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Welcome to our final project for the CSE 111: Database Systems course!.
Browse through the vast history of games and how successful they were through this website. Data goes up to 2016, with more on the way as we get it.
If there's a game you're thinking belongs on here but you can't find it through the search function, feel free to send it in to us over here. We will review the submission and, if appropriate, add it to our database. You can also send in any bugs or errors you find over there.
Hello! I am a third year Computer Science and Engineering Major with an English Minor. I was responsible for the frontend of the website, as well as finding appropriate data for the database. My favorite games include NieR: Automata, Disco Elysium, and Outer Wilds.
My name is Francesco Gnerre, a fourth year computer science major with a minor in applied math. I worked on the backend of this site. My favorite games include Minecraft and League of Legends.


--- layout: default title: Using Plugins ---
To use a plugin you need two things:
Include it when you instantiate List.js
new List('list-id', {
plugins: [ ListPagination(), NameOfOtherPlugin(options) ]
});
It is also possible to add options and load multiple instances of the same plugins (if the plugin itself allows it).
If the property name
is added in the plugin option
parameter does the plugin become
accessible through listObj.namePropertyValue.This is useful when having multiple instances of the same plugin.
var myList = new List('list-id', {
plugins: [
ListPagination({ name: "topPagination", paginationClass: "top-pagination" }),
ListPagination({ name: "bottomPagination", paginationClass: "bottom-pagination" })
]
});
console.log(myList.topPagination);
console.log(myList.bottomPagination);
Go to: /docs/plugins/build


THE APOSTLE By Harvey Karten, Ph.D. October films Director: Robert Duvall Writer: Robert Duvall Cast: Robert Duvall, Farrah Fawcett, Miranda Richardson, Todd Allen, John Beasley, June Carter Cash, Walton Goggins, Billy Joe Shaver, Billy Bob Thornton, Rick Dial According to Robert Duvall, who turns in a tour de force performance as a dancing, shouting, hell-raising Pentecostal- type preacher, Hollywood is generally afraid of making movies with religious issues. When such a picture does get a released, the fundamentalist preacher fellas are either patronized or ridiculed. During the press conference following a screening at the New York Film Festival, he might have mentioned, as an example, Fredric March's depiction of the William Jennings Bryan character in "Inherit the Wind," based on the 1925 Scopes trial, in which the prosecuting attorney is reduced to babbling by the more rational Paul Muni, succumbing to a heart attack after the trial. In "The Apostle," by contrast, which is Robert Duvall's third directorial effort, the director plays the role of preacher with such flair, such authenticity, such downright dramatic impact, that we in the audience are tempted to rise out of our seats and join the congregation, whatever our religious convictions. Though religion appears to be its center, this story of an individual who is a good person, however eccentric--who goes bad for a spell and later restores his dignity and generosity--is more about a man than about a creed. Seldom does a movie penetrate the soul of an individual in such depth, but Duvall does exactly that, spending a solid two and one-half hours in virtually every scene to gave his viewers a firm examination of his every dimension. He seems a bit different in each vignette, experiencing a lifetime of a episodes in a short period as he goes through his paces tending to a backwater parish in the Louisiana bayous (actually filmed in that state's Lafayette) which goes by the name Bayou Boutte. Eulis "Sonny" Dewey's whimsically erratic but essentially conscientious and generous nature comes through in the very opening scene in Texas where, driving with his mother (played by singing legend June Carter Cash) he comes upon a serious auto accident. Hopping down a hill, he runs into a mortally injured young man and, without regard to the victim's creed or wishes, asks him to accept Jesus, whom he will "soon meet." Back at the ranch, however, things are not going so well with his wife Jessie (Farrah Fawcett), who tires of his absences and has taken up with a young minister, Horace (Todd Allen). Begging her not to ask for a divorce, he gets drunk and, at a Little League game, clubs the boy friend fatally in the head with a baseball bat. Discarding his identity, he flees Texas for the Louisiana bayou where he becomes friendly with Rev. Blackwell (John Beasley), who left his congregation because of health problems, and talks him into helping re-create his church.  You're in for a surprise if you think that a man in his position leads a sedentary, relaxed life in a remote region, inhabited by good-natured but essentially poorly-educated people who take to gospel song and fire-and-brimstone preaching like a terrier to a buried bone. One day he raises a voluntary crew of children to clean, paint, and assemble a large sign for the old abandoned shack. At another time, he must fend off an attack by a racist provocateur (Billy Bob Thornton) who, furious at the idea of an integrated house of worship, is about to demolish the building with a bulldozer. He wins the hostile man over in a remarkable way. At another time he romances a southern belle, Toosie (Miranda Richardson), who is separated from her husband and fascinated by his company and distinct aura. Duvall excels not only in the movie's principal role but as its director, building each scene skillfully on its preceding one while welding a group of ordinary Louisiana citizens with a cast of remarkable actors. Miranda Richardson has particular merit, knowing just how to play a fascinating "date," mixing a come-hither twinkle with a maybe-next-time prospect. The picture is photographed in a straightforward way by cameraman Barry Markowitz, featuring a soundtrack of glorious gospel song, and allegedly has been pared down with the use of a computer in Mr. Duvall's Virginia farmhouse. The picture is too long and could stand a shave, particularly of Duvall's final sermon before his inevitable arrest for his crime of passion. Thirteen years in the making, this five million dollar production is an obvious showcase for Duvall, who will doubtless garner an Oscar nomination after the picture is released--for that purpose--in L.A. this December, to open nationally on January 30, 1988. Not Rated. Running Time: 148 minutes. (C) 1997 Harvey Karten
The review above was posted to the
rec.arts.movies.reviews newsgroup (de.rec.film.kritiken for German reviews).
The Internet Movie Database accepts no responsibility for the contents of the
review and has no editorial control. Unless stated otherwise, the copyright
belongs to the author.
Please direct comments/criticisms of the review to relevant newsgroups.
Broken URLs inthe reviews are the responsibility of the author.
The formatting of the review is likely to differ from the original due
to ASCII to HTML conversion.
Related links: index of all rec.arts.movies.reviews reviews


Key Storage Providers¶
By default the data protection system employs a heuristic to determine where cryptographic key material should be persisted. The developer can override the heuristic and manually specify the location.
Note
If you specify an explicit key persistence location, the data protection system will deregister the default key encryption at rest mechanism that the heuristic provided, so keys will no longer be encrypted at rest. It is recommended that you additionally specify an explicit key encryption mechanism for production applications.
The data protection system ships with two in-box key storage providers.
File system¶
We anticipate that the majority of applications will use a file system-based key repository. To configure this, call the PersistKeysToFileSystem configuration routine as demonstrated below, providing a DirectoryInfo pointing to the repository where keys should be stored.
sc.ConfigureDataProtection(configure =>
{
// persist keys to a specific directory
configure.PersistKeysToFileSystem(new DirectoryInfo(@"c:\temp-keys\"));
});
The DirectoryInfo can point to a directory on the local machine, or it can point to a folder on a network share. If pointing to a directory on the local machine (and the scenario is that only applications on the local machine will need to use this repository), consider using Windows DPAPI to encrypt the keys at rest. Otherwise consider using an X.509 certificate to encrypt keys at rest.
Registry¶
Sometimes the application might not have write access to the file system. Consider a scenario where an application is running as a virtual service account (such as w3wp.exe’s app pool identity). In these cases, the administrator may have provisioned a registry key that is appropriate ACLed for the service account identity. Call the PersistKeysToRegistry configuration routine as demonstrated below to take advantage of this, providing a RegistryKey pointing to the location where cryptographic key material should be stored.
sc.ConfigureDataProtection(configure =>
{
// persist keys to a specific location in the system registry
configure.PersistKeysToRegistry(Registry.CurrentUser.OpenSubKey(@"SOFTWARE\Sample\keys"));
});
If you use the system registry as a persistence mechanism, consider using Windows DPAPI to encrypt the keys at rest.
Custom key repository¶
If the in-box mechanisms are not appropriate, the developer can specify his own key persistence mechanism by providing a custom IXmlRepository.


