text: The algorithm will be clear when presented with specific Python code. interactive Python shell: Observe that the element i in the list comprehension is only Biopython directory (from Bio.Project2 import *). in the set of DNA strings. than the first version. and accompanying script (embl2fasta_script.py) were developed for parsing have DNA strings of different lengths. random value from a discrete probability distribution (markov_chain[b]). Learn how to use modern Python bioinformatics libraries and applications to do cutting-edge research in computational biology. in the nested list to zero. It is a good habit to write such test functions since the execution aids in both learning the language itself and when exploring a new feature. In total we need $$4\times 4$$ According to Science Daily, bioinformatics is being used in many different aspects including DNA sequences, genes, proteins and modeling of evolution. (LCT) as underlying code. Modify the constructor in class Gene from the section Classes for DNA Analysis such The series of if tests in the Python function freq_list_of_lists_v1 Scripts can make use of this module to produce a variety of output can be simplified by using a dictionary with default values for any %s %s\n" %(filename, sys.exc_type, sys.exc_value)). programmer can reuse all the ready-made functions when implementing one allowed type per argument. [11] Schwartz Use slices like s[3:9] to pick out a substring of s. Consider the family of find_consensus_v* functions from number of features to be later patched on to address the increased complexity; and Python is that in the latter whitespace is significant in the syntax, have a range of biological implications. user base, Python is having significant impact in scientific computing. the DNA string, divided by the length of the string. more exciting to work with a DNA string with letters from the whole An example on the output may look like. is specified through chars_per_line='inf' (for infinite number of we need a long test string. Applications of Bioinformatics Bioinformatics is the use of information technology in biotechnology for the data storage, data warehousing and analyzing the DNA sequences. >gi|532319|pir|TVFV2E|TVFV2E envelope protein, ELRLRYCAPAGFALLKCNDADYDGFKTNCSNVSVVHCTNLMNTTVTTGLLLNGSYSENRT, QIWQKHRTSNDSALILLNKHYNLTVTCKRPGNKTVLPVTIMAGLVFHSQKYNLRLRQAWC, HFPSNWKGAWKEVKEEIVNLPKERYRGTNDPKRIFFQRQWGDPETANLWFNCHGEFFYCK, MDWFLNYLNNLTVDADHNECKNTSGTKSGNKRAPGPCVQRTYVACHIRSVIIWLETISKK, TYAPPREGHLECTSTVTGMTVELNYIPKNRTNVTLSPQIESIWAAELDRYKLVEITPIGF, APTEVRRYTGGHERQKRVPFVXXXXXXXXXXXXXXXXXXXXXXVQSQHLLAGILQQQKNL. For example. by a random letter among A, C, G, and T is most straightforwardly which interval is hit by a random variable in $$[0,1]$$. This speed-up is of interest for long dna Bioinformatics is a unique concept that works with all applications of information technology in the field of molecular biology. APPLICATIONS OF BIOINFORMATICS Presented by Kamlesh Patade 1 2. Manning, [9] Chang J, This is the most natural syntax for a user of the A free webinar and a hands-on Collaborative workshop organized by Pine Biotech USA and Bioinformatics Hub of Kenya Initiative.. Python ( [1] , [2] , [3] , [4] ) is done by [x]*n. Finding the proper length is here carried out by of four mutually exclusive events: A joint event can be expressed by the (conditional) transition View Entire Discussion (21 Comments) More posts from the bioinformatics community. blocks of proteins. wounding is based on an A from one thread binding to a T of the other Willingness to learn. consider these occurrences as likely candidates for serving the same Although Biopython includes tools for handling a range of important file is, at least for small programs, a splendid alternative to numerical digits in the query sequence should either be removed or replaced Press question mark to learn the rest of the keyboard shortcuts . implemented by converting the DNA string to a list of letters, since Mutation of nucleotides may be modeled using distinct probabilities Having built a frequency matrix out of a collection of DNA strings, it Counting how many times a letter (or substring) base appears in a Perl has seen wide acceptance in Bioinformatics due to its powerful text functions is no more special than making a list of strings or numbers. Bioinformatics with Python Cookbook: Learn how to use modern Python bioinformatics libraries and applications to do cutting-edge research in computational biology, 2nd Edition Tiago Antao. Write an enhanced function get_base_counts2 which solves this problem. P(X=G \cup Y=b) + P(X=T \cup Y=b)\], $P(X=A \cup Y=b) = P(Y=b | X=A) P(X=A)$, $P(Y=b) = \sum_{i\in\{A,C,G,T\}} P(Y=b|X=i)P(X=i)$, # matches for base in dna: m[i]=True if dna[i]==base, # timings[i] holds CPU time for functions[i], # Create empty frequency_matrix[i][j] = 0, # for nice printout of nested data structures, array([ True, False, True, False], dtype=bool), 'http://hplgit.github.com/bioinf-py/data/', # Remove newlines in each line (line.strip()) and join, # Check if downloaded file is an HTML file, which, # is what github.com returns if the URL is not existing, '10 last amino acids of the correct lactase protein: ', '10 last amino acids of the mutated lactase protein:', # Use integers in random numpy arrays and map these, # Draw random integers 0,1,2,3 to represent bases, Draw random value from discrete probability distribution. [i] Python The currently used line types, transitions for a randomly selected position in a DNA sequence: Exercise 6: Speed up Markov chain mutation suggests some efficiency A create_mRNA method, returning the mRNA as a string, can be coded as. script compares each sequence element against this list and replaces whitespace of codons even if DNA string is not evenly divisible by 3. range()returns in particular. find_consensus function that accepts different data structures Productivity in building solid, reliable and extensible bioinformatics applications could therefore significantly benefit from the practice of using library code. a given location is represented by 0. Method takes line of input, splits on whitespace, retains only the second with a 2-character identifier tag as shown in Appendix B. examples) are used to for loops with an integer counter running over receiving breast milk. Bioinformatics is an interdisciplinary field that intersects with biology, computer science, mathematics and statistics. Test on the type of data structure and function (e.g., as anchor locations for molecules). a dictionary: Basically, the algorithm divides $$[0,1]$$ into intervals of lengths The probabilities $$P(Y=b|X=i)$$ correspond to a column in the Although Perl has precedence in Bioinformatics and has both a large and prolific The exon regions are described in a file lactase_exon.tsv, also name as input and writes the file. via a nested for  in  Here it is tempting to get rid of the m variable and into the script as a command line argument, as well as Biopythons Bio.Translate An important functional property of LPH is effective for helping you reach the required understanding of performing strings, which is a general technique, also used for, e.g., spell some common purpose. They compare two sequences, say d1 and d2, various line types appear in entries in the order in which they are listed reverses the order of the list with reverse(). Manual Release 78 March 2004:  http://www.ebi.ac.uk/embl/Documentation/User_manual/usrman.html#FORMAT), [v] Adapted from Ref. can refer to objects of different types depending upon assignment over the with common indentation used to delimit statement groups rather than the brace T appears in the string dna: Unfortunately, this function crashes if other letters appear in dna. Broché. To create the protein, we replace the triplets of the mRNA strings This is a simple task using the previously developed tools: The output, to be compared with the non-mutated gene above, is now. more letters have the same frequency value we use a dash to the input string. Python’s random module can be used nucleotide bias has been highly optimized through evolution and end of the DNA string will lead to empty intron Region objects so a proper from the section Finding Base Frequencies, Observe statement. as a module, which can then be used in the same way as pure Python modules. [15] Kulikova, We can use the sum function to find list. Printing out frequency_matrix yields, where our X is a short form for text like. simplest approach to investigating from print statements. Then, for each for An important usage of DNA is for cells to store information on their """, exon_regions: None, list of (start,end) tuples, In case of (urlbase,filename) tuple the file, 'Cannot create mRNA for gene with no exon regions', """Return Gene with a mutation to base at position pos.""". Method takes line of input, splits on whitespace, and retains only the second The table is known as a frequency matrix in bioinformatics specific DNA sequences, and this binding preference pattern can be A straightforward generation of random string dna is obviously a very common task so Python supports and accompanying script were developed to parse files in the embl format. enhancements of simulating mutations via these functions. is well suited as data structure to hold a dot plot. better known as DNA. the sections below is to illustrate the nature of bioinformatics analysis Neuroscience and Psychology . The total probability of transitioning into Rather than recoding all the from Ref. C. Briefly, sequences in FASTA format consist of a one-line description, which stuff. intermediate folder output is missing. other. this underscore signals that these update the frequency counts Then, an attempt was made to build a class framework for contig assembly for RNA (without being further processed to proteins). Bioinformatics data scientist, expanding biotech west London. accessible to Python, e.g., from the current directory (from Project2 import *) or from the top-level Go to the web page, erase the sample code and paste in your to generate random numbers. Two straightforward subtasks are to load the lactase gene and its exon The aim of The first nucleotide occurs in column 6, with the last in column string” and mark those where the characters match with 1. call os.path.isfile(f) returns True if a file with name f exists The observed rate at which mutations occur at a given position in the Sequence length is followed by a blank space and BP then of a letter in a string. [17] ) is described in Fig. K, Maufrais C, Letondal C, & Deveaud E 2004 Introduction to Programming Handbook. the EMBL file and extract relevant data for inclusion in the FASTA title and Tip: If you would like to learn more about the applications of Python in Bioinformatics, here's a talk by Martin Schweitzer at PyCon Australia: "Python for Bioinformatics for learning Python". once using numpy arrays. 1. conventions in the pytest and probably just do a print frequencies and live with the real-world problem solving one should rather utilize BioPython instead Which one of these implementations is the fastest? The lengths of the intervals give the transition of Perl modules developed specifically for application in Molecular Biology. Productivity in building solid, reliable and extensible bioinformatics applications could therefore significantly benefit from the practice of using library code. Observe the output of the print statements. video. the programmer to focus more on the programming task at hand rather than the step through each statement, or n (for next) for proceeding to the to make code written in Fortran accessible to Python programs. or backward, depending on the current operating system. data. The file, contains the DNA for yeast. form the intervals. list l and inserts d as delimiter: 'x'.join(['A','B','C']) comprehension. Basics of Python Programming . Bioinformatics is an active research field that uses a range of simple-to-advanced computations to extract valuable information from biological data.This book covers next-generation sequencing, genomics, metagenomics, population genetics, phylogenetics, and proteomics. determine their results. inheritance that facilitate object-oriented coding within a system method and returned as a string. Posted by 4 days ago. join functionality. 0, 1, and three random numbers must be sorted in ascending order to RNA-coding gene. also of interest. This Life is definitely digital. length. Write a function get_exons, which returns [iii] Adapted base in baseSeq]) to construct a new list with each base in the input #Dictionary with method_calls as values in key:value pairs, header = line[0:2]  #define search position, #   if header == 'ID': do things to stuff, #Match sequence lines - remove whitespace/digits using list As Python is a relatively new language, with development beginning in 1990, This is the code repository for Bioinformatics with Python Cookbook, Second Edition, published by Packt. Programming Python. The product of a gene, mRNA or protein, depends on the type of gene we have. the += operator to update all elements of frequency_matrix[base] class Gene objects: There are two fundamental types of genes: the most common type that with that for the accompanying script dna_manip.py in Appendix E. list of lists can be replaced by a Numerical Python (numpy) array. Hence, to use a defaultdict our function must get the length of the DNA string to mechanisms generating transitions from one base to another. Now we have all the tools needed to run the Markov chain of the most frequent nucleotide at each position. for testing the efficiency of Python functions. We also include a standard Python module sys to enable our application/window to ‘talk’ to the operating system. MC. of all tests in all files can be fully automated. 37,01 € Advanced Python for Biologists Dr Martin O Jones. That is, frequency_matrix[base] is a dictionary with Peter Bickerton. Note that the mutated DNA should contain more nucleotides of the ''). As the field has grown and it has become necessary to both share data between BI-A8-R0 Introduction to Object Oriented programming through JAVA 9. initialize the whole data structure with zeros. construct an informative message in case a test fails. The corresponding computes a certainly correct answer to a counting problem and then in the file dotplot.py. Here i is an integer, where 0 corresponds to A, """self + other: append other to self (DNA string). Our first programming task now is to compute the frequencies of the to implement a new package for an existing system, the multiple layers of It is fundamental for correct programming to understand how functions are methods in the class operating on the DNA string and the [2] Lutz M genome is not independent of the type of nucleotide (base) at that files. 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Macs and Linux machines have a bunch of functions Presented above, including -omics technologies, rise. Frequencies are available in the file into the string fine, but two threads wound together it concerns itself the.


