text:
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## Overview

Teaching: 30 min
Exercises: 20 min
Questions
• How can I describe the quality of my data?

Objectives
• Explain how a FASTQ file encodes per-base quality scores.

• Interpret a FastQC plot summarizing per-base quality across all reads.

• Use for loops to automate operations on multiple files.

## Bioinformatic workflows

When working with high-throughput sequencing data, the raw reads you get off the sequencer must pass through several different tools to generate your final desired output. The execution of this set of tools in a specified order is commonly referred to as a workflow or a pipeline.

An example of the workflow we will be using for our analysis is provided below, with a brief description of each step.

1. Quality control - Assessing quality using FastQC and Trimming and/or filtering reads (if necessary)
2. Assembly of metagenome
3. Binning
4. Taxonomic assignation

These workflows in bioinformatics adopt a plug-and-play approach in that the output of one tool can be easily used as input to another tool without any extensive configuration. Having standards for data formats is what makes this feasible. Standards ensure that data is stored in a way that is generally accepted and agreed upon within the community. Therefore, the tools used to analyze data at different workflow stages are
built, assuming that the data will be provided in a specific format.

## Quality control

We will now assess the quality of the sequence reads contained in our FASTQ files.

### Details on the FASTQ format

Although it looks complicated (and it is), we can understand the FASTQ format with a little decoding. Some rules about the format include the following:

Line Description
1 Always begins with ‘@’ followed by the information about the read
2 The actual DNA sequence
3 Always begins with a ‘+’ and sometimes contains the same info as in line 1
4 Has a string of characters which represent the quality scores; must have same number of characters as line 2

We can view the first complete read in one of the files from our dataset using head to look at the first four lines. But we have to decompress one of the files first.

$cd ~/dc_workshop/data/untrimmed_fastq/$ gunzip JP4D_R1.fastq.gz

head -n 4 JP4D_R1.fastq  @MISEQ-LAB244-W7:156:000000000-A80CV:1:1101:12622:2006 1:N:0:CTCAGA CCCGTTCCTCGGGCGTGCAGTCGGGCTTGCGGTCTGCCATGTCGTGTTCGGCGTCGGTGGTGCCGATCAGGGTGAAATCCGTCTCGTAGGGGATCGCGAAGATGATCCGCCCGTCCGTGCCCTGAAAGAAATAGCACTTGTCAGATCGGAAGAGCACACGTCTGAACTCCAGTCACCTCAGAATCTCGTATGCCGTCTTCTGCTTGAAAAAAAAAAAAGCAAACCTCTCACTCCCTCTACTCTACTCCCTT + A>>1AFC>DD111A0E0001BGEC0AEGCCGEGGFHGHHGHGHHGGHHHGGGGGGGGGGGGGHHGEGGGHHHHGHHGHHHGGHHHHGGGGGGGGGGGGGGGGHHHHHHHGGGGGGGGHGGHHHHHHHHGFHHFFGHHHHHGGGGGGGGGGGGGGGGGGGGGGGGGGGGFFFFFFFFFFFFFFFFFFFFFBFFFF@F@FFFFFFFFFFBBFF?@;@####################################  Line 4 shows the quality of each nucleotide in the read. Quality is interpreted as the probability of an incorrect base call (e.g., 1 in 10) or, equivalently, the base call accuracy (e.g., 90%). Each nucleotide’s numerical score’s value is converted into a character code where every single character represents a quality score for an individual nucleotide. This conversion allows the alignment of each individual nucleotide with its quality score. For example, in the line above, the quality score line is: A>>1AFC>DD111A0E0001BGEC0AEGCCGEGGFHGHHGHGHHGGHHHGGGGGGGGGGGGGHHGEGGGHHHHGHHGHHHGGHHHHGGGGGGGGGGGGGGGGHHHHHHHGGGGGGGGHGGHHHHHHHHGFHHFFGHHHHHGGGGGGGGGGGGGGGGGGGGGGGGGGGGFFFFFFFFFFFFFFFFFFFFFBFFFF@F@FFFFFFFFFFBBFF?@;@####################################  The numerical value assigned to each character depends on the sequencing platform that generated the reads. The sequencing machine used to generate our data uses the standard Sanger quality PHRED score encoding, using Illumina version 1.8 onwards. Each character is assigned a quality score between 0 and 41, as shown in the chart below. Quality encoding: !"#%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJ
|         |         |         |         |
Quality score:    01........11........21........31........41

Each quality score represents the probability that the corresponding nucleotide call is incorrect. These probability values are the results of the base calling algorithm and depend on how much signal was captured for the base incorporation. This quality score is logarithmically based, so a quality score of 10 reflects a base call accuracy of 90%, but a quality score of 20 reflects a base call accuracy of 99%. In this link you can find more information about quality scores.

@MISEQ-LAB244-W7:156:000000000-A80CV:1:1101:12622:2006 1:N:0:CTCAGA
CCCGTTCCTCGGGCGTGCAGTCGGGCTTGCGGTCTGCCATGTCGTGTTCGGCGTCGGTGGTGCCGATCAGGGTGAAATCCGTCTCGTAGGGGATCGCGAAGATGATCCGCCCGTCCGTGCCCTGAAAGAAATAGCACTTGTCAGATCGGAAGAGCACACGTCTGAACTCCAGTCACCTCAGAATCTCGTATGCCGTCTTCTGCTTGAAAAAAAAAAAAGCAAACCTCTCACTCCCTCTACTCTACTCCCTT
+
A>>1AFC>DD111A0E0001BGEC0AEGCCGEGGFHGHHGHGHHGGHHHGGGGGGGGGGGGGHHGEGGGHHHHGHHGHHHGGHHHHGGGGGGGGGGGGGGGGHHHHHHHGGGGGGGGHGGHHHHHHHHGFHHFFGHHHHHGGGGGGGGGGGGGGGGGGGGGGGGGGGGFFFFFFFFFFFFFFFFFFFFFBFFFF@F@FFFFFFFFFFBBFF?@;@####################################

We can now see that there is a range of quality scores but that the end of the sequence is very poor (# = a quality score of 2).

## Exercise 1: Looking at specific reads

In the terminal, how would you show the ID and quality of the last read JP4D_R1.fastq?
a) tail JP4D_R1.fastq
b) head -n 4 JP4D_R1.fastq
c) more JP4D_R1.fastq
d) tail -n4 JP4D_R1.fastq
e) tail -n4 JP4D_R1.fastq | head -n2

Do you trust the sequence in this read?

## Solution

a) It shows the ID and quality of the last read but also unnecessary lines from previous reads.
b) No. It shows the first read's info.
c) It shows the text of the entire file.
d) This option is the best answer as it only shows the last read's information.
e) It does show the ID of the last read but not the quality.

@MISEQ-LAB244-W7:156:000000000-A80CV:1:2114:17866:28868 1:N:0:CTCAGA

CCCGTTCTCCACCTCGGCGCGCGCCAGCTGCGGCTCGTCCTTCCACAGGAACTTCCACGTCGCCGTCAGCCGCGACACGTTCTCCCCCCTCGCATGCTCGTCCTGTCTCTCGTGCTTGGCCGACGCCTGCGCCTCGCACTGCGCCCGCTCGGTGTCGTTCATGTTGATCTTCACCGTGGCGTGCATGAAGCGGTTCCCGGCCTCGTCGCCACCCACGCCATCCGCGTCGGCCAGCCACTCTCACTGCTCGC

+

AA11AC1>3@DC1F1111000A0/A///BB#############################################################################################################################################################################################################################

This read has more consistent quality at its first than at the end but still has a range of quality scores, most of them are low. We will look at variations in position-based quality in just a moment.

In real life, you won’t be assessing the quality of your reads by visually inspecting your FASTQ files. Instead, you’ll use a software program to assess read quality and filter out poor reads. We’ll first use a program called FastQC to visualize the quality of our reads. Later in our workflow, we’ll use another program to filter out poor-quality reads.

First, let’s make available our metagenomics software:

## Activating an environment

Environments are part of a bioinformatic tendency to do reproducible research; they are a way to share and maintain our programs in their needed versions used for a pipeline with our colleagues and our future self. FastQC has not been activated in the (base) environment, but this AWS instance came with an environment called metagenomics. We need to activate it in order to start using FastQC.

We will use Conda as our environment manager. Conda is an open-source package and environment management system that runs on Windows, macOS and Linux. Conda environments are activated with the conda activate direction:

$conda activate metagenomics  After the environment has been activated, a label is shown before the $ sign.

(metagenomics) $ Now, if we call FastQC, a long help page will be displayed on our screen. $ fastqc -h

            FastQC - A high throughput sequence QC analysis tool

SYNOPSIS

fastqc seqfile1 seqfile2 .. seqfileN

fastqc [-o output dir] [--(no)extract] [-f fastq|bam|sam]
[-c contaminant file] seqfile1 .. seqfileN

DESCRIPTION

FastQC reads a set of sequence files and produces from each one a quality
control report consisting of many different modules, each one of
which will help to identify a different potential type of problem in your
data.
.
.
.

If FastQC is not installed, then you would expect to see an error like

The program 'fastqc' is currently not installed. You can install it by typing:
sudo apt-get install fastqc

If this happens, check with your instructor before trying to install it.

## Assessing quality using FastQC

FastQC has several features that can give you a quick impression of any problems your data may have, so you can consider these issues before moving forward with your analyses. Rather than looking at quality scores for each read, FastQC looks at quality collectively across all reads within a sample. The image below shows one FastQC-generated plot that indicates a very high-quality sample:

The x-axis displays the base position in the read, and the y-axis shows quality scores. In this example, the sample contains reads that are 40 bp long. This length is much shorter than the reads we are working on within our workflow. For each position, there is a box-and-whisker plot showing the distribution of quality scores for all reads at that position. The horizontal red line indicates the median quality score, and the yellow box shows the 1st to 3rd quartile range. This range means that 50% of reads have a quality score that falls within the range of the yellow box at that position. The whiskers show the whole range covering the lowest (0th quartile) to highest (4th quartile) values.

The quality values for each position in this sample do not drop much lower than 32, which is a high-quality score. The plot background is also color-coded to identify good (green), acceptable (yellow) and bad (red) quality scores.

Now let’s look at a quality plot on the other end of the spectrum.

The FastQC tool produces several other diagnostic plots to assess sample quality and the one plotted above. Here, we see positions within the read in which the boxes span a much more comprehensive range. Also, quality scores drop pretty low into the “bad” range, particularly on the tail end of the reads.

## Running FastQC

We will now assess the quality of the reads that we downloaded. First, make sure you’re still in the untrimmed_fastq directory.