Agility
Years of training allows the Occultist to act deftly in combat and execute maneuvers with speed and precision.
${value}% Chance to Avoid Projectiles
6 7.5 9 10.5 12 13.5 15 16.5 18 19.5
+${value}% Casting Speed
22 30 38 46 54 65 73 81 89 100
+${value}% Defensive Ability
6 7.5 9 10.5 12 13.5 15 16.5 18 19.5
MaxLevel
6
UltimateLevel
10


Alviz is a fun way to learn algorithm and visualization in Javascript.
It is pre-loaded with Javascript libraries to support drawing in Turtle mode or
Artist mode. Read more below to learn how to use the commands.
Help Wanted:
I found it fascinating on recursive algorithms (a.k.a 'recipes') which make beautiful arts
such as fractals, snowflakes, etc. If you have an interesting recipe to share (ie, recursively drawing a cat),
please send me a pull request or email me at hoangdangninh[at]gmail[dot]com.
Note: Make sure to turn off Vimium (Chrome) or Vimperator (Firefox) if you are an avid Vim user.


Ocean [aw58], ocean, in The Inner Sea
Last updated Turn #64
Ocean [aw58], ocean, in The Inner Sea
------------------------------------------------------------------------
14: Fast Winds [ak2q], ship, 2 hulls, 1 rowing port, 6 sails, 8% damaged,
14: arrived from Ocean [aw59], owner:
14: Dorothy [v1b], Pen, with eight sailors
15: Fast Winds [ak2q], ship, 2 hulls, 1 rowing port, 6 sails, 8% damaged,
15: departed for Ocean [aw57].
Routes leaving Ocean:
North, plain, to Buffalo Plains [av58], Imperia, 2 days
East, to Ocean [aw59], 6 days
South, to Ocean [ax58], 6 days
West, to Ocean [aw57], 6 days
Inner locations:
Island [ac1z], island, 1 day, owner:
Cyclops [hv3u], cyclops, number: 23, with five winged horses
It is raining.


Round Up
Second week into the Pre-Course and things are running smoothly. After the pains I went through setting up all the software (on windows) last week, this week has been a breeze. Having completed the Git / Github work for week 2 already, I moved on to finishing off Chris Pine's Learn to Program. I've completed up till chapter 12 and so only two more to go! With the exception of chapter 10, on recursion, things haven't been too confusing. Recursion was a rather strange concept to wrap my head around at first, but after completing the exercises and playing around you just learn by seeing what does and doesn't work.
Recursion is a technique where you create a method that calls itself. I'll try explain through an example I definitely didn't steal off stackoverflow:
How do you empty a vase containing N flowers?

This code checks first if the number of flowers in a vase is greater than 0. If that is false then the program will simply return 'The vase is empty' But if it is true it'll call the method 'empty' on the (numberFlowersInVase - 1). This will keep going and going until the vase is empty.
I know your asking the same thing I did, "Why can't you just use a loop?" and to be honest you could, but this is a lot more 'elegant'. Now I don't know about elegant but it is a lot tidier... and cooler.
On Friday, week 2's challenge was released, the Octocat challenge. This involved the basic uses of Git and Github commands and involved the creation of a repository containing files which themselves contained questions. The task was to then create a separate branch of this repository which would be made up of the same files, but instead of the files containing questions they would contain the answers. Essentially creating two versions of the files, one with questions, one with answers. We also had to fork a provided repository, add our adaptations to it and then create a pull request. This is just editing a file on someone else's repository and then submitting your changes for them to review. All very easy but absolutely essential things to know and understand.
Only 2 weeks to go now until the real work begins and I can't wait.
Chris


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Head back home here


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GOVERNMENT OF THE DISTRICT OF COLUMBIA Department of Behavioral Health Open Government Report 2014 To institutionalize a culture of transparent and open government, accountability, and to expand opportunities for resident participation and collaboration, this Report describes how the Department of Behavioral Health has and will continue to develop and enhance transparency, public participation, and collaboration. In accordance with Mayor’s Order 2014-170, this report addresses the following topics: 1) Transparency Describe the steps your agency has taken or plans to take to be more transparent. Please include a description of: How and to what extent your agency shares information with the public, e.g. publication of information in the District register and on the agency website, press releases, and documents in the agency’s FOIAXpress reading room. The Department of Behavioral Health (DBH) communicates with the public through timely posting of information on its website, including a twice yearly report on service costs, utilization and access to mental health services, a monthly report on key performance measures at Saint Elizabeths Hospital, and regular news releases describing services, new initiatives, or public events/meetings. DBH also posts a monthly calendar of meetings for providers. The website is used to provide notice to the public of contract solicitations and funding availability for grant initiatives. The DBH Training Institute which is open to the public posts a calendar of classes for online registration. DBH posts notice of proposed rules and regulations in the District Register. How the agency meets its obligations pursuant to the District’s Freedom of Information Act and Open Meetings Act. DBH responds in a timely manner to all Freedom of Information Act (FOIA) requests. DBH uses the new FOIAXpress software to process FOIA requests and share information. Steps your agency will take to increase public access to information. DBH will begin in January issuing a news release that calls attention to a proposed rule or regulation published in the District Register and to solicit public comment. Steps your agency will take, including an implementation timeline, to webcast live and archive on the internet board or commission meetings. (This question only applies to agencies that are overseen or advised by a board or commission that is subject to the Open Meetings Act.) NA 1 How your agency has taken or plans to take steps to make more of its data available to the public. DBH is working closely with the Office of the City Administrator (OCA) and the Office of the Chief Technology Officer (OCTO) to identify a significant number of datasets that OCA and OCTO expect to publish online later this year. If no dataset from DBH is selected to be included in this 2014 release, we will continue to work with OCA and OCTO to identify datasets appropriate to publish in 2015. In the future, requirements for open data reporting will be provided by the Chief Data Officer. 2) Public Engagement and Participation Describe the steps your agency has taken or plans to take to enhance or expand opportunities for the public to participate in agency decision-making. Please include a description of: How your agency provides online access to proposed rules and regulations and secures public input on them. Please list links to specific websites. How your agency shares information and resources to keep the public properly informed, e.g. community meetings, public hearings, FAQs, and ways the public can provide input such as Twitter, grade.dc.gov, email contacts. NA DBH regularly participates in community meetings, health fairs and other special events to share information about available services. DBH funds four Prevention Centers that service all eight wards. DBH also maintains two websites aimed at youth about drug prevention including one about the dangers of synthetic marijuana, and the Prevention Resource Clearinghouse which provides educational materials on maintaining healthy living free of alcohol, tobacco, and other drugs and reducing factors that place youth, families and communities at risk. How your agency identifies stakeholders and invites their participation. DBH works closely with advocates, providers, consumer/clients and their families, and government agencies with similar functions. However, we use multiple ways to share information with the general public and solicit input. Steps your agency will take to improve public engagement and participation including any new feedback tools or mechanisms the agency is considering. DBH will begin in January issuing a news release that calls attention to a proposed rule or regulation published in the District Register and to solicit public comment. In early 2015, DBH is establishing the Behavioral Health Council made up of government agencies, providers, advocates, consumers/clients and their families to advise the Director on policies and services. 2 3) Collaboration Describe the steps your agency has taken or plans to take to enhance or expand cooperation among departments, other governmental agencies, the public, and nonprofit and private entities in fulfilling its obligations. Please include a description of: How your agency currently collaborates with the above parties. Please list links to specific websites if appropriate. DBH spearheaded last year the establishment of the Creating Community Solutions Task Force which is now supported by the DC Children and Youth Investment Trust Corporation to call attention to mental health needs and available services with a focus on children and young adults. DBH also participates in the DC Collaborative for Mental Health in Pediatric Primary Care to increase collaboration between pediatric primary care providers and child behavioral health specialists. DBH coordinates the Prevention Policy Consortium, which consists of 15 different District agencies, that provides guidance on the development and implementation of a strategic substance use prevention plan. Steps your agency will take to improve collaboration with the above parties including any new feedback tools or mechanisms the agency is considering, e.g. prizes, competitions, and other innovative methods. DBH will continue to create opportunities to strengthen collaboration. For example, as part of its annual Children’s Mental Health Awareness Day celebration, DBH routinely offers prizes to increase participation among public school students. 3