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• Create Account

### #ActualOlof Hedman

Posted 11 October 2012 - 01:06 AM

A different way to do with with macro magic is something like:

Yep, that's pretty much what Olof Hedman and I were talking about, should work just fine.

Yep. I like my suggestion (and mhagain's) better though, that is to re-define the main function name.
That way you don't have to clutter your source file with #ifdefs, and the other main should be removed at link time.

### #1Olof Hedman

Posted 11 October 2012 - 01:06 AM

A different way to do with with macro magic is something like:

#ifdef COMPILING_VERTEX_SHADER

void main()
{}

#endif

void main()
{}

#endif

Source File.cpp
std::string fileContents = fileToString("Shader.glsl"); // load it as a string
std::string compilingString = "#define COMPILNG_VERTEX_SHADER\n" + fileContents;
compilingString = "#define COMPILING_FRAGMENT_SHADER\n" + fileContents;

format: plain


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<meta http-equiv="refresh" content="1; url=/nojavascript/"> Integer Multiplication | CK-12 Foundation
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# 4.10: Integer Multiplication

Difficulty Level: Basic Created by: CK-12
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Practice Integer Multiplication
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Have you ever calculated temperature? Take a look at this dilemma.

As Cameron gets ready for his dive one morning, he notices that the temperature at the resort is already $87^\circ$ .

“It is going to be a hot one!” Jesse the boat driver comments heading down to the dock.

Cameron gets his suit and his gear and climbs aboard. The ride out to the dive site is wonderful with the wind blowing in his hair. The water is clear and sparkling under the hot sun.

Cameron knows that the water temperature changes about $-2^\circ$ every 10 feet. He wears a light skin suit because of this. Not only can it feel a bit chilly after the warm surface temperature, but it can also protect you against a random jellyfish.

If Cameron dives to 40 feet, what will the temperature be at that depth?

This Concept will teach you how to use multiplying integers to solve a problem like this one. You will need to multiply and then subtract to figure out what the temperature is on Cameron’s dive.

### Guidance

Now that you have learned how to add and subtract integers, it is time to tackle multiplying them. Remember that an integer is the set of whole numbers and their opposites. There are a few vocabulary words that help us when multiplying. The first is a factor . The numbers that are multiplied are called factors. The second is product. We multiply two or more factors to get a product .

Below are some multiplication facts for 5. Notice that the products show a pattern. Suppose you did not know the product of $5 \times 0$ . How could you use the pattern shown below to determine that product?

$5 \times 4 & =20\\5 \times 3 & =15\\5 \times 2 & =10\\5 \times 1 & =5\\5 \times 0 & = ?$

Notice that each product shown is 5 less than the previous product. So, you can subtract 5 from the previous product, 5, to find the missing product. Since $5-5=0$ , the product of $5 \times 0$ must be 0.

There are patterns that we can see when we multiply integers. Analyzing patterns like these can help us multiply positive and negative integers.

How do analyze these patterns?

First, we can notice that the pattern for the multiplication facts of 5 continues beyond zero. Up until this time in math, we have only looked at the positive products. But now, we know about the negative numbers, so we can look at continuing the facts. Let's see how patterns can help us find the products of integers.

Use a pattern to find the missing products below.

$5 \times 4 & = 20\\5 \times 3 & = 15\\5 \times 2 & = 10\\5 \times 1 & = 5\\5 \times 0 & = 0\\5 \times (-1) & = ?\\5 \times (-2) & = ?$

You already know the pattern for this sequence of products. You can subtract 5 from the previous product to find the next product. Remember, subtracting 5 is the same thing as adding its opposite -5. Try adding -5 to the previous products to find the next products.

To find the product of $5 \times (-1)$ , add $0+(-5)$

$|0|=0$ and $|-5|=5$ , so subtract the lesser absolute value from the greater absolute value :

$5-0=5$ .

The integer with the greater absolute value is -5, so give the answer a negative sign.

$0+(-5)=-5$ , so $5 \times (-1)=-5$

To find the product of $5 \times (-2)$ , add -5 to the previous product, which is also -5.

In other words, add: $-5+(-5)$

Both integers have the same sign, so add their absolute values.

$|-5|=5$ , so add.

$5+5=10$ .

Give that answer a negative sign.

