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Engineering Projects/Car Hack/Howard Community College/Spring2012/p1501kmd 2

Problem Statement
Increase gas mileage and read tire pressure using an arduino on vehicle

Team Members

 * Kimberly
 * Martin
 * David

Summary
For this project, our team decided to stay with the process of hacking a vehicle (2002 Toyota Corolla) in order to better the miles per gallon. First, we brainstormed the different routes we could take in order to reach our goal, and decided it would be best if each of us took on "sub-projects" instead of all of us working on one whole project so that we could get more work done within the time we were given. We all split up, Martin making a MPGuino, David writing codes to control the fuel injection, and Kim trying to tamper with the tire pressure on the car.

We all did research on our separate projects and started getting to work. Martin researched how to solder and started making the MPGuino from scratch while Kim and David used past projects of other teams as a tutorial/guideline to write codes and figure out of to read the tire pressure from certain devices. With all of us being by ourselves for the small projects, we had limited resources and had to order parts for some of our projects then wait for those to come in, but once we all had what we need by the third and fourth week, we started but our projects back into one whole project.

For example, Martin's MPGuino could be used in David's controlling the fuel injection and could also be used to hack the wireless tire pressure sensors in Kim's project giving us more control over the car as a whole through just one item. We also decided to make the project an open-ended project by trying to write a program for the LCD screen in the MPGuino to signal either or red backlight or blue backlight to tell if the program we wrote to control the Fuel injection and tire pressure bettered or worsened the initial/factory MPG for the vehicle.

Story
Week 1-This week was more of brainstorming than actual hands on work. We decided to better the MPG within a vehicle, but each took on "sub-projects" within the overall idea.

Week 2- We all started to go our separate ways with our sub-projects. Martin trying to build the MPGuino, David writing codes to control the fuel injection within the vehicle, and Kim trying to figure out how to tamper with the tire pressure on the car.

Week 3- This week was about collecting efficient data on our vehicle and learning what to use and how to go about getting the results we wanted in each of our experiments, this week also consisted of gaining those materials, installing software, and actually doing hands on work once we had received the materials we all needed.

Week 4- The last week of this project, all of us started finishing up our own projects to bring them back together within one project, our overall goal, and make them all merge. For example, David's project of controlling the fuel injection valve with the arduino played into Martin's MPGuino which then played into to wireless signals within Kim's tire pressure project allowing us to control a whole range of items in one thing, the arduino. We also made the project an open-end project by using the LCD screen in the MPGuino to signal if the MPG on the vehicle gets better or worse than the original after the codes we send to the arduino.

Individual Work

 * David's Story and work completed
 * Kim's Story and work completed
 * Martin's Story and work completed

Decision List
The decisions we had to make during this project was in which way were we going to lower the MPG within the vehicle. Being that there were so many, we all decided to take on "sub-projects" meaning that each of us decided to due our own project that could be done within the timeframe given, but that was also still relevant to the overall outcome of the project. This way we could get a lot more done in the given timeframe rather than each of us working on the same project that could've been done by one person in the first place. So Martin decided to take on the project of creating a way to read the MPG in the vehicle once we tampered with it. David decided to take Martin's idea of creating a MPGuino to read the MPG on the car and make the LCD screen on the arduino tell us if the MPG on the vehicle was better or worse than the vehicle's manufactured MPG by signaling red or blue lights. David also decided to use the arduino to create a program allowing us to control the fuel injection within the vehicle making the whole MPGuino idea into a "game" for engineers; "who could create the best program to better the MPG".Kim decided to take another approach and try to better the MPG on the vehicle by tampering with the tire pressure on the car.

Material List

 * Arduino Uno USB, 1, $12
 * 02 Toyota Corolla Fuel Injector, 1, $20 (junkyard) $196.02 ( Factory )
 * SSR, 1, $13 ( Amazon )
 * Arduino LCD Screen, 1, $24 ( Adafruit )
 * ATEQ VT55 TPMS trigger tool, 1, $1300 ( Orange Electronic )
 * Tire pressure sensors, 2, $30 ( Amazon )
 * Low noise amplifier (LNA), 1, ?
 * Screwdrivers, 4, $7 ( Amazon )
 * Soldering Iron, 1, $45 ( Amazon )

Software List
We used a software package called Arduino to aid us in creating codes and processing them from the arduino to control the fuel injection timing, and to create an custom LCD screen for the arduino from a printer LCD screen.

We also used GNU Radio to read off the tire pressure from the sensors

Time
In total, all of us spent a collective: 25.10 hours on the project

Tutorials
Creating the LCD screen for the arduino: MPGuino Assemble and Wiring the arduino

Creating the codes for the MPGuino Screen: Arduino Codes for LCD Screen

Next Steps
The successors will have to finish writing the program to make the LCD screen on the MPGuino signal the red backlight or the blue backlight to tell if the MPG in the vehicle is getting better or worse. Once this is done, one can continue using arduino to write own programs to control the fuel injection and tire pressure to better the MPG in the vehicle while using the MPGuino to tell if they are making the MPG better or worse.


text: File:Cordic rtl.vhdl.20120323.pdf

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title: File:Sampling.2.B.BandPass.20120323.pdf


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Category:User ko

Users in this category indicate they have knowledge of language Korean.

title: Category:User ko


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Category:User ko-1

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Category:User ja-0

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User:Shujenchang

Projects: Meta | Wikipedia | Wiktionary | Wikibooks | Wikiversity | Wikiquote | Wikinews | Commons | Wikisource | Wikispecies | MediaWiki | Incubator | Labs

title: User:Shujenchang


title: File:VHDL.Signal.1.A.20120323.pdf


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User:Egm6936.f10/Zeros of polynomial equations

Polynomial:

Polynomial

Discriminant:

Discriminant

Linear function: Linear function

Quadratic function: Quadratic function

Solving quadratic equations with continued fractions: Solving quadratic equations with continued fractions

Cubic function: Cubic function

Quartic function Quartic function

Quintic function Quintic function

Sextic function Sextic function

Septic function Septic function

Galois theory Galois theory


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UTPA STEM/Template2

Course Title:

Lecture Topic:

Instructor:

Institution:

Backwards Design
Course Objectives

 * Primary Objectives- By the next class period students will be able to:
 * text
 * text
 * text

 * Sub Objectives- The objectives will require that students be able to:
 * text
 * text

 * Difficulties- Students may have difficulty:
 * text
 * text

 * Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
 * text
 * text

Model of Knowledge

 * Concept Map
 * text
 * text
 * text

 * Content Priorities
 * Enduring Understanding
 * text
 * text
 * text
 * Important to Do and Know
 * text
 * text
 * text
 * Worth Being Familiar with
 * text
 * text
 * text

Assessment of Learning

 * Formative Assessment
 * In Class (groups)
 * text
 * text
 * Homework (individual)
 * text
 * text
 * Summative Assessment
 * text
 * text

Legacy Cycle
OBJECTIVE

By the next class period, students will be able to: The objectives will require that students be able to:
 * text
 * text
 * text
 * text
 * text
 * text

THE CHALLENGE

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GENERATE IDEAS

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MULTIPLE PERSPECTIVES

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RESEARCH & REVISE

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TEST YOUR METTLE

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GO PUBLIC

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Pre-Lesson Quiz

 * 1) question
 * 2) question
 * 3) question
 * 4) question
 * 5) question
 * 6) question
 * 7) question
 * 8) question
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Test Your Mettle Quiz

 * 1) question
 * 2) question
 * 3) question
 * 4) question
 * 5) question
 * 6) question
 * 7) question
 * 8) question
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 * 10) question
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title: UTPA STEM/Template2


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User:Lsun224/enes100/My Work 2

Write problem/project Goal
Construct a Rube Goldberg Machine

My First Task
Look up examples of Rube Goldberg Machines. Continue work on Stepper Motor/Arduino combo for use in moving the gears

Summary of actual work over first weekend
Work went according to plan

Week1 Narrative
Tried playing with the arduino again over the weekend. The arduino itself still works and software is running correctly, but the motor is not working. Copied code from the arduino website but it did not work. Soldering may still be faulty on the wire connections. Will try to fix on Wednesday. It is also possible that the code I used is incorrect for the type of motor i have. Found a site that has some interesting info on contests and winning designs, etc.

My Second Task
Record what you are planning on doing for the team during the second weekend.

1. Experiment with new stepper motor

2. Map out motor's wiring

3. Try attaching to arduino

Summary of actual work over second weekend
How is what you did different than what you planned?

For the most part, everything went according to plan

Week2 Narrative
Tell a detailed story describing what you did for your team over the weekend.

Began by mapping out the wires. With the pointed end up, from right to left, the pins are:

1+2: Ground

3: Doubles the speed of the motors rotation (only if plugged into the -12v)

4: If any voltage is applied, will stop the motors rotation. If touched to ground, will do nothing

5+6: Power (12v?)

Note: if you reverse the ground and power wires by accident, the motor will not spin in reverse. Rather it will spark and it will hurt. Also it will overload the power supply. Luckily it has overload safeties and still works

Closeup of the wire connections

Video of the wire test

Aside from stopping the motors rotation, still not sure what wire 4 does. According to the tutorial it is likely one of three things: touch to ground to make motor spin, touch to ground to cut speed in half, or hook to scope to see motor output. Touched it to ground and it did neither of the first two, so by elimination one possibility is that it shows the motor output. Cannot check this until returning to the lab next week. Wires keep falling apart. Need to maybe solder them togethor.

As for attaching the motor to the arduino, I believe Zach was working on hooking them up. He said something about being able to wire the two together and control the spin of the motor with the blink command. Will contact him about his progress. For now I will continue experimenting on my own with the motor and arduino as well as wire combinations with the power box

My Third task
Record what you are planning on doing for the team during the third weekend.

Continue testing wire connections and maybe incorporate arduino into tests

Summary of actual work over third weekend
How is what you did different than what you planned?

For the most part went according to plan

Week3 Narrative
Tell a detailed story describing what you did for your team over the weekend.

Experimented with plugging the motors various wires into arduino ports. On the left side of the arduino you can plug the wire 1 into 5v or 3.3v and wire 5 into the right side where it says ground. The motor should spin. However, 3.3v is not enough power so plugging wire 1 into 3.3v will not spin the motor. Some other configurations i found: reversing the above setup wire 1 to ground and wire 5 to the 5v will turn the arduino off (we know it didn't short it out because as soon as you pull the wire it turns back on). Plugging wire 1 to 13 and wire 5 to ground will turn the LED on (even with the blink command, the LED will not turn off, but simply grow slightly dimmer). If i remember from the beginning of the semester Zach said he found a way to reverse the spin of the motor, ill ask him when i see him again. Will upload a video of weekend tests as soon as youtube decides it wants to work.

Update: Looked at Zach's page. No word on reversing motor spin with arduino. However, he says that you can control the spin of the motor with the blink command. Should be interesting to see

My Fourth task
Record what you are planning on doing for the team during the final weekend of project work.

Map out wires of new motor

Summary of actual work over fourth weekend
How is what you did different than what you planned?

Followed according to plan

Week4 Narrative
Tell a detailed story describing what you did for your team over the weekend.

Tested the wires with the motor, results are as follows:

From left to right with point facing up....

Pin 1+2: Ground

Pin 3: Does nothing, motor still spins normally

Pin 4+5: Power (12v? maybe 24)

Pin 6: Nothing...

Pin 7: Halves the speed when touched to ground

Not sure what pins 3 and 6 do, but the rest make sense, since they were approximately the same as with the other motor

Update: So I just remembered something....there was a tutorial here. Ok so judging from the tutorial, pin 3 is supposed to turn the motor on (which makes no sense) and pin 6 is meant to hook up to a scope to see the output of the motor

Complete Team Page
Fill out the Team Form (should have already copied the form, created the team page, linked to it and started filling it out).

title: User:Lsun224/enes100/My Work 2


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UTPA STEM/CBI Courses/Probability and Statistics/Binomial Distribution (Anahit Galstyan)

Course Title: Precalculus and Trigonometry

Lecture Topic: Probability and Statistics

Instructor: Anahit Galstyan

Institution:University of Texas-Pan American

Backwards Design
Course Objectives

 * Primary Objectives- By the next class period students will be able to:
 * Find probabilities of events related to binomial random variables
 * Use the binomial probability tables and the TI 84 calculator to calculate binomial probability
 * Analyze, describe and summarize their findings

 * Sub Objectives- The objectives will require that students be able to:
 * Understand Bernoulli Scheme.
 * Understand the relationship between the mean and the binomial parameters.

 * Difficulties- Students may have difficulty:
 * Setting up the problem.
 * Identifying the parameters of the binomial distribution.

 * Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
 * Finding the probability between failure and success.
 * Building models based on binomial distribution.

Model of Knowledge

 * Concept Map
 * Understand the concept of a random variable.
 * Understand the concept of expected value.
 * Analyze, describe and summarize the data.

 * Content Priorities
 * Enduring Understanding
 * Understand the binomial model
 * Understand why to use binomial probability
 * Advantages and disadvantages of using binomial probability
 * Important to Do and Know
 * Express probabilities of events in terms of random variables.
 * Know how to use the binomial probability tables.
 * Know how to use the TI 84 to find binomial probabilities.
 * Worth Being Familiar with
 * Parameters
 * Binomial distribution formula
 * Mean, Variance and Standard Deviation formulas

Assessment of Learning

 * Formative Assessment
 * In Class (Groups)
 * Work solving problems
 * Helping each other to solve the review
 * Homework (individual)
 * Pre-lecture quiz after student has read the section from the book
 * Recommended problems from the book.
 * Summative Assessment
 * Tests
 * Final Exam

Legacy Cycle
OBJECTIVE

By the next class period, students will be able to:
 * Find probabilities of events related to binomial random variables
 * Use the binomial probability tables and the TI 84 calculator to calculate binomial probability
 * Analyze, describe and summarize their findings

The objectives will require that students be able to:
 * Understand Bernoulli Scheme.
 * Understand the relationship between the mean and the binomial parameters.

THE CHALLENGE

65% of the college students who are in a relationship achieve better grades than the college students who are not in a relationship. You randomly select 10 college students and ask each to name if he or she achieve better grades when they are in a relationship. Find the probability that the number of college students who say that when they are in a relationship, they achieve better grades. If convenient, use technology to find the probabilities.

GENERATE IDEAS

We should be in a relationship.

MULTIPLE PERSPECTIVES

We need to know the probability of success.

RESEARCH & REVISE

Students will be able to use their book and calculator. Students will work in groups.

TEST YOUR METTLE

The groups will present their ideas to the class and will receive feedback. Students will take a quiz at the beginning of the following class.

GO PUBLIC

Students will be able to answer similar questions presented at the beginning of the lesson and would be asked to turn in a summary of their findings.

Pre-Lesson Quiz

 * 1) What is a binomial random variable?
 * 2) What does the Bernoulli Scheme assume?
 * 3) What does μ, n, p, and q represent?
 * 4) What are the advantages of using Binomial Probability?
 * 5) What are the disadvantages of using Binomial Probability?

Test Your Mettle Quiz
65% of the college students who are in a relationship achieve better grades than the college students who are not in a relationship. You randomly select 10 college students and ask each to name if he or she achieve better grades when they are in a relationship. Find the probability that the number of college students who say that when they are in a relationship, they achieve better grades. Find the binomial probability of exactly two, more than two and between two and five inclusive. If convenient, use technology to find the probabilities.


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User:Zbennett/enes100/My Work (Rube Goldberg)

Write problem/project Goal
Rube Goldberg projects are best illustrated by the band OK Go video "This too shall pass." The idea is to take apart something like a laser printer and expose all the parts that move, blink, smell, etc. Then make an art exhibit like this Honda Accord commercial in an art gallary.

My First Task
Construct a support structure for the Rube Goldberg machine to be mounted on.

Summary of actual work over first weekend
Over the weekend I constructed two supports that would be able to support the Rube Goldberg machine freestanding.

Week1 Narrative
They were made using spare 2X4's that I found in my garage. With these, the machine can be placed anywhere in a room, and does not need the support of a wall to remain upright. I was sure to make them tall so that they can accommodate a large Rube Goldberg machine if necessary. Unfortunately, as can be seen in the picture, the diagonal supports on the top structure are not exactly lined up, but the structure will most likely not have to support enough weight for this to make a noticeable difference.

My Second Task
Disassemble printer motor's wiring and attach to arduino.

Summary of actual work over second weekend
I found, through trial and error, that the first and the fifth wire can be attached to the arduino to power the printer motor.

Week2 Narrative
Through trial and error, I discovered that when the first and fifth wires of the printer motor are attached to the GND and 13th slot, respectively, of the arduino, the motor will spin. Using this, a simple program can be written that turns the motor on and off, using a timer to control the "blink" speed.

My Third task
Record what you are planning on doing for the team during the third weekend.

Summary of actual work over third weekend
How is what you did different than what you planned?

Week3 Narrative
Tell a detailed story describing what you did for your team over the weekend.

My Fourth task
''Record what you are planning on doing for the team during the final weekend of project work. ''

Summary of actual work over fourth weekend
How is what you did different than what you planned?

Week4 Narrative
Tell a detailed story describing what you did for your team over the weekend.

Complete Team Page
Fill out the Team Form (should have already copied the form, created the team page, linked to it and started filling it out).


text:
Car Hack2/Howard Community College/Spring2012/p1501kmd

Problem Statement
Preform car hacks such as increasing the cars gas mileage using an arduino

Team Members

 * Martin
 * David
 * Kimberly

Summary
Put an overall, short one paragraph summary here.

Poster
''Put a graphic in wikimedia, include the graphic here or link to it here. The graphic should be suitable for creating a traditional project poster.''

Story
''Tell a detailed story of the project. Describe how split up, what the obstacles were, what testing was done, what informal decisions were made, what assumptions were made, what the results were.''

Decision List
List all formal decisions made with links to their documentation such as a decision tree or decision matrix.

Material List

 * 1. List materials used, quantity, size, cost.
 * 2. Describe what needs to be purchased in the future to continue working on this project.

Software List
''You installed different software packages or used already installed software. Describ the programs here.''

Time
''Put one number here. Hours. Total hours that everyone worked on the project. Do not estimate. Do not talk about it. ADD.''

Tutorials
''All projects create new tutorials of technical details future participants are going to want to know. They are going to be separate pages that are linked to here.''

Next Steps
List specific details, advice, describe the current problems that the next team faces associated with the project here.


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User:Marshallsumter/Radiation astronomy/Cyans

The color cyan is used often as an indicator of features in a diagram. Cyan astronomy is astronomy in the blue-green portion of the visible spectrum, a wavelength range of 476–495 nm.

Perhaps the most prominent cyan planetary source is Uranus, which has only been visited by the space probe Voyager 2. More recent images come from the Hubble Space Telescope in orbit around Earth.

Cyan blue is the color of several cyanide (CN) containing materials, including CN detected in comet haloes.

Astronomy
"The spiral galaxy, Messier 83, NGC 5236 [is shown in the image at right]. [The t]ri-colour photograph, exposures of 30, 30 and 35 minutes on hypersensitised plates, [is] in blue, green and red light respectively".

Cyan radiation
Cyan light has a wavelength of between 490 and 520 nanometers, between the wavelengths of blue and green.

Planck's equation describes the amount of [spectral radiance at] a certain wavelength radiated by a black body in thermal equilibrium.

In terms of ... wavelength (λ), Planck's [equation] is written:[ as]
 * $$B_\lambda(T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda k_\mathrm{B}T}} - 1}$$

where B is the spectral radiance, T is the absolute temperature of the black body, kB is the Boltzmann constant, h is the Planck constant, and c is the speed of light.

This form of the equation contains several constants that are usually not subject to variation with wavelength. These are h, c, and kB. They may be represented by simple coefficients: c1 = 2h c2 and c2 = h c/kB.

By setting the first partial derivative of Planck's equation in wavelength form equal to zero, iterative calculations may be used to find pairs of (λ,T) that to some significant digits represent the peak wavelength for a given temperature and vice versa.

 * $$\frac{\partial B}{\partial \lambda} = \frac{c1}{\lambda^6}\frac{1}{ e^{\frac{c2}{\lambda T}} - 1}[\frac{c2}{\lambda T}\frac{1}{ e^{\frac{c2}{\lambda T}} - 1}e^{\frac{c2}{\lambda T}} - 5] = 0.$$

Or,

 * $$\frac{c2}{\lambda T}\frac{1}{ e^{\frac{c2}{\lambda T}} - 1}e^{\frac{c2}{\lambda T}} - 5 = 0.$$

 * $$\frac{c2}{\lambda T}\frac{1}{ e^{\frac{c2}{\lambda T}} - 1}e^{\frac{c2}{\lambda T}} = 5.$$

Use c2 = 1.438833 cm K.

For a star to have a peak in the cyan, iterative calculations using the last equation yield the pairs: approximately (476 nm, 6300 K) and (495 nm, 6100 K).

Although Planck's equation is not an exact fit to a star's spectral radiance, it may be close enough to suggest if a star is an astronomical cyan source.

Planetary sciences
"The location of a newly discovered moon, designated S/2004 N 1, orbiting Neptune, is seen in this composite Hubble Space Telescope handout image [at right] taken in August 2009. The new moon is the 14th known moon to be circling the distant blue-green planet."

"Estimated to be about 12 miles (20 km) in diameter, the moon is located about 65,400 miles (105,251 km) from Neptune [left image]."

"Images taken by NASA's Voyager 2 spacecraft unveiled the second largest moon, Proteus, and five smaller moons, Naiad, Thalassa, Despina, Galatea and Larissa."

"Ground-based telescopes found Halimede, Laomedeia, Sao and Nestor in 2002. Sister moon Psamathe turned up a year later."

"The newly found moon, designated S/2004 N 1, is located between Larissa and Proteus. It orbits Neptune in 23 hours."

Colors
Electric blue is a color close to cyan that is a representation of the color of lightning, an electric spark, and argon sign.

The electric blue glow of electricity results from the spectral emission of the excited ionized atoms (or excited molecules) of air (mostly oxygen and nitrogen) falling back to unexcited states, which happens to produce an abundance of electric blue light. This is the reason electrical sparks in air, including lightning, appear electric blue. It is a coincidence that the color of Cherenkov radiation and light emitted by ionized air are a very similar blue despite their very different methods of production.

Aero blue is a fluorescent cyan color.

The word cerulean is probably derived from the Latin word caeruleus, "dark blue, blue or blue-green", which in turn probably derives from caelulum, diminutive of caelum, "heaven, sky".

Natural gas (methane) has a cyan colored flame when burned with a mixture of air.

Cyan minerals
The gem-gravel placer deposits of Sri Lanka contain aquamarine.

The deep blue version of aquamarine is called maxixe. Maxixe is commonly found in the country of Madagascar. Its color fades to white when exposed to sunlight or is subjected to heat treatment, though the color returns with irradiation.

"The pale blue color of aquamarine is attributed to Fe2+. The Fe3+ ions produce golden-yellow color, and when both Fe2+ and Fe3+ are present, the color is a darker blue as in maxixe. Decoloration of maxixe by light or heat thus may be due to the charge transfer Fe3+ and Fe2+.   Dark-blue maxixe color can be produced in green, pink or yellow beryl by irradiating it with high-energy particles (gamma rays, neutrons or even X-rays).

Turquoise at right is an opaque, blue-to-green mineral that is a hydrous phosphate of copper and aluminium, with the chemical formula CuAl6(PO4)4(OH)8·4H2O.

Halogen minerals
Although fluorite usually appears violet or purple in color, the crystals at left are cyan with some blue or violet fluorite mixed in suggesting slight variations in composition.

Objects
The image at right contains IC 2497, the galaxy near the image top, and "an unusual, ghostly green blob of gas [that] appears to float near a normal-looking spiral galaxy."

"The bizarre object, dubbed Hanny's Voorwerp (Hanny's Object in Dutch), is the only visible part of a 300,000-light-year-long streamer of gas stretching around the galaxy, called IC 2497. The greenish Voorwerp is visible because a searchlight beam of light from the galaxy's core illuminated it. This beam came from a quasar, a bright, energetic object that is powered by a black hole. The quasar may have turned off about 200,000 years ago."

"This Hubble view uncovers a pocket of star clusters, the yellowish-orange area at the tip of Hanny's Voorwerp. The star clusters are confined to an area that is a few thousand light-years wide. The youngest stars are a couple of million years old. The Voorwerp is the size of our Milky Way galaxy, and its bright green color is from glowing oxygen."

"Hubble also shows that gas flowing from IC 2497 may have instigated the star birth by compressing the gas in Hanny's Voorwerp. The galaxy is located about 650 million light-years from Earth."

"What appears to be a gaping hole in Hanny's Voorwerp actually may be a shadow cast by an object in the quasar's light path. The feature gives the illusion of a hole about 20,000 light-years wide. Hubble reveals sharp edges but no other changes in the gas around the apparent opening, suggesting that an object close to the quasar may have blocked some of the light and projected a shadow on the Voorwerp. This phenomenon is similar to a fly on a movie projector lens casting a shadow on a movie screen."

"An interaction between IC 2497 and another galaxy about a billion years ago may have created Hanny's Voorwerp and fueled the quasar. The Hubble image shows that IC 2497 has been disturbed, with complex dust patches, warped spiral arms, and regions of star formation around its core. These features suggest the aftermath of a galaxy merger. The bright spots in the central part of the galaxy are star-forming regions. The small, pinkish object to the lower right of IC 2497 is an edge-on spiral galaxy in the background."

"The image was made by combining data from the Advanced Camera for Surveys (ACS) and the Wide Field Camera 3 (WFC3). The ACS exposures were taken April 12, 2010; the WFC3 data, April 4, 2010."

Strong forces
"Hubble astronomers have found an unexpected surprise while surveying more than 100 planetary nebulae in the central bulge of our Milky Way galaxy. Those nebulae that are butterfly-shaped or hourglass-shaped tend to be mysteriously aligned such that their rotation axis is perpendicular to the plane of our galaxy."

"Astronomers have used the NASA/ESA Hubble Space Telescope and ESO's New Technology Telescope to explore more than 100 planetary nebulae in the central bulge of our galaxy. They have found that butterfly-shaped members of this cosmic family tend to be mysteriously aligned — a surprising result given their different histories and varied properties."

"Planetary nebulae are the expanding gaseous shrouds encircling dying stars. A subset of this population has bipolar outflows that align to the star's rotation axis. Such nebulae formed in different places and have different characteristics and so it is a puzzle why they should always point on the same sky direction, like bowling pins set up in an alley."

"All these nebulae formed in different places and have different characteristics. Neither the individual nebulae, nor the stars that formed them, interact with other planetary nebulae. However, a new study by astronomers from the University of Manchester, UK, now shows surprising similarities between some of these nebulae: many of them line up in the sky in the same way. The "long axis" of a bipolar planetary nebula slices though the wings of the butterfly, whilst the "short axis" slices through the body."

"The astronomers looked at 130 planetary nebulae in the Milky Way's central bulge. They identified three different types, and peered closely at their characteristics and appearance. The shapes of the planetary nebula images were classified into three types, following conventions: elliptical, either with or without an aligned internal structure, and bipolar."

"This really is a surprising find and, if it holds true, a very important one, [...] Many of these ghostly butterflies appear to have their long axes aligned along the plane of our galaxy. By using images from both Hubble and the NTT we could get a really good view of these objects, so we could study them in great detail."

"While two of these populations were completely randomly aligned in the sky, as expected, we found that the third — the bipolar nebulae — showed a surprising preference for a particular alignment, [...] While any alignment at all is a surprise, to have it in the crowded central region of the galaxy is even more unexpected."

"Planetary nebulae are thought to be sculpted by the rotation of the star system from which they form. This is dependent on the properties of this system — for example, whether it is a binary [A binary system consists of two stars rotating around their common centre of gravity.], or has a number of planets orbiting it, both of which may greatly influence the form of the blown bubble. The shapes of bipolar nebulae are some of the most extreme, and are thought to be caused by jets blowing mass outwards from the star system perpendicular to its orbit."

"The alignment we're seeing for these bipolar nebulae indicates something bizarre about star systems within the central bulge, [...] For them to line up in the way we see, the star systems that formed these nebulae would have to be rotating perpendicular to the interstellar clouds from which they formed, which is very strange."

"While the properties of their progenitor stars do shape these nebulae, this new finding hints at another more mysterious factor. Along with these complex stellar characteristics are those of our Milky Way; the whole central bulge rotates around the galactic centre. This bulge may have a greater influence than previously thought over our entire galaxy — via its magnetic fields. The astronomers suggest that the orderly behaviour of the planetary nebulae could have been caused by the presence of strong magnetic fields as the bulge formed."

"Researchers suggest that there is something bizarre about star systems within the central hub of our galaxy. They would all have to be rotating perpendicular to the interstellar clouds from which they formed. At present, the best guess is that the alignment is caused by strong magnetic fields that were present when the galactic bulge formed billions of years ago."

"As such nebulae closer to home do not line up in the same orderly way, these fields would have to have been many times stronger than they are in our present-day neighbourhood. Very little is known about the origin and characteristics of the magnetic fields that were present in our galaxy when it was young, so it is unclear how they have changed over time."

"We can learn a lot from studying these objects, [...] If they really behave in this unexpected way, it has consequences for not just the past of individual stars, but for the past of our whole galaxy."

Continua
"This unusual image [at right of the Crab Nebula] is the result of long exposures in the Red, Blue, Green [including Hα], and a separate set of exposures on the inner continuum radiation with RGB and polarizers crossed 120 degrees for each color. The result is an inner region that is mapped in polarization according to color. The outer filaments are primarily HII and OIII regions and have no polarization. The Object: The Crab Nebula in Taurus is a super nova remnant that exploded in the year 1084 AD and has been rapidly expanding ever since. It is located a degree from the easternmost star in the Bulls horns, and glows dimly at a magnitude of 8.4. While small at 6 arc minutes, it is typical of the [telescope image] size of many galaxies".

Meteor showers
Of some 670 Perseids examined for colors from 1985, 1988, and 1989, 128 were blue meteors, 3 were multi-colored yellow-blue and one was blue-green [cyan]. The average pre-atmospheric velocity is 59.9 km/s.

The Perseids are a prolific meteor shower associated with the comet Swift-Tuttle. The Perseids are so-called because the point from which they appear to come, called the radiant, lies in the constellation Perseus.

Of 225 Geminids observed in 1990 some 38 were blue in color, with one yellow-blue and one blue-green [cyan]. The average pre-atmospheric velocity is 36.2 km/s.

Neutrals
"A giant cosmic necklace glows brightly in this Nasa Hubble Space Telescope image."

"The object, aptly named the Necklace Nebula, is a recently discovered planetary nebula, the glowing remains of an ordinary, sun-like star."

"The nebula consists of a bright ring, measuring 12trillion miles wide, dotted with dense, bright knots of gas that resemble diamonds in a necklace."

"Newly discovered: The Necklace Nebula glows brightly in this composite image taken by the Hubble Space Telescope last month. The glow of hydrogen, oxygen, and nitrogen are shown by the colours blue, green and red respectively".

"It is located 15,000 light-years away in the constellation Sagitta."

"A pair of stars orbiting close together produced the nebula, also called PN G054.2-03.4."

"About 10,000 years ago, one of the ageing stars ballooned to the point where it engulfed its companion star. The smaller star continued orbiting inside its larger companion, increasing the giant’s rotation rate. The bloated companion star spun so fast that a large part of its gaseous envelope expanded into space. Due to centrifugal force, most of the gas escaped along the star’s equator, producing a ring. The embedded bright knots are dense gas clumps in the ring. The pair is so close, only a few million miles apart, that they appear as one bright dot in the centre. The stars are furiously whirling around each other, completing an orbit in a little more than a day."

Ultraviolets
"ESO’s La Silla Observatory has snapped a new image [at right] of the famous Helix planetary nebula, revealing a rich — and rarely photographed — background of distant galaxies."

"The Helix Nebula, NGC 7293, about 700 light-years away in the constellation of Aquarius, is a Sun-like star in its final explosion before retirement as a white dwarf."

"The blue-green glow in the centre of the Helix comes from oxygen atoms shining under effects of the intense ultraviolet radiation of the 120 000 degree Celsius central star and the hot gas. Further out from the star and beyond the ring of knots, the red colour from hydrogen and nitrogen is more prominent."

"Shells of gas are blown off from the surface of such stars, often in intricate and beautiful patterns, and shine under the harsh ultraviolet radiation from the faint, hot central star. The main ring of the Helix Nebula is about two light-years across, or half the distance between the Sun and its nearest stellar neighbour."

"Despite being photographically spectacular, the Helix is hard to see visually as its light is thinly spread over a large area of sky."

Blues
"Another [...] supernova remnant is the Veil Nebula. Near the constellation Cygnus, the portion of the Eastern Veil shown gloriously [at right], spans only about 1/2 degree in the sky, about the size of the Moon."

"In this composite [...] image of the Veil Nebula, data are recorded through narrow band filters. Emission from hydrogen atoms in the remnant is shown in red with strong emission from oxygen atoms in blue-green hues".

Greens
At right is a visual astronomy image of IC 1396 using narrowband filters: sulfur is red, oxygen blue and hydrogen in green. The image was captured using a Ritchey Chretien 12.5" with a 2180 mm focal length.

"This intriguing picture from ESO’s Very Large Telescope shows the glowing green planetary nebula IC 1295 [at left] surrounding a dim and dying star. It is located about 3300 light-years away in the constellation of Scutum (The Shield). This is the most detailed picture of this object ever taken." Three filters are used in this image: the blue (B), visual (V) in green, and red (R) optical filters. IC 1295 is at RA 18 54 37.25, Dec 39.41", the image is 6.82 x 6.82 arcminutes.

Yellows
"NGC 3132 [imaged at right] is a striking example of a planetary nebula. This expanding cloud of gas, surrounding a dying star, is known to amateur astronomers in the southern hemisphere as the "Eight-Burst" or the "Southern Ring" Nebula."

"The name "planetary nebula" refers only to the round shape that many of these objects show when examined through a small visual telescope. In reality, these nebulae have little or nothing to do with planets, but are instead huge shells of gas ejected by stars as they near the ends of their lifetimes. NGC 3132 is nearly half a light year in diameter, and at a distance of about 2000 light years is one of the nearer known planetary nebulae. The gases are expanding away from the central star at a speed of 9 miles per second."

"This image, captured by NASA's Hubble Space Telescope, clearly shows two stars near the center of the nebula, a bright white one, and an adjacent, fainter companion to its upper right. (A third, unrelated star lies near the edge of the nebula.) The faint partner is actually the star that has ejected the nebula. This star is now smaller than our own Sun, but extremely hot. The flood of ultraviolet radiation from its surface makes the surrounding gases glow through fluorescence. The brighter star is in an earlier stage of stellar evolution, but in the future it will probably eject its own planetary nebula."

"In the Heritage Team's rendition of the Hubble image, the colors were chosen to represent the temperature of the gases. Blue represents the hottest gas [the oxygen 500.9 nm line], which is confined to the inner region of the nebula. Red represents the coolest gas [hydrogen Hα line], at the outer edge. The Hubble image also reveals a host of filaments, including one long one that resembles a waistband, made out of dust particles which have condensed out of the expanding gases. The dust particles are rich in elements such as carbon. Eons from now, these particles may be incorporated into new stars and planets when they form from interstellar gas and dust. Our own Sun may eject a similar planetary nebula some 6 billion years from now."

The yellow line, or band, used as an intermediate temperature is due to the overlap between the oxygen cyan line and the Hα line.

Plasma objects
In spacecraft propulsion, a Hall thruster is a type of ion thruster in which the propellant is accelerated by an electric field. Hall thrusters trap electrons in a magnetic field and then use the electrons to ionize propellant, efficiently accelerate the ions to produce thrust, and neutralize the ions in the plume. Hall thrusters are sometimes referred to as Hall effect thrusters or Hall current thrusters. Hall thrusters are often regarded as a moderate specific impulse (1,600 s) space propulsion technology. Hall thrusters operate on a variety of propellants, the most common being xenon. Other propellants of interest include krypton, argon, bismuth, iodine, magnesium, and zinc.

At left are two images of the plasma associated with and a part of NGC 7662. The color of the nebula is very blue-green where the dominant light source is the 500.7 nm oxygen emission.

The second image is from the Hubble Space Telescope through three filters: F502N (blue), F555W (green), and F658N (red). The object is a planetary nebula (NGC 7662). A small star in the center has produced the nebula.

The aurora borealis on the right, third down, is probably the usual green aurora but appears greenish-blue or cyan. This cyan aurora, partially corroborated by the image third down on the left is the only total cyan aurora found so far.

Gaseous objects
"In the western part of the Veil [Nebula] lies another seasonal apparition, the Witch’s Broom, [a] portion of the same remnant, this time resembling, you guessed it, a witch’s broom, with the sweeping end of the broom facing bottom right of the image [at right.] [... The Broom is imaged] with narrow band filters[. The] glowing filaments are like long ripples in a sheet seen almost edge-on, remarkably well separated into atomic hydrogen (red) and oxygen (blue-green) gas".

Liquid objects
Cyan is the color of clear water over a sandy beach, as here at Cala Macaralleta, Menorca.

Rocky objects
Slate is a fine-grained, foliated, homogeneous metamorphic rock derived from an original shale-type sedimentary rock composed of clay or volcanic ash through low-grade regional metamorphism. It is the finest grained foliated metamorphic rock. Foliation may not correspond to the original sedimentary layering, but instead is in planes perpendicular to the direction of metamorphic compression. Slate is frequently grey in color, especially when seen, en masse, covering roofs. However, slate occurs in a variety of colors even from a single locality; for example, slate from North Wales can be found in many shades of grey, from pale to dark, and may also be purple, green or cyan.

Hydrogens
The hydrogen H-beta line (Hβ) has a wavelength of 486.1 nm.

On July 1, 1957, "Following the intense auroral display of the previous night, ... The variation in Hβ emission ... shows quite clearly that the sudden transition from an [auroral] arc to rays coincides with a decrease in the intensity of the hydrogen emission and an inversion of the polarity of the magnetic disturbance."

"IC 5148 is a beautiful planetary nebula located some 3000 light-years away in the constellation of Grus (The Crane). The nebula has a diameter of a couple of light-years, and it is still growing at over 50 kilometres per second — one of the fastest expanding planetary nebulae known. The term “planetary nebula” arose in the 19th century, when the first observations of such objects — through the small telescopes available at the time — looked somewhat like giant planets. However, the true nature of planetary nebulae is quite different."

"The ESO Faint Object Spectrograph and Camera (EFOSC2) on the New Technology Telescope at La Silla gives a somewhat more elegant view of this object. Rather than looking like a spare tyre, the nebula resembles ethereal blossom with layered petals."

The color bands and filters used for the IC 5158 image are blue (optical), Hβ (blue, optical), visual (V, green optical), yellow (R, optical), and Hα (red, optical).

The purple coloration results from a combination of blue and red.

Heliums
The spectral lines from the atmospheres of spectral type O and B stars "show a large number of isolated and overlapping He I lines, the strongest of which are the spectral lines at 447.1 and 492.2 nm".

Helium has a line at 493 nm in the cyan.

"The Wide Field Camera 3 (WFC3), a new camera aboard NASA's Hubble Space Telescope, snapped this image of the planetary nebula, catalogued as NGC 6302, but more popularly called the Bug Nebula or the Butterfly Nebula. WFC3 was installed by NASA astronauts in May 2009, during the servicing mission to upgrade and repair the 19-year-old Hubble telescope."

"What resemble dainty butterfly wings are actually roiling cauldrons of gas heated to more than 36,000 degrees Fahrenheit. The gas is tearing across space at more than 600,000 miles an hour—fast enough to travel from Earth to the Moon in 24 minutes!"

"NGC 6302 lies within our Milky Way galaxy, roughly 3,800 light-years away in the constellation Scorpius. The glowing gas is the star's outer layers, expelled over about 2,200 years. The "butterfly" stretches for more than two light-years, which is about half the distance from the Sun to the nearest star, Alpha Centauri."

"The central star itself cannot be seen, because it is hidden within a doughnut-shaped ring of dust, which appears as a dark band pinching the nebula in the center. The thick dust belt constricts the star's outflow, creating the classic "bipolar" or hourglass shape displayed by some planetary nebulae."

"The star's surface temperature is estimated to be about 400,000 degrees Fahrenheit, making it one of the hottest known stars in our galaxy. Spectroscopic observations made with ground-based telescopes show that the gas is roughly 36,000 degrees Fahrenheit, which is unusually hot compared to a typical planetary nebula."

"The WFC3 image reveals a complex history of ejections from the star. The star first evolved into a huge red-giant star, with a diameter of about 1,000 times that of our Sun. It then lost its extended outer layers. Some of this gas was cast off from its equator at a relatively slow speed, perhaps as low as 20,000 miles an hour, creating the doughnut-shaped ring. Other gas was ejected perpendicular to the ring at higher speeds, producing the elongated "wings" of the butterfly-shaped structure. Later, as the central star heated up, a much faster stellar wind, a stream of charged particles traveling at more than 2 million miles an hour, plowed through the existing wing-shaped structure, further modifying its shape."

"The image also shows numerous finger-like projections pointing back to the star, which may mark denser blobs in the outflow that have resisted the pressure from the stellar wind."

"The nebula's reddish outer edges are largely due to light emitted by nitrogen, which marks the coolest gas visible in the picture. WFC3 is equipped with a wide variety of filters that isolate light emitted by various chemical elements, allowing astronomers to infer properties of the nebular gas, such as its temperature, density, and composition."

"The white-colored regions are areas where light is emitted by sulfur. These are regions where fast-moving gas overtakes and collides with slow-moving gas that left the star at an earlier time, producing shock waves in the gas (the bright white edges on the sides facing the central star). The white blob with the crisp edge at upper right is an example of one of those shock waves."

"NGC 6302 was imaged on July 27, 2009, with Hubble's Wide Field Camera 3 in ultraviolet and visible light. Filters that isolate emissions from oxygen, helium, hydrogen, nitrogen, and sulfur from the planetary nebula were used to create this composite image."

The filters used for this image are F373N ([O II], purple), F469N (He II, blue), F502N ([O III], cyan), F656N (Hα, brown), F658N ([N II], orange), and F673N ([S II], white).

Berylliums
Beryllium has a line at the small wavelength end of the cyan.

Borons
Boron has a line in the cyan.

Carbons
Carbon cyan lines as C I occur at 476.667, 477.000, 477.588, and 493.205 nm.

Carbon in carbon cluster molecules may have an absorption line at 492 nm. "These are: C; (311 and 348 nm), C-, (447 and 492 nm), and Cy (586 and 643 nm)."

Nitrogens
Nitrogen appears to have lines near to the cyan.

"M2-9 [in the image at right] is a striking example of a "butterfly" or a bipolar planetary nebula. Another more revealing name might be the "Twin Jet Nebula." If the nebula is sliced across the star, each side of it appears much like a pair of exhausts from jet engines. Indeed, because of the nebula's shape and the measured velocity of the gas, in excess of 200 miles per second, astronomers believe that the description as a super-super-sonic jet exhaust is quite apt. Ground-based studies have shown that the nebula's size increases with time, suggesting that the stellar outburst that formed the lobes occurred just 1,200 years ago."

"The central star in M2-9 is known to be one of a very close pair which orbit one another at perilously close distances. It is even possible that one star is being engulfed by the other. Astronomers suspect the gravity of one star pulls weakly bound gas from the surface of the other and flings it into a thin, dense disk which surrounds both stars and extends well into space."

"The disk can actually be seen in shorter exposure images obtained with the Hubble telescope. It measures approximately 10 times the diameter of Pluto's orbit. Models of the type that are used to design jet engines ("hydrodynamics") show that such a disk can successfully account for the jet-exhaust-like appearance of M2-9. The high-speed wind from one of the stars rams into the surrounding disk, which serves as a nozzle. The wind is deflected in a perpendicular direction and forms the pair of jets that we see in the nebula's image. This is much the same process that takes place in a jet engine: The burning and expanding gases are deflected by the engine walls through a nozzle to form long, collimated jets of hot air at high speeds."

"M2-9 is 2,100 light-years away in the constellation Ophiucus. The observation was taken Aug. 2, 1997 by the Hubble telescope's Wide Field and Planetary Camera 2. In this image, neutral oxygen is shown in red, once-ionized nitrogen in green, and twice-ionized oxygen in blue."

Oxygens
Oxygen lines appear to be in the cyan.

"The breathtaking butterfly-like planetary nebula NGC 6881 is visible here [at right] in an image taken by the NASA/ESA Hubble Space Telescope. Located in the constellation of Cygnus, it is formed of an inner nebula, estimated to be about one fifth of a light-year across, and symmetrical “wings” that spread out about one light-year from one tip to the other. The symmetry could be due to a binary star at the nebula’s centre."

"NGC 6881 has a dying star at its core which is about 60% of the mass of the Sun. It is an example of a quadrupolar planetary nebula, made from two pairs of bipolar lobes pointing in different directions, and consisting of four pairs of flat rings. There are also three rings in the centre."

"A planetary nebula is a cloud of ionised gas, emitting light of various colours. It typically forms when a dying star — a red giant — throws off its outer layers, because of pulsations and strong stellar winds."

"The image was taken through three filters which isolate the specific wavelength of light emitted by nitrogen (N II, 658 nm, shown in red), hydrogen (Hα, 656 nm, shown in green) and oxygen (O III, 502 nm, shown in blue)."

Fluorines
Fluorine has two emission lines that occur in plasmas at 479.132 and 479.45 nm from F IX.

The emission and absorption spectra of fluorine contains at least eight lines or bands from the cyan to the ultraviolet.

Neons
Like fluorine, neon has at least fourteen emission and absorption lines or bands from the cyan to the violet.

Neon lines in the cyan are very weak.

Argons
Argon has several emission lines that occur in an electron cyclotron resonance (ECR) heated plasmas: 480.602, 484.781, and 493.321 nm from Ar II.

Irons
Fe I has a series of lines 477.282, 478.966, 480.289, 488.542, 488.631, 488.719, 489.076, 490.332, 491.003, 491.055, 491.724, 491.802, 491.900, 492.478, 492.743, 493.882, and 493.922 in the cyan.

There is an Fe I line at 485.97 nm.

Fe II also has lines in the cyan: 483.321 and 489.381.

Mercuries
Mercury has a line at 491.6 nm in the cyan.

Earth
"Harmful algal blooms, once considered mainly a problem in salt water, have been appearing with increasing severity in the Madison lakes [...] No longer just a smelly, unsightly nuisance, the masses of blue-green algae can also exude toxins that attack the liver or nervous system."

"Given the conditions today, we estimate an 85 percent chance of toxic blue-green algae tomorrow."

"Toxic blue-green algae are also becoming more problematic in many inland waters, including the Great Lakes."

"They are naturally present in lakes that get a lot of nutrients, and are not an invasive species. But we think an increase in phosphorus from the agricultural and urban landscapes is contributing to more severe blooms."

"[S]atellite remote sensing [is being used with] bacteriology and limnology — [to] study [the] lakes".

Moon
"[A]ll the blue basalt types (high in UV/VIS ratio [0.40/0.56 µm]) are also the darkest mare soils.” Both Luna 24 and Apollo 12 soil samples are from mare soils that reflect primarily cyan that is likely due to the presence of TiO2 in the soils.

"Previous work has suggested that a cyan color in the multispectral frame represents highland material, and that yellows and greens are freshly excavated basalts. However, we have recently found that a cyan color can also result from a freshly excavated high-Ti basalt."

Mars
"The [true] color images of Mars [at right] were taken in 1999, across almost 60 million miles (!) by a talented amateur astronomer in Oeiras, Portugal – Antonio Cidadao."

"They were acquired with a modest 10-inch "Schmidt-Cassegrain" reflecting telescope, and a commercially available CCD (charge coupled device) camera. Mr. Cidadao’s total investment in his "Mars imaging system"—commercial telescope and electronic camera, plus computer to process the images, and the appropriate software—was approximately three thousand American dollars."

"In 1997, before the arrival of the Mars Pathfinder spacecraft (the first NASA Lander sent to Mars since Viking), the Hubble Telescope was tasked to acquire a series of "weather forecast Mars images" prior to the landing [at left]."

"This long-distance reconnaissance detected a small dust storm less than a month before the Pathfinder arrival, which (with its potentially high winds) could have posed a serious threat to the Pathfinder entry and landing."

"If dust diffuses to the landing site, the sky could turn out to be pink like that seen by Viking... otherwise [based on the Hubble images - above], Pathfinder will likely show blue sky with bright clouds."

Europa
"View of a small region of the thin, disrupted, ice crust in the Conamara region of Jupiter's moon Europa showing the interplay of surface color with ice structures. The white and blue colors outline areas that have been blanketed by a fine dust of ice particles ejected at the time of formation of the large (26 kilometer in diameter) crater Pwyll some 1000 kilometers to the south. A few small craters of less than 500 meters or 547 yards in diameter can be seen associated with these regions. These were probably formed, at the same time as the blanketing occurred, by large, intact, blocks of ice thrown up in the impact explosion that formed Pwyll. The unblanketed surface has a reddish brown color that has been painted by mineral contaminants carried and spread by water vapor released from below the crust when it was disrupted. The original color of the icy surface was probably a deep blue color seen in large areas elsewhere on the moon. The colors in this picture have been enhanced for visibility."

"North is to the top of the picture and the sun illuminates the surface from the right. The image, centered at 9 degrees north latitude and 274 degrees west longitude, covers an area approximately 70 by 30 kilometers (44 by 19 miles), and combines data taken by the Solid State Imaging (CCD) system on NASA's Galileo spacecraft during three of its orbits through the Jovian system. Low resolution color (violet, green, and infrared) data acquired in September 1996, were combined with medium resolution images from December 1996, to produce synthetic color images. These were then combined with a high resolution mosaic of images acquired on February 20th, 1997 at a resolution of 54 meters (59 yards) per picture element and at a range of 5340 kilometers (3320 miles)."

At left is another "image of Jupiter's icy satellite Europa shows surface features such as domes and ridges, as well as a region of disrupted terrain including crustal plates which are thought to have broken apart and "rafted" into new positions. The image covers an area of Europa's surface about 250 by 200 kilometer (km) and is centered at 10 degrees latitude, 271 degrees longitude. The color information allows the surface to be divided into three distinct spectral units. The bright white areas are ejecta rays from the relatively young crater Pwyll, which is located about 1000 km to the south (bottom) of this image. These patchy deposits appear to be superposed on other areas of the surface, and thus are thought to be the youngest features present. Also visible are reddish areas which correspond to locations where non-ice components are present. This coloring can be seen along the ridges, in the region of disrupted terrain in the center of the image, and near the dome-like features where the surface may have been thermally altered. Thus, areas associated with internal geologic activity appear reddish. The third distinct color unit is bright blue, and corresponds to the relatively old icy plains."

"This product combines data taken by the Solid State Imaging (SSI) system on NASA's Galileo spacecraft during three separate flybys of Europa. Low resolution color data (violet, green, and 1 micron) acquired in September 1996 were combined with medium resolution images from December 1996, to produce synthetic color images. These were then combined with a high resolution mosaic of images acquired in February 1997."

Comet Bennett 1970 II
The velocities of the cyan molecule as produced in the head of comet Bennett 1970 II have been measured.

Comet Halley
“During the Halley Monitoring Program at La Silla from Feb.17 to Apr.17,1986 ... In the light of the neutral CN-radical a continuous formation and expansion of [cyan] gas-shells could be observed.” “The gas-expansion velocity decreases with increasing heliocentric distance from 1 km/s in early March to 0.8 km/s in April.”

Comet Holmes
In addition to the usually cyan color of the plasma around the comet nucleus is a blue ion tail leading away to the right.

Comet Kohoutek 1973 XII
The neutral cyan coma of comet Kohoutek 1973 XII is measured.

Comet Lulin
Shown at top "Lulin's green color comes from the gases that make up its Jupiter-sized atmosphere. Jets spewing from the comet's nucleus contain cyanogen (CN: a poisonous gas found in many comets) and diatomic carbon (C2). Both substances glow green when illuminated by sunlight".

Comet Swan
"Comet Swan recently made a swing through the inner solar and emerged in the evening sky. Astronomy enthusiast Ginger Mayfield recorded the blue-green color of the comet's nucleus and a tenuous tail in this composite created from multiple images taken on October 26 from Divide, Colorado."

Comet West 1976 VI
The physical parameters of the neutral cyan coma of comet West (1975n) have been measured.

Saturn
"The view [at right] from Hubble [of Saturn], taken on March 22, 2004, is so sharp that many individual Saturnian ringlets can be seen."

"Hubble's exquisite optics, coupled with the high resolution of its Advanced Camera for Surveys, allow it to take pictures of Saturn which are nearly as sharp as Cassini's, even though Hubble is nearly a billion miles farther from Saturn than Cassini."

"Camera exposures in four filters (blue, blue-green, green, and red) were combined to form the Hubble image, to render colors similar to what the eye would see through a telescope focused on Saturn. The subtle pastel colors of ammonia-methane clouds trace a variety of atmospheric dynamics. Saturn displays its familiar banded structure, and haze and clouds of various altitudes. Like Jupiter, all bands are parallel to Saturn's equator. Even the magnificent rings, at nearly their maximum tilt toward Earth, show subtle hues, which indicate the trace chemical differences in their icy composition."

Uranus
In larger amateur telescopes with an objective diameter of between 15 and 23 cm, the planet appears as a pale cyan disk with distinct limb darkening.

"Methane possesses prominent absorption bands in the visible and near-infrared (IR) making Uranus aquamarine or cyan in color."

In 1986 Voyager 2 found that the visible southern hemisphere of Uranus can be subdivided into two regions: a bright polar cap and dark equatorial bands (see figure on the right). Their boundary is located at about -45 degrees of latitude. A narrow band straddling the latitudinal range from -45 to -50 degrees is the brightest large feature on the visible surface of the planet. It is called a southern "collar". The cap and collar are thought to be a dense region of methane clouds located within the pressure range of 1.3 to 2 bar (see above). Besides the large-scale banded structure, Voyager 2 observed ten small bright clouds, most lying several degrees to the north from the collar. In all other respects Uranus looked like a dynamically dead planet in 1986. Unfortunately Voyager 2 arrived during the height of the planet's southern summer and could not observe the northern hemisphere. At the beginning of the 21st century, when the northern polar region came into view, the Hubble Space Telescope (HST) and Keck telescope initially observed neither a collar nor a polar cap in the northern hemisphere. So Uranus appeared to be asymmetric: bright near the south pole and uniformly dark in the region north of the southern collar. In 2007, when Uranus passed its equinox, the southern collar almost disappeared, while a faint northern collar emerged near 45 degrees of latitude.

On August 23, 2006, researchers at the Space Science Institute (Boulder, CO) and the University of Wisconsin observed a dark spot on Uranus's surface, giving astronomers more insight into the planet's atmospheric activity. Why this sudden upsurge in activity should be occurring is not fully known, but it appears that Uranus's extreme axial tilt results in extreme seasonal variations in its weather. Determining the nature of this seasonal variation is difficult because good data on Uranus's atmosphere have existed for less than 84 years, or one full Uranian year. A number of discoveries have been made. Photometry over the course of half a Uranian year (beginning in the 1950s) has shown regular variation in the brightness in two spectral bands, with maxima occurring at the solstices and minima occurring at the equinoxes. A similar periodic variation, with maxima at the solstices, has been noted in microwave measurements of the deep troposphere begun in the 1960s. Stratospheric temperature measurements beginning in the 1970s also showed maximum values near the 1986 solstice. The majority of this variability is believed to occur owing to changes in the viewing geometry.

There are some reasons to believe that physical seasonal changes are happening in Uranus. While the planet is known to have a bright south polar region, the north pole is fairly dim, which is incompatible with the model of the seasonal change outlined above. During its previous northern solstice in 1944, Uranus displayed elevated levels of brightness, which suggests that the north pole was not always so dim. This information implies that the visible pole brightens some time before the solstice and darkens after the equinox. Detailed analysis of the visible and microwave data revealed that the periodical changes of brightness are not completely symmetrical around the solstices, which also indicates a change in the meridional albedo patterns. Finally in the 1990s, as Uranus moved away from its solstice, Hubble and ground based telescopes revealed that the south polar cap darkened noticeably (except the southern collar, which remained bright), while the northern hemisphere demonstrated increasing activity, such as cloud formations and stronger winds, bolstering expectations that it should brighten soon. This indeed happened in 2007 when the planet passed an equinox: a faint northern polar collar arose, while the southern collar became nearly invisible, although the zonal wind profile remained slightly asymmetric, with northern winds being somewhat slower than southern.

"These are among the first clear images, taken from the distance of Earth, to show aurorae on the planet Uranus. Aurorae are produced when high-energy particles from the Sun cascade along magnetic field lines into a planet's upper atmosphere. This causes the planet's atmospheric gasses to fluoresce. The ultraviolet images were taken at the time of heightened solar activity in November 2011 that successively buffeted the Earth, Jupiter, and Uranus with a gusher of charged particles from the Sun. Because Uranus' magnetic field is inclined 59 degrees to its spin axis, the auroral spots appear far from the planet's north and south poles. This composite image combines 2011 Hubble observations of the aurorae in visible and ultraviolet light, 1986 Voyager 2 photos of the cyan disk of Uranus as seen in visible light, and 2011 Gemini Observatory observations of the faint ring system as seen in infrared light."

Neptune
On July 12, 2011, Neptune "has arrived at the same location in space where it was discovered nearly 165 years ago. To commemorate the event, NASA's Hubble Space Telescope has taken these "anniversary pictures" of the blue-green giant planet."

"Neptune is the most distant major planet in our solar system. German astronomer Johann Galle discovered the planet on September 23, 1846. At the time, the discovery doubled the size of the known solar system. The planet is 2.8 billion miles (4.5 billion kilometers) from the Sun, 30 times farther than Earth. Under the Sun's weak pull at that distance, Neptune plods along in its huge orbit, slowly completing one revolution approximately every 165 years."

"These four Hubble images of Neptune were taken with the Wide Field Camera 3 on June 25-26, during the planet's 16-hour rotation. The snapshots were taken at roughly four-hour intervals, offering a full view of the planet. The images reveal high-altitude clouds in the northern and southern hemispheres. The clouds are composed of methane ice crystals."

"The giant planet experiences seasons just as Earth does, because it is tilted 29 degrees, similar to Earth's 23-degree-tilt. Instead of lasting a few months, each of Neptune's seasons continues for about 40 years."

"The snapshots show that Neptune has more clouds than a few years ago, when most of the clouds were in the southern hemisphere. These Hubble views reveal that the cloud activity is shifting to the northern hemisphere. It is early summer in the southern hemisphere and winter in the northern hemisphere."

"In the Hubble images, absorption of red light by methane in Neptune's atmosphere gives the planet its distinctive aqua color. The clouds are tinted pink because they are reflecting near-infrared light."

"A faint, dark band near the bottom of the southern hemisphere is probably caused by a decrease in the hazes in the atmosphere that scatter blue light. The band was imaged by NASA's Voyager 2 spacecraft in 1989, and may be tied to circumpolar circulation created by high-velocity winds in that region."

"The temperature difference between Neptune's strong internal heat source and its frigid cloud tops, about minus 260 degrees Fahrenheit, might trigger instabilities in the atmosphere that drive large-scale weather changes."

Interstellar medium
The cyanide radical CN- has been identified in interstellar space. The cyanide radical (called cyanogen) is used to measure the temperature of interstellar gas clouds.

"This helmet-shaped cosmic cloud [in the image at right] with wing-like appendages is popularly called Thor's Helmet. Heroically sized even for a Norse god, Thor's Helmet is about 30 light-years across. In fact, the helmet is actually more like an interstellar bubble, blown as a fast wind from the bright, massive star near the bubble's center sweeps through a surrounding molecular cloud. Known as a Wolf-Rayet star, the central star is an extremely hot giant thought to be in a brief, pre-supernova stage of evolution. Cataloged as NGC 2359, the nebula is located about 15,000 light-years away in the constellation Canis Major. The sharp image, made using broadband and narrowband filters, captures striking details of the nebula's filamentary structures. It shows off a blue-green color from strong emission due to oxygen atoms in the glowing gas."

Beta Virginis
Beta Virginis has a surface Teff = 6,132 ± 26. According to SIMBAD, beta Virginis is a high proper-motion star with spectral type equal to F9V. Beta Virginis is also known as LHS 2465, GJ 449, HD 102870, and HR 4540. It is an X-ray source per 2E,RBS,RX,1RXS, and is a double star.

Theta Ursae Majoris
Theta Ursae Majoris is a spectral type F7V star. It has a surface temperature of 6300 ± 33 K. Such an effective surface temperature has a Planckian black body peak wavelength of 476 nm which places this star at the high temperature end of the cyan band.

Planetary nebulas
NGC 7048 is a planetary nebula in the constellation of Cygnus. The bright star to the lower left of the nebula is a magnitude 10.5 star. The nebula is slightly brighter along the west and east sides. This planetary nebula is rated at magnitude 12.1. NGC 7048 was discovered by Jean Marie Edouard Stephan on October 1878 using a 31.5-inch reflector. "A planetary nebula (PN) is an expanding ionized circumstellar cloud that was ejected during the asymptotic giant branch (AGB) phase of the stellar progenitor." In recent years, Hubble Space Telescope images have revealed many planetary nebulae to have extremely complex and varied morphologies. About a fifth are roughly spherical, but the majority are not spherically symmetric.

"The Hubble Space Telescope has imaged striking details of the famed planetary nebula designated NGC 2818 [at left], which lies in the southern constellation of Pyxis (the Compass). The spectacular structure of the planetary nebula contains the outer layers of a star that were expelled into interstellar space. The glowing gaseous shrouds in the nebula were shed by the central star after it ran out of fuel to sustain the nuclear reactions in its core."

"This Hubble image was taken in November 2008 with the Wide Field Planetary Camera 2. The colors in the image represent a range of emissions coming from the clouds of the nebula: red represents nitrogen, green represents hydrogen, and blue represents oxygen."

Recent history
The recent history period dates from around 1,000 b2k to present.

"The Hubble community bids farewell to the soon-to-be decommissioned Wide Field Planetary Camera 2 (WFPC2) onboard the Hubble Space Telescope. In tribute to Hubble's longest-running optical camera, a planetary nebula [at right] has been imaged as WFPC2's final "pretty picture.""

"This planetary nebula is known as Kohoutek 4-55 (or K 4-55). It is one of a series of planetary nebulae that were named after their discoverer, Czech astronomer Lubos Kohoutek. A planetary nebula contains the outer layers of a red giant star that were expelled into interstellar space when the star was in the late stages of its life. Ultraviolet radiation emitted from the remaining hot core of the star ionizes the ejected gas shells, causing them to glow."

"In the specific case of K 4-55, a bright inner ring is surrounded by a bipolar structure. The entire system is then surrounded by a faint red halo, seen in the emission by nitrogen gas. This multi-shell structure is fairly uncommon in planetary nebulae."

"This Hubble image was taken by WFPC2 on May 4, 2009. The colors represent the makeup of the various emission clouds in the nebula: red represents nitrogen, green represents hydrogen, and blue represents oxygen. K 4-55 is nearly 4,600 light-years away in the constellation Cygnus."

"The WFPC2 instrument, which was installed in 1993 to replace the original Wide Field/Planetary Camera, will be removed to make room for Wide Field Camera 3 during the upcoming Hubble Servicing Mission."

"During the camera's amazing, nearly 16-year run, WFPC2 provided outstanding science and spectacular images of the cosmos. Some of its best-remembered images are of the Eagle Nebula pillars, Comet P/Shoemaker-Levy 9's impacts on Jupiter's atmosphere, and the 1995 Hubble Deep Field—the longest and deepest Hubble optical image of its time."

"The scientific and inspirational legacy of WFPC2 will be felt by astronomers and the public alike, for as long as the story of the Hubble Space Telescope is told."

Hubble Space Telescope
At right, the Hubble Space Telescope is seen from space shuttle Columbia in 2002. Each of the three camera systems aboard the Hubble has used narrow and wide filters covering the cyan or blue-green band of the visible spectrum.

Stardust
To produce 3D images of Comet Wild 2, the Stardust spcecraft at right takes images in red and cyan.

STEREO
To produce 3D images of the Sun, the STEREO spacecraft at right take images in red and cyan.

Voyager 2
In January 1986, the Voyager 2 spacecraft flew by Uranus at a minimal distance of 107,100 km providing the first close-up images and spectra of its atmosphere. They generally confirmed that the atmosphere was made of mainly hydrogen and helium with around 2% methane. The atmosphere appeared highly transparent and lacking thick stratospheric and tropospheric hazes. Only a limited number of discrete clouds were observed.

Hypotheses

 * 1) Cyans and CN are closely related.
 * 2) The cyan color source in minerals is the same as from astronomical objects.


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Partition related number triangles

Set partitions by number of blocks
All partitions of a set with n elements into k blocks are enumerated by the number triangle called Stirling numbers of the second kind. When some partitions are left out or treated as equivalent the new set of partitions is enumerated by a new number triangle. The number triangles for noncrossing partitions are symmetric.

In the two illustrations above some set partitions, sharing the same colors, are grouped together. They always correspond to the same integer partition, shown in the following illustration:

Lah numbers
Counting not only the partitions (which gives the Stirling2 numbers shown above), but also the possible permutations within the blocks gives the triangle of unsigned Lah numbers (with the row sums ).

Also here one could define a noncrossing version (where T(4;2) would be 32 instead of 36).

Set partitions by type and by number of singletons

Subsequences
The by type triangles have some interesting subsequences which are highlighted by colors in the files linked below. 1 * n + n * 1 stands for the integer partition with one addend n and n addends 1, etc.

N  =  0, 1, 2,  3,  4,   5,   6...  (N)  =  1, 2, 6, 20, 70, 252, 924... (central binomial coefficients)
 * The sequence 1,6,20,70,252,924... appears as All, All ■ and All, Noncrossing ■ :
 * There is one partition of a 2-set that has a 1-block (i.e. a singleton) and another singleton.
 * For n>1 there are (n) partitions of a 2n-set that have an n-block and n singletons.

N  =  0, 1,  2,  3,   4,   5...  (N)  =  1, 3, 10, 35, 126, 462...
 * All, All □ :
 * There are (n+1) partitions of a 2n-set that have two n-blocks.

 * All, All ■ and All, Noncrossing  ■ :
 * There is one partition of a 3-set that has a 1-block (i.e. a singleton) and two other singletons.
 * For n>1 there are (n) partitions of a (2n+1)-set that have an n-block and n+1 singletons.

 * All, All ■ and All, Noncrossing ■ :
 * There are (n) partitions of a (2n+1)-set that have an (n+1)-block and n singletons.

 * All, All □ :
 * There are (n) partitions of a (2n+1)-set that have an (n+1)-block and an n-block.

N+  =  1, 2, 3,  4,  5,  6...  (N+)  =  1, 2, 4, 10, 26, 80...
 * Up to rot, All ■ and Up to rot, Noncrossing ■ :
 * There are (n) necklaces of length 2n with n black and n white beads.

 * Up to rot & ref, All ■ ■  □ and Up to rot & ref, Noncrossing ■  ■ :
 * There are (n) free necklaces (or bracelets) of length 2n+1 with n black and n+1 white beads.

title: Partition related number triangles


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User:Egm4313.s12.team14.kimiagarov/HW 5

= Problem R5.1 =

Given

 * $$\displaystyle{1. \quad r(x) = \sum^\infin_{k=0}(k+1)kx^k}$$

 * $$\displaystyle{2. \quad r(x) = \sum^\infin_{k=0}{(-1)^k \over \gamma^k}x^{2k}}$$

 * $$\displaystyle{\gamma = constant}$$

 * $$\displaystyle{3. \quad \sin x\ at\ \hat x = 0}$$

 * $$\displaystyle{4. \quad \log (1+x)\ at\ \hat x = 0}$$

 * $$\displaystyle{5. \quad \log (1+x)\ at\ \hat x = 1}$$

Find

 * $$\displaystyle{R_c}$$

R5.1.1
$$\displaystyle{Using\ the\ ratio\ test;}$$

$$\displaystyle{\lim_{k \to \infty}\left| {a_{k+1} \over a_k} \right|<1}$$

$$\displaystyle{Where:}$$

$$\displaystyle{a_k = (k+1)kx^k}$$

$$\displaystyle{Plug\ Eqn.2\ into\ Eqn.1:}$$

$$\displaystyle{\lim_{k \to \infty}\left| {(k+2)(k+1)x^{k+1} \over (k+1)kx^k} \right|<1}$$

$$\displaystyle{\left| x \right|<1}$$

$$\displaystyle{R=1}$$

R5.1.2
$$\displaystyle{Using\ the\ ratio\ test;}$$

$$\displaystyle{\lim_{k \to \infty} \left| {a_{k+1} \over a_k} \right|}$$

$$\displaystyle{Where:}$$

$$\displaystyle{a_k ={(-1)^k \over \gamma^k}x^{2k}}$$

$$\displaystyle{\lim_{k \to \infty}\left| {{(-1)^{k+1} \over \gamma^{k+1}}x^{2(k+1)} \over {(-1)^k \over \gamma^k}x^{2k}} \right|<1}$$

$$\displaystyle{\left| {x^2 \over \gamma} \right|<1}$$

$$\displaystyle{\left| x \right| < \sqrt{ \left| \gamma  \right|}}$$

$$\displaystyle{R = \sqrt{ \left| \gamma \right|}}$$

R5.1.3
$$\displaystyle{Using\ Taylor\ series\ expansion;}$$

$$\displaystyle{\sum^\infin_{k=0}{f^{(n)}(\hat x) \over n!}(x- \hat x)^n\ at\ \hat x=0}$$

$$\displaystyle{{\sin0 \over 0!}x^0+{\cos0 \over 1!}x^1-{\sin0 \over 2!}x^2-{\cos0 \over 3!}x^3...}$$

$$\displaystyle{{1 \over 1!}x^1-{1 \over 3!}x^3 ...}$$

$$\displaystyle{\sum^\infin_{k=0}{(-1)^k \over (2k+1)!}x^{k+1}}$$

$$\displaystyle{Using\ (2)p.7-31;}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {d_{k+1} \over d_k} \right| \right]^{-1}}$$

$$\displaystyle{Where\ d_k = {(-1)^k \over (2k+1)!}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{(-1)^{k+1} \over (2(k+1)+1)!} \over {(-1)^k \over (2k+1)!}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{1 \over (2k+3)!} \over {1 \over (2k+1)!}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {(2k+1)! \over (2k+3)(2k+2)(2k+1)!} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {1 \over (2k+3)(2k+2)} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ 0 \right]^{-1}}$$

$$\displaystyle{R_c = \infty}$$

R5.1.4
$$\displaystyle{Using\ Taylor\ series\ expansion;}$$

$$\displaystyle{\sum^\infin_{n=0}{f^{(n)}(\hat x) \over n!}(x- \hat x)^n\ at\ \hat x=0}$$

$$\displaystyle{{log(1+0) \over 0!}x^0+{(1+0)^{-1} \over 1!}x^1-{(1+0)^{-2} \over 2!}x^2+{2(1+0)^{-3} \over 3!}x^3-{6(1+0)^{-4} \over 4!}x^4 ...}$$

$$\displaystyle{0+x^1-{x^2 \over 2!}+{2x^3 \over 3!}-{6x^4 \over 4!} ...}$$

$$\displaystyle{\sum^\infin_{k=0}{(-1)^kx^{k+1} \over (k+1)}}$$

$$\displaystyle{Using\ (2)p.7-31;}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {d_{k+1} \over d_k} \right| \right]^{-1}}$$

$$\displaystyle{Where\ d_k = {(-1)^k \over (k+1)}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{(-1)^{k+1} \over (k+1+1)} \over {(-1)^k \over (k+1)}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{1 \over (k+2)} \over {1 \over (k+1)}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ 1 \right]^{-1}}$$

$$\displaystyle{R_c = 1}$$

R5.1.5
$$\displaystyle{Using\ Taylor\ series\ expansion;}$$

$$\displaystyle{\sum^\infin_{n=0}{f^{(n)}(\hat x) \over n!}(x- \hat x)^n\ at\ \hat x=0}$$

$$\displaystyle{{log(1+1) \over 0!}(x-1)^0+{(1+1)^{-1} \over 2!}(x-1)^1-{(1+1)^{-2} \over 3!}(x-1)^2+{2(1+1)^{-3} \over 4!}(x-1)^3-{6(1+1)^{-4} \over 5!}(x-1)^4 ...}$$

$$\displaystyle{log(2)+(x-1)^1 2^{-1}-{(x-1)^2 2^{-2} \over 2!}+{2(x-1)^3 2^{-3} \over 3!}-{6(x-1)^4 2^{-4} \over 4!} ...}$$

$$\displaystyle{log(2) + \sum^\infin_{k=0}{(-1)^k(x-1)^{k+1} 2^{-(k+1)} \over (k+1)}}$$

$$\displaystyle{Note:\ log(2)\ will\ not\ affect\ R_c}$$

$$\displaystyle{Using\ (2)p.7-31;}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {d_{k+1} \over d_k} \right| \right]^{-1}}$$

$$\displaystyle{Where\ d_k = {(-1)^k 2^{-(k+1)} \over (k+1)}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{(-1)^{k+1} 2^{-(k+1+1)} \over (k+1+1)} \over {(-1)^k 2^{-(k+1)} \over (k+1)}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {(k+1) \over 2(k+2)} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ {1 \over 2} \right]^{-1}}$$

$$\displaystyle{R_c = 2}$$

title: User:Egm4313.s12.team14.kimiagarov/HW 5


title: User:Egm4313.s12.team12/R5