$cd ~/dc_workshop/data/untrimmed_fastq/  ## Exercise 2: Looking at metadata about the untrimmed-files To know which files have more data, you need to see metadata about untrimmed files. In files, metadata includes owners of the file, state of the write, read, and execute permissions, size, and modification date. Using the ls command, how would you get the size of the files in the untrimmed_fastq\ directory? (Hint: Look at the options for the ls command to see how to show file sizes.) a) ls -a b) ls -S c) ls -l d) ls -lh e) ls -ahlS ## Solution  a) No. The flag -a shows all the contents, including hidden files and directories, but not the sizes. b) No. The flag -S shows the content Sorted by size, starting with the most extensive file, but not the sizes. c) Yes. The flag -l shows the contents with metadata, including file size. Other metadata are permissions, owners, and modification dates. d) Yes. The flag -lh shows the content with metadata in a human-readable manner. e) Yes. The combination of all the flags shows all the contents with metadata, including hidden files, sorted by size.  ls -ahls  -rw-r--r-- 1 dcuser dcuser 24M Nov 26 21:34 JC1A_R1.fastq.gz -rw-r--r-- 1 dcuser dcuser 24M Nov 26 21:34 JC1A_R2.fastq.gz -rw-r--r-- 1 dcuser dcuser 616M Nov 26 21:34 JP4D_R1.fastq -rw-r--r-- 1 dcuser dcuser 203M Nov 26 21:35 JP4D_R2.fastq.gz  Four FASTQ files oscillate between 24M (24MB) to 616M. The largest file is JP4D_R1.fastq with 616M. FastQC can accept multiple file names as input, and on both zipped and unzipped files, so we can use the \*.fastq*wildcard to run FastQC on all FASTQ files in this directory. $ fastqc *.fastq*

You will see an automatically updating output message telling you the progress of the analysis. It will start like this:

Started analysis of JC1A_R1.fastq.gz
Approx 5% complete for JC1A_R1.fastq.gz
Approx 10% complete for JC1A_R1.fastq.gz
Approx 15% complete for JC1A_R1.fastq.gz
Approx 20% complete for JC1A_R1.fastq.gz
Approx 25% complete for JC1A_R1.fastq.gz
Approx 30% complete for JC1A_R1.fastq.gz
Approx 35% complete for JC1A_R1.fastq.gz

It should take around five minutes for FastQC to run on all four of our FASTQ files. When the analysis completes, your prompt will return. So your screen will look something like this:

Approx 80% complete for JP4D_R2.fastq.gz
Approx 85% complete for JP4D_R2.fastq.gz
Approx 90% complete for JP4D_R2.fastq.gz
Approx 95% complete for JP4D_R2.fastq.gz
Analysis complete for JP4D_R2.fastq.gz
$ The FastQC program has created several new files within our data/untrimmed_fastq/ directory. $ ls

JC1A_R1_fastqc.html  JC1A_R2_fastqc.html  JP4D_R1.fastq        JP4D_R2_fastqc.html  TruSeq3-PE.fa
JC1A_R1_fastqc.zip   JC1A_R2_fastqc.zip   JP4D_R1_fastqc.html  JP4D_R2_fastqc.zip
JC1A_R1.fastq.gz     JC1A_R2.fastq.gz     JP4D_R1_fastqc.zip   JP4D_R2.fastq.gz

For each input FASTQ file, FastQC has created a .zip file and a .html file. The .zip file extension indicates that this is a compressed set of multiple output files. We’ll be working with these output files soon. The .html file is a stable webpage displaying the summary report for each of our samples.

We want to keep our data files and our results files separate, so we will move these output files into a new directory within our results/ directory.

$mkdir -p ~/dc_workshop/results/fastqc_untrimmed_reads$ mv *.zip ~/dc_workshop/results/fastqc_untrimmed_reads/
$mv *.html ~/dc_workshop/results/fastqc_untrimmed_reads/  Now we can navigate into this results directory and do some closer inspection of our output files. $ cd ~/dc_workshop/results/fastqc_untrimmed_reads/

## Viewing the FastQC results

If we were working on our local computers, we’d be able to look at each of these HTML files by opening them in a web browser. However, these files are currently sitting on our remote AWS instance, where our local computer can’t see them. If we are working with the terminal provided by r-studio we can either: select the html files and with the secondary click chose the option open in a browser or export the files to our local computer as we learned in the Introduction to the Command Line lesson.

Suppose you only have access to a terminal and there is not a web browser available at the AWS remote machine. The following are the steps needed to download your files to your computer. Observe the third step and answer false/true of each question about the scp command.

1. Open a new terminal on your local computer.
2. Make a new local directory on our computer to store the HTML files $mkdir -p ~/Desktop/fastqc_html 3. Transfer our HTML files to our local computer using scp. $ scp dcuser@ec2-34-238-162-94.compute-1.amazonaws.com:~/dc_workshop/results/fastqc_untrimmed_reads/*.html ~/Desktop/fastqc_html
A) dcuser is your local user
B) ec2-34-238-162-94.compute-1.amazonaws.com is the adress of your remote machine
C) the current adress of the file goes after the second space in the scp command.
D) ~/dc_workshop/results/fastqc_untrimmed_reads/*.html is the path of the file you want to download E) ~/Desktop/fastqc_html is a remote direction. F) : Divides the direction of your local computer and the path of the file.

## Solution

A) False. dcuser is your remote user.
B) True. ec2-34-238-162-94.compute-1.amazonaws.com is the adress of your remote machine
C) False. The current adress of the file goes after the first space in the scp command.
D) True. ~/dc_workshop/results/fastqc_untrimmed_reads/*.html is the path of the file you want to download in the remote machine. E) False. ~/Desktop/fastqc_html is a local direction where your file will be downloaded. F) False. : Divides the direction of your remote computer and the path of the file.

You should see a status output like this:

 JC1A_R1_fastqc.html     100%  253KB 320.0KB/s   00:00
JC1A_R2_fastqc.html     100%  262KB 390.1KB/s   00:00
JP4D_R1_fastqc.html     100%  237KB 360.8KB/s   00:00
JP4D_R2_fastqc.html     100%  244KB 385.2KB/s   00:00

Now we can open the 4 HTML files.

Depending on your system, you should be able to select and open them all at once via a right-click menu in your file browser.

## Exercise 4: Discuss the quality of sequencing files

Discuss your results with a neighbor. Which sample(s) looks the best per base sequence quality? Which sample(s) look the worst?

## Solution

All of the reads contain usable data, but the quality decreases toward the end of the reads. File JC1A_R2_fastqc shows the lowest quality.

## Decoding the other FastQC outputs

We’ve now looked at quite a few “Per base sequence quality” FastQC graphs, but there are nine other graphs that we haven’t talked about! Below we have provided a brief overview of interpretations for each plot. For more information, please see the FastQC documentation here

• Per tile sequence quality: the machines that perform sequencing are divided into tiles. This plot displays patterns in base quality along these tiles. Consistently low scores are often found around the edges, but hot spots could also occur in the middle if an air bubble was introduced during the run.
• Per sequence quality scores: a density plot of quality for all reads at all positions. This plot shows what quality scores are most common.
• Per base sequence content: plots the proportion of each base position over all of the reads. Typically, we expect to see each base roughly 25% of the time at each position, but this often fails at the beginning or end of the read due to quality or adapter content.
• Per sequence GC content: a density plot of average GC content in each of the reads.
• Per base N content: the percent of times that ‘N’ occurs at a position in all reads. If there is an increase at a particular position, this might indicate that something went wrong during sequencing.
• Sequence Length Distribution: the distribution of sequence lengths of all reads in the file. If the data is raw, there is often a sharp peak; however, if the reads have been trimmed, there may be a distribution of shorter lengths.
• Sequence Duplication Levels: a distribution of duplicated sequences. In sequencing, we expect most reads to only occur once. If some sequences are occurring more than once, it might indicate enrichment bias (e.g. from PCR). This might not be true if the samples are high coverage (or RNA-seq or amplicon).
• Overrepresented sequences: a list of sequences that occur more frequently than would be expected by chance.
• K-mer Content: a graph showing any sequences which may show a positional bias within the reads.

## Working with the FastQC text output

Now that we’ve looked at our HTML reports getting a feel for the data, let’s look more closely at the other output files. Go back to the tab in your terminal program that is connected to your AWS instance (the tab label will start with dcuser@ip) and make sure you’re in our results subdirectory.

$cd ~/dc_workshop/results/fastqc_untrimmed_reads/$ ls

JC1A_R1_fastqc.html           JP4D_R1_fastqc.html
JC1A_R1_fastqc.zip            JP4D_R1_fastqc.zip
JC1A_R2_fastqc.html           JP4D_R2_fastqc.html
JC1A_R2_fastqc.zip            JP4D_R2_fastqc.zip

Our .zip files are compressed files. Each contains multiple different types of output files for a single input FASTQ file. To view the contents of a .zip file, we can use the program unzip to decompress these files. Let’s try doing them all at once using a wildcard.

$unzip *.zip  Archive: JC1A_R1_fastqc.zip caution: filename not matched: JC1A_R2_fastqc.zip caution: filename not matched: JP4D_R1_fastqc.zip caution: filename not matched: JP4D_R2_fastqc.zip  This decompresion didn’t work. It identified the first file and got a warning message for the other .zip files. This is because unzip expects to get only one zip file as input. We could go through and unzip each file one at a time, but this is very time-consuming and error-prone. Someday you may have 500 files to unzip! A more efficient way is to use a for loop like we learned in the Command Line lesson to iterate through all of our .zip files. Let’s see what that looks like, and then we’ll discuss what we’re doing with each line of our loop. $ for filename in *.zip
> do
> unzip $filename > done  In this example, the input is the four filenames (one filename for each of our .zip files). Each time the loop iterates, it will assign a file name to the variable filename and run the unzip command. The first time through the loop, $filename is JC1A_R1_fastqc.zip. The interpreter runs the command unzip on JC1A_R1_fastqc.zip. For the second iteration, $filename becomes JC1A_R2_fastqc.zip. This time, the shell runs unzip on JC1A_R2_fastqc.zip. It then repeats this process for the other .zip files in our directory. When we run the for loop, you will see an output that starts like this: Archive: JC1A_R1_fastqc.zip creating: JC1A_R1_fastqc/ creating: JC1A_R1_fastqc/Icons/ creating: JC1A_R1_fastqc/Images/ inflating: JC1A_R1_fastqc/Icons/fastqc_icon.png inflating: JC1A_R1_fastqc/Icons/warning.png inflating: JC1A_R1_fastqc/Icons/error.png inflating: JC1A_R1_fastqc/Icons/tick.png inflating: JC1A_R1_fastqc/summary.txt inflating: JC1A_R1_fastqc/Images/per_base_quality.png inflating: JC1A_R1_fastqc/Images/per_tile_quality.png inflating: JC1A_R1_fastqc/Images/per_sequence_quality.png inflating: JC1A_R1_fastqc/Images/per_base_sequence_content.png inflating: JC1A_R1_fastqc/Images/per_sequence_gc_content.png inflating: JC1A_R1_fastqc/Images/per_base_n_content.png inflating: JC1A_R1_fastqc/Images/sequence_length_distribution.png inflating: JC1A_R1_fastqc/Images/duplication_levels.png inflating: JC1A_R1_fastqc/Images/adapter_content.png inflating: JC1A_R1_fastqc/fastqc_report.html inflating: JC1A_R1_fastqc/fastqc_data.txt inflating: JC1A_R1_fastqc/fastqc.fo  The unzip program is decompressing the .zip files and creates a new directory (with subdirectories) for each of our samples, to store all of the different output that is produced by FastQC. There are a lot of files here. We’re going to focus on the summary.txt file. If you list the files in our directory, now you will see the following: $ ls

JC1A_R1_fastqc                  JP4D_R1_fastqc
JC1A_R1_fastqc.html             JP4D_R1_fastqc.html
JC1A_R1_fastqc.zip              JP4D_R1_fastqc.zip
JC1A_R2_fastqc                  JP4D_R2_fastqc
JC1A_R2_fastqc.html             JP4D_R2_fastqc.html
JC1A_R2_fastqc.zip              JP4D_R2_fastqc.zip

The .html files and the uncompressed .zip files are still present, but now we also have a new directory for each sample. We can see that it’s a directory if we use the -F flag for ls.