Individual: Tuesday
http://www.w3.org/2006/time#Tuesday
Types (1)
DayOfWeek
Usage (1)
DayOfWeek
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{
Wednesday
,
Friday
,
Saturday
,
Tuesday
,
Monday
,
Sunday
,
Thursday
}
OWL HTML inside


After the Internet of People, the next best revolution is the Internet of Things. My Passion towards IoT drove me to take a Masters in Computer Systems Engineering Specialization in IoT.
I have been doing Cloud and Full-stack Development for the past two years. My real strength is my ability to communicate and fully understand my client's needs and build solutions that solve interesting problems while meeting their business objectives. What I am looking for is a company that I could add value to and collaborate as a team to gain valuable experience that will help grow in my career and contribute back to my society.
MQTT, CoAP
AWS: EC2, IAM, S3, SNS, SES, Dynamo DB, Circle CI, CodeDeploy ELB, Lambda, CloudWatch
HTTP, DNS, Google Cloud, TCP/IP, UDP
Golang, Kops, Kubernetes, Terraform, Ansible, CI/CD, Jenkins, Docker
Logging with Elasticsearch,
Fluentd, and Kibana
Monitoring/Metric: Prometheus and Grafana
Front-end: AngularJS, React
NodeJS, .NET, JSON, HTML, CSS, git, MEAN & REST, SQL, MysSQL,
PostgreSQL, MongoDB, Unit testing
Java, JavaScript, Python, C++, Agile SCRUM, Git, Maven, Jenkins
Web Chair (IoT Connect Club)
Application Developer Intern
As an intern, Harsha worked for us in the IT Section, for the Sec. of the Commonwealth of MA. He did a good job learning new skills, writing code independently for a new project, and collaborating with me on the re-design of an old legacy project. He was good at taking direction, researching how to achieve certain results, and coming to me when stuck. Easy to work with; should fit in well in different work environments
I had the opportunity to have Harsha Kalyanaraman intern on my team for 7 months, in collaboration with Northeastern University’s Co-op program. We worked together on a web application for the 2020 Census; Front End, Back End and SQL Database. Here at SOC we have a fairly strict set of standards both in coding and design, which tends to make a bit of a learning curve for our interns, and a small office with a group that works closely together each day. However, Harsha showed strong interpersonal skills and being able to learn on the fly. He also has a keen eye for detail as well as an eagerness to learn.Over the 7 months interning, Harsha was always willing to put in an extra effort and showed great improvement in his coding skills. I am confident in Harsha, he has a promising future in the tech industry ahead and comes highly recommended.


Working Capital, Leasing, Term Loans, and Equipment Financing. Get your funding in as fast as 24 hours.
You are unique and so is your business. Lenders compete to offer you the best cost of funds. At the rate you deserve!
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Funding options from $1,000 to $50,000,000. Equipment financing is a powerful fiscal instrument that nearly 80% of all US business owners take advantage of. Maximize your business tax deducations and write off up to 100% of your monthly payment


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Ph.D Student
SUNY University at Buffalo Department of Economics
Research Interests
Labor Economics, Applied Microeconomics, Applied Econometrics
Contact Information
E-mail<EMAIL_ADDRESS>Phone: (+1)716-580-000*
Writing Samples:
The impact of family involvement in the internationalization of Chinese family businesses
The population growth forecast model
=========================
News and Information:
UB website=========================
The web address of my page: https://sites.google.com/view/liqiangliu
Creation Date: Sep.7th, 2019


Lesson Plan Title: Shopping Cart Concurrency Flaw
Concept / Topic To Teach:
Web applications can handle many HTTP requests simultaneously. Developers often use variables that are not thread safe. Thread safety means that the fields of an object or class always maintain a valid state when used concurrently by multiple threads. It is often possible to exploit a concurrency bug by loading the same page as another user at the exact same time. Because all threads share the same method area, and the method area is where all class variables are stored, multiple threads can attempt to use the same class variables concurrently.General Goal(s):
For this exercise, your mission is to exploit the concurrency issue which will allow you to purchase merchandise for a lower price.


I have started to strongly dislike reviews. If I am on the fence about a game I either ask for opinions on gamefaqs or watch some videos on youtube. But there is nothing that convinces me to buy a game more than a well written article on it. Most of the reviews that I read on gaming websites today just don't cut it. For me the reviews are like vapid walls of text written to justify the score that the reviewer had in mind when he was playing the game.
One of the reasons that I loved 1up so much was because there were detailed dissertations on games and the work that goes into creating them. I had come to love the writers on 1up. But since it was closed down this year I have not found a good replacement website for reading on video games. The closest I have come to finding one is Jermy's blog which he is kind enough to update frequently. Why are there not more quality blogs on video games?
I have been asking this question for a long time now. Just about every topic that can be discussed has a blog dedicated to it. But when it comes to video games they are really hard to find. I am not talking about big networks where they cover just about every game released under the sun. No. I am talking about a small place on the internet where an enthusiast talks about his passion for the game. Where scores, page views and provocative articles have no place.
I understand why ign decided to shut down 1up. The cost of running a website is quite high and if the returns are not good it does not make any business sense to keep it running. Running a blog is a different story however.  I find it a bit strange that one of the most passionate communities has such a small blogging presence. Don't you?