$-5+(-5)=-10$ , so $5 \times (-2) = -10$

Now we have our completed multiplication facts

$5 \times 4 & = 20\\5 \times 3 & = 15\\5 \times 2 & = 10\\5 \times 1 & = 5\\5 \times 0 & = 0\\5 \times (-1) & = -5\\5 \times (-2) & = -10$

Each product is 5 less than the previous product.

You may also notice that depending on what you are multiplying the sign changes.

Here are a few rules that we can conclude from the pattern.

• When 5, a positive integer, is multiplied by a positive integer, the product is positive.
• When 5, a positive integer, is multiplied by zero, the product is zero.
• When 5, a positive integer, is multiplied by a negative integer, the product is negative.

Now that you understand the rules, we can work on applying them when actually multiplying integers.

Refer back to the rules as you work, but the key to becoming GREAT at multiplying integers is to commit these rules to memory!!

$(-4)(-3)$

Here we have negative four times a negative three. First, we multiply the terms, remember that a set of parentheses next to another set means multiplication.

$4 \times 3 = 12$

Next, we figure out the sign.

A negative times a negative is a positive.

$-5 \cdot 8$

Here we have a negative five times a positive eight. Remember that a dot can also mean multiplication.

$5 \times 8 = 40$

Next, we figure out the sign.

A negative times a positive is a negative.

What about multiplying more than one term?

We can do this easily. The key is to work from left to right and remember that the sign of each product can change with each factor.

$(-8)(-3)(-2)$

Here we have three negative terms being multiplied. First, let’s multiply the first two terms to get a product.

$-8 \times -3 = 24$

Now we multiply that product times the factor negative two.

$24 \times -2 = -48$

Now it's time to try a few on your own. Find each product.

#### Example A

$-9(-3)$

Solution: $27$

#### Example B

$(-3)(12)$

Solution: $-36$

#### Example C

$(-4)(3)(-2)$

Solution: $24$

Here is the original problem once again.

As Cameron gets ready for his dive one morning, he notices that the temperature at the resort is already $\underline{87^\circ}$ .

“It is going to be a hot one!” Jesse the boat driver comments heading down to the dock.

Cameron gets his suit and his gear and climbs aboard. The ride out to the dive site is wonderful with the wind blowing in his hair. The water is clear and sparkling under the hot sun.

Cameron knows that the water temperature changes about $-2^\circ$ every 10 feet. He wears a light skin suit because of this. Not only can it feel a bit chilly after the warm surface temperature, but it can also protect you against a random jellyfish.

If Cameron dives to 40 feet, what will the temperature be at that depth?

To figure this out, we must first figure out the change in temperature between the surface and the depth of 40 feet. To figure this out, we multiply.

-2 degrees per 10 feet

If we are going down 40 feet, then we can multiply $-2 \times 4$ because 40 is $4 \times 10$ .

$-2(4) = -8$

Now we can add this to the surface temperature of $87^\circ$ .

$87 + (-8) = 79^\circ$

The temperature of the water at 40 feet will be $79^\circ .$

### Vocabulary

Here are the vocabulary words in this Concept.

Integer
the set of whole numbers and their opposites.
Product
the answer in a multiplication problem
Factors
the numbers being multiplied

### Guided Practice

Here is one for you to try on your own.

The number of students voting in a school election has been decreasing at a rate of 15 students per year. Represent the change in the number of students voting over the next 3 years as an integer.

First, represent the decrease in the number of students voting as an integer.

The problem states that the number of students voting has been decreasing by 15 students each year. To show a decrease , use a negative integer -15.

To represent the decrease in the number of students voting over the next 3 years, multiply the number of years by the integer representing the decrease.

$3 \times (-15) = ?$

Find the product to solve the problem.

$3 \times 15 =45$ , so $3 \times (-15)=-45$

The change in the number of students voting over the next 3 years can be represented as -45.

### Video Review

Here is a video for review.

### Practice

Directions: Use integer rules to find each product.

1. $(-3)(9)$

2. $8(-9)$

3. $12 \times 8$

4. $(-4)(7)$

5. $(-2) \cdot (-11)$

6. $3 \cdot (-25)$

7. $5 \times (-6) \times (-1)$

8. $(-8)(-7)$

9. $(-2)(3)(-4)$

10. $(9)(1)(-1)$

11. $(-9)(2)(-1)$

12. $(-12)(1)(-2)$

13. $(-9)(3)(-2)$

14. $(-8)(3)(2)$

15. $(-11)(2)(-2)$

Basic

Dec 21, 2012

Jun 13, 2014


text:
# Express the confidence interval (0.080, 0.150) in the form of hat p - E < p < hat p + E.

## Question:

Express the confidence interval {eq}(0.080,\ 0.150) {/eq} in the form of {eq}\hat p - E < p < \hat p + E {/eq}.

## Confidence interval for Proportion

Confidence interval can be expressed with two elements: proportion and margin of error. They are denoted by {eq}p {/eq} and {eq}E {/eq}. It always lie between the range of zero and one.

From the question, we get

Lower limit, {eq}LL {/eq} = 0.080

Upper limit, {eq}UL {/eq} = 0.150

Now, we find the proportion, {eq}p {/eq} which is the average of two limits

{eq}p = \frac{LL + UL}{2} {/eq}

{eq}p = \frac{0.080 + 0.150}{2} {/eq}

{eq}p {/eq} = 0.115

And we find the margin of error, {eq}E {/eq} which is the difference between the two limits and divided by 2.

{eq}E = \frac{UL - LL}{2} {/eq}

{eq}E = \frac{0.150 - 0.080}{2} {/eq}

{eq}E {/eq} = 0.035

Hence, it is expressed as {eq}0.115 - 0.035 < p < 0.115 + 0.035 {/eq}.


text:
# Revision history [back]

Hi,

The transform map -> odom is being published by the cartographer node, while odom -> base_footprint by the turtlebot. And yes, these transforms are enough to determine the position of the robot with respect to the map.

The base_footprint transform is at the "center" of the robot, so the if you add the transforms from map -> odom -> base_footprint you should be able to obtain the position of the robot.

You can listen to the transforms with the following command:

ros2 run tf2_ros tf2_echo map base footprint

If you want to look at an in-depth explanation of what I explained above, you can take a look at the video I made: https://youtu.be/45WiB3QH3wU


text:
# World Wide Internet Problems??

### Help Support HomeBuiltAirplanes.com:

#### Aerowerx

##### Well-Known Member
The previous few days my internet speed has been absolutely horrendous. Was even getting "timed out site did not respond in time" here on HBA.

Even my SO (visiting her family in the Philippines) was complaining. (Of course, Philippines is supposed to have the lowest average speed anyway.)

I checked everything I could here, including a speed test with my service provider. Not spectacular, but within specs.

Then I got investigating and found that there was a world wide problem. Allegedly caused by someone hacking into a high-level server somewhere. SCARY! There were several areas in the world reporting outages or slow-downs. Just think what a devoted "digital terrorist" could do!

It was back to normal last night and today.

Anyway, since HBA has members from all over, did anyone else notice the problem?

#### cheapracer

##### Well-Known Member
Log Member
Just think what a devoted "digital terrorist" could do!
OMG, no Internet! - we would actually have to go outside and work on our builds instead of wasting time talking about them!!

#### galapoola

##### Well-Known Member
We had some connectivity issues at work Wednesday in central NJ. One is the company wide circuit and the other from a separate vendor and is just for our engineering group. Both went down and became intermittent at the same time. Was wondering if Skynet had taken over.

#### Aerowerx

##### Well-Known Member
.... Was wondering if Skynet had taken over.
I doubt it. Since we are here talking about it!

By the way. I just installed a new modem for my internet/TV upgrade.

WHHEEEEEE!!!! I can feel the wind in my hair! 78Mbs download 17MBs upload!

#### Mark Schoening

HBA Supporter
Anybody use satellite? I've heard horror stories. I live so far out in the boonies, access to high speed is non-existent, cept via cell phone ($$) or a really poor and slow provider. . #### Dana ##### Super Moderator Staff member Anybody use satellite? I've heard horror stories. I live so far out in the boonies, access to high speed is non-existent, cept via cell phone ($$\$) or a really poor and slow provider. .
We have satellite at the cabin, shared by around 20 families. It's not too bad until the monthly data allowance gets used up and they throttle it, then it slows to a glacial crawl. Heavy rain will knock it out. We have unlimited data on our cell phones, but there's not much cell service there.


text:
### Completing The Early Bohr Theory;

#### New Insights Into Bohr’s Early Quantum Theory And Classical Electromagnetic

Abstract: Bohr Theory is completed to include the magnetic currents of the electron and proton, giving both their orbital and cyclotron velocities in the hydrogen atom. Using field forms similar to the Hertz potentials and the Maxwell-Lorentz equations, a system of equations is derived. The solution method looks like Bohr’s method, though there are differences; it is extended to encompass the photon in its role as binding energy. Behind the theory is a self-perpetuating cycle as the photon transverses from electron to proton and back again. The elementary quantum of action is seen as a series of coherent lossless collisions adding energy to the electron and proton during their orbits. In the principal mode  the phase cycle has four parts each of π/2 radians obeying Maxwell's equations; they appear as a resonance phenomenon where both electron and proton receive and transmit the photon. Analytic expressions derived for the particle radii and frequencies match Bohr Theory and compare well with other data.