$ls -F  JC1A_R1_fastqc/ JP4D_R1_fastqc/ JC1A_R1_fastqc.html JP4D_R1_fastqc.html JC1A_R1_fastqc.zip JP4D_R1_fastqc.zip JC1A_R2_fastqc/ JP4D_R2_fastqc/ JC1A_R2_fastqc.html JP4D_R2_fastqc.html JC1A_R2_fastqc.zip JP4D_R2_fastqc.zip  Let’s see what files are present within one of these output directories. $ ls -F JC1A_R1_fastqc/

fastqc_data.txt  fastqc.fo  fastqc_report.html	Icons/	Images/  summary.txt

Use less to preview the summary.txt file for this sample.

$grep FAIL fastqc_summaries.txt  FAIL Per base sequence quality JC1A_R1.fastq.gz FAIL Per sequence GC content JC1A_R1.fastq.gz FAIL Sequence Duplication Levels JC1A_R1.fastq.gz FAIL Adapter Content JC1A_R1.fastq.gz FAIL Per base sequence quality JC1A_R2.fastq.gz FAIL Per sequence GC content JC1A_R2.fastq.gz FAIL Sequence Duplication Levels JC1A_R2.fastq.gz FAIL Adapter Content JC1A_R2.fastq.gz FAIL Per base sequence content JP4D_R1.fastq FAIL Adapter Content JP4D_R1.fastq FAIL Per base sequence quality JP4D_R2.fastq.gz FAIL Per base sequence content JP4D_R2.fastq.gz FAIL Adapter Content JP4D_R2.fastq.gz  ## Quality of large datasets Explore MultiQC if you want a tool that can show the quality of many samples at once. ## Quality Encodings Vary Although we’ve used a particular quality encoding system to demonstrate the interpretation of read quality, different sequencing machines use different encoding systems. This means that depending on which sequencer you use to generate your data, a # may not indicate a poor quality base call. This mainly relates to older Solexa/Illumina data. However, it’s essential that you know which sequencing platform was used to generate your data to tell your quality control program which encoding to use. If you choose the wrong encoding, you run the risk of throwing away good reads or (even worse) not throwing away bad reads! ## Bonus Exercise: Automating a quality control workflow If you lose your FastQC analysis results. How would you do it again but faster than the first time? As we have seen in a previous lesson, making scripts for repetitive tasks is a very efficient practice during bioinformatic pipelines. ## Solution Make a new script with nano nano quality_control.sh  Paste inside the commands that we used along with echo commands that shows you how the script is running. set -e # This will ensure that our script will exit if an error occurs cd ~/dc_workshop/data/untrimmed_fastq/ echo "Running FastQC ..." fastqc *.fastq* mkdir -p ~/dc_workshop/results/fastqc_untrimmed_reads echo "Saving FastQC results..." mv *.zip ~/dc_workshop/results/fastqc_untrimmed_reads/ mv *.html ~/dc_workshop/results/fastqc_untrimmed_reads/ cd ~/dc_workshop/results/fastqc_untrimmed_reads/ echo "Unzipping..." for filename in *.zip do unzip$filename
done

echo "Saving summary..."
mkdir -p ~/dc_workshop/docs
cat */summary.txt > ~/dc_workshop/docs/fastqc_summaries.txt

If we were to run this script, it would ask us for confirmation to redo several steps because we already did all of them. If you want to, you can run it to check that it works, but it is not necessary if you already completed every step of the previous episode.

## Key Points

• It is important to know the quality of our data to make decisions in the subsequent steps.

• FastQC is a program that allows us to know the quality of FASTQ files.

• for loops let you perform the same operations on multiple files with a single command.


text:
John, Matthew: I think there is a mild problem with your construction of a non-trivial functor \$$F\colon \mathbf{Set}\to \mathbb{N}\$$. You have the map \$$f\colon \emptyset\to \mathbb{R}\$$, with \$$F(f)=n\$$, and with all other functions going to \$$\mathrm{id}\$$. Let \$$t \colon \mathbb{R}\to \\{\ast\\}\$$ be the function that send every number to \$$\ast\$$. Of course with that domain and that codomain you have no choice. Then we have the composite \$$t\circ f\colon \emptyset \to \(\ast$$\\). We should then have \$$n=\mathrm{id}\circ n = F(t)\circ F(f)= F(t\circ f) =\mathrm{id} \$$.

We need to send *all* initial functions to \$$n\$$. I guess we have a functor from \$$\mathbf{Set}\$$ to the category with two objects \$$a\$$ and \$$b\$$ and with precisely one non-identity morphism \$$a\to b\$$: the empty set gets sent to \$$a\$$, everything else goes to \$$b\$$. The functor you want from \$$\mathbf{Set}\$$ to \$$\mathbb{N}\$$ factors through this.

format: plain


text:
# Correctly defining model

Hi,
I am new to PyMC3 and am trying to implement a rather simple model, but cannot seem to get it working.
I have observations \textbf{B} with shape (m\times1). Model parameters \textbf{m} of shape (n\times1). And a data kernel \textbf{G} with shape (m\times n).
Forward problem, \textbf{B}=\textbf{Gm}.
In this specific example m=270000 and n=3.

Below my intent was to define each element in \textbf{m} to have a normal prior distribution with mean zero and a standard deviation determined from another place. And that the residuals between should have a standard normal distribution.
When running NUTS with the settings below the problem does not converge. And I do not understand why. Can anyone tell me if the above model indeed defines the prior and likelihood I intended?
And is the matrix vector multiplication correct? I would normally use G @ m in python.
Some might ask if the problem might be with \textbf{B} or \textbf{G}.
The reason I do not think that’s the problem is because I have used the exact same observations and data kernel in STAN’s NUTS implementation and samples it successfully within a minute.

basic_model = pm.Model()

with basic_model:

# Prior distribution
m = pm.Normal('m', mu=0, sd=limits/2, shape=(3))

# Predictions
pred = pm.math.dot(G,m)

# Likelihood
Y_obs = pm.Normal('Y_obs', mu=pred, sd=1, observed=B)

with basic_model:
step = pm.NUTS()
chains = 4
njobs=2
draws = 1000

trace = pm.sample(draws=draws, step=step, chains=chains, njobs=njobs, tune=2000)

I had a similar issue and followed the trick in this post to reparameterize the Normal for your Y_obs, which worked beautifully.

Thanks you for the suggestion. But unfortunately this did not solve the problem.
Below I have appended the reparameterized version.
I also define the prediction as deterministic.
I normally start with a sample rate of around 10 draws/s with within the first 10% of the draws is reduced to 10 s/draw. Eventually a chain fails.

basic_model = pm.Model()

with basic_model:

# Priors for unknown model parameters
m_offset = pm.Normal('m_offset', mu=0, sd=1, shape=(3))
m = pm.Deterministic('m', m_offset*(limits/2))

# Expected value of outcome
pred = pm.Deterministic('pred', pm.math.dot(G,m))

# Likelihood (sampling distribution) of observations
Y_obs = pm.Normal('Y_obs', mu=pred, sd=1, observed=B)

Looks like this code only has the the reparameterization for the m draw, not the Y_obs draw too. It would require drawing some Y_obs_offset as a standard normal and then forming some Deterministic equal to pm.math.dot(G,m) + sd * Y_obs_offset. What I’m not sure about at this point and haven’t tried before is if Deterministic variables can be observed?

P.s. I remember reading in some deep documentation that I can’t find anymore that NUTS starts of with a tree_depth of 8, and then switches to whatever max_treedepth is passed in, which defaults to 10 I believe. So that’s why you observe the dropoff in speed (and in my case I observed the same drastic dropoff and it was fixed just by reparameterizing).

P.p.s. How many times am I going to hit tab to indent code (which just my mouse focus to the reply button) and then space or enter and accidentally submit my post??


text:
## Positive periodic solutions of Hill’s equations with singular nonlinear perturbations.(English)Zbl 1148.34025

The paper proves existence and multiplicity of positive periodic solutions of the perturbation of Hill’s equation $x''+a(t)x=f(t,x)+e(t), (1)$ where $$a(t),e(t)$$ are continuous, $$T$$-periodic functions. The nonlinearity $$f(t,x)$$ is continuous in $$(t,x)$$ and $$T$$-periodic in $$t$$ and has a singularity at $$x=0$$. The case of a strong singularity as well as that of a weak singularity is considered, and $$e$$ does not need to be positive. The proofs are based on Krasnoselskii’s fixed point theorem in cones and on the Leray-Schauder alternative together with a truncation technique. Some recent results in the literature are generalized and improved.

