| PostgreSQL 9.6.15 Documentation | |||
|---|---|---|---|
| Prev | Up | Chapter 15. Parallel Query | Next |
There are several settings which can cause the query planner not to generate a parallel query plan under any circumstances. In order for any parallel query plans whatsoever to be generated, the following settings must be configured as indicated.
max_parallel_workers_per_gather must be set to a value which is greater than zero. This is a special case of the more general principle that no more workers should be used than the number configured via max_parallel_workers_per_gather.
dynamic_shared_memory_type must be set to a value other than none. Parallel query requires dynamic shared memory in order to pass data between cooperating processes.
In addition, the system must not be running in single-user mode. Since the entire database system is running in single process in this situation, no background workers will be available.
Even when it is in general possible for parallel query plans to be generated, the planner will not generate them for a given query if any of the following are true:
The query writes any data or locks any database rows. If a query contains a data-modifying operation either at the top level or within a CTE, no parallel plans for that query will be generated. This is a limitation of the current implementation which could be lifted in a future release.
The query might be suspended during execution. In any situation in which the system thinks that partial or incremental execution might occur, no parallel plan is generated. For example, a cursor created using DECLARE CURSOR will never use a parallel plan. Similarly, a PL/pgsql loop of the form FOR x IN query LOOP .. END LOOP will never use a parallel plan, because the parallel query system is unable to verify that the code in the loop is safe to execute while parallel query is active.
The query uses any function marked PARALLEL UNSAFE. Most system-defined functions are PARALLEL SAFE, but user-defined functions are marked PARALLEL UNSAFE by default. See the discussion of Section 15.4.
The query is running inside of another query that is already parallel. For example, if a function called by a parallel query issues an SQL query itself, that query will never use a parallel plan. This is a limitation of the current implementation, but it may not be desirable to remove this limitation, since it could result in a single query using a very large number of processes.
The transaction isolation level is serializable. This is a limitation of the current implementation.
Even when parallel query plan is generated for a particular query, there are several circumstances under which it will be impossible to execute that plan in parallel at execution time. If this occurs, the leader will execute the portion of the plan below the Gather node entirely by itself, almost as if the Gather node were not present. This will happen if any of the following conditions are met:
No background workers can be obtained because of the limitation that the total number of background workers cannot exceed max_worker_processes.
The client sends an Execute message with a non-zero fetch count. See the discussion of the extended query protocol. Since libpq currently provides no way to send such a message, this can only occur when using a client that does not rely on libpq. If this is a frequent occurrence, it may be a good idea to set max_parallel_workers_per_gather in sessions where it is likely, so as to avoid generating query plans that may be suboptimal when run serially.
A prepared statement is executed using a CREATE TABLE .. AS EXECUTE .. statement. This construct converts what otherwise would have been a read-only operation into a read-write operation, making it ineligible for parallel query.
The transaction isolation level is serializable. This situation does not normally arise, because parallel query plans are not generated when the transaction isolation level is serializable. However, it can happen if the transaction isolation level is changed to serializable after the plan is generated and before it is executed.


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We Students study Graphics & Multimedia to understand the concept of basic 2D & 3D transformations and how some complex graphics problems can be solved using different algorithms.
Graphics Designing is a process of uniting creative thinking with technological thinking. We humans normally understand graphics and videos better rather than raw text.
So the main objective is to simplify the raw text into images and videos for better understanding. It involves creative thinking which is essential to produce graphically communicative ideas. It means the visual communication that conveys a message with a concept and a purpose
Importance of Graphics Designing:
1. There is a lot of importance of Graphics Designing because it is used in numerous fields like education, applications, entertainment, videos etc.
2. You can communicate the ideas easily and effectively which is useful in creating a strong impression.
3. It supports in brand recognition which is essential for a business. It helps you to stand out from the competitors and aids any business in having their own identity.
4. It is important to establish a story for business which can interest prospects to buy any product or take your services.
5. We are living in a digital world where everything single thing can be accessed online. You won’t be able to convey your business objective to your potential audience if you are working on a badly made interactive design.
6. It also plays an important role in business, marketing & sales.
Unit wise description of this Subject is as given below:-
UNIT I OUTPUT PRIMITIVES
Basic − Line − Curve and ellipse drawing algorithms − Examples – Applications - Attributes − Two- Dimensional geometric transformations − TwoDimensional clipping and viewing – Input techniques.
UNIT II THREE-DIMENSIONAL CONCEPTS
Three-Dimensional object representations − Three-Dimensional geometric and modeling transformations − Three-Dimensional viewing − Hidden surface elimination − Color models – Virtual reality - Animation.
UNIT III MULTIMEDIA SYSTEMS DESIGN
Multimedia basics − Multimedia applications − Multimedia system architecture − Evolving technologies for multimedia − Defining objects for multimedia systems − Multimedia data interface standards − Multimedia databases
UNIT IV MULTIMEDIA FILE HANDLING
Compression and decompression − Data and file format standards − Multimedia I/O technologies − Digital voice and audio − Video image and animation − Full motion video − Storage and retrieval technologies.
UNIT V HYPERMEDIA
Multimedia authoring and user interface − Hypermedia messaging − Mobile messaging − Hypermedia message component − Creating hypermedia message − Integrated multimedia message standards − Integrated document management − Distributed multimedia systems.
There are a variety of fields where the applications of computer graphics and multimedia are being used nowadays. This includes medical, engineering, education & training, entertainment, science etc. The applications of computer graphics and multimedia are growing day by day. Let’s see the major applications of computer graphics and multimedia which we are using in our daily life.
• BUSINESS
• SCHOOL
• CAD
• RATIONAL DRUG DESIGN
• MULTIPLAYER GAMES
• VIRTUAL REALITY
• WEB DESIGN
• INFOGRAPHICS
• COMPUTER SIMULATION
The Following are the Major Positions which are apt for this Subject
• LOGO DESIGN, NEWSPAPERS, MAGAZINES ETC.
• WEBSITE DESIGNER
• ART DIRECTOR
• ADVERTISING
• BOOK DESIGNER
• ILLUSTRATIONS DESIGNER
• BROCHURES FOR THE MARKETING FIELD
• CORPORATE REPORTS
Students who choose to have their careers in graphics designing must have deep knowledge in shapes , lines , texture and colour etc,. and should have knowledge in using the graphics software such as,
• Adobe Illustrator
• Adobe Photoshop
• CorelDraw
• Adobe InDesign
• QuarkXPress
Four major steps that are involved in a good graphic design are:
• Briefing
• Design
• Artwork
• Production
Some Marketable Certifications For Graphic Designers Include The Following:
• ACA: Adobe Certified Associate
• ACE: Adobe Certified Expert


The COMBAT-TB Workbench provides easy to use tools for analysing M. tuberculosis genomic sequence samples including lineage typing, drug resistance and the phylogentic relationship between samples.
It was developed at part of the COMBAT-TB project, a SAMRC and DST/NRF Research chair funded project undertaken by researchers an d software developers at SANBI. The COMBAT-TB Workbench is developed using technology from the IRIDA and Galaxy projects and incorporates leading tools developed for M. tuberculosis and microbial bioinformatics.


Sian Williams didn't know 'emotional' Bill Turnbull cancer doc moment was filmed
Sian Williams has opened up about her appearance on Bill Turnbull’s documentary Staying Alive, where she spoke about living with cancer.
And it turns out the most emotional moment of the film was captured when she thought the cameras had stopped rolling.
The honest conversation saw both journalists in tears as they discussed the sad death of their show’s editor Alison Ford in 2013 and Williams’ own breast cancer diagnosis, which saw her undergo a double mastectomy.
In the documentary, the journalist can be seen telling Turnbull that ‘even if you think [the cancer] is sorted, it comes tapping on the window.’
Speaking to Metro.co.uk at the Channel 5 2020 showcase, on Tuesday night, the 54 year-old admitted that the conversation wasn’t initially meant to be caught on camera.
‘We thought they’d turned the cameras off,’ she said: ‘They kept the cameras rolling and neither of us were sure whether to keep that bit in.
Advertisement Advertisement
‘[Bill] rang me and said, “Listen, they’re keeping the bit when we thought the cameras stopped rolling, and I think it’s okay.”‘
The friends were the co-anchors of BBC Breakfast for more than a decade, but have known each other for well over 25 years.
Turnbull – who was recently diagnosed with terminal prostate cancer – also discusses the after-effects of the illness with the Channel 5 newsreader, and asked if her blood is still checked monthly.
Williams continued: ‘I saw it and I thought…no, it has to stay. It’s honest. It’s honest about how awful [cancer] can be and the affection between the two of us.’
The storied journalist – who has been a newscaster for Channel 5 for four years now – also opened about feeling ‘well ensconced’ as a reporter for the next general election.
Tell us about it!
‘Who doesn’t love a General Election campaign?’ She asked: ‘I’ve been in news for 30 years and this one is particularly…it’s different. Different to any of the others that I’ve covered.’
Turnbull himself had previously opened up about the documentary being a ‘blubathon’ ahead of its premiere on Channel 4 last month.
‘I’m a bit embarrassed,’ the BBC news presenter said to Metro.co.uk and other press: ‘It’s a bit of a blubathon really. “He’s crying again!” But it’s a very emotional business.’
Got a showbiz story?
If you've got a story, video or pictures get in touch with the Metro.co.uk Entertainment team by emailing us<EMAIL_ADDRESS>calling 020 3615 2145 or by visiting our Submit Stuff page - we'd love to hear from you.


Three Leaves is a retail store focusing on sales of premium organic loose leaf teas and take out beverages. The primary goal for this project was to create a diverse and adaptive brand that could stretch across multiple platforms from in-store retail to online, and in-store product to take-away. Designing a spectrum of packaging elements based around the artwork from Köhler's Medicinal Plants became the central theme that tied each piece together. The artwork is featured in some way in each piece, creating the style of the overall brand — mainly in that images corresponding to main ingredients in each blend of tea were chosen to feature on that tea’s blend specific packaging.
The second element that I focused on was creating the effect of handwritten packaging and labels. The vintage imagery tied well into the simplicity of hand drawn and written design, and so I incorporated a handwritten font into the packaging and highlighted it with the colour that corresponded to whatever blend of tea it was. Drawing more inspiration from the images from Köhler's Medicinal Plants, I chose colours from the image for Camellia Sinensis — the plant which tea comes from — to correspond to each class of tea featured; black, green, white, and herbal.


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Being John Malkovich (1999) Reviewed by Eugene Novikov http://www.ultimate-movie.com/ Member: Online Film Critics Society
"Don't stand in the way of my actualization as a man."
Starring John Cusack, Cameron Diaz, Catherine Keener, Orson Bean, John Malkovich Rated R.
As many of us have long suspected, and as Being John Malkovich proves, the people behind the enigma we know as Hollywood really should be institutionalized.  The result is an exercise in bona fide originality, a wacky, weird film that is surreal and straight-forward at the same time. "Offbeat" doesn't even begin to describe it.
An abstract of the plot will give some of you the temptation to dive for your browser's "Back" button, but stay with me. The film stars John Cusack as Craig Schwartz, an unhappily married (to Cameron Diaz, whom I didn't even recognize until a third of the way into the movie) puppeteer who is forced to get a day job at Lester Corp., a filing business on the 7 1/2th floor of a skyscraper where the ceilings are so low, the employees have to stoop the whole time. The company is headed by 105 year-old Dr. Lester, whose odd secretary has convinced him that he is suffering from a speech impediment even though he speaks clearly.
So one sunny afternoon, as Craig is merrily filing away, a stack of papers falls behind a cabinet. Craig moves the cabinet away from the wall and is shocked to discover a little door, which opens to a passageway just big enough for him to crawl through. Curious, he goes in.  Let me clarify. This rabbit hole that Craig finds is -- yes -- a portal into the head of John Malkovich, the actor. For 15 minutes, you get to experience the world through John Malkovich's eyes, until you are thrown in a ditch by the New Jersey Turnpike.
It gets weirder yet. You see, Craig falls for his sexy co-worker Maxine (Catherine Keener)... and so does his wife, who all of a sudden realizes that she wants to be a man.  Meanwhile, Craig and Maxine start a service: for some $200 a pop, you can be John Malkovich for 15 minutes.
Alright, you can catch your breath. I can wait.
All done? Good. My first order of business, then, is to inform you that even though you may have been positively repulsed by my plot synopsis, Being John Malkovich is a masterpiece of lunacy. Love it or hate it, it's the most inventive thing to come about since The Truman Show. The idea itself seems inconceivable. I'm not sure who thought of it, but whoever did has a great future in Hollywood. Even more importantly, Hollywood needs him.
The brilliance of Being John Malkovich is in the execution as well as the concept. First time director Spike Jonze made one genius move: his movie never admits just how weird it is. The script is way out there, but everything is laid out in a matter-of-fact style. In effect, the film is the epitome of deadpanning, which is exactly what makes it so effective. There is a point in the middle of the film where Cameron Diaz tells John Cusack not to "stand in the way of her actualization as a man." The line is hilarious, the funniest thing I've heard all year. And yet unlike any gag the Austin Powers sequel, for instance, it doesn't feel like it was intended to elicit gales of laughter. So we laugh in retrospect rather than right then and there. And ya know what? It's been a week and the joke only gets funnier.
Kudos to the real two-time Oscar nominee John Malkovich for agreeing to play this slightly exaggerated version of himself. The role stops just short of self-parody; the actor is never jeered or mocked. Indeed, this is probably Malkovich's best role and best performance to date. I strongly believe that the hardest role to play is yourself. John Malkovich's work here should not be ignored come award time.
I saw Being John Malkovich immediately after Mansfield Park, which was unoriginal and mostly uninteresting. I appreciated Malkovich that much more. It's a film to treasure.
Grade: A-
©1999 Eugene Novikov
Sent via Deja.com http://www.deja.com/ Before you buy.
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


Trading Reimagined
A Non-Rent-Seeking Open Protocol for Decentralized Lending that Enables Margin Trading and Short Selling of ERC20 Tokens
A Non-Rent-Seeking Open Protocol for Decentralized Lending that Enables Margin Trading and Short Selling of ERC20 Tokens
Lendroid seeks to overcome on-chain limitations of latency and impractical gas cost by creating a symbiotic off-chain infrastructure supported by incentivized participants. The native token keeps the network operational, fuels the utility layer of the protocol and offers security to community governance. The ecosystem nurtures a decentralized, global, shared lending pool.
01
Lendroid is non-rent-seeking, where all transaction throughput goes to the off-chain utility actors - the 0x based Relayers and the newly introduced Wranglers - who coordinate to make the network stable.
02
Smart contracts ensure no custodianship of user funds, thus mitigating counterparty risks. This lets users and off-chain keepers manage their operations without being controlled by a central authority.
03
Supports multiple exchanges and is compatible with exchange protocols, thus contributing to a global shared liquidity pool. Lenders facilitate a global shared lending pool thereby enhancing lending liquidity.
• LST is the native token of the Lendroid protocol.
• It is non-rent seeking, meaning it does not collect any fee for itself.
• Protocol users (Lenders, Margin Traders) pay keepers/enablers (Relayers, Wranglers) in LST.
• The fee amounts are negotiated offline.
• Fee amounts are negotiated offline.
EXPLORE
Email: connect@lendroid.com


Forecast
menu
Verify data
Choose your data
{% for o in menu %}
{{ o }}
{% endfor %}
Select data for Forecasting
How far into the future do you want to predict?
Number of Periods
Forecast
send


(if test then-clause else-clause)
Executes and returns the value of then-clause if test is not zero; otherwise it executes else-clause if it is present.
Example: Tab to column 33 if dot is currently at the start of a line:
(if (bolp) (to-col 33) )You can also have multiple tests, which gives you the equivalent of switch or case statements in other programming languages:
(defun (month response (setq response (get-tty-string "Month number: ")) (if (= response 1) (insert-string "January") (= response 2) (insert-string "February") (= response 3) (insert-string "March")) ) ; if ) )
(illegal-operation) All keys not currently bound
illegal-operation is bound to those keys that do not have a defined interpretation. Executing illegal-operation is an error.
Start incremental search in forward search mode. See extension incsearch for details.
Locates the current C language procedure and passes it as the standard input to the INDENT program. The resulting indented C procedure will replace the initial C procedure. The old text is temporarily stored in Kill buffer.
Inserts its numeric argument into the buffer as a single character prefix-argument times.
Example: Each of the following inserts a zero into the current buffer:
(insert-character 48) (insert-character '0')
Inserts the specified file's contents at dot in the current buffer.
See also:
Inserts the string that results from evaluating the given expression. insert-string concatenates multiple string arguments.
Examples:
Insert the current date and time into the current buffer:
(insert-string (current-time))Insert the current time and buffer name:
(insert-string "Time: " (current-time) "\nBuffer: " (current-buffer-name) "\n")Sample output from this line:
Time: Tue Sep 28 10:48:41 1999 Buffer: fred
Returns 1 if the invoking MLisp function was called interactively.
When bound to a single key causes any current activity within Emacs to be aborted, and forces Emacs to accept commands from the keyboard when that key is struck.
Binding interrupt-key to a sequence longer than one character does not allow the key sequence to interrupt Emacs. Only single key sequences can be used for interrupting.
Returns 1 if all of its variable name arguments are bound to some storage.
Checks that its string parameter is a defined function. It returns 1 if it is defined and 0 otherwise.
You can use this function to determine whether a function is defined before attempting to execute it.


Boris MUD
Boris MUD source
Boris MUD client
Developer Documentation & API documentation
See Also
MCP 2.1
- MUD Client Protocol Specification
MCP
- MUD Client Protocol (another spec)
MXP
- MUD eXtension Protocol
MSP
- MUD Sound Protocol
MSSP
- MUD Server Status Protocol
Intermud Protocols
- Info on many Intermud protocols.
Intermud-3
- Intermud protocol (MudOS LP driver)
IMC2(InterMud Communication 2)
- specification fo connecting through IMC2.
InterMUD
- wikipedia article
wxMUD
- a cross-platform MUD client.
MudWalker
- A simple client for OSX.
Cantrip
- A MUD client for Mac OS X
The Mud Connector
- Mud and RPG game index, mudlist and reviews.
Top Mud Sites
- MUD and RPG Rankings
A Small MUD
- A C++ code base
NakedMud
- A C code base with excellent docs (fork of SocketMUD)
SocketMUD
- A C framework for writing your own MUD.
The Grendal Project
- A Windows / Linux MUD Server (Pascal/Delphi)
Copyright 2009-2019 Jon Mayo


Course Syllabus – CMSC 478
| Instructor: | Manas Gaur<EMAIL_ADDRESS>|
| Office Hours: | Tuesday/Thursday (4 PM - 5 PM) |
| Teaching Assistant (TA): | Ali Mohammadi<EMAIL_ADDRESS>|
| TA Office Hours: | 12 PM - 2 PM (Thursday) |
| Grader | Vamshi Krishna Ginna<EMAIL_ADDRESS>|
| Room: | ITE Building 231 |
| Class Days & Times: | Tuesday/Thursday (2:30 PM - 3:45 PM) |
Course Overview
You'll discover how to use and implement machine learning techniques. This course focuses on giving you the practical skills you need to make these algorithms function. You will gain knowledge of frequently employed learning strategies such as supervised learning algorithms (such as logistic regression, linear regression, SVM, neural networks/deep learning), unsupervised learning algorithms (such as k-means), as well as particular applications like anomaly detection and creating recommender systems.
Advice on Machine Learning: Please review SLIDESCourse Syllabus:
Required Course Materials:
Required Course Materials:
Blackboard Submission
We use Blackboard for assignment submission. Blackboard access will be granted after enrollment to the class as we periodically synchronize with the official course roster. If you enrolled in the class but do not have access to blackboard, it should come within two days. If it has been more than that, send TA an email.
Grading Policy
There will be four assignments, one midterm, and a final project. The assignments will contain written questions and questions that require some Python programming. The grading breakdown is as follows: assignments are collectively worth 40%, the midterm is worth 30%, and the final project is worth 30%. The two quizzes would be supportive to compensate for missing/late assignments. Class attendance is worth 5% and Class interaction is worth 5%, which would be used to provide bonus marks.
Blackboard Discussion (Students, TA, and Instructor)
To review material from the prerequisites, you can arrange an hour-long session with the TA. The Date and Time will be updated soon. Making such an appointment with the TA is optional but encouraged. Further, there will also be optional weekly discussion sections led by TAs. These sessions are meant to be interactive and in a small, traditional classroom setting. They will largely involve working through problems that are similar to the homeworks. We will have one discussion section in two weeks. The timings, schedule and location will be updated soon.
Late Assignments
Each student will have a total of three free late (calendar) days to use for homeworks. Once these late days are exhausted, any assignments turned in late will be penalized 20% per late day. However, no assignment will be accepted more than three days after its due date. Each 24 hours or part thereof that a homework is late uses up one full late day. Please note that late days are applied individually. Please keep in mind that late days cannot be applied to the final project report and poster deadlines due to constraints in the grading deadline.
Regrade Requests
Students will have one week to submit regrade requests on Gradescope for assignments starting from when the grades are released.
Course Logistics
(A)Prerequisites:
Students are expected to have the following background:
-
(a) Knowledge of basic computer science principles and skills, at a level sufficient to write a reasonably non-trivial computer program in Python/NumPy.
-
(b) Familiarity with probability theory, multivariable calculus and linear algebra.
(B) Academic Misconduct:
We strongly encourage students to form study groups. Students may discuss and work on homework problems in groups. However, each student must write down the solution independently, and without referring to written notes from the joint session. Each student must understand the solution well enough in order to reconstruct it by him/herself.
It is an academic integrity violation to copy, refer to, or look at written or code solutions from a previous year, including but not limited to: official solutions from a previous year, solutions posted online, solutions you or someone else may have written up in a previous year, and solutions for related problems. In particular, you may not compare your solutions and code to other students’ or to resources you can find online. If you discuss a problem with others or use an online resource for help, you must cite this in your submission.
Furthermore, it is an honor code violation to post your assignment solutions online, such as on a public git repo. We run plagiarism-detection software on your code against past solutions as well as student submissions from previous years. Additionally, you may not screen share your Overleaf doc or code to other students (including during office hours). Please take the time to familiarize yourself with UMBC Academic Integrity Policies.
(C) Students with Accommodations:
The office of Student Disability Services (SDS) works to ensure that students can access and take advantage of UMBC’s educational environment, regardless of disability. From the SDS, UMBC is committed to eliminating discriminatory obstacles that may disadvantage students based on disability. Services for students with disabilities are provided for all students qualified under the Americans with Disabilities Act (ADA) of 1990, the ADAAA of 2009, and Section 504 of the Rehabilitation Act who request and are eligible for accommodations. The Office of Student Disability Services (SDS) is the UMBC department designated to coordinate accommo- dations that would allow students to have equal access and inclusion in all courses, programs, and activities at the University. If you have a documented disability and need to request academic accommodations, please refer to the SDS website for registration information and to begin the process, or alternatively you may visit the SDS office in the Math/Psychology Building, Room 212. For questions or concerns, you may contact us through email or phone<PHONE_NUMBER>. If you require accommodations for this class, make an appointment to meet with me to discuss your SDS-approved accommodations.
(D) UMBC policy for COVID-19:
While face masks are recommended in many areas of the campus, they are still mandatory for laboratory and classroom contexts. If students are having difficulties hearing and understanding a masked individual, instructors can submit an RT ticket with DOIT to discuss microphone and other AV options. We remind you to be respectful of one’s decision to mask or not outside of classroom contexts.


Creating Views
In this chapter, we will examine how to use routing to create different views in our app. We will also clean up our app a bit by moving functionality out of our ever-growing main.dart and into more appropriate locations. Lastly, We will create a more sophisticated query service that shows you a little bit more about Futures. When you are finished, you will be able to create views within your app that are bookmarkable. You will also have a deeper understanding of how to write robust application code so that it works well in the asynchronous world of modern web apps.
The code
Here is the code for the next version of our Recipe Book app. Run the code in Dart Editor. Click on a recipe to view it. The first thing you’ll notice is some new buttons. In addition to viewing a recipe, we can now add and edit them as well. Notice how the URL changes to reflect your specific view when you navigate from recipe to recipe. Notice also that the back button now works as you would expect.
Application structure and organization
Before we dive into routing and views, lets look at the structural changes we made to the app. The first thing we’ve done is to make subdirectories for specific purposes like components, filters, services, and views. Next, we removed all of our custom components, filters, and services from the main.dart file to their own files, leaving only our main Controller, the RecipeBookController
inside main.dart.
Encapsulating view logic into components
We also created two more components: SearchRecipeComponent
and ViewRecipeComponent
. These are examples of how components can be used to encapsulate view specific functionality. The RatingComponent
is a little bit different in that it is truly generic and can be used anywhere. For this reason, we put it in a separate directory (lib/rating).
Our two new components aren’t generic. They contain logic that is specific to the view they control and so they live in the view directory.
Lets look at SearchRecipeComponent
and the HTML that goes with it. The component takes two attributes that should look familiar: nameFilterString
and categoryFilterMap
. The component contains just enough logic to be able to control the portion of the view responsible for searching and filtering. Note that both attributes are declared as bidirectional. This is because the SearchRecipeComponent
clears the search filters by resetting these two attributes.
Now our app’s main index.html file is a lot simpler. Instead of containing all the markup to set up the search and filter views, it now just contains the call to the component. The details of the view elements are hidden away in the SearchRecipeComponent
’s HTML template.
<search-recipe name-filter-string="ctrl.nameFilter" category-filter-map="ctrl.categoryFilterMap"> </search-recipe>
Using routing to create views
In earlier versions of our app, we had only one view. No matter where you were in the app, the URL never changed to reflect what you were doing. This is generally undesirable and user unfriendly. If one of our users found a recipe they liked, they had no way of bookmarking it for later. Using routing to create views lets us address this problem. Routing makes it easier to build large single-page applications. It lets you map the browser address bar to the semantic structure of your application, and keeps them in sync.
Configuring the router
Let’s look at recipe_book_router.dart. There we define all of our app’s routes and map them to views. To take advantage of routing, just create a class that implements the RouteInitializer
interface. Angular will look for an instance of RouteInitializer
when it instantiates the router and will use it to configure the routes. RouteInitializer
has a single method: init
. It takes two parameters, a Router
, and a ViewFactory
, both of which are created by the dependency injector for you.
class RecipeBookRouteInitializer implements RouteInitializer { init(Router router, ViewFactory view) {
All you need to do to set up routing is to implement the init
method, and then make the RouteInitializer
available in the your app’s module configuration:
type(RouteInitializer, implementedBy: RecipeBookRouteInitializer);
Here is a simple route configuration. Let’s examine the call to addRoute
, and all of its parameters in detail:
router.root ..addRoute( name: 'add', path: '/add', enter: view('view/addRecipe.html'))
name
Is the name you give to the route. You will use this name in the template to tell Angular which route to use for the view.
path
Is the URL pattern that maps to this route. More than one route can match a given pattern, so patterns should go from more specific to more general. (Hint: if your routes are misbehaving, this is probably the place to start looking for misconfiguration). If you find you need to debug your routing configuration, import Dart’s logging library and add this magic to your main()
:
Logger.root.level = Level.FINEST; Logger.root.onRecord.listen((LogRecord r) { print(r.message); });
enter
A RouteEventHandler
that tells the router what to do when the route is entered. Our example code above asks the ViewFactory
to create a RouteEventHandler
that points our route to a specific html file. You can also write your own custom RouteEventHandlers
.
leave
A RouteEventHandler
that tells the router what to do when the route is left. You can use this for a number of purposes. For example, you can determine if a view is leavable, and what to do if it shouldn’t be left (for example, if there are unsaved changes on a view, you might want to prevent the route from leaving, or warn the user that their changes will be lost).
defaultRoute
Specifies whether this route is the default route.
Routes can do more than just load a view
The configuration below is an example of a more complex enter handler. It uses the defaultRoute
and a special RouteEventHandler
to redirect users to something sane in the event that they entered an invalid URL. In our case, it routes them back to the “view recipe” view. The replace: true
causes the invalid route to be removed from the history, so going back doesn’t cause your users to go through an invalid route.
..addRoute( name: 'view_default', defaultRoute: true, enter: (_) => router.go('view', {'recipeId': ':recipeId'}, startingFrom: route, replace: true))
Routes are hierarchical
The mount
parameter allows you to define nested routes. The URL for each nested route is created by appending it’s path to its parents’ paths. Here is an example of nested paths in our app:
..addRoute( name: 'recipe', path: '/recipe/:recipeId', mount: (Route route) => route ..addRoute( name: 'view', path: '/view', enter: view('recipeView.html'))
Our app’s nested paths are a little more complicated, because they also add a parameter to the path. In the example above, the all the routes underneath the recipe subtree require a recipeId parameter.
Here is how our routing configuration will build the paths we’ve defined:
...# ...#/add ...#/recipe/6/view ...#/recipe/6/edit
Connecting a route to a view
Now lets look at the how the routes and views are connected in our app. There are two pieces involved: the router configuration, and the ng-view
Directive. Open up the index.html file and look at the details section. It contains only an ng-view
Directive.
<section id="details"> <ng-view></ng-view> </section>
The ng-view
Directive causes the rendered template of the current route to be included into the layout where the ng-view
appears. Whenever the current route changes, the ng-view
changes the view.
Next we’ll look at how to make the route parameters available inside our ViewRecipeComponent
.
We’ve added a parameter for the RouteProvider
to its constructor, which we use to extract the parameters for the current route. In our case, we extract the recipeId
to look up the recipe being viewed:
ViewRecipeComponent(RouteProvider routeProvider) { _recipeId = routeProvider.parameters['recipeId']; }
More on Futures
In this version of our app, we also separated the query layer out from our main app by creating a QueryService
. The QueryService
does basically the same thing that our old _loadData
method did, but with a couple of enhancements.
First, it provides getters for common queries. Notice that these getters don’t return concrete objects. They return Futures. Our application code still has to plan for the possibility that the service will be slow, so our app’s _loadData
method isn’t too different from the last version. It still determines if the data is loaded and sets appropriate view messages. It just delegates to the QueryService
to do the actual query and JSON mapping.
Second, we’ve built some caching into the QueryService
. This (overly simplistic) cache determines if the data has already been loaded, and if so, serves it from the cache. The service guarantees that data is loaded by using Future chaining. Let’s look at the QueryService
constructor:
QueryService(Http this._http) { _loaded = Future.wait([_loadRecipes(), _loadCategories()]); }
It wraps the calls to _loadRecipes
and _loadCategories
in a call to Future.wait
. Future.wait
also returns a Future
which completes only when all the Futures in the list complete.
The getters in the service (getRecipeById
, getAllRecipes
, getAllCategories
) first check to see if the cache has finished being loaded. If not, it returns a Future
that will wait until _loaded
is complete. If the cache has been populated, it still returns a Future
- a new Future
with the value of the cached data.
Here’s an example:
Future<Recipe> getRecipeById(String id) { if (_recipesCache == null) { return _loaded.then((_) { return _recipesCache[id]; }); } return new Future.value(_recipesCache[id]); }
Now it's your turn
We have set up a stub for adding and editing recipes. Try building the view to edit an existing recipe and write it back out to the mock data store (the JSON file). First you’ll need to create a view component for editing a recipe. Next, you will need to create a method on the QueryService
to update the recipe data store, and then reload the recipe cache from the data store.
Extra credit
Try to use the router leave()
callback to prevent a user from leaving a page with unsaved edits.


--- layout: page title: "Visualizations" ---
Some visualizations I've made for one purpose or another


When we determine how much space our elements will take on a page, and how far from other elements it will stand,we will need to set such element's properties as margin, border and padding. Let's figure out what each of this properties does.
Padding is the distance between the content of an element (like text) and its border. By extending the padding we push the border away from element. We set padding like this: "padding: 10px 10px 10px 10px" , where we start from the top padding ( how far the border will go from element and up) , and go clockwise. Or we can set it like this: "padding: 10px"; in this case padding is set to 10px for each side of an element.
Margin determines how far the element will stay from other closest elements on the page. We set margin same way as we set padding. Margin is transparent , which means that background-color of an element does not apply to it.
We use border property when we want to to see not only the content, but as well some kind of a "frame" around it. We can set border to be different color, thickness, we can set border-radius, make it solid or dashed - there are lot's of options.


class: left, middle background-image: url(../../images/master-folie.png) .title[ ## Classes
### Oleg Varaksin ####<EMAIL_ADDRESS>] --- # Class-like structures in ES 5. ```js function Car () { this.fuel = 0; this.distance = 0; } Car.prototype.move = function () { if (this.fuel < 1) { throw new RangeError('Fuel tank is depleted') } this.fuel-- this.distance += 2 } Car.prototype.addFuel = function () { if (this.fuel >= 60) { throw new RangeError('Fuel tank is full') } this.fuel++ } ``` ```js var car = new Car(); car.addFuel(); car.move(); ``` --- # Class declaration in ES6 (1/2). ES6 classes are syntactic sugar over the prototype-based OO pattern. The class pattern is easier to use than class-like structures. Classes support prototype-based inheritance, super calls, instance and static methods and constructors. ```js class Car { constructor () { this.fuel = 0 this.distance = 0 } move () { if (this.fuel < 1) { throw new RangeError('Fuel tank is depleted') } this.fuel-- this.distance += 2 } addFuel () { if (this.fuel >= 60) { throw new RangeError('Fuel tank is full') } this.fuel++ } } ``` --- # Class declaration in ES6 (2/2). __Important notes:__ - Class declarations begin with the `class` keyword followed by the name of the class. - Commas are invalid between methods in a class. - Properties that occur on the instance rather than the prototype, can only be created inside a class constructor or method. In the example these are `fuel` and `distance`. - All methods are non-enumerable. They don't show up when you do a `for...in` loop over the object or use `Object.keys()` to get an array of property names. But you can iterate over class methods by `Object.getOwnPropertyNames(MyClass.prototype)`. - Calling the class constructor without `new` throws an error. - Attempting to overwrite the class name within a class method throws an error. --- # Class expressions (1/3) Classes and functions are similar in that they have two forms: declarations and expressions. - Function and class __declarations__ begin with an appropriate keyword (`function` or `class`, respectively) followed by an identifier. - Functions have an __expression form__ that doesn't require an identifier after `function`, and similarly, classes have an expression form that doesn't require an identifier after `class`. ```js let PersonClass = class { constructor(name) { this.name = name; } sayName() { console.log(this.name); } }; let person = new PersonClass("Nicholas"); person.sayName(); // outputs "Nicholas" ``` --- # Class expressions (2/3) The previous example used an __anonymous class expression__. The `name` property of `PersonClass` (`PersonClass.name`) is an empty string. When using a class declaration, `PersonClass.name` would be `"PersonClass"`. A __named class expression__ includes an identifier after the `class` keyword. ```js let PersonClass = class PersonClass2 { constructor(name) { this.name = name; } sayName() { console.log(this.name); } }; console.log(typeof PersonClass); // "function" console.log(typeof PersonClass2); // "undefined" ``` The `PersonClass2` identifier exists only within the class definition. It is defined for use only inside the class. Outside the class, `typeof PersonClass2` is `undefined` because no `PersonClass2` binding exists there. --- # Class expressions (3/3) Another interesting use of class expressions is creating singletons by immediately invoking the class constructor. To do so, you must use `new` with a class expression and include parentheses at the end. ```js let person = new class { constructor(name) { this.name = name; } sayName() { console.log(this.name); } }("Nicholas"); person.sayName(); // "Nicholas" ``` --- # Inheritance (1/4). Classes make inheritance easier to implement by using the familiar `extends` keyword. ```js class Warrior { constructor(name) { this.name = name; this.health = 100; } heal(amount) { this.health = Math.min(this.health + amount, 100); } } class Ninja extends Warrior { constructor(name) { super(name); } heal(amount) { super.heal(amount * 1.2); } } ``` --- # Inheritance (2/4). Derived classes require you to use `super()` if you specify a constructor; if you don't, an error will occur. If you choose not to use a constructor, then `super()` is automatically called for you with all arguments upon creating a new instance of the class. For instance, the following two classes are identical: ```js class Square extends Rectangle { // no constructor } // Is equivalent to class Square extends Rectangle { constructor(...args) { super(...args); } } ``` __Important notes:__ - You can only use `super()` in a derived class. If you try to use it in a non-derived class (a class that doesn't use extends), it will throw an error. - You must call `super()` before accessing `this` in the constructor. Attempting to access `this` before calling `super()` results in an error. --- # Inheritance (3/4). The methods on derived classes always shadow methods of the same name on the base class. ```js class Square extends Rectangle { constructor(length) { super(length, length); } // override and shadow Rectangle.prototype.getArea() getArea() { return this.length * this.length; } } ``` You can also call the base class version of the method by using the `super`. E.g. ```js class Square extends Rectangle { ... // override, shadow, and call Rectangle.prototype.getArea() getArea() { return super.getArea(); } } ``` --- # Inheritance (4/4). In ES5 and earlier, it was not possible to inherit from built-ins like `Array`, `RegExp`, etc. It is possible in ES6. ```js class MyArray extends Array { // empty } var colors = new MyArray(); colors[0] = "red"; console.log(colors.length); // 1 ``` Any method that returns an instance of the built-in will automatically return a derived class instance instead. So, if you have a derived class called `MyArray` that inherits from `Array`, methods such as `slice()` return an instance of `MyArray`. ```js let items = new MyArray(1, 2, 3, 4), subitems = items.slice(1, 3); console.log(items instanceof MyArray); // true console.log(subitems instanceof MyArray); // true ``` --- # Accessors (1/2). We have the ability to define getters and setters for our fields. This feature allows to access the fields as any other object property. ```js class Warrior { constructor(name) { this._name = name; this._health = 100; } get health() { return this._health; } set health(newHealth){ this._health = newHealth; } } let donatello = new Warrior("Donatello"); console.log(donatello.health); // 100 donatello.health = 80; ``` --- # Accessors (2/2). When we define a getter and no setter, the variable with the getter / setter name becomes read-only. ```js class Warrior { constructor(name) { this._name = name; this._health = 100; } get health() { return this._health; } } let donatello = new Warrior("Donatello"); console.log(donatello.health); // 100 donatello.health = 80; // TypeError: Cannot set property health... ``` --- # Static methods (1/3). Adding additional methods directly onto constructors to simulate static members is a common pattern in ES5 and earlier. ```js function PersonType(name) { this.name = name; } // static method PersonType.create = function(name) { return new PersonType(name); }; // instance method PersonType.prototype.sayName = function() { console.log(this.name); }; var person = PersonType.create("Nicholas"); ``` --- # Static methods (2/3). ES6 classes simplify the creation of static members by using the `static` annotation before the method or accessor property name. ```js class PersonClass { constructor(name) { this.name = name; } sayName() { console.log(this.name); } static create(name) { return new PersonClass(name); } } let person = PersonClass.create("Nicholas"); ``` --- # Static methods (3/3). One thing you can do in JavaScript but can't in Java: from the derived class you can override and call your parents static method. ```js class Foo { static classMethod() { return 'hello'; } } class Bar extends Foo { static classMethod() { return super.classMethod() + ', too'; } } Bar.classMethod(); // 'hello, too' ``` --- # Static fields. There are no static fields in ES6, but there is a discussion about it for ES7. Right now you can simulate the functionality like this: ```js class Foo { constructor() {} } const staticNumber = 5; const staticObj = {prop: 5}; Foo.staticNumber = staticNumber; Foo.staticObj = staticObj; ``` --- # Abstract classes. This concept is not implemented in ES6 or ES7, but we can simulate abstract classes. We can use `new.target` in class constructors to determine the constructor the class is being instantiated with. ```js // abstract base class class Shape { constructor() { if (new.target === Shape) { throw new Error("This class cannot be instantiated directly.") } } } class Rectangle extends Shape { constructor(length, width) { super(); this.length = length; this.width = width; } } var x = new Shape(); // throws error because new.target is Shape var y = new Rectangle(3, 4); // no error, new.target is Rectangle ``` --- # Abstract methods. This concept is not implemented too, but we can mimic abstract methods. ```js class Abstract { constructor() { if (this.method === undefined) { throw new TypeError("Must override method"); } } } class Derived1 extends Abstract {} class Derived2 extends Abstract { method() {} } const a = new Abstract(); // this.method is undefined; error const b = new Derived1(); // this.method is undefined; error const c = new Derived2(); // this.method is // Derived2.prototype.method; no error ``` --- # Private fields. Private fields are not supported in any ES right now, but there are a couple of new structures in ES6 that let you emulate the private scope. ```js const private = new WeakMap(); class Warrior { constructor(name) { this._name = name; private.set(this, { health: 100 }); } get health(){ return private.get(this).health; } } ``` __Warning__: The above `WeakMap` is shared among all Warrior instances, so potentially you can access another warrior's health. --- # Functions overloading. This functionality have never existed in JavaScript and there is no plan on supporting it. If you have e.g. three methods of the same name `say` ```js class Person { say(what) { //string console.log(thing); } say(things) { //array things.forEach(function(thing){ console.log(thing); }); } say() { console.log("blah") } } ``` the last `say()` method wins (overwrites all other). JavaScript has no type checking on arguments or required quantity of arguments, so you can just have one implementation of `say()`. --- # Interfaces. There is no concept of interfaces in any version of JavaScript. You can find interfaces in TypeScript. --- class: center, middle, inverse # That's all folks, thanks for your attention! .back[[Back to the homepage](http://ova2.github.io/es6-presentations/)] Slideshow created with [remark](https://github.com/gnab/remark)


BART Trip
{{#if tripData}} {{#if realTimeEstimates}}Real Time Train Estimates
| {{abbreviation}} | {{getRealTimeEstimateMinutes estimate[0].minutes[0]}} {{getRealTimeCountdown estimate[0].minutes[0]}} | Platform {{estimate[0].platform}} |
{{tripTitle this}}
{{$.fare}}
{{tripMarkup this}}
CO2 Emission
{{co2emission}}
Special Schedule
{{specialSchedule}}
Bikes are allowed on this train.
This shows how full the train is at this time.
Normal Transfer.
Timed Transfer. Connecting trains will wait up to five minutes for transferring passengers.
Scheduled Transfer. Connecting trains will NOT wait for transferring passengers if there is a delay.


Westchester Square, a condominium in Ansley Park, was looking for a way for residents to access shared documents as well as provide a forward facing web site for their community. This PHP based application provides admin roles to manage the site as well as standard users roles to view content without making changes. Users can login and download association documents, view community contact information, provide notifications to residents, and use contact forms for residents to get in touch with association managers and maintenance contractors.


viewResources(resources: any): any
Decorator: Provides the ability to add resources to the related View Same as: <require from="..."></require>
Parameters
resources: any
Either: strings with moduleIds, Objects with 'src' and optionally 'as' properties or one of the classes of the module to be included.


Is Nana Aba Anamoah Leaving GHOne To UTV?
by BB JonesPoaching and defection are very common in the media landscape and it’s something that will continue for a long time.
Just recently, the likes of Berla Mundi, Giovanni Caleb, Vim Lady, Captain Smart and other great media personalities have crossed carpet to the opposing end, a situation which was trailed by so much controversies.
And it’s like Nana Aba Anamoah, who is one of the hottest cakes in the media fraternity, could follow suit.
How is it possible that Nana Aba Anamoah who is still with GHOne TV congratulated a competitor for winning an award over them?
That’s what she did and although some might think it’s harmless, it sends loads of signals from our perspective.
She sent a congratulatory message to United TV (UTV) for emerging as the TV station of the year (Akan) at the just ended 24th Ghana Journalists Association Awards on her Instagram page.
She wrote;
“Congratulations to the @utvghana team too for emerging as the TV station of the year (Akan). Mo y3 gam!”


{% include 'project/partials/filters/_filter_by_keyword.html' %}
{% include 'project/partials/filters/_filter_by_data_sender.html' %}
{% include 'project/partials/filters/_filter_by_submission_date.html' %}
{% if first_filterable_field %}
{% if first_filterable_field.type == 'unique_id' %}
{% include 'project/partials/filters/_filter_by_subject.html' with entity_type=first_filterable_field.entity_type%}
{% else %}
{% include 'project/partials/filters/_filter_by_date_question.html' with date=first_filterable_field %}
{% endif %}
{% endif %}
{% include 'project/partials/filters/_filter_base_for_duplicates.html' with duplicates_for=duplicates_filter_list %}
{% if filterable_fields %}
{% for field in filterable_fields %}
{% if field.type == 'unique_id' %}
{% include 'project/partials/filters/_filter_by_subject.html' with entity_type=field.entity_type%}
{% else %}
{% include 'project/partials/filters/_filter_by_date_question.html' with date=field %}
{% endif %}
{% endfor %}


--- layout: post title: no title date: '2016-02-19T06:16:44+01:00' categories: - Sketches - Storyboards picture_url: https://film-Storyboards.tumblr.com/post/139596922546 ---


Driven oscillations and resonances with physical examples.
Summary of analytical expressions from February 27-March 3 (see slides at https://mhjensen.github.io/Physics321/doc/pub/week9/html/week9-reveal.html. Discussion of resonances and other examples like the mathematical pendulum and other systems.
Reading suggestion: Taylor sections 5.6-5.8.
Examples of oscillations and Fourier analysis applied to harmonic oscillations.
Reading suggestion: See lecture notes at https://mhjensen.github.io/Physics321/doc/pub/week10/html/week10-reveal.html and Taylor sections 5.6-5.8.
Work on homework 6 and discussions of harmonic oscillation examples and systems.
Solving the problem of driven oscillations numerically gives us much more flexibility to study different types of driving forces. We can reuse our earlier code by simply adding a driving force. If we stay in the \( x \)-direction only this can be easily done by adding a term \( F_{\mathrm{ext}}(x,t) \). Note that we have kept it rather general here, allowing for both a spatial and a temporal dependence.
Before we dive into the code, we need to briefly remind ourselves about the equations we started with for the case with damping, namely
$$
m\frac{d^2x}{dt^2} + b\frac{dx}{dt}+kx(t) =0,
$$
with no external force applied to the system.
Let us now for simplicty assume that our external force is given by
$$
F_{\mathrm{ext}}(t) = F_0\cos{(\omega t)},
$$
where \( F_0 \) is a constant (what is its dimension?) and \( \omega \) is the frequency of the applied external driving force. Small question: would you expect energy to be conserved now?
Introducing the external force into our lovely differential equation and dividing by \( m \) and introducing \( \omega_0^2=\sqrt{k/m} \) we have
$$
\frac{d^2x}{dt^2} + \frac{b}{m}\frac{dx}{dt}+\omega_0^2x(t) =\frac{F_0}{m}\cos{(\omega t)},
$$
Thereafter we introduce a dimensionless time \( \tau = t\omega_0 \) and a dimensionless frequency \( \tilde{\omega}=\omega/\omega_0 \). We have then
$$
\frac{d^2x}{d\tau^2} + \frac{b}{m\omega_0}\frac{dx}{d\tau}+x(\tau) =\frac{F_0}{m\omega_0^2}\cos{(\tilde{\omega}\tau)},
$$
Introducing a new amplitude \( \tilde{F} =F_0/(m\omega_0^2) \) (check dimensionality again) we have
$$
\frac{d^2x}{d\tau^2} + \frac{b}{m\omega_0}\frac{dx}{d\tau}+x(\tau) =\tilde{F}\cos{(\tilde{\omega}\tau)}.
$$
Our final step, as we did in the case of various types of damping, is to define \( \gamma = b/(2m\omega_0) \) and rewrite our equations as
$$
\frac{d^2x}{d\tau^2} + 2\gamma\frac{dx}{d\tau}+x(\tau) =\tilde{F}\cos{(\tilde{\omega}\tau)}.
$$
This is the equation we will code below using the Euler-Cromer method.
# 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')
from pylab import plt, mpl
plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'
DeltaT = 0.001
#set up arrays
tfinal = 20 # in dimensionless time
n = ceil(tfinal/DeltaT)
# set up arrays for t, v, and x
t = np.zeros(n)
v = np.zeros(n)
x = np.zeros(n)
# Initial conditions as one-dimensional arrays of time
x0 = 1.0
v0 = 0.0
x[0] = x0
v[0] = v0
gamma = 0.2
Omegatilde = 0.5
Ftilde = 1.0
# Start integrating using Euler-Cromer's method
for i in range(n-1):
# Set up the acceleration
# Here you could have defined your own function for this
a = -2*gamma*v[i]-x[i]+Ftilde*cos(t[i]*Omegatilde)
# update velocity, time and position
v[i+1] = v[i] + DeltaT*a
x[i+1] = x[i] + DeltaT*v[i+1]
t[i+1] = t[i] + DeltaT
# Plot position as function of time
fig, ax = plt.subplots()
ax.set_ylabel('x[m]')
ax.set_xlabel('t[s]')
ax.plot(t, x)
fig.tight_layout()
save_fig("ForcedBlockEulerCromer")
plt.show()
In the above example we have focused on the Euler-Cromer method. This method has a local truncation error which is proportional to \( \Delta t^2 \) and thereby a global error which is proportional to \( \Delta t \). We can improve this by using the Runge-Kutta family of methods. The widely popular Runge-Kutta to fourth order or just RK4 has indeed a much better truncation error. The RK4 method has a global error which is proportional to \( \Delta t \).
Let us revisit this method and see how we can implement it for the above example.
Runge-Kutta (RK) methods are based on Taylor expansion formulae, but yield in general better algorithms for solutions of an ordinary differential equation. The basic philosophy is that it provides an intermediate step in the computation of \( y_{i+1} \).
To see this, consider first the following definitions
$$
\begin{equation}
\frac{dy}{dt}=f(t,y),
\tag{1}
\end{equation}
$$
and
$$
\begin{equation}
y(t)=\int f(t,y) dt,
\tag{2}
\end{equation}
$$
and
$$
\begin{equation}
y_{i+1}=y_i+ \int_{t_i}^{t_{i+1}} f(t,y) dt.
\tag{3}
\end{equation}
$$
To demonstrate the philosophy behind RK methods, let us consider the second-order RK method, RK2. The first approximation consists in Taylor expanding \( f(t,y) \) around the center of the integration interval \( t_i \) to \( t_{i+1} \), that is, at \( t_i+h/2 \), \( h \) being the step. Using the midpoint formula for an integral, defining \( y(t_i+h/2) = y_{i+1/2} \) and \( t_i+h/2 = t_{i+1/2} \), we obtain
$$
\begin{equation}
\int_{t_i}^{t_{i+1}} f(t,y) dt \approx hf(t_{i+1/2},y_{i+1/2}) +O(h^3).
\tag{4}
\end{equation}
$$
This means in turn that we have
$$
\begin{equation}
y_{i+1}=y_i + hf(t_{i+1/2},y_{i+1/2}) +O(h^3).
\tag{5}
\end{equation}
$$
However, we do not know the value of \( y_{i+1/2} \). Here comes thus the next approximation, namely, we use Euler's method to approximate \( y_{i+1/2} \). We have then
$$
\begin{equation}
y_{(i+1/2)}=y_i + \frac{h}{2}\frac{dy}{dt}=y(t_i) + \frac{h}{2}f(t_i,y_i).
\tag{6}
\end{equation}
$$
This means that we can define the following algorithm for the second-order Runge-Kutta method, RK2.
$$
\begin{equation}
k_1=hf(t_i,y_i),
\tag{7}
\end{equation}
$$
and
$$
\begin{equation}
k_2=hf(t_{i+1/2},y_i+k_1/2),
\tag{8}
\end{equation}
$$
with the final value
$$
\begin{equation}
y_{i+i}\approx y_i + k_2 +O(h^3).
\tag{9}
\end{equation}
$$
The difference between the previous one-step methods is that we now need an intermediate step in our evaluation, namely \( t_i+h/2 = t_{(i+1/2)} \) where we evaluate the derivative \( f \). This involves more operations, but the gain is a better stability in the solution.
The fourth-order Runge-Kutta, RK4, has the following algorithm
$$
k_1=hf(t_i,y_i) \hspace{0.5cm} k_2=hf(t_i+h/2,y_i+k_1/2)
$$
and
$$
k_3=hf(t_i+h/2,y_i+k_2/2)\hspace{0.5cm} k_4=hf(t_i+h,y_i+k_3)
$$
with the final result
$$
y_{i+1}=y_i +\frac{1}{6}\left( k_1 +2k_2+2k_3+k_4\right).
$$
Thus, the algorithm consists in first calculating \( k_1 \) with \( t_i \), \( y_1 \) and \( f \) as inputs. Thereafter, we increase the step size by \( h/2 \) and calculate \( k_2 \), then \( k_3 \) and finally \( k_4 \). The global error goes as \( O(h^4) \).
However, at this stage, if we keep adding different methods in our main program, the code will quickly become messy and ugly. Before we proceed thus, we will now introduce functions that enbody the various methods for solving differential equations. This means that we can separate out these methods in own functions and files (and later as classes and more generic functions) and simply call them when needed. Similarly, we could easily encapsulate various forces or other quantities of interest in terms of functions. To see this, let us bring up the code we developed above for the simple sliding block, but now only with the simple forward Euler method. We introduce two functions, one for the simple Euler method and one for the force.
Note that here the forward Euler method does not know the specific force function to be called. It receives just an input the name. We can easily change the force by adding another function.
def ForwardEuler(v,x,t,n,Force):
for i in range(n-1):
v[i+1] = v[i] + DeltaT*Force(v[i],x[i],t[i])
x[i+1] = x[i] + DeltaT*v[i]
t[i+1] = t[i] + DeltaT
def SpringForce(v,x,t):
# note here that we have divided by mass and we return the acceleration
return -2*gamma*v-x+Ftilde*cos(t*Omegatilde)
It is easy to add a new method like the Euler-Cromer
def ForwardEulerCromer(v,x,t,n,Force):
for i in range(n-1):
a = Force(v[i],x[i],t[i])
v[i+1] = v[i] + DeltaT*a
x[i+1] = x[i] + DeltaT*v[i+1]
t[i+1] = t[i] + DeltaT
and the Velocity Verlet method (be careful with time-dependence here, it is not an ideal method for non-conservative forces))
def VelocityVerlet(v,x,t,n,Force):
for i in range(n-1):
a = Force(v[i],x[i],t[i])
x[i+1] = x[i] + DeltaT*v[i]+0.5*a*DeltaT*DeltaT
anew = Force(v[i],x[i+1],t[i+1])
v[i+1] = v[i] + 0.5*DeltaT*(a+anew)
t[i+1] = t[i] + DeltaT
Finally, we can now add the Runge-Kutta2 method via a new function
def RK2(v,x,t,n,Force):
for i in range(n-1):
# Setting up k1
k1x = DeltaT*v[i]
k1v = DeltaT*Force(v[i],x[i],t[i])
# Setting up k2
vv = v[i]+k1v*0.5
xx = x[i]+k1x*0.5
k2x = DeltaT*vv
k2v = DeltaT*Force(vv,xx,t[i]+DeltaT*0.5)
# Final result
x[i+1] = x[i]+k2x
v[i+1] = v[i]+k2v
t[i+1] = t[i]+DeltaT
Finally, we can now add the Runge-Kutta2 method via a new function
def RK4(v,x,t,n,Force):
for i in range(n-1):
# Setting up k1
k1x = DeltaT*v[i]
k1v = DeltaT*Force(v[i],x[i],t[i])
# Setting up k2
vv = v[i]+k1v*0.5
xx = x[i]+k1x*0.5
k2x = DeltaT*vv
k2v = DeltaT*Force(vv,xx,t[i]+DeltaT*0.5)
# Setting up k3
vv = v[i]+k2v*0.5
xx = x[i]+k2x*0.5
k3x = DeltaT*vv
k3v = DeltaT*Force(vv,xx,t[i]+DeltaT*0.5)
# Setting up k4
vv = v[i]+k3v
xx = x[i]+k3x
k4x = DeltaT*vv
k4v = DeltaT*Force(vv,xx,t[i]+DeltaT)
# Final result
x[i+1] = x[i]+(k1x+2*k2x+2*k3x+k4x)/6.
v[i+1] = v[i]+(k1v+2*k2v+2*k3v+k4v)/6.
t[i+1] = t[i] + DeltaT
The Runge-Kutta family of methods are particularly useful when we have a time-dependent acceleration. If we have forces which depend only the spatial degrees of freedom (no velocity and/or time-dependence), then energy conserving methods like the Velocity Verlet or the Euler-Cromer method are preferred. As soon as we introduce an explicit time-dependence and/or add dissipitave forces like friction or air resistance, then methods like the family of Runge-Kutta methods are well suited for this. The code below uses the Runge-Kutta4 methods.
DeltaT = 0.001
#set up arrays
tfinal = 20 # in dimensionless time
n = ceil(tfinal/DeltaT)
# set up arrays for t, v, and x
t = np.zeros(n)
v = np.zeros(n)
x = np.zeros(n)
# Initial conditions (can change to more than one dim)
x0 = 1.0
v0 = 0.0
x[0] = x0
v[0] = v0
gamma = 0.2
Omegatilde = 0.5
Ftilde = 1.0
# Start integrating using the RK4 method
# Note that we define the force function as a SpringForce
RK4(v,x,t,n,SpringForce)
# Plot position as function of time
fig, ax = plt.subplots()
ax.set_ylabel('x[m]')
ax.set_xlabel('t[s]')
ax.plot(t, x)
fig.tight_layout()
save_fig("ForcedBlockRK4")
plt.show()
Let us end our discussion of oscillations with another classical case, the pendulum. You will find the material here useful for homework 7.
The angular equation of motion of the pendulum is given by Newton's equation and with no external force it reads
$$
\begin{equation}
ml\frac{d^2\theta}{dt^2}+mgsin(\theta)=0,
\tag{10}
\end{equation}
$$
with an angular velocity and acceleration given by
$$
\begin{equation}
v=l\frac{d\theta}{dt},
\tag{11}
\end{equation}
$$
and
$$
\begin{equation}
a=l\frac{d^2\theta}{dt^2}.
\tag{12}
\end{equation}
$$
We do however expect that the motion will gradually come to an end due a viscous drag torque acting on the pendulum. In the presence of the drag, the above equation becomes
$$
\begin{equation}
ml\frac{d^2\theta}{dt^2}+\nu\frac{d\theta}{dt} +mgsin(\theta)=0, \tag{13}
\end{equation}
$$
where \( \nu \) is now a positive constant parameterizing the viscosity of the medium in question. In order to maintain the motion against viscosity, it is necessary to add some external driving force. We choose here a periodic driving force. The last equation becomes then
$$
\begin{equation}
ml\frac{d^2\theta}{dt^2}+\nu\frac{d\theta}{dt} +mgsin(\theta)=Asin(\omega t), \tag{14}
\end{equation}
$$
with \( A \) and \( \omega \) two constants representing the amplitude and the angular frequency respectively. The latter is called the driving frequency.
We define
$$
\omega_0=\sqrt{g/l},
$$
the so-called natural frequency and the new dimensionless quantities
$$
\hat{t}=\omega_0t,
$$
with the dimensionless driving frequency
$$
\hat{\omega}=\frac{\omega}{\omega_0},
$$
and introducing the quantity \( Q \), called the quality factor,
$$
Q=\frac{mg}{\omega_0\nu},
$$
and the dimensionless amplitude
$$
\hat{A}=\frac{A}{mg}
$$
We have
$$
\frac{d^2\theta}{d\hat{t}^2}+\frac{1}{Q}\frac{d\theta}{d\hat{t}}
+sin(\theta)=\hat{A}cos(\hat{\omega}\hat{t}).
$$
This equation can in turn be recast in terms of two coupled first-order differential equations as follows
$$
\frac{d\theta}{d\hat{t}}=\hat{v},
$$
and
$$
\frac{d\hat{v}}{d\hat{t}}=-\frac{\hat{v}}{Q}-sin(\theta)+\hat{A}cos(\hat{\omega}\hat{t}).
$$
These are the equations to be solved. The factor \( Q \) represents the number of oscillations of the undriven system that must occur before its energy is significantly reduced due to the viscous drag. The amplitude \( \hat{A} \) is measured in units of the maximum possible gravitational torque while \( \hat{\omega} \) is the angular frequency of the external torque measured in units of the pendulum's natural frequency.
We need to define a new force, which we simply call the pendulum force. The only thing which changes from our previous spring-force problem is the non-linearity introduced by angle \( \theta \) due to the \( \sin{\theta} \) term. Here we have kept a generic variable \( x \) instead. This makes our codes very similar. Feel free to use these examples for homework seven.
def PendulumForce(v,x,t):
# note here that we have divided by mass and we return the acceleration
return -gamma*v-sin(x)+Ftilde*cos(t*Omegatilde)
DeltaT = 0.001
#set up arrays
tfinal = 20 # in years
n = ceil(tfinal/DeltaT)
# set up arrays for t, v, and x
t = np.zeros(n)
v = np.zeros(n)
theta = np.zeros(n)
# Initial conditions (can change to more than one dim)
theta0 = 1.0
v0 = 0.0
theta[0] = theta0
v[0] = v0
gamma = 0.2
Omegatilde = 0.5
Ftilde = 1.0
# Start integrating using the RK4 method
# Note that we define the force function as a PendulumForce
RK4(v,theta,t,n,PendulumForce)
# Plot position as function of time
fig, ax = plt.subplots()
ax.set_ylabel('theta[radians]')
ax.set_xlabel('t[s]')
ax.plot(t, theta)
fig.tight_layout()
save_fig("PendulumRK4")
plt.show()
If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular solution for the entire driving force is
$$
\begin{equation}
x_p(t)=\sum_nx_{pn}(t).
\tag{15}
\end{equation}
$$
This is known as the principal of superposition. It only applies when the homogenous equation is linear. If there were an anharmonic term such as \( x^3 \) in the homogenous equation, then when one summed various solutions, \( x=(\sum_n x_n)^2 \), one would get cross terms. Superposition is especially useful when \( F(t) \) can be written as a sum of sinusoidal terms, because the solutions for each sinusoidal (sine or cosine) term is analytic, as we saw above.
Driving forces are often periodic, even when they are not sinusoidal. Periodicity implies that for some time \( \tau \)
$$
\begin{eqnarray}
F(t+\tau)=F(t).
\end{eqnarray}
$$
One example of a non-sinusoidal periodic force is a square wave. Many components in electric circuits are non-linear, e.g. diodes, which makes many wave forms non-sinusoidal even when the circuits are being driven by purely sinusoidal sources.
The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \).
import numpy as np
import math
from scipy import signal
import matplotlib.pyplot as plt
# number of points
n = 500
# start and final times
t0 = 0.0
tn = 1.0
# Period
t = np.linspace(t0, tn, n, endpoint=False)
SqrSignal = np.zeros(n)
SqrSignal = 1.0+signal.square(2*np.pi*5*t)
plt.plot(t, SqrSignal)
plt.ylim(-0.5, 2.5)
plt.show()
For the sinusoidal example studied in the previous week the period is \( \tau=2\pi/\omega \). However, higher harmonics can also satisfy the periodicity requirement. In general, any force that satisfies the periodicity requirement can be expressed as a sum over harmonics,
$$
\begin{equation}
F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau).
\tag{16}
\end{equation}
$$
We can write down the answer for \( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv 2\pi/\tau \),
$$
\begin{equation}
\tag{17}
F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t).
\end{equation}
$$
The solutions for \( x(t) \) then come from replacing \( \omega \) with \( n\omega \) for each term in the particular solution,
$$
\begin{eqnarray}
x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\
\nonumber
\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\
\nonumber
\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\
\nonumber
\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right).
\end{eqnarray}
$$
Because the forces have been applied for a long time, any non-zero damping eliminates the homogenous parts of the solution, so one need only consider the particular solution for each \( n \).
The problem will considered solved if one can find expressions for the coefficients \( f_n \) and \( g_n \), even though the solutions are expressed as an infinite sum. The coefficients can be extracted from the function \( F(t) \) by
$$
\begin{eqnarray}
\tag{18}
f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\
\nonumber
g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau).
\end{eqnarray}
$$
To check the consistency of these expressions and to verify Eq. (18), one can insert the expansion of \( F(t) \) in Eq. (17) into the expression for the coefficients in Eq. (18) and see whether
$$
\begin{eqnarray}
f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{
\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t)
\right\}\cos(n\omega t).
\end{eqnarray}
$$
Immediately, one can throw away all the terms with \( g_m \) because they convolute an even and an odd function. The term with \( f_0/2 \) disappears because \( \cos(n\omega t) \) is equally positive and negative over the interval and will integrate to zero. For all the terms \( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition formulas to see that \( \cos(m\omega t)\cos(n\omega t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate to zero unless \( m=n \). In that case the \( m=n \) term gives
$$
\begin{equation}
\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2},
\tag{19}
\end{equation}
$$
and
$$
\begin{eqnarray}
f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\
\nonumber
&=&f_n~\checkmark.
\end{eqnarray}
$$
The same method can be used to check for the consistency of \( g_n \).
Consider the driving force:
$$
\begin{equation}
F(t)=At/\tau,~~-\tau/2 < t < \tau/2,~~~F(t+\tau)=F(t).
\tag{20}
\end{equation}
$$
Find the Fourier coefficients \( f_n \) and \( g_n \) for all \( n \) using Eq. (18).
Only the odd coefficients enter by symmetry, i.e. \( f_n=0 \). One can find \( g_n \) integrating by parts,
$$
\begin{eqnarray}
\tag{21}
g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2}dt~\sin(n\omega t) \frac{At}{\tau}\\
\nonumber
u&=&t,~dv=\sin(n\omega t)dt,~v=-\cos(n\omega t)/(n\omega),\\
\nonumber
g_n&=&\frac{-2A}{n\omega \tau^2}\int_{-\tau/2}^{\tau/2}dt~\cos(n\omega t)
+\left.2A\frac{-t\cos(n\omega t)}{n\omega\tau^2}\right|_{-\tau/2}^{\tau/2}.
\end{eqnarray}
$$
The first term is zero because \( \cos(n\omega t) \) will be equally positive and negative over the interval. Using the fact that \( \omega\tau=2\pi \),
$$
\begin{eqnarray}
g_n&=&-\frac{2A}{2n\pi}\cos(n\omega\tau/2)\\
\nonumber
&=&-\frac{A}{n\pi}\cos(n\pi)\\
\nonumber
&=&\frac{A}{n\pi}(-1)^{n+1}.
\end{eqnarray}
$$
More text will come here, chpater 5.7-5.8 of Taylor are discussed during the lectures. The code here uses the Fourier series discussed in chapter 5.7 for a square wave signal. The equations for the coefficients are are discussed in Taylor section 5.7, see Example 5.4. The code here visualizes the various approximations given by Fourier series compared with a square wave with period \( T=0.2 \), witth \( 0.1 \) and max value \( F=2 \). We see that when we increase the number of components in the Fourier series, the Fourier series approximation gets closes and closes to the square wave signal.
import numpy as np
import math
from scipy import signal
import matplotlib.pyplot as plt
# number of points
n = 500
# start and final times
t0 = 0.0
tn = 1.0
# Period
T =0.2
# Max value of square signal
Fmax= 2.0
# Width of signal
Width = 0.1
t = np.linspace(t0, tn, n, endpoint=False)
SqrSignal = np.zeros(n)
FourierSeriesSignal = np.zeros(n)
SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
a0 = Fmax*Width/T
FourierSeriesSignal = a0
Factor = 2.0*Fmax/np.pi
for i in range(1,500):
FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
plt.plot(t, SqrSignal)
plt.plot(t, FourierSeriesSignal)
plt.ylim(-0.5, 2.5)
plt.show()
Consider a particle at rest in the bottom of an underdamped harmonic oscillator, that then feels a sudden impulse, or change in momentum, \( I=F\Delta t \) at \( t=0 \). This increases the velocity immediately by an amount \( v_0=I/m \) while not changing the position. One can then solve the trajectory by solving the equations with initial conditions \( v_0=I/m \) and \( x_0=0 \). This gives
$$
\begin{equation}
x(t)=\frac{I}{m\omega'}e^{-\beta t}\sin\omega't, ~~t>0.
\tag{22}
\end{equation}
$$
Here, \( \omega'=\sqrt{\omega_0^2-\beta^2} \). For an impulse \( I_i \) that occurs at time \( t_i \) the trajectory would be
$$
\begin{equation}
x(t)=\frac{I_i}{m\omega'}e^{-\beta (t-t_i)}\sin[\omega'(t-t_i)] \Theta(t-t_i),
\tag{23}
\end{equation}
$$
where \( \Theta(t-t_i) \) is a step function, i.e. \( \Theta(x) \) is zero for \( x < 0 \) and unity for \( x>0 \). If there were several impulses linear superposition tells us that we can sum over each contribution,
$$
\begin{equation}
x(t)=\sum_i\frac{I_i}{m\omega'}e^{-\beta(t-t_i)}\sin[\omega'(t-t_i)]\Theta(t-t_i)
\tag{24}
\end{equation}
$$
Now one can consider a series of impulses at times separated by \( \Delta t \), where each impulse is given by \( F_i\Delta t \). The sum above now becomes an integral,
$$
\begin{eqnarray}label{eq:Greeny}
x(t)&=&\int_{-\infty}^\infty dt'~F(t')\frac{e^{-\beta(t-t')}\sin[\omega'(t-t')]}{m\omega'}\Theta(t-t')\\
\nonumber
&=&\int_{-\infty}^\infty dt'~F(t')G(t-t'),\\
\nonumber
G(\Delta t)&=&\frac{e^{-\beta\Delta t}\sin[\omega' \Delta t]}{m\omega'}\Theta(\Delta t)
\end{eqnarray}
$$
The quantity \( e^{-\beta(t-t')}\sin[\omega'(t-t')]/m\omega'\Theta(t-t') \) is called a Green's function, \( G(t-t') \). It describes the response at \( t \) due to a force applied at a time \( t' \), and is a function of \( t-t' \). The step function ensures that the response does not occur before the force is applied. One should remember that the form for \( G \) would change if the oscillator were either critically- or over-damped.
When performing the integral in Eq. \eqref{eq:Greeny} one can use angle addition formulas to factor out the part with the \( t' \) dependence in the integrand,
$$
\begin{eqnarray}
\tag{25}
x(t)&=&\frac{1}{m\omega'}e^{-\beta t}\left[I_c(t)\sin(\omega't)-I_s(t)\cos(\omega't)\right],\\
\nonumber
I_c(t)&\equiv&\int_{-\infty}^t dt'~F(t')e^{\beta t'}\cos(\omega't'),\\
\nonumber
I_s(t)&\equiv&\int_{-\infty}^t dt'~F(t')e^{\beta t'}\sin(\omega't').
\end{eqnarray}
$$
If the time \( t \) is beyond any time at which the force acts, \( F(t'>t)=0 \), the coefficients \( I_c \) and \( I_s \) become independent of \( t \).
Consider an undamped oscillator (\( \beta\rightarrow 0 \)), with characteristic frequency \( \omega_0 \) and mass \( m \), that is at rest until it feels a force described by a Gaussian form,
$$
\begin{eqnarray*}
F(t)&=&F_0 \exp\left\{\frac{-t^2}{2\tau^2}\right\}.
\end{eqnarray*}
$$
For large times (\( t>>\tau \)), where the force has died off, find \( x(t) \).\\ Solve for the coefficients \( I_c \) and \( I_s \) in Eq. (25). Because the Gaussian is an even function, \( I_s=0 \), and one need only solve for \( I_c \),
$$
\begin{eqnarray*}
I_c&=&F_0\int_{-\infty}^\infty dt'~e^{-t^{\prime 2}/(2\tau^2)}\cos(\omega_0 t')\\
&=&\Re F_0 \int_{-\infty}^\infty dt'~e^{-t^{\prime 2}/(2\tau^2)}e^{i\omega_0 t'}\\
&=&\Re F_0 \int_{-\infty}^\infty dt'~e^{-(t'-i\omega_0\tau^2)^2/(2\tau^2)}e^{-\omega_0^2\tau^2/2}\\
&=&F_0\tau \sqrt{2\pi} e^{-\omega_0^2\tau^2/2}.
\end{eqnarray*}
$$
The third step involved completing the square, and the final step used the fact that the integral
$$
\begin{eqnarray*}
\int_{-\infty}^\infty dx~e^{-x^2/2}&=&\sqrt{2\pi}.
\end{eqnarray*}
$$
To see that this integral is true, consider the square of the integral, which you can change to polar coordinates,
$$
\begin{eqnarray*}
I&=&\int_{-\infty}^\infty dx~e^{-x^2/2}\\
I^2&=&\int_{-\infty}^\infty dxdy~e^{-(x^2+y^2)/2}\\
&=&2\pi\int_0^\infty rdr~e^{-r^2/2}\\
&=&2\pi.
\end{eqnarray*}
$$
Finally, the expression for \( x \) from Eq. (25) is
$$
\begin{eqnarray*}
x(t>>\tau)&=&\frac{F_0\tau}{m\omega_0} \sqrt{2\pi} e^{-\omega_0^2\tau^2/2}\sin(\omega_0t).
\end{eqnarray*}
$$


{pos ?= {}}
From now on we'll ignore common. Lets introduce placeholders. We want to put some content at the top to the page (the chide) that we calculate later. Of course, this could be solved in other ways! But it's a good introduction point.
Lets introduce {pos.next.value.title ?: "this"} next. {builder::as_code('examples/placeholders.html')}


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Modeltime is built for scale, and as such many users may want to use a distributed backend that allows for scale beyond their local machine. One such tool is Spark, a unified engine for large-scale data analytics. In this tutorial, we’ll show you how to use the Modeltime Spark Backend for high-performance computing.
One of the most common situations that parallel computation is required is when doing iterative forecasting where the data scientist needs to experiment with 10+ models across 10,000+ time series. It’s common for this large-scale, high-performance forecasting exercise to take days.
We’ll show you how we can combine Modeltime Nested Forecasting and it’s Parallel Spark Backend to scale this computation with distributed parallel Spark execution. Let’s go!
This tutorial requires:
sparklyr: Used to register the spark foreach
adapter via registerDoSpark()
.
Java: Spark installation depends on Java being installed.
Spark Installation: Can be accomplished via sparklyr::spark_install()
provided the user has sparklyr
and Java.
Next, we set up a Spark connection via sparklyr
. For this tutorial, we use the “local” connection. But many users will use Databricks to scale the forecasting workload.
Ro run Spark locally:
sc <- spark_connect(master = "local")
If using Databricks, you can use:
sc <- spark_connect(method = "databricks")
Next, we register the Spark Backend using parallel_start(sc, .method = "spark")
. This is a helper to set up the registerDoSpark()
foreach adaptor. In layman’s terms, this just means that we can now run parallel using Spark.
parallel_start(sc, .method = "spark")
We’ll run through a minimal nested forecasting example. Refer to the Nested Forecasting Tutorial for further information on how to perform iterative forecasting with the “nested” data structure in modeltime
.
The dataset we’ll be forecasting is the walmart_sales_weekly
, which we modify to just include 3 columns: “id”, “date”, “value”.
walmart_sales_weekly %>%
select(id, Date, Weekly_Sales) %>%
set_names(c("id", "date", "value")) %>%
group_by(id) %>%
plot_time_series(date, value, .facet_ncol = 2, .interactive = F)
We prepare as nested data using the Nested Forecasting preparation functions.
nested_data_tbl <- walmart_sales_weekly %>%
select(id, Date, Weekly_Sales) %>%
set_names(c("id", "date", "value")) %>%
extend_timeseries(
.id_var = id,
.date_var = date,
.length_future = 52
) %>%
nest_timeseries(
.id_var = id,
.length_future = 52
) %>%
split_nested_timeseries(
.length_test = 52
)
nested_data_tbl
#> # A tibble: 7 × 4
#> id .actual_data .future_data .splits
#> <fct> <list> <list> <list>
#> 1 1_1 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]>
#> 2 1_3 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]>
#> 3 1_8 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]>
#> 4 1_13 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]>
#> 5 1_38 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]>
#> 6 1_93 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]>
#> 7 1_95 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]>
We’ll create two unfitted models: XGBoost and Prophet. Then we’ll use modeltime_nested_fit()
to iteratively fit the models to each of the time series using the Spark Backend.
We create the XGBoost model on features derived from the date column.
rec_xgb <- recipe(value ~ ., extract_nested_train_split(nested_data_tbl)) %>%
step_timeseries_signature(date) %>%
step_rm(date) %>%
step_zv(all_predictors()) %>%
step_dummy(all_nominal_predictors(), one_hot = TRUE)
wflw_xgb <- workflow() %>%
add_model(boost_tree("regression") %>% set_engine("xgboost")) %>%
add_recipe(rec_xgb)
wflw_xgb
#> ══ Workflow ════════════════════════════════════════════════════════════════════
#> Preprocessor: Recipe
#> Model: boost_tree()
#>
#> ── Preprocessor ────────────────────────────────────────────────────────────────
#> 4 Recipe Steps
#>
#> • step_timeseries_signature()
#> • step_rm()
#> • step_zv()
#> • step_dummy()
#>
#> ── Model ───────────────────────────────────────────────────────────────────────
#> Boosted Tree Model Specification (regression)
#>
#> Computational engine: xgboost
Next, we create a prophet workflow.
rec_prophet <- recipe(value ~ date, extract_nested_train_split(nested_data_tbl))
wflw_prophet <- workflow() %>%
add_model(
prophet_reg("regression", seasonality_yearly = TRUE) %>%
set_engine("prophet")
) %>%
add_recipe(rec_prophet)
wflw_prophet
#> ══ Workflow ════════════════════════════════════════════════════════════════════
#> Preprocessor: Recipe
#> Model: prophet_reg()
#>
#> ── Preprocessor ────────────────────────────────────────────────────────────────
#> 0 Recipe Steps
#>
#> ── Model ───────────────────────────────────────────────────────────────────────
#> PROPHET Regression Model Specification (regression)
#>
#> Main Arguments:
#> seasonality_yearly = TRUE
#>
#> Computational engine: prophet
Now, the beauty is that everything is set up for us to perform the nested forecasting with Spark. We simply use modeltime_nested_fit()
and make sure it uses the Spark Backend by setting control_nested_fit(allow_par = TRUE)
.
Note that this will take about 30-seconds because we have a one-time cost to move data, libraries, and environment variables to the Spark clusters. But the good news is that when we scale up to 10,000+ time series, that the one-time cost is minimal compared to the speed up from distributed computation.
nested_modeltime_tbl <- nested_data_tbl %>%
modeltime_nested_fit(
wflw_xgb,
wflw_prophet,
control = control_nested_fit(allow_par = TRUE, verbose = TRUE)
)
#> Using existing parallel backend with 12 clusters (cores)...
#> Beginning Parallel Loop | 0.004 seconds
#> Finishing parallel backend. Clusters are remaining open. | 20.393 seconds
#> Close clusters by running: `parallel_stop()`.
#>
#> Finished in: 20.39446 secs.
The nested modeltime object has now fit the models using Spark.
nested_modeltime_tbl
#> # Nested Modeltime Table
#> # A tibble: 7 × 5
#> id .actual_data .future_data .splits .modeltime_tables
#> <fct> <list> <list> <list> <list>
#> 1 1_1 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]> <mdl_time_tbl [2 ×…
#> 2 1_3 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]> <mdl_time_tbl [2 ×…
#> 3 1_8 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]> <mdl_time_tbl [2 ×…
#> 4 1_13 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]> <mdl_time_tbl [2 ×…
#> 5 1_38 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]> <mdl_time_tbl [2 ×…
#> 6 1_93 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]> <mdl_time_tbl [2 ×…
#> 7 1_95 <tibble [143 × 2]> <tibble [52 × 2]> <split [91|52]> <mdl_time_tbl [2 ×…
We can observe the results. First, we can check the accuracy for each model
nested_modeltime_tbl %>%
extract_nested_test_accuracy() %>%
table_modeltime_accuracy(.interactive = F)
| Accuracy Table | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| id | .model_id | .model_desc | .type | mae | mape | mase | smape | rmse | rsq |
| 1_1 | 1 | XGBOOST | Test | 4552.91 | 18.14 | 0.90 | 17.04 | 7905.80 | 0.39 |
| 1_1 | 2 | PROPHET | Test | 4078.01 | 17.53 | 0.80 | 17.58 | 5984.31 | 0.65 |
| 1_3 | 1 | XGBOOST | Test | 1556.36 | 10.51 | 0.60 | 11.12 | 2361.35 | 0.94 |
| 1_3 | 2 | PROPHET | Test | 2126.56 | 13.15 | 0.83 | 13.91 | 3806.99 | 0.82 |
| 1_8 | 1 | XGBOOST | Test | 3573.49 | 9.31 | 1.52 | 9.87 | 4026.58 | 0.13 |
| 1_8 | 2 | PROPHET | Test | 3068.24 | 8.00 | 1.31 | 8.38 | 3639.60 | 0.00 |
| 1_13 | 1 | XGBOOST | Test | 2626.57 | 6.53 | 0.97 | 6.78 | 3079.24 | 0.43 |
| 1_13 | 2 | PROPHET | Test | 3367.92 | 8.26 | 1.24 | 8.73 | 4006.58 | 0.11 |
| 1_38 | 1 | XGBOOST | Test | 7865.35 | 9.62 | 0.67 | 10.14 | 10496.79 | 0.24 |
| 1_38 | 2 | PROPHET | Test | 16120.88 | 19.73 | 1.38 | 22.49 | 18856.72 | 0.05 |
| 1_93 | 1 | XGBOOST | Test | 6838.18 | 8.49 | 0.69 | 9.05 | 8860.59 | 0.48 |
| 1_93 | 2 | PROPHET | Test | 7304.08 | 10.00 | 0.74 | 9.57 | 9024.07 | 0.03 |
| 1_95 | 1 | XGBOOST | Test | 5678.51 | 4.59 | 0.68 | 4.72 | 7821.85 | 0.57 |
| 1_95 | 2 | PROPHET | Test | 5856.69 | 4.87 | 0.71 | 4.81 | 7540.48 | 0.49 |
Next, we can examine the test forecast for each of the models.
nested_modeltime_tbl %>%
extract_nested_test_forecast() %>%
group_by(id) %>%
plot_modeltime_forecast(.facet_ncol = 2, .interactive = F)
We can close the Spark adapter and shut down the Spark session when we are finished.
# Unregisters the Spark Backend
parallel_stop()
# Disconnects Spark
spark_disconnect_all()
#> [1] 1
We’ve now successfully completed a Nested Forecast using the Spark Backend. You may find this challenging, especially if you are not familiar with the Modeltime Workflow, terminology, or tidy-modeling in R. If this is the case, we have a solution. Take our high-performance forecasting course.
Become the forecasting expert for your organization
High-Performance Time Series Course
Time series is changing. Businesses now need 10,000+ time series forecasts every day. This is what I call a High-Performance Time Series Forecasting System (HPTSF) - Accurate, Robust, and Scalable Forecasting.
High-Performance Forecasting Systems will save companies by improving accuracy and scalability. Imagine what will happen to your career if you can provide your organization a “High-Performance Time Series Forecasting System” (HPTSF System).
I teach how to build a HPTFS System in my High-Performance Time Series Forecasting Course. You will learn:
Modeltime
- 30+ Models (Prophet, ARIMA, XGBoost, Random Forest, & many more)GluonTS
(Competition Winners)Become the Time Series Expert for your organization.


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