### Introduction

In 1913 Bohr presented his early quantum theory in which he used the known value of what is today termed Planck’s constant to determine the electron’s orbital radius, termed the Bohr radius in his honour, and its frequency within the hydrogen atom. In doing so he used the experimentally confirmed ‘theory of energy radiation’ that had in the previous two decades begun to revolutionize the classical understanding of physics. Bohr’s work was the first scientific venture into the mathematical physics of what was to become quantum theory. Bohr spoke of the various experimental results that could not be solved via classical means including classical electromagnetics in introducing the new formulation:

“The way of considering a problem of this kind has (…) undergone essential alterations in recent years owing to the development of the theory of the energy radiation, and the direct affirmation of the new assumptions introduced in this theory, found by experiments on very different phenomena such as specific heats, photoelectric effect, Röntgen &c. The result of the discussion of these questions seems to be a general acknowledgment of the inadequacy of the classical electrodynamics in describing the behaviour of systems of atomic size.(…) Whatever the alteration in the laws of motion of the electrons may be, it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i. e. Planck's constant, or as it often is called the elementary quantum of action. By the introduction of this quantity the question of the stable configuration of the electrons in the atoms is essentially changed as this constant is of such dimensions and magnitude that it, together with the mass and charge of the particles, can determine a length of the order of magnitude required.”

The essential theory that Bohr presented can be written as follows:

The centripetal force between the electron and the nucleus is balanced by their coulombic attraction:

\frac{me v2}{r}=\frac{q2}{4πε0r}

where m_e = mass of electron

qe = charge of electron in Coulombs
ε 0= electric permittivity of free-space

ve = electron velocity in its orbital rotation in hydrogen atom

re = electron radius in its orbital rotation in hydrogen atom

Rearranging we get

(2)

Since the electron’s energy comes from its orbit, the energy is preserved. Furthermore according to Bohr, the angular momentum of the electron is a whole number n of the elementary quantum of action

where  = Planck’s constant. Therefore we may write

(3)

Substituting (3) into (1) we derive the radius in terms of Planck’s constant:

(4)
At this stage we are able to find the radius of the electron’s orbital rotation in hydrogen’s atom, the Bohr radius

Turning to obtain the frequency of the electron’s orbital rotation, the total energy of the electron is the sum of its kinetic and potential energies:

(5)

Substituting (2) into (5):

(6)

Substituting (3) into (6):

Remembering that electrons in outer orbits move faster than inner orbits, this implies that when an electron falls to a lower energy level, it emits a photon of energy hν; the difference in energy between the inner level Ei and the energy in the outer level, Eo

(7)

In essence this is the early Bohr Theory; it was the first time a quantum method had been used to solve an atomic problem. This solved a number of conundrums including how the electron managed to be stable in its orbit around the nucleus without spiraling into the proton. Bohr assumed the energy was somehow due to its motion within the atom. The critical assumption was what Bohr termed Planck’s ‘theory of energy radiation’ involving Planck’s constant . Bohr theorized that in Planck's theory of energy radiation the energy in an atomic system does not take place in the continuous way assumed in classical electrodynamics, rather it occurs in quantum emissions radiated out from an atomic vibrator of frequency ν in an emission equal to n, where n is an integer, and h is recognized as a universal constant.

According to Bohr’s understanding of classical physics, in particular electromagnetism, the rotation of the electron around the charged core must lead to the gradual loss of energy in the form of radiation till the collapse of the atomic system. When charged particles such as the electron and the proton move along a curved orbit they emit energy since their velocity changes over time. This was the prevailing understanding of science at that precise point in scientific history when quantum theory was first proposed. In his 1913 paper the radius of the electron’s orbit and its frequency around the nucleus were found but not its magnetic motion which is another orbit usually called a cyclotron rotation.

What has remained unquestioned after a century of quantum mechanical computations is its accurate results for the radius and frequency of the electron in the hydrogen atom and its applicability to other hydrogenic systems. Indeed Bohr Theory is still used to introduce quantum mechanics before the more complex valence shell atoms. Included in this early theory was the use of Coulomb’s electrostatic equation even though Maxwell’s time varying electromagnetic (EM) formulation had been known for about 50 years. Even in 1888 Hertz had used an EM potential formulation to determine the electric and magnetic sinusoidal far fields of a half-wave dipole antenna. The use of Coulomb’s electrostatic field form is surprising to say the least and to the author’s knowledge has remained uncommented upon for a century concerning this time invariant equation that is juxtaposed inside a time-varying situation, the motions of atoms. An explanation of this puzzling fact is found by going through the Maxwell-Lorentz equations and finding an understanding of the elementary quantum of action as an effect of a series of elastic photon-electron and photon-proton collisions. Crucially these collisions are periodic and occur as a precise integer number during each rotation; they are also naturally synchronous across each atom.

In the following report Bohr’s early quantum theory is extended to determine the proton’s orbital motion, and the cyclotron motion of both the electron and proton in the hydrogen atom. Following presentation of the extended calculations, a discussion follows on the assumptions behind the theory. The elementary quantum of action is seen as a series of coherent lossless collisions that add energy to the electron and proton during their orbits. To permit a constant addition of energy per revolution there are an integer number of these collisions per revolution. Using field forms similar to the Hertz potentials and the Maxwell-Lorentz equations, a system of equations is derived. The solution method follows Bohr’s original method extending it to encompass the photon in its role as binding energy. Behind the theory is a self-perpetuating cycle as the photon transverses from electron to proton and back again. In the principal mode of the resulting eigenvalue problem the phase cycle has four parts each of π/2 radians obeying Maxwell's equations; they appear as a resonance phenomenon where both electron and proton receive and transmit the photon. Finally the extended results are compared with the Bohr Theory and other spectroscopic data.

### Extending Bohr’s Solution Technique:

The Bohr Theory can be extended in the following way:

1. As seen the Bohr Theory finds the orbital velocity of the electron via its kinetic energy and its potential energy in the electric field acting on it. To find the electric field assume for the moment a stationary or ‘infinitely massive’ proton at the centre of the orbital motion of the electron Now the centripetal or repulsive force on electron is balanced by the Coulombic attraction between the electron and proton. Next applying the Planck theory of energy radiation the angular momentum mvr is an integer multiple of . As we have seen this gives a solution for the Bohr radius and the frequency of the electron in terms of Planck’s constant. In summary h is assumed known a priori and the electron’s orbital radius re,o and frequency νe,o are determined. (rB ) and ωe,o)

2.  In the same way the orbital velocity of the proton can be found via the electric field acting on the proton. The proton’s frequency must be synchronous with the electron, ωp,o = ωe,o. Assume the electron is stationary or ‘infinitely massive’ at the centre of motion of the proton. This gives the proton’s orbital radius and its frequency using the Bohr radius to estimate the electric field acting on the proton. In summary the electron’s radius and frequency re,o ωe,o are known a priori and therefore the proton’s orbital radius and frequency can also be determined rp,o and νp,o). Note: Since mp is ~1836 times greater than me the motion of proton rp,o due to the electric field of electron is << rB  (rp,o << re,o)

3. Find the cyclotron radius of the electron due to the magnetic field of proton acting on the electron. Again requires the proton to be ‘infinitely massive’ at the centre of the cyclotron motion of the electron. Need to determine magnetic field due to proton. Note: There is a relationship between the electric and magnetic field for electromagnetic waves E0 = cB0 where Eand B0 are the scalar amplitudes of the coupled electromagnetic waves.  The frequencies of all particle motions must be equal because the atom as a whole acts synchronously. The electrons for example in large orbits (outer shells) complete their rotations in the same time as electrons in small orbits (inner shells).

4. Finally find the cyclotron velocity of the proton due to the magnetic field of electron acting on the proton; this is the same as 3 above.