### MSC:

 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Bonheure, D.; De Coster, C., Forced singular oscillators and the method of lower and upper solutions, Topol. methods nonlinear anal., 22, 297-317, (2003) · Zbl 1108.34033 [2] Chu, J.; Lin, X.; Jiang, D.; O’Regan, D.; Agarwal, R.P., Multiplicity of positive solutions to second order differential equations, Bull. austral. math. soc., 73, 175-182, (2006) · Zbl 1096.34518 [3] De Coster, C.; Habets, P., Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, (), 1-78 · Zbl 0889.34018 [4] del Pino, M.A.; Manásevich, R.F.; Montero, A., $$T$$-periodic solutions for some second order differential equations with singularities, Proc. roy. soc. Edinburgh sect. A, 120, 231-243, (1992) · Zbl 0761.34031 [5] del Pino, M.A.; Manásevich, R.F., Infinitely many $$T$$-periodic solutions for a problem arising in nonlinear elasticity, J. differential equations, 103, 260-277, (1993) · Zbl 0781.34032 [6] Franco, D.; Webb, J.R.L., Collisionless orbits of singular and nonsingular dynamical systems, Discrete contin. dyn. syst., 15, 747-757, (2006) · Zbl 1120.34029 [7] D. Franco, P.J. Torres, Periodic solutions of singular systems without the strong force condition, Proc. Amer. Math. Soc. (in press) · Zbl 1129.37033 [8] Gordon, W.B., Conservative dynamical systems involving strong forces, Trans. amer. math. soc., 204, 113-135, (1975) · Zbl 0276.58005 [9] Habets, P.; Sanchez, L., Periodic solution of some Liénard equations with singularities, Proc. amer. math. soc., 109, 1135-1144, (1990) [10] Jiang, D.; Chu, J.; Zhang, M., Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. differential equations, 211, 282-302, (2005) · Zbl 1074.34048 [11] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen [12] Lazer, A.C.; Solimini, S., On periodic solutions of nonlinear differential equations with singularities, Proc. amer. math. soc., 99, 109-114, (1987) · Zbl 0616.34033 [13] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (), 74-142 · Zbl 0798.34025 [14] Rachunková, I.; Tvrdý, M.; Vrkoc˘, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations, 176, 445-469, (2001) · Zbl 1004.34008 [15] Torres, P.J., Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. methods appl. sci., 23, 1139-1143, (2000) · Zbl 0966.34038 [16] Torres, P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations, 190, 643-662, (2003) · Zbl 1032.34040 [17] Torres, P.J., Weak singularities may help periodic solutions to exist, J. differential equations, 232, 277-284, (2007) · Zbl 1116.34036 [18] Yan, P.; Zhang, M., Higher order nonresonance for differential equations with singularities, Math. methods appl. sci., 26, 1067-1074, (2003) · Zbl 1031.34040 [19] Zhang, M., A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. roy. soc. Edinburgh sect. A., 128, 1099-1114, (1998) · Zbl 0918.34025 [20] Zhang, M.; Li, W., A Lyapunov-type stability criterion using $$L^\alpha$$ norms, Proc. amer. math. soc., 130, 3325-3333, (2002) · Zbl 1007.34053 [21] Zhang, M., Periodic solutions of equations of ermakov – pinney type, Adv. nonlinear stud., 6, 57-67, (2006) · Zbl 1107.34037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
Substantial Breadth

# Substantial Breadth

Commens
Digital Companion to C. S. Peirce
Substantial Breadth
1867 | Upon Logical Comprehension and Extension | W 2:79-81; CP 2.409-14

The informed breadth and depth suppose a state of information which lies somewhere between two imaginary extremes. These are, first, the state in which no fact would be known, but only the meaning of terms; and, second, the state in which the information would amount to an absolute intuition of all there is, so that the things we should know would be the very substances themselves, and the qualities we should know would be the very concrete forms themselves. This suggests two other sorts of breadth and depth corresponding to these two states of information, and which I shall term respectively the essential and the substantial breadth and depth.

[—]

Substantial breadth is the aggregate of real substances of which alone a term is predicable with absolute truth.


text:
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# Go Duration, PostgreSQL Interval

Building a web application in Go and using PostgreSQL as the database is a real joy. Both of these tools are produced by incredibly vibrant open source projects. These projects represent some of the best attributes of the open source movement. When used in combination, occasionally idioms from Go don't …

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# Decrypting Vault Ciphertext with a Context

In a previous post, I described a use case for customer provided keys with Vault. One of the implications of this was the need for decryption after a bulk data export. In that post, I gave a concrete example of decrypting Vault ciphertext directly with a customer provided key. However …

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# Atomically Idempotent

Recently, I was analyzing some initialization code in Go with a teammate. The value being initialized was meant to be used in concurrent Go, so initialization had some requirement of atomicity. The code essentially boiled down to:

func (t *T) Start() {
return // Early Exit …
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# Wrapping Behavior of context.WithValue()

### Motivation

Throughout the Go monorepo we use context.WithValue() to "stash" a global value on a root context. For example

ctx = logger.WithLogger(ctx, log)
// ... later ...
log := logger.GetLogger(ctx)

The implementations for stashing a logger.Log are in the same general form as most context wrapping helpers:

type loggerKey …
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# Setting Per-Connection Timeouts with TypeORM

### PostgreSQL Statement Timeout

For most applications that use a database, user-facing queries must complete in a reasonable amount of time. In order to ensure a maximum query time, PostgreSQL supports a statement_timeout which will cause a query to be cancelled if it exceeds the timeout:

\$ psql
monsters_inc=> SHOW statement_timeout;
statement_timeout …


text:
# American Institute of Mathematical Sciences

2016, 10: 33-79. doi: 10.3934/jmd.2016.10.33

## Invariant distributions for homogeneous flows and affine transformations

 1 UMR CNRS 8524, UFR de Mathématiques, Université de Lille 1, F59655 Villeneuve d’Asq CEDEX 2 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States 3 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received  May 2013 Revised  December 2015 Published  March 2016

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.
Citation: Livio Flaminio, Giovanni Forni, Federico Rodriguez Hertz. Invariant distributions for homogeneous flows and affine transformations. Journal of Modern Dynamics, 2016, 10: 33-79. doi: 10.3934/jmd.2016.10.33
##### References:
 [1] A. Avila, B. Fayad and A. Kocsard, On manifolds supporting distributionally uniquely ergodic diffeomorphisms,, J. Differential Geom., 99 (2015), 191. [2] L. Auslander and L. W. Green, $G$-induced flows,, Amer. J. Math., 88 (1966), 43. doi: 10.2307/2373046. [3] A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms,, Duke Math. J., 158 (2011), 501. doi: 10.1215/00127094-1345662. [4] _________, Private communication,, in preparation, (2013). [5] L. Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory,, Bull. Amer. Math. Soc., 79 (1973), 227. doi: 10.1090/S0002-9904-1973-13134-9. [6] _________, An exposition of the structure of solvmanifolds. II. $G$-induced flows,, Bull. Amer. Math. Soc., 79 (1973), 262. doi: 10.1090/S0002-9904-1973-13139-8. [7] W. Chen and M. Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $T^ n$,, Commun. Partial Differential Equations, 25 (2000), 337. doi: 10.1080/03605300008821516. [8] P. Collet, H. Epstein and G. Gallavotti, Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties,, Comm. Math. Phys., 95 (1984), 61. doi: 10.1007/BF01215756. [9] S. G. Dani, Spectrum of an affine transformation,, Duke Math. J., 44 (1977), 129. doi: 10.1215/S0012-7094-77-04407-6. [10] S. G. Dani, A simple proof of Borel's density theorem,, Math. Z., 174 (1980), 81. doi: 10.1007/BF01215084. [11] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,, J. Reine Angew. Math., 359 (1985), 55. doi: 10.1515/crll.1985.359.55. [12] _________, Bounded orbits of flows on homogeneous spaces,, Comment. Math. Helv., 61 (1986), 636. doi: 10.1007/BF02621936. [13] R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for Livšic cohomology equation,, Ann. Math. (2), 123 (1986), 537. doi: 10.2307/1971334. [14] D. Dolgopyat, Livsič theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55. [15] B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305. doi: 10.1007/s00222-004-0408-x. [16] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8. [17] _________, On the cohomological equation for nilflows,, Journal of Modern Dynamics, 1 (2007), 37. [18] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus,, Ann. of Math. (2), 146 (1997), 295. doi: 10.2307/2952464. [19] _________, On the Greenfield-Wallach and Katok conjectures in dimension three,, in Geometric and Probabilistic Structures in Dynamics, (2008), 197. doi: 10.1090/conm/469/09167. [20] L. Flaminio and M. Paternain, Linearization of cohomology-free vector fields,, Discrete Contin. Dyn. Syst., 29 (2011), 1031. doi: 10.3934/dcds.2011.29.1031. [21] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,, Topology, 19 (1980), 301. doi: 10.1016/0040-9383(80)90015-4. [22] V. V. Gorbatsevich, Splittings of Lie groups and their application to the study of homogeneous spaces,, Math. USSR, 15 (1980), 441. doi: 10.1070/IM1980v015n03ABEH001257. [23] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. doi: 10.1016/0040-9383(73)90011-6. [24] S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084. [25] N. Jacobson, Lie Algebras,, Republication of the 1962 original, (1962). [26] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in collaboration with E. A. Robinson, (1999), 107. doi: 10.1090/pspum/069/1858535. [27] __________, Combinatorial Constructions in Ergodic Theory and Dynamics,, University Lecture Series, (2003). doi: 10.1090/ulect/030. [28] A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems,, Math. Res. Lett., 3 (1996), 191. doi: 10.4310/MRL.1996.v3.n2.a6. [29] D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sina\u\i's Moscow Seminar on Dynamical Systems, (1996), 141. [30] A. Kocsard, Cohomologically rigid vector fields: The Ktwo-formatok conjecture in dimension 3,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165. doi: 10.1016/j.anihpc.2008.07.005. [31] D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory,, in Handbook of Dynamical Systems, (2002), 813. doi: 10.1016/S1874-575X(02)80013-3. [32] A. N. Livšic, Some homology properties of U-systems,, Mat. Zametki, 10 (1971), 555. [33] R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8. [34] S. Matsumoto, The parameter rigid flows on 3-manifolds,, in Foliations, (2009), 135. doi: 10.1090/conm/498/09746. [35] S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth type interval exchange maps,, J. Amer. Math. Soc., 18 (2005), 823. doi: 10.1090/S0894-0347-05-00490-X. [36] D. W. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005). [37] A. L. Onishchik and E. B. Vinberg, eds., Lie Groups and Lie Algebras. III. Structure of Lie Groups and Lie Algebras,, A translation of Current Problems in Mathematics. Fundamental Directions, (1990). doi: 10.1007/978-3-662-03066-0. [38] W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds,, Amer. J. Math., 91 (1969), 757. doi: 10.2307/2373350. [39] M. S. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972). [40] F. Rodriguez Hertz and J. Rodriguez Hertz, Cohomology free systems and the first Betti number,, Continuous and Discrete Dynam. Systems, 15 (2006), 193. doi: 10.3934/dcds.2006.15.193. [41] F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle,, Invent. Math., 172 (2008), 353. doi: 10.1007/s00222-007-0100-z. [42] A. N. Starkov, On a criterion for the ergodicity of $G$-induced flows,, Uspekhi Mat. Nauk, 42 (1987), 197. [43] ________, Dynamical Systems on Homogeneous Spaces,, Transl. Math. Monogr., (2000). [44] W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms,, Ergodic Theory and Dynam. Systems, 6 (1986), 449. doi: 10.1017/S0143385700003606. [45] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Astérisque, 358 (2013), 75.

