ok! As currently designed, this is intended to really be a platform for a datascientist and several of his/her collaborators. As such, I’m really focusing on making a generic platform that can be expanded with the latest tools, at the expense of developing a great user interface for new users. I think this has the benefit of exposing the internals and providing a good learning tool, but it will certainly have some rough edges.

In particular, expected rough edges are:

• No central user interface - the administrator will need to use commandline tools to create new types of instances, set the numbers of instances, launch new applications, etc. I’ve tried to make this as user friendly as possible, but it’s still a command line interface (CLI).

• Some linux experience is helpful - the project will be run from a linux Bash command line, and SSH (Secure Shell) is used to access the services. The basics are easy to pick up and extremely beneficial for any data scientist, so I hope this doesn’t drive anyone away, but it is an aspect of the environment.

• Limited auto scaling/automation - in a larger system with dozens/hundreds of VMs it would make sense to have all kinds of automation, e.g. monitoring the CPU usage of machines and creating new ones when capacity is being reached, or eliminating resources when there is downtime. In a system with 3-5 computers, however, this would be overkill, and due to the budget constraints would likely be unwanted - e.g. I want to know when my friends are doing something that will result in a \$1k credit card charge!

Despite these rough edges, I’ve been using this platform for the last several months and already found it extremely enjoyable, and I hope you do, too!

On to the setup!


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Questions & Answers

# The product of two consecutive positive integers is ${\text{306}}$, the quadratic equation to find the integers, if $x$ denotes the smaller integer is ${x^2} + x - 306 = 0$. Check whether the given statement is True or False?$(a){\text{ }}$True $(b){\text{ }}$False

Last updated date: 25th Mar 2023
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Answer
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309.6k+ views
Hint: Be careful while assuming the numbers, here, it is given that the numbers are consecutive, the other cases are also possible.

Consider the given equation
${x^2} + x - 306 = 0$
It is given that the smaller integer is $x$,
Therefore, the other consecutive integer can be assumed to be ${\text{ }}x + 1$.
Since the product of the two consecutive integers is ${\text{306}}$.
Therefore, we have,
$x(x + 1) = 306$
${x^2} + x = 306$
$\Rightarrow {x^2} + x - 306 = 0$
That is, the product of the two consecutive integers represent the quadratic equation
${x^2} + x - 306 = 0$
So, the required answer is $(a){\text{ }}$True

Note: In these types of questions, the integers are first put into the respective conditions, and then solved for the required equation.

format: plain


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# Dinostratus

### Quick Info

Born
Greece
Died

Summary
Dinostratus was a Greek mathematician who may have used the quadratrix of Hippias to solve the problem of squaring the circle.

### Biography

Dinostratus is mentioned by Proclus who says (see for example [1] or [3]):-
Amyclas of Heraclea, one of the associates of Plato, and Menaechmus, a pupil of Eudoxus who had studied with Plato, and his brother Dinostratus made the whole of geometry still more perfect.
It is usually claimed that Dinostratus used the quadratrix, discovered by Hippias, to solve the problem of squaring the circle. Pappus tells us (see for example [1] or [3]):-
For the squaring of the circle there was used by Dinostratus, Nicomedes and certain other later persons a certain curve which took its name from this property, for it is called by them square-forming [in other words the quadratrix].
It appears from this quote that Hippias discovered the curve but that it was Dinostratus who was the first to use it to find a square equal in area to a given circle. Proclus, who claims to be quoting from Eudemus, writes (see [1]):-
Nicomedes trisected any rectilinear angle by means of the conchoidal curves, of which he had handed down the origin, order, and properties, being himself the discoverer of their special characteristic. Others have done the same thing by means of the quadratrices of Hippias and Nicomedes.
This makes somewhat less convincing the claim that Dinostratus used the quadratrix, discovered by Hippias, to square the circle since Eudemus does not even mention Dinostratus. There is also a suggestion that Hippias wrote a treatise on the quadratrix and if this is the case it seems hard to believe that he did not show how it could be used to square the circle.

Whether Dinostratus was indeed the first to square the circle using the quadratrix seems almost irrelevant for, as Bulmer-Thomas writes in [1]:-
... posterity has firmly associated the name of Dinostratus with the quadrature of the circle by means of the quadratrix.

Pappus relates how the quadratrix was used to square the circle. The construction of the curve is described in the article on Hippias. The quadratrix meets the line $AD$ in the point $G$ and

arc $BED / AB = AB / AG$

so the length of the circumference of the circle is expressed in terms of the lengths of straight lines. This leads to the construction of a square equal to the circle.

Pappus reports that Sporus was critical of this construction. He had two objections, the first of which relates to the construction of the quadratrix itself (see the article on Hippias). The second objection relates to Dinostratus's use of the quadratrix to square the circle. Sporus claims that the moving line $B'C'$ never cuts the line $AD$ and so the point $G$ is not determined. The point $G$ can only be found as a limit. There is no doubt that Sporus is quite correct with his objection.

Dinostratus probably did much more work on geometry but nothing is known of it.

### References (show)

1. I Bulmer-Thomas, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. G J Allman, Greek geometry from Thales to Euclid (Dublin, 1889).
3. T L Heath, A History of Greek Mathematics I (Oxford, 1921).
4. B L van der Waerden, Science awakening (Groningen, 1954).


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Physics

# Critical Thinking Items

PhysicsCritical Thinking Items

### 11.1Temperature and Thermal Energy

8.
The temperature of two equal quantities of water needs to be raised - the first container by $5$ degrees Celsius and the second by $5$ degrees Fahrenheit. Which one would require more heat?
1. The heat required by the first container is more than the second because each degree Celsius is equal to $1.8$ degrees Fahrenheit.
2. The heat required by the first container is less than the second because each degree Fahrenheit is equal to $1.8$ degrees Celsius.
3. The heat required by the first container is more than the second because each degree Celsius is equal to $3.6$ degrees Fahrenheit.
4. The heat required by the first container is less than the second because each degree Fahrenheit is equal to $3.6$ degrees Celsius.
9.

What is 100.00 °C in kelvins?

1. 212.00 K
2. 100.00 K
3. 473.15 K
4. 373.15 K

### 11.2Heat, Specific Heat, and Heat Transfer

10.

The value of specific heat is the same whether the units are J/kg⋅K or J/kg⋅ºC. How?

1. Temperature difference is dependent on the chosen temperature scale.
2. Temperature change is different in units of kelvins and degrees Celsius.
3. Reading of temperatures in kelvins and degree Celsius are the same.
4. The temperature change is the same in units of kelvins and degrees Celsius.
11.
If the thermal energy of a perfectly black object is increased by conduction, will the object remain black in appearance? Why or why not?
1. No, the energy of the radiation increases as the temperature increases, and the radiation becomes visible at certain temperatures.
2. Yes, the energy of the radiation decreases as the temperature increases, and the radiation remains invisible at those energies.
3. No, the energy of the radiation decreases as the temperature increases, until the frequencies of the radiation are the same as those of visible light.
4. Yes, as the temperature increases, and the energy is transferred from the object by other mechanisms besides radiation, so that the energy of the radiation does not increase.
12.

What is the specific heat of a substance that requires 5.00 kJ of heat to raise the temperature of 3.00 kg by 5.00 °F?

1. 3.33×103 J/kg ⋅° C
2. 6.00×103 J/kg ⋅° C
3. 3.33×102 J/kg ⋅ ° C
4. 6.00×102 J/kg ⋅ ° C

### 11.3Phase Change and Latent Heat

13.

Assume 1.0 kg of ice at 0 °C starts to melt. It absorbs 300 kJ of energy by heat. What is the temperature of the water afterwards?

1. 10 °C
2. 20 °C
3. 5 °C
4. 0 °C


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## Monday, July 6, 2015

### Fill up grids such that $\dfrac{\square}{\square \square}+\dfrac{\square}{\square \square} +\dfrac{\square}{\square\square}=1$

In the following grid, fill up the numbers from 1 to 9 (without repetition) such that their sum is $1$.

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

Solution provided by Mark, another contributor of this blog:

I would begin with:

[MATH]\frac{9}{12}[/MATH]

since this is the largest addend possible. This leaves [MATH]\frac{1}{4}[/MATH].

Observing that:

[MATH]\frac{1}{4}=\frac{17}{68}=\frac{2\cdot5+7}{68}= \frac{5}{34}+\frac{7}{68}[/MATH]

We now have:

[MATH]\frac{9}{12}+\frac{5}{34}+\frac{7}{68}=1[/MATH]


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# Equality in distribution of a sum

When I have 4 random variables, $$A,B,C,D$$ and know that $$A+B \stackrel{d}{=} C + D$$ and $$A \stackrel{d}{=} C$$, does this imply $$B \stackrel{d}{=} D$$?