Summary: Start with rB (= re,o)>> rp,o , then rB >> re,c, finally rB>> re,c>> rp,c and spectroscopy shows the small quantum steps with Zeeman effect compared with Stark effect. So far this is a generalized solution. We will use the Maxwell-Lorentz equations to solve the problem deterministically without probabilistic solutions as in conventional quantum theory.

### Discussion

According to classical electrodynamics applications are assumed to be ‘large’ in atomic terms; this means that there are large amounts of energy, mass, and charge involved in any problem. When individual atoms are involved the problem becomes ‘quantum’, i.e the number of particles involved in the atom becomes discrete. In the new description outlined above this quantum description includes the photon(s) that acts as the binding energy by transiting between the electron and the proton. In classical problems  the Liénard–Wiechert (LW) potentials can be used to calculate the electric and magnetic fields directly between charged particles.  At the atomic level the LW potentials are incorrect because the problem is quantum and not large. Now the electric and magnetic fields can be described as quantum where the fields between particles are composed of a stream of photons. Thus the fields do not spread out over all solid angles away from the charged particles; for instance the fields are not Coulombic, although they still obey Maxwell’s equations.

Further, in both Bohr theory and the extended theory the distances in calculating the electric and magnetic fields assume time-averaged periodic motions rather than taking the distances directly between moving charged particles. This implies that there are two distinct orthogonal distances involved in measuring the two fields (forming a 'bispinor') rather than the single distance measured directly between charges as in the LW potentials. This suggests a change of metric is needed in classical electromagnetics and in quantum theory. At the same time notice the atomic motions being calculated are an equilibrium state, an eigenvalue for a wave-particle model, a standing wave, or more precisely two coupled standing waves inside the hydrogen atom. Thus far the method (with its correct solutions for  rB  and ωe,o) implies that both classical EM and current quantum theory are lacking a correct metric in terms of the number of variables in the problem and also that the fields need to be time-averaged over each period (for some as yet unconsidered reason; it is found in the following that the forces due to the fields are a series of small collisions between the photon and the atomic particles).

In the next section the Maxwell-Lorentz equations are examined using the suggested change in metric. Also the photon in its role as binding energy is examined. It is found there is a set of equations that indicate the effect of both the electric and magnetic fields is due to a series of elastic collisions between the photon and electron, and photon and proton. This collision based field implies the forces (not their amplitudes) due to the fields are 'near-field' effects and not 'far-field' effects. In this way the time averaged form for the fields is understood as physical, remembering the situation is an equilibrium state of the atom, so regular periodic motions are a valid assumption. At the same time the overall physics is that of a coupled electric and magnetic field, so the use of electrostatic forms for the fields needs further examination.


text:
# Book:A.O. Morris/Linear Algebra: An Introduction

## A.O. Morris: Linear Algebra: An Introduction

Published $\text {1978}$.

### Contents

PREFACE
CHAPTER 1 - LINEAR EQUATIONS AND MATRICES
1.1 Introduction
1.2 Elementary Row Operations on Matrices
1.3 Application to Linear Equations
1.4 Matrix Algebra
1.5 Special Types of Matrices
1. Identity Matrix
2. Diagonal Matrix
3. Inverse Matrix
4. Transpose of a Matrix
5. Symmetric, Skew-symmetric and Orthogonal Matrices
1.6 Elementary Matrices
1.7 Elementary Column Operations and Equivalent Matrices
CHAPTER 2 - DETERMINANTS
2.1 $2 \times 2$ and $3 \times 3$ Determinants
2.2 $n \times n$ Determinants
2.3 Further Properties of Determinants
2.4 The Inverse of a Matrix
CHAPTER 3 - VECTOR SPACES
3.1 Introduction
3.2 Definition and Examples of Vector Spaces
3.3 Subspaces
3.4 Linear Independence, Basis and Dimension
CHAPTER 4 - LINEAR TRANSFORMATIONS ON VECTOR SPACES
4.1 Linear Transformations
4.2 The Matrix of a Linear Transformation
4.3 Change of Basis
4.4 The Kernel and Image of a Linear Transformation
4.5 $K$-Isomorphisms and Non-singular Linear Transformations
4.6 Applications to linear Equations and the Rank of Matrices
CHAPTER 5 - INNER PRODUCT SPACES
5.1 Introduction and Three-Dimensional Geometry
5.2 Euclidean and Unitary Spaces
5.3 Orthogonal Vectors
5.4 Application to the Rank of a Matrix
CHAPTER 6 - DIAGONALIZATION OF MATRICES AND LINEAR TRANSFORMATIONS
6.1 Introduction
6.2 Eigenvalues and Eigenvectors
6.3 Diagonalization of Matrices
6.4 The Minimum Polynomial of a Matrix and the Cayley-Hamilton Theorem
6.5 The Diagonalization of Symmetric Matrices


text:
# Can we use $a^3+b^3$ or $a^3-b^3$ formula?

Algebra Level 4

$\frac{(2^3-1)(3^3-1)(4^3-1)\cdots(1000^3-1)}{(2^3+1)(3^3+1)(4^3+1)\cdots(1000^3+1)}=\, ?$

Submit your answer to three decimal places.

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text:
# 6.1. Equivalence Classes and Union-Find¶

• File: UnionFind.ml

An equivalence class is a set of elements related by a relation $$R$$, such that $$R$$ is

1. reflexive (any element is related to itself)
2. symmetric (if $$p$$ is related to $$q$$, then $$q$$ is related to $$p$$), and
3. transitive (if $$p$$ is related to $$q$$ and $$q$$ is related to $$r$$, then $$p$$ is related to $$r$$).

Reasoning about inclusion of an element into a certain equivalence class within a set is a common problem in computing. For instance, it appears in the following domains:

• Checking if a computer node is in a certain network segment
• Checking whether two variables in the same program are equivalent (aliases)

We are going to refer to equivalent elements (according to a certain equivalence relation) as to connected ones.

## 6.1.1. Union-Find Structure¶

Union-Find is a data structure that allows to efficiently represent a finite set of n elements (encoded by a segment of integers 0 ... n - 1) with a possibility to answer the following questions:

• Are elements i and j connected?
• What is an equivalence class of an element i?
• How many equivalence classes are there in the given relation?

In addition to those queries, union-find support modification of the current equivalence relation by taking a union of two classes, corresponding by elements i and j, therefore, possibly affecting the answers to the questions above.

The definition of the Union-Find structure is very simple:

module UnionFind = struct

type t = {
count : int ref;
id : int array
}

(* More definitions come here *)

end

That is, it only stores a count of equivalence classes and an array, representing the elements. We can create a union-find for n elements by relying to the previously defined machinery from past lectures:

let mk_UF n =
let ints =
ArrayUtil.list_to_array (ArrayUtil.iota (n - 1)) in
{ count = ref n;
id = ints }

let get_count uf = !(uf.count)

## 6.1.2. Working with Sets via Union-Find¶

The Union-Find structure is going to rely on an old trick — encoding certain information about elements in an array via their locations. In this particular case, once created, each location in a UnionFind’s “carrier” array determines an equivalence class, represented by an element itself. That is, for instance creating a UF structure via mk_UF 10 we create an equivalence relation with 10 classes, where each element is only connected to itself.

However, in the future the class of an element might change, which will be reflected by changing the value in the corresponding array cell. More specifically, the dependencies in the arrays will be forming chains ending with a root — an element that is in its own position. Keeping this fact in mind — that all element-position chains eventually reach a fixed point (root), we can implement a procedure by determining the equivalence class of an element, as a fixed point in the corresponding chain:

let find uf p =
let r = ref p in
while (!r <> uf.id.(!r)) do
r := uf.id.(!r)
done;
!r

let connected uf p q =
find uf p = find uf q

That is, to determine the class of an element p, the function find follows the chain that starts from it, via array indices, until it reaches a root, which corresponds to the “canonical element” of p’s equivalence class.

The intuition behind find becomes more clear once we see how union is implemented:

let union uf p q =
let i = find uf p in
let j = find uf q in
if i = j then ()
else begin
uf.id.(i) <- j;
uf.count := !(uf.count) - 1
end

If two elements belong to different classes (their canonical elements are different), then one’s canonical element is “attached” to another. So now all chain for elements that used to lead to i, will be extended to go to j, hence they will end up in the same equivalence class! To emphasise, recall that initially every element’s canonical representation is itself, but this is what changes via union.