show all references

##### References:
 [1] A. Avila, B. Fayad and A. Kocsard, On manifolds supporting distributionally uniquely ergodic diffeomorphisms,, J. Differential Geom., 99 (2015), 191. [2] L. Auslander and L. W. Green, $G$-induced flows,, Amer. J. Math., 88 (1966), 43. doi: 10.2307/2373046. [3] A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms,, Duke Math. J., 158 (2011), 501. doi: 10.1215/00127094-1345662. [4] _________, Private communication,, in preparation, (2013). [5] L. Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory,, Bull. Amer. Math. Soc., 79 (1973), 227. doi: 10.1090/S0002-9904-1973-13134-9. [6] _________, An exposition of the structure of solvmanifolds. II. $G$-induced flows,, Bull. Amer. Math. Soc., 79 (1973), 262. doi: 10.1090/S0002-9904-1973-13139-8. [7] W. Chen and M. Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $T^ n$,, Commun. Partial Differential Equations, 25 (2000), 337. doi: 10.1080/03605300008821516. [8] P. Collet, H. Epstein and G. Gallavotti, Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties,, Comm. Math. Phys., 95 (1984), 61. doi: 10.1007/BF01215756. [9] S. G. Dani, Spectrum of an affine transformation,, Duke Math. J., 44 (1977), 129. doi: 10.1215/S0012-7094-77-04407-6. [10] S. G. Dani, A simple proof of Borel's density theorem,, Math. Z., 174 (1980), 81. doi: 10.1007/BF01215084. [11] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,, J. Reine Angew. Math., 359 (1985), 55. doi: 10.1515/crll.1985.359.55. [12] _________, Bounded orbits of flows on homogeneous spaces,, Comment. Math. Helv., 61 (1986), 636. doi: 10.1007/BF02621936. [13] R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for Livšic cohomology equation,, Ann. Math. (2), 123 (1986), 537. doi: 10.2307/1971334. [14] D. Dolgopyat, Livsič theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55. [15] B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305. doi: 10.1007/s00222-004-0408-x. [16] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8. [17] _________, On the cohomological equation for nilflows,, Journal of Modern Dynamics, 1 (2007), 37. [18] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus,, Ann. of Math. (2), 146 (1997), 295. doi: 10.2307/2952464. [19] _________, On the Greenfield-Wallach and Katok conjectures in dimension three,, in Geometric and Probabilistic Structures in Dynamics, (2008), 197. doi: 10.1090/conm/469/09167. [20] L. Flaminio and M. Paternain, Linearization of cohomology-free vector fields,, Discrete Contin. Dyn. Syst., 29 (2011), 1031. doi: 10.3934/dcds.2011.29.1031. [21] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,, Topology, 19 (1980), 301. doi: 10.1016/0040-9383(80)90015-4. [22] V. V. Gorbatsevich, Splittings of Lie groups and their application to the study of homogeneous spaces,, Math. USSR, 15 (1980), 441. doi: 10.1070/IM1980v015n03ABEH001257. [23] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. doi: 10.1016/0040-9383(73)90011-6. [24] S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084. [25] N. Jacobson, Lie Algebras,, Republication of the 1962 original, (1962). [26] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in collaboration with E. A. Robinson, (1999), 107. doi: 10.1090/pspum/069/1858535. [27] __________, Combinatorial Constructions in Ergodic Theory and Dynamics,, University Lecture Series, (2003). doi: 10.1090/ulect/030. [28] A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems,, Math. Res. Lett., 3 (1996), 191. doi: 10.4310/MRL.1996.v3.n2.a6. [29] D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sina\u\i's Moscow Seminar on Dynamical Systems, (1996), 141. [30] A. Kocsard, Cohomologically rigid vector fields: The Ktwo-formatok conjecture in dimension 3,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165. doi: 10.1016/j.anihpc.2008.07.005. [31] D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory,, in Handbook of Dynamical Systems, (2002), 813. doi: 10.1016/S1874-575X(02)80013-3. [32] A. N. Livšic, Some homology properties of U-systems,, Mat. Zametki, 10 (1971), 555. [33] R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8. [34] S. Matsumoto, The parameter rigid flows on 3-manifolds,, in Foliations, (2009), 135. doi: 10.1090/conm/498/09746. [35] S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth type interval exchange maps,, J. Amer. Math. Soc., 18 (2005), 823. doi: 10.1090/S0894-0347-05-00490-X. [36] D. W. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005). [37] A. L. Onishchik and E. B. Vinberg, eds., Lie Groups and Lie Algebras. III. Structure of Lie Groups and Lie Algebras,, A translation of Current Problems in Mathematics. Fundamental Directions, (1990). doi: 10.1007/978-3-662-03066-0. [38] W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds,, Amer. J. Math., 91 (1969), 757. doi: 10.2307/2373350. [39] M. S. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972). [40] F. Rodriguez Hertz and J. Rodriguez Hertz, Cohomology free systems and the first Betti number,, Continuous and Discrete Dynam. Systems, 15 (2006), 193. doi: 10.3934/dcds.2006.15.193. [41] F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle,, Invent. Math., 172 (2008), 353. doi: 10.1007/s00222-007-0100-z. [42] A. N. Starkov, On a criterion for the ergodicity of $G$-induced flows,, Uspekhi Mat. Nauk, 42 (1987), 197. [43] ________, Dynamical Systems on Homogeneous Spaces,, Transl. Math. Monogr., (2000). [44] W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms,, Ergodic Theory and Dynam. Systems, 6 (1986), 449. doi: 10.1017/S0143385700003606. [45] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Astérisque, 358 (2013), 75.
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Home / Basic Electrical / Open-Circuit Impedance Parameters

Open-Circuit Impedance Parameters

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To define the composition of a four-terminal, two-port network, we need four parameters. The test circuit of Figure 1 gives a set of parameters called the open-circuit impedance parameters (z-parameters) of the network.

Figure 1 Determining open-circuit impedance parameters

We start by opening the right-hand switch in Figure 1 so that I2 = 0. Leaving this switch open prevents the output circuit from affecting the measurement of input impedance. When we close the left-hand switch, the readings from the voltmeter measuring V1 and the ammeter measuring I1 give us a V/I ratio that represents only the open-circuit input impedance of the network.

We now open the left-hand switch and close the right-hand switch. I1 becomes zero, making the voltage drop across Zp zero. However, there is a reading on the voltmeter measuring V1 as a part of I2 is coupled back to the input side of the network. The ratio between the voltage, V1, at the input terminals (port 1) and the current, I2, flowing into the output terminals (port 2) is an impedance (in ohms) that indicates the amount of output-to-input (reverse-transfer) coupling in the network.

The same technique also produces parameters for the output impedance and input-to-output coupling. Thus, there are four open-circuit impedance parameters:

Open-circuit input impedance:

$\begin{matrix}{{\text{z}}_{\text{11}}}\text{=}\frac{{{\text{E}}_{\text{1}}}}{{{\text{I}}_{\text{1}}}}\left( with\text{ }{{\text{I}}_{\text{2}}}=0 \right) & {} & \left( 1 \right) \\\end{matrix}$

Open-circuit reverse-transfer impedance:

$\begin{matrix}{{\text{z}}_{\text{12}}}\text{=}\frac{{{\text{V}}_{\text{1}}}}{{{\text{I}}_{\text{2}}}}\left( with\text{ }{{\text{I}}_{\text{1}}}=0 \right)Open-Circuit & {} & \left( 2 \right) \\\end{matrix}$

Open-circuit forward-transfer impedance:

$\begin{matrix}{{\text{z}}_{21}}\text{=}\frac{{{\text{V}}_{\text{2}}}}{{{\text{I}}_{\text{1}}}}\left( with\text{ }{{\text{I}}_{\text{2}}}=0 \right)Open-Circuit & {} & \left( 3 \right) \\\end{matrix}$

Open-circuit forward-transfer impedance:

$\begin{matrix}{{\text{z}}_{\text{22}}}\text{=}\frac{{{\text{E}}_{\text{2}}}}{{{\text{I}}_{\text{2}}}}\left( with\text{ }{{\text{I}}_{\text{1}}}=0 \right) & {} & \left( 4 \right) \\\end{matrix}$

Having determined the four z-parameters, we now close both switches in Figure 1 so that the network actually does couple the output and input circuits. Consequently, both z11 and z12 are now present in the input circuit, and V1 consists of two voltages in series: the IZ drop due to I1 flowing in z11, plus a coupled voltage representing I2 flowing through the reverse-transfer impedance z12.

Applying a Kirchhoff’s voltage-law equation to the input terminals (port 1) of the network gives

$\begin{matrix}{{\text{z}}_{\text{11}}}{{\text{I}}_{\text{1}}}\text{+}{{\text{z}}_{\text{12}}}{{\text{I}}_{\text{2}}}\text{=}{{\text{E}}_{\text{1}}} & {} & \left( 5 \right) \\\end{matrix}$

Similarly, for the output terminals (port 2),

$\begin{matrix}{{\text{z}}_{\text{21}}}{{\text{I}}_{\text{1}}}\text{+}{{\text{z}}_{\text{22}}}{{\text{I}}_{\text{2}}}\text{=}{{\text{E}}_{2}} & {} & \left( 6 \right) \\\end{matrix}$

The next step is to determine the equivalent circuit represented by these two Kirchhoff’s voltage-law equations. The sum of two voltages at the input terminals represents a series circuit.

With the left-hand switch open and the right-hand switch closed in Figure 1, we discovered a voltage, V1, across the input port due to the output current I2. Hence, we can argue that we are looking into a Thévenin-equivalent voltage source that is dependent on the current in some other part of the circuit. The open-circuit voltage of this current-controlled voltage source is z12I2, and z11 appears to be the internal impedance of the source.

Looking into the output terminals, we see a similar Thévenin-equivalent dependent voltage source. Hence, Figure 2 is the z-parameter equivalent circuit for the open-circuit impedance parameters of a four-terminal coupling network.

Figure 2 z-parameter equivalent circuit

We can now find the open-circuit impedance parameters for the T-network shown by dashed lines in Figure 1. With the left-hand switch closed and the right-hand switch open,

${{\text{I}}_{\text{1}}}\text{=}\frac{{{\text{E}}_{\text{1}}}}{{{\text{Z}}_{\text{p}}}\text{+}{{\text{Z}}_{\text{m}}}}$

And

$\begin{matrix}{{\text{Z}}_{\text{p}}}\text{+}{{\text{Z}}_{\text{m}}}\text{=}\frac{{{\text{E}}_{\text{1}}}}{{{\text{I}}_{\text{1}}}}\text{=}{{\text{z}}_{\text{11}}} & {} & \left( 7 \right) \\\end{matrix}$

At the same time, the voltmeter measuring V2 reads

${{\text{V}}_{\text{2}}}\text{=}{{\text{Z}}_{\text{m}}}{{\text{I}}_{\text{1}}}$

Hence,

$\begin{matrix}{{\text{Z}}_{\text{m}}}\text{=}\frac{{{\text{V}}_{\text{2}}}}{{{\text{I}}_{\text{1}}}}={{z}_{21}} & {} & \left( 8 \right) \\\end{matrix}$

Similarly, with the left-hand switch in Figure 1 open and the right-hand switch closed,

$\begin{matrix}{{\text{Z}}_{\text{s}}}\text{+}{{\text{Z}}_{\text{m}}}\text{=}\frac{{{\text{E}}_{\text{2}}}}{{{\text{I}}_{\text{2}}}}\text{=}{{\text{z}}_{22}} & {} & \left( 9 \right) \\\end{matrix}$

And

$\begin{matrix}{{\text{Z}}_{\text{m}}}\text{=}\frac{{{\text{V}}_{\text{1}}}}{{{\text{I}}_{\text{2}}}}\text{=}{{\text{z}}_{12}} & {} & \left( 10 \right) \\\end{matrix}$

Rearranging Equations 7 to 10 gives the three T-network components in terms of the z-parameters:

\begin{align}& \begin{matrix}{{Z}_{m}}={{z}_{12}}={{z}_{21}} & {} & \left( 11 \right) \\\end{matrix} \\& \begin{matrix}{{Z}_{p}}={{z}_{11}}-{{z}_{12}} & {} & \left( 12 \right) \\\end{matrix} \\& \begin{matrix}{{Z}_{s}}={{z}_{22}}-{{z}_{21}} & {} & \left( 13 \right) \\\end{matrix} \\\end{align}

If we know the z-parameters for a passive coupling network, we can readily find the equivalent T-network, or vice versa. For a T-network, the forward-transfer and reverse-transfer impedances must be equal. This equality is characteristic of any passive network containing only resistance and reactance elements. Consequently, we call Zm the mutual impedance of the coupling network.