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Since methods for reading DNA sequences became available, DNA has been used in research on species, particularly
their genetic relationships and identification. Because DNA accumulates changes (mutations) over time, DNA sequences
between species become more different from each other the longer the species have been distinct from each other.
It was only in 2003 that researchers brought worldwide attention to the idea that a specific standardized gene region
for animals could be used, therefore, as a new tool for identifying species. This came to be known as "DNA Barcoding",
translated to "DNA Barkodning" or "DNA streckkoding" in Swedish. The initial gene selected as a barcode region for the
animal kingdom was the mitochondrial gene cytochrome c oxidase subunit I (COI). When it was demonstrated that this gene
did not perform as well as a barcode for all organism groups, additional genes were chosen as additional barcode regions
for plants (chloroplast genes RbCI and Matk) and fungi (nuclear gene ITS).
Since 2003, researchers across the globe have been testing how effectively these gene regions serve as species identification
tools for different organisms and in different regions of the world. They have investigated questions such as: how large can
sequence variation be within species, are there always many differences between species but few differences within a species,
do these genes works equally well in all animal groups, how many individuals of each species should be represented in a "reference
library" of sequences, how widespread are false genes (called pseudogenes) in these regions, what is the effect of species hybridization,
which of the several methods developed to match an unknown sequence against a reference library provides the greatest reliability for
species determinations, and is there any difference between using regional rather than a global reference libraries?
These studies have led to several interesting results. First, we are now more aware of the difference between delimiting species
(= what we call taxonomy) and identifying specimens to species (= determination). The early years involved a lot of confusion between
these two activities with assertions that DNA barcoding could replace them, methods for one objective could be used for the other, etc.
Today we also have a rather good perspective concerning what DNA barcoding is good at accomplishing, under what conditions, and its limitations.
For example, the COI gene does not perform well for distinguishing closely-related species from marine animal populations, fungi or cnidarians.
In other plant and animal groups, some recently diverged species are indistinguishable from related species and can even be more similar to each
other in barcode genes than individuals within a single species.
Equipped with this knowledge, we are now able to improve methods using a reference library for determining a species with improved reliability
from an unknown sequence. An important insight has been that a regional, geographically-limited reference library like the “Swedish DNA key”
can be a well-functioning species identification tool for groups where problems have occured when referencing a global database. The reason is
exactly the same as why a traditional determination key is easier to make for species found only in Sweden (or Fennoscandia: Sweden, Norway, Finland)
than to all the world's species - pick out every other leaf on the tree of life and the differences become larger (longer genetic distances) between
the remaining leaves, which are, therefore, easier to distinguish.
To identify species from an unknown sample, we first need to sequence the standardized gene region that is represented in the Swedish DNA key reference
library. If you do not have your own molecular laboratory, then you can send the sample to a lab that can do so for you. The cost of DNA sequencing has
decreased greatly during the past few years, especially if you process many samples simultaneously. The DNA sequence that one receives afterwards is a
plain text file, a text string approximately 650 letters long with the letters A, C, T, and G in various combinations and repetitions. The letters represent
four elements, the so-called nitrogenous bases (adenine, guanine, cytosine, and thymine), which binds together the DNA structure (the double helix). This text
string of letters can be copied and pasted into the Swedish DNA key to query the web portal, compared with all sequences in the reference library. From this,
a species identification is delivered, or a species list if the sample included more than one species, provided that the organism group for your unknown sequence
is actually represented in the reference database.
For a web portal like the Swedish DNA key to function well, a well-stocked reference library of known DNA sequences is needed for comparison to unknown sequences.
First, the reference library must contain sequences of at least all the species in a group of organisms from which an unknown sequence may have originated.
The Swedish DNA key started as a
-funded project to build a reference library for Swedish vertebrates (birds, mammals, fish, amphibians, and reptiles). For this organism group, the current Swedish
DNA key is relatively comprehensive and has known reference sequences for 75-82% of those in Sweden commonly occurring species. Here you often get an accurate result
if you are looking for in the Web Sites. For other groups of animals, plants and fungi, the Swedish DNA key is in an initial development phase and will be improved
during the upcoming years.
Second, there are high quality standards for the sequences compiled in a reference database. These quality standards have been developed by an international working
group of international cooperation in CBOL (Consortium for the Barcode of Life). The most important requirement, and what distinguishes a barcode database from
,
is that the sequences are linked to a "voucher" or reference specimen. The reference specimen is the individual animal, plant or fungus from which the gene has been
sequenced. This specimen is stored, curated and made available at a public institution where researchers can borrow or go study it in order to reassess the determination.
Swedish DNA key sequences are linked to records presented in the Swedish web portal,
naturarv
,
with detailed information on voucher specimens housed at Swedish natural history collections. When you get a hit and determination from a reference sequence whose voucher
specimen is stored at a Swedish natural history collection, you can follow the link to this record on naturarv where you can find collecting data for and, often,
photographs of the individual.
#{msg.dnatext6}
CTAATTTTTGGGGCCTGAGCGGGCATGGTTGGAACCGCCCTCAGCCTGCTCATTCGCGCAGAAC
TCGGTCAACCAGGAACCCTACTAGGAGACGACCAGATCTACAACGTAATCGTCACTGCCCATGC
CTTCGTAATAATCTTCTTTATAGTCATACCAATCATAATCGGGGGCTTTGGAAACTGACTAGTC
CCCCTTATAATCGGCGCCCCAGACATAGCGTTCCCCCGTATAAATAACATAAGCTTCTGACTAC
TCCCCCCATCCTTCCTTCTCCTACTAGCCTCCTCCACCGTAGAAGCTGGTGCTGGTACAGGGTG
AACAGTCTACCCCCCTCTAGCTGGTAATCTGGCCCACGCCGGAGCTTCTGTAGACCTAGCCATC
TTCTCCCTACACCTCGCTGGAGTCTCATCAATCCTAGGTGCAATCAACTTCATCACTACTGCCA
TTAACATAAAGCCCCCGGCCCTATCTCAATACCAAACCCCCCTATTCGTATGATCCGTACTAAT
CACAGCCGTCCTACTATTACTTTCACTGCCCGTCCTCGCTGCCGGCATTACAATGCTCCTCACA
GACCGAAACCTAAACACCACATTCTTTGACCCAGCCGGAGGAGGAGACCCAGTCCTATACCAAC
ACCTATTC
#{msg.dnatext7}
CTAATCTTCGGTGCATGAGCTGGCATGGTCGGAACCGCCCTCAGCCTGCTTATTCGTGCAGAAC
TAGGCCAACCAGGAACCCTCTTGGGAGATGACCAAATCTACAATGTAATCGTTACTGCTCATGC
ATTCGTAATAATTTTCTTCATAGTTATACCAATCATGATCGGAGGATTTGGAAATTGACTAGTC
CCACTCATAATCGGCGCCCCCGACATAGCATTTCCTCGTATAAACAATATAAGCTTCTGACTAC
TCCCCCCATCATTCCTATTATTACTAGCATCCTCTACAGTAGAAGCTGGAGCTGGCACAGGATG
AACAGTATATCCACCCCTCGCCGGCAACCTAGCCCACGCAGGAGCCTCAGTAGACCTAGCTATT
TTCTCCCTCCATTTAGCAGGTGTCTCCTCCATCCTAGGTGCCATTAACTTTATCACCACTGCCA
TTAACATAAAACCACCAGCCCTGTCCCAATACCAAACACCCCTATTTGTATGATCAGTACTCAT
TACCGCCGTCTTACTGCTACTCTCACTCCCAGTCCTTGCTGCCGGCATCACCATGCTATTAACA
GATCGTAATCTAAACACCACATTCTTTGACCCAGCCGGAGGAGGAGACCCAGTCCTATACCAAC
ATCTCTTC
One can represent the four nitrogen bases A, C, T and G of a gene's DNA sequence with four differently colored lines, which provides
a more effective visual overview. This visualization clearly emphasizes the analogy with product barcodes and why this method has been
called "DNA barcoding".
European Roller top, bottom Woodcock: the same sequences as above. Do you see the differences? (hint: look, for example, at the very beginning of the barcode)
#{msg.dnatext8}
#{msg.dnatext9}
#{msg.dnatext10}
Now then, for what purposes can DNA barcoding be used? There is a wide range of applications for DNA barcoding in our society. The major advantage of DNA-based
species identification is that it works equally well for all forms or life stages of an organism - whether you have samples from eggs, larvae, nymphs, larvae,
juveniles, pupae, or adults. Forensic detectives, for example, need to identify different insects as eggs or larvae in a body that has been in the woods in order
to determine the time of death.
Samples from environmental monitoring can, in many cases, consist of organisms in larval stages that are difficult to identify to species level.
DNA-based species identification works equally well whether you have the whole or only part of an organism, for example, a leaf, a feather, a seed, a blood
sample, a fillet, a steak, or a non-fruiting fungal mycelium. Airport security officers may need to identify which species of bird which led to an incident
involving a damaged motor based on blood or feather residues. In North America and Europe, DNA identification of fish fillets has revealed widespread consumer
deception at both restaurants and markets.
DNA-based species identification also works on partially processed products as half-cooked lasagna, frozen fish fillets, leather bags, fur coats, feces, or stomach
contents. Food inspection is an example of where one can identify the raw material used even after the product is no longer possible to identify with the naked eye –
the horse lasagna scandal is one example. Similarly, customs officers have to verify whether leather or fur products such as clothes, bags or shoes crossing the border
are illegal when they are suspected of having been made from a CITES protected species.
That even feces or gut content samples can be used in obtaining species determinations opens up new doors in diet analysis, wildlife management, ecosystem research,
food web reconstruction, and biological control. In an EU-funded project, gray seal diet in the Baltic Sea was analyzed by DNA sequencing of stomach contents. It is
also possible to determine which plants are eaten by animals by sequencing stomach contents when there is a reference library of relevant plant species for the
comparisons. With a reference library of angiosperms can find out which species of bees, bumble bees and flies that pollinate crops and orchards by sequencing pollen
on their legs.
DNA-based species identification has begun to be used by government agencies in the US and Canada for the analysis of environmental monitoring samples, such as in
aquatic environments. This provides a higher taxonomic resolution because often a greater proportion of the sample can be identified to species. Even in Sweden DNA-based
identification is now being tested. In environmental monitoring of Baltic Sea and Gulf of Bothnia bottom sediment, the percentage of spring mosquito larvae that can be
identified at species increases from 10-15% to 90%, leading to a far more accurate environmental description. One can imagine that the DNA-based identification can open
up for environmental monitoring of a wider representation of Sweden's diversity, even for groups that were previously considered too difficult to identify.
#{msg.dnatext13}
Because organisms emit DNA into the environment, one can even detect species' presence without seeing any of the organisms just by taking, for example, a sample of water or
soil. This technique is starting to be used in programs for early detection of invasive species, ballast water inspection on ships, field inventories for endangered species,
and more.
There is a long list of additional applications that could be mentioned. Several countries have collaborative programs at schools, where there are many possibilities to
conduct engaging science lab experiments that can apply DNA-based species identification to samples collected by the students themselves.
DNA barcoding has come so far that it is an available tool for all kinds of agencies and organizations. It is now often neither more difficult or expensive to use DNA
to identify species or individuals than classical studies of the species' appearance. In addition, DNA analyzes provide information on species in situations when classical
methods are insufficient.
What is critical for the tool to be useful is that there must be a comprehensive reference library for the organism groups of interest for all these different applications.
The reference library in the current Swedish DNA key is in the construction phase, and we have relatively good coverage for some groups, such as Swedish vertebrates, but
limited coverage for other groups such as plants. Intense efforts are ongoing to secure funding for build a reference library for several other groups of organisms that are
important for applications in society.


Work, Energy, Momentum and Conservation laws¶
Energy conservation is most convenient as a strategy for addressing problems where time does not appear. For example, a particle goes from position \(x_0\) with speed \(v_0\), to position \(x_f\); what is its new speed? However, it can also be applied to problems where time does appear, such as in solving for the trajectory \(x(t)\), or equivalently \(t(x)\).
Work and Energy¶
Material to be added here.
Energy Conservation¶
Energy is conserved in the case where the potential energy, \(V(\boldsymbol{r})\), depends only on position, and not on time. The force is determined by \(V\),
The net energy, \(E=V+K\) where \(K\) is the kinetic energy, is then conserved,
The same proof can be written more compactly with vector notation,
Inverting the expression for kinetic energy,
allows one to solve for the one-dimensional trajectory \(x(t)\), by finding \(t(x)\),
Note this would be much more difficult in higher dimensions, because you would have to determine which points, \(x,y,z\), the particles might reach in the trajectory, whereas in one dimension you can typically tell by simply seeing whether the kinetic energy is positive at every point between the old position and the new position.
Consider a simple harmonic oscillator potential, \(V(x)=kx^2/2\), with a particle emitted from \(x=0\) with velocity \(v_0\). Solve for the trajectory \(t(x)\),
Here \(E=mv_0^2/2\) and \(x_{\rm max}\) is defined as the maximum displacement before the particle turns around. This integral is done by the substitution \(\sin\theta=x/x_{\rm max}\).
Conservation of Momentum¶
Newton’s third law which we met earlier states that For every action there is an equal and opposite reaction, is more accurately stated as If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.
This means that for two bodies \(i\) and \(j\), if the force on \(i\) due to \(j\) is called \(\boldsymbol{F}_{ij}\), then
Newton’s second law, \(\boldsymbol{F}=m\boldsymbol{a}\), can be written for a particle \(i\) as
where \(\boldsymbol{F}_i\) (a single subscript) denotes the net force acting on \(i\). Because the mass of \(i\) is fixed, one can see that
Now, one can sum over all the particles and obtain
The last step made use of the fact that for every term \(ij\), there is an equivalent term \(ji\) with opposite force. Because the momentum is defined as \(m\boldsymbol{v}\), for a system of particles,
By “isolated” one means that the only force acting on any particle \(i\) are those originating from other particles in the sum, i.e. “no external” forces. Thus, Newton’s third law leads to the conservation of total momentum,
Consider the rocket of mass \(M\) moving with velocity \(v\). After a brief instant, the velocity of the rocket is \(v+\Delta v\) and the mass is \(M-\Delta M\). Momentum conservation gives
In the second step we ignored the term \(\Delta M\Delta v\) because it is doubly small. The last equation gives
Integrating the expression with lower limits \(v_0=0\) and \(M_0\), one finds
Because the total momentum of an isolated system is constant, one can also quickly see that the center of mass of an isolated system is also constant. The center of mass is the average position of a set of masses weighted by the mass,
The rate of change of \(\bar{x}\) is
Thus if the total momentum is constant the center of mass moves at a constant velocity, and if the total momentum is zero the center of mass is fixed.
Conservation of Angular Momentum¶
Consider a case where the force always points radially,
where \(\hat{r}\) is a unit vector pointing outward from the origin. The angular momentum is defined as
The rate of change of the angular momentum is
The first term is zero because \(\boldsymbol{v}\) is parallel to itself, and the second term is zero because \(\boldsymbol{F}\) is parallel to \(\boldsymbol{r}\).
As an aside, one can see from the Levi-Civita symbol that the cross product of a vector with itself is zero. Here, we consider a vector
For any term \(i\), there are two contributions. For example, for \(i\) denoting the \(x\) direction, either \(j\) denotes the \(y\) direction and \(k\) denotes the \(z\) direction, or vice versa, so
This is zero by the antisymmetry of \(\epsilon\) under permutations.
If the force is not radial, \(\boldsymbol{r}\times\boldsymbol{F}\ne 0\) as above, and angular momentum is no longer conserved,
where \(\boldsymbol{\tau}\) is the torque.
For a system of isolated particles, one can write
where the last step used Newton’s third law, \(\boldsymbol{F}_{ij}=-\boldsymbol{F}_{ji}\). If the forces between the particles are radial, i.e. \(\boldsymbol{F}_{ij} ~||~ (\boldsymbol{r}_i-\boldsymbol{r}_j)\), then each term in the sum is zero and the net angular momentum is fixed. Otherwise, you could imagine an isolated system that would start spinning spontaneously.
One can write the torque about a given axis, which we will denote as \(\hat{z}\), in polar coordinates, where
to find the \(z\) component of the torque,
One can use the chain rule to write the partial derivative w.r.t. \(\phi\) (keeping \(r\) and \(\theta\) fixed),
Combining the two equations,
Thus, if the potential is independent of the azimuthal angle \(\phi\), there is no torque about the \(z\) axis and \(L_z\) is conserved.
Symmetries and Conservation Laws¶
When we derived the conservation of energy, we assumed that the potential depended only on position, not on time. If it depended explicitly on time, one can quickly see that the energy would have changed at a rate \(\partial_tV(x,y,z,t)\). Note that if there is no explicit dependence on time, i.e. \(V(x,y,z)\), the potential energy can depend on time through the variations of \(x,y,z\) with time. However, that variation does not lead to energy non-conservation. Further, we just saw that if a potential does not depend on the azimuthal angle about some axis, \(\phi\), that the angular momentum about that axis is conserved.
Now, we relate momentum conservation to translational invariance. Considering a system of particles with positions, \(\boldsymbol{r}_i\), if one changed the coordinate system by a translation by a differential distance \(\boldsymbol{\epsilon}\), the net potential would change by
Thus, if the potential is unchanged by a translation of the coordinate system, the total momentum is conserved. If the potential is translationally invariant in a given direction, defined by a unit vector, \(\hat{\epsilon}\) in the \(\boldsymbol{\epsilon}\) direction, one can see that
The component of the total momentum along that axis is conserved. This is rather obvious for a single particle. If \(V(\boldsymbol{r})\) does not depend on some coordinate \(x\), then the force in the \(x\) direction is \(F_x=-\partial_xV=0\), and momentum along the \(x\) direction is constant.
We showed how the total momentum of an isolated system of particle was conserved, even if the particles feel internal forces in all directions. In that case the potential energy could be written
In this case, a translation leads to \(\boldsymbol{r}_i\rightarrow \boldsymbol{r}_i+\boldsymbol{\epsilon}\), with the translation equally affecting the coordinates of each particle. Because the potential depends only on the relative coordinates, \(\delta V\) is manifestly zero. If one were to go through the exercise of calculating \(\delta V\) for small \(\boldsymbol{\epsilon}\), one would find that the term \(\nabla_i V(\boldsymbol{r}_i-\boldsymbol{r}_j)\) would be canceled by the term \(\nabla_jV(\boldsymbol{r}_i-\boldsymbol{r}_j)\).
The relation between symmetries of the potential and conserved quantities (also called constants of motion) is one of the most profound concepts one should gain from this course. It plays a critical role in all fields of physics. This is especially true in quantum mechanics, where a quantity \(A\) is conserved if its operator commutes with the Hamiltonian. For example if the momentum operator \(-i\hbar\partial_x\) commutes with the Hamiltonian, momentum is conserved, and clearly this operator commutes if the Hamiltonian (which represents the total energy, not just the potential) does not depend on \(x\). Also in quantum mechanics the angular momentum operator is \(L_z=-i\hbar\partial_\phi\). In fact, if the potential is unchanged by rotations about some axis, angular momentum about that axis is conserved. We return to this concept, from a more formal perspective, later in the course when Lagrangian mechanics is presented.
Bulding a code for the Earth-Sun system¶
We will now venture into a study of a system which is energy conserving. The aim is to see if we (since it is not possible to solve the general equations analytically) we can develop stable numerical algorithms whose results we can trust!
We solve the equations of motion numerically. We will also compute quantities like the energy numerically.
We start with a simpler case first, the Earth-Sun system in two dimensions only. The gravitational force \(F_G\) on the earth from the sun is
where \(G\) is the gravitational constant,
the mass of Earth,
the mass of the Sun and
is the distance between Earth and the Sun. The latter defines what we call an astronomical unit AU. From Newton’s second law we have then for the \(x\) direction
and
for the \(y\) direction.
Here we will use that \(x=r\cos{(\theta)}\), \(y=r\sin{(\theta)}\) and
We can rewrite
and
for the \(y\) direction.
 M A T H _ B L O C K
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Building a code for the solar system, final coupled equations¶
The four coupled differential equations
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can be turned into dimensionless equations or we can introduce astronomical units with \(1\) AU = \(1.5\times 10^{11}\).
Using the equations from circular motion (with \(r =1\mathrm{AU}\))
we have
and using that the velocity of Earth (assuming circular motion) is \(v = 2\pi r/\mathrm{yr}=2\pi\mathrm{AU}/\mathrm{yr}\), we have
Building a code for the solar system, discretized equations¶
The four coupled differential equations can then be discretized using Euler’s method as (with step length \(h\))
5 4
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5 5
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Code Example with Euler’s Method¶
The code here implements Euler’s method for the Earth-Sun system using a more compact way of representing the vectors. Alternatively, you could have spelled out all the variables \(v_x\), \(v_y\), \(x\) and \(y\) as one-dimensional arrays.
%matplotlib inline
# Common imports
import numpy as np
import pandas as pd
from math import *
import matplotlib.pyplot as plt
import os
# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"
if not os.path.exists(PROJECT_ROOT_DIR):
os.mkdir(PROJECT_ROOT_DIR)
if not os.path.exists(FIGURE_ID):
os.makedirs(FIGURE_ID)
if not os.path.exists(DATA_ID):
os.makedirs(DATA_ID)
def image_path(fig_id):
return os.path.join(FIGURE_ID, fig_id)
def data_path(dat_id):
return os.path.join(DATA_ID, dat_id)
def save_fig(fig_id):
plt.savefig(image_path(fig_id) + ".png", format='png')
DeltaT = 0.001
#set up arrays
tfinal = 10 # in years
n = ceil(tfinal/DeltaT)
# set up arrays for t, a, v, and x
t = np.zeros(n)
v = np.zeros((n,2))
r = np.zeros((n,2))
# Initial conditions as compact 2-dimensional arrays
r0 = np.array([1.0,0.0])
v0 = np.array([0.0,2*pi])
r[0] = r0
v[0] = v0
Fourpi2 = 4*pi*pi
# Start integrating using Euler's method
for i in range(n-1):
# Set up the acceleration
# Here you could have defined your own function for this
rabs = sqrt(sum(r[i]*r[i]))
a = -Fourpi2*r[i]/(rabs**3)
# update velocity, time and position using Euler's forward method
v[i+1] = v[i] + DeltaT*a
r[i+1] = r[i] + DeltaT*v[i]
t[i+1] = t[i] + DeltaT
# Plot position as function of time
fig, ax = plt.subplots()
#ax.set_xlim(0, tfinal)
ax.set_ylabel('x[m]')
ax.set_xlabel('y[m]')
ax.plot(r[:,0], r[:,1])
fig.tight_layout()
save_fig("EarthSunEuler")
plt.show()
Problems with Euler’s Method¶
We notice here that Euler’s method doesn’t give a stable orbit. It means that we cannot trust Euler’s method. In a deeper way, as we will see in homework 5, Euler’s method does not conserve energy. It is an example of an integrator which is not symplectic.
Here we present thus two methods, which with simple changes allow us to avoid these pitfalls. The simplest possible extension is the so-called Euler-Cromer method. The changes we need to make to our code are indeed marginal here. We need simply to replace
r[i+1] = r[i] + DeltaT*v[i]
in the above code with the velocity at the new time \(t_{i+1}\)
r[i+1] = r[i] + DeltaT*v[i+1]
By this simple caveat we get stable orbits. Below we derive the Euler-Cromer method as well as one of the most utlized algorithms for sovling the above type of problems, the so-called Velocity-Verlet method.
Deriving the Euler-Cromer Method¶
Let us repeat Euler’s method. We have a differential equation
and if we truncate at the first derivative, we have from the Taylor expansion
which when complemented with \(t_{i+1}=t_i+\Delta t\) forms the algorithm for the well-known Euler method. Note that at every step we make an approximation error of the order of \(O(\Delta t^2)\), however the total error is the sum over all steps \(N=(b-a)/(\Delta t)\) for \(t\in [a,b]\), yielding thus a global error which goes like \(NO(\Delta t^2)\approx O(\Delta t)\).
To make Euler’s method more precise we can obviously decrease \(\Delta t\) (increase \(N\)), but this can lead to loss of numerical precision. Euler’s method is not recommended for precision calculation, although it is handy to use in order to get a first view on how a solution may look like.
Euler’s method is asymmetric in time, since it uses information about the derivative at the beginning of the time interval. This means that we evaluate the position at \(y_1\) using the velocity at \(v_0\). A simple variation is to determine \(x_{n+1}\) using the velocity at \(v_{n+1}\), that is (in a slightly more generalized form)
and
The acceleration \(a_n\) is a function of \(a_n(y_n, v_n, t_n)\) and needs to be evaluated as well. This is the Euler-Cromer method.
Exercise: go back to the above code with Euler’s method and add the Euler-Cromer method.
Deriving the Velocity-Verlet Method¶
Let us stay with \(x\) (position) and \(v\) (velocity) as the quantities we are interested in.
We have the Taylor expansion for the position given by
The corresponding expansion for the velocity is
Via Newton’s second law we have normally an analytical expression for the derivative of the velocity, namely
If we add to this the corresponding expansion for the derivative of the velocity
and retain only terms up to the second derivative of the velocity since our error goes as \(O(h^3)\), we have
We can then rewrite the Taylor expansion for the velocity as
The velocity Verlet method¶
Our final equations for the position and the velocity become then
and
Note well that the term \(a_{i+1}\) depends on the position at \(x_{i+1}\). This means that you need to calculate the position at the updated time \(t_{i+1}\) before the computing the next velocity. Note also that the derivative of the velocity at the time \(t_i\) used in the updating of the position can be reused in the calculation of the velocity update as well.
Adding the Velocity-Verlet Method¶
We can now easily add the Verlet method to our original code as
DeltaT = 0.01
#set up arrays
tfinal = 10
n = ceil(tfinal/DeltaT)
# set up arrays for t, a, v, and x
t = np.zeros(n)
v = np.zeros((n,2))
r = np.zeros((n,2))
# Initial conditions as compact 2-dimensional arrays
r0 = np.array([1.0,0.0])
v0 = np.array([0.0,2*pi])
r[0] = r0
v[0] = v0
Fourpi2 = 4*pi*pi
# Start integrating using the Velocity-Verlet method
for i in range(n-1):
# Set up forces, air resistance FD, note now that we need the norm of the vecto
# Here you could have defined your own function for this
rabs = sqrt(sum(r[i]*r[i]))
a = -Fourpi2*r[i]/(rabs**3)
# update velocity, time and position using the Velocity-Verlet method
r[i+1] = r[i] + DeltaT*v[i]+0.5*(DeltaT**2)*a
rabs = sqrt(sum(r[i+1]*r[i+1]))
anew = -4*(pi**2)*r[i+1]/(rabs**3)
v[i+1] = v[i] + 0.5*DeltaT*(a+anew)
t[i+1] = t[i] + DeltaT
# Plot position as function of time
fig, ax = plt.subplots()
ax.set_ylabel('x[m]')
ax.set_xlabel('y[m]')
ax.plot(r[:,0], r[:,1])
fig.tight_layout()
save_fig("EarthSunVV")
plt.show()
You can easily generalize the calculation of the forces by defining a function which takes in as input the various variables. We leave this as a challenge to you.
Studying Energy Conservation¶
In order to study the conservation of energy, we will need to perform a numerical integration, unless we can integrate analytically. Here we present the Trapezoidal rule as a the simplest possible approximation.
Numerical Integration¶
It is also useful to consider methods to integrate numerically. Let us consider the following case. We have classical electron which moves in the \(x\)-direction along a surface. The force from the surface is
The constant \(b\) represents the distance between atoms at the surface of the material, \(F_0\) is a constant and \(x\) is the position of the electron. Using the work-energy theorem we can find the work \(W\) done when moving an electron from a position \(x_0\) to a final position \(x\) through the integral
which results in
Numerical Integration¶
There are several numerical algorithms for finding an integral numerically. The more familiar ones like the rectangular rule or the trapezoidal rule have simple geometric interpretations.
Let us look at the mathematical details of what are called equal-step methods, also known as Newton-Cotes quadrature.
Newton-Cotes Quadrature or equal-step methods¶
The integral
has a very simple meaning. The integral is the area enscribed by the function \(f(x)\) starting from \(x=a\) to \(x=b\). It is subdivided in several smaller areas whose evaluation is to be approximated by different techniques. The areas under the curve can for example be approximated by rectangular boxes or trapezoids.
Basic philosophy of equal-step methods¶
In considering equal step methods, our basic approach is that of approximating a function \(f(x)\) with a polynomial of at most degree \(N-1\), given \(N\) integration points. If our polynomial is of degree \(1\), the function will be approximated with \(f(x)\approx a_0+a_1x\).
Simple algorithm for equal step methods¶
The algorithm for these integration methods is rather simple, and the number of approximations perhaps unlimited!
Choose a step size \(h=(b-a)/N\) where \(N\) is the number of steps and \(a\) and \(b\) the lower and upper limits of integration.
With a given step length we rewrite the integral as
The strategy then is to find a reliable polynomial approximation for \(f(x)\) in the various intervals. Choosing a given approximation for \(f(x)\), we obtain a specific approximation to the integral.
With this approximation to \(f(x)\) we perform the integration by computing the integrals over all subintervals.
Simple algorithm for equal step methods¶
One possible strategy then is to find a reliable polynomial expansion for \(f(x)\) in the smaller subintervals. Consider for example evaluating
which we rewrite as
We have chosen a midpoint \(x_0\) and have defined \(x_0=a+h\).
The rectangle method¶
A very simple approach is the so-called midpoint or rectangle method. In this case the integration area is split in a given number of rectangles with length \(h\) and height given by the mid-point value of the function. This gives the following simple rule for approximating an integral
where \(f(x_{i-1/2})\) is the midpoint value of \(f\) for a given rectangle. We will discuss its truncation error below. It is easy to implement this algorithm, as shown below
Truncation error for the rectangular rule¶
The correct mathematical expression for the local error for the rectangular rule \(R_i(h)\) for element \(i\) is
and the global error reads
where \(R_h\) is the result obtained with rectangular rule and \(\xi \in [a,b]\).
Codes for the Rectangular rule¶
We go back to our simple example above and set \(F_0=b=1\) and choose \(x_0=0\) and \(x=1/2\), and have
The code here computes the integral using the rectangle rule and \(n=100\) integration points we have a relative error of \(10^{-5}\).
from math import sin, pi
import numpy as np
from sympy import Symbol, integrate
# function for the Rectangular rule
def Rectangular(a,b,f,n):
h = (b-a)/float(n)
s = 0
for i in range(0,n,1):
x = (i+0.5)*h
s = s+ f(x)
return h*s
# function to integrate
def function(x):
return sin(2*pi*x)
# define integration limits and integration points
a = 0.0; b = 0.5;
n = 100
Exact = 1./pi
print("Relative error= ", abs( (Rectangular(a,b,function,n)-Exact)/Exact))
Relative error= 4.112453549290521e-05
The trapezoidal rule¶
The other integral gives
and adding up we obtain
which is the well-known trapezoidal rule. Concerning the error in the approximation made, \(O(h^3)=O((b-a)^3/N^3)\), you should note that this is the local error. Since we are splitting the integral from \(a\) to \(b\) in \(N\) pieces, we will have to perform approximately \(N\) such operations.
This means that the global error goes like \(\approx O(h^2)\). The trapezoidal reads then
with a global error which goes like \(O(h^2)\).
Hereafter we use the shorthand notations \(f_{-h}=f(x_0-h)\), \(f_{0}=f(x_0)\) and \(f_{h}=f(x_0+h)\).
Error in the trapezoidal rule¶
The correct mathematical expression for the local error for the trapezoidal rule is
and the global error reads
where \(T_h\) is the trapezoidal result and \(\xi \in [a,b]\).
Algorithm for the trapezoidal rule¶
The trapezoidal rule is easy to implement numerically through the following simple algorithm
Choose the number of mesh points and fix the step length.
calculate \(f(a)\) and \(f(b)\) and multiply with \(h/2\).
Perform a loop over \(n=1\) to \(n-1\) (\(f(a)\) and \(f(b)\) are known) and sum up the terms \(f(a+h) +f(a+2h)+f(a+3h)+\dots +f(b-h)\). Each step in the loop corresponds to a given value \(a+nh\).
Multiply the final result by \(h\) and add \(hf(a)/2\) and \(hf(b)/2\).
Trapezoidal Rule¶
We use the same function and integrate now using the trapoezoidal rule.
import numpy as np
from sympy import Symbol, integrate
# function for the trapezoidal rule
def Trapez(a,b,f,n):
h = (b-a)/float(n)
s = 0
x = a
for i in range(1,n,1):
x = x+h
s = s+ f(x)
s = 0.5*(f(a)+f(b)) +s
return h*s
# function to integrate
def function(x):
return sin(2*pi*x)
# define integration limits and integration points
a = 0.0; b = 0.5;
n = 100
Exact = 1./pi
print("Relative error= ", abs( (Trapez(a,b,function,n)-Exact)/Exact))
Relative error= 8.224805627923717e-05
Simpsons’ rule¶
Instead of using the above first-order polynomials approximations for \(f\), we attempt at using a second-order polynomials. In this case we need three points in order to define a second-order polynomial approximation
Using again Lagrange’s interpolation formula we have
Inserting this formula in the integral of Eq. (18) we obtain
which is Simpson’s rule.
Simpson’s rule¶
Note that the improved accuracy in the evaluation of the derivatives gives a better error approximation, \(O(h^5)\) vs.\ \(O(h^3)\) . But this is again the local error approximation. Using Simpson’s rule we can easily compute the integral of Eq. (17) to be
with a global error which goes like \(O(h^4)\).
Mathematical expressions for the truncation error¶
More formal expressions for the local and global errors are for the local error
and for the global error
with \(\xi\in[a,b]\) and \(S_h\) the results obtained with Simpson’s method.
Algorithm for Simpson’s rule¶
The method can easily be implemented numerically through the following simple algorithm
Choose the number of mesh points and fix the step.
calculate \(f(a)\) and \(f(b)\)
Perform a loop over \(n=1\) to \(n-1\) (\(f(a)\) and \(f(b)\) are known) and sum up the terms \(4f(a+h) +2f(a+2h)+4f(a+3h)+\dots +4f(b-h)\). Each step in the loop corresponds to a given value \(a+nh\). Odd values of \(n\) give \(4\) as factor while even values yield \(2\) as factor.
Multiply the final result by \(\frac{h}{3}\).
Code example¶
from math import sin, pi
import numpy as np
from sympy import Symbol, integrate
# function for the trapezoidal rule
def Simpson(a,b,f,n):
h = (b-a)/float(n)
sum = f(a)/float(2);
for i in range(1,n):
sum = sum + f(a+i*h)*(3+(-1)**(i+1))
sum = sum + f(b)/float(2)
return sum*h/3.0
# function to integrate
def function(x):
return sin(2*pi*x)
# define integration limits and integration points
a = 0.0; b = 0.5;
n = 100
Exact = 1./pi
print("Relative error= ", abs( (Simpson(a,b,function,n)-Exact)/Exact))
Relative error= 5.412252157986472e-09
We see that Simpson’s rule gives a much better estimation of the relative error with the same amount of points as we had for the Rectangle rule and the Trapezoidal rule.


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Write a function that does each of the following:
Create an empty list and print it.
Create a list containing several strings and integers. Print it.
Append a value to one of these lists. Print it.
Insert a value at the beginning of one of these lists. Print it.
Print the length of the list.
Select (index) an element from one of your lists. Print it.
Select and print a slice from one of your lists.
Add two lists together. Print the result.
Add a list to this list using the augment operator (+=). Print it.
Sort the list. Print it.
Use the for: statement to print each item in one of your lists.
Create a tuple with one element. Print it.
Create a tuple with more than one element. Print it.
Write a function apply(fn, numbers) that takes a function and a list of numbers. apply should return a new list that contains the result of applying fn to each element of numbers.
Given a list data1 (with an even number of items), create a list of 2-tuples data2 where:
data2[i] = (data1[i] , data1[i+1])
For example, given this list:
data1 = ['aa', 11, 'bb', 22, 'cc', 33, 'dd', 44]
create this list:
data2 = [('aa', 11), ('bb', 22), ('cc', 33), ('dd', 44)]
Create a list of integers, some of which are odd and some of which are even. Sort this list using the sort method so that all the odd numbers are at the beginning. Hint: the sort method takes an additional, optional argument, a function cmp(a, b) which returns:
What you will learn:


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