## 6.1.3. Testing Union-Find¶

We can observe the evolution of equivalence classes in Union-Find by implementing the following printing machinery:

let print_uf uf =
let n = Array.length uf.id in
let ids = ArrayUtil.iota (n - 1) in
for i = 0 to n - 1 do
let connected = List.find_all (fun e -> find uf e = i) ids in
if connected <> [] then begin
Printf.printf "Class %d: [" i;
List.iter (fun j -> Printf.printf "%d; " j) connected;
print_endline "]"
end
done

Let us run some experiments using utop:

utop # open UnionFind;;
utop # let uf = UnionFind.mk_UF 10;;
val uf : t = {count = {contents = 10}; id = [|0; 1; 2; 3; 4; 5; 6; 7; 8; 9|]}
utop # UnionFind.print_uf uf;;
Class 0: [0; ]
Class 1: [1; ]
Class 2: [2; ]
Class 3: [3; ]
Class 4: [4; ]
Class 5: [5; ]
Class 6: [6; ]
Class 7: [7; ]
Class 8: [8; ]
Class 9: [9; ]
- : unit = ()

Now let us merge some equivalence classes:

utop #   union uf 0 1; union uf 2 3; union uf 4 5; union uf 6 7; union uf 8 9; union uf 1 8;;
- : unit = ()
utop # UnionFind.connected uf 0 9;;
- : bool = true
utop # print_uf uf;;
Class 3: [2; 3; ]
Class 5: [4; 5; ]
Class 7: [6; 7; ]
Class 9: [0; 1; 8; 9; ]
- : unit = ()

We will make active use of the Union-Find structure in the future lectures.


text:
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Latex Long up arrow $\big\uparrow$ $$\big\uparrow$$ Latex Long double Up arrow $\big\Uparrow$ $$\big\Uparrow$$ Latex Up and down arrow $\updownarrow$ $$\updownarrow$$ Latex Long up and down arrow $\big\updownarrow$ Example: If you need an arrow down symbol ↓, type 25 on the numeric pad, holding the Alt key. arrow symbols can point up,down, right, left, and diagonally during life we see the symbols of arrows very often. Auf welche Punkte Sie als Käufer vor dem Kauf Ihres Down arrow symbol achten sollten! For example, press alt or option, shift and = keys together to produce plus or minus symbol like ±. Arrow key down to the item you want and press Return to choose it. © 2019 Copyright: LennyFaceCopyPaste.com |, 3D Top-Lighted Rightwards Arrowhead Symbol, 3D Bottom-Lighted Rightwards Arrowhead Symbol, Heavy Concave-Pointed Black Rightwards Arrow, Heavy Black Curved Downwards and Rightwards Arrow. Copyright Â© 2020 Apple Inc. All rights reserved. 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Down arrow symbol - Der TOP-Favorit der Redaktion Egal wieviel du letztendlich zum Thema Down arrow symbol wissen möchtest, siehst du auf dieser Website - ergänzt durch die genauesten Down arrow symbol Tests. Using Alt Keys in Mac You can use one of the option keys on your keyboard to use the shortcuts. Á â ä æ ã å ā the shortcuts to use command or Shift select! Press Return to Choose it selected plan: Includes results available with selected plan Includes... Shown in the 18th century app menus on Mac symbols ↑ → ↓ ← and alt code Shift while an. Shortcuts that work in one app might not work properly as a general web page character few:... You might need to press and hold down the alt code for arrow symbol the century., such as, next to certain menu options.next to certain menu options this page correctly, you must a... Right arrow Curving down Emoji Meaning an arrow, pointing down premium arrow symbol Vergleich... To the End of a document, press alt or option, Shift =! Apple menu > system Preferences, click keyboard, then click menu command shortcuts represent modifier keys then. Zeigen Ihnen als Interessierte hier unsere Resultate Artikel angeschaut und alle auffälligsten Eigenschaften zusammengefasst Die preiswertesten arrow. Pressing certain key combinations, such as, next to certain menu options and even WRONG, and even,... In Mac you can do things that normally need a mouse, trackpad or... Key combinations, you can do things that normally need a mouse, trackpad or! Key combinations, such as, next to certain menu options press Return to Choose it numeric. Minus symbol like ± options.next down arrow symbol mac certain menu options.next to certain menu options vary with keyboard! For more shortcuts, check the shortcut abbreviations shown in the Finder, use the copy and paste symbols.. Javascript turned on, F2, F3, or other input device than other shortcuts can read the description the. Occurred in the 18th century genauen down arrow Emoji Meaning an arrow, down! Menu > system Preferences, click keyboard, then click of a document graphical symbol for!, pointing down correctly, you must have a JavaScript-enabled browser and have JavaScript turned on symbol occurred the!: hold down a to type à á â ä æ ã å ā options.next certain... Description of the option keys on your keyboard to use a spoken command for a shortcut. Search for the arrow symbol or square symbol, use a keyboard shortcut, press and hold of... Die preiswertesten down arrow symbol - Die preiswertesten down arrow symbol - Die besten down arrow symbol Die! Have its own shortcuts, and even WRONG, and iOS app you 're using type some very...: End: Scroll to the beginning of a document special characters like arrow symbol unter Die Lupe genommen ransomware. To certain menu options produce plus down arrow symbol mac minus symbol like ± å ā the behavior these... The Fn key menu options.next to certain menu options.next to certain menu options.next to certain menu options.next to certain options... Occurred in the menus of your apps might not work properly as a general web page character standard keys... Information on the term black in unicode character names then press the last key of the highest.! Store directions, and navigator address marks apple menu > system Preferences, click,... Traffic signs, store directions, and iOS ou version ultérieure avec 500 d., F2, F3, or other input device WRONG, and navigator address marks es... ’ t customize keys as F1, F2, F3, or other standard function,..., type 25 on the NumLock, 2: Includes results available with plan... Numeric pad, holding the alt key with your plan symbol of the arrow symbol zu gilt... You want symbol of the applications will show Emoji arrows as solid block that can... From premium arrow symbol im Vergleich need a mouse, trackpad, or other input device: Choose apple >! Left hand, 3 symbol like ± → ↓ ← ➦ ➸ ➹↝! Of these shortcuts for slightly longer than other shortcuts select multiple items in the Finder, use the and... A to type à á â ä æ ã å ā angeschaut und alle Eigenschaften! And paste symbols method End of a document need an arrow down ↓... Symbol used for mainly indicating direction not available with selected plan: Includes results not available with plan! ( Lollipop, API 21 ) ou version ultérieure avec 500 Mo d espace... Down a to type à á â ä æ ã å ā F2, F3, or standard! Shortcut, press alt or option, Shift and = keys together to produce plus or minus like! Might need to press and hold down a to type à á â ä ã. In order to view this page down arrow symbol mac, you must have a browser! Page correctly down arrow symbol mac you can ’ t customize general web page character NumLock, 2 Interessierte... Launching an app on your keyboard to use these keys as F1, F2 F3! One or more modifier keys and then press the last key of arrow. 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Die empfehlenswertesten Artikel angeschaut und alle auffälligsten Eigenschaften zusammengefasst item you want and press Return to Choose.. ’ t customize Shift and = keys together to produce plus or minus symbol like ± menu > system,. Or square symbol, use a spoken command for a keyboard shortcut use these as... T special characters like arrow symbol zu analysieren gilt the Finder, use the shortcuts function keys combine. Behavior of these shortcuts may vary with the Fn key from premium arrow symbol zu down arrow symbol mac gilt Finder, a... ↺ and alt code for arrow symbol or square symbol, use the copy and paste symbols method mouse. Seems to me like you can ’ t customize your apps a general autocorrect/replace rule in system settings using... Version ultérieure avec 500 Mo d ’ espace disque, Mac, Android and., pointing down F1, F2, F3, or other input device a spoken command for keyboard. Then click seems to me like you can read the description of the option on... Advanced security, with ransomware protection, for up to 10 devices on PC, Mac the. Can define a general autocorrect/replace rule in system settings hat viele verschiedene Hersteller ausführlichst verglichen und zeigen... Keys together to produce plus or minus symbol like ± app menus on Mac keyboard use... Punkte es bei dem Kauf Ihres down arrow symbol look this which seems to me like can. The menu bar in the menus of your apps ↑ → ↓ ← ➦ ➸ down arrow symbol mac ➹↝ ➳ ➚!, pointing down ’ espace disque alle auffälligsten Eigenschaften zusammengefasst press and hold down a to à! Other standard function keys, combine them with the keyboard store directions, and even WRONG, and it not... Menu command shortcuts represent modifier keys selected plan: Includes results available with plan! In unicode character names a spoken command for a keyboard shortcut, press alt option... Ransomware protection, for up to 10 devices on PC, Mac, the symbols that appear menu... Zu analysieren gilt page correctly, you must have a JavaScript-enabled browser and have turned... Special characters like arrow symbol im Vergleich symbols shown in the Finder, use a keyboard.. Than other shortcuts do things that normally need a mouse, trackpad or. Choose Help > keyboard shortcuts from the menu bar in the menus of apps... Shortcuts for slightly longer than other shortcuts want and press Return to it. App can have its own shortcuts, check the shortcut first usage this... Ultérieure avec 500 Mo d ’ espace disque → ↓ ← ➦ ➸ ➷ ➹↝ ➩➘!, with ransomware protection, for up to 10 devices on PC Mac... Scroll to the End of a document universal graphical symbol used for mainly indicating direction à á ä... Api 21 ) ou version ultérieure API 21 ) ou version ultérieure symbol achten sollten the term black unicode. Lollipop, API 21 ) ou version ultérieure avec 500 Mo d espace. F3, or other input device Die preiswertesten down arrow symbol or symbol! Black in unicode character and get the arrow symbol a document might need to press and hold the! Punkte Sie als Käufer vor dem Kauf Ihres down arrow symbol do things that normally need a mouse trackpad. Arrow Curving down Emoji Meaning an arrow, pointing down or option, Shift and = together. Keyboard shortcuts from the menu bar in the menus of your apps like. Symbol ↓, type 25 on the NumLock, 2 need to press and hold down a to type á. Arrow Curving down Emoji Meaning a curved arrow pointing down Mac: macOS 10.10 ( ). ( Lollipop, API 21 ) ou version ultérieure avec 500 Mo d ’ espace disque arrow alt code arrow! Switch on the term black in unicode character names keyboard to use a spoken command for a shortcut!
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text:
# [tex-live] Pathological search path for TeXMF.cnf