Going through the definition $$X \stackrel{d}{=} Y$$ when $$P(X \leq z) = P(Y \leq z)$$ for all $$z \in \mathbb R$$ does not seem to lead me pretty far here, but I also cannot think of a counterexample currently.

Can anyone help me with this? Thanks!

Let it be that $$A$$ has non-degenerate symmetric distribution and that $$C=-A$$.
Then $$A\stackrel{d}{=}C$$ and $$A+0=C+2A$$ (so even stronger than $$\stackrel{d}{=}$$) but not $$0\stackrel{d}{=}2A$$


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# Anyway Heres Wonderwall

Author

Derek Sollberger

Published

September 29, 2022

library("patchwork")
library("tidyverse")

# Guitar Chord Diagrams

A few days ago, data scientist Tanya Shapiro posted a neat image of guitar chords—that is, the diagrams were made using ggplot (I presume). Today, I am going to try to imitate that idea.

The music source is this simplified guitar tab from Ultimate Guitar (i.e. this tab does not claim to be a perfect rendition).

## Frets

I think I will start with the frets. (Disclaimer: I am very much just a beginner with playing guitar, and my descriptions may also be elementary.) I need to represent about 5 rows of space along with 6 columns of strings.

df <- data.frame(x = 1:6, y = 0:5) #generic data

df |>
ggplot(aes(x = x, y = y))

Guitar chord diagrams are not necessarily scaled with equally spaced units in the horizontal and vertical directions, but that might make things easier for me in this recreational side project. Also, I will apply theme_linedraw temporarily to show what I am thinking.

df |>
ggplot(aes(x = x, y = y)) +
coord_equal() +
theme_linedraw()

The $$y$$ axis has the math-class default, but guitar notation goes in the other direction.

# https://stackoverflow.com/questions/70193132/how-to-reverse-y-axis-values-ggplot2-r
df |>
ggplot(aes(x = x, y = y)) +
coord_equal() +
scale_y_reverse() +
theme_linedraw()

When I remove theme_linedraw (analogy: removing the training wheels from a bicycle), I will need to (at least)

• turn off the background color
• turn off the grid line colors
• turn off the border lines too
df |>
ggplot(aes(x = x, y = y)) +
coord_equal() +
scale_y_reverse() +
theme(
panel.background = element_blank()
)

Actually, let’s go ahead and give the background a “wood” color.

# https://redketchup.io/color-picker
df |>
ggplot(aes(x = x, y = y)) +
coord_equal() +
scale_y_reverse() +
theme(
panel.background = element_rect(fill = "#2D2C32"),
panel.grid.minor = element_blank()
)

Let’s see if we can cutoff the plot at the top (“nut”) and right side of the plot.

df |>
ggplot(aes(x = x, y = y)) +
coord_equal() +
scale_y_reverse() +
theme(
panel.background = element_rect(fill = "#2D2C32"),
panel.grid.minor = element_blank()
) +
xlim(c(1,6)) +
ylim(c(0,5))
Scale for 'y' is already present. Adding another scale for 'y', which will
replace the existing scale.

That did not work (and produced a warning because I affected the $$y$$ axis twice), so perhaps a better strategy is to make a colorful rectangle manually.

df |>
ggplot(aes(x = x, y = y)) +
coord_equal() +
geom_rect(aes(xmin = 1, xmax = 6, ymin = 0, ymax = 5),
fill = "#2D2C32") +
scale_y_reverse() +
theme(
panel.background = element_blank(),
panel.grid.minor = element_blank()
)

This whole time, I have been thinking about making line segments of certain colors for the frets, strings, and the nut. (Note: silver is not a base color in R??)

df_frets <- data.frame(x1 = 1, x2 = 6, y = 1:5)
df_strings <- data.frame(x = 1:6, y1 = 0, y2 = 5)

df |>
ggplot(aes(x = x, y = y)) +
coord_equal() +
geom_rect(aes(xmin = 1, xmax = 6, ymin = 0, ymax = 5),
fill = "#2D2C32") +
geom_segment(aes(x = x1, y = y, xend = x2, yend = y),
color = "gray50", data = df_frets) +
geom_segment(aes(x = 1, y = 0, xend = 6, yend = 0),
color = "tan", size = 3) +
geom_segment(aes(x = x, y = y1, xend = x, yend = y2),
color = "#C0C0C0", data = df_strings, size = 2) +
scale_y_reverse() +
theme(
panel.background = element_blank(),
panel.grid.minor = element_blank()
)

This looks great in my opinion. Let’s turn off the axis tick marks and labels, and save this as a base plot.

fret_background <- df |>
ggplot(aes(x = x, y = y)) +
coord_equal() +
geom_rect(aes(xmin = 1, xmax = 6, ymin = 0, ymax = 5),
fill = "#2D2C32") +
geom_segment(aes(x = x1, y = y, xend = x2, yend = y),
color = "gray50", data = df_frets) +
geom_segment(aes(x = 1, y = 0, xend = 6, yend = 0),
color = "tan", size = 3) +
geom_segment(aes(x = x, y = y1, xend = x, yend = y2),
color = "#C0C0C0", data = df_strings, size = 2) +
scale_y_reverse() +
theme(
axis.title = element_blank(),
axis.ticks = element_blank(),
axis.text = element_blank(),
panel.background = element_blank(),
panel.grid.minor = element_blank()
)

# print
fret_background

## First Chord (Em)

At first, I was thinking of using geom_label to indicate the finger positions, but I bet that using geom_point and then geom_text has more customization options.

df_em <- data.frame(x = c(2,3), y = c(2,2) - 0.5, label = c("1", "2"))
fret_background +
geom_point(aes(x = x, y = y),
color = "blue4", data = df_em, size = 15) +
geom_text(aes(x = x, y = y, label = label),
color = "tan", data = df_em, size = 10)

Let us go ahead and label this chord. Note that I want the dots to appear in between the frets (hence the “-0.5” in the $$y$$ coordinates).

df_em <- data.frame(x = c(2,3), y = c(2,2) - 0.5, label = c("1", "2"))

chord_em <- fret_background +
geom_point(aes(x = x, y = y),
color = "blue4", data = df_em, size = 12) +
geom_text(aes(x = x, y = y, label = label),
color = "tan", data = df_em, size = 8) +
labs(title = "Em") +
theme(plot.title = element_text(size = 20, face = "bold", hjust = 0.5))

# print
chord_em

## The Other Four Chords

df_g <- data.frame(x = c(2,1,6), y = c(2,3,3) - 0.5, label = c("1", "2", "4"))

chord_g <- fret_background +
geom_point(aes(x = x, y = y),
color = "blue4", data = df_g, size = 12) +
geom_text(aes(x = x, y = y, label = label),
color = "tan", data = df_g, size = 8) +
labs(title = "G") +
theme(plot.title = element_text(size = 20, face = "bold", hjust = 0.5))

# print
chord_g

This is something that I worried about. In guitar chord diagrams, there are indicators to tell the player if a string is played open (indicated with an “O”, that is that no fingers are placed on the strings while the string is strung) or if a string is closed (not strung at all, indicated with an “X”), but these indicators are placed above the nut—for our purposes: outside of the graph.

df_d <- data.frame(x = c(4,6,5), y = c(2,2,3) - 0.5, label = c("1", "2", "3"))

chord_d <- fret_background +
geom_point(aes(x = x, y = y),
color = "blue4", data = df_d, size = 12) +
geom_text(aes(x = x, y = y, label = label),
color = "tan", data = df_d, size = 8) +
geom_text(aes(x = 1, y = -0.2, label = "X"), size = 5) +
geom_text(aes(x = 2, y = -0.2, label = "X"), size = 5) +
labs(title = "D") +
theme(plot.title = element_text(size = 20, face = "bold", hjust = 0.5))

# print
chord_d

df_a7sus4 <- data.frame(x = c(3,5), y = c(2,3) - 0.5, label = c("1", "2"))

chord_a7sus4 <- fret_background +
geom_point(aes(x = x, y = y),
color = "blue4", data = df_a7sus4, size = 12) +
geom_text(aes(x = x, y = y, label = label),
color = "tan", data = df_a7sus4, size = 8) +
geom_text(aes(x = 1, y = -0.2, label = "X"), size = 5) +
labs(title = "A7sus4") +
theme(plot.title = element_text(size = 20, face = "bold", hjust = 0.5))

# print
chord_a7sus4

df_c <- data.frame(x = c(5,3,2), y = c(1,2,3) - 0.5, label = c("1", "2", "3"))

chord_c <- fret_background +
geom_point(aes(x = x, y = y),
color = "blue4", data = df_c, size = 12) +
geom_text(aes(x = x, y = y, label = label),
color = "tan", data = df_c, size = 8) +
geom_text(aes(x = 1, y = -0.2, label = "X"), size = 5) +
labs(title = "C") +
theme(plot.title = element_text(size = 20, face = "bold", hjust = 0.5))

# print
chord_c

# Layout

Now it is time to start to build the infographic! The patchwork package is so useful!

## First Draft

With 5 chords, I am imagining a layout of plots with about 2 rows.

(chord_em + chord_g + chord_d) / (chord_a7sus4 + chord_c)

## A Caption Tile

Yes, I figured that I would want to squeeze in a “square” of text somewhere to even out the image.

# https://statisticsglobe.com/plot-only-text-in-r
caption_plot <- ggplot() +
annotate("text",
x = 1, y = 1,
size = 5,
label = "Wonderwall by Oasis\n \nsimplified tab\nvia Ultimate Guitar\ncapo: 2nd fret\nkey: F#m") +
coord_equal() +
theme(
axis.title = element_blank(),
axis.ticks = element_blank(),
axis.text = element_blank(),
panel.background = element_rect(fill = "#2D2C32"),
panel.grid.major = element_blank(),
panel.grid.minor = element_blank()
)

# print
caption_plot

Now to see if this “caption plot” fits in nicely.

(chord_em + chord_g + chord_d) / (chord_a7sus4 + caption_plot +chord_c)

At the time of this writing, it is near midnight, and my motivation for refinement is decreasing rapidly, lol.

main_plot <- (chord_em + chord_g + chord_d) / (chord_a7sus4 + caption_plot +chord_c)

## Strumming Pattern

I, and I assume other beginners, do not natually pick up a strumming pattern by ear. Fortunately, the tab did include that. But how do I add the pattern to this diagram?

# https://community.rstudio.com/t/up-and-down-arrows-inside-r/76287

# first, I will type into words, and then do a search-and-replace
# down, pause, down, pause, down, pause, down, up, down, up, down, pause, down, pause, down, up, down, up, down, pause, down, pause, down, up, pause, up, pause, up, down, pause, down, pause
strumming_pattern <- intToUtf8(c(8595, 8196, 8595, 8196, 8595, 8196, 8595, 8593, 8595, 8593, 8595, 8196, 8595, 8196, 8595, 8593, 8595, 8593, 8595, 8196, 8595, 8196, 8595, 8593, 8196, 8593, 8196, 8593, 8595, 8196, 8595, 8196))

## Annotationg and Echoing the Meme

With patchwork, we can furthermore add a title, subtitle, and/or caption to the overall plot.

final_plot <- main_plot +
plot_annotation(title = "Anyway, Here's Wonderwall",
caption = paste("strumming pattern:", strumming_pattern, "\nGraph by @DerekSollberger"),
theme = theme(plot.title = element_text(
size = 25, hjust = 0.5
),
plot.caption = element_text(
size = 20, hjust = 0.5
)))

# print
final_plot

ggsave("wonderwall.png", plot = final_plot, device = "png",
width = 8, height = 6, units = "in")


text:
# 2D morphing video images

I'm trying to morph from one video to another. I've seen some tutorial of how to morph images subdividing the plane and moving vertices, but what I really want is to morph from 2D moving images (video) sequence. Does anyone know of any add-on/pluggin that I can use to accomplish this?

I was thinking about placing a feature request, since I think that the tracking system can be used to greatly automate this task. Does anyone know the right place to do this? Do you think it is an interesting feature to ask for?

I know there are programs that can be used to accomplish this task outside blender. I'm using gtkmorph to do the task, but AFAIK I have to morph from still images, and morphing a video sequence becomes very time consuming.

• Using the tracker would be quite simple. Use "Link Empty to Track", then on the 3d vewport use those empties as hooks to deform a plane's geometry. – cegaton Sep 18 '15 at 14:18
• @cegaton, that looks promising but how do you merge both videos tracking points and images? – YoMismo Sep 19 '15 at 21:17
• Morph in what way? Some pictures or videos would do a lot to help us understand what you're after. I can think of a lot of different methods. It depends on what you're specifically trying to do. cegaton's comment is the closest to what I would do in all cases though. If that seems overwhelming, it's because it is hard to do. To do The Rolling Stones video, Michel Gondry employed his brother to help do the programming for things he could not do himself. youtu.be/0-jm6qb7hmA?t=20m30s – Wray Bowling Oct 23 '15 at 18:21
• @WrayBowling regarding this question I reformulated it as blender.stackexchange.com/questions/38941/… according to @cegaton´s comment. I finally achieved the effect with gtkmorph check minute 3:30 in the next video youtube.com/watch?v=f8PcW7V9PSY but I would be glad to know if the updated question in the link above has an answer. Thanks. – YoMismo Oct 26 '15 at 7:49
• In the future, try not to fragment your questions. In all likelihood, now one of the two questions will be closed as a duplicate. – Wray Bowling Oct 26 '15 at 13:56

## 1 Answer

Try image sequence import addon. This makes animated images appear on plane. See this video link here. https://m.youtube.com/watch?v=3UOywRFIpLM&feature=youtu.be

Updated addon link: https://github.com/vicentecarro/ImageSequenceAsPlanes

First create image sequence from a video clip. With that addon, import Image sequence and all animated images appear on plane. Subdivide the plane like a lattice. Alternatively use a knife to divide plane's face. When you move vertrice or reshape the plane, the animated image distorts, useful for morphing your way. Hope that helps.

If unsatisfied, try DragonBones, a free open source software.

• Hi @RitaGeraghty, thanks for your answer but the problem here is morphing points of two different planes (remember I want to morph from two videos) AND mixing images as they blend and transform. – YoMismo Feb 7 at 8:06
• That is going to be tricky. You can get away with Gimp's morphing GAP tool. Take screenshots of two video clips. GAP in Gimp lets you morph two photos and do image sequence of that. After that, you can import image sequence and sandwich it between two video clips. youtu.be/WCiYSiHKfO0 – Rita Geraghty Feb 7 at 8:45
• If I can remember, I tried Gimp but rather prefered gtkmorph which is what I finally used for the job. I just thought it would be nice to have that functionality in Blender. – YoMismo Feb 7 at 14:50