In active networks containing transistors, z12 and z21 must differ for amplification to take place. The z-parameter equivalent circuit of Figure 2 can represent both passive and active coupling networks.

Summary

• The open-circuit impedance parameters of a two-port network can be determined by representing the network by its equivalent T-network.

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# convert non linear function to linear

 0 1 Hi Dear Are there anybody that know how to convert this function to linear? min〖∑(k=1)^K▒∑(i=3)^I▒∑(t=1)^T▒〖(( i * N_it^k )-(i^' * 〖N'〗( i^' t)^k))〗^2 〗 N_it^k and 〖N'〗_( i^' t)^k) are binary variable asked 13 Nov '16, 02:01 coolman 15●2 accept rate: 0% Rob Pratt 1.2k●2●6

 0 Your formula is a bit mangled, but if I understand correctly, you can linearize as follows. Expand the summand to get an expression of the form $$a x^2 + b y^2 + c x y$$, where $$x$$ and $$y$$ are binary variables. Then replace $$x^2$$ and $$y^2$$ with $$x$$ and $$y$$, respectively. Finally, perform the usual linearization of the product of binary variables to replace $$x y$$ with binary variable $$z$$, yielding $$a x + b y + c z$$. answered 13 Nov '16, 08:27 Rob Pratt 1.2k●2●6 accept rate: 28% thank you about your answer and excuse me for not clearing of formula, most recently i find this site and i have problem with writing formula. (13 Nov '16, 23:45) coolman
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# Intrinsic carrier concentration formula

1. Apr 16, 2012

### aarnes

In the ni=$\sqrt{Nc*Nv}*e^{\frac{-Eg}{2kT}}$ formula for intrinsic carrier concentration, what values Nc,Nv and Eg take in relation to temperature? Are they each calculated for that temperature or, for example, is Eg always taken for 300K? My guess is they should all be calculated for given temperature but that would mean some solved problems I got my hands on are wrong. In them, Eg is always taken to be 1.196eV.

2. Apr 20, 2012

### reddvoid

Yes, Nc ,Nv and Eg are functions of temperature, you should take into account changes in Nc Nv when temperature is not 300k
but I think you can neglect change in Eg, you can consider it too if you want more accurate answer


text:
Bitcoin’s epic weekend pump has continued into Monday as gains are holding. BTC is still above $9k but has hit resistance once again. Analysts and traders are looking for its next move and some are now eyeing altcoins for bigger gains. Bitcoin surged through resistance yesterday to record a new 13 month high of just north of$9,300. The move resulted in 8 percent on the day for BTC has it pumped from around $8,600 to its highest price of 2019. A small pullback occurred but it retested this price a few hours later forming a double top on the day. A larger pullback then took Bitcoin back down to$8,850 but it has since recovered again and is heading upwards during the morning’s Asian trading session. At the time of writing BTC was trading at $9,100, marginally higher than yesterday morning. Daily volume has also pumped back up to$23 billion taking Bitcoin’s market capitalization to $162 billion, higher than that of the total crypto market back in March. Bitcoin market cap has not been this high since May 2018. ## What is Next For Bitcoin? Traders and analysts have been looking at the Ichimoku Cloud, which is a collection of technical indicators that show support and resistance levels, as well as momentum and trend direction. The weekly candle has opened inside the cloud which is bullish and could result in further upwards momentum; “$BTC – found resistance right at the lower cloud and has opened the weekly candle inside the cloud. This represents the highest time frame that #bitcoin has breached into the cloud…with resistance at just above $10K…”$BTC - found resistance right at the lower cloud and has opened the weekly candle inside the cloud. This represents the highest time frame that #bitcoin has breached into the cloud...with resistance at just above $10K... 174 8:05 AM - Jun 17, 2019 39 people are talking about this Twitter Ads info and privacy Looking at next moves for BTC, the analyst added; “one more step up the mountain as #bitcoin hits right at the 38.2% fib resistance making it the next target for the bulls to close over and also makes maintaining the 23.6 fib support that much more important for overall continuation…” The 38.2% Fib level is pretty much were BTC reached during May last year when it started to fall from over$9k so this does form a natural resistance level. It is an obvious take profit zone so could well be tested again with a bounce to the low side.

Current chart patterns are mirroring those from previous market cycles which could also give indication as to where Bitcoin will go next. Trader ‘CryptoHamster’ pulled out a chart from the 2013 bull run which shows similarities with what is occurring now;

CryptoHamster@CryptoHamsterIO

When you try to find similarities between the current price action of #bitcoin and what happened in history...
(Don't really think that it will play out, but if it will, short-term bears will be super-happy, i guess.)$BTC$BTCUSD

20

10:27 AM - Jun 17, 2019

See CryptoHamster's other Tweets

Many have been waiting for a correction back to $6k but this is looking more distant as the days go by and Bitcoin continues to make new 2019 highs. A big 30 percent plus dump from current levels will drop BTC back to$6,400 which, coincidentally, was the most traded price in 2018. But as we have seen, BTC has done the opposite several times this year, as predicted late last week the consolidation cracked with Bitcoin’s latest move.

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## Logic Seminar

Siddharth Bhaskar
IU Bloomington
Finitary stability
Abstract: We introduce a novel measure of complexity of set systems called thicket density which can be seen as an analogue to both VC density and Cantor- Bendixson rank of topological spaces. The analogue to VC theory is quite exact: we have "thicket" versions of dimension, shatter function, and density. In this case, infinite dimensionality captures exactly those set systems which satisfy the order property, as opposed to the independence property. These quantities satisfy several identities (including the Sauer-Shelah dichotomy) that hold verbatim in the VC case. Curiously, the proofs of these identities seem quite different, and we cannot infer the "thicket" versions from the VC versions and vice versa.
On the other hand, thicket density can be seen as a finitary version of Cantor-Bendixson rank; in particular, the polynomial/exponential growth dichotomy from "thicket Sauer-Shelah" corresponds to the countable/continuum cardinality dichotomy in the perfect set theorem. (This justifies the name "Finitary Stability.") We even believe there is a notion of "derivative of a set system" that corresponds to the Cantor derivative and induces something like the actual discrete deriative on shatter functions.
Finitary stability has at least one application in pure model theory, namely a short proof of the Erdös-Hajnal property for stable graphs, which we might go over, time permitting. Curiously enough, I discovered it not through model theory, but implicitly hidden in a 1989 paper---about something totally different---by Polish computer scientist Jerzy Tiuryn.
This talk should be widely accessible, being relatively close to first principles. It describes work that is very current, and I would like to present one proof (of "thicket Sauer-Shelah").
Tuesday November 29, 2016 at 4:00 PM in SEO 427
UIC LAS MSCS > seminars >


text:
REU Main Page

### Math Application Server

This server is an alternative to using ssh/Zeno to run math applications. You can "remote" to the server from home or in the office like ssh/Zeno, but with a graphical interface.
Applications available to you are:
• Maple 15
• Matlab 2012a
• Mathematica 8
• Python 3.2
• R
• MS Office 2010
Details on how to connect.

### Sage Notebook

This notebook will be open for the duration of the 2008 REU, and a few weeks afterwards: https://eight.math.washington.edu:7744

### Matlab and LaTeX help

• Ernie and Ryan's Matlab and LaTeX examples (ps, pdf, tex)
• Sam, Jeff, and Ernie's LaTeX starter template (starter.tex)
• Sam's advanced REU Linux/LaTeX guide, with fig/bib files and lots of references to documentation (all files, compressed, pdf)
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• Nearly comprehensive list of latex symbols (pdf)
• PSTricks Manual (pdf)
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• Not so short guide to latex (pdf)
• Jack Lee's handout on writing mathematical proofs (pdf)
• Math into LATEX (pdf)

### Graphics in LaTeX

Drawing pictures using LaTeX's picture environment is very time consuming. For graphics with any significant complexity it is probably best to import them using the graphicx package and the \includegraphics command. Example code that does this is as follows:

\documentclass{whatever}
\usepackage{graphicx}
\begin{document}

...

\begin{figure}[h]
\begin{center}
\includegraphics[width=10cm]{graphic.eps}
\caption{This caption is self-referential \label{some_label}}
\end{center}
\end{figure}

...

\end{document}

This then requires that the graphic file is in .eps format (encapsulated postscript). Adobe Illustrator is a good program to use to produce .eps graphics. It is installed on the computers in the Odegaard computer lab. You might also want to try Xfig. It is installed on Hilbert and is another drawing program that can output eps graphics. To use Xfig, just type xfig at the prompt with X-Windows running in the background.

It's also possible to convert other formats to .eps (although probably not very efficiently) with image conversion software such as Imagemagick. For example, to do this in UNIX you might type, convert mypic.jpg mypic.eps. Most mathematical software can also produce some eps graphics. For example, plots in MATLAB can be saved as .eps files and therefore be easily included in a LaTeX document.

MATLAB also has a latex command that gives the LaTeX representation of a symbolic expression, which could then be pasted directly into a LaTeX document. This could potentially be a time-saver for some.


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# Constant in Komlos conjecture

Given $n$ vectors $v_1,\dots,v_n\in\Bbb R^N$ with $\|v_i\|_2^2\leq1$ at every $i\in\{1,\dots,n\}$, Komlos conjecture states that, there is a $c\in\Bbb R$ (independent of $n,N$) such that at some $\epsilon\in\{-1,+1\}^n$, $$\Big\|\sum_{i=1}^n\epsilon_iv_i\Big\|_\infty<c.$$ What is best lower bound known for $c$?

Best upper bound is $c=O(\sqrt{\log n})$.

Is there a set of examples for $c=1+\delta$ for any $\delta>0$ where Komlos conjecture fails?