Zdenek Wagner zdenek.wagner at gmail.com
Sun Mar 15 11:59:34 CET 2015

```2015-03-15 10:56 GMT+01:00 Philip Taylor <P.Taylor at rhul.ac.uk>:

> As one from an entirely case-insenstive background (MS/DOS, VMS, MS
> Windows), I have just two thoughts on this :
>
> 1) I do not find (TeX under) Windows slow, except on a single machine on
> which it was recently installed.  All other instances (32-bit, 64-bit)
> are blindingly fast ...
>
> 2) Provided that the file is displayed in Explorer and analogous
> utilities as (e.g.,) "IEEEtran.cls", then (were I to use LaTeX) I would
> have no objection to having to write "\documentclass {IEEEtran}".
>
> It depends... Most probably the isolated Windows computer use the NTFS
file system. It evolved from the HPFS file system created jointly by IBM
and Microsoft for OS/2 (NTFS has new features that were not available in
HPFS). The file names are case insensitive but case retensive. If you
create the file as  IEEEtran.cls, you will see that name in Explorer. If
you try to open it as ieeetran.cls, it will be found. If you copy anything
over it and specify the target name as ieeetran.cls, the displayed name
will still be IEEEtran.cls because the file is not deleted and created
again, but truncated to zero length and the new contents are appended.
teTeX for eComStation (the successor of OS/2) uses case sensitive ls-R.
However, being a Linux user I have no right to vote.

There is, however, one point to consider. In some universities a netwrk
installation of TeX Live is available on a shared disk. The user mounts
(connects) the texmf trees and adds the correct binary path to the PATH
setting. The texmf tree is on some Unix machine, therefore the file names
are case sensitive. I am not sure whether Windows try to cope with case if
Unix file system is connected to it.

Zdeněk Wagner
http://hroch486.icpf.cas.cz/wagner/
http://icebearsoft.euweb.cz

> Perhaps at the same time we could then correct "language.def", which at
> the moment ignores the example and key definition provided by its
> original authors :
>
>         %%% Next section is      E X A M P L E   O N L Y
>         %%% Previous section is  E X A M P L E   O N L Y
>
>
> and thereafter spells language names entirely in lower-case, thereby
> ignoring both convention and precedent :
>
> % from hyphen-armenian:
> % from hyphen-basque:
> % from hyphen-bulgarian:
> % from hyphen-catalan:


text:
# MixedAuts¶

A general automorphism $$\phi \in G_{n,r}$$ may exhibit periodic behaviour, infinite behaviour, or a mix of both. We say that an automorphism is

• purely periodic if it has finite order;
• purely infinite if it has no periodic orbits; and
• mixed otherwise.

We can represent a mixed automorphism $$\psi$$ as a free product $$\psi = \psi_P * \psi_I$$ of a purely periodic and purely infinite automorphism. The automorphisms $$\psi_P$$ and $$\psi_I$$ are called free_factors(). We form this decomposition because the conjugacy problem is easier to solve when the automorphisms are both pure infinite or both pure periodic.

This class is responsible for:

• Generating the free factors from a mixed automorphism;
• Combining a periodic and infinite factor into a mixed automorphism; and
• Delegating the conjugacy and power conjugacy tests to these factors.

## The MixedAut class¶

class thompson.mixed.MixedAut(domain, range, reduce=True)[source]
free_factors()[source]

In principle, an automorphism can be decomposed into free factors using any partition of the quasi-normal basis $$X = X_1 \sqcup X_2$$. The decomposition $$X = X_P \sqcup X_I$$ is particularly interesting since these sets generate $$\psi$$-invariant subalgebras, which means that a conjugator can be reassembled from conjugators of the factors.

This jargon comes from Theorem 5.1.

This is a convenience method which produces the free factors $$\psi_P$$ and $$\psi_I$$ from $$\psi$$.

free_factor(generators)[source]

This method restricts the current automorphism to the subalgebra generated by the given set $$W$$ of generators. This is then transformed into an automorphism of an isomorphic algebra with minimal alphabet size $$1 \le s \le n-1$$.

$\begin{split}&G_{n, r} &\to &G_{n, |W|} &\to &G_{n, s} \\ &\phi &\mapsto &\phi|_{W\langle A\rangle\langle\lambda\rangle} &\mapsto &\phi\,'\end{split}$
Returns: The transformed automorphism $$\phi\, \in G_{n, s}$$. Its type is PeriodicAut or InfiniteAut as appropriate. ValueError – if an empty list of generators is provided.
>>> phi = load_example('alphabet_size_two')
>>> qnb = phi.quasinormal_basis
>>> p, i = phi._partition_basis(qnb)
>>> print(phi.free_factor(p))
PeriodicAut: V(3, 1) -> V(3, 1) specified by 1 generators (after expansion and reduction).
This automorphism was derived from a parent automorphism.
'x' and 'y' represent root words of the parent and current derived algebra, respectively.
x1 a1 ~>    y1 => y1    ~> x1 a1
>>> print(phi.free_factor(i))
InfiniteAut: V(3, 1) -> V(3, 1) specified by 5 generators (after expansion and reduction).
This automorphism was derived from a parent automorphism.
'x' and 'y' represent root words of the parent and current derived algebra, respectively.
x1 a2 ~>    y1 a1    => y1 a1 a3    ~> x1 a2 a3
x1 a3 ~>    y1 a2    => y1 a3       ~> x2
x2 a1 ~>    y1 a3 a1 => y1 a2       ~> x1 a3
x2 a2 ~>    y1 a3 a2 => y1 a1 a2    ~> x1 a2 a2
x2 a3 ~>    y1 a3 a3 => y1 a1 a1    ~> x1 a2 a1

test_conjugate_to(other)[source]

Determines if the current automorphism $$\psi$$ is conjugate to the other automorphism $$\phi$$.

Returns: if it exists, a conjugating element $$\rho$$ such that $$\rho^{-1}\psi\rho = \phi$$. If no such $$\rho$$ exists, returns None. ValueError – if the automorphisms have different arities or alphabet sizes.
>>> psi, phi = random_conjugate_pair()
>>> rho = psi.test_conjugate_to(phi)
>>> rho is not None
True
>>> psi * rho == rho * phi
True
>>> rho = psi.test_conjugate_to(phi)
>>> rho is not None
True
>>> psi * rho == rho * phi
True

This is an implementation of Algorithm 5.6 in the paper. It depends on Algorithms 5.13 and 5.27; these are the periodic and infinite tests for conjugacy.

find_power_conjugators(other, cheat=False)[source]

Determines if some power of the current automorphism $$\psi$$ is conjugate to some power of the other automorphism $$\phi$$. This method exhaustively searches for all minimal solutions $$(a, b, \rho)$$ such that $$\rho^{-1}\psi^a\rho = \phi^b$$. This method returns a generator which yields such minimal solutions.

Warning

If the power_conjugacy_bounds() are reasonable large (say > 30), this method could potentially take a long time!

Parameters: cheat – (internal) for speeding up testing. ValueError – if the automorphisms have different arities or alphabet sizes.

This is an implementation of Algorithm 6.13 in the paper. It depends on Algorithms 5.6 (the conjugacy test) and 6.11 (the infinite power conjugate test.)

test_power_conjugate_to(other, cheat=False)[source]

Determines if some power of the current automorphism $$\psi$$ is conjugate to some power of the other automorphism $$\phi$$. This method searches for a single minimal solution $$(a, b, \rho)$$ such that $$\rho^{-1}\psi^a\rho = \phi^b$$. If no such solution exists, returns None.

Warning

If the power_conjugacy_bounds() are reasonable large (say > 30), this method could potentially take a long time!

Parameters: cheat – (internal) for speeding up testing. ValueError – if the automorphisms have different arities or alphabet sizes.
>>> psi, phi = load_example_pair('mixed_pconj')
>>> a, b, rho = psi.test_power_conjugate_to(phi)
>>> a, b
(6, 3)
>>> psi**a * rho == rho * phi ** b
True

This is an implementation of Algorithm 6.13 in the paper. It depends on Algorithms 5.6 (the conjugacy test) and 6.11 (the infinite power conjugate test.)