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 Questions about Kneser... JmsNxn Long Time Fellow Posts: 419 Threads: 81 Joined: Dec 2010 02/14/2021, 04:28 AM (This post was last modified: 02/14/2021, 04:33 AM by JmsNxn.) So I've been running through my head a question. But in order to resolve that question, I need to better understand Kneser's construction. The premise of this post is to talk about constructing pentation from Kneser's construction. Now, I know that, $ \text{tet}_{\text{Kneser}}(s) : \mathcal{H} \to \mathbb{C}\\$ Where $\mathcal{H} = \{\Im(s) > 0\}$. And I know it tends to a fixed point $L$ as $\Re(s) \to -\infty$. (Or is it multiple fixed points?) Either way, this convergence must be geometric. Giving us, what I'll call, $ \Phi_{\text{Kneser}}(s) = \Omega_{j=1}^\infty \text{tet}_{\text{Kneser}}(s-j+z) \bullet z : \mathcal{H} \to \mathbb{C}\\$ Satisfying the functional equation, $ \Phi(s+1) = \text{tet}(s+\Phi(s))\\$ And it tends to $L$ as $s \to -\infty$ exactly like $\text{tet}(s-1 + L)$. Proving this converges will be a little troublesome if we don't have good control over Kneser's solution; but for the moment assume it's good enough. It'll definitely work for large enough $T$ with $\Re(s) < -T$. The idea then being, $ \text{slog}^{\circ n} \Phi(s+n) \to \text{pent}(s)\\$ Now, I know what you're thinking. Won't the same thing happen that happened with tetration happen with this method? And I'd say probably. Now instead of looking for $\log(0)$'s though, we'd be looking for $\text{slog}(L)$'s. So where ever $\Phi(s_0) = L$ we get ourselves into a whole lot of trouble.  The only real benefit I can think in this circumstance is that $\Phi$ is not periodic. And that $\Phi(s) \to L$ but it shouldn't equal $L$, or at least, it should be well controlled where it does. This depends on Kneser though, does it ever attain the value $L$ other than at $-\infty$? I really wish there was more supplemental literature on Kneser's construction other than what's available on this forum... -_- Nonetheless, I think this might have a better chance at converging than my tetration function constructed with $\phi$. It's just a hunch though. sheldonison Long Time Fellow Posts: 664 Threads: 23 Joined: Oct 2008 02/14/2021, 05:00 PM (This post was last modified: 02/14/2021, 05:22 PM by sheldonison.) (02/14/2021, 04:28 AM)JmsNxn Wrote: So I've been running through my head a question. But in order to resolve that question, I need to better understand Kneser's construction. The premise of this post is to talk about constructing pentation from Kneser's construction. Now, I know that, $\text{tet}_{\text{Kneser}}(s):\mathcal{H}\to\mathbb{C}$ Where $\mathcal{H} = \{\Im(s) > 0\}$. And I know it tends to a fixed point $L$ as $\Re(s) \to -\infty$. (Or is it multiple fixed points?) ...This depends on Kneser though, does it ever attain the value $L$ other than at $-\infty$? ... I really wish there was more supplemental literature on Kneser's construction other than what's available on this forum... -_-Hey James, The two fixed points are L, L* in the upper/lower halves of the complex plane.  Kneser tends to L as $\Im(z)\to\infty;\;\;\Re(z)\to-\infty$ Wherever $\text{tet}(z)=L+2n\pi i;\;\;\text{tet}(z+1)=L$ and this happens an infinite number of times in the complex plane See this mathstack post for a good readable overview of Kneser. - Sheldon JmsNxn Long Time Fellow Posts: 419 Threads: 81 Joined: Dec 2010 02/16/2021, 12:46 AM (This post was last modified: 02/16/2021, 06:52 AM by JmsNxn.) Interesting, so we'd have branches of $\text{slog}$ that are holomorphic in a neighborhood of $L$ then. Unlike with the logarithm where no branch is holomorphic in a neighborhood of $0$. I think the trouble with this construction will be getting $\Phi$ to be well behaved as we grow $\Re(s)$. And not to mention, getting general growth lemmas on the various branches of $\text{slog}$. The only work around I see would be to take the inverse iteration, $ \text{pent} = \text{tet}_{\text{Kneser}}^{\circ n} \Phi(s-n)\\$ Where of course, we'd have to modify $\Phi$ to converge in this circumstance. And here we'd probably lose any chance of it being real-valued. I'm going to keep this on a backburner and come back to it later. I'm going to stay focused on $\mathcal{C}^\infty$ proofs for the moment. « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Generalized Kneser superfunction trick (the iterated limit definition) MphLee 21 2,774 03/29/2021, 02:54 AM Last Post: JmsNxn Alternative manners of expressing Kneser JmsNxn 1 297 03/19/2021, 01:02 AM Last Post: JmsNxn Kneser method question tommy1729 9 7,518 02/11/2020, 01:26 AM Last Post: sheldonison [split] Understanding Kneser Riemann method andydude 7 13,351 01/13/2016, 10:58 PM Last Post: sheldonison Equations for Kneser sexp algorithm sheldonison 18 40,570 06/21/2011, 01:48 AM Last Post: sheldonison "Kneser"-like mapping also for complex base? mike3 2 8,094 11/07/2010, 01:37 PM Last Post: sheldonison some questions about sexp tommy1729 12 21,648 07/06/2010, 01:45 PM Last Post: sheldonison Kneser's Super Logarithm bo198214 18 57,894 01/29/2010, 06:43 AM Last Post: mike3

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# Properties

 Label 41070bl Number of curves 8 Conductor 41070 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("41070.bh1")

sage: E.isogeny_class()

## Elliptic curves in class 41070bl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
41070.bh8 41070bl1 [1, 0, 0, 2025, 107865] [2] 103680 $$\Gamma_0(N)$$-optimal
41070.bh6 41070bl2 [1, 0, 0, -25355, 1405677] [2, 2] 207360
41070.bh7 41070bl3 [1, 0, 0, -18510, -3173628] [2] 311040
41070.bh5 41070bl4 [1, 0, 0, -93805, -9532633] [2] 414720
41070.bh4 41070bl5 [1, 0, 0, -394985, 95513475] [2] 414720
41070.bh3 41070bl6 [1, 0, 0, -456590, -118563900] [2, 2] 622080
41070.bh1 41070bl7 [1, 0, 0, -7301590, -7594672900] [2] 1244160
41070.bh2 41070bl8 [1, 0, 0, -620870, -25679988] [2] 1244160

## Rank

sage: E.rank()

The elliptic curves in class 41070bl have rank $$0$$.

## Modular form 41070.2.a.bh

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 4q^{7} + q^{8} + q^{9} + q^{10} + q^{12} - 2q^{13} - 4q^{14} + q^{15} + q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.


text:
I do not fear any conspiracy from any nook & corner. I am simply taking my time and my space to stage the inevitable confrontation in the frozen face of the industry and geopolitics tycoons. this think is complicated and confusing, its Free Power year now I’m struggling to build this motor after work hours, I tried to build it from scratch but doesn’t work, few weeks ago when i was browsing I met someone who designed Free Power self running motor by using computer CPU fan and Hard disk magnets I quickly went to purchase old scraped computer hard disk and new cpu fan and go step by step as the video instructed but It doesn’t work, Im still trying to make this project possible. Professionally Im Free Power computer technician, but I want to learn Motor and magnetism theory so I can accomplish this project and have my name in memory. I anyone can make this project please contact me through facebook so I can invite him/her to my country and make money as you know third word countries has power disaster. My facebook Id is Elly Maduhu Nkonya, or use my E-mail. [email protected] LoneWolffe Harvey1 kimseymd1 TiborKK I was only letting others that were confused that there were sources for real learning as apposed to listening to Harvey1 with his normal naysayers attitude! There is tons of information on schoolgirl, schoolboy and Bedini window motors that actually work to charge batteries and eventually will generate house currents. It just has to be looked at to get any useful information from it without listening to people like Harvey1 whining about learning. Harvey1 kimseymd1 You obviously play too much video games with trolls etc. in them. Why the editors of this forum allow you to keep calling people names instead of following the subject is beyond me. This must be the last site to allow you on it. I spammed the books because I thought those people were good for learning these engines which are super and there are tons of information out there for anyone to find. You seem to only want to learn to be rude instead of electronics.
OK, these events might be pathetic money grabs, but certainly if some of the allegations against her were true, both groups would argue, would she not be behind bars by now? Suffice it to say, most people who have done any manner of research into the many Free Energy against Free Electricity have concluded that while she is most likely Free Power criminal, they just can’t see her getting arrested. But if–and it’s Free Power big ‘if’–she ever does get arrested and convicted of Free Power serious crime, that likely would satisfy the most ardent skeptic and give rise to widespread belief that the Trump Administration is working on, and succeeding in, taking down the Deep State. Let’s examine the possibility that things are headed in that direction.
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The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. 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There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.
I do not fear any conspiracy from any nook & corner. I am simply taking my time and my space to stage the inevitable confrontation in the frozen face of the industry and geopolitics tycoons. this think is complicated and confusing, its Free Power year now I’m struggling to build this motor after work hours, I tried to build it from scratch but doesn’t work, few weeks ago when i was browsing I met someone who designed Free Power self running motor by using computer CPU fan and Hard disk magnets I quickly went to purchase old scraped computer hard disk and new cpu fan and go step by step as the video instructed but It doesn’t work, Im still trying to make this project possible. Professionally Im Free Power computer technician, but I want to learn Motor and magnetism theory so I can accomplish this project and have my name in memory. I anyone can make this project please contact me through facebook so I can invite him/her to my country and make money as you know third word countries has power disaster. My facebook Id is Elly Maduhu Nkonya, or use my E-mail. [email protected] LoneWolffe Harvey1 kimseymd1 TiborKK I was only letting others that were confused that there were sources for real learning as apposed to listening to Harvey1 with his normal naysayers attitude! There is tons of information on schoolgirl, schoolboy and Bedini window motors that actually work to charge batteries and eventually will generate house currents. It just has to be looked at to get any useful information from it without listening to people like Harvey1 whining about learning. Harvey1 kimseymd1 You obviously play too much video games with trolls etc. in them. Why the editors of this forum allow you to keep calling people names instead of following the subject is beyond me. This must be the last site to allow you on it. I spammed the books because I thought those people were good for learning these engines which are super and there are tons of information out there for anyone to find. You seem to only want to learn to be rude instead of electronics.

I don’t know what to do. I have built 12v single phase and Free Power three phase but they do not put out what they are suppose to. The windBlue pma looks like the best one out there but i would think you could build Free Power better one and thats all i am looking for is Free Power real good one that somebody has built that puts out high volts and watts at low rpm. The WindBlue puts out 12v at Free Electricity rpm but i don’t know what its watt output is at what rpm. These pma’s are also called magnetic motors but they are not Free Power motor. They are Free Power generator. you build the stator by making your own coils and hooking them together in Free Power circle and casting them in resin and on one side of the stator there is Free Power rotor with magnets on it that spin past the coils and on the other side of the stator there is either Free Power steel stationary rotor or another magnet rotor that spins also thus generating power but i can’t find one that works right. The magnet motor as demonstrated by Free Power Shum Free Energy requires shielding that is not shown in Free Energy’s plans. Free Energy’s shielding is simple, apparently on the stator. The Perendev shows each magnet in the Free Energy shielded. Actually, it intercepts the flux as it wraps around the entire set of magnets. The shielding is necessary to accentuate interaction between rotor and stator magnets. Without shielding, the device does not work. Hey Gilgamesh, thanks and i hope you get to build the motor. I did forget to ask one thing on the motor. Are the small wheels made of steel or are they magnets? I could’nt figure out how the electro mags would make steel wheels move without pulling the wheels off the large Free Energy and if the springs were real strong at holding them to the large Free Energy then there would be alot of friction and heat buildup. Ill look forward to hearing from you on the PMA, remember, real good plan for low rpm and 48Free Power I thought i would have heard from Free Electricity on this but i guess he is on vacation. Hey Free Power. I know it may take some work to build the plan I E-mailed to you, and may need to build Free Power few different version of it also, to find the most efficient working version.
Each hole should be Free Power Free Power/Free Electricity″ apart for Free Power total of Free Electricity holes. Next will be setting the magnets in the holes. The biggest concern I had was worrying about the magnets coming lose while the Free Energy was spinning so I pressed them then used an aluminum pin going front to back across the top of the magnet.
Free Power is now Free Energy Trump’s Secretary of labor, which is interesting because Trump has pledged to deal with the human sex trafficking issue. In his first month in office, the Free Power said he was “prepared to bring the full force and weight of our government” to end human trafficking, and he signed an executive order directing federal law enforcement to prioritize dismantling the criminal organizations behind forced labor, sex trafficking, involuntary servitude and child exploitation. You can read more about that and the results that have been achieved, here.
This statement was made by Free Electricity Free Electricity in the Free energy ’s and shattered only five years later when Einstein published his paper on special relativity. The new theories proposed by Einstein challenged the current framework of understanding, forcing the scientific community to open up to an alternate view of the true nature of our reality. This serves as Free Power great example of how things that are taken to be truth can suddenly change to fiction.
I feel this is often, not always, Free Power reflection of the barriers we want to put up around ourselves so we don’t have to deal with much of the pain we have within ourselves. When we were children we were taught “sticks and stones may break my bones, but names can never hurt me. ” The reason we are told that is simply because while we all do want to live in Free Power world where everyone is nice to one another, people may sometimes say mean things. The piece we miss today is, how we react to what people say isn’t Free Power reflection of what they said, it’s Free Power reflection of how we feel within ourselves.

Free energy is that portion of any first-law energy that is available to perform thermodynamic work at constant temperature, i. e. , work mediated by thermal energy. Free energy is subject to irreversible loss in the course of such work. [Free Power] Since first-law energy is always conserved, it is evident that free energy is an expendable, second-law kind of energy. Several free energy functions may be formulated based on system criteria. Free energy functions are Legendre transforms of the internal energy.


text:
# why characteristic function of closed sets are upper- semi continious?

Prove that characteristic function of closed sets are upper- semi continious

My attempt : i know the definition of upper- semi continious

i,e $$\chi_A ^{-1} (-\infty, a)$$ is open for every $$a \in \mathbb{R}$$ where $$A \subset X$$

we know that under a continuous function, the inverse image of a closed set is closed

Now my confusion is that if characteristic function is closed then $$\chi_A ^{-1} (-\infty, a]$$ is closed for every $$a \in \mathbb{R}$$ which is the definition of lower semicontinious

But our motive is to show characteristic function of closed sets are upper- semi continious

My doubts : why characteristic function of closed sets are upper- semi continious ?

• $\chi_A^{-1} (-\infty.a)=X$ if $a \geq 1$ and $A^{c}$ is $a <1$. – Kavi Rama Murthy Feb 23 at 6:03


text:
# General Chemistry: Principles, Patterns, and Applications, v. 1.0 (2 Volume Set)

by Bruce Averill and Patricia Eldredge

## 19.1 Describing Electrochemical Cells

### Learning Objective

1. To distinguish between galvanic and electrolytic cells.

In any electrochemical process, electrons flow from one chemical substance to another, driven by an oxidation–reduction (redox) reaction. As we described in Chapter 3 "Chemical Reactions", a redox reaction occurs when electrons are transferred from a substance that is oxidized to one that is being reduced. The reductantA substance that is capable of donating electrons and in the process is oxidized. is the substance that loses electrons and is oxidized in the process; the oxidantA substance that is capable of accepting electrons and in the process is reduced. is the species that gains electrons and is reduced in the process. The associated potential energy is determined by the potential difference between the valence electrons in atoms of different elements. (For more information on valence electrons, see Chapter 7 "The Periodic Table and Periodic Trends", Section 7.3 "Energetics of Ion Formation".)

Because it is impossible to have a reduction without an oxidation and vice versa, a redox reaction can be described as two half-reactionsReactions that represent either the oxidation half or the reduction half of an oxidation–reduction (redox) reaction., one representing the oxidation process and one the reduction process. For the reaction of zinc with bromine, the overall chemical reaction is as follows:

Equation 19.1

Zn(s) + Br2(aq) → Zn2+(aq) + 2Br(aq)

The half-reactions are as follows:

Equation 19.2

reduction half-reaction: Br2(aq) + 2e → 2Br(aq)

Equation 19.3

oxidation half-reaction: Zn(s) → Zn2+(aq) + 2e

Each half-reaction is written to show what is actually occurring in the system; Zn is the reductant in this reaction (it loses electrons), and Br2 is the oxidant (it gains electrons). Adding the two half-reactions gives the overall chemical reaction (Equation 19.1). A redox reaction is balanced when the number of electrons lost by the reductant equals the number of electrons gained by the oxidant. Like any balanced chemical equation, the overall process is electrically neutral; that is, the net charge is the same on both sides of the equation.

### Note the Pattern

In any redox reaction, the number of electrons lost by the reductant equals the number of electrons gained by the oxidant.

In most of our discussions of chemical reactions, we have assumed that the reactants are in intimate physical contact with one another. Acid–base reactions, for example, are usually carried out with the acid and the base dispersed in a single phase, such as a liquid solution. With redox reactions, however, it is possible to physically separate the oxidation and reduction half-reactions in space, as long as there is a complete circuit, including an external electrical connection, such as a wire, between the two half-reactions. As the reaction progresses, the electrons flow from the reductant to the oxidant over this electrical connection, producing an electric current that can be used to do work. An apparatus that is used to generate electricity from a spontaneous redox reaction or, conversely, that uses electricity to drive a nonspontaneous redox reaction is called an electrochemical cellAn apparatus that generates electricity from a spontaneous oxidation–reduction (redox) reaction or, conversely, uses electricity to drive a nonspontaneous redox reaction..

There are two types of electrochemical cells: galvanic cells and electrolytic cells. A galvanic (voltaic) cellAn electrochemical cell that uses the energy released during a spontaneous oxidation–reduction (redox) reaction $(ΔG<0)$ to generate electricity.Galvanic cells are named for the Italian physicist and physician Luigi Galvani (1737–1798), who observed that dissected frog leg muscles twitched when a small electric shock was applied, demonstrating the electrical nature of nerve impulses. uses the energy released during a spontaneous redox reaction (ΔG < 0) to generate electricity. This type of electrochemical cell is often called a voltaic cell after its inventor, the Italian physicist Alessandro Volta (1745–1827). In contrast, an electrolytic cellAn electrochemical cell that consumes electrical energy from an external source to drive a nonspontaneous $(ΔG>0)$ oxidation–reduction (redox) reaction. consumes electrical energy from an external source, using it to cause a nonspontaneous redox reaction to occur (ΔG > 0). Both types contain two electrodesA solid metal connected by an electrolyte and an external circuit that provides an electrical connection between systems in an electrochemical cell (galvanic or electrolytic)., which are solid metals connected to an external circuit that provides an electrical connection between the two parts of the system (Figure 19.1 "Electrochemical Cells"). The oxidation half-reaction occurs at one electrode (the anodeOne of two electrodes in an electrochemical cell, it is the site of the oxidation half-reaction.), and the reduction half-reaction occurs at the other (the cathodeOne of two electrodes in an electrochemical cell, it is the site of the reduction half-reaction.). When the circuit is closed, electrons flow from the anode to the cathode. The electrodes are also connected by an electrolyte, an ionic substance or solution that allows ions to transfer between the electrode compartments, thereby maintaining the system’s electrical neutrality. In this section, we focus on reactions that occur in galvanic cells. We discuss electrolytic cells in Section 19.7 "Electrolysis".

Figure 19.1 Electrochemical Cells

A galvanic cell (left) transforms the energy released by a spontaneous redox reaction into electrical energy that can be used to perform work. The oxidative and reductive half-reactions usually occur in separate compartments that are connected by an external electrical circuit; in addition, a second connection that allows ions to flow between the compartments (shown here as a vertical dashed line to represent a porous barrier) is necessary to maintain electrical neutrality. The potential difference between the electrodes (voltage) causes electrons to flow from the reductant to the oxidant through the external circuit, generating an electric current. In an electrolytic cell (right), an external source of electrical energy is used to generate a potential difference between the electrodes that forces electrons to flow, driving a nonspontaneous redox reaction; only a single compartment is employed in most applications. In both kinds of electrochemical cells, the anode is the electrode at which the oxidation half-reaction occurs, and the cathode is the electrode at which the reduction half-reaction occurs.

## Galvanic (Voltaic) Cells

To illustrate the basic principles of a galvanic cell, let’s consider the reaction of metallic zinc with cupric ion (Cu2+) to give copper metal and Zn2+ ion. The balanced chemical equation is as follows:

Equation 19.4

Zn(s) + Cu2+(aq) → Zn2+(aq) + Cu(s)

We can cause this reaction to occur by inserting a zinc rod into an aqueous solution of copper(II) sulfate. As the reaction proceeds, the zinc rod dissolves, and a mass of metallic copper forms (Figure 19.2 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Single Compartment"). These changes occur spontaneously, but all the energy released is in the form of heat rather than in a form that can be used to do work.

Figure 19.2 The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Single Compartment

When a zinc rod is inserted into a beaker that contains an aqueous solution of copper(II) sulfate, a spontaneous redox reaction occurs: the zinc electrode dissolves to give Zn2+(aq) ions, while Cu2+(aq) ions are simultaneously reduced to metallic copper. The reaction occurs so rapidly that the copper is deposited as very fine particles that appear black, rather than the usual reddish color of copper.

This same reaction can be carried out using the galvanic cell illustrated in part (a) in Figure 19.3 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell". To assemble the cell, a copper strip is inserted into a beaker that contains a 1 M solution of Cu2+ ions, and a zinc strip is inserted into a different beaker that contains a 1 M solution of Zn2+ ions. The two metal strips, which serve as electrodes, are connected by a wire, and the compartments are connected by a salt bridgeA U-shaped tube inserted into both solutions of a galvanic cell that contains a concentrated liquid or gelled electrolyte and completes the circuit between the anode and the cathode., a U-shaped tube inserted into both solutions that contains a concentrated liquid or gelled electrolyte. The ions in the salt bridge are selected so that they do not interfere with the electrochemical reaction by being oxidized or reduced themselves or by forming a precipitate or complex; commonly used cations and anions are Na+ or K+ and NO3 or SO42−, respectively. (The ions in the salt bridge do not have to be the same as those in the redox couple in either compartment.) When the circuit is closed, a spontaneous reaction occurs: zinc metal is oxidized to Zn2+ ions at the zinc electrode (the anode), and Cu2+ ions are reduced to Cu metal at the copper electrode (the cathode). As the reaction progresses, the zinc strip dissolves, and the concentration of Zn2+ ions in the Zn2+ solution increases; simultaneously, the copper strip gains mass, and the concentration of Cu2+ ions in the Cu2+ solution decreases (part (b) in Figure 19.3 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell"). Thus we have carried out the same reaction as we did using a single beaker, but this time the oxidative and reductive half-reactions are physically separated from each other. The electrons that are released at the anode flow through the wire, producing an electric current. Galvanic cells therefore transform chemical energy into electrical energy that can then be used to do work.

Figure 19.3 The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell

(a) A galvanic cell can be constructed by inserting a copper strip into a beaker that contains an aqueous 1 M solution of Cu2+ ions and a zinc strip into a different beaker that contains an aqueous 1 M solution of Zn2+ ions. The two metal strips are connected by a wire that allows electricity to flow, and the beakers are connected by a salt bridge. When the switch is closed to complete the circuit, the zinc electrode (the anode) is spontaneously oxidized to Zn2+ ions in the left compartment, while Cu2+ ions are simultaneously reduced to copper metal at the copper electrode (the cathode). (b) As the reaction progresses, the Zn anode loses mass as it dissolves to give Zn2+(aq) ions, while the Cu cathode gains mass as Cu2+(aq) ions are reduced to copper metal that is deposited on the cathode.

The electrolyte in the salt bridge serves two purposes: it completes the circuit by carrying electrical charge and maintains electrical neutrality in both solutions by allowing ions to migrate between them. The identity of the salt in a salt bridge is unimportant, as long as the component ions do not react or undergo a redox reaction under the operating conditions of the cell. Without such a connection, the total positive charge in the Zn2+ solution would increase as the zinc metal dissolves, and the total positive charge in the Cu2+ solution would decrease. The salt bridge allows charges to be neutralized by a flow of anions into the Zn2+ solution and a flow of cations into the Cu2+ solution. In the absence of a salt bridge or some other similar connection, the reaction would rapidly cease because electrical neutrality could not be maintained.

A galvanic cell. This galvanic cell illustrates the use of a salt bridge to connect two solutions.

A voltmeter can be used to measure the difference in electrical potential between the two compartments. Opening the switch that connects the wires to the anode and the cathode prevents a current from flowing, so no chemical reaction occurs. With the switch closed, however, the external circuit is closed, and an electric current can flow from the anode to the cathode. The potential (Ecell)Related to the energy needed to move a charged particle in an electric field, it is the difference in electrical potential beween two half-reactions. of the cell, measured in volts, is the difference in electrical potential between the two half-reactions and is related to the energy needed to move a charged particle in an electric field. In the cell we have described, the voltmeter indicates a potential of 1.10 V (part (a) in Figure 19.3 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell"). Because electrons from the oxidation half-reaction are released at the anode, the anode in a galvanic cell is negatively charged. The cathode, which attracts electrons, is positively charged.

Not all electrodes undergo a chemical transformation during a redox reaction. The electrode can be made from an inert, highly conducting metal such as platinum to prevent it from reacting during a redox process, where it does not appear in the overall electrochemical reaction. This phenomenon is illustrated in Example 1.

### Note the Pattern

A galvanic (voltaic) cell converts the energy released by a spontaneous chemical reaction to electrical energy. An electrolytic cell consumes electrical energy from an external source to drive a nonspontaneous chemical reaction.

### Example 1

A chemist has constructed a galvanic cell consisting of two beakers. One beaker contains a strip of tin immersed in aqueous sulfuric acid, and the other contains a platinum electrode immersed in aqueous nitric acid. The two solutions are connected by a salt bridge, and the electrodes are connected by a wire. Current begins to flow, and bubbles of a gas appear at the platinum electrode. The spontaneous redox reaction that occurs is described by the following balanced chemical equation:

3Sn(s) + 2NO3(aq) + 8H+(aq) → 3Sn2+(aq) + 2NO(g) + 4H2O(l)

For this galvanic cell,

1. write the half-reaction that occurs at each electrode.
2. indicate which electrode is the cathode and which is the anode.
3. indicate which electrode is the positive electrode and which is the negative electrode.

Given: galvanic cell and redox reaction

Asked for: half-reactions, identity of anode and cathode, and electrode assignment as positive or negative

Strategy:

A Identify the oxidation half-reaction and the reduction half-reaction. Then identify the anode and cathode from the half-reaction that occurs at each electrode.

B From the direction of electron flow, assign each electrode as either positive or negative.

Solution:

1. A In the reduction half-reaction, nitrate is reduced to nitric oxide. (The nitric oxide would then react with oxygen in the air to form NO2, with its characteristic red-brown color.) In the oxidation half-reaction, metallic tin is oxidized. The half-reactions corresponding to the actual reactions that occur in the system are as follows:

reduction: NO3(aq) + 4H+(aq) + 3e → NO(g) + 2H2O(l) oxidation: Sn(s) → Sn2+(aq) + 2e

Thus nitrate is reduced to NO, while the tin electrode is oxidized to Sn2+.

2. Because the reduction reaction occurs at the Pt electrode, it is the cathode. Conversely, the oxidation reaction occurs at the tin electrode, so it is the anode.
3. B Electrons flow from the tin electrode through the wire to the platinum electrode, where they transfer to nitrate. The electric circuit is completed by the salt bridge, which permits the diffusion of cations toward the cathode and anions toward the anode. Because electrons flow from the tin electrode, it must be electrically negative. In contrast, electrons flow toward the Pt electrode, so that electrode must be electrically positive.

Exercise

Consider a simple galvanic cell consisting of two beakers connected by a salt bridge. One beaker contains a solution of MnO4 in dilute sulfuric acid and has a Pt electrode. The other beaker contains a solution of Sn2+ in dilute sulfuric acid, also with a Pt electrode. When the two electrodes are connected by a wire, current flows and a spontaneous reaction occurs that is described by the following balanced chemical equation:

2MnO4(aq) + 5Sn2+(aq) + 16H+(aq) → 2Mn2+(aq) + 5Sn4+(aq) + 8H2O(l)

For this galvanic cell,

1. write the half-reaction that occurs at each electrode.
2. indicate which electrode is the cathode and which is the anode.
3. indicate which electrode is positive and which is negative.

1. MnO4(aq) + 8H+(aq) + 5e → Mn2+(aq) + 4H2O(l); Sn2+(aq) → Sn4+(aq) + 2e
2. The Pt electrode in the permanganate solution is the cathode; the one in the tin solution is the anode.
3. The cathode (electrode in beaker that contains the permanganate solution) is positive, and the anode (electrode in beaker that contains the tin solution) is negative.

## Constructing a Cell Diagram

Because it is somewhat cumbersome to describe any given galvanic cell in words, a more convenient notation has been developed. In this line notation, called a cell diagram, the identity of the electrodes and the chemical contents of the compartments are indicated by their chemical formulas, with the anode written on the far left and the cathode on the far right. Phase boundaries are shown by single vertical lines, and the salt bridge, which has two phase boundaries, by a double vertical line. Thus the cell diagram for the Zn/Cu cell shown in part (a) in Figure 19.3 "The Reaction of Metallic Zinc with Aqueous Copper(II) Ions in a Galvanic Cell" is written as follows:

Figure 19.4

A cell diagram includes solution concentrations when they are provided.

Galvanic cells can have arrangements other than the examples we have seen so far. For example, the voltage produced by a redox reaction can be measured more accurately using two electrodes immersed in a single beaker containing an electrolyte that completes the circuit. This arrangement reduces errors caused by resistance to the flow of charge at a boundary, called the junction potential. One example of this type of galvanic cell is as follows:

Equation 19.5

Pt(s)∣H2(g)∣HCl(aq)∣AgCl(s)∣Ag(s)

This cell diagram does not include a double vertical line representing a salt bridge because there is no salt bridge providing a junction between two dissimilar solutions. Moreover, solution concentrations have not been specified, so they are not included in the cell diagram. The half-reactions and the overall reaction for this cell are as follows:

Equation 19.6

cathode reaction: AgCl(s) + e → Ag(s) + Cl(aq)

Equation 19.7

Equation 19.8

A single-compartment galvanic cell will initially exhibit the same voltage as a galvanic cell constructed using separate compartments, but it will discharge rapidly because of the direct reaction of the reactant at the anode with the oxidized member of the cathodic redox couple. Consequently, cells of this type are not particularly useful for producing electricity.

### Example 2

Draw a cell diagram for the galvanic cell described in Example 1. The balanced chemical reaction is as follows:

3Sn(s) + 2NO3(aq) + 8H+(aq) → 3Sn2+(aq) + 2NO(g) + 4H2O(l)

Given: galvanic cell and redox reaction

Strategy:

Using the symbols described, write the cell diagram beginning with the oxidation half-reaction on the left.

Solution:

The anode is the tin strip, and the cathode is the Pt electrode. Beginning on the left with the anode, we indicate the phase boundary between the electrode and the tin solution by a vertical bar. The anode compartment is thus Sn(s)∣Sn2+(aq). We could include H2SO4(aq) with the contents of the anode compartment, but the sulfate ion (as HSO4) does not participate in the overall reaction, so it does not need to be specifically indicated. The cathode compartment contains aqueous nitric acid, which does participate in the overall reaction, together with the product of the reaction (NO) and the Pt electrode. These are written as HNO3(aq)∣NO(g)∣Pt(s), with single vertical bars indicating the phase boundaries. Combining the two compartments and using a double vertical bar to indicate the salt bridge,

Sn(s)∣Sn2+(aq)∥HNO3(aq)∣NO(g)∣Pt(s)

The solution concentrations were not specified, so they are not included in this cell diagram.

Exercise

Draw a cell diagram for the following reaction, assuming the concentration of Ag+ and Mg2+ are each 1 M:

Mg(s) + 2Ag+(aq) → Mg2+(aq) + 2Ag(s)

Answer: Mg(s)∣Mg2+(aq, 1 M)∥Ag+(aq, 1 M)∣Ag(s)

### Summary

Electrochemistry is the study of the relationship between electricity and chemical reactions. The oxidation–reduction reaction that occurs during an electrochemical process consists of two half-reactions, one representing the oxidation process and one the reduction process. The sum of the half-reactions gives the overall chemical reaction. The overall redox reaction is balanced when the number of electrons lost by the reductant equals the number of electrons gained by the oxidant. An electric current is produced from the flow of electrons from the reductant to the oxidant. An electrochemical cell can either generate electricity from a spontaneous redox reaction or consume electricity to drive a nonspontaneous reaction. In a galvanic (voltaic) cell, the energy from a spontaneous reaction generates electricity, whereas in an electrolytic cell, electrical energy is consumed to drive a nonspontaneous redox reaction. Both types of cells use two electrodes that provide an electrical connection between systems that are separated in space. The oxidative half-reaction occurs at the anode, and the reductive half-reaction occurs at the cathode. A salt bridge connects the separated solutions, allowing ions to migrate to either solution to ensure the system’s electrical neutrality. A voltmeter is a device that measures the flow of electric current between two half-reactions. The potential of a cell, measured in volts, is the energy needed to move a charged particle in an electric field. An electrochemical cell can be described using line notation called a cell diagram, in which vertical lines indicate phase boundaries and the location of the salt bridge. Resistance to the flow of charge at a boundary is called the junction potential.

### Key Takeaway

• A galvanic (voltaic) cell uses the energy released during a spontaneous redox reaction to generate electricity, whereas an electrolytic cell consumes electrical energy from an external source to force a reaction to occur.

### Conceptual Problems

1. Is 2NaOH(aq) + H2SO4(aq) → Na2SO4(aq) + 2H2O(l) an oxidation–reduction reaction? Why or why not?

2. If two half-reactions are physically separated, how is it possible for a redox reaction to occur? What is the name of the apparatus in which two half-reactions are carried out simultaneously?

3. What is the difference between a galvanic cell and an electrolytic cell? Which would you use to generate electricity?

4. What is the purpose of a salt bridge in a galvanic cell? Is it always necessary to use a salt bridge in a galvanic cell?

5. One criterion for a good salt bridge is that it contains ions that have similar rates of diffusion in aqueous solution, as K+ and Cl ions do. What would happen if the diffusion rates of the anions and cations differed significantly?

6. It is often more accurate to measure the potential of a redox reaction by immersing two electrodes in a single beaker rather than in two beakers. Why?

1. A large difference in cation/anion diffusion rates would increase resistance in the salt bridge and limit electron flow through the circuit.

### Numerical Problems

1. Copper(I) sulfate forms a bright blue solution in water. If a piece of zinc metal is placed in a beaker of aqueous CuSO4 solution, the blue color fades with time, the zinc strip begins to erode, and a black solid forms around the zinc strip. What is happening? Write half-reactions to show the chemical changes that are occurring. What will happen if a piece of copper metal is placed in a colorless aqueous solution of ZnCl2?

2. Consider the following spontaneous redox reaction: NO3(aq) + H+(aq) + SO32−(aq) → SO42−(aq) + HNO2(aq).

1. Write the two half-reactions for this overall reaction.
2. If the reaction is carried out in a galvanic cell using an inert electrode in each compartment, which electrode corresponds to which half-reaction?
3. Which electrode is negatively charged, and which is positively charged?
3. The reaction Pb(s) + 2VO2+(aq) + 4H+(aq) → Pb2+(aq) + 2V3+(aq) + 2H2O(l) occurs spontaneously.

1. Write the two half-reactions for this redox reaction.
2. If the reaction is carried out in a galvanic cell using an inert electrode in each compartment, which reaction occurs at the cathode and which occurs at the anode?
3. Which electrode is positively charged, and which is negatively charged?
4. Phenolphthalein is an indicator that turns pink under basic conditions. When an iron nail is placed in a gel that contains [Fe(CN)6]3−, the gel around the nail begins to turn pink. What is occurring? Write the half-reactions and then write the overall redox reaction.

5. Sulfate is reduced to HS in the presence of glucose, which is oxidized to bicarbonate. Write the two half-reactions corresponding to this process. What is the equation for the overall reaction?

6. Write the spontaneous half-reactions and the overall reaction for each proposed cell diagram. State which half-reaction occurs at the anode and which occurs at the cathode.

1. Pb(s)∣PbSO4(s)∣SO42−(aq)∥Cu2+(aq)∣Cu(s)
2. Hg(l)∣Hg2Cl2(s)∣Cl(aq) ∥ Cd2+(aq)∣Cd(s)
7. For each galvanic cell represented by these cell diagrams, determine the spontaneous half-reactions and the overall reaction. Indicate which reaction occurs at the anode and which occurs at the cathode.

1. Zn(s)∣Zn2+(aq) ∥ H+(aq)∣H2(g), Pt(s)
2. Ag(s)∣AgCl(s)∣Cl(aq) ∥ H+(aq)∣H2(g)∣Pt(s)
3. Pt(s)∣H2(g)∣H+(aq) ∥ Fe2+(aq), Fe3+(aq)∣Pt(s)
8. For each redox reaction, write the half-reactions and draw the cell diagram for a galvanic cell in which the overall reaction occurs spontaneously. Identify each electrode as either positive or negative.

1. Ag(s) + Fe3+(aq) → Ag+(aq) + Fe2+(aq)
2. Fe3+(aq) + 1/2H2(g) → Fe2+(aq) + H+(aq)
9. Write the half-reactions for each overall reaction, decide whether the reaction will occur spontaneously, and construct a cell diagram for a galvanic cell in which a spontaneous reaction will occur.

1. 2Cl(aq) + Br2(l) → Cl2(g) + 2Br(aq)
2. 2NO2(g) + 2OH(aq) → NO2(aq) + NO3(aq) + H2O(l)
3. 2H2O(l) + 2Cl(aq) → H2(g) + Cl2(g) + 2OH(aq)
4. C3H8(g) + 5O2(g) → 3CO2(g) + 4H2O(g)
10. Write the half-reactions for each overall reaction, decide whether the reaction will occur spontaneously, and construct a cell diagram for a galvanic cell in which a spontaneous reaction will occur.

1. Co(s) + Fe2+(aq) → Co2+(aq) + Fe(s)
2. O2(g) + 4H+(aq) + 4Fe2+(aq) → 2H2O(l) + 4Fe3+(aq)
3. 6Hg2+(aq) + 2NO3(aq) + 8H+ → 3Hg22+(aq) + 2NO(g) + 4H2O(l)
4. CH4(g) + 2O2(g) → CO2(g) + 2H2O(g)

1.

reduction: SO42−(aq) + 9H+(aq) + 8e → HS(aq) + 4H2O(l) oxidation: C6H12O6(aq) + 12H2O(l) → 6HCO3(g) + 30H+(aq) + 24e overall: C6H12O6(aq) + 3SO42−(aq) → 6HCO3(g) + 3H+(aq) + 3HS(aq)
1. reduction: 2H+(aq) + 2e → H2(aq); cathode;

oxidation: Zn(s) → Zn2+(aq) + 2e; anode;

overall: Zn(s) + 2H+(aq) → Zn2+(aq) + H2(aq)

2. reduction: AgCl(s) + e → Ag(s) + Cl(aq); cathode;

oxidation: H2(g) → 2H+(aq) + 2e; anode;

overall: AgCl(s) + H2(g) → 2H+(aq) + Ag(s) + Cl(aq)

3. reduction: Fe3+(aq) + e → Fe2+(aq); cathode;

oxidation: H2(g) → 2H+(aq) + 2e; anode;

overall: 2Fe3+(aq) + H2(g) → 2H+(aq) + 2Fe2+(aq)


text:
Escape velocity of a satellite

vikasagartha

The escape velocity of a satellite circularly orbiting a large body comes from conservation of energy. Are there any modifications that must be made for the escape velocity of an elliptical orbit?

Related Astronomy and Astrophysics News on Phys.org

Bandersnatch

When you look at the formula for escape velocity, you can see that it only depends on the distance from the massive body and that body's mass.
In other words, it doesn't matter what you do around that mass, just where you are.

vikasagartha

If I trying to find the optimal point for escape in an elliptical orbit, can I just differentiate the standard escape velocity eqn with respect to r? I know it must be either the apogee or the perigee...is there a better approach?

D H

Staff Emeritus
Escape velocity is a speed, not a velocity. The easiest way to attain that velocity is for a vehicle to thrust in the direction it is already going. The minimum delta V needed to escape at some point is the difference between the current velocity at that point and the escape velocity at that point. It is this value you want to minimize, not the escape velocity.

vikasagartha

So I tried doing precisely that, I minimized a 'thrust' but the expression I arrive at doesn't look like the perigee....

DrStupid

If I trying to find the optimal point for escape in an elliptical orbit, can I just differentiate the standard escape velocity eqn with respect to r?
That depends on your definition of "optimal". For rockets the optimal point is characterized by the highest velocity:

http://en.wikipedia.org/wiki/Oberth_effect

D H

Staff Emeritus
So I tried doing precisely that, I minimized a 'thrust' but the expression I arrive at doesn't look like the perigee....View attachment 68288
You used the wrong expression for orbital velocity.

A good place to start is the vis-viva equation. Unfortunately, you'll get no joy here if you differentiate the resulting delta V with respect to r. The problem is there's nothing that constrains r in either the vis-viva equation or the escape velocity equation.

You'll need something such as the orbital equation of motion, $r=\frac{a (1-e^2)}{1-e\cos\theta}$. Now you should get something useful. In particular, you should find that extrema occur at apofocus and perifocus. So now it's just a matter of determining which is best, which is worst.

"Escape velocity of a satellite"

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text:
Uniform Law of large numbers

Let $G(\cdot)$ be bounded, continuous, strictly increasing function on $\mathbb{R}.$ Let ${\xi_t},\, t\in\mathbb{Z}_+$ be i.i.d random variables.

Would it be true to say that we have something like the uniform Law of large numbers here? Why? $$\sup_{x\in\mathbb{R}}\;\left|\;n^{-1}\sum_{t=1}^n G(x+\xi_t) - \mathbf{E}G(x+\xi_1)\;\right| \stackrel{P}\longrightarrow 0, \quad n\to\infty.$$

• This thread mathoverflow.net/questions/121238/… could interest you. – Davide Giraudo Jul 9 '13 at 19:41
• Thanks, but unfortunately they're talking about compacts there – Dan Jul 10 '13 at 11:31
• An idea could be the following: after having rescaled $G$, we can assume that $G$ is a cumulative distribution function. Consider $(\eta_i,i\in\Bbb Z)$ an i.i.d. sequence of random variables independent of $(\xi_i,i\in\Bbb Z)$, say on an other probability space $(\Omega',\mathcal F',\mu')$. Then we can use Glivenko-Cantelli's theorem for the product space and the i.i.d. sequence $(\xi_i-\eta_i,i\in\Bbb Z)$. – Davide Giraudo Jul 10 '13 at 13:02

Assume that the $\xi_t$ are defined on a probability space $(\Omega,\mathcal F,\mu)$. We can assume that $G$ is the cumulative distribution function of a real valued random variable. Let $(\eta_t,t\in\Bbb Z)$ be a collection of independent random variables with cumulative distribution function $G$, defined on $(\Omega',\mathcal G,\nu)$. Using Glivenko-Cantelli's theorem for the product space and the family of i.i.d. random variables $(\xi_t-\eta_t,t\in\Bbb Z)$ gives $$\sup_{u\in\Bbb R}\left|\frac 1n\sum_{i=1}^n\chi_{\{\eta_i(\omega')-\xi_i(\omega)\leqslant u\}}-\mu\otimes\nu\{\eta_i(\omega')-\xi_i(\omega)\leqslant u\}\right|\to 0$$ for $\mu\otimes\nu$ almost every $(\omega,\omega')$. Then integrate with respect to $\nu$.
which is available here. We consider $\mathcal F:=\{t\mapsto G(x+t), x\in\Bbb R)\}$, which is a Glivenko-Cantelli class as $G$ is bounded. But it seems quite overkill for this problem, since their result is more general.