• An easy way to see that $c$ must be at least $\frac{3}{2}$ is to consider the vectors $\frac{1}{2}(1,1,1,1)$,$\frac{1}{2}(1,1,-1,-1)$, and $\frac{1}{2}(1,-1,1,-1)$. Not sure if this warrants an answer. Aug 18, 2015 at 17:17
• @KlausDraeger Yes it does if you can also tell the logic behind the example. I can see how you might have constructed this. Aug 19, 2015 at 0:52

A simple way of obtaining a lower bound $c\ge\sqrt{2}$ is to consider pairs of vectors $u,v\in\mathbb{R}$. First of all, it makes sense to focus on pairs of unit vectors for which all $\{-1,1\}$-linear combinations are as long as possible (note that this is just an interesting special case, I'm not saying that it is opotimal in any way). This is achieved when $u,v$ are orthogonal, and by checking the possible rotations we find that $u=\frac{1}{\sqrt{2}}(1,1), v=\frac{1}{\sqrt{2}}(1,-1)$ show that $c$ must be at least $\sqrt{2}$.

This example can be generalized to the sets of vectors $V_k=\{2^{-\frac{k}{2}}v_{i,k}\ |\ 0\le i\le k\}\subseteq\mathbb{R}^{2^k}$, where the $j$-th coefficient $(v_{i,k})_j$ of $v_{i,k}$ is $1$ if the $i-th$ binary digit in $j$ is $0$, and $-1$ otherwise.

The $\infty$-norm of any $\{-1,1\}$-linear combination of the vectors in $V_k$ is $(k+1)2^{-\frac{k}{2}}$, which reaches its maximum $\frac{3}{2}$ at $k=2$, with the set of vectors

$V_2=\{\frac{1}{2}(1,1,1,1),\frac{1}{2}(1,1,-1,-1),\frac{1}{2}(1,-1,1,-1)\}$.

There may be better lower bounds, but this is a start.

• It seems to me that your example is already pretty good. Could $c$ be much more than this? Aug 20, 2015 at 9:41

Taking the $v_i$ to be the columns of this matrix shows $c \geq \frac{4}{\sqrt{6}} \approx 1.633$ (I found and verified the matrix by computer experiment):

$$M = \frac{1}{\sqrt{6}}\begin{pmatrix} 1 & 1 & 1 & -1 & 1 & -1\\ 1 & 1 & -1 & 1 & -1 & -1\\ -1 & 1 & 1 & 1 & 1 & 1\\ -1 & 1 & 1 & -1 & -1 & 1\\ -1 & -1 & 1 & 1 & -1 & -1 \\ -1 & 1 & -1 & -1 & 1 & -1 \end{pmatrix}$$

There is no best lower bound, because c is an upper bound. The best known upper bound is $O(\sqrt{\log n})$, as proved by Banaszczyk in 1998.

• There is no best lower bound - I don't understand what you mean by this. I interpret the question as "what is the largest $c$ for which the conjecture statement is known to be false?" which seems well-defined.
– usul
Aug 17, 2015 at 21:16
• Turbo: Ok, so you have corrected your formuation, but it still lacks mathematical precision, because for all we know, the Komlos conjecture may be false, so that there might not be "smallest such value" for the c's, because the set is not bounded below. Now if your question is something like "Assuming that the Komlos conjecture is true, what is the infimum of the c's"? then the best currently known answer is "I don't know." usul: That's a nice interpretation. I don't think anyone looked into it with much depth. In some ad-hoc experiments I did, I found that there are counterexamples for c = 2. Aug 18, 2015 at 7:17
• @Turbo Why don't you delete the whole post? Aug 18, 2015 at 8:43


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### ImageProcessing.Compute.Path.RenderMultiPath

This node generates an image from multiple SVG path data inputs and styling information. In the SVG format a <path> tag contains both path data and styling information. The path data is given in the "d" attribute and styling is given in additional attributes, e.g.,

<path d="M 72 182 L 286 288" stroke='#000000' stroke-width='4'/>

The RenderMultiPathNode requires this input in separate text vectors. The path data can be defined as a vector of text nodes via the paths input slot. The styling information can be given as a vector of text nodes via the styles input slot. The advantage is that styling information can be re-used for different paths.

E.g., to represent the 4-pixel wide black line from the example above, one would need to add a Text node with content "M 72 182 L 286 288" to the vector at the paths input slot and a Text node with " stroke='#000000' stroke-width='4' " to the vector at the styles input slot.

The two tables below describe the path data format and styling information format. However, these are provided only for reference. Typically, there are other nodes that generate the required input. For example, path data can be created interactively with the ImageProcessing.Compute.Path.Construct node. For generating styling information the node ImageProcessing.Compute.Path.PathStyle is available.

The example MultiplePaths demonstrates the usage of these nodes.

#### Path data format

 Command character Number of numeric values Example Example semantic M 2 M 30 40 Move to absolute position $x=30$, $y=40$ m 2 m 10 20 Move to relative position $x=\tilde{x}+20$, $y=\tilde{y}+10$, where $\tilde{x}$ and $\tilde{y}$ define the position after execution of the previous command L 2 L 50 60 Line to absolute position $x=50$, $y=60$ l 2 l 20 10 Line to relative position $x=\tilde{x}+20$, $y=\tilde{y}+10$ H 1 H 50  Horizontal line to absolute $x$-position $x=50$, $y=\tilde{y}$ h 1 h 20  Horizontal line to relative $x$-position $x=\tilde{x}+20$, $y=\tilde{y}$ V 1 V 10  Vertical line to absolute $y$-position $x=\tilde{x}$, $y=10$ v 1 v 10  Vertical line to relative $y$-position $x=\tilde{x}$, $y=\tilde{y}+10$ C 6 C 10 20 30 40 50 60 Cubic Bézier curve to absolute position $x=50$, $y=60$, where $x_1=10$, $y_1=20$ and $x_2=30$, $y_2=40$ define the absolute positions of the curve's control points c 6 c 10 20 30 40 50 60 Cubic Bézier curve to relative position $x=\tilde{x}+50$, $y=\tilde{y}+60$, where $x_1=\tilde{x}+10$, $y_1=\tilde{y}+20$ and $x_2=\tilde{x}+30$, $y_2=\tilde{y}+40$ define the positions of the curve's control points S 4 C 30 40 50 60 Cubic Bézier curve to absolute position $x=50$, $y=60$, where $x_1=2\tilde{x}-\tilde{x}_2$, $y_1=2\tilde{y}-\tilde{y}_2$ and $x_2=30$, $y_2=40$ define the positions of the curve's control points s 4 c 30 40 50 60 Cubic Bézier curve to relative position $x=\tilde{x}+50$, $y=\tilde{y}+60$, where $x_1=2\tilde{x}-\tilde{x}_2$, $y_1=2\tilde{y}-\tilde{y}_2$ and $x_2=\tilde{x}+30$, $y_2=\tilde{y}+40$ define the positions of the curve's control points Q 4 Q 10 20 50 60 Quadratic Bézier curve to absolute position $x=50$, $y=60$, where $x_1=10$, $y_1=20$ define the absolute position of the curve's control point q 4 c 10 20 50 60 Quadratic Bézier curve to relative position $x=\tilde{x}+50$, $y=\tilde{y}+60$, where $x_1=\tilde{x}+10$, $y_1=\tilde{y}+20$ define the position of the curve's control point T 2 T 50 60 Quadratic Bézier curve to absolute position $x=50$, $y=60$, where $x_1=2\tilde{x}-\tilde{x}_1$, $y_1=2\tilde{y}-\tilde{y}_1$ define the position of the curve's control point t 2 t 50 60 Quadratic Bézier curve to relative position $x=\tilde{x}+50$, $y=\tilde{y}+60$, where $x_1=2\tilde{x}-\tilde{x}_1$, $y_1=2\tilde{y}-\tilde{y}_1$ define the position of the curve's control point Z 0 Z closes the current path z 0 z closes the current path

Detailed explanation of the SVG path data format can be found in the SVG specification. The elliptical arc commands "A" and "a" are currently not fully supported.

#### Path style format

 Attribute name Attribute value Example Example semantic stroke color or "none" stroke="black" Stroke path with black color stroke="#000000" Stroke path with black color stroke="#FF0000" Stroke path with red color stroke="none" Not stroke the path stroke-opacity float value in range [0.0, 1.0] stroke-opacity="0.5" Stroke path with 50% opacity stroke-width float value stroke-width="4" Width of stroke in pixels stroke-linecap "butt" or "square" or "round" stroke-linecap="round" Stroke starts and ends with a round cap stroke-linejoin "miter" or "round" or "bevel" stroke-linejoin="round" Subpaths are joined with a round edge stroke-dasharray array of dash values stroke-dasharray="5,5" Draw a dashed path by toggling between stroke and blank every 5 pixels fill color or "none" fill="white" Fill path with white color fill="#FFFFFF" Fill path with white color fill="#00FF00" Fill path with green color fill="none" Not fill the path fill-opacity float value in range [0.0, 1.0] fill-opacity="0.25" Fill path with 24% opacity

Detailed explanation of the SVG painting style attributes can be found in the SVG specification. Only the attributes that are listed above are supported.

### ImageProcessing.Compute.Path.PathStyle

The PathStyle node outputs a text node with path style information formatted as described in the table above. It is designed to work in combination with the ImageProcessing.Compute.Path.RenderMultiPath node and the ImageProcessing.Compute.Path.ToSVG node.

The example MultiplePaths demonstrates the usage of this node.


text:
# What algorithm is used for AudioLocalMeasurements: FundamentalFrequency?

GROUPS:
 Frank Scherbaum 1 Vote I want to determine the "pitch"-tracks for a number of monophonic audio files. Does anyone know what algorithm is used for AudioLocalMeasurements[a, "FundamentalFrequency"] ?
Answer
10 months ago
2 Replies
 Sean Clarke 1 Vote Hi. I asked the developers in charge of this. I'm not personally familiar with this, but it's apparently a variation on the YIN algorithm:https://en.wikipedia.org/wiki/Pitch_detection_algorithm
Answer
10 months ago
 Thanks, is there more information about the tuning parameters?
Answer
10 months ago


text:
# Understanding CircleCI

The official about us page of CircleCI states the following:

CircleCI allows teams to rapidly build quality projects, at scale. Our mission is to give people everywhere the power to build and deliver software at the speed of imagination.

Their package allows us as a developer to develop on the principles of continuous integration.

Continuous integration is a software development strategy that increases the speed of development while ensuring the quality of the code. Developers continually commit small increments of code (at least daily, or even several times a day), which is then automatically built and tested before it is merged with the shared repository.

The official repo of FEniCS uses CircleCI to test the commits by different developers working on the repository before merging them into the master branch. In this post, I would like to introduce you to the concept of CircleCI and how I have incorporated it into my GSoC repository.

The first step is to create a “.circleci” folder in the root directory of your project. Inside that directory, we have to create a “config.yml” file. Inside “config.yml” we would load our commands that would then be utilized by CircleCI to build and test our code. After configuring our “config.yml” we will connect our repository with our CircleCi account. More on it here. That’s it. Now every time we commit our code to our Github repo, CircleCI would automatically build and run tests on our code.

CircleCI can build your code inside a docker image. You can read more about docker image here. When you ask CircleCI to build and test your repository it essentially performs the following steps:

• Loads the specified docker image.
• Loads the contents of your repository into the root of the docker image.
• Build and run your repository inside docker image.

First, we start with specifying which image we would like to use for our build. For my own repository, I am using the official dolfinx image. This command alone would load the docker image and my repository into the root of that image.

version: 2
jobs:
build:
docker:
- image: quay.io/fenicsproject/dolfinx:real

Next, we could just simply specify the commands that we wish to execute inside the image. Here, we first get inside the python directory inside directory Scripts and then run the program “understanding_mesh_workflow.py”.

steps:
- checkout # check out the code in the project directory
- run: |
cd Scripts/python
python3 ./understanding_mesh_workflow.py

This simple config file is all that was required to test the code using the concepts of CI. You can see the build output here.

Categories: FEniCS


text:
# how is cosmological constant used in equations? [closed]

People say that they use cosmological constant in an equation to remove a non required value. For example, the cosmological constant $\Lambda$ appears in Einstein's field equation in the form of:

$$R_{\mu \nu} -\frac{1}{2}R\,g_{\mu \nu} + \Lambda\,g_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu}$$

But then how can we just put a value in an equation and remove some other value like in the case of the cosmological constant?

For example, if I build a house and after building I notice that the dimensions of a certain room in the building is not right I just cant write in my blue print that my dimensions are wrong. I have to actually go and make it correct. But here we are just putting a value and eliminating some other value. So if anyone can tell me how this thing happens I will really appreciate that.

## closed as unclear what you're asking by Ben Crowell, Brandon Enright, JamalS, ACuriousMind♦, NeuneckNov 14 '14 at 14:02

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• "...*remove some other value like in the case of the cosmological constant*": what do you believe has been removed here? – WetSavannaAnimal Nov 14 '14 at 10:47

Einstein's equation is a mathematical model devised to provide an approximate description of the universe. We know it's approximate because it generically predicts singularities that we believe to be unphysical, and it takes no account of quantum mechanics.

So the question is whether the equation:

$$R_{\mu \nu} -\frac{1}{2}R\,g_{\mu \nu} + \Lambda\,g_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu} \tag{1}$$

is a better approximation to the real world than the equation:

$$R_{\mu \nu} -\frac{1}{2}R\,g_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu} \tag{2}$$

The only way to tell which is the better approximation is by experiment. Prior to the observation of cosmic acceleration equation (2) seemed to be a perfectly good description of the universe, but once the acceleration had been measured equation (1) is a better approximation than equation (2).

Your analogy of the house is misleading because a blueprint is used to construct the house, but Einstein's equation wasn't used to construct the universe. A better analogy would be to make a plan of an existing house. Suppose you make a plan of a house, but a bit later you notice that the size of a room on the plan isn't the same as the size of the actual room. Obviously you made a mistake when drawing up the plan, so you change the plan. You can think of Einstein's equation as a plan of the universe. When cosmic acceleration was observed this showed the plan was wrong so we had to modify it by adding the cosmological constant.

• The cosmological constant is not a constant value, it's what we use to describe the propulsion effect of dark energy on the expansion of the universe, in comparison to the gravity pulling the universe back together. Dark energy itself may be a form of energy that has it's own gravity, changing the equations for how much propulsion it takes to overcome what amount of gravity, all forms of propulsion have limits, and it is impossible to know how much dark energy there is, because if there was more, there would also be more gravity, and the only reason we know it's there at all, is it must be. – rowanman28 Mar 3 '15 at 4:56


text:
# Assessing fit with identity line in Q-Q plot

I'm using a QQ plot to asses the similarity between two observed distributions. I need a number that quantifies how much the quantile-quantile plot separates itself (is there a better word for this concept?) from the identity line y=x (IL).

The R^2 coefficient is of no use since it compares the points with the best fit line, not the IL.

I was thinking of simply summing the absolute distances of each quantile to the identity line so that the larger this number the more I can say they are separated from the IL.

Is there a better/recommended way to go about this? I'm just looking for a number that can be used to tell at a glance how far away my quantiles are from the IL.

Example of one of my QQ plots in graph below. In this example, I should get a somewhat extreme (whether it is high or low) value of whatever statistic I end up using (the data set for the plot can be downloaded here).

• You could look at area between the curves. I am not sure how to do this, but it is surely do-able. – Peter Flom Dec 5 '13 at 12:35
• @PeterFlom I could do that. Is that a better estimate then simply summing the absolute distances points-IL? – Gabriel Dec 5 '13 at 12:37
• It's likely to be quite similar, but I've heard more discussion of area than summed distances. Of course, the area is just the limit of the average distances as the number of distances approaches infinity (integral calculus) – Peter Flom Dec 5 '13 at 12:39
• Summarizing whether variables agree is answered by concordance correlation. See e.g. stats.stackexchange.com/questions/3061/… – Nick Cox Dec 5 '13 at 12:44
• "I was thinking of simply summing the absolute distances of each quantile to the identity line so that the larger this number the more I can say they are separated from the IL" -- well, that works as a measure of discrepancy, but the sum of squared distances would also work. There are other measures, such as the Kolmogorov-Smirnov-distance ($D$ in the two sample test) – Glen_b Dec 5 '13 at 23:39

The question underlines a simple fact often overlooked even in statistical circles: for two variables, correlation (and its square) quantify linearity of relationship (does $y = a + bx$?), not agreement (does $y = x$?). A trivial example is that an exact proportionality $y = bx$ implies correlation of 1 for any $b > 0$, but as $b$ moves away from 1 agreement (with ideal as equality of corresponding values) certainly diminishes.

Comparison of distributions quantile to quantile, i.e. comparing corresponding order statistics, imparts one or two twists to comparisons that might be made between paired variables. The first twist is required if two distributions are being compared through different-sized samples, say sizes of $m, n, m < n$. Then the larger set would usually be reduced to $m$ by interpolation. The second twist is always present: sets of quantiles are weakly increasing, so measures of agreement should be interpreted in that light of the constraint that implies.

Concordance correlation (a measure most associated with L. I-K. Lin, but invented earlier) is a correlation-like measure of agreement. If (and only if) two variables have identical values, the concordance correlation will be 1. Note that the question makes no sense unless the variables have the same units of measurement or more generally are recorded in the same way. The concordance correlation can be factored into the usual Pearson correlation and a bias correction. Conversely, Pearson correlation and concordance correlation both return 1 when $y = x$ and departures from that condition will reduce concordance correlation even if they don't reduce Pearson correlation.

For an informal introduction to this area, see

Cox, N.J. 2006. Assessing agreement of measurements and predictions in geomorphology. Geomorphology 76: 332-346. http://www.sciencedirect.com/science/article/pii/S0169555X05003740

Here "in geomorphology" indicates the field of the examples, not a restriction of statistical scope. However, use of concordance correlation for paired quantiles is not covered there.

Using the data for the example above these calculations in Stata show what is possible. Stata users would need to download concord after identifying locations using search concord, sj. A complete explanation is not given here. The main point for this example is that concordance correlation at 0.259 is quite different from the Pearson correlation at 0.989. (Incidentally, that implies $R^2$ of 0.978.) No adjustments are made here for the effects of monotonicity constraints on confidence or significance results.

. concord y x

Concordance correlation coefficient (Lin, 1989, 2000):

rho_c   SE(rho_c)   Obs    [   95% CI   ]     P        CI type
---------------------------------------------------------------
0.259     0.027      96     0.205  0.312    0.000   asymptotic
0.204  0.311    0.000  z-transform

Pearson's r =  0.989  Pr(r = 0) = 0.000  C_b = rho_c/r =  0.262
Reduced major axis:   Slope =     3.133   Intercept =     0.008

Difference = y - x

Difference                 95% Limits Of Agreement
Average     Std Dev.             (Bland & Altman, 1986)
---------------------------------------------------------------
0.286       0.170                 -0.048      0.620

Correlation between difference and mean = 0.994

Bradley-Blackwood F =  1.6e+04 (P = 0.00000)


text:
## Wiener-Khintchine Theorem

Recall the definition of the Autocorrelation function of a function ,

 (1)

Also recall that the Fourier Transform of is defined by
 (2)

giving a Complex Conjugate of
 (3)

Plugging and into the Autocorrelation function therefore gives

 (4)

so, amazingly, the Autocorrelation is simply given by the Fourier Transform of the Absolute Square of ,
 (5)

The Wiener-Khintchine theorem is a special case of the Cross-Correlation Theorem with .


text:
The search of higher multipole radiation in gravitational waves from compact binary coalescences by a minimally-modelled pipeline
Abstract The coherent WaveBurst (cWB) pipeline implements a minimally-modelled search to find a coherent response in the network of gravitational wave detectors of the LIGO-Virgo Col-laboration in the time-frequency domain. In this manuscript, we provide a timely introduction to an extension of the cWB analysis to detect spectral features beyond the main quadrupolar emission of gravitational waves during the inspiral phase of compact binary coalescences; more detailed discussion will be provided in a forthcoming paper [1]. The search is performed by defining specific regions in the time-frequency map to extract the energy of harmonics of main quadrupole mode in the inspiral phase. This method has already been used in the GW190814 discovery paper (Astrophys. J. Lett. 896 L44). Here we show the procedure to detect the (3, 3) multipole in GW190814 within the cWB framework.
Authors:
; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
Award ID(s):
Publication Date:
NSF-PAR ID:
10344397
Journal Name:
Journal of Physics: Conference Series
Volume:
2156
Issue:
1
Page Range or eLocation-ID:
012081
ISSN:
1742-6588
5. Abstract We discuss observational strategies to detect prompt bursts associated with gravitational wave (GW) events using the Australian Square Kilometre Array Pathfinder (ASKAP). Many theoretical models of binary neutron stars mergers predict that bright, prompt radio emission would accompany the merger. The detection of such prompt emission would greatly improve our knowledge of the physical conditions, environment, and location of the merger. However, searches for prompt emission are complicated by the relatively poor localisation for GW events, with the 90% credible region reaching hundreds or even thousands of square degrees. Operating in fly’s eye mode, the ASKAP field of view can reach $\sim1\,000$ deg $^2$ at $\sim$ $888\,{\rm MHz}$ . This potentially allows observers to cover most of the 90% credible region quickly enough to detect prompt emission. We use skymaps for GW170817 and GW190814 from LIGO/Virgo’s third observing run to simulate the probability of detecting prompt emission for GW events in the upcoming fourth observing run. With only alerts released after merger, we find it difficult to slew the telescope sufficiently quickly as to capture any prompt emission. However, with the addition of alerts released before merger by negative-latency pipelines, we find that it should be possible to searchmore »