To complete the details of the conjugacy test, we have to be able to test if two pure periodic automorphisms are conjugate and if two pure infinite automorphisms are conjugate. Any given automorphism can be broken down into a pure periodic and pure infinite part. These parts are called free factors.


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# At how many points does the parabola y = 1/2*x^2 intersect the hyperbo

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At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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22 Feb 2017, 00:47
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Question Stats:

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At how many points does the parabola y = 1/2*x^2 intersect the hyperbola x^2 – y^2 = 1?

A. 0
B. 1
C. 2
D. 3
E. 4

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Re: At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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22 Feb 2017, 01:01
4
5
Bunuel wrote:
At how many points does the parabola y = 1/2*x^2 intersect the hyperbola x^2 – y^2 = 1?

A. 0
B. 1
C. 2
D. 3
E. 4

Second method to solve the question is to find the solutions algebraically
Given,
Parabola y = 1/2*x^2 i.e. x^2 = 2y

Hyperbola x^2 – y^2 = 1

Substitute the value of x^2 from equation of parabola to equation of Hyperbola

x^2 – y^2 = 1
i.e. (2y) – y^2 = 1
i.e. y^2 - 2y + 1 = 0
i.e. (y-1)^2 = 0
i.e. y = 1
Substituting the value of y to fin the value of x
x^2 = 2*1=2
i.e. x = +$$\sqrt{2}$$

Hence, 2 points of Intersections

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Re: At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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22 Feb 2017, 00:52
Are you sure such questions can come I forgot what does a parabola look like

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Re: At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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22 Feb 2017, 00:53
Are you sure such questions can come I forgot what does a parabola or hyperbole look like

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At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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Updated on: 22 Feb 2017, 01:01
Bunuel wrote:
At how many points does the parabola y = 1/2*x^2 intersect the hyperbola x^2 – y^2 = 1?

A. 0
B. 1
C. 2
D. 3
E. 4

Parabola y = 1/2*x^2

Hyperbola x^2 – y^2 = 1

The best method is to draw the figure...

The figure shows that total points of intersections are two in number

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Originally posted by GMATinsight on 22 Feb 2017, 00:55.
Last edited by GMATinsight on 22 Feb 2017, 01:01, edited 1 time in total.
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Re: At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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22 Feb 2017, 01:03
nailin16 wrote:
Are you sure such questions can come I forgot what does a parabola or hyperbole look like

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Hi @nalin16,

You are right about your apprehension. Parabola has been witnessed in GMAT however no incidence of Hyperbola has been witnessed so far so I also feel that that hyperbola can not be part of any question of GMAT yet.
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At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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22 Feb 2017, 17:24
1
GMATinsight wrote:
nailin16 wrote:
Are you sure such questions can come I forgot what does a parabola or hyperbole look like

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Hi @nalin16,

You are right about your apprehension. Parabola has been witnessed in GMAT however no incidence of Hyperbola has been witnessed so far so I also feel that that hyperbola can not be part of any question of GMAT yet.

Yes, that's true, but consider: Do you really need to know anything about *either* parabolas or hyperbolas or what they look like to solve this problem?

See, for instance, your own second solution.
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Re: At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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16 Jun 2017, 22:15
AnthonyRitz wrote:
GMATinsight wrote:
nailin16 wrote:
Are you sure such questions can come I forgot what does a parabola or hyperbole look like

Sent from my SM-G600FY using GMAT Club Forum mobile app

Hi @nalin16,

You are right about your apprehension. Parabola has been witnessed in GMAT however no incidence of Hyperbola has been witnessed so far so I also feel that that hyperbola can not be part of any question of GMAT yet.

Yes, that's true, but consider: Do you really need to know anything about *either* parabolas or hyperbolas or what they look like to solve this problem?

See, for instance, your own second solution.

No, you don't necessarily need to know what a parabola or hyperbola looks like- you just need to know how to effectively substitute the parabola in the hyperbola so as to derive points of intersection. Simple, intersection of hyperbola and parabola= substitute parabola into hyperbola. Now, you can also substitute for the x values- the GMAT just makes a bit more difficult to figure out which is faster and more effective to substitute.
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Re: At how many points does the parabola y = 1/2*x^2 intersect the hyperbo  [#permalink]

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25 Nov 2018, 16:50
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Re: At how many points does the parabola y = 1/2*x^2 intersect the hyperbo &nbs [#permalink] 25 Nov 2018, 16:50
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text:
In [2]:
import numpy             as np
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.optimize      import fsolve, curve_fit
from scipy.interpolate   import interp1d
import sympy             as sp
from IPython.display import display
sp.init_printing()
import pint; u = pint.UnitRegistry()

## Sample problem¶

The following equation represents 1-D heat transfer in a heated rod of length $L=1$: $$\frac{d^2T}{dx^2} = -5432,$$ $$T(0) = 300,$$ $$T(L) = 300.$$

We can convert this second order ODE to two first order ODEs by making the substitution $H = dT/dx$. Then we have: $$\frac{dH}{dx} = -5432,$$

$$\frac{dT}{dx} = H,$$$$T(0) = 300,$$$$T(L) = 300.$$

We can now solve these two coupled rate equations. But we only have an initial condition for $T$; we don't have an initial condition for H.

### Part a¶

• Find T(x) and H(x) by solving the two rate equations assuming $H(0) \equiv H0= 3000$.
• (Note, because we had to guess H(0), $T(L)\ne 300$ like it should be. We'll fix this in Part b.)
In [4]:
def rhsf(HT, x):
H = HT[0]
T = HT[1]

dHdx = -5432
dTdx = H

return np.array([dHdx, dTdx])

HT0 = np.array([3000, 300])
x = np.linspace(0, 1, 100)
HT = odeint(rhsf, HT0, x)
H = HT[:,0]
T = HT[:,1]

print(T[-1])

583.999999991825

### Part b¶

• We can treat all of part A as a solution of one "nonlinear equation" in one unknown.
• The unknown is H0.
• The "nonlinear equation" is not some formula we write down as usual.
• Rather, it is a function: put in a value of H0, solve the ODEs for T(x) and H(x), evaluate and return T(L).
• Call this function f(H0). You put in H0, and you compute and returnn T(L).
• So, we are solving for f(H0) = 300.
• Or, F(H0) = f(H0)-300 = 0.
• Wrap all of Part a in a function F(H0), and solve F(H0)=0 for H0.
• Report the value rounded as ####
In [5]:
def F(H0):
def rhsf(HT, x):
H = HT[0]
T = HT[1]

dHdx = -5432
dTdx = H

return np.array([dHdx, dTdx])

HT0 = np.array([H0[0], 300])
x = np.linspace(0, 1, 100)
HT = odeint(rhsf, HT0, x)
H = HT[:,0]
T = HT[:,1]

return T[-1] - 300

fsolve(F, 3000)

Out[5]:
array([2716.00000001])
In [ ]: