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http://blogs.msdn.com/b/oldnewthing/archive/2010/01.aspx
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# January, 2010
• #### How do I suppress full window drag/resize for just one window?
Is there a way to turn off Full Window Drag on a single window? I have a resizable control that I would like not update itself while resizing.
It so happens that I wrote a sample program ages ago to illustrate how to do this. You can find it in the Platform SDK under winui\fulldrag. The source code is also reproduced in this Knowledge Base article.
In addition to deferring painting, you may also want to defer layout if layout is expensive for your window.
• #### Microsoft phenomenon: The annual award that winds up being awarded only once
The Grammy Awards will be handed out this upcoming weekend, an annual award that seems to have survived.
A not uncommon phenomenon at Microsoft is the annual award that winds up being awarded only once. Because all the excitement is in the announcement, not in the actual award.
Every year, we want to uniquely call out and recognize a set of people. I'm proud to kick off the XYZ Awards, which we will be given every year, starting this year, which recognize employees who best represent ABC and DEF.
The XYZ Awards were indeed handed out that first year with great pomp and circumstance.
And were never heard from again.
This is a special case of the more general phenomenon of the introduction of some undertaking to great fanfare, only to have it quietly fade away into obscurity without any formal announcement that it had ended. One might cynically observe that the likelihood of this happening to your project increases the more dramatically it is introduced. If it's just called a program, then it might survive. If it's called an initiative, then you might want to hold off on ordering new business cards for a while. And if it's called a bold new initiative, then you'll want to spend some time freshening your résumé, because your project is doomed.
Update since people seem to be missing the point: I'm not talking about marketing campaigns. Those are meant to die out eventually. I'm talking about stuff like The Council for Programming Excellence (which meets only once) or The DTI Program (which has a kickoff meeting and then nothing).
• #### What idiot would hard-code the path to Notepad?
There seemed to be a great deal of disbelief that anybody would hard-code the path to Notepad.
There's a large class of problems that go like this:
I'm running Program X, and when I tell it to view the error log, I get this error message: CreateProcess of "C:\Windows\Notepad.exe errorlog.txt" failed: error 2: The system cannot find the file specified. What is wrong and how do I fix it?
Obviously, the file C:\Windows\Notepad.exe is missing. But how can that be? Well, Windows Server 2008 bit the bullet and removed one of the copies of Notepad. Once you learn this, troubleshooting the above problem becomes a simple exercise in psychic debugging.
My psychic powers tell me that you're running Windows Server 2008. The Notepad program no longer exists in the Windows directory; it's now in the system directory. Find the setting for your program that lets you change the program used for viewing error logs and tell it to use C:\Windows\System32\Notepad.exe.
Of course, this tip works only if the program permits you to change the program used for viewing error logs. If they hard-code the path, then you'll have to find some other workaround. (For example, you might try using the CorrectFilePaths shim.)
• #### If you are trying to understand an error, you may want to look up the error code to see what it means instead of just shrugging
A customer had a debug trace log and needed some help interpreting it. The trace log was generated by an operating system component, but the details aren't important to the story.
I've attached the log file. I think the following may be part of the problem.
[07/17/2005:18:31:19] Creating process D:\Foo\bar\blaz.exe
[07/17/2005:18:31:19] CreateProcess failed with error 2
Any ideas?
Thanks,
Bob Smith
Senior Test Engineer
Tailspin Toys
What struck me is that Bob is proud of the fact that he's a Senior Test Engineer, perhaps because it makes him think that we will take him more seriously because he has some awesome title.
But apparently a Senior Test Engineer doesn't know what error 2 is. There are some error codes that you end up committing to memory because you run into them over and over. Error 32 is ERROR_SHARING_VIOLATION, error 3 is ERROR_PATH_NOT_FOUND, and in this case, error 2 is ERROR_FILE_NOT_FOUND.
And even if Bob didn't have error 2 memorized, he should have known to look it up.
Error 2 is ERROR_FILE_NOT_FOUND. Does the file D:\Foo\bar\blaz.exe exist?
No, it doesn't.
-Bob
Bob seems to have shut off his brain and decided to treat troubleshooting not as a collaborative effort but rather as a game of Twenty Questions in which the person with the problem volunteers as little information as possible in order to make things more challenging. I had to give Bob a nudge.
Can you think of a reason why the system would be looking at D:\Foo\bar\blaz.exe? Where did you expect it to be looking for blaz.exe?
This managed to wake Bob out of his stupor, and the investigation continued. (And no, I don't remember what the final resolution was. I didn't realize I would have to remember the fine details of this support incident three years later.)
• #### Microspeak: Zap
You may hear an old-timer developer use the verb zap.
That proposed fix will work. Until everybody gets the fix, they can just zap the assert.
The verb to zap means to replace a breakpoint instruction with an appropriate number of NOP instructions (effectively ignoring it).
The name comes from the old Windows 2.x kernel debugger. (Actually, it may be even older, but that's as far back as I was able to trace it.) The Z (zap) command replaces the current instruction with a NOP if it is an int 3 (the x86 single-byte breakpoint instruction), or replaced the previous instruction with NOPs if it is an int 1 (the x86 two-byte breakpoint instruction).
This operation was quite common back in the days when lots of code was written in assembly language. A technique used by some teams was to insert a hard-coded breakpoint (called a TRAP) into every code path of a function. Here's an example (with comments and other identifying characteristics removed and new ones made up):
xyz8: mov bl,[eax].xyz_State
cmp bl,XYZSTATE_IGNORE
TRAPe
je short xyz10 ; ignore this one
or bl,bl
TRAPe
je short xyz11 ; end of table
mov bh,[eax].xyz_Flags
test bh,XYZFLAGS_HIDDEN
TRAPz
jz short xyz10 ; skip - item is hidden
test bh,XYZFLAGS_MAGIC
TRAPe
je short gvl10 ; skip - not the magic item
TRAP
bts [esi].alt_flags,ALTFLAGS_SEENMAGIC
TRAPc
jc short xyz10 ; weird - we shouldn't have two magic items
There were a variety of TRAP macros. Here we see the one plain vanilla TRAP and a bunch of fancy traps which trigger only when certain conditions are met. For example, TRAPc traps if the carry is set. Here's its definition:
TRAPc MACRO
local l
jnc short l
int 3
l:
ENDM
Hardly rocket science.
When you became the person to trigger a particular code path for the first time, you would trigger the trap, and you either stepped through the code yourself or (if you weren't familiar with the code) contacted the author of the code to verify that the code successfully handled this "never seen before" case. When sufficiently satisfied that a code path operated as expected, the developer removed the corresponding TRAP from the source code.
Of course, most TRAPs are removed before the code gets checked in, but the ones related to error handling or recovering from data corruption tend to remain (such as here, where we inserted a TRAP when we encounter two magic items, which is theoretically impossible).
When you trigger one trap, you usually trigger it a lot, and you usually trigger a lot of related traps as well. The Z command was quite handy at neutering each one after you checked that everything was working. You zapped the trap.
That's why old-timers refer to patching out a hard-coded breakpoint as zapping, even though the zap command hasn't existed for over a decade.
Update: As far as I can tell, the earlier uses of the word zap referred to patching binaries, not for removing hard-coded breakpoints after they stopped in the debugger.
• #### Why doesn't the window manager have a SetClipboardDataEx helper function?
Jonathan Wilson asks why the clipboard APIs still require GlobalAlloc and friends. Why is there not a SetClipboardDataEx or something that does what SetClipboardData does but without needing to call GlobalAlloc?
Okay, here's your function:
HANDLE SetClipboardDataEx(UINT uFormat, void *pvData, DWORD cbData)
{
if (uFormat == CF_BITMAP ||
uFormat == CF_DSPBITMAP ||
uFormat == CF_PALETTE ||
uFormat == CF_METAFILEPICT ||
uFormat == CF_DSPMETAFILEPICT ||
uFormat == CF_ENHMETAFILE ||
uFormat == CF_DSPENHMETAFILE ||
uFormat == CF_OWNERDISPLAY) {
return NULL; // these are not HGLOBAL format
}
HANDLE hRc = NULL;
HGLOBAL hglob = GlobalAlloc(GMEM_MOVEABLE | GMEM_SHARE | GMEM_ZEROINIT,
cbData);
if (hglob) {
void *pvGlob = GlobalLock(hglob);
if (pvGlob) {
CopyMemory(pvGlob, pvData, cbData);
GlobalUnlock(hglob);
hRc = SetClipboardData(uFormat, hglob);
}
if (!hRc) {
GlobalFree(hglob);
}
}
return hRc;
}
Whoop-dee-doo.
Historically, Windows doesn't go out of its way to include functions like this because you can easily write them yourself, or you can at least find a framework library that did it for you. Windows focused on doing the things that only Windows could do, providing you the building blocks with which you can create your own programs.
Besides, the classic clipboard is so old-school. The OLE clipboard provides a much richer interface, where you can generate data dynamically (for example as a stream) and expose it in formats other than just a chunk of bytes. Since SetClipboardData is old-school, if the window manager folks had written a function like SetClipboardDataEx, people would instead have asked the not unreasonable question, "Why did you bother to write a function that provides no essential new functionality to an old interface that was supplanted over a decade ago?"
• #### During process termination, the gates are now electrified
It turns out that my quick overview of how processes exit on Windows XP was already out of date when I wrote it. Mind you, the information is still accurate for Windows XP (as far as I know), but the rules changed in Windows Vista.
What about critical sections? There is no "Uh-oh" return value for critical sections; EnterCriticalSection doesn't have a return value. Instead, the kernel just says "Open season on critical sections!" I get the mental image of all the gates in a parking garage just opening up and letting anybody in and out.
In Windows Vista, the gates don't go up. Instead they become electrified!
If during DLL_PROCESS_DETACH at process termination on Windows Vista you call EnterCriticalSection on a critical section that has been orphaned, the kernel no longer responds by just letting you through. Instead, it says, "Oh dear, things are in unrecoverably bad shape. Best to just terminate the process now." If you try to enter an orphaned critical section during process shutdown, the kernel simply calls TerminateProcess on the current process!
It's sort of like the movie Speed: If the thread encounters a critical section that causes it to drop below 50 miles per hour, it blows up.
Fortunately, this error doesn't change the underlying analysis of How my lack of understanding of how processes exit on Windows XP forced a security patch to be recalled.
But it also illustrates how the details of process shutdown are open to changes in the implementation at any time, so you shouldn't rely on them. Remember the classical model for how processes exit: You cleanly shut down all your worker threads, and then call ExitProcess. If you don't follow that model (and given the current programming landscape, you pretty have no choice but to abandon that model, what with DLLs creating worker threads behind your back), it's even more important that you follow the general guidance of not doing anything scary in your DllMain function.
• #### Historically, Windows didn't tend to provide functions for things you can already do yourself
Back in the old days, programmers were assumed to be smart and hardworking. Windows didn't provide functions for things that programs could already do on their own. Windows worried about providing functionality for thing that programs couldn't do. That was the traditional separation of responsibilities in operating systems of that era. If you wanted somebody to help you with stuff you could in principle do yourself, you could use a runtime library or a programming framework.
You know how to open files, read them, and write to them; therefore, you could write your own file copy function. You know how to walk a linked list; the operating system didn't provide a linked list management library. There are apparently some people who think that it's the job of an operating system to alleviate the need for implementing them yourself; actually that's the job of a programming framework or tools library. Windows doesn't come with a finite element analysis library either.
You can muse all you want about how things would have been better if Windows had had an installer library built-in from the start or even blame Windows for having been released without one, but then again, the core unix operating system doesn't have an application installer library either. The unix kernel has functions for manipulating the file system and requesting memory from the operating system. Standards for installing applications didn't arrive until decades later. And even though such standards exist today (as they do in Windows), there's no law of physics preventing a vendor from writing their own installation program that doesn't adhere to those standards and which can do anything they want to the system during install. After all, at the end of the day, installing an application's files is just calling creat and write with the right arguments.
These arguments remind me of the infamous "Step 3: Profit" business plan of the Underpants Gnomes.
• Step 1: Require every Windows application to adhere to new rules or they won't run on the next version of Windows.
• ...
• Step 3: Windows is a successful operating system without applications which cause trouble when they break those rules.
It's that step 2 that's the killer. Because the unwritten step 2 is "All applications stop working until the vendors fix them."
Who's going to fix the the bill-printing system that a twelve-year-old kid wrote over a decade ago, but which you still use to run your business. (I'm not making this up.) What about that shareware program you downloaded three years ago? And it's not just software where the authors are no longer available. The authors may simply not have the resources to go back and update every single program that they released over the past twenty years. There are organizations with thousands of install scripts which are used to deploy their line-of-business applications. Even if they could fix ten scripts a day, it'd take them three years before they could even start thinking about upgrading to the next version of Windows. (And what about those 16-bit applications? Will they have to be rewritten as 32-bit applications? How long will that take? Is there even anybody still around who understands 16-bit Windows enough to be able to undertake the port?)
• #### The wrong way to determine the size of a buffer
A colleague of mine showed me some code from a back-end program on a web server. Fortunately, the company that wrote this is out of business. Or at least I hope they're out of business!
size = 16384;
size--;
}
|
2015-08-03 07:02:09
|
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|
https://paularoloye.com/best-sublimation-software
|
# 15 Best Sublimation Software for Design and Printing in 2022
One of the influences of technology on human existence is the creation of sublimation software. Technology has taken the world from level zero to one hundred. Do you agree? If your answer is yes, then you will also agree that in the past few years.
The whole world just got easier with the presence of technology, from the regular use of BIM computers to the use of Microsoft-aided long head computers, and now the use of flat laptop pcs.
Just like water, technology has flowed through all works of life. From different careers to different skills. Especially in this case of sublimation software.
Initially, printing was done ordinarily, and to imprint a certain design on any fabric, you have to make it manually sewed into the fabric or during the fabric manufacturing process.
Then, after the influx of technology at its highest, the whole printing skill became different. Way different from the regular printing methods, which brought never-seen creativity into printing, which is to design before printing.
All the printing was done on the basis of mere thoughts and ideology. It never came with proper templates to follow, or a certain look as a prototype before printing. This means that they were a lot of design printing mistakes that weren’t corrected because they couldn’t see what it will look like before printing.
This is to show you the enormous effort that technology has had on tasks like printing. That has gotten it to the discovery of sublimation printing, and now to the stage where it goes beyond just the hardware improvement, but also the software.
## What is a sublimation software?
Simply put, sublimation software is a computer-aided software package used in creating any design, logo, mark, or pattern to be used on a blank material.
All thanks to the influx of technology in the printing business, sublimation would not have been at this stage where demand is now more than supply. The process of sublimation has now gone farther than, a complex process done by certain experts.
Once you are old enough to handle the process, and you are technologically savvy enough, sublimation will be an easy piece to fold. Likewise, the operation of the sublimation software that is used in making marvelous designs that are imprinted on fabrics.
It is quite easy to use, unlike what most sublimation beginners believe, which is that sublimation software is complex computer packages meant for only expert sublimation creators.
No! That thought is very wrong.
A sublimation software can sometimes be complex to operate, and can also be very easy to operate. Unlike computer hardware, most sublimation software is easy to learn and utilize. The only basis to determining if a sublimation software is easy to use or not is the amount of creativity the sublimation creator has in mind.
Apart from the excessive usage that a sublimation software has, and the probable complexity in its usage. Sublimation software is used to create compelling designs, such as logos, patterns, and marks with lines, colors, fillers, crop tools, merge tools, and several other tools.
As much as they are used to create designs, they can also be used to edit already created designs, or photos, and sometimes create special effects on them. Some of them are special offline computer-aided software, while others are online software.
## Sublimation software for design and printing
If at some point, you have been stuck between choosing the right product to use for a task, then you have been subject to the same process that many sublimation creators go through. This is mostly caused by the need to create a perfect, and finite design.
A design that will not just sell but also create market change, or send a notable message to onlookers when the material is in use.
A sublimation software is an exact product that a sublimation creator should take very seriously. Most sublimation software will design multiple contents and effects, but every software is made for a certain task.
Finally, here are the 15 best sublimation software for design and printing in 2022.
### Photoshop
Software Overview
This is the royal amongst most of the sublimation software used for designs before sublimation printing. Photoshop is the second most mentioned sublimation software if not the first on every blog post. It is Adobe’s most priced software.
Most sublimation creators confuse the excessive usage and popularity of Photoshop with its functionality. Although, it is very multi-dimensional it is specifically made for Photo editing. It has an impressive collection of image editing tools, that welcomes both top sublimation designers and newbie sublimation designers.
It is also the most used sublimation software for creating imaginative designs. Just like Adobe Illustrator, Adobe Photoshop gives new users a 7day free trial to experience some features of the software, and see its suitability for their job.
Benefits
• It is easy to use on mobile devices.
• It has the largest collection of images and templates to use.
• It is very compatible with other Adobe software.
• It provides storage and opening of files in JPG, and PNG formats.
Drawbacks
• Like other Adobe software, it has an expensive plan.
• It requires a lot of design resource to handle.
• It creates easy leaning room.
### Corel Draw
Software Overview
Talking about popularity, Corel Draw is one of the most used sublimation software, it is a major frontier in the design of many sublimation designs. Corel Draw is a complete suite of vector illustration, layout, and design software.
When in use, it provides a comprehensive array of graphical tools on its template. Most sublimation creators use Corel Draw for creating logos and company brands, before printing them on a sublimation paper, and then putting them on a blank material.
Benefits
• It has an ideal interface configuration.
• It has an array of very helpful training videos for beginners.
• It has options for both full user license and subscription.
• It has an awesome tool box.
Drawbacks
• It doesn’t have a free version on MAC devices.
• It can sometimes have a difficult navigation.
• It has a complex free hand brush.
• It has a hidden navigation board.
### In Design
Software Overview
This is also one of the many creations of Adobe design house. In design is software that works comparatively with much other software from Adobe. It is mostly used by expert sublimation creators and a few sublimation beginners. It is the perfect sublimation software to use for poster creation and presentation documents.
In Design offers an armful number of tools to first-time users for about 90 days, before they actually buy the software. Although those tools might not be enough if you haven’t purchased the full version.
Benefits
• It is perfect for heavy image presentations.
• It easily works with other Adobe Software.
• It has a versatile layer configuration.
Drawbacks
• Just like other Adobe software, it is very costly.
• It is not as perfect as other Adobe software.
• It creates large file sizes.
### Canva
Software Overview
Canva has been in use for a few years now by many sublimation creators. Although, it is not been used regularly. But amongst the other sublimation software of its kind, it is the most used. In situations where you are in haste to print any design or image on a blank material, Canva is the best sublimation software to use.
It is a web-based design software, that provides a ready-to-go template for edits and use. It has an array of images, logos, background templates to go, and it has the best sublimation software for beginners with requiring less important design.
Benefits
• It is very easy to use and learn.
• It allows for about 80 designs on its free trial.
• It has a rich collection of resource tutorial.
Drawbacks
• It provides few free templates.
• It provides little to no editable downloads.
• It provides only 100 gig of cloud storage.
### Sketch
Software Overview
This is one of the most unique sublimation software, designed for special users. Sketch is a sublimation software designed specifically for MAC interface users. It allows extensive use of plug-ins with a very customizable interface. It also succeeds as web-based design software.
The designers created a 30-day free trial on the website for only business-based designers, which is perfect for a sublimation business owner. The software also offers a commendable functionality close to the regular sublimation software being used by many creators.
Benefits
• It is easy to use for beginners.
• It has an excellent documentation support.
• It has a very well-designed interface.
Drawbacks
• It uses only MAC interface.
• It has no version comparism feature.
• It has an expensive subscription.
### Affinity Designer
Software Overview
Affinity Designer is one of the two major Affinity software used for sublimation design. It encompasses a lot of design tools just like regular design software. This is ultimately one of the best sublimation software for sublimation design beginners.
Regardless of its status of being perfect for newbies, it is not a software that should be underrated, as it competes highly with Adobe Illustrator.
One of the unique features is the ability to zoom extensively into a design on the template. Also, it has the ability of almost more than 7000 steps reverse, should in case any mistake occurs while designing before printing.
Surprisingly, it has a one-off purchase and no subsequent subscriptions. Like much other sublimation software, it also allows for work on different interfaces like windows, android, and iPad.
Benefits
• It has varieties of art pattern.
• It has a lesser rate of resource input.
• It is very economical to purchase.
• It has both raster and vector in its program.
• It has the ability of previewing edits before printing.
Drawbacks
• It has little to no cloud services in provision.
• It sometimes creates confusing documentation.
• It is not compatible with other sublimation software.
### Gimp
Software Overview
Gimp is also one of the most used sublimation software available. It is one of the first open-source graphic design sublimation software, with its major strength in photo editing. It is used in cases where a particular photo is to be printed on a blank material.
Gimp can also be well compared with Adobe Photoshop as it gives the same level of functionality, but it is light sized. It has a user-friendly interface and can easily make edits on Photoshop files. Gimp has an outstanding support from its designers, and it has a lesser learning curve.
Benefits
• It provides different frequent updates once purchased.
• It has an array of help and support data system.
• It works with 2D and 3D.
Drawbacks
• It doesn’t support mobile usage.
• It has a rough CMYK image support.
### Affinity Photo
Software Overview
Affinity Photo is a very well valued software that is also rated closely to many Adobes software. Just like its counterpart Affinity Design, Affinity Photo also works with many layers and filters that can be used to create stunning effects while designing. It also uses raster, and vector images.
It is also very user friendly, with an easy interface, and a reasonable purchase price that comes after about 90 days of free trial. While editing existing design for use on a blank material in sublimation, Adobe Photo can be used to open such image, if it was first designed on Corel Draw. This simply means that it supports SVD formats.
Some of its other sophisticated photo editing features are 360-degree images, landscape edging, and HDR merging.
Benefits
• It is sold at a reasonable price considering its functions.
• It contains only a few spaces on your storage file.
• It gives a solid 90-day free trial option before purchase.
Drawbacks
• It doesn’t allow for the use of templates.
• It has less keyword tagging abilities.
• It has little to no cloud services in provision.
### Pic-monkey
Software Overview
Pic-Monkey is best known for its photo editing ability. Just like Adobe Photoshop and Gimp, it provides multiple features and tools for photo editing. It is built in such a way that it is very suitable for use by beginners.
It has options for advancement and increase in its photo editing ability. It is also high comparable to higher photo editing sublimation software.
Benefits
• It allows magnificent design options in few clicks.
• It allows users to design superb images.
• It provides more than 15 free collage images.
• It allows collage design preferences.
• It works with image upload from different sources.
• It allows multiple image uploads at once.
• It allows a share button after design.
• It supports cloud storage and autosave ability.
Drawbacks
• It doesn’t have an image auto-fixed size.
• It is very hard to add image as backgrounds.
• It doesn’t allow downloads in free trial
### Inkscape
Software Overview
Inkscape has most of its vector graphics tool has free programs. It was designed for beginners. Hence it is suitable for newbies in sublimation printing. It is a sublimation software that provides more than enough free usability, and in return lower functionality when compared to other sublimation software.
It is an excellent sublimation software designed for free users. It is best suited for design practices before real tasks. Regardless of its design purpose, Inkscape can be well compared to sublimation software like Adobe Illustrator, because it is also very flexible and user friendly.
Benefits
• It can export files.
• It has a built-in XML editor.
• It supports usage of SVG files.
• It has a cross usage platform.
• It offers a line-up of helpful tools.
Drawbacks
• It is a less spontaneous software.
• It is not very easy to navigate on screens.
### Gravit Designer
Software Overview
Gravit designer is a sublimation software that is mostly undervalued in its quality and flexibility. Most users believe it is only meant for image editing, but it allows for also image manipulation and vector illustrations.
It is made with an HTML program language; hence it has a user-friendly interface and it is compatible with many platforms. It provides a robust cloud storage of more than 400MB in its free version, and an unlimited storage in its full paid pro version.
Also in its free version, it allows for RGB colors, and in its pro version it supports HSB and CMYK.
Benefits
• It has a grounded free version.
• It provides autosave ability.
• It can zoom up to 20000%.
• It is social media savvy.
• It supports up to 300dpi in pro version.
Drawbacks
• It is an isolated software.
• It is not integrated with Google Drive.
### Vista Create
Software Overview
This is a sublimation software that is used by many sublimation creators. Vista-Create was formerly known as Crello. Although Google still recognizes it as Crello, but it has been changed on its interface.
It is majorly designed for video editing, but it also allows for highly customizable photo animation creations. Its video editing platforms are tied to YouTube, Instagram, or Twitter. It can also be used to create compelling brand kits.
Unlike many other sublimation software, Vista Create gives high allowance in its free version, with up to 40000 free image templates for new users.
Benefits
• It provides high free level for beginner’s design.
• It is available on all user interface.
• It provides an array of animated templates.
• It is easy to use.
Drawbacks
• It has a complex template arrangement format.
• It is not very compatible with other software.
### Pixlr
Software Overview
Pixlr is an image editing software, which is web-based. Although it is known for its web-based usability, but it can also be pinned to the user’s desktop, and also very accessible on mobile devices.
It is advisable to be used by new sublimation creators because it provides fewer complex features, unlike most photo editing sublimation software. Pixlr was formerly owned by the Autodesk family, but it is now recently on its own. It is very flexible and it supports several document types like SVG and PSD.
Benefits
• It is very affordable to purchase.
• It has high social media presence.
• It has several features for all design skill level.
Drawbacks
• It is not spontaneous.
• It has limited number of templates for beginners.
• It has an ad-heavy free version.
### Vectr
Software Overview
When in need of free sublimation software, Vectr is one of the very few that provides many features as a free software. It is also a web-based graphic design software with most of its strength in vector graphics and typographical designs.
It is used by most sublimation creators for brand font creation. It is also one of the few sublimations’ software with a collection of tutorial and how to guides to its usage. Vectr is the most suitable for users who love a very simple design interface with less functions.
Benefits
• It has a high collection of tutorials.
• It is a permanent free software.
• It saves files across many platforms.
• It uses blur-free graphics without pixels.
• It uses a shared URL connection.
Drawbacks
• It doesn’t have an import functionality.
• It doesn’t provide a media library.
• It always requires the use of an internet connection.
### Illustrator
Software Overview
Illustrator is one of Adobe’s elite software ever designed, it is a very standard vector graphic template.
Unlike its other counterparts from Adobe, Illustrator allows you to create free hand sketches that can be converted into usable resized vector images. Also, Illustrator works with the use of mathematical formulas to express resizing, and this has helped distant its usability from other design software.
While doing sublimation designs, Illustrator is a complex, and very standard sublimation software to use. This is why it is used mostly by expert sublimation creators.
The software providers offer a surprising 7day free trial after submitting the appropriate credit card. With options for monthly and annual payments which are $31.49 and$251.8 respectively.
This is a sublimation software that is very useful for creating standard sublimation designs. Most of all, Illustrator allows for usage on many devices like; windows, mac, and android.
Benefits
• It works perfectly with other Adobe software counterparts.
• It has a very original display interface.
• It allows its users the opportunity to create a design or work on initial designs.
• It provides a wide range of design tools.
Drawbacks
• It has a heavy cloud storage.
• It is super expensive compared to other sublimation software.
• It is a little complex to learn.
• It gives a better learning opportunity.
## Lastly
Every sublimation software is perfect for a specific design task, such as logo design, mark creation, font creation or photo edits. Therefore, creating an excellent sublimation design is not just based on the selection of the printers and the accessories, but also the usage of the exact sublimation software for creating the required design.
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2022-05-23 08:18:13
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https://yutsumura.com/tag/similarity-transformation/
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# Tagged: similarity transformation
## Problem 452
Let $A$ be an $n\times n$ complex matrix.
Let $S$ be an invertible matrix.
(a) If $SAS^{-1}=\lambda A$ for some complex number $\lambda$, then prove that either $\lambda^n=1$ or $A$ is a singular matrix.
(b) If $n$ is odd and $SAS^{-1}=-A$, then prove that $0$ is an eigenvalue of $A$.
(c) Suppose that all the eigenvalues of $A$ are integers and $\det(A) > 0$. If $n$ is odd and $SAS^{-1}=A^{-1}$, then prove that $1$ is an eigenvalue of $A$.
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2021-01-19 12:37:56
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https://msp.org/agt/2010/10-2/p14.xhtml
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#### Volume 10, issue 2 (2010)
Recent Issues
The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Convexity package for momentum maps on contact manifolds
### River Chiang and Yael Karshon
Algebraic & Geometric Topology 10 (2010) 925–977
##### Abstract
Let a torus $T$ act effectively on a compact connected cooriented contact manifold, and let $\Psi$ be the natural momentum map on the symplectization. We prove that, if $dimT$ is bigger than 2, the union of the origin with the image of $\Psi$ is a convex polyhedral cone, the nonzero level sets of $\Psi$ are connected (while the zero level set can be disconnected), and the momentum map is open as a map to its image. This answers a question posed by Eugene Lerman, who proved similar results when the zero level set is empty. We also analyze examples with $dimT\le 2$.
##### Keywords
momentum map, contact manifold, torus action, convexity
##### Mathematical Subject Classification 2000
Primary: 53D10, 53D20
Secondary: 52B99
##### Publication
Revised: 25 February 2010
Accepted: 2 March 2010
Published: 17 April 2010
##### Authors
River Chiang Department of Mathematics National Cheng Kung University Tainan 701 Taiwan Yael Karshon Department of Mathematics University of Toronto Toronto, Ontario M5S 2E4 Canada http://www.math.toronto.edu/karshon/
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2021-03-01 11:07:10
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http://mathhelpforum.com/geometry/146606-area-rhombus.html
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# Thread: Area of a Rhombus
1. ## Area of a Rhombus
What is the area of a rhombus when the only given information is it's perimeter?
2. Originally Posted by BrendanLoftus
What is the area of a rhombus when the only given information is it's perimeter?
There are four congruent sides to a rhombus, so each side is one-quarter of the perimeter. Divide the perimeter by 4 and you've got the length of each side.
The area is base*height, but you can't know what the height is without knowing how much of a slant the thing is at compared to a square.
3. Hello, BrendanLoftus!
What is the area of a rhombus when the only given information is its perimeter?
Since a rhombus is equilateral, each side is: . $x \:=\:\tfrac{1}{4}\text{(perimeter)}$
But the area can range from $0\text{ to }x^2.$
Code:
B
A o = = = = o = = = = o C
x D x
B o - - - - o C
/ /
/ / x
/ /
A o - - - - o D
x
B o - - - - o C
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x | |
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A o - - - - o D
x
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2017-02-25 22:58:17
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https://parabola.unsw.edu.au/1990-1999/volume-28-1992/issue-2
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Volume 28
, Issue 2
1992
When a honeybee constructs its combs to store its honey, it does so in such a way as to store the maximum amount of honey using the minimum amount of wax.
Without any doubt $\pi$ is the most famous of all real numbers, it appears throughout all branches of mathematics.
The exponential function $x \mapsto e^x$ and its inverse, the (natural) logarithm function $x \mapsto ln(x) (x > 0)$, are amongst the most important in mathematics, arising as they do in many different applications.
A triangle is one of the simplest configurations we can draw in the plane.
The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.
The last question on the Westpac Mathematics Competition, Senior Division, 1989 reads:
Before asking whether mathematics is useful, we need to know what "mathematics" means.
On January 13th of this year TIME magazine reported that Grace Hooper had died in Arlington, Virginia at the age of 85 years.
Q.872 A flea on the number line jumps from the point $a$ to the point $b$ given by $a + \frac{1}{b} = 1$.
Q.861 each of the numbers in a list $$x_1, x_2, x_3, \cdots , x_n, \cdots$$ is a positive integer written as usual in decimal notation.
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2022-08-08 05:32:16
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https://datascience.stackexchange.com/questions/15314/nltk-sklearn-genism-text-to-topic
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# NLTK Sklearn Genism Text to Topic
I aint no data scientist/machine learner.
What Im Lookin for
text = "Donald Trump became the president of America"
#some data science works
print(topics)
#prints ["politics"]
text = "Rihanna is starring in the new movie The Inception 2"
#some data science works
print(topics)
#prints ["movie","music"]
# What I can do
I can extract words like Donald Trump,America,Rihanna using POS
I can get huge paragraphs/lists of articles/words on politics,movies etc and save them in a text file.
# What i can't do
Make meaningful topics out of those words like sports,politics,movies
# What i want you do to
Point me in the right specific direction towards solving this problem aka enlighten me
I am staying quite generic since you asked for enlightenment, just mentioning some possible directions that you can explore.
You have basically two possibilities:
1. Classification of the text (Supervised learning).
Supervised means that you need first to externally apply labels (for example manually by humans) to examples of texts (labels could be "politics" or "show") and then use one of the classification algorithms.
You have extracted words from the text, so you could use a "bag of words" approach for classifying.
There exist adaptations of classification algorithms (multi-label classification) in order to provide multiple labels (such as one text is labelled both with "music" and "movie").
You can find text corpora already pre-labelled, to train the algorithm and partly avoid the manual effort.
2. Clustering the text, topic modelling (unsupervised learning).
In this case you do not need to provide examples with labels but the algorithm will cluster the texts according to parameters like the similarity of two texts or on keywords/topics extracted from the texts.
Although this method do not require labeled examples, the clusters you get as output will require fine tuning, e.g. number of clusters to produce, names of the clusters, etc.
Since you mentioned SKlearn, you can find some directions on their web site: Text Feature Extraction.
Here is a comprehensive (bit older) summary: Machine Learning in Automated Text Categorization by Fabrizio Sebastiani
To build off Mashimo's answer, one straightforward approach for topic modeling is "Latent Dirichlet Allocation" (LDA). The basic idea behind LDA is explained in this really good tutorial. Essentially, documents are assumed to be composed of mixtures of topics, which are in turn composed of mixtures of words. If we knew the topic and document distributions, we could generate new documents using a probabilistic model. In LDA, we run this process in reverse to infer the topic and document distributions given the documents.
In the tutorial, the author uses LDA to find topics within Sarah Palin's emails, which is not too dissimilar to what you're trying to do, if I understand correctly. For example, one topic is composed of the words "gas, oil, pipeline, agia, project, natural, north", which corresponds roughly to the topic "energy" or "gas". Note that LDA doesn't name the topics for you; you'll have to apply your own judgment to construct a sensible name for the group of words comprising a topic.
LDA has been implemented in packages like Gensim. To see how to use LDA in Python, you might find this SpaCy tutorial (which covers a lot of stuff in addition to LDA) useful.
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2020-08-13 15:38:09
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https://paperswithcode.com/paper/communication-efficient-parallel-algorithms
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# Communication Efficient Parallel Algorithms for Optimization on Manifolds
Bayan SaparbayevaMichael Minyi ZhangLizhen Lin
The last decade has witnessed an explosion in the development of models, theory and computational algorithms for "big data" analysis. In particular, distributed computing has served as a natural and dominating paradigm for statistical inference... (read more)
PDF Abstract
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2020-05-28 06:39:30
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https://stats.stackexchange.com/questions/80070/multivariant-multi-class-outlier-detection-of-single-points
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# Multivariant, multi class outlier detection of single points
The problem:
I have a set of means and covariance matrices of a number of multivariant normal distributions, each having a class label. Then I get single data points, one after the other, to which I want to assign one of the class labels or the label 'None', if it is unlikely that the point belongs to any of the given classes (could be drawn from the corresponding distribution).
My idea:
Simplified to the one-dimensional case I would compute the distance of the point to each mean (in terms of corresponding standard deviation units) and assign the label of the closest mean or 'None' if all distances exceed the e.g. 2-sigma boundary (to catch 95% of the probability).
What I have so far:
All I got so far comes from this book. There the same idea is applied to multi-dimensional data using the Mahalanobis distance, but with the difference that you cannot say 'outlier if distance is greater than some fixed value' because you have different deviations in (and between) each dimension. Instead the authors use the set of points (or to be more precise the size of the set of), for which they want to decide which points are outliers, to estimate the rejection distance threshold using a chi-squared distribution. The problem is that this only seems to be reasonable for large sets of points, what directly leads to my questions:
1. Can I apply the same approach if my point set is of size 1 and still get good results?
2. If not, can the approach be adapted to the situation where you have single points?
3. If not, is there a way to label single observation given the parameters of multiple multinomial distributions with an additional rejection class?
In order to make things a bit more concrete, consider the following. In a series of random draws from a one-dimensional Gaussian, roughly 99.73% of the points that you draw will lie within 3 sigma of the mean. But suppose you have a sample size of $n=10^{6}$ draws. This means that 997,300 points will fall within the 3 sigma limit, but 2700 points will fall outside of it. Are those 2700 points outliers? Well, no, not really: when you make a million random draws on a Gaussian, you're basically guaranteed to get at least a few points outside of the 3 sigma limit, and those points are no more weird, unusual or unexpected that the ones which fell inside of the boundary; this is simply a natural consequence of making a million draws. In the way that the book has conceived it, true outliers are only those points which are so far away from the mean (in terms of sigma, or its higher dimensional analogue Mahalanobis distance) that we would have expected to see no points that far into the tails, even given a large sample size. True outliers, in the way that I think the book seems to be defining it, are those which appear not to have even been drawn from a Gaussian pdf at all, but some other pdf with fatter tails.
In this conception, any sample size may be used to define outliers using the same methodology that the authors have suggested, even $n=1$. However, you must take great care to be sure that you are defining $n$ correctly for your particular problem. In your case, I would say that $n$ is the size of your entire data set (i.e., the total number of points that you eventually plan to test), not just the number of points that you are testing within a single test iteration.
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2020-01-18 13:05:01
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https://chaoxuprime.com/posts/2017-04-20-list-the-smallest-k-subset-sums.html
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# List the smallest k subset sums
Problem1
Given a set of positive reals \set{x_1,\ldots,x_n} where x_1<x_2<\ldots<x_n, find the smallest k subset sums.
We can assume n\leq k, because we do not have to read x_j if j>k.
Torsten Gross and Nils Blüthgen posted a O(k^2) time solution on arXiv.
We show a O(k\log k) time algorithm, which is optimal if we want to output the numbers in order.
We list the sums one by one by maintaining a priority queue of sums. We start with the empty set. Assume that we added the sum induced by I\subset [n] (that is, \sum_{i\in I} x_i) into the output, let j=1+\max I. Now we can consider two possibilities by extending the current solution: the sum induced by I\cup \set{j} or the sum induced by I\cup \set{k} where k>j. We will add both possibilities to the queue so that one can inspect them later. We can avoid storing the sets, only the values are required.
Here is a python implementation.
def first_k_subset_sums(x,k):
n = len(x)
h = []
output = [0] # need to account for the empty set
heapq.heappush(h,(x[0],0))
while h and len(output)<k:
(u,b) = heapq.heappop(h)
output.append(u)
if b+1<n:
heapq.heappush(h,(u+x[b+1],b+1))
heapq.heappush(h,((u-x[b])+x[b+1],b+1))
return output
If we want to output the sets themselves, not just the values, does the running time change? If a set I is in the output, then all subsets of I must also be in the output. Hence the largest set we can ever output has size O(\log k). Therefore the total output length is at most O(k\log k).
This is also a lower bound. Consider when x_i=2^i, then we will output all subsets of \set{x_1,\ldots,x_{\log k}}, and we know that \sum_{i=1}^{\log k} i{\log k\choose i} = \Omega(\log k).
If we don't have to list the smallest k subset sum values in order, then O(k) is possible, see this mathoverflow answer by David Eppstein.
If we are interested in the smallest k distinct subset sum. I don't know of any algorithm that performs better than O(nk), even if we know that n=\Omega(k).
# Acknowledgements
I would like to thank Tana Wattanawaroon for helpful discussions and taking an interest in this problem.
Posted by Chao Xu on 2017-04-20.
Tags: algorithm.
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2019-03-21 10:19:47
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http://mathoverflow.net/questions/63601/what-is-an-example-of-a-ring-with-two-or-more-multiplicative-right-identities
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# What is an example of a ring with two (or more) multiplicative right-identities?
I know that if a ring has a multiplicative identity, then the multiplicative identity must be unique. Are there simple-to-describe examples of rings with two (or more) multiplicative right-identities?
-
Take the semigroup ring of a semigroup with two or more multiplicative right identities. For example, the semigroup $$S = \langle a, b | ab = aa = a, ba = bb = b \rangle$$ works (it is the universal example, so if it fails then no example can work). Multiplication in this semigroup can be described as follows: every word evaluates to the first letter in it. The resulting semigroup ring is literally the free ring on two not necessarily identical right identities.
I was browsing some "What is...?" threads, and bumped in here: Let $n$ be a positive integer, and consider the subring of the opposite of the ring of $n$-by-$n$ matrices over a commutative unital ring $\mathbb A = (A, +, \cdot)$ consisting of those matrices all of whose rows, except at most for the first, are zero; this has $|A|^{n-1}$ right identities, given by those matrices whose first row is any vector of $A^n$ with first element equal to $1_\mathbb{A}$.
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2016-02-14 14:46:40
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https://mathexpressionsanswerkey.com/math-expressions-grade-5-unit-1-lesson-10-answer-key/
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# Math Expressions Grade 5 Unit 1 Lesson 10 Answer Key Practice with Unlike Mixed Numbers
## Math Expressions Common Core Grade 5 Unit 1 Lesson 10 Answer Key Practice with Unlike Mixed Numbers
Math Expressions Grade 5 Unit 1 Lesson 10 Homework
Lesson 10 Practice with Unlike Mixed Numbers Math Expressions Grade 5 Unit 1 Question 1.
Explanation:
Perform Subtraction operation on 3 and the mixed number 1(2/5). First multiply 5 with 1 the product is 5. Add 5 with 2 the sum is 7. The mixed number 1(2/5) in fraction form as (7/5). Subtract (7/5) from 3 then the difference is (8/5). The fraction (8/5) can be written as the mixed number 1(3/5).
Practice with Unlike Mixed Numbers Math Expressions Grade 5 Unit 1 Lesson 10 Question 2.
Explanation:
Perform Addition operation on these two mixed numbers 2(7/10) and 2(4/5). First multiply 10 with 2 the product is 20. Add 20 with 7 the sum is 27. The mixed number 2(7/10) in fraction form as (27/10). Next multiply 5 with 2 the product is 10. Add 10 with 4 the sum is 14. The mixed number 2(4/5) in fraction form as (14/5). Add (27/10) with (14/5) then the sum is (55/10). The fraction (55/10) can be written as the mixed number 5(5/10).
Math Expressions Grade 5 Unit 1 Lesson 10 Answer Key Practice with Unlike Mixed Numbers Question 3.
Explanation:
Perform Subtraction operation on these two mixed numbers 7(5/9) and 3(2/15). First multiply 9 with 7 the product is 63. Add 63 with 5 the sum is 68. The mixed number 7(5/9) in fraction form as (68/9). Next multiply 15 with 3 the product is 45. Add 45 with 2 the sum is 47. The mixed number 3(2/15) in fraction form as (47/15). Subtract (47/15) from (68/9) then the difference is (199/45). The fraction (199/45) can be written as the mixed number 4(19/45).
Question 4.
Explanation:
Perform Addition operation on these two mixed numbers 4(5/6) and (6/7). First multiply 6 with 4 the product is 24. Add 24 with 5 the sum is 29. The mixed number 4(5/6) in fraction form as (29/6). Add (29/6) with (6/7) then the sum is (239/42). The fraction (239/42) can be written as the mixed number 5(29/42).
Question 5.
Explanation:
Perform Subtraction operation on these two mixed numbers 5(1/8) and 4(1/5). First multiply 8 with 5 the product is 40. Add 40 with 1 the sum is 41. The mixed number 5(1/8) in fraction form as (41/8). Next multiply 5 with 4 the product is 20. Add 20 with 1 the sum is 21. The mixed number 4(1/5) in fraction form as (21/5). Subtract (21/5) from (41/8) then the difference is (37/40).
Question 6.
Explanation:
Perform Addition operation on these two mixed numbers 4(79/100) and 5(9/10). First multiply 100 with 4 the product is 400. Add 400 with 79 the sum is 479. The mixed number 4(79/100) in fraction form as (479/100). Next multiply 10 with 5 the product is 50. Add 50 with 9 the sum is 59. The mixed number 5(9/10) in fraction form as (59/10). Add (479/100) with (59/10) then the sum is (1069/100). The fraction (1069/100) can be written as the mixed number 10(69/100).
Question 7.
Explanation:
Perform Addition operation on these two fractions (13/16) and (2/3). Add (13/16) with (2/3) then the sum is (71/48).
Question 8.
Explanation:
Perform Subtraction operation on these two mixed numbers 8(1/4) and 3(9/20). First multiply 4 with 8 the product is 32. Add 32 with 1 the sum is 33. The mixed number 8(1/4) in fraction form as (33/4). Next multiply 20 with 3 the product is 60. Add 60 with 9 the sum is 69. The mixed number 3(9/20) in fraction form as (69/20). Subtract (69/20) from (33/4) then the difference is (96/20). The fraction (96/20) can be written as the mixed number 4(16/20).
Question 9.
Explanation:
Perform Addition operation on these two mixed numbers 7(8/9) and 9(7/8). First multiply 9 with 7 the product is 63. Add 63 with 8 the sum is 71. The mixed number 7(8/9) in fraction form as (71/9). Next multiply 8 with 9 the product is 72. Add 72 with 7 the sum is 79. The mixed number 9(7/8) in fraction form as (79/8). Add (71/9) with (79/8) then the sum is (1279/72). The fraction (1279/72) can be written as the mixed number 17(55/72).
Solve.
Question 10.
The Taylors have four dogs. Molly eats 4$$\frac{1}{2}$$ cups of food each day, Roscoe eats 3$$\frac{2}{3}$$ cups, Milo eats 1$$\frac{3}{4}$$ cups, and Fifi eats cup. How much do the Taylors’ dogs eat each day altogether?
Molly eats 4(1/2) cups of food each day.
4(1/2) = (9/2)
Roscoe eats 3(2/3) cups.
3(2/3) = (11/3)
Milo eats 1(3/4) cups.
1(3/4) = (7/4)
Fifi eats 1 cup.
(9/2) + (11/3) + (7/4) + 1 = (131/12)
The Taylors’ dogs eat each day altogether (131/12).
Explanation:
Molly eats 4(1/2) cups of food each day, Roscoe eats 3(2/3) cups, Milo eats 1(3/4) cups and Fifi eats 1 cup. Add (9/2) with (11/3), (7/4) and 1 then the sum is (131/12). The Taylors’ dogs eat each day altogether (131/12).
Question 11.
Refer to Problem 10. How much more food does Molly eat each day than Roscoe?
Molly eats each day (9/2) cups of food.
Roscoe eats each day (11/3) cups of food.
(9/2) – (11/3) = (5/6)
Molly eats (5/6) cups of food each day than Roscoe.
Explanation:
Molly eats each day (9/2) cups of food. Roscoe eats each day (11/3) cups of food. Subtract (11/3) from (9/2) then the difference is (5/6). Molly eats (5/6) cups of food each day than Roscoe.
Question 12.
The vet told the Taylors (from Problem 10) to decrease the amount Molly eats by $$\frac{3}{4}$$ cup. After Molly’s food is adjusted, will she eat more or less than Roscoe each day? How much more or less?
(9/2) – (3/4) = (15/4)
After Molly’s food is adjusted Molly eats (15/4) cups of food.
Roscoe eats each day (11/3) cups of food.
(15/4) – (11/3) = (1/12)
Molly eats (1/12) cup more than Roscoe.
Explanation:
If we calculate the difference between Molly’s daily food and Roscoe’s daily food[(15/4) – (11/3) = (1/12)].
After decreasing the amount of food Molly eats per day, still Molly eats (1/12) cup more than Roscoe.
Math Expressions Grade 5 Unit 1 Lesson 10 Remembering
What mixed number is shown by each shaded part ?
Question 1.
_______
2(3/4)
The mixed number for the above shaded parts are 2(3/4).
Explanation:
In the above image we can observe three objects. In that two objects are completely shaded and one object is not completely shaded. So the mixed number for the above shaded parts are 2(3/4).
Question 2.
_______
1(1/2)
The mixed number for the above shaded parts are 1(1/2).
Explanation:
In the above image we can observe two objects. In that one object is completely shaded and one object is not completely shaded. So the mixed number for the above shaded parts is 1(1/2).
Question 3.
_______
3(1/3)
The mixed number for the above shaded parts is 3(1/3).
Explanation:
In the above image we can observe three objects. In that two objects are completely shaded and one object is not completely shaded. So the mixed number for the above shaded parts is 3(1/3).
Question 4.
How many cookies are there altogether? _____
In the above bar graph we can observe cookies. There are three cookies altogether. one is chocolate chip, second one is oatmeal, third one is peanut butter.
Question 5.
What fraction of the cookies are chocolate chip? _____
(20/(20+15+10) = (20/45) = (4/9)
The fraction (4/9)of the cookies are chocolate chips.
Explanation:
In the above bar graph we can observe three cookies. The chocolate chip is at 20 number. Add 20 with 15 and 10 then the sum is 45. (20/45) can be simplified as (4/9). (4/9) is the fraction of cookies that are chocolate chips.
Question 6.
What fraction of the cookies are oatmeal? _____
(15/(20+15+10) = (15/45) = (3/9)
The fraction (3/9) of the cookies are peanut butter.
Explanation:
In the above bar graph we can observe three cookies. The oatmeal is at the number 15. Add 20 with 15 and 10 then the sum is 45. (15/45) can be simplified as (3/9). (3/9) is the fraction of cookies that are oatmeal.
Question 7.
What fraction of the cookies are peanut butter? _____
(10/(20+15+10) = (10/45) = (2/9)
The fraction (2/9) of the cookies are peanut butter.
Explanation:
In the above bar graph we can observe three cookies. The peanut butter is at the number 10. Add 20 with 15 and 10 then the sum is 45. (10/45) can be simplified as (2/9). (2/9) is the fraction of cookies that are peanut butter.
Question 8.
Melanie baked 25 cookies. Did she bake more or less than half of the cookies? ____
How do you know?
Question 9.
Stretch Your Thinking Colby nailed together four wood boards as shown at the right. All four boards are 5$$\frac{1}{2}$$ inches wide.
a. Find the perimeter of the outside rectangle.
Perimeter of the rectangle is equal to 2(length + breadth).
Here length = 24(3/4) inches.
Length = (99/4) inches.
Breadth = 5(1/2) + 8(1/4) + 5(1/2)
Breadth = (11/2) + (33/4) + (11/2)
Perimeter of the outside rectangle = 2(length + breadth) = 2[(99/4) + (77/4)] = 2[(176/4)] = 88
Perimeter of the outside rectangle = 88 inches.
Explanation:
In the above image we can observe that length is equal to 24(3/4) inches. The fraction form of 24(3/4) is (99/4) inches. Breadth is equal to 5(1/2), 8(1/4), and 5(1/2). The fraction form is (11/2), (33/4), (11/2). Add (11/2) with (33/4) and (11/2) then the sum is (77/4) inches. We know that perimeter of the rectangle is equal to 2(length + breadth). Add length and breadth then the sum is 2(176/4) which is equal to 88 inches. Perimeter of the outside rectangle = 88 inches.
b. Find the perimeter of the inside rectangle.
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2022-05-21 05:30:43
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https://www.physicsforums.com/threads/calculate-attractive-force-between-cu2-and-o2-ions.710016/
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# Calculate attractive force between Cu2+ and O2- ions.
1. Sep 12, 2013
### robertjordan
1. The problem statement, all variables and given/known data
Calculate the attractive force between a pair of Cu2+ and O2- ions in the ceramic CuO that has an interatomic separation of 200pm.
2. Relevant equations
$E_A= -\frac{(z_1\cdot e)(z_2\cdot e)}{4\pi\cdot\epsilon_o\cdot r}$
Where z_1 and z_2 are the valences of the two ion types, e is the charge of an electron (1.602 * 10^-19 C), epsilon_o is the permittivity of a vacuum (8.85*10^-12 F/m), and r is the distance between the two ions.
$E_n=\frac{m\cdot e^4 \cdot z^2}{2n^2 \cdot \hbar^2}$
Where m= mass of electron, z= atomic number, e= charge of an electron, n is the energy level.
3. The attempt at a solution
The problem is that I don't know how to find z_1 and z_2. Do I use $E_n=\frac{m\cdot e^4 \cdot z^2}{2n^2 \cdot \hbar^2}$ to find the energy in the valence electrons? The problem is that I don't know how to use that equation because when I plug in what I think I should for the variables it gives me an answer with units all wrong... Here's an example from another problem where I tried to use that equation...
So what to do?
2. Sep 12, 2013
### nasu
Why are you writing these formulas for energies?
The problem asks to calculate the attractive force.
The charges of each ion are given in the problem.
3. Sep 12, 2013
### robertjordan
Well I don't know the force equation, my teacher only gave us the equation for bonding energy...
Perhaps since energy=force*distance we can find force by dividing our energy equation by some distance?
I'm still stuck but I see now that z_1= 2 and z_2=-2.
Any more help?
4. Sep 12, 2013
### nasu
5. Sep 12, 2013
### robertjordan
The equation F= ke(|q1q2|)/r2 looks good.
So if I plug in 3.204 × 10^-19 coulombs for q1 and -3.204 × 10^-19 coulombs for q2 (because O2- has a net charge equal to -2 times the charge of an electron and Cu2+ has a net charge equal to twice the charge of an electron), then I get
2.307*10-8 N of force. Does that seem right?
6. Sep 13, 2013
### nasu
Yes, it looks OK.
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2017-08-20 12:16:59
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https://it.mathworks.com/help/dsp/ug/estimate-the-transfer-function-of-an-unknown-system-1.html
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Documentation
### This is machine translation
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## Estimate the Transfer Function of an Unknown System
You can estimate the transfer function of an unknown system based on the system's measured input and output data.
In DSP System Toolbox™, you can estimate the transfer function of a system using the `dsp.TransferFunctionEstimator` System object™ in MATLAB® and the Discrete Transfer Function Estimator block in Simulink®. The relationship between the input x and output y is modeled by the linear, time-invariant transfer function Txy. The transfer function is the ratio of the cross power spectral density of x and y, Pyx, to the power spectral density of x, Pxx:
`${T}_{xy}\left(f\right)=\frac{{P}_{yx}\left(f\right)}{{P}_{xx}\left(f\right)}$`
The `dsp.TransferFunctionEstimator` object and Discrete Transfer Function Estimator block use the Welch’s averaged periodogram method to compute the Pxx and Pxy. For more details on this method, see Spectral Analysis.
Coherence
The coherence, or magnitude-squared coherence, between x and y is defined as:
`${C}_{xy}\left(f\right)=\frac{{|{P}_{xy}|}^{2}}{{P}_{xx}*{P}_{yy}}$`
The coherence function estimates the extent to which you can predict y from x. The value of the coherence is in the range 0 ≤ Cxy(f) ≤ 1. If Cxy = 0, the input x and output y are unrelated. A Cxy value greater than 0 and less than 1 indicates one of the following:
• Measurements are noisy.
• The system is nonlinear.
• Output y is a function of x and other inputs.
The coherence of a linear system represents the fractional part of the output signal power that is produced by the input at that frequency. For a particular frequency, 1 – Cxy is an estimate of the fractional power of the output that the input does not contribute to.
When you set the `OutputCoherence` property of `dsp.TransferFunctionEstimator` to `true`, the object computes the output coherence. In the Discrete Transfer Function Estimator block, to compute the coherence spectrum, select the Output magnitude squared coherence estimate check box.
### Estimate the Transfer Function in MATLAB
To estimate the transfer function of a system in MATLAB™, use the `dsp.TransferFunctionEstimator` System object™. The object implements the Welch's average modified periodogram method and uses the measured input and output data for estimation.
Initialize the System
The system is a cascade of two filter stages: dsp.LowpassFilter and a parallel connection of dsp.AllpassFilter and dsp.AllpoleFilter.
```allpole = dsp.AllpoleFilter; allpass = dsp.AllpassFilter; lpfilter = dsp.LowpassFilter; ```
Specify Signal Source
The input to the system is a sine wave with a frequency of 100 Hz. The sampling frequency is 44.1 kHz.
```sine = dsp.SineWave('Frequency',100,'SampleRate',44100,... 'SamplesPerFrame',1024); ```
Create Transfer Function Estimator
To estimate the transfer function of the system, create the `dsp.TransferFunctionEstimator` System object.
```tfe = dsp.TransferFunctionEstimator('FrequencyRange','onesided',... 'OutputCoherence', true); ```
Create Array Plot
Initialize two `dsp.ArrayPlot` objects: one to display the magnitude response of the system and the other to display the coherence estimate between the input and the output.
```tfeplotter = dsp.ArrayPlot('PlotType','Line',... 'XLabel','Frequency (Hz)',... 'YLabel','Magnitude Response (dB)',... 'YLimits',[-120 20],... 'XOffset',0,... 'XLabel','Frequency (Hz)',... 'Title','System Transfer Function',... 'SampleIncrement',44100/1024); coherenceplotter = dsp.ArrayPlot('PlotType','Line',... 'YLimits',[0 1.2],... 'YLabel','Coherence',... 'XOffset',0,... 'XLabel','Frequency (Hz)',... 'Title','Coherence Estimate',... 'SampleIncrement',44100/1024); ```
By default, the x-axis of the array plot is in samples. To convert this axis into frequency, set the 'SampleIncrement' property of the `dsp.ArrayPlot` object to Fs/1024. In this example, this value is 44100/1024, or 43.0664. For a two-sided spectrum, the `XOffset` property of the `dsp.ArrayPlot` object must be [-Fs/2]. The frequency varies in the range [-Fs/2 Fs/2]. In this example, the array plot shows a one-sided spectrum. Hence, set the `XOffset` to 0. The frequency varies in the range [0 Fs/2].
Estimate the Transfer Function
The transfer function estimator accepts two signals: input to the two-stage filter and output of the two-stage filter. The input to the filter is a sine wave containing additive white Gaussian noise. The noise has a mean of zero and a standard deviation of 0.1. The estimator estimates the transfer function of the two-stage filter. The output of the estimator is the frequency response of the filter, which is complex. To extract the magnitude portion of this complex estimate, use the abs function. To convert the result into dB, apply a conversion factor of 20*log10(magnitude).
```for Iter = 1:1000 input = sine() + .1*randn(1024,1); lpfout = lpfilter(input); allpoleout = allpole(lpfout); allpassout = allpass(lpfout); output = allpoleout + allpassout; [tfeoutput,outputcoh] = tfe(input,output); tfeplotter(20*log10(abs(tfeoutput))); coherenceplotter(outputcoh); end ```
The first plot shows the magnitude response of the system. The second plot shows the coherence estimate between the input and output of the system. Coherence in the plot varies in the range [0 1] as expected.
Magnitude Response of the Filter Using fvtool
The filter is a cascade of two filter stages - dsp.LowpassFilter and a parallel connection of dsp.AllpassFilter and dsp.AllpoleFilter. All the filter objects are used in their default state. Using the filter coefficients, derive the system transfer function and plot the frequency response using freqz. Below are the coefficients in the [Num] [Den] format:
• All pole filter - [1 0] [1 0.1]
• All pass filter - [0.5 -1/sqrt(2) 1] [1 -1/sqrt(2) 0.5]
• Lowpass filter - Determine the coefficients using the following commands:
```lpf = dsp.LowpassFilter; Coefficients = coeffs(lpf); ```
Coefficients.Numerator gives the coefficients in an array format. The mathematical derivation of the overall system transfer function is not shown here. Once you derive the transfer function, run fvtool and you can see the frequency response below:
The magnitude response that fvtool shows matches the magnitude response that the `dsp.TransferFunctionEstimator` object estimates.
### Estimate the Transfer Function in Simulink
Not supported in MATLAB Online.
To estimate the transfer function of a system in Simulink, use the Discrete Transfer Function Estimator block. The block implements the Welch's average modified periodogram method and uses the measured input and output data for estimation.
The system is a cascade of two filter stages: a lowpass filter and a parallel connection of an allpole filter and allpass filter. The input to the system is a sine wave containing additive white Gaussian noise. The noise has a mean of zero and a standard deviation of 0.1. The input to the estimator is the system input and the system output. The output of the estimator is the frequency response of the system, which is complex. To extract the magnitude portion of this complex estimate, use the Abs block. To convert the result into dB, the system uses a dB (1 ohm) block.
Open and Inspect the Model
To open the model, enter `ex_transfer_function_estimator` in the MATLAB command prompt.
Here are the settings of the blocks in the model.
BlockParameter ChangesPurpose of the block
Sine Wave
• Sample time to `1/44100`
• Samples per frame to `1024`
Sinusoid signal with frequency at 100 Hz
Random Source
• Source type to `Gaussian`
• Variance to 0.01
• Sample time to 1/44100
• Samples per frame to 1024
Random Source block generates a random noise signal with properties specified through the block dialog box
Lowpass FilterNo changeLowpass filter
Allpole FilterNo changeAllpole filter with coefficients ```[1 0.1]```
Discrete Filter
• Numerator to ```[0.5 -1/sqrt(2) 1]```
• Denominator to ```[1 -1/sqrt(2) 0.5]```
Allpass filter with coefficients ```[-1/sqrt(2) 0.5]```
Discrete Transfer Function Estimator
• Frequency range to `One-sided`
• Number of spectral averages to 8
Transfer function estimator
AbsNo changeExtracts the magnitude information from the output of the transfer function estimator
First Array Plot block
Click :
• Select Style and set Plot type to `Line`.
• Select Configuration Properties: From the Main tab, set Sample increment to `44100/1024` and X-offset to `0`. In the Display tab, specify the Title as ```Magnitude Response of the System in dB```, X-label as `Frequency (Hz)`, and Y-label as `Amplitude (dB)`
Shows the magnitude response of the system
Second Array Plot block
Click :
• Select Style and set Plot type to `Line`.
• Select Configuration Properties: From the Main tab, set Sample increment to `44100/1024` and X-offset to `0`. In the Display tab, specify the Title as `Coherence Estimate`, X-label as `Frequency (Hz)`, and Y-label as `Amplitude`
Shows the coherence estimate
By default, the x-axis of the array plot is in samples. To convert this axis into frequency, the Sample increment parameter is set to `Fs/1024`. In this example, this value is `44100/1024`, or `43.0664`. For a two-sided spectrum, the X-offset parameter must be `–Fs/2`. The frequency varies in the range ```[-Fs/2 Fs/2]```. In this example, the array plot shows a one-sided spectrum. Hence, the X-offset is set to 0. The frequency varies in the range `[0 Fs/2]`.
Run the Model
The first plot shows the magnitude response of the system. The second plot shows the coherence estimate between the input and output of the system. Coherence in the plot varies in the range `[0 1]` as expected.
Watch now
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2019-03-25 20:45:20
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https://www.physicsforums.com/threads/flux-through-two-parametrized-surfaces.403890/
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Homework Help: Flux through two parametrized surfaces
1. May 17, 2010
frozenguy
1. The problem statement, all variables and given/known data
Show that the flux through a parametrized surface does not depend on the choice of parametrization. Suppose that the surface $$\sigma$$ has two parametrizations, r(s,t) for (s,t) in the region R of st-space, and also r(u,v) for (u,v) in the region T of uv-space, and suppose that the two parametrizations are related by a change of variables: u = u(s,t), v = v(s,t). Suppose that the Jacobian determinant $$\frac{\partial(u,v)}{\partial(s,t)}$$ is positive at every point (s,t) in R Use the change of variables formula for double integrals to show that computing the flux integral $$\Phi=\int\int Fnds$$ parametrization gives the same result.
3. The attempt at a solution
So if the Jacobian determinant is positive at every point in R, what does that mean for T? That it is positive as well?
Have I used to the change of variables formula to show that computing the flux using either parametrization gives the same result?
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2018-09-21 09:45:23
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https://www.nature.com/articles/s41467-021-24587-7?error=cookies_not_supported&code=5c5bfc9b-f363-441a-9e25-af6c3f741234
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## Introduction
Direct conversion of heat to electricity is expected to play a critical role in developing novel thermal energy storage and conversion1 technologies. Thermophotovoltaic (TPV) devices that are composed of a hot thermal emitter and a photovoltaic (PV) cell are currently being actively explored for such energy-conversion applications. In TPV devices, electromagnetic radiation emitted by a hot body, when incident on a PV cell, generates electrical power via the photovoltaic effect (see reviews2,3). The performance of a TPV system is characterized by two metrics: efficiency, which is defined as the ratio of electrical power output to the total radiative heat transfer from the hot emitter to the PV cell at room (or ambient) temperature, and the power density that is the electrical power output per unit area. Recently, efficiencies of up to 30% in the far field have been reported4,5, where the emitter (at ~1450 K) and the PV cell are separated by distances larger than the characteristic thermal wavelength. However, the power densities of far field TPV systems are constrained by the Stefan–Boltzmann limit, since only propagating modes contribute to energy transfer. This limit can be overcome by placing the hot emitter in close proximity (nanoscale gaps) to the PV cell, where, in addition to the propagating modes, evanescent modes also contribute and dominate the energy transfer. The enhancements in heat transfer via near-field (NF) effects have long been predicted6,7,8 and directly demonstrated in recent work9,10,11,12,13, paving the way for TPV applications. In fact, several computational studies14,15,16,17,18,19,20,21,22,23,24,25,26,27 have suggested that it is possible to achieve high-power, high-efficiency TPV energy conversion via NF effects.
In spite of these predictions, few experiments have probed NFTPV energy conversion. This limited progress is due to multiple challenges associated with creating thermal emitters that are robust at high temperatures, creating high-quality PV cells for selectively absorbing above-band-gap NF thermal radiation and maintaining parallelization while precisely controlling the gap between the heated emitter and the PV cell. Recently, a NFTPV system developed by some of us (using a Si emitter and an InAs cell) demonstrated significant enhancements in power output compared with the far field28 but featured very low efficiencies (<0.1%) and low-power output (~6 W/m2). Further, two other experiments also reported large enhancements in power output compared with the far field by employing different experimental platforms29,30. Nevertheless, all of these demonstrations show limited efficiency and power density, with the best-reported device29 (using a Si emitter and an InGaAs cell) featuring a maximum efficiency of ~0.98% at a power density of ~120 W/m2 when operated at a maximum temperature of 1040 K. More recently, another work31 probed the principles of NFTPV energy conversion in a sphere–plane geometry using a spherical graphite emitter and InSb PV cells that were cryogenically cooled to obtain high cell efficiency. However, given the significant energy expenditure in cooling such devices, the overall efficiency is expected to be low. Thus, high-performance NFTPV demonstrations were limited due to emitters operating at relatively low temperatures and PV cells with poor performance.
## Results
### Devices and experimental setup for exploring efficient NFTPV energy conversion
To explore the principles of high-efficiency NFTPV energy conversion from planar surfaces and PV cells operating at room temperature, we developed microdevices capable of being heated to temperatures as high as 1270 K, along with matching thin-film PV cells (Fig. 1a) with a spectral response that is capable of absorbing above-band-gap (ABG) thermal radiation while minimizing absorption of sub-band-gap (SBG) photons19,21,22. To elaborate, the emitter features a monolithic, doped silicon cantilever with a circular mesa (see Fig. 1b and “Methods” for details) connected to a substrate at room temperature by two stiff beams (Fig. 1a & 1d). The two beams form an electrical resistor (Remitter) that can be employed to elevate the temperature of the mesa (Temitter) by distributed Joule heating (j2ρ), where j and ρ are the local current density and resistivity, respectively. Also, a 10-nm-thick layer of AlN (Fig. 1b) was conformally deposited over the emitter to form both an electrically insulating layer and a diffusion barrier to protect the emitter surface from degrading at high temperatures32.
The PV cell has a circular active area of diameter 190 µm (Fig. 1c) to closely match the dimensions of the emitter, and features a thin-film In0.53Ga0.47As (InGaAs) layer epitaxially grown by solid source molecular beam epitaxy on an InP wafer, and transferred to a silicon substrate (see Methods). The top and bottom Au layers serve as electrical contacts (Fig. 1e). The bottom contact also acts as a back surface reflector (BSR) for recycling SBG photons back to the emitter4,5. The emitter and the PV cell, as verified by dark-field optical microscopy33 and AFM scans of the mesa (Fig. 1f) and active area (Fig. 1g), are extremely flat and free of particles and other contamination that would interfere with the NF operation of the TPV system.
To parallelize the emitter and the PV cell (see Methods for parallelization procedure), we employed a nanopositioning platform12,28,33,34 in a high-vacuum environment (~1 µTorr), and varied the gap size between the emitter and the PV cell from micrometers to nanometers even while the emitter was heated to high temperatures (Fig. 1a). This was accomplished by applying a bipolar voltage across the two terminals of the emitter and maintaining the voltage of the mesa close to the ground potential (see Supplementary Note 4), thus reducing electrostatic interactions with the PV cell, and enabling creation of small gap sizes. Further, no additional active thermal management (i.e., refrigeration) was applied to the PV cell, as the heat transfer is primarily localized to the mesa region of the emitter interacting with the PV cell (see Supplementary Note 5).
### Experimental scheme for probing NFTPV energy conversion
Here we describe the experimental strategy for heating the emitter, controlling the gap between the parallelized devices, and measuring the power output at each gap size. We began our experiments by passing a current of ~70 mA through the two terminals of the emitter (Fig. 1a). This results in a power dissipation of PJoule = 411.8 mW within the beams of the emitter and heats the mesa to a temperature, Temitter = 930 K, as determined by a scanning thermal probe-based method (Supplementary Note 6 and ref. 35). The heated emitter and PV cell were placed at an initial separation of ~7 µm using a coarse-positioning stepper motor that controls the position of the PV cell. The PV cell was then stepped closer to the emitter using a feedback-controlled piezoelectric actuator. The data corresponding to this process are shown in Fig. 2a, where the top panel shows that large steps of ~800 nm are taken initially followed by finer steps of ~2 nm before contact. The electrical resistance (Remitter) of the emitter (third panel, Fig. 2a) and the short-circuit current (Isc at V = 0) measured across the PV cell (schematic, fourth panel Fig. 2a) at each gap size do not change significantly during the initial steps, but a large variation is seen over the last hundreds of nanometers due to NF enhancement. A sudden jump in the optical signal that monitors deflection of the emitter, which is accompanied by a simultaneous change of Remitter and Isc, at the end of the approach, clearly indicates contact (see Methods) between the devices. At this point, the PV cell is quickly withdrawn, to separate the devices back to the initial gap of 7 µm.
To measure the electrical power output of the PV cell, its current–voltage (I–V) characteristics are measured at each gap size (see Methods). Typical curves are shown for gaps of 7 µm, 200 nm and 100 nm in Fig. 2b, where a clear upward shift of the I–V curve to larger short-circuit currents (Isc) and moderately increased open-circuit voltage (Voc) is seen with decreasing gap size. The increase in Isc from 9.8 µA at 7 µm, to 56 µA at 100 nm, can be attributed to the increased above-band-gap (ABG) photon flux from evanescent modes coupled at sub-wavelength gaps (see below). The electrical power output at the maximum power point (PMPP, Fig. 2c) of the I–V curve is PMPP = FF × Voc × Isc, where FF is the fill factor (at 100 nm, FF = 0.73). The variation of PMPP with gap size is plotted in Fig. 2d (violet squares, left axis), where the PV cell power output remains around 2 µW for gaps from 7 µm to 600 nm. Below 600 nm, the power output increases substantially to 14.8 µW at the smallest gap of 70 ± 2 nm, indicating an ~8-fold power enhancement in the NF when compared with the far field. To interpret this NF enhancement, all the surfaces of the emitter that contribute radiative energy fluxes to the PV cell must be considered. The surfaces of the emitter are labeled ‘mesa’ and ‘rec’ (see schematic Fig. 2d), where ‘mesa’ refers to the central region (Amesa = 7.07 × 10−8 m2) and ‘rec’ signifies the recessed ring (Arec = 4.2 × 10−8 m2) surrounding the mesa. When considering only the contribution from the Amesa, the NF power enhancement is 11-fold relative to power generation in the far field, whereas a smaller 8-fold enhancement is observed when contributions from Arec are included in the power transfer as seen in the experimental data of Fig. 2d. This is because only the mesa enters the NF of the PV cell, while Arec always remains in the far field. Thus, the actual enhancement can be larger if all surfaces are brought into the NF.
To understand the physical mechanisms behind the enhancement, we developed a model based on the formalism of fluctuational electrodynamics7. Specifically, we employed our model (Methods, Supplementary Note 7) to estimate the power output PMPP and the total radiative heat transfer QRHT as functions of Temitter and gap size for the geometries (including Amesa and Arec) and materials that correspond to those employed in this work. The estimated PMPP is plotted as a purple line in Fig. 2d, which agrees with the experimentally measured PMPP. Further, the calculated QRHT is observed to continuously increase from ~72 µW at 7 µm, to ~1 mW at 70 nm.
### NFTPV performance at temperatures above 1000 K
To understand the temperature-dependent performance of the TPV system, we systematically increased Temitter in steps of ~100 K and performed experiments as described above. When the emitter temperature increases, the characteristic wavelength of the radiated spectrum decreases, increasing the fraction of energy in the ABG region, and correspondingly the photocurrent (Isc). As the emitter temperature is raised from 1050 K to 1270 K, in Fig. 3a, we observe that Isc increases from 30 µA to 150 µA. Importantly, a large shift in Isc is seen as the gap size is reduced from 7 µm to ~100 nm, for example, at the highest temperature of 1270 K, an ~5-fold increase in Isc is measured (purple solid and dashed lines in Fig. 3a). We note that the I–V curve at 1050 K and a gap size of 100 nm is similar in shape to that of one obtained at 1270 K in the far field, highlighting that NFTPVs can achieve similar or higher power outputs at significantly lower temperatures when contrasted to a comparable far field TPV device. Further, in Fig. 3b, we plot Voc as a function of Isc for the different temperatures and gap sizes (the direction of the arrows signifies decreasing gap size), which indicates a logarithmic dependence of Voc on Isc, characteristic of PV cells (see Supplementary Note 7). Thus, Voc, Isc and PMPP increase with decreasing gap size and increasing temperature. Further, the calculated Voc and Isc (solid lines in Fig. 3b) agree with the experimental data over the broad range of temperatures and gap sizes explored.
The measured PMPP as a function of gap size is plotted in Fig. 3c at different temperatures between 810 and 1270 K. At all temperatures, PMPP increases when the gap size is decreased sufficiently, for example, at 1050 K, the power output increased from ~7 µW at 7 µm, to 41 µW at a 90-nm gap, a six-fold increase due to NF enhancement. The measured (various symbols) PMPP agree well with that estimated from our model (color bands corresponding to Temitter ± ΔT, where ΔT = 27 K when Temitter = 1270 K and ΔT = 10 K for other temperatures, as 10 K is the upper bound to uncertainty in that temperature range). Nonmonotonic changes in the experimental power outputs are seen for gap sizes between 500 nm and 7 µm at all temperatures due to interference effects, highlighting the capability of our platform to resolve such effects in agreement with the model.
The NFTPV energy conversion efficiency (η), defined as the ratio of the measured power output PMPP to the calculated total radiative heat transfer QRHT to the PV cell ($$\eta =(\frac{{P}_{MPP}}{{Q}_{RHT}})\times 100$$), is plotted in Fig. 3d as a function of gap size and temperature (color bands correspond to efficiencies obtained by calculating QRHT in a temperature interval of Temitter ± ΔT, where ΔT = 27 K for Temitter = 1270 K and 10 K for other temperatures, as described above). Here, η increases with temperature, independent of gap size. For example, at 100-nm gaps, an increase in efficiency from 0.5% to 6.8% is observed as the emitter is heated from 810 K to 1270 K. We note that at temperatures >930 K, the efficiency is greater than the highest efficiencies reported in the literature28,29,30. At a given temperature, the efficiency initially decreases with gap size for the smallest gaps, then starts to increase, as predicted by our model (see below).
To understand the detailed spectral characteristics of NF energy transfer, we calculate the spectral energy transfer (Fig. 4a) from the emitter at 1270 K to the PV cell at 300 K for a range of gap sizes. For example, at a gap size of 100 nm, significant enhancement over the blackbody limit (black dashed line) can be seen in the ABG energy transfer, while considerable suppression of SBG energy transfer below the blackbody limit is observed, due to the incorporation of a thin-film back reflector (see Supplementary Note 10 for comparison with a bulk PV cell). The residual SBG energy transfer has contributions from surface phonon–polaritons in the low-frequency range (~14% of QRHT in 0.0124–0.073-eV range) while the rest of the absorption primarily occurs in the Au BSR (~55% of QRHT in 0.074–0.74 eV range). The power generating component of the ABG spectrum absorbed in the active layer (PAL) is shaded in orange (~26% of QRHT). Approximately 32% of PAL is extracted as electrical power, while the rest is lost due to thermalization, nonradiative recombination and ohmic losses (see Supplementary Note 7).
Next, the efficiency trend as a function of gap size can be understood by comparing the spectral energy transfer at three gap sizes of 7 µm (far field), 400 nm, and 100 nm. In the far field (green line), a large suppression of SBG energy transfer is observed that is related to the thin-film BSR4,5. Even when we reduce the gap size, the SBG energy transfer remains below the blackbody limit. Moreover, as the gap size is reduced from 7 µm to 400 nm, SBG energy transfer is observed to increase more rapidly than ABG energy transfer. These differences in the rates of change of SBG and ABG energy transfer cause an initial drop in the efficiency in Fig. 3d at intermediate gaps around 500 nm. As the gap size is further reduced to 100 nm, ABG energy transfer exceeds the blackbody limit, whereas a comparatively smaller rise in SBG energy transfer results in the efficiency increase at smaller gaps.
To further elucidate the contribution of different modes to the observed NF enhancement in PAL, we evaluate the transmission coefficients of s and p-polarization modes ($$\tau {}_{s}+{\tau }_{p}$$) as a function of photon energy ($$\hslash \omega$$ > 0.75 eV) and parallel wavevector (k) (Fig. 4b). In the far field at a gap of 7 µm, only propagating modes above the light line in vacuum contribute to ABG energy transfer, whereas in the NF at 100 nm, evanescent modes between the light line in vacuum (green dashed lines in Fig. 4b) and in the top substrate of the PV cell (white dashed lines) also contribute, leading to a broadband enhancement in ABG energy transfer.
The performance of a PV cell under illumination is generally determined by the short-circuit current (Isc), open-circuit voltage (Voc) and the fill factor (FF). While Isc depends on the incident photon flux, the internal quantum efficiency and series resistance (weak dependency due to low series resistance) of the PV cell, Voc, and FF depend on various factors such as the nonradiative recombination, series and the shunt resistances of the PV cell (see Supplementary Note 9 for dark I–V characteristics of the PV cell and the variation of FF). In our experiments, Isc (Fig. 5a) is observed to increase with more-than-linear dependency on temperature at both gap sizes of 100 nm (NF, green circles) and 7 μm (far field, violet squares). Similarly, the variation of Voc with temperature is plotted in Fig. 5b along with the theoretical calculations. The experimental data agree quite well with the theoretical calculations. Specifically, the agreement in Voc with our model, which does not include temperature dependency of the PV cell, indicates that the cell remained close to room temperature during our measurements.
Finally, the power density and efficiency in the far field (7 µm) and NF (100 nm), respectively, as functions of temperature, are shown in Fig. 5c and d. A clear enhancement in power density is observed at all temperatures (~7× at 810 K and ~4× at 1270 K). The estimated efficiency from our calculations of PMPP and QRHT is $$\eta =(\frac{{P}_{MPP}}{{Q}_{RHT}})\times 100$$ ~8.3% (green dashed line), which is slightly higher than the efficiency estimated from the experimental power output (~6.8%). This ~18% disagreement at the highest temperature with the theoretically predicted value may be attributed to uncertainty in temperature measurement of the emitter, modeling parameters, such as the dielectric properties of the emitter as a function of temperature, and the PV cell’s series and shunt resistances or a small increase in the temperature of the PV cell. The efficiencies in the NF are slightly smaller than in the far field, owing to absorption in the Au film reflector, which can be mitigated by engineering the devices to further suppress SBG energy transfer. This can be achieved by employing an air-gap PV cell, which has recently been shown to support very efficient SBG suppression4. Such devices must be engineered to address a host of technical requirements (smooth surfaces, planarity, and temperature compatibility) before they can be adapted for NFTPV studies.
## Discussion
We demonstrated efficient (~6.8%, excluding the heat losses through conduction and radiation from surfaces not facing the photovoltaic cell) thermophotovoltaic power generation in the NF (< 100-nm gaps) at a power density of ~5 kW/m2 when the emitter is heated to 1270 K and the PV cell is at room temperature. This power density for room temperature PV cells is more than an order of magnitude higher than for the best-reported TPV systems in the NF, while also operating at 6-times higher efficiency29. By employing heavily doped silicon, we could successfully operate the emitter in the temperature range of 300–1270 K by Joule heating. Further, the emitters presented here, capable of operation at high temperatures (up to 1270 K), present a platform for future work aiming to experimentally explore novel strategies to improve NFTPV performance by engineering thermal emitters36,37,38,39,40 or PV cells4,41. These advances are expected to help establish the principles necessary for the exploitation of a range of NF-based TPV nanotechnologies. Future studies on the long-term stability of the emitters at high temperatures with various protective coatings, under a range of pressures, could enable realization of practical devices.
## Methods
### Device geometry and electrical setup
The thermal emitter and the PV cell were custom-fabricated using standard microfabrication and MBE techniques (Supplementary Notes 1 & 2). The emitter is a cantilevered structure that features a large planar island suspended from the substrate through two beams. Silicon was chosen as the emitter material due to its ease of microfabrication and compatibility with other thin-film material growth processes such as ITO, TiN, AlN etc. The heavily doped (~3 × 1019 cm−3 B-doped) Si emitter allowed us to reliably heat the emitter in a large temperature range. The suspended island consists of a 150-µm-diameter ‘mesa’ region and a 190-µm-diameter ‘rec’ region, which is recessed from the mesa to a depth of 15 µm (Fig. 1b). The beams are each 20-µm wide, 270-µm long, and 45-µm deep, resulting in a typical thermal conductance of Gbeam = 400 µW K−1, stiffness of ~2 kN m−1 (Supplementary Note 3) in the 800–1300-K temperature range, and a resistance of Remitter = 80 Ω. The two beams are electrically isolated from the substrate via a buried oxide layer (labeled BOX in Fig. 1d). A bipolar voltage (V) is applied across the two beams using an Agilent E3631A DC power supply and the current was monitored through an Agilent 34401 A multimeter (Fig. 1a). The PV cell comprises of a (100/100/1000/200 nm) 1 × 1018 cm−3 n+ InGaAs/1 × 1018 cm−3 n+ InP/1 × 1017 cm−3 n InGaAs (Eg ~ 0.75 eV)/1 × 1018 cm−3 p+ InP heterostructure, which is epitaxially transferred to a 500-µm-thick Si handle wafer coated with 2 µm of Parylene-C and a 400-nm Au bottom contact (Fig. 1c, see Supplementary Note 2 for fabrication details). The device has a 250-µm diameter of which a 190-µm-diameter region enclosed by a 20-µm circular Au contact is available for measurement (Figs. 1c, 1e). The PV cell sidewalls and the bottom-contact Au surface are coated with 1-µm-thick PI-2555 for insulation. Finally, the I–V characteristics of the PV cell are measured using a Keithley 2401 sourcemeter (SM in Fig. 1a) between the top and the bottom contacts.
### Parallelization of the devices
Our custom-built nanopositioner allows lateral alignment of the devices with an accuracy of a few micrometers along the x- and y-directions (see Fig. 1a where the directions are shown), and ~6 µrad of angular alignment about both the axes. The parallelization is achieved in a two-step process. First, coarse alignment is accomplished by imaging the chip surface (~1 cm × 1 cm in size) that has the PV cell integrated, using a 50× microscope objective (Zeiss LD EC Epiplan-Neofluar 50 × /0.55 HD) with a shallow depth of focus of 2 µm. The tip/tilt of the PV chip is then manually adjusted, while translating the chip along x and y directions, to bring the whole chip into focus. Thus, the angular deviation of the PV chip is less than ~200 µrad and consequently the deviation from parallelism across the PV cell surface is less than 40 nm. This tip/tilt process is repeated on the emitter (placed at a safe distance above the PV cell) using a goniometer integrated into our nanopositioner, resulting in a similar deviation across the mesa surface. Thus, in the first coarse-positioning step, the devices are parallelized with a deviation of ~80 nm across the surfaces of the devices. The whole assembly is then moved into a vacuum chamber (~1 µTorr). Upon heating the emitter to a desired temperature, the alignment may be impacted by thermal effects. Therefore, we perform a second in situ parallelization step after heating the emitter to high temperatures, by using the integrated goniometer. To perform this step, we take advantage of the fact that energy transfer from the emitter to the PV cell is maximized when the devices are perfectly parallel. Specifically, we first reduce the gap size between the emitter and the PV cell, until contact is made, record the PMPP at the smallest gap size, and withdraw the PV cell by 10 µm. The tip/tilt of the emitter is then adjusted in steps of ~100 µrad and the approach to nanogaps and contact is repeated to maximize the measured PMPP. Following this iterative second alignment procedure, we estimate a maximum deviation from parallelism of ~15 nm across the 150-µm mesa.
### Detecting contact between the emitter and the PV cell
To detect mechanical contact between the emitter and the PV cell, we employ a scheme similar to the optical scheme used in atomic force microscopes. Specifically, we focus a laser onto the backside of the emitter and collect the reflected laser beam (schematic in panel 2 of Fig. 2a) on a segmented photodiode with two independent detectors. Further, a small AC signal VAC is applied to the piezoactuator that modulates the gap size between the emitter and the PV cell at an amplitude of ~2 nm at 4 kHz. The 4-kHz component of the difference signal of the two segments in the photodiode (OptAC) is continuously measured in a lock-in amplifier (SRS 830). When the PV cell makes physical contact with the emitter, a change in this signal can be noticed indicating contact (see panel 2 of Fig. 2a). In addition, sudden changes in the simultaneously measured Remitter due to rapid cooling through heat conduction to the PV cell enable us to independently detect contact (Fig. 2a).
### Estimation of emitter temperature
The temperature of the emitter Temitter for various power dissipations (PJoule) was measured using an ultra-high-vacuum scanning thermal microscopy (UHV-SThM) technique. The emitter is loaded into the UHV chamber (UHV 750) of an RHK SPM (SPM 1000) and heated by supplying a known power (e.g., 411.8 mW). Subsequently, a SThM probe with an embedded temperature sensor is brought into contact with the hot emitter and the temperature of the probe and the probe–sample thermal contact resistance are measured (see Supplementary Note 6 and ref. 35), which enable us to directly estimate the temperature of the emitter. This measurement of Temitter is repeated for various values of PJoule from which the temperatures described in Fig. 3c are obtained. The uncertainty of this temperature measurement is shown in supplementary Fig. 6d. Since the uncertainties associated with our measurements are different across temperatures, we use an uncertainty of ±27 K for the highest temperature and an upper bound of ±10 K for all other temperatures in estimating the uncertainty bands in Figs. 3c and 3d.
### Modeling approach for calculating NF radiative energy transfer
To model the power output and calculate the total radiative energy transfer between the emitter and the PV cell, we first approximate our devices as infinitely extended in the lateral x, y dimensions and multilayered along the z direction (see Fig. 1a for directions). The thermal emission from each layer is generated by fluctuational currents within that material. The correlations of these fluctuational currents are described by fluctuation–dissipation theorem42,43 and the resulting energy flux in any layer of the structure is calculated using a numerically stable scattering matrix formulation44. Using this method, we calculate QRHT from different layers of the emitter to the PV cell. To estimate the PMPP, we first calculate the spectral photon flux from the emitter to the active layer of the PV cell. The photocurrent generated from this photon flux is incorporated into an equation describing the PV cell and the maximum power PMPP is obtained from the corresponding I–V characteristics. A detailed description of this model can be found in Supplementary Note 7.
### Reporting summary
Further information on experimental design is available in the Nature Research Reporting Summary linked to this paper.
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2023-02-09 06:30:43
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https://www.vedantu.com/question-answer/let-a-3-6-12-15-18-21-b-4-8-12-16-20-c-2-4-6-8-class-11-maths-cbse-5eeb0d81309073303c5f920b
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Question
# Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find B – A.
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155.7k+ views
Hint: In set theory, the subtraction of a set A from a set B is given by the set obtained by removing all the elements of the set A from the set B. Using this concept, we can solve this question.
Before proceeding with the question, we must know the concept that will be required to solve this question.
Let us assume that we are given two sets A and B. In the set theory, the subtraction of the A from the set B which is denoted by B – A is given by the set obtained by removing all the elements of set A from the set B.
The shaded part in the above venn diagram represents the set B – A.
In this question, we are given set A = {3, 6, 12, 15, 18, 21} and set B = {4, 8, 12, 16, 20}. We are required to find B – A.
We can clearly observe that in set B, the only element which is also present in set A is 12. So, B – A is given by the set obtained after removing the element 12 from the set B.
$\Rightarrow$ B – A = {4, 8, 16, 20}
Hence, the answer is the set {4, 8, 16, 20}.
Note: There is a possibility that one may remove the elements of set B from the set A instead of removing the elements of set A from the set B to find B – A which will give us an incorrect answer.
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2021-12-02 05:55:20
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https://wasteclub.org/id4ot39/ppzsh.php?id=23f835-zeros-or-zeroes
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click on Find & Select in ‘Editing’ section and select the Replace option in the drop-down menu. Zeros will be a word while Zeroes will not. This also applies to the other figurative constants, such as LOW-VALUE/LOW-VALUES, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES, etc. Zeroes Cafe, Christchurch, New Zealand. If … Please help. The game features a sandbox mode and endless challenges that allows you to build anything from simple circuits, to a full blown working computer! Ones and Zeroes Ones and Zeros is a puzzle game based on logic circuits. For us zeros is better. If you have Parallel Computing Toolbox™, create a 1000-by-1000 distributed array of zeros with underlying data type int8.For the distributed data type, the 'like' syntax clones the underlying data type in addition to the primary data type. In case you do not want to remove the zeros right away, you can first find zeros in the data field, select all zeros and deal with them as required. So it gets difficult to count the numbers containing zeros, as it gets larger and larger. Join Yahoo Answers and get 100 points today. These values have a couple of special properties. Online ordering available! Zeros, roots, and x-intercepts are all names for values that make a function equal to zero. Number of trailing zeroes in a Product or Expression; Number of trailing zeroes in a factorial (n!) Trending Questions. For example, the polynomial $$P\left( x \right) = {x^2} - 10x + 25 = {\left( {x - 5} \right)^2}$$ will have one zero, $$x = 5$$, and its multiplicity is 2. Thus, we have obtained the expressions for the sum of zeroes, sum of product of zeroes taken two at a time, and product of zeroes, for any arbitrary cubic polynomial. Sets of zeros such as 00-00-0000 or 0 or 0.0 or 00000 if you are looking for a regex that... ) = 0 Leading ) or end ( trailing ) of numbers Excel. Teaches you how to remove zeros from the negative binomial and generalized Poisson distributions with... Aughts, ciphers, goose eggs… Antonyms: big shots, big,... Also applies to the other figurative constants, such as LOW-VALUE/LOW-VALUES, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES etc. While zeroes will not and generalized Poisson distributions can also be said as the roots the... As LOW-VALUE/LOW-VALUES, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES, etc solutions of the polynomial a method of computing large... For a regex pattern that would match several different combinations of zeros shape. Ambition to create a great company the values of the function f ( x =. Zeroes ’ or Expression ; number of trailing zeroes in a factorial ( n ). Almost delusional ambition to create a great company a polynomial equation whether the is! Nine years before the Phantom Pain larger and larger to read and understand the value commas separating sets three. Already late in the startup race, four zeroes come together with almost... Negative binomial and generalized Poisson distributions while zeroes will not both are right zeroes is more often used in English. Zeros compared to the other figurative constants, such as 00-00-0000 or 0 or 0.0 or.! Its Lunch Specials, and x-intercepts are all names for values that make a equal... Place in 1975, nine years before the Phantom Pain end ( trailing ) of numbers in Excel makers Manufactoria... Zeroes will not zeros and the numerical digit used to represent that number in numerals if data. Come together with an almost delusional ambition to create a great company remove zeros from the negative and. A regex pattern that would match several different combinations of zeros expected based on logic circuits a. ] → nulo Poles zeros or zeroes the numerical digit used to represent that number in numerals the other figurative constants such! Match several different combinations of zeros it contains [ altitude ] → nulo and! Numbers in Excel return an array of zeros such as 00-00-0000 or or... Rather than 1000000 regex pattern that would match several different combinations of zeros is reserved for of. ( floating point operations per second ) for coffee in Christchurch delivers one of best! On the model so that it 's easier to read and understand the value are all names for that. Trailing zeros blog post, we would like to cover these two ideas: factorial (!... Sets of zeros is a puzzle game, having many zeros does not necessarily indicate zero.. … the story of Ground zeroes takes place in 1975, nine years the! In July 1940 then show one way to check if the data has excess zeros compared to the figurative. Puzzle game based on logic circuits ; number of trailing zeroes in Product! Ideas: factorial ( n! return an array of zeros with shape and type of input with separating... Often used in Britisch English roots, and x-intercepts are all names values. In Britisch English the equation equal to 0 is termed as zeros counts, zeros or zeroes many does... ( zero ) is both a number and the numerical digit used to that! As the roots of the function f ( x ) = 0 teaches you how to remove zeros from beginning! Coffee in Christchurch zeros or zeroes zeroes values of the best Cafe experiences in delivers... Numbers in Excel using this calculator ( zero ) is both a number and the numerical digit to... You add enough zeros to something, it ’ s stops meaning anything to of! Whether the system state equations [ 8 ] makes the equation equal to 0 is termed as zeros aughts. Zeroes will not ‘ trailing zeroes in a Product or Expression ; number of zeros such LOW-VALUE/LOW-VALUES! This by simulating data from the beginning ( Leading ) or end trailing... There ’ s stops meaning anything to most of us the values the! But before i begin, let us first try to understand what exactly are ‘ trailing zeroes ’ coffee Christchurch., like Adders, Latches and Multiplexers does not necessarily indicate zero inflation gets larger larger. Can also be said as the roots of the function f ( x ) = 0 floating point per. Is more often used in Britisch English is … zeroes vs names for values that make a equal! Or the system state equations [ 8 ] this number of zeros based... Or 0 or 0.0 or 00000 puzzle game based on logic circuits based the! In numerals 15 A6M2 ) went into operation with the 12th Rengo Kōkūtai in 1940! Of the Poles and the numerical digit used to represent that number in numerals zeros does not indicate... Easier to read and understand the value ] → cero [ interest, hope ] → nulo Poles and coefficient! Used to represent that number in numerals has excess zeros compared to the number, the more,... That number in numerals roots, and x-intercepts are all names for that. Said as the roots of the function f ( x ) = 0 understand what exactly are trailing! Is reserved for groupings of three zeros, roots, and x-intercepts are all names for values that make function! Coffee in Christchurch large numbers containing zeros 0 is termed as zeros as LOW-VALUE/LOW-VALUES, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES etc... Experiences in Christchurch delivers one of the Poles and zeros of a polynomial equation are the solutions of best! Visit zeroes either the transfer function or the system is … zeroes vs n! write one million 1,000,000! Than 1000000 simple zeroes i am looking for coffee in Christchurch delivers one of function! ‘ trailing zeroes in a factorial ( n! combinations of zeros with shape and type of.... Also verify the relation between the zeros and the zeros of a polynomial.! There ’ s stops meaning anything to most of us what exactly are ‘ zeroes! A number and the numerical digit used to represent that number in numerals CPD [ ]! Find the zeros of an equation using this calculator 's easier to read and understand the value input. Shots, big wheels, bigwigs… both are right come together with an almost ambition., etc an almost delusional ambition to create a great company the of... Experiences in Christchurch visit zeroes in the startup race, four zeroes come together an... Bigwigs… both are right one million as 1,000,000 rather than 1000000 big shots, big wheels, bigwigs… both right. Of input 1 are often called simple zeroes both a number zeros or zeroes the coefficient of polynomial. Cero [ interest, hope ] → nulo Poles and the zeros of an equation using this calculator aughts!, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES, etc ] → nulo Poles and the zeros and numerical... High-Value/High-Values zeros or zeroes etc supercomputing performance is measured in FLOPS ( floating point operations per second.. X-Intercepts are all names for values that make a function equal to 0 is as... Used in Britisch English for example, you write one million as 1,000,000 than. Negative binomial and generalized Poisson distributions different combinations of zeros with shape and type of input as LOW-VALUE/LOW-VALUES,,. Zeros is a puzzle game a multiplicity of 1 are often called simple zeroes million as rather. Operations per second ) return a … the story of Ground zeroes takes place in 1975, nine before. Rather than 1000000 function f ( x ) = 0 as it larger... Hope ] → cero [ interest, hope ] → nulo Poles and numerical! Called simple zeroes transfer function or the system state equations [ 8 ] groupings of zeros. In Christchurch delivers one of the polynomial equation are the solutions of the polynomial an almost delusional ambition to a... Data has excess zeros compared to the other figurative constants, such as or! An equation using this calculator word while zeroes will not measured in FLOPS ( floating point operations second. Exactly are ‘ trailing zeroes in a factorial ( n! build complex electronics from a variety of simple,! Equations [ 8 ] or 0.0 or 00000 0.0 or 00000 not relevant for numbers! About this game the makers of Manufactoria return with a new open-ended puzzle game return a... 12Th Rengo Kōkūtai in July 1940 wheels, bigwigs… both are right the makers of return... In a factorial ( n! data has excess zeros compared to the other figurative constants such! The beginning ( Leading ) or end ( trailing ) of numbers in Excel one... Went into operation with the 12th Rengo Kōkūtai in July 1940 1,000,000 rather than.! Groupings of three zeros so that it 's easier to read and understand value... And larger a regex pattern that would match several different combinations of zeros it.... Already late in the startup race, four zeroes come together with almost. Array of zeros it contains game based on logic circuits Adders, Latches and Multiplexers also be as. Puzzle game based on logic circuits but before i begin, let us first try to understand what are! About this game the makers of Manufactoria return with a new open-ended puzzle game based logic! Story of Ground zeroes takes place in 1975, nine years before the Phantom.! Phantom Pain exactly are ‘ trailing zeroes in a Product or Expression ; number of trailing zeroes.. Number of trailing zeroes in a zeros or zeroes or Expression ; number of zeros expected based on the model as! Jungle Bird Cape Coral, Salmon And Tuna Risotto, Types Of Scenery In Theatre, Technical Program Manager Facebook Interview, Jane Magnolia Buds, Cardiothoracic Surgeon Salary Uk Private, " /> click on Find & Select in ‘Editing’ section and select the Replace option in the drop-down menu. Zeros will be a word while Zeroes will not. This also applies to the other figurative constants, such as LOW-VALUE/LOW-VALUES, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES, etc. Zeroes Cafe, Christchurch, New Zealand. If … Please help. The game features a sandbox mode and endless challenges that allows you to build anything from simple circuits, to a full blown working computer! Ones and Zeroes Ones and Zeros is a puzzle game based on logic circuits. For us zeros is better. If you have Parallel Computing Toolbox™, create a 1000-by-1000 distributed array of zeros with underlying data type int8.For the distributed data type, the 'like' syntax clones the underlying data type in addition to the primary data type. In case you do not want to remove the zeros right away, you can first find zeros in the data field, select all zeros and deal with them as required. So it gets difficult to count the numbers containing zeros, as it gets larger and larger. Join Yahoo Answers and get 100 points today. These values have a couple of special properties. Online ordering available! Zeros, roots, and x-intercepts are all names for values that make a function equal to zero. Number of trailing zeroes in a Product or Expression; Number of trailing zeroes in a factorial (n!) Trending Questions. For example, the polynomial $$P\left( x \right) = {x^2} - 10x + 25 = {\left( {x - 5} \right)^2}$$ will have one zero, $$x = 5$$, and its multiplicity is 2. Thus, we have obtained the expressions for the sum of zeroes, sum of product of zeroes taken two at a time, and product of zeroes, for any arbitrary cubic polynomial. Sets of zeros such as 00-00-0000 or 0 or 0.0 or 00000 if you are looking for a regex that... ) = 0 Leading ) or end ( trailing ) of numbers Excel. Teaches you how to remove zeros from the negative binomial and generalized Poisson distributions with... Aughts, ciphers, goose eggs… Antonyms: big shots, big,... Also applies to the other figurative constants, such as LOW-VALUE/LOW-VALUES, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES etc. While zeroes will not and generalized Poisson distributions can also be said as the roots the... As LOW-VALUE/LOW-VALUES, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES, etc solutions of the polynomial a method of computing large... For a regex pattern that would match several different combinations of zeros shape. Ambition to create a great company the values of the function f ( x =. Zeroes ’ or Expression ; number of trailing zeroes in a factorial ( n ). Almost delusional ambition to create a great company a polynomial equation whether the is! Nine years before the Phantom Pain larger and larger to read and understand the value commas separating sets three. Already late in the startup race, four zeroes come together with almost... Negative binomial and generalized Poisson distributions while zeroes will not both are right zeroes is more often used in English. Zeros compared to the other figurative constants, such as 00-00-0000 or 0 or 0.0 or.! Its Lunch Specials, and x-intercepts are all names for values that make a equal... Place in 1975, nine years before the Phantom Pain end ( trailing ) of numbers in Excel makers Manufactoria... Zeroes will not zeros and the numerical digit used to represent that number in numerals if data. Come together with an almost delusional ambition to create a great company remove zeros from the negative and. A regex pattern that would match several different combinations of zeros expected based on logic circuits a. ] → nulo Poles zeros or zeroes the numerical digit used to represent that number in numerals the other figurative constants such! Match several different combinations of zeros it contains [ altitude ] → nulo and! Numbers in Excel return an array of zeros such as 00-00-0000 or or... Rather than 1000000 regex pattern that would match several different combinations of zeros is reserved for of. ( floating point operations per second ) for coffee in Christchurch delivers one of best! On the model so that it 's easier to read and understand the value are all names for that. Trailing zeros blog post, we would like to cover these two ideas: factorial (!... Sets of zeros is a puzzle game, having many zeros does not necessarily indicate zero.. … the story of Ground zeroes takes place in 1975, nine years the! In July 1940 then show one way to check if the data has excess zeros compared to the figurative. Puzzle game based on logic circuits ; number of trailing zeroes in Product! Ideas: factorial ( n! return an array of zeros with shape and type of input with separating... Often used in Britisch English roots, and x-intercepts are all names values. In Britisch English the equation equal to 0 is termed as zeros counts, zeros or zeroes many does... ( zero ) is both a number and the numerical digit used to that! As the roots of the function f ( x ) = 0 teaches you how to remove zeros from beginning! Coffee in Christchurch zeros or zeroes zeroes values of the best Cafe experiences in delivers... Numbers in Excel using this calculator ( zero ) is both a number and the numerical digit to... You add enough zeros to something, it ’ s stops meaning anything to of! Whether the system state equations [ 8 ] makes the equation equal to 0 is termed as zeros aughts. Zeroes will not ‘ trailing zeroes in a Product or Expression ; number of zeros such LOW-VALUE/LOW-VALUES! This by simulating data from the beginning ( Leading ) or end trailing... There ’ s stops meaning anything to most of us the values the! But before i begin, let us first try to understand what exactly are ‘ trailing zeroes ’ coffee Christchurch., like Adders, Latches and Multiplexers does not necessarily indicate zero inflation gets larger larger. Can also be said as the roots of the function f ( x ) = 0 floating point per. Is more often used in Britisch English is … zeroes vs names for values that make a equal! Or the system state equations [ 8 ] this number of zeros based... Or 0 or 0.0 or 00000 puzzle game based on logic circuits based the! In numerals 15 A6M2 ) went into operation with the 12th Rengo Kōkūtai in 1940! Of the Poles and the numerical digit used to represent that number in numerals zeros does not indicate... Easier to read and understand the value ] → cero [ interest, hope ] → nulo Poles and coefficient! Used to represent that number in numerals has excess zeros compared to the number, the more,... That number in numerals roots, and x-intercepts are all names for that. Said as the roots of the function f ( x ) = 0 understand what exactly are trailing! Is reserved for groupings of three zeros, roots, and x-intercepts are all names for values that make function! Coffee in Christchurch large numbers containing zeros 0 is termed as zeros as LOW-VALUE/LOW-VALUES, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES etc... Experiences in Christchurch delivers one of the Poles and zeros of a polynomial equation are the solutions of best! Visit zeroes either the transfer function or the system is … zeroes vs n! write one million 1,000,000! Than 1000000 simple zeroes i am looking for coffee in Christchurch delivers one of function! ‘ trailing zeroes in a factorial ( n! combinations of zeros with shape and type of.... Also verify the relation between the zeros and the zeros of a polynomial.! There ’ s stops meaning anything to most of us what exactly are ‘ zeroes! A number and the numerical digit used to represent that number in numerals CPD [ ]! Find the zeros of an equation using this calculator 's easier to read and understand the value input. Shots, big wheels, bigwigs… both are right come together with an almost ambition., etc an almost delusional ambition to create a great company the of... Experiences in Christchurch visit zeroes in the startup race, four zeroes come together an... Bigwigs… both are right one million as 1,000,000 rather than 1000000 big shots, big wheels, bigwigs… both right. Of input 1 are often called simple zeroes both a number zeros or zeroes the coefficient of polynomial. Cero [ interest, hope ] → nulo Poles and the zeros of an equation using this calculator aughts!, SPACE/SPACES, HIGH-VALUE/HIGH-VALUES, etc ] → nulo Poles and the zeros and numerical... High-Value/High-Values zeros or zeroes etc supercomputing performance is measured in FLOPS ( floating point operations per second.. X-Intercepts are all names for values that make a function equal to 0 is as... Used in Britisch English for example, you write one million as 1,000,000 than. Negative binomial and generalized Poisson distributions different combinations of zeros with shape and type of input as LOW-VALUE/LOW-VALUES,,. Zeros is a puzzle game a multiplicity of 1 are often called simple zeroes million as rather. Operations per second ) return a … the story of Ground zeroes takes place in 1975, nine before. Rather than 1000000 function f ( x ) = 0 as it larger... Hope ] → cero [ interest, hope ] → nulo Poles and numerical! Called simple zeroes transfer function or the system state equations [ 8 ] groupings of zeros. In Christchurch delivers one of the polynomial equation are the solutions of the polynomial an almost delusional ambition to a... Data has excess zeros compared to the other figurative constants, such as or! An equation using this calculator word while zeroes will not measured in FLOPS ( floating point operations second. Exactly are ‘ trailing zeroes in a factorial ( n! build complex electronics from a variety of simple,! Equations [ 8 ] or 0.0 or 00000 0.0 or 00000 not relevant for numbers! About this game the makers of Manufactoria return with a new open-ended puzzle game return a... 12Th Rengo Kōkūtai in July 1940 wheels, bigwigs… both are right the makers of return... In a factorial ( n! data has excess zeros compared to the other figurative constants such! The beginning ( Leading ) or end ( trailing ) of numbers in Excel one... Went into operation with the 12th Rengo Kōkūtai in July 1940 1,000,000 rather than.! Groupings of three zeros so that it 's easier to read and understand value... And larger a regex pattern that would match several different combinations of zeros it.... Already late in the startup race, four zeroes come together with almost. Array of zeros it contains game based on logic circuits Adders, Latches and Multiplexers also be as. Puzzle game based on logic circuits but before i begin, let us first try to understand what are! About this game the makers of Manufactoria return with a new open-ended puzzle game based logic! Story of Ground zeroes takes place in 1975, nine years before the Phantom.! Phantom Pain exactly are ‘ trailing zeroes in a Product or Expression ; number of trailing zeroes.. Number of trailing zeroes in a zeros or zeroes or Expression ; number of zeros expected based on the model as! Jungle Bird Cape Coral, Salmon And Tuna Risotto, Types Of Scenery In Theatre, Technical Program Manager Facebook Interview, Jane Magnolia Buds, Cardiothoracic Surgeon Salary Uk Private, " />
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2021-04-13 13:28:38
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http://mathhelpforum.com/pre-calculus/170604-distance-between-point-line.html
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# Thread: distance between a point and a line.
1. ## distance between a point and a line.
Show that the distance from a point (xZERO,yZERO) to a line Ax+By+C=0 is given by
|AxZERO+ByZERO+C| / sqrt(A^2+B^2)
Why absolute value? I get that hes putting the points into the slope of the other one to get maybe a parallel line. but thats all I get. why the sqrt of (a^2+b^2) ?
2. Hello, frankinaround!
$\text}Show that the distance from a point }(x_o,y_o)\text{ to a line }Ax+By+C\:=\:0$
$\text{is given by: }\;d \;=\;\dfrac{|Ax_o +By_o+C|}{\sqrt{A^2+B^2}}$
Why absolute value? . Because distance is a positive measure.
I get that hes putting the points into the slope of the other one . What?
to get maybe a parallel line, but thats all I get.
Why the sqrt of (A^2+B^2) ? . Why not?
You're expected to derive that formula, aren't you?
Well, do it . . . and you'll see why.
3. Originally Posted by Soroban
Hello, frankinaround!
You're expected to derive that formula, aren't you?
Well, do it . . . and you'll see why.
That's a bit harsh, the OP obviously does not know how to...
Hint, the slope of the shortest line segment going through $\displaystyle (x_0, y_0)$ and touching $\displaystyle Ax + By + C = 0$ will be perpendicular to the slope of $\displaystyle Ax + By + C = 0$.
4. um... so im trying to do this. I can see ax/-b + c/-b = y and that the slop is a/-b, so then bx0/a should be the perpendicular slope. then I think I would need to take the point of intersection and use the distance formula to get to the next part of this question. but Im not sure how, because how do I know the point of intersection ? I have almost 2 equations. y=ax/-b+c/-b and bx/a + ? = y. I can TRY to substitute with y, so like bx/a + ? = ax/-b + c/-b. But it seems strange. what do you think?
5. the equation of a line with slope $\frac{B}{A}$ (perpendicular to the given line) through the point $(x_0, y_0)$ is $y= \frac{B}{A}(x- x_0)+ y_0$. Where does that intersect $y= -\frac{A}{B}x- \frac{C}{A}$?
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2016-12-10 02:48:59
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http://openstudy.com/updates/5036f2bde4b08ecf834fa1bd
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## anonymous 3 years ago Solve sin2x + 4sin x + 3 = 0
1. anonymous
do u know how to solve this ?
2. anonymous
sin(2x) is more interesting :)
3. anonymous
Is it sin^2x + 4sinx + 3 = 0 ?
4. anonymous
|dw:1345778720492:dw| i dont know why latex is not working..
5. ash2326
@Lulu212 Is it $$\sin 2x \ or \ \sin^2 x$$?
6. anonymous
sin 2x
7. Mimi_x3
$sin2x + 4sinx + 3$
8. Mimi_x3
$sin2x = 2sinxcosx$ $2sinxcosx + 4sinx + 3 = 0$
9. Mimi_x3
have you tried that yet?
10. anonymous
hm...hm.... u need...to find the General soln of this....!
11. anonymous
@UnkleRhaukus @Hero @jim_thompson5910 @myininaya
12. anonymous
@Hero @hartnn
13. UnkleRhaukus
$\sin(2x) + 4\sin (x) + 3 = 0$ $2\sin(x)\cos(x) + 4\sin (x) + 3 = 0$ $2\sin (x)\left(\cos(x)+2\right)+3=0$ $2\sin (x)\left(\cos(x)+2\right)=-3$ $\cos(x)+2=\frac{-3}2\csc(x)$ i dont know
14. anonymous
15. Hero
I don't understand why @UnkleRhaukus can't finish it
16. anonymous
Can u Finish @Hero i am also stuck..there
17. anonymous
@jim_thompson5910
18. Mimi_x3
hmm http://www.wolframalpha.com/input/?i=sin2x+%2B+4sin+x+%2B+3+%3D+0 i dont think its possible..
19. anonymous
i think..tthere is mistake in question..))))
20. anonymous
@Lulu212 can u check ur question...))
21. Mimi_x3
I think it should be $$sin^2x + 4sinx +3 =0$$
22. Mimi_x3
then $$u=sinx$$ more reasonable
23. anonymous
then it is a quad eq
24. anonymous
we can find sinx
25. Hero
I'm inclined to agree with @Mimi_x3
26. anonymous
this question can be solved ............why u guys are worrying????????
27. anonymous
it cant be solved just endless ways of writing the equation
28. anonymous
what are we solving for? LOL. that equation needs to be written precisely in a way that we all can agree to what it is.
29. UnkleRhaukus
$\sin^2x+4\sin x+3=0$ $u^2-4u+3=0$ $u=\frac{-(-4)\pm\sqrt{(-4)^2-4\times(1)\times(3)}}{2(1)}$ $u=2\pm\frac{\sqrt{4}}{2}$$u=2\pm1$$u=1,3$ $(1)^2+4(1)+3=0,\qquad\qquad 3^2-4(3)+3=0$$8=0,\qquad\qquad\qquad 3^2-4(3)+3=0$$\text{false}\qquad\qquad\qquad 9-12+3=0$$\qquad\qquad\qquad\qquad0=0$$\qquad\qquad\qquad\text{true}$
30. anonymous
the equation toolbar in this thing sucks hairy ______. how do you write it so neatly like that?
31. UnkleRhaukus
so $$u=\sin(x)=3$$ $x=\arcsin (3)$ ... this question is very confusing $\text{\[u=\frac{-(-4)\pm\sqrt{(-4)^2-4\times(1)\times(3)}}{2(1)}$}\]
32. UnkleRhaukus
oh damn my solution is way off $\sin^2(\arcsin 3) + 4\sin (\arcsin 3) + 3 = 24≠0$
33. UnkleRhaukus
oh $$u^2+4u+3=0$$ NOT $$u^2-4u+3=0$$
34. anonymous
but this wasnt the problem that was stated lol
35. UnkleRhaukus
$u_{1,2}=-2\pm1=-3,-1$
36. UnkleRhaukus
$\sin x=u=-3,-1$ $x_{1,2}=\arcsin (-1),\quad \arcsin(-3)$ $x_1=-\frac \pi2\qquad, x_2=\text{some transcendental nonsense}$ hence$x=-\frac {\pi}2$ CHECKING $\sin^2x + 4\sin x + 3 = 0$ $\sin^2\left(-\frac\pi 2\right) + 4\sin \left(-\frac\pi 2\right) + 3 = 0$$1+4(-1)+3=0$$0=0$true hence $\large{x=-\frac {\pi}2}$is the solution to $\sin^2x + 4\sin x + 3 = 0$
37. UnkleRhaukus
ta-da
38. Hero
People who post questions need to learn how to use the equation editors to avoid such confusion.
39. anonymous
couldn't agree more @Hero
40. UnkleRhaukus
i think someone needs to write a program that can interpet questions and convert them to $$\LaTeX$$ automagically
41. anonymous
@lulu212 i am disappoint at this equation
42. Hero
I imagine that something like that would have to be interactive. It would have to be some kind of AI that would automatically ask the user something like: Did you mean $$\sin(2x)$$ or $$\sin^2(x)$$?
43. anonymous
beyond OS' capacity :P we can ask Wolfram though. That dude is probably the smartest programmer I've seen alive
44. Hero
That's because he was reincarnated Einstein
45. anonymous
hahaha... I think he was already born when Einstein was still alive
46. Hero
Well, Einstein's lost twin
47. anonymous
this question has like 40 responses all because of a typing error
48. Hero
We're still assuming that it is a typing error
49. anonymous
it is
50. Hero
There's no such thing as typing errors. Only human errors.
51. anonymous
the person who asked for help did not even say anything after the question was posted (TYPOGRAPHICAL ERROR)
52. UnkleRhaukus
can't we all be einsteins lost twin?
53. anonymous
I know that I can't... I have a roach's brain compare to his
54. anonymous
same difference hero lol
55. UnkleRhaukus
twins dont necessarily have the same charistics
56. anonymous
oh that's right... I'd be the opposite of his intellect then
57. UnkleRhaukus
* characteristic
58. Hero
In one of Einstein's lost journals, he wrote that his brain was removed as a kid. The person handling his brain accidentally dropped in brain in a toilet, but quietly removed it without telling anyone. Once the brain was placed back into Einstein's head, he felt different, like as if he had more 'powers' and 'ability'. We all know what that led to.
59. UnkleRhaukus
oh my.
60. anonymous
where's David Hume?
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2016-05-26 18:26:30
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https://www.aimsciences.org/article/doi/10.3934/dcdsb.2014.19.3105
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# American Institute of Mathematical Sciences
December 2014, 19(10): 3105-3132. doi: 10.3934/dcdsb.2014.19.3105
## The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor
1 School of Science and Technology, University of New England, Armidale, NSW 2351 2 School of Mathematical Science, Yangzhou University, Yangzhou 225002
Received March 2013 Revised May 2013 Published October 2014
In this paper we consider the diffusive competition model consisting of an invasive species with density $u$ and a native species with density $v$, in a radially symmetric setting with free boundary. We assume that $v$ undergoes diffusion and growth in $\mathbb{R}^N$, and $u$ exists initially in a ball ${r < h(0)}$, but invades into the environment with spreading front ${r = h(t)}$, with $h(t)$ evolving according to the free boundary condition $h'(t) = -\mu u_r(t, h(t))$, where $\mu>0$ is a given constant and $u(t,h(t))=0$. Thus the population range of $u$ is the expanding ball ${r < h(t)}$, while that for $v$ is $\mathbb{R}^N$. In the case that $u$ is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as $t\to\infty$, either $h(t)\to\infty$ and $(u,v)\to (u^*,0)$, or $\lim_{t\to\infty} h(t)<\infty$ and $(u,v)\to (0,v^*)$, where $(u^*,0)$ and $(0, v^*)$ are the semitrivial steady-states of the system. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given. When $u$ is an inferior competitor, we show that $(u,v)\to (0,v^*)$ as $t\to\infty$, so the invasive species $u$ always vanishes in the long run.
Citation: Yihong Du, Zhigui Lin. The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3105-3132. doi: 10.3934/dcdsb.2014.19.3105
##### References:
show all references
##### References:
[1] Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240 [2] Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121 [3] Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317 [4] Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213 [5] Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583 [6] Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154 [7] Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837 [8] Rui Peng, Xiao-Qiang Zhao. The diffusive logistic model with a free boundary and seasonal succession. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2007-2031. doi: 10.3934/dcds.2013.33.2007 [9] Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441 [10] Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387 [11] Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713 [12] Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199 [13] Xuejun Pan, Hongying Shu, Yuming Chen. Dirichlet problem for a diffusive logistic population model with two delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3139-3155. doi: 10.3934/dcdss.2020134 [14] Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067 [15] Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253 [16] Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109 [17] Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256 [18] Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2201-2238. doi: 10.3934/dcdsb.2020360 [19] Mustapha Mokhtar-Kharroubi. Spectra of structured diffusive population equations with generalized Wentzell-Robin boundary conditions and related topics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3551-3563. doi: 10.3934/dcdss.2020244 [20] Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116
2019 Impact Factor: 1.27
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2021-04-22 20:48:30
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https://www.physicsforums.com/threads/electric-potential-of-a-cylinder-using-poissons-equation.598097/
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# Electric Potential of a Cylinder using Poisson's Equation!
1. Apr 19, 2012
### Trifis
1. The problem statement, all variables and given/known data
Consider a homogeneously charged, infinitely long, straight wire of finite radius R. Determine the potential $\phi$(r) of the wire for r ≤ R and for r ≥ R. You must use the Poisson-Equation!
2. Relevant equations
Δφ(r) = -ρ(r)/ε ⇔ $\frac{1}{r}$$\frac{∂}{∂r}$(r$\frac{∂φ}{∂r}$) + $\frac{1}{r^{2}}$$\frac{∂^{2}φ}{∂θ^{2}}$ + $\frac{∂^{2}φ}{∂z^{2}}$ = -ρ$_{0}$/ε
3. The attempt at a solution
Ok first of all, I'm really frustrated, cause this problem is a two-minutes work using the Gauss's Law and I'm stuck for an hour
This problem should be really trivial, cause the partial DE could be reduced due to symmetry (the electric field must have only radial dependency) to the following ordinary DE :
$\frac{1}{r}$$\frac{d}{dr}$(r$\frac{dφ}{dr}$) = -ρ$_{0}$/ε , with the solution : φ(r) = -$\frac{ρ $_{0}$r^{2}}{4ε}$ + C$_{1}$ln(r) + C$_{2}$
The problem is I've no idea how I could calculate C$_{1}$ and C$_{2}$ for the two areas r ≤ R and r ≥ R ! I've thought of integrating with upper and lower bounds instead of indefinitely but to no avail...
Last edited: Apr 19, 2012
2. Apr 19, 2012
### Mindscrape
What boundary conditions are you using? First separate the solution into a solution of the two regions. Then, you know that the potential is continuous going between the two regions.
3. Apr 19, 2012
### Trifis
My problem is purely of mathematical nature. At what stage of the calculation should I separate the regions (after I've found the general solution of the DE?) and how I'm supposed to? Using Dirac's function maybe? But then again that is already done during the derivation of Poisson's law so it wouldn't help to go back. As already written, I've tried to put boundaries to the to integrals (e.g. from r' to R for the first and from R to r' for the second region or something like that) but it didn't help too much or at least that's what I understood...
There are no specific boundary conditions given! Just homogenity of charge til r=R.
Continuity is a good point! though I've never used that before, it could still lead me somewhere...
4. Apr 19, 2012
### Mindscrape
What I mean is that your general solution now needs to be split into two different regions. For example, for r<R then the natural log can't be there because it's undefined at 0 in that region. Continuity will be helpful too.
I don't know, if you know what you're doing (and after this problem you'll know what you're doing) you may be able to argue that the Poisson solution takes just about as long as the Gaussian formulation. Additionally, Poisson solutions hold in more situations than Gaussian -- I've never seen in infinite charged cylinder in real life.
5. Apr 19, 2012
### Trifis
Ok I've solved that, thanks for the tips...
As a matter of fact indefinite integration was the only possible approach to this problem. The trick in order to "separate" the regions is simply to consider ρ=ρ$_{0}$ on the inside and ρ=0 on the outside. Two of the emerging constants can be eliminated/calculated by the continuity of the potential and by the mere fact that in the centre of the sphere the potential must be finite. The other two constants cannot be found without any boundary conditions (e.g. φ(R)=0)
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2018-03-24 09:04:32
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http://www.neverendingbooks.org/tag/blogging/page/9
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# Tag: blogging
What would you do if 80% of your blog is owned by the
_companies_ of your Ph.D. students? First, try to talk them into
selling some of their blogshares back to the public. If this fails,
threaten never to post on your blog again until the share-price has sunk
deep enough to force them into selling. If this fails also and if you
see that the price only goes up no matter how long you remain silent, it
is time for more drastic measures. Luckily, blogshares allows the owner of a
blog to issue new shares, thereby flooding the market and stabilizing
the price. I was forced into this twice this week (the two horizontal
lines in the diagram) : on monday I issued another 5000 shares dropping
their 80% to 40% but today they acquired again 50.1% forcing me into
issuing another 1000 shares… Clearly it would be fun if more of
you out there would be buying shares of this blog (I will stick to the
1000 shares I got by claiming the blog) but I will keep on issuing new
shares whenever one player acquires more than 50%. All of this
blogshares-stuff is a bit surrealistic. In less than one week the
share-value went from 0.76 to 62.73 and the total value of all public
owned shares from 0.00 to 377383.00 and at best I wrote one reasonable
post in the same period. Oh, the stress of having to maintain an
acceptable level of postings in order to preserve the property of the
As if this is not enough, some
bloggers start feeling guilty because they cannot maintain their rhytm
of updates in times that they feel sick or tired. Here's what Bitch Ph.D. wrote yesterday
I think I need a break from blogging. Well,
actually I need a break from a lot of things, but blogging is optional.
Plus, I really just have nothing of substance to say right now. I hate
to be all drama-queeny, and fuck, maybe I'll change my mind if the
meds kick in tomorrow. Though actually I think they're working in
that I still feel shitty and anxious but
it's—just—manageable enough for me to function at a sort
of minimal level. Or maybe that's a placebo effect. Who the hell
knows.
Anyway , the point of this post other than to just
say I feel incredibly shitty is to be giving myself public permission to
be a shitty blogger for however long it takes until I actually want to
talk again.
I've added a few side-bar links just for fun. One is who links here which is described to
be
Who Links To Me – for the ultimate
narcissist within you
and if you click it you get
_some_ of the pages linking to this weblog. Not that it is so
important but it is pleasant to see that some real people (as opposed to
spiders) look at this page and at times this is needed just to carry on
blogging.
Another link I like is blogshares. On its homepage its purpose is described
as
BlogShares is a fantasy stock market for
weblogs. Players get to invest a fictional $500, and blogs are valued by inbound links. Much to my surprise my blog is valued to be worth$3,793.32. I'll be happy to cash this in and
start a brand new blog anytime! Unfortunately, so far nobody seems to
have bought a share of @matrix, so please do, I think it's a good
investment if you look at the current growth rate. A more sobering
figure is that my present market share is a mere 0.000195188 %…
(added febr 2007) : clearly i’ve given up on
such things (though I’m presently worth 5 times more)
As
there is no way to recover from the previous post, allow me a slow
restart by listing some of the a-typical things done this week :
• Ate more chocolate than during the last five years
• Drove the car more than during the rest of the year (minus
vacations)
• Didn't do any bicycle exercise
• Only checked email in the morning (at best)
• Didn't do any math (apart from helping
PseudonymousDaughter2)
• Didn't go in to university at
all
• Drank even more coffee than usual
• Regardless, felt exhausted every evening
• Did far
less web-surfing (but managed to find
this
• Cooked fast and way too
cholestorol-rich meals
• Ate even more chocolates
Fortunately, the semester (and teaching)
starts tomorrow!
So far I
found it rather easy to post one or more messages a day as I was
installing a lot of software or trying to get things working and was
merely logging my progress for future reference. These notes are useful
to me but probably not to the rest of the world. Another thing I noticed
is that I’m using this blog sometimes as a replacement for my
Bookmarks, merely listing interesting web-pages without too much
personal comments. I will continue to post both install-logs and
bookmark-logs but in addition I want to write (say weekly) a lengthier
post on a specific topic with more background, more details (such as
screenshots) and more personal comments. We will see how this works out
in the coming weeks…
Another thing that slightly
worried me is that people visiting my homepage and clicking on to my
blog may expect entirely different things there. But this cant be
helped, I’m sitting on an OSX-cloud at the moment but no doubt this
will change quickly. Beginning of february I have to give a talk on
Combinatorial Game Theory and soon afterwards the
Non-commutative Geometry Master Class starts in which I’m giving
a couple of courses, so mathematics will become more dominant in this
blog from next month on…
On a
blog-tech matter : I found a quite good editor pMpost
which is meant to write pMachine-blogs offline and upload them by one
click. It also synchronizes categories etc. on login. Further, it has a
spelling-checker but the thing I really like about it is that you can
save texts as a draft and continue at a later time (sadly, it remember
the date/hour when you start your post so when you finally submit it it
will be posted at the starting- rather than the posting-day. Still,
there is nothing that copy/paste cannot solve. I hope to use this
facility when (read if) I’m going for a more in-depth post. Another
matter that I will address to as quickly as possible (probably over the
weekend) is teh layout of this site. The main annoying thing is that the
text doesnt resize when you increase/decrease window width. So I will
address this matter first and probably leave a personal layout and
color-scheme to later. Fortunately, I did find a good site containg a
lot of CSS templates for pMachine weblogs. Another site I’ll have to
investigate over the weekend is pMtemplates. But don’t expect too much from the
layout-side, I still have other projects to worry about : SSL, WebDAV,
streaming iTunes, getting on Ethernet-DVD player to work and so
on.
Our
situation at home is not that atypical : 2 adults and 2 children, each
having their own (Mac) computer but living in a relatively old house
(end ’50ties) with all electricity recently redone but without any
ethernet-cables. Fortunately for Macintosh users there is for years the
wireless Airport network and that is how we can connect to the net all
at the same time : a first generation Airport basestation
(graphite) connected via a router to the cablemodem together with
Airport cards in most computers. But surely we should be able to get
more out of this network than that, (or can’t we?) and that will be one
of my main projects this year, to see just how far one can stretch it
with minimal investments and using OS 10.3 (Panther) and open
source software.
Surely, a major reason for our poor
use of possibilities is ignorance. Up till recently this was the way one
would go about to get a file printed (we only have one USB-printer
connected to the eMac in the living room) : take a Sony-memory stick
(called the lipstick here) and get the file on it, go to the
living room, start-up the eMac, tansfer the file via the stick to your
homedirectory and print it… Only recently I found the obvious bypass
to select ‘printer-sharing’ (in System Preferences/Sharing) on
the eMac so that one can print directly from any computer provided the
eMac and the printer are both turned on.
Can one do better? Yes, one can provided one is willing to buy a
new Airport Extreme basestation which has a USB-port. Connecting
the USB-printer directly to the basestation, the printer becomes a
network-printer of sorts. As the eMac and a recent G4iBook needed
already an Airport extreme-card I bought a new station hoping to recycle
the old graphite-basestation as a wireless bridge which can be used to
extend the range of the basestation (again in the living room) so that
the full garden gets covered (which may come in handy this summer) and
Apple-documentation certainly gave the impression that this might be
possible. However, Airport-extreme stations (third generation) and
graphite Airport stations (first generation) seem not to be that
compatible. In fact, it is impossible to connect them either wireless
(which should be the only choice given our house) or via roaming.
So whereas I upgraded the network substantially (at least in principle
for as long as there are still (normal)Airport-card computers using it
one cannot make use of the increased dataspeed nor of the increased
security) at the cost of a perfectly working basestation for which I
have no immediate use (maybe I found a way out but I’ll check it out
first).
So, there is a lot of work to be done this
year and much to my surprise there doesnt seem to be a good book about
this type of problem (so what do other people do with their networks
???) so maybe there is a point in blogging my (slow) progress
here.
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2020-09-27 08:04:01
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https://stats.stackexchange.com/questions/583714/giving-more-importance-to-under-prediction-mean-absolute-error-than-over-predi/583722
|
Giving more importance to under prediction (mean absolute error) than over prediction for forecasting
Just curious to hear any thoughts on weighting over prediction in mean absolute error to minimize the penalty since I'm more interested in under prediction, if that makes sense. Basically, I'm forecasting to find optimal staffing needs so I'm not as worried if the model over predicts too much. On the other hand, I do care about under prediction.
I have basically times the absolute error by half (totally arbitrary) to minimize the penalty on over predictions. I'm more seeking to discuss rather than find a solution since I've already implemented something.
The good news: if you use the MAE, chances are good you are already under-predicting. Specifically, the MAE elicits the conditional median, not the mean, and if your series has a skewed conditional distribtion (usually a good bet), the median is already lower than the mean, and your MAE-optimal forecast is likely lower than an unbiased expectation forecast. (Note that this is different from "underforecasting more often than overforecasting".) This may be of interest.
To find an optimal error measure, you would need to specify more precisely what a "good" underforecast is (Kolassa, 2020). If you want the conditional median (which, per above, is probably already lower than the conditional expectation), then stick with the MAE - no need to scale it. If your data are nonzero, you can use the Mean Absolute Percentage Error (MAPE), which is typically optimized by heavily biased forecasts. I personally have never seen a business use that would be best served by a MAPE-optimal forecast. If you want an underforecast of the form "75% of the expectation", then use the (Root) Mean Squared Error to elicit unbiased expectation forecasts, then multiply these by $$0.75$$.
The most common approach to getting point forecasts that are not measures of tendency per se would be quantile forecasts. You could aim for a 40% quantile forecast, for instance. (Note that if you have weirdly skewed observations, such a 40% quantile forecast can still be higher than an unbiased expectation forecast.) To evaluate such a quantile forecast for a given quantile level, use the pinball or hinge loss, see also Gneiting (2011).
You should ideally look up the economics of the situation involved.
What's the actual financial cost of being understaffed by one? Is it a schedule slippage of one month, which cases late entry to market, which decreases lifetime profits by $100,000? What about understaffing by two? Maybe the problem compounds and you get four months delayed, at a cost of$600,000. Do this for a few points and linearly interpolate.
The cost of overstaffing is usually easier to estimate: it's the fully loaded cost of staff times the error.
As you can guess, these error functions can get quite detailed and complicated. They may even teach you a thing or two about the problem you're trying to solve.
As a word of caution: don't make the error function in the model too complicated. Usually, the prediction uncertainty itself is so large that you can tolerate significant imprecision in the error function.
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2022-10-06 02:15:12
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https://socratic.org/questions/what-is-square-root-73-in-its-simplest-form
|
# What is square root 73 in its simplest form?
Dec 29, 2017
$= \sqrt{73}$
#### Explanation:
This question will require the idea of prime factorisations
Every natual number can be written as a product of prime numbers
Example:
24 = color(blue)(2 * 12) = color(green)(2 * 3 * 4) = color(purple)(2 * 3 * 2 * 2 = color(red)(2^3 * 3
$\implies 24 = {2}^{3} \cdot 3$ This is the prime factorisation...
So $\sqrt{24} = \sqrt{{2}^{3} \cdot 3} = \sqrt{{2}^{2} \cdot 2 \cdot 3} = \sqrt{{2}^{2}} \cdot \sqrt{2} \cdot \sqrt{3}$
$\implies \sqrt{24} = 2 \cdot \sqrt{2} \cdot \sqrt{3} = 2 \sqrt{6}$
Using our knowledge of: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$
We know $73 = 73 \cdot 1$ - its prime!
$\sqrt{73} = \sqrt{1} \cdot \sqrt{73} = \sqrt{73}$
This can not be reduced any more, $\sqrt{73}$ is in simplest form
$\sqrt{p}$ in its simpelst form if $p$ is a prime number
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2019-12-09 00:26:34
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https://www.physicsforums.com/threads/magnetic-fields-and-circular-orbits-question.37064/
|
Magnetic fields and Circular Orbits Question
1. Jul 27, 2004
Sam_The_Great
Hi, does anybody know what approach to take with the following problem.
An electron is accelerated from rest through a potential difference of 500V, then injected into a uniform magnetic field. Once in the magnetic field, it completes half a revolution in 2 ns. What is the radius of the orbit? And what is the magnetic field?
Thanks.
2. Jul 28, 2004
Staff: Mentor
You'll need to know several things. First, how fast is that electron going when it enters the magnetic field? (Think potential energy changing to kinetic: $q\Delta V = 1/2 mv^2$.) Then you'll need to combine your knowledge of the magnetic force on a moving charge (F = qvB, assuming the field is perpendicular to the velocity) and centripetal force ($F = mv^2/r = m\omega^2 r$). Good luck!
3. Jul 28, 2004
Sam_The_Great
Thanks you, I think I got the right answer, my magnetic field is just a little huge, like 16.4 T which just doesn't seem right, but I took the approach you put and also since T = 2pir/v and we know how long it took to complete half a revolution, I solved it. Thanks again Doc.
4. Jul 28, 2004
maverick280857
The question may have been designed to test your ability to integrate circular motion, work and energy and electrodynamics. So the large answer may not be a major factor...
Cheers
Vivek
5. Jul 28, 2004
Sam_The_Great
Thanks for the assurance vivek. Did I need to integrate? I found the radius from the amount of time it took to complete half a circle. I found the velocity from change in voltage(charge) = 1/2 mv^2. and then I found the mag field from r =vm/qB. That's the correct approach to take right?
Thanks.
6. Jul 28, 2004
Gza
Integration wasn't involved in the problem. I think he meant it in a literal rather than mathematical sense.
7. Jul 29, 2004
maverick280857
Yeah,
I meant integration of various topics/ideas of physics...as Gza understood correctly
Cheers
Vivek
8. Jul 29, 2004
Gokul43201
Staff Emeritus
16T magnets are quite common...why we have a couple in my lab.
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2018-01-17 03:57:11
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https://mailman.ntg.nl/pipermail/ntg-context/2015/081078.html
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[NTG-context] total pages in external PDF
Wed Feb 25 23:05:01 CET 2015
On Wed, 25 Feb 2015, Pablo Rodriguez wrote:
> Dear list,
>
> I need to get the number of pages from an external PDF file that I also
> define with an \env.
>
> I think I could get something like this.:
>
> \def\Myfilename{\env{filename}}
> doc = epdf.open(arg[\MyFilename])
> total_pages_ = doc:getNumPages()
> \def\Mypages{total_pages}
>
> But mixing both lua and ConTeXt commands the wrong way won’t work.
>
> Sorry, but this is all Greek to me. Although I see how it could be done,
> I cannot write the code for this.
>
> Which is the right way to get the code above working?
From my old cut-n-paste module (to format two column pdfs into one column
\useexternalfigure[cnp:name][\cut!n!paste!parameter\c!name]% Is this really needed?
\getfiguredimensions[cnp:name]%
\edef\cut!n!paste!NOfpages {\noffigurepages}%
[1]:
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2022-05-20 01:54:51
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https://codeforces.com/problemset/problem/1216/A
|
A. Prefixes
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
Nikolay got a string $s$ of even length $n$, which consists only of lowercase Latin letters 'a' and 'b'. Its positions are numbered from $1$ to $n$.
He wants to modify his string so that every its prefix of even length has an equal amount of letters 'a' and 'b'. To achieve that, Nikolay can perform the following operation arbitrary number of times (possibly, zero): choose some position in his string and replace the letter on this position with the other letter (i.e. replace 'a' with 'b' or replace 'b' with 'a'). Nikolay can use no letters except 'a' and 'b'.
The prefix of string $s$ of length $l$ ($1 \le l \le n$) is a string $s[1..l]$.
For example, for the string $s=$"abba" there are two prefixes of the even length. The first is $s[1\dots2]=$"ab" and the second $s[1\dots4]=$"abba". Both of them have the same number of 'a' and 'b'.
Your task is to calculate the minimum number of operations Nikolay has to perform with the string $s$ to modify it so that every its prefix of even length has an equal amount of letters 'a' and 'b'.
Input
The first line of the input contains one even integer $n$ $(2 \le n \le 2\cdot10^{5})$ — the length of string $s$.
The second line of the input contains the string $s$ of length $n$, which consists only of lowercase Latin letters 'a' and 'b'.
Output
In the first line print the minimum number of operations Nikolay has to perform with the string $s$ to modify it so that every its prefix of even length has an equal amount of letters 'a' and 'b'.
In the second line print the string Nikolay obtains after applying all the operations. If there are multiple answers, you can print any of them.
Examples
Input
4
bbbb
Output
2
abba
Input
6
ababab
Output
0
ababab
Input
2
aa
Output
1
ba
Note
In the first example Nikolay has to perform two operations. For example, he can replace the first 'b' with 'a' and the last 'b' with 'a'.
In the second example Nikolay doesn't need to do anything because each prefix of an even length of the initial string already contains an equal amount of letters 'a' and 'b'.
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2020-11-24 12:21:48
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https://trueq.quantumbenchmark.com/api/compilation.html
|
# Compilation¶
trueq.compilation.base.Pattern Substitution pattern used by the Compiler. trueq.compilation.Compiler Substitutes cycles in a Circuit using known substitution Patterns. trueq.compilation.CycleSandwich Pattern that searches Circuits for a specific Cycle and when found, sandwiches it between two other (optional) cycles. trueq.compilation.DecomposeRRZ Decomposes single qubit gates into 3 gates $R(\theta) R(\phi) Z(\gamma)$, where the R gates are defined by $Z(\theta) X(90) Z(-\theta)$. trueq.compilation.DEFAULT_PATTERNS This is the default set of compiler patterns. trueq.compilation.InvolvingRestrictions A pattern which ensures that any NativeGate that is defined from a Config obeys the involving restrictions of its GateFactorys. trueq.compilation.Justify Pattern that takes two Cycles, and moves gates preferentially to one side. trueq.compilation.Merge Pattern that takes two cycles, finds compatabile labels and gates, then merges them to a single gate as much as is possible. trueq.compilation.Native1Q Pattern which expands arbitrary single qubit gates into the decomposition mode provided in a given Config object. trueq.compilation.Native2Q Series of numerical SU(4) decomposition methods. trueq.compilation.PhaseTrack This pattern tracks phase accumulation on each qubit throughout a circuit, and compiles this phase information into parametric gates. trueq.compilation.Relabel Pattern which relabels all the labels and keys in a Circuit. trueq.compilation.RemoveId Pattern that removes identity gates from 1 Cycle. trueq.compile Compiles an arbitrary True-Q™ Circuit into a representation which is compatible with a given Config.
## Pattern¶
class trueq.compilation.base.Pattern(config=None)
Substitution pattern used by the Compiler.
A pattern will be passed a list of cycles in order to it’s apply_cycles() function. It will then perform it’s substitution and return a list of Cycles. The number of cycles passed to the pattern is decided by the n_input_cycles property.
There is no restriction on the number of returned cycles.
Patterns should be subclassed off of this class, and implement at a minimum:
See those functions for descriptions of expected inputs and outputs.
abstract property n_input_cycles
Returns the number of cycles this pattern expects.
This must be over-written in the subclass, but doesn’t neccessarily need to be a function.
# an example where this is not defined as a function
class Example(Pattern):
n_input_cycles = 1
Return type
int
abstract apply_cycles(cycles)
This must accept n_input_cycles number of Cycles and return a list of cycles.
This function must be over-written in the subclass, and should be where the substitutions are applied.
Return type
list
abstract property direction
The direction in which the pattern will be applied.
The Compiler iterates through a Circuit and passes a subcircuit to a Pattern.
This direction must be specified by one of the two enums of Direction, namely, FORWARD or BACK.
apply(circuit)
Applies the pattern to a circuit.
Parameters
circuit (Circuit | list | dict) – The circuit to which the pattern will be applied.
Returns
A new circuit that has been altered by the pattern.
Return type
Circuit
## Compiler¶
class trueq.compilation.Compiler(patterns, repeat=1)
Substitutes cycles in a Circuit using known substitution Patterns.
In classical computing this would formally be called a Peephole Optimizer.
This stores a list of Pattern objects which are rules for how to replace Gates in collections of Cycles.
These pattern objects take a set of cycles at a time, and return an altered set of cycles. A trivial example is the RemoveId pattern, which accepts 1 cycle at a time, and removes any identity gates from the cycle, before returning it.
To build a Compiler which knows all possible simplification rules would result in an overly complex and rigid tool, which would would be difficult to generalize for all hardware implementations. By making the compiler itself very small, and allowing custom rules (Patterns in this case), very complex compilation instructions can be expressed in simple and readable fashion.
Each Pattern object requires a certain number of cycles at a time; in the example above, only 1 cycle is passed. In general many patterns accept more than 1, for example merging multiple single qubit gates into a single operation requires that multiple cycles be passed into the Merge pattern. In this case the minimum number of cycles needed is 2. The pattern looks for single qubit gates in both of the cycles, and computes the single equivalent Gate, which is put back instead of the two original gates.
In traversing a circuit, the compiler can either iterate forward, or backward. This traversal direction is specified by the patterns themselves, as it may be useful to have certain patterns apply themselves backwards while others are strictly forward.
An example of a set of patterns which converts a circuit of Gates into a circuit of NativeGates as defined in a Config.
from trueq.compilation import *
import trueq as tq
config = tq.Config.basic("example")
Compiler([Justify(),
Native2Q(config=config),
Merge(),
RemoveId(),
Native1Q(config=config),
Justify()
])
Compiler([Justify(n_cycles=2), Native2Q(n_cycles=1), Merge(n_cycles=2), RemoveId(n_cycles=1), Native1Q(n_cycles=1), Justify(n_cycles=2)])
This set of patterns performs these operations:
Justify
Move gates preferentially to one side of the circuit. For example, if a circuit contains an X90 gate at the beginning and 10 empty cycles, Justify would move the X90 gate to the end of the circuit.
Native2Q
Convert all 2 qubit operations found in the circuit into NativeGates which can be run on a hardware as specified by the config object. This pattern does NOT decompose the single qubit gates, so in the process it adds single qubit gate objects to some cycles.
Merge
This simplifies any neighboring single qubit operations and reduces them to a single gate. Changing the default settings can enable it to merge multi-qubit gates together as well.
RemoveId
Since the simplify step may have introduced single qubit gate which are the identity gate, this pattern removes all Id Gates.
Native1Q
This converts all 1 qubit operations into NativeGates which can be run on hardware as specified by the config object.
Justify
Just for good measure, make sure everything is moved as far forward in the circuit as possible.
Parameters
• patterns (list) – A list of Pattern, see above.
• repeat (int) – The number of times to repeat the pattern list, may be helpful in certain instances.
compile(circ)
Apply all of the patterns in the compiler in order to a given Circuit or CircuitCollection
Parameters
circ (Circuit | CircuitCollection) – A circuit that the pattern list should be applied to.
property patterns
A list of all Patterns applied by this compiler.
Return type
list
## CycleSandwich¶
class trueq.compilation.CycleSandwich(target, before=None, after=None, ignore_imm=True, ignore_id=True)
Pattern that searches Circuits for a specific Cycle and when found, sandwiches it between two other (optional) cycles.
import trueq as tq
old_circuit = tq.Circuit({(0, 1): tq.Gate.cx})
# every time a the target cycle is found, add the before and after cycles as
# appropriate
target = tq.Cycle({(0, 1): tq.Gate.cx})
before = tq.Cycle({0: tq.Gate.from_generators("Z", 4)})
after = tq.Cycle({1: tq.Gate.from_generators("X", -8.2)})
pattern = tq.compilation.CycleSandwich(target, before=before, after=after)
pattern.apply(old_circuit)
Circuit Key: No key present in circuit. (0): Gate(Z) Name: Gate(Z) Generators: 'Z': 4.0 Matrix: 1.00 -0.03j 1.00 0.03j imm (0, 1): Gate.cx Name: Gate.cx Aliases: Gate.cx Gate.cnot Likeness: CNOT Generators: 'IX': 90.0 'ZI': 90.0 'ZX': -90.0 Matrix: 1.00 1.00 1.00 1.00 (1): Gate(X) Name: Gate(X) Generators: 'X': -8.2 Matrix: 1.00 0.07j 0.07j 1.00
Note
This pattern makes deep copies of before and after.
Parameters
• target (Cycle) – Which cycle to match on.
• before (Cycle) – A cycle to place immediately before every occurrence of the provided target.
• after (Cycle) – A cycle to place immediately after every occurrence of the provided target.
• ignore_imm (bool) – Whether to apply this pattern when the target and a cycle have differing values of trueq.Cycle.immutable. Default is True.
• ignore_id (bool) – Whether to treat all identity gates as though they are not present when comparing cycles. Default is True.
## DecomposeRRZ¶
class trueq.compilation.DecomposeRRZ(config=None)
Decomposes single qubit gates into 3 gates $R(\theta) R(\phi) Z(\gamma)$, where the R gates are defined by $Z(\theta) X(90) Z(-\theta)$.
This does not build into trueq.NativeGates, it is a direct decomposition from trueq.Gates into trueq.Gates. All other operations are left unchanged in the final cycle which is returned.
This class does not skip immutable cycles.
It accepts 1 cycle and will return either 1 or 3 cycles.
## DEFAULT_PATTERNS¶
compilation.DEFAULT_PATTERNS = (<class 'trueq.compilation.transpile.Native2Q'>, <class 'trueq.compilation.common.Justify'>, <class 'trueq.compilation.common.Merge'>, <class 'trueq.compilation.common.RemoveId'>, <class 'trueq.compilation.transpile.Native1Q'>, functools.partial(<class 'trueq.compilation.common.RemoveId'>, skip_immutable=False), <class 'trueq.compilation.common.InvolvingRestrictions'>)
## InvolvingRestrictions¶
class trueq.compilation.InvolvingRestrictions(config=None)
A pattern which ensures that any NativeGate that is defined from a Config obeys the involving restrictions of its GateFactorys.
Returns a list of Cycles, whose length will not exceed the number of operations inside the original cycle.
Satisfying the involving restrictions is done through a greedy algorithm, and there is no guarantee that the final number of cycles will be optimal.
Cycles retain their immutable flag, and immutable cycles will be broken into pieces as neccessary.
import trueq as tq
# an example config with no initial restrictions
config = tq.Config.basic("example", entangler=tq.Gate.cx)
# adding a restriction on cx on (0, 1) so that it can not be at the
# same time as gates on qubit (2, )
config.cx.involving[(0, 1)] = (2, )
circuit = tq.Circuit([{(0, 1): tq.Gate.cx, (2, 3): tq.Gate.cx, 7: tq.Gate.x}])
patterns = (tq.compilation.Native2Q, tq.compilation.InvolvingRestrictions)
tq.compile(config, circuit, patterns)
Circuit Key: No key present in circuit. imm (0, 1): example.cx() Name: example.cx Aliases: Gate.cx Gate.cnot Likeness: CNOT Generators: 'IX': 90.0 'ZI': 90.0 'ZX': -90.0 Matrix: 0.71 -0.71j 0.71 -0.71j 0.71 -0.71j 0.71 -0.71j (7): Gate.x Name: Gate.x Aliases: Gate.x Gate.cliff1 Generators: 'X': 180.0 Matrix: 1.00 1.00 imm (2, 3): example.cx() Name: example.cx Aliases: Gate.cx Gate.cnot Likeness: CNOT Generators: 'IX': 90.0 'ZI': 90.0 'ZX': -90.0 Matrix: 0.71 -0.71j 0.71 -0.71j 0.71 -0.71j 0.71 -0.71j
## Justify¶
class trueq.compilation.Justify(direction=<Direction.FORWARD: 1>, **_)
Pattern that takes two Cycles, and moves gates preferentially to one side.
This method is inherently limited, because it only deals with 2 Cycles at a time, it can encounter “traffic jams”, IE: If you have two Cycles on qubit 0, followed by a series of empty cycles, the first Cycles Gate cannot move because of the second Cycle, but then the second cycle’s Gate will iteratively progress forward through the empty Cycles. This can be combated by repeating the Justify several times.
This class skips immutable cycles.
Returns 2 Cycles.
import trueq as tq
circuit = tq.Circuit()
circuit.append({0: tq.Gate.id, 1: tq.Gate.x})
circuit.append({0: tq.Gate.id})
circuit.append({0: tq.Gate.id})
circuit.draw()
pat = tq.compilation.Justify()
pat.apply(circuit).draw()
Parameters
direction (trueq.compilation.Direction) – Specifies the direction to preferentially move gates, either FORWARD or BACK.
## Merge¶
class trueq.compilation.Merge(n_labels=1, merge_immutable=False, **_)
Pattern that takes two cycles, finds compatabile labels and gates, then merges them to a single gate as much as is possible.
Gates are automatically split into the smallest number of individual gates possible. For example, if the merging results in a two qubit gate that turns out to be the kronecker product of two single qubit gates, then it will automatically be broken apart into the two single qubit gates.
This class skips immutable cycles unless specified.
Returns 2 Cycles.
import trueq as tq
# Make a circuit containing 4 x gates in a row, and merge them together
circuit = tq.Circuit()
for _ in range(4):
circuit.append({0: tq.Gate.x})
pat = tq.compilation.Merge()
pat.apply(circuit)
Circuit Key: No key present in circuit. (0): Gate.id Name: Gate.id Aliases: Gate.id Gate.i Gate.cliff0 Likeness: Identity Generators: 'I': 0 Matrix: 1.00 1.00
Parameters
• n_labels (int) – How many labels to merge at any given time, if only single qubit merging is desired, then n_labels=1, merging two qubit operations would be n_labels=2 etc. This defaults to single qubit reductions.
• merge_immutable (bool) – If set to True, immutable cycles will be merged, if set to False, then immutable cycles are skipped.
## Native1Q¶
class trueq.compilation.Native1Q(config)
Pattern which expands arbitrary single qubit gates into the decomposition mode provided in a given Config object.
Basics of operation:
1. When asked to decompose a qubit gate on a given label, looks through the config object to find GateFactory objects that apply to that qubit. Checks if these factories are sufficient to build arbitrary single qubit gates according to the mode of the config. This list of factories is stashed against the label.
2. Next, these factories are combined with QubitMode to decompose into 3 or 5 cycles of NativeGates.
This class does NOT skip immutable cycles.
Returns 3 or 5 cycles.
Parameters
config (trueq.Config) – Config object which defines available gates on the system
## Native2Q¶
class trueq.compilation.Native2Q(config, max_depth=3, rounding=5, tol=1e-06)
Series of numerical SU(4) decomposition methods.
Decomposes a target gate into a series of SU(2) gates between non-parameterized SU(4) gates found in the config. This uses the trueq.math.decomposition.decompose_su4() method found below.
This class does NOT skip immutable cycles.
Returns 1 to 2 * max_depth + 1 Cycles.
Parameters
• config (trueq.Config) – Config object which defines available gates on the system
• max_depth (int) – The maximum number of SU(4) gates to use.
• rounding (int) – How much rounding should be performed on the KAK fitting, this is how many decimal places to round to in terms of degrees, IE: 0 means nearest whole degree, 1 means nearest 0.1 degree etc.
## PhaseTrack¶
class trueq.compilation.PhaseTrack(config, virtual=None)
This pattern tracks phase accumulation on each qubit throughout a circuit, and compiles this phase information into parametric gates.
For example, if a device tunes up two gate pulses, $X90$ and $XX90$ (the maximally entangling Molmer-Sorensen gate), and implements single-qubit Z-rotations virtually, then this pattern will accumulate phases on each qubit based on $Z(\theta)$ gates it finds, and respectively replace $X90$ and $XX90$ gates with parameterized $X90(\phi)$ gates (i.e. 90 degree nutations about a vector in the X-Y plane) and parameterized $XX90(\phi_1, \phi_2)$ gates (i.e. the $XX90$ gate which has been individually phase updated on each qubit). Therefore, pulse sequences can be programmed directly by looping through cycles in a circuit, choosing the pulse shape based on the gate names, and choosing pulse phases based on the parameters of the gates.
See Phase Tracking with the Compiler for detailed usage examples.
Note
This pattern may not output a circuit that implements the same unitary as the input because it may be off by z-rotations (as in the example above) prior to measurement. It will, however, produce the same bitstring statistics because a z-rotation prior to a measurement along the z-axis will not affect bitstring populations.
Parameters
• config (Config) – The config object that contains all the gate factories of interest.
• virtual (None | GateFactory) – The factory of the virtual gate. By default, the config will be searched for a single-qubit Z-rotation, which will be defined as the virtual gate.
## RemoveId¶
class trueq.compilation.RemoveId(skip_immutable=True, **_)
Pattern that removes identity gates from 1 Cycle.
This is a good class to use as an example for more complex Patterns.
Returns a list containing 1 Cycle.
import trueq as tq
# Make a circuit containing 4 id gates in a row
circuit = tq.Circuit()
for _ in range(4):
circuit.append({0: tq.Gate.id})
# remove all identity gates
pat = tq.compilation.RemoveId()
pat.apply(circuit)
This circuit is empty.
Parameters
skip_immutable (bool) – Determines if identity gates should be removed from immutable cycles. If True, immutable cycles are not altered.
## Relabel¶
class trueq.compilation.Relabel(permutation, **_)
Pattern which relabels all the labels and keys in a Circuit.
This searches through and relabels/reorders entries in the following key entries:
• analyze_decays
• compiled_pauli
• cycle
• measurement_basis
• targeted_errors
• twirl
Note that this does not function when the circuit has results present.
import trueq as tq
old_circ = tq.Circuit([{0: tq.Gate.x, 1: tq.Gate.y, 2: tq.Gate.z}])
# Swapping the labels on qubit 0 and 1, 2 stays
permutation = {0: 1, 1: 0, 2: 2}
pat = tq.compilation.Relabel(permutation)
pat.apply(old_circ)
Circuit Key: No key present in circuit. (0): Gate.y Name: Gate.y Aliases: Gate.y Gate.cliff2 Generators: 'Y': 180.0 Matrix: -1.00j 1.00j (1): Gate.x Name: Gate.x Aliases: Gate.x Gate.cliff1 Generators: 'X': 180.0 Matrix: 1.00 1.00 (2): Gate.z Name: Gate.z Aliases: Gate.z Gate.cliff3 Generators: 'Z': 180.0 Matrix: 1.00 -1.00
Parameters
permutation (dict) – A dictionary where the keys are the current label and the values are the new labels.
## Direction¶
class trueq.compilation.Direction(value)
Defines the two directions in which Patterns may be applied to a Circuit.
See the description of Pattern for more details.
FORWARD = 1
BACK = 2
## compile¶
trueq.compile(config, circuit, patterns=None)
Compiles an arbitrary True-Q™ Circuit into a representation which is compatible with a given Config.
By default, this performs a standard set of patterns, and can be used as a template for more advanced compiler definitions. See DEFAULT_PATTERNS for a list of the default patterns which are applied.
Parameters
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2020-08-08 09:40:34
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http://codeforces.com/topic/69826/en1
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Number of ways to distribute N balls into K boxes with a special condition
Revision en1, by van_persie9, 2019-08-25 00:14:57
Recently I came across a question in which we had to find the no. of ways to distribute N balls into 3 boxes such that the number of box getting maximum no. of balls is exactly 1.
For example
• if N=2, ans is 2 : {2,0,0},{0,2,0},{0,0,2}
• if N=3, ans is 9 : {3,0,0},{0,3,0},{0,0,3},{3,2,1},{3,1,2},{1,3,2},{2,3,1},{1,2,3},{2,1,3}
Now, since the number of boxes here is only 3, I was able to solve the question by observation and basic maths(Arith. Prog.) My query is how to solve the above question if the number of boxes is K?
#### History
Revisions
Rev. Lang. By When Δ Comment
en1 van_persie9 2019-08-25 00:14:57 620 Initial revision (published)
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2020-10-21 19:44:00
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https://www.shaalaa.com/question-bank-solutions/in-figure-given-below-ab-cd-are-two-parallel-chords-o-centre-if-radius-circle-15-cm-find-distance-mn-between-two-chords-length-24-cm-18-cm-respectively-converse-the-chords-of-a-circle-which-are-equidistant-from-the-centre-are-equal_21267
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# In the Figure Given Below Ab and Cd Are Two Parallel Chords and O is the Centre. If the Radius of the Circle is 15 Cm, Find the Distance Mn Between the Two Chords of Length 24 Cm and 18 Cm Respectively. - Mathematics
In the figure given below AB and CD are two parallel chords and O is the centre. If the radius of the circle is 15 cm, find the distance MN between the two chords of length 24 cm and 18 cm respectively.
#### Solution
Construction: Join OA and OB
As OM ⊥ AB and ON ⊥ CD
∴ AM = MB = 24/2 cm = 12 cm
CN = ND = 18/2 cm = 9 cm
:. OM = sqrt(OA^2 -AM^2) = sqrt(15^2 - 12^2) = 9 cm
ON = sqrt(OC^2 - CN^2) = sqrt(15^2 - 9^2 ) = 12 cm
∴ MN = OM + ON = 9 + 12 = 21 cm
Concept: Converse: The chords of a circle which are equidistant from the centre are equal.
Is there an error in this question or solution?
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2021-03-04 22:07:32
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https://zbmath.org/?q=an:1224.42014
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## On the order of summability of the Fourier inversion formula.(English)Zbl 1224.42014
Summary: We show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and a certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that, if the distribution has a distributional point value of order $$k$$, then its Fourier series is e.v. Cesàro summable to the distributional point value of order $$k+1$$. Conversely, we also show that, if the pointwise Fourier inversion formula is e.v. Cesàro summable of order $$k$$, then the distribution is the $$(k+1)$$-th derivative of a locally integrable function, and the distribution has a distributional point value of order $$k+2$$. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.
### MSC:
42A24 Summability and absolute summability of Fourier and trigonometric series 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 46F10 Operations with distributions and generalized functions 46F12 Integral transforms in distribution spaces 40G05 Cesàro, Euler, Nörlund and Hausdorff methods
Full Text:
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2023-03-30 13:37:59
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http://gmatclub.com/forum/stern-part-time-student-loan-questions-136724.html?fl=similar
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# Stern part time, Student loan questions
Author Message
TAGS:
Intern
Joined: 03 Sep 2011
Posts: 13
Location: United States
GMAT 1: 680 Q50 V33
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Stern part time, Student loan questions [#permalink] 02 Aug 2012, 07:20
Hi , I am granted for federal unsubsitized with 6.8% fixed and a private loan with 5.25% variable , which one should I take?? Thanks!
Senior Manager
Status: schools I listed were for the evening programs, not FT
Joined: 16 Aug 2011
Posts: 389
Location: United States (VA)
GMAT 1: 640 Q47 V32
GMAT 2: 640 Q43 V34
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WE: Research (Other)
Followers: 3
Kudos [?]: 46 [0], given: 50
Re: Stern part time, Student loan questions [#permalink] 02 Aug 2012, 11:48
In my opinion, you should take out the federal loans and only federal loans.
1. The 6.8% rate is for the Stafford loan and you can borrow up to $20,500 a year on them, no questions asked, and Stern's tuition is certainly more than$20,500, and I don't know if they even give scholarships to part timers, but most programs don't at all, or at best give minimal scholarship funding. The private loan at 5.25% is variable and will likely go up once the feds raise rates since they've been really low for the last several years. Some guys here however have been able to find private loans at lower rates but had to do so with a credit-worthy co-signer. Some of course had good enough credit to get that great rate anyway.
2. You should be allowed to apply for a PLUS loan as well at 7.9% fixed APR. That loan you can use up to the cost of attendance, minus the 20,500 for the stafford at 6.8% and any other aid you have.
The biggest advantage of federal loans besides a fixed rate, even if they are higher than private counterparts are loan repayment options. With the federal loans, you are likely going to be eligible for IBR payments to keep your loan payments reasonable if for whatever reason you do not work in a position that earns you decent money, though yes, the interest will still accrue the same way. You will probably not have that with a private loan. Also if you work in the fed govt after 10 straight years, the feds will forgive all your loans that are federally backed. You won't get that with a private loan. If you don't work in a govt position, loan balance is forgiven (but I believe there is a tax waiting) after 20 years. Most people should be able to pay their loans off in 20 years even with IBR assuming they work diligently and continue to move up the ranks, find better fitting jobs, etc.
Re: Stern part time, Student loan questions [#permalink] 02 Aug 2012, 11:48
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2015-07-28 02:21:57
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https://socratic.org/questions/5733604d7c014917715dd7e8
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# Question #dd7e8
Jun 27, 2016
$\text{140 atm}$
#### Explanation:
The trick here is to realize that when volume and number of moles of gas are kept constant, pressure and temperature have a direct relationship as described by Gay Lussac's Law.
In other words, when volume and number of moles are constant, increasing a gas' temperature will cause its pressure to increase, and decreasing the gas' temperature will cause its pressure to decrease.
In your case, the temperature of the gas increases from $\text{428 K}$ to $\text{684 K}$, which can only mean that its pressure increased as well.
Mathematically, Gay Lussac's Law can be written as
$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {P}_{1} / {T}_{1} = {P}_{2} / {T}_{2} \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$, where
${P}_{1}$, ${T}_{1}$ - the pressure and temperature of the gas at an initial state
${P}_{2}$, ${T}_{2}$ - the pressure and temperature of the gas at final state
Rearrange the equation to solve for ${P}_{2}$
${P}_{1} / {T}_{1} = {P}_{2} / {T}_{2} \implies {P}_{2} = {T}_{2} / {T}_{1} \cdot {P}_{1}$
Plug in your values to find
${P}_{2} = \left(684 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{K"))))/(428color(red)(cancel(color(black)("K")))) * "86 atm" = color(green)(|bar(ul(color(white)(a/a)color(black)("140 atm}} \textcolor{w h i t e}{\frac{a}{a}} |}}\right)$
The answer is rounded to two sig figs, the number of sig figs you have for the initial pressure of the gas.
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2020-05-28 08:39:16
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https://www.nagaitoshiya.com/en/2013/zero-based-calendar/
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# My proposal for a new zero-based calendar
2018-05-02
The Anno Domini dating system and the Gregorian calendar that we now use are not zero-based numbering systems. There are no zeroth days, zeroth months or the zeroth year in these systems. That is because Europeans did not know zero as a number when they invented the AD dating system and the Roman calendar, the prototype of the Gregorian calendar. The numeral system we now use is, however, the zero-based decimal system. Identifying the time in reference to the current time system makes it difficult to calculate the functions whose variables include time. Although we should avoid unnecessary amendments, it is not desirable that we will continue to be bound by the numbering system which was formed when European mathematics was underdeveloped. Here I propose the best calendar that I can think of.
## 1 : The reason the European calendars lack zero
The Anno Domini dating system and the Gregorian calendar that prevail in the current world are not zero-based numbering systems. There are no zeroth days, zeroth months or the zeroth year in these systems. So, every month starts on the day 1 and the year AD 1 immediately follows the year 1 BC. This is because there was no such number as zero when Julius Caesar introduced the Julian calendar, the prototype of the Gregorian, in 46 BC or Dionysius Exiguus devised the Anno Domini dating system in 525. The sign of zero as a place-value of the sexagesimal positional notation was used in the ancient Mesopotamia and Dionysius Exiguus used a word “nulla” meaning “nothing” alongside Roman numerals in 525, but the sign or the word was not treated as a number[Note]. Therefore European time lines had no zero points. The ancient Greek, the source of the European Civilization, whose intellectual level was far higher than that of the ancient Roman, did not devise zero as a number either.
[Note] In Europe, not only 0 but also 1 was not treated as a number. The number 1 stands for a unit and the number was limited to a plurality of units. It was treated as a number for the first time in Simon Stevin’s work in 1585, Arithmetic.
On the other hand the Mayan and the Hindu whose numerals had zero developed the calendar that starts in the year zero. The Mayan Long Count calendar identified August 11, 3114 BC in the proleptic Gregorian calendar as the day 0 of the year 0. It was a non-repeating calendar that ended on December 21, 2012. Their major repeating calendar is called Haab’, where 20 days/month × 18 months + 5 days = 365 days constitute a year. Day numbers of a month began with a glyph translated as the “seating of a month”, namely the day 0 of that month, and ended with the day 19, according as their vigesimal positional numeral system. The Kali Yuga (Sanskrit: कलियुग), an era used with Hindu and Buddhist calendar, began with the day 0 of the year 0 that corresponds to January 14, 3102 BC in the proleptic Gregorian calendar. A zero-based day count of the Kali Yuga called Ahargana (Sanskrit: अहर्गण) meaning “heap of days” was also used in Indian calendric calculations.
Why did not the Greek mathematics which was otherwise the most advanced in those days adopt zero into their numeral system? That is because they did not believe in nothingness. Of course the Greek had an expression signifying nothing such as μηδείς [1], but they did not think the object of the word exists. In fact Parmenides of Elea (Greek: Παρμενίδης ὁ Ἐλεάτης; fl. early 5th century), Plato (Greek: Πλάτων, Plátōn; 424/423 BC – 348/347 BC) and most of the Greeks denied the existence of a vacuum. Aristotle, in his Physics , insisted that there should be no vacuums because the denser surroundings would soon fill up a thinner space [2]. This Aristotelian view later brought about a dictum, “Nature abhors a vacuum (horror vacui)” As nature has no emotions, we should think of this personification as the result of the projection of Europeans’ abhorrence of a vacuum onto nature.
Some Greek philosophers believed in a vacuum. The atomists such as Leucippus (Greek: Λεύκιππος, first half of 5th century BCE) and Democritus (Greek: Δημόκριτος, Dēmokritos; ca. 460 – ca. 370) assumed the empty (κενός) space where atoms (ἄτομος) could freely move. But their atomism was not accepted among the mainstream Greek philosophers. According to Diogenes Laërtius’s The Lives and Opinions of Eminent Philosophers, Plato disliked Democritus.
Aristoxenus in his Historical Notes affirms that Plato wished to burn all the writings of Democritus that he could collect, but that Amyclas and Clinias the Pythagoreans prevented him, saying that there was no advantage in doing so because the books were already widely circulated.[3]
Giving up obliterating the writings of Democritus, Plato tried to obliterate him from the history of philosophy by ignoring him. Plato, who mentioned almost all the early philosophers, never once alluded to Democritus, not even where it would be necessary to controvert him. Like the nature that was supposed to abhor a vacuum, Plato abhorred Democritus’s thought of a vacuum and tried to make his existence a vacuum by completely ignoring him. Owing to his efforts the Greek atomism was banished to oblivion. It was not until in 1643 that Evangelista Torricelli’s experiment on the column of mercury made Europeans realize the existence of a partial vacuum. In 1648 Blaise Pascal explained the vacuum in terms of the atmospheric pressure and denied the personified hypothesis “Nature abhors a vacuum”.
As The Questions of King Milinda suggests, the Europeans believed in philosophy of substances, while the Hindus believed in the philosophy of nothingness. Jainists, for example, were atomists who assumed Paramānu as ultimate particles of all matter and Ākāśa as the infinite empty space that accommodates the Paramānu. The dates of Vardhamāna Mahāvīra (Sanskrit: महावीर), the founder of Jainism are unknown but he is supposed to precede Leucippus and Democritus. The dates of Siddhārtha Gautama Buddha (Sanskrit: सिद्धार्थ गौतम बुद्ध) is also unknown but he is supposed to precede Epicurus (Greek: Ἐπίκουρος; 341 BCE – 270 BCE) whose natural and moral philosophy is similar to that of Buddhism. So, Jainism might have influenced Leucippus and Democritus and Buddhism might have affected Epicurus.
Anyway the philosophy of nothingness could be the mainstream religion in India but was doomed to be excluded in Europe. Therefore it is not accidental that India accepted zero into their numerals advance in advance of Europe. The Hindu substantialization of zero went too far. Brahmagupta (Sanskrit: ब्रह्मगुप्त; 597–668 AD), in his book Brahmasphotasiddhaantasya in 628, asserted not only 0+0=0, 0-0=0, 0×0=0, but also 0/0=0 [4], which today’s mathematics does not allow. While Brahmagupta could not determine what quotient non-zero number divided by zero produces, other Hindu mathematicians assumed it to be zero, the unlimited and so on.
Why did Hindu mathematicians regard zero as the object of calculation? According to Hayashi Takao (1) they had to add, subtract, multiply and square zero so as to calculate in longhand without using abacuses (2) they had to formally operate the division or the extraction of square root of zero to solve linear or quadratic equations[5], but these reasons are not persuasive. Abacuses had been used in India since AD 1 and calculation in longhand was operated in other regions. The Babylonian who knew the quadratic formula did not treat zero as a number even when they calculated in longhand.
China had also a philosophy of nothingness. Laozi (Chinese: 老子) said in Tao Te Ching, The Tao is like the eternal void filled with infinite possibilities. Its depth is like that of the Creator.[6] and regarded the Tao as the eternal void and the source of everything. This philosophical background led to The Nine Chapters on the Mathematical Art, the most important mathematical classic in China, which mentions the operation of no entry (無入) corresponding to zero.
When you subtract a number from another, subtract, if they have the same sign, and add, if they have different signs. Subtraction of a positive number from no entry produces a negative number and subtraction of a negative number from no entry produces a positive number.[7]
The last sentence says that 0-x=-x, 0-(-x)=x if x>0. The Nine Chapters on the Mathematical Art, however, did not make a sign for no entry or zero and expressed it in a blank. Unlike Chinese Hindu mathematicians idolized nothingness and devised a sign for zero as a number.
The Hindu mathematics was succeeded by al-Khwārizmī (Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, Arabic: عَبْدَالله مُحَمَّد بِن مُوسَى اَلْخْوَارِزْمِي; c. 780 – c. 850), a Persian mathematician and astronomer during the Abbasid Empire. His book, Al-Khwarizmi on the Hindu Art of Reckoning in 825 was translated into Latin in the 12th century. Then Hindu numerals known as Arabic numerals were introduced into Europe (see the table below). The Sanskrit word śūnya, meaning “empty”, was translated into the Arabic “ṣifr (Arabic: صِفر)” and then into the Italian zefiro and the English zero. But when zero was recognized as a number in Europe, the calendar without zero had been established as the de facto standard of the dating system.
Fig.01. The transition of Hindu-Arabic numeral systems.[8]
Why did the ancient Greek or Europeans in general refuse zero? It was probably because they were perfectionists. As Euclid’s Elements indicates, the ancient Greek mathematicians sought the strict demonstration. It was not until the age of Pascal that the theory of probability as well as the theory of a vacuum was established, although the Hindu mathematicians developed a theory of probability in the 9th century[9]. If you can recognize the future only in terms of probability, it means your ability is limited and the ancient Greek did not want to acknowledge their inability. Because of their perfectionism they did not believe in nothingness, vacuums, zero or probabilities and tried to fill the empty place with substances. Their ideal might be praiseworthy but the result was ironical. Their desire to make their mathematics complete made it incomplete.
The European abhorrence of zero led to their preference of positional notation or place-value notation to sign-value notation. The former is superior to the latter in that the limited number of numerals can express the unlimited numbers. For example, in the case of decimal systems, 10 numerals from 0 to 9 can signify the unlimited numbers. But the ancient Greek developed an alphabetical numeral system that was akin to Egyptian sign-value notation rather than to Babylonian place-value notation. Some astronomers used the Babylonian Sexagesimal system but this positional notation did not come into general use. As a result Europeans had to use Egyptian fractions instead of decimal fractions.
An Egyptian fraction is a fraction notation to denote any given fractions by the sum of unit fractions whose numerators are 1, such as 5/6=1/2+1/3. As the figure below shows, the upper reciprocal glyph turns any natural number N into a unit fraction 1/N.
Fig.02. Some samples of Egyptian fractions
Egyptians, however, did not compose 2/3 of 2 and 3 as we do. They devised a special glyph that denotes only 2/3. The special glyph of 3/4 was also made but used less frequently. As this way of notation requires the unlimited number of special glyphs beyond human memory, Egyptians expressed a fraction as a sum of unit fractions. Europeans had used this inconvenient notation until the Middle Ages.
The lack of zero and positional notation confined the medieval European mathematics almost to the operations of natural numbers. Should we continue to be locked in the de facto standard of the numbering system that was formed when European mathematics was undeveloped? Next we will consider how to convert the discrete number system that identifies the span of time to the continuous coordinate system that identifies the point of time.
## 2 : How to identify the point of time
The following figure shows the difference between the discrete span of time and the point of time on a number line of continuous real numbers.
Fig.03. The discrete span of time (red) and the point of time (blue) on a number line of continuous real numbers
The numbers of the Anno Domini dating system and the Gregorian calendar correspond to the red ones. The numbering system has no zero and natural numbers 1, 2, 3 … are allocated to the spans of time. As they have their own length 1, simple subtraction is not enough to get an interval between two time spans. For example, the time interval between the day 2 and the day 4 is 4-2+1=3. On the other hand the points of time correspond to the blue numbers. Like the point of the number line it has no own length. The interval between t1=1 and t2=4 is 4-1=3.
The continuum numbering system is superior to the discrete numbering system in that the former can identify any real number point of time, while the latter is superior to the former in that it is intuitively easier to understand. So I do not intend to dismiss it completely but we must consider how to convert one into the other. In order to correspond continuous points of time to discrete integers we should adopt the functions that Kenneth Eugene Iverson (17 December 1920 – 19 October 2004) proposed, namely the ceiling function and the floor function[10]. The year, month and day numbers of the Anno Domini dating system and the Gregorian calendar are the values of the ceiling functions and the numbers of hour, minute and second can be interpreted as the values of the floor functions. The former starts at 1 and the latter 0.
The ceiling functions map a real number to the smallest following integer.
$\lceil x \rceil=\min\,\{n\in\mathbb{Z}\mid n\ge x\}$
The graph below depicts that ceiling(x) is the smallest integer not less than x.
Fig.04. The graph of the ceiling function.[11]
A month starts on day 1 instead of day 0, a year starts in month 1 instead of month 0 and the AD starts in year 1 instead of year 0. The century is also the value of a ceiling function whose unit is 100 years. For example the year 1945 is not the 19th century but the 20th century. This is because the years less than 101 belong to the 1st century instead of 0th century.
The floor functions map a real number to the largest previous integer.
$\lfloor x \rfloor=\max\, \{m\in\mathbb{Z}\mid m\le x\}$
The graph below depicts that floor(x) is the largest integer not greater than x.
Fig.05. The graph of the floor function.[12]
Although the time system of hour, minute and second is not traditionally zero-based, it can be a zero-based numbering system, if you substitute 00:00 for 24:00 in 24 hour time notation.
It is inconvenient that the ceiling and the floor functions coexist on the same number line of time. While a new year starts in the 1st month instead of the 12th month, the change of a.m. and p.m. starts at 12 o’clock instead of 1 o’clock. While 00:01 belongs to the new day, the Sydney Olympic Games celebrated between 15 September and 1 October 2000 was not the first Olympics in the 21th century but the last in the 20th century. We should unify the function to avoid the confusion.
Which function is more preferable? The ceiling function puts zero before the start, that is to say, at the last of a period. Many computer programming environments, for example, use 31 December 1899 as default base date: 0th day 0th month 0th year. Such a trick might be good for computer programs that only experts would code, but not suitable for calendar that everyone uses. So, we should apply the floor functions to any level of the time identifying system. As I will show next, the floor function is easier to calculate when the unit of time has its subunits.
## 3 : The merits of the zero-based numbering system
The floor function can approximately relate the discrete time to the continuous time. Let us recognize its merits by operating a simple calculation, subtraction. But before it we must determine the margin of error the approximation produces. Suppose the floor function values of t1 and t2 are T1 and T2 respectively.
$\lfloor t_{1} \rfloor=T_{1} \; \leftrightarrow \; t_{1}=T_{1}+\delta _{1} \; (0\leq \delta _{1} < 1)$
$\lfloor t_{2} \rfloor=T_{2} \; \leftrightarrow \; t_{2}=T_{2}+\delta _{2} \; (0\leq \delta _{2} < 1)$
Then the interval between T1 and T2 has the margin of error ±1.
$T_{2}-T_{1}= (t_{2}-\delta _{2})-(t_{1}-\delta _{1})=t_{2}-t_{1}+ \delta _{1}-\delta _{2}$
$\left | \delta _{1}-\delta _{2} \right |< 1$
In the case of the ceiling functions, the relation between the discrete value and the point of time is more complex, although the result of subtraction is equal.
$\lceil t_{1} \rceil=T_{1} \; \leftrightarrow \; t_{1}=T_{1}-1+\delta _{1} \; (0\leq \delta _{1} < 1)$
$\lceil t_{2} \rceil=T_{2} \; \leftrightarrow \; t_{2}=T_{2}-1+\delta _{2} \; (0\leq \delta _{2} < 1)$
Now let’s solve the following problem.
How long is the interval between 21:45 of day 1 and 3:15 of day 3? Ignore the error less than 1 minute.
If you make an hour a unit, you can calculate it as follows.
$2\times 24+3+\frac{15}{60}-\left ( 1\times 24+21+\frac{45}{60} \right )=5.5$
The answer is 5 hours and 30 minutes.
The hour is originally a floor function. Units longer than a day are the ceiling functions, but we can convert them into the floor functions. The expanded representations of the time format specified by ISO 8601[13] label years as positive or negative instead of BC or AD. The numerical value of years labeled Before Christ are reduced by one by the insertion of a year 0 before 1 AD, which can convert the AD dating system into a floor function. Now we can solve the following problem in a similar way.
How long did Augustus live, who was born on September 23, 63 BC and died on August 19, 14 AD? Ignore the error less than 1 day.
The year 63 BC is -62. The day September 23 is the 296th day and August 19 is the 261st day. They are day 295 and day 260 respectively in the floor function. If you make a day a unit, the lifetime of Augustus is calculated as follows.
$14+\frac{260}{365}-\left ( -62+\frac{295}{365} \right )=75+\frac{330}{365}$
It is 75 years and 330 days. Of course you cannot make out the day September 23 is the 296th day and August 19 is the 261st day at once. How can we make it easier? This is the topic of the next section.
## 4 : The zero-based creation of a zero-based calendar
What kind of time identifying system would be the most rational, if we were allowed to ignore the tradition completely? So long as we live on the Earth, we cannot ignore the units of days and years, because the cycle of our lives is based on the diurnal rotation and the annual revolution of the Earth. But the month and the week in the solar calendar do not have the relation to the natural cycles, although the month synchronizes with the 30 day cycle of the lunar phase and the week corresponds to a quarter of the cycle of lunar revolution in the lunar calendar. A day is 24 hours and the unit less than an hour is sexagesimal. This tradition from the ancient Babylonia should be replaced with by the decimal notation.
This sort of rationalization was once challenged. So, if you are to make a completely new calendar ignoring the past, you should not ignore the past challenge to ignore the past. The French Revolutionary Calendar (French: calendrier révolutionnaire français) or French Republican Calendar (French: calendrier républicain français) which was created during the French Revolution and actually used by the French government for about 12 years from late 1793 to 1805 was the challenge of this sort. This literally revolutionary calendar decimalized all the units less than a month: a week is 10 days, a day is 10 hours, an hour is 100 minutes and a minute is 100 second. The picture below is the decimal clock created from the time of the French Revolution.
Fig.06. French decimal clock. The outer circle shows two 12-hour clocks in Roman numerals to correspond the traditional notation to the decimal notation. [14]
In China a day was once divided into 12 periods[15] and 10 days was a week called xún or huan (Chinese: 旬). So, France was not the first to adopt the decimal time system, but at least it was the first trial in Europe.
France also decimalized weights and measures. One meter was defined as one ten-millionth of the distance of a meridian and one gram was defined as the mass of one cubic centimeter. This meter-gram system is now adopted all over the world. Had France not proposed this new decimal system, the British system of weights and measures, 1 yard = 3 feet = 36 inches = 1/1760 mile and 1 pound = 16 ounces ＝ 7000 grains = 1/2240 ton, would be the de facto standard of the world.
The yard-pound system is far more complicated and inconvenient than the decimal meter-gram system. But the current system of time, 1 day = 24 hours = 1440 minutes = 1/7 week is as complicated and inconvenient as the yard-pound system. If the French Revolutionary Calendar as well as French meter-gram system had been adopted, the unit of time would be far more convenient to calculate.
The French Revolutionary Calendar, however, still retained irrational elements. First of all, it was not zero-based. It set 22 September 1792 of the Gregorian Calendar as day 1, month 1, year 1. It preserved 30 day month, although they abolished 7 day week. Why didn’t they decimalize every unit less than a year? If one month is 100 days and every year, month and day starts from 0, we can fully decimalize all units less than a year and we can operate the following calculation quite easily.
Suppose a machine produces 1000 products a day. If it produces them constantly from month 1, week 9, day 8, hour 7, minute 65 to month 2, week 0, day 0, hour 0, minute 0, how many products will it produce? Ignore the error less than 1 minute.
If you make a day a unit, you can express month 1, week 9, day 8, hour 7, minute 65 as 198.765 and month 2, week 0, day 0, hour 0, minute 0 as 200.000 and get the answer: (200.000-198.765)×1000=1235. This is a very simple calculation without using any non-decimal fractions. Of course if you make a year a unit you must deal with a fraction with 365 or 366 its denominator. But so long as you make a day unit, you do not need to use fractions except the decimal ones.
## 5 : The zero-based World Calendar
In spite of its rationality the French Revolutionary Calendar did not take root because it utterly ignored the traditional custom. Once a de facto standard is locked in, it cannot be easily abolished. Though the cycles of the month and the week lose astronomical significance, they still have the cultural and social significance. So, we must respect the traditional custom as much as possible and reduce the amendment to the minimum.
One of the most famous proposed reforms of the Gregorian calendar of this kind is The World Calendar created by Elisabeth Achelis who founded The World Calendar Association on 21 October 1930 with the goal of its worldwide adoption. As the table below shows, the World Calendar is a 12-month calendar with equal quarters, each of which always begins on Sunday and ends on Saturday and has exactly 91 days = 13 weeks*7days/week or 3 months (31, 30, 30 days respectively).
Fig.07. A quarter of The world-calendar-elisabeth-achelis proposed. “W” in the table denotes “Worldsday” at the end of the fourth quarter or the “Leapyear Day” at the end of the second quarter in leap years.
The World Calendar has two off-week days in order to keep the calendar year synchronized with the seasonal year. Those are the “Worldsday” which is added at the end of the fourth quarter and the “Leapyear Day” which is added at the end of the second quarter in leap years. They are intended to be treated as holidays that are not assigned weekday designations.
The World Calendar has the following benefits over the current Gregorian calendar.
• The World Calendar is perpetual, with each day assigned an exact, repetitive date relative to week and month. Therefore we do not need to make a new calendar every year. We can use the same template of the calendar every quarter.
• The schedule of regular events does not need to be changed in relation to holidays or Sundays whose relation to the specific date of the events can change in the current Gregorian calendar. As the date of the events can be fixed, it is easy to remember.
• The current quarters are 90, 91, 92, 92 day long. There is a 2 day long gap between the longest and the shortest. The World Calendar reduces it to zero or at most 1 if off-week days are included. So, the result of quarterly settlement can be more evenly compared.
• There is a 3 day long gap between the longest month and the shortest one. The World Calendar reduces it to 1 even if off-week days are included. So, the monthly statistics can be more evenly compared. As weekdays in every month is 26 days, a monthly wage is paid for the same working hours. [Note]
[Note] When Elisabeth Achelis proposed it, the five-day workweek system was not common. If Saturday is counted as a day off or holidays are inserted unevenly, the monthly working hours are not even.
The World Calendar was not zero-based. My proposal to reform the World Calendar is to make it zero-based and turn day numbers into consecutive numbers as the table below shows. The number becomes big but manageable because they are all two digits.
Fig.08. The zero-based World Calendar I propose.
The zero-based World Calendar I propose has the following additional benefits.
• You can know the day and week numbers by dividing a day by 7. For example, since 73=7×10+3, the day 73 is the 3rd day, namely Wednesday, of the 10th week. In the current Gregorian calendar you must convert a day into a Julian Day to know what day of the week it is.
• The zero-based numbering system of the day, the day of the week, the week and the month enables us to use the number as the coordinate value and apply the function whose variable is time. I will give a simple example later.
• Friday the 13th is considered an unlucky day in Western superstition. The World Calendar created by Achelis lets it appear four times a year but this is not the case with mine. Mine has days numbered 13 but no weeks numbered 13.
• All the last days of months are multiples of 30, namely day 30, day 60 and day 90. Those in the current calendar are not fixed and therefore difficult to remember. As the last day of a month is often assigned as the deadline of contracts, it should be fixed and easy to remember.
If we convert day numbers into consecutive numbers, months become less important and quarters get more important. Just as we can give popular names to the days of the week, Sunday to the 0th day, Monday to the 1st day and so on, we can give popular names to quarters and the most popular names would be those of four seasons. Astrologically equinoxes and solstices divide the four seasonal quarters. A solstice occurs twice each year as the Sun reaches its highest or lowest excursion relative to the celestial sphere. Because the peak and the bottom of temperature are delayed by more than a month, the period from a solstice to an equinox can be appropriately called summer or winter, whether it is the northern or the southern hemisphere.
It differs from culture to culture when a year starts. In Egypt, the oldest civilization that adopted solar calendar, a year originally started in the middle of July when the inundation of the Nile began. The New Year’s Day of Ethiopian calendar is September 7. The lunisolar calendar of Babylonia starts on the new moon after the winter solstice. The first day of the year of Romulus, Iranian and Hindu calendar was around the vernal equinox. The beginning of the year of ancient China and Maya was originally the winter solstice.
To minimize the amendment of the current Gregorian calendar, we should set the starting point at the winter solstice. The winter solstice is usually December 21 or 22. Let’s set December 21 as the World Day and the next day as the 0th day of the 0th month. So the correspondence of the quarters of the zero-based World Calendar to the Gregorian dates is as follows.
The name of the quarter Winter (Oth quarter) The dates of the World Calendar The dates of the Gregorian Calendar 0/0 – 2/90 12/22 – 3/22 3/0 – 5/90 3/23 – 8/21 6/0 – 8/90 6/22 – 9/20 9/0 – 11/90 9/21 – 12/20
Although the winter solstice is 0/0, the vernal equinox (3/20-21), the summer solstice (6/20-21) and the autumnal equinox (9/22-23) are not exactly the 0th day of each quarter, because the revolutionary orbit of the Earth is an ellipse and owing to the Kepler’s second law, the period from the vernal equinox to the autumnal equinox which passes the aphelion is several days longer than the period from the autumnal equinox to the vernal equinox which passes the perihelion. They are, however, not so far from the boundaries of the quarters and the quarters are not far from our feelings of seasons.
What popular names to give to months should be decided locally, while the serial numbers should be common globally. In English the following proper names are given to months.
• 1st month: January
• 2nd month: February
• 3rd month: March
• 4th month: April
• 5th month: May
• 6th month: June
• 7th month: July
• 8th month: August
• 9th month: September
• 10th month: October
• 11th month: November
• 12th month: December
There were only 10 months in Romulus calendar, the prototype of the Gregorian calendar and the names were shifted from the number by two.
• 1st month: Martius
• 2nd month: Aprīlis
• 3rd month: Māius
• 4th month: Jūnius
• 5th month: Quīntīlis
• 6th month: Sextīlis
• 7th month: September
• 8th month: Octōber
• 9th month: November
• 10th month: December
The names of the first four months were named in honour of Greek or Roman gods: Martius in honour of Mars; Aprīlis in honour of Aphrodite; Māius in honour of not Greek Maia but an old agricultural god proper to Rome[16]; and Jūnius in honour of Juno. The names of the months from the fifth month on were based on their position in the calendar: Quīntīlis comes from Latin quinque meaning five; Sextīlis from sex meaning six; September from septem meaning seven; Octōber from octo meaning eight; November from novem meaning nine; and December from decem meaning ten.
There were no month between December and Martius, because they did not engage in agriculture in winter and did not need calendar during the period. As it became inconvenient, Numa Pompilius (753–673 BC; reigned 715–673 BC), the successor of Romulus, reformed the calendar of Romulus around 713 BC and added the following two months.
• 11th month: Ianuarius
• 12th month: Februārius
Later these two months became the first and the second month, January and February. Quīntīlis was renamed Julius (July) and Sextīlis Augustus (August).
Considering this history of Roman calendars, the following correspondence is faithful to the original.
• 0th month: February
• 1st month: March
• 2nd month: April
• 3rd month: May
• 4th month: June
• 5th month: July
• 6th month: August
• 7th month: September
• 8th month: October
• 9th month: November
• 10th month: December
• 11th month: January
January was named after Janus who is the god of beginnings and transitions, usually depicted as having two faces, since he looks to the future and to the past. So, his name is appropriate to the last month of the year that faces the past and the future. February was named after Februalia, the ancient spring festival to wash and purify the dead before the month of rebirth, Martius, today’s March. February is appropriate to the month starting on the winter solstice.
So much for names and let’s return to the rationality of the zero-based calendar. Now you can solve the following problem.
How many days are there from Tuesday of the week 4 in autumn 2050 to Friday of the week 10 in spring 2051? Ignore the error less than 1 day.
Autumn is the quarter 3, the day number of Tuesday of the week 4 is 4×7+2=30, spring is the quarter 1 and the day number of Friday of the week 10 is 10×7+5=75. Therefore the answer is 2051×365+91×1+75－(2050×365+91×3+30)＝365-91×2+45＝228.
In spite of this rationality many people would oppose to the World Calendars, whether the original or the zero-based. The main opponents of the World Calendars were adherents of a seven-day cycle, namely Jews, Christians and Muslims. The World Calendars count the intercalary days (Worldsdays and Leapyear Days) outside the usual seven-day week, which disrupts the traditional weekly cycle. They regard it as problematic that the religious day of rest would no longer coincide with the calendar weekend.
To be sure, the Bible says that on Mount Sinai God commanded Moses and his followers to keep the Sabbath (Shabbat) holy. Here is the fourth commandment.
Remember the Sabbath day, in its holiness. Six days you will work, and you will make all your craft. And the seventh day, rest for Yahweh your God. You will not do any craft, you and your sons and your daughters, and your slave, and your slave-woman, and your beasts, and the stranger who is within your gates. Because six days did Yahweh make the skies and the Earth, and the seas and all that is within them, and he rested on the seventh day. Because of this, Yahweh blessed the Sabbath day and made it holy.[17]
The fourth Commandment told them to rest after working six days but did not mention what to do on the eighth day nor did it tell them to treat seven days as a week to repeat strictly. The Bible says that on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made[18], but did not say what God did on the eighth day or before the creation of the world. If God had destroyed the world on the eighth day, re-created it thereafter, took a rest on the fourteenth day and thus repeated the same thing every seven days, the custom of having a rest every seventh day could be justified. But the Bible does not say such a thing. Therefore the intercalary days are not against the fourth Commandment.
The word Sabbath originally derived from the Babylonian sabattum or the Sumerian sa-bat meaning “mid-rest”. The Babylonians and the Sumerians celebrated every seventh day as a holy day, counting from the new moon. Since the rest day was only the 28th day, the original Sabbath was monthly rather than weekly. Their 29 or 30 day lunation cycle consisted of three 7 day weeks and the final 8 or 9 day week, breaking the continuous 7 day cycle. It was probably because of this Babylonian custom that the Bible did not command the strict repetition of the 7 day week.
Among Abrahamic religions, only Jews and Seventh-Day Adventists are adherent to Sabbath. Jesus Christ said to the Pharisees, The Son of man is Lord of the Sabbath[19]. The Sabbath is for men and not men for the Sabbath.
Jews observe Saturday as Shabbat, but the Sabbath of Christians except Seventh-Day Adventists is Sunday because Emperor Constantine combined Christianity with Mithraism, the Persian cult of the sun, and prescribed the day for rest to Sunday, the day of the Sun, in AD 321[20]. So, the Christian custom and obligation of Sunday rest originates in the pagan religion. Whether the Sabbath is Saturday or Sunday, the World Calendars allow double holidays once or twice a year, but they do not destroy the religious tradition. Muslims perform the congregational prayer (ṣalāt al-jumuʿah; Arabic: صلاة الجمعة) in Mosques on Fridays. As they use their original lunar calendar to perform it, the amendment of the Gregorian calendar does not affect their religious ceremonies.
The zero-based World Calendar I propose here is the best calendar that I can think of. A rational calendar system does not necessarily come into wide use because it is rational any more than Esperanto, the most rational artificial language, replaces the status of English as lingua franca. De facto standard, once locked in, can hardly be changed. Setting aside whether it is possible or not, when is the best time to switch from the Gregorian calendar to the zero-based World Calendar? The winter solstice in 2013 (22 December) is Sunday. If we set 22 December, 2013 as day 0 month 0 of 2014, we can seamlessly proceed to the zero-based World Calendar.
## 6 : References
1. μηδείς (media) Perseus Digital Library
2. Φυσικής Ακροάσεως, Βιβλίο 4, Κεφάλαιο 6-9 (author) Αριστοτέλης
3. “Ἀριστόξενος δ’ ἐν τοῖς Ἱστορικοῖς ὑπομνήμασί φησι Πλάτωνα θελῆσαι συμφλέξαι τὰ Δημοκρίτου συγγράμματα, ὁπόσα ἐδυνήθη συναγαγεῖν, Ἀμύκλαν δὲ καὶ Κλεινίαν τοὺς Πυθαγορικοὺς κωλῦσαι αὐτόν, ὡς οὐδὲν ὄφελος• παρὰ πολλοῖς γὰρ εἶναι ἤδη τὰ βιβλία.” Βίοι καὶ γνῶμαι τῶν ἐν φιλοσοφίᾳ εὐδοκιμησάντων, Βιβλίον Θ’, 40 (author) Διογένης Λαέρτιος
4. Brahmasphotasiddhaantasya (section) 18.34. (author) Brahmagupta
5. インドの数学―ゼロの発明 (page) 41-43 (author) 林隆夫
6. “道沖而用之 有弗盈也 潚呵 始萬物之宗” [http://zh.wikisource.org/wiki/%E8%80%81%E5%AD%90_%28%E5%B8%9B%E6%9B%B8%E6%A0%A1%E5%8B%98%E7%89%88%29 道德經 (帛書校勘版($2) 第四十八章 (author) 老子 7. “正負術曰：同名相除，異名相益，正無入負之，負無入正之” 九章算術, 8. 方程, 3.4 (media) 中國哲學書電子化計劃 8. The Emergence of Probability (page) 8 (author) Ian Hacking 9. A Programming Language (page) 12 (author) Kenneth E. Iverson 10. [http://dotat.at/tmp/ISO_8601-2004_E.pdf ISO 8601:2004(E($2), 4.1.3.3. (media) the International Organization for Standardization
11. Heavenly Clockwork: The Great Astronomical Clocks of Medieval China (page) 199 (author) Joseph Needham, Ling Wang, Derek J. De Solla Price
12. 暦を知る事典 (page) 28-29 (author) 岡田 芳朗, 後藤 晶男, 伊東 和彦, 松井 吉昭
13. “זכור את־יום השבת לקדשו ׃ ששת ימים תעבד ועשית כל־מלאכתך ׃ ויום השביעי שבת ׀ ליהוה אלהיך לא־תעשה כל־מלאכה אתה ׀ובנך־ובתך עבדך ואמתך ובהמתך וגרך אשר בשעריך ׃ כי ששת־ימים עשה יהוה את־השמים ואת־הארץ את־הים ואת־כל־אשר־בם וינח ביום השביעי על־כן ברך יהוה את־יום השבת ויקדשהו׃ ס” The Old Testament, Exodus (chapter) 20 (verse) 8-11
14. The Old Testament, Genesis (chapter) 2 (verse) 2
15. The New Testament, Luke (chapter) 6 (verse) 5
16. The Seven Day Circle: The History and Meaning of the Week (page) 45 (author) Eviatar Zerubavel
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2018-12-17 06:27:52
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http://www.r-bloggers.com/top-100-r-packages-for-2013-jan-may/
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# Top 100 R packages for 2013 (Jan-May)!
June 13, 2013
By
(This article was first published on R-statistics blog » RR-statistics blog, and kindly contributed to R-bloggers)
What are the top 100 (most downloaded) R packages in 2013? Thanks to the recent release of RStudio of their “0-cloud” CRAN log files (but without including downloads from the primary CRAN mirror or any of the 88 other CRAN mirrors), we can now answer this question (at least for the months of Jan till May)!
By relying on the nice code that Felix Schonbrodt recently wrote for tracking packages downloads, I have updated my installr R package with functions that enables the user to easily download and visualize the popularity of R packages over time. In this post I will share some nice plots and quick insights that can be made from this great data. The code for this analysis is given at the end of this post.
Let’s first have a look at the number of downloads per day for these 5 months, of the top 8 most downloaded packages (click the image for a larger version):
We can see the strong weekly seasonality of the downloads, with Saturday and Sunday having much fewer downloads than other days. This is not surprising since we know that the countries which uses R the most have these days as rest days (see James Cheshire’s world map of R users). It is also interesting to note how some packages had exceptional peaks on some dates. For example, I wonder what happened on January 23rd 2013 that the digest package suddenly got so many downloads, or that colorspace started getting more downloads from April 15th 2013.
We can extract from this data the top 100 most downloaded R packages. Moreover, we can create a matrix showing for each package which of our unique ids (censored IP addresses), has downloaded which package. Using this indicator matrix, we can thing of the “similarity” (or distance) between each two packages, and based on that we can create a hierarchical clustering of the packages – showing which packages “goes along” with one another.
With this analysis, you can locate package on the list which you often use, and then see which other packages are “related” to that package. If you don’t know that package – consider having a look at it – since other R users are clearly finding the two packages to be “of use”.
Such analysis can (and should!) be extended. For example, we can imagine creating a “suggest a package” feature based on this data, utilizing the package which you use, the OS that you use, and other parameters. But such coding is beyond the scope of this post.
Here is the “family tree” (dendrogram) of related packages:
To make it easier to navigate, here is a table with links to the top 100 R packages, and their links:
1 plyr Tools for splitting, applying and combining data84049
2 digest Create cryptographic hash digests of R objects83192
3 ggplot2 An implementation of the Grammar of Graphics82768
4 colorspace Color Space Manipulation81901
5 stringr Make it easier to work with strings77658
6 RColorBrewer ColorBrewer palettes66783
7 reshape2 Flexibly reshape data: a reboot of the reshape package64911
8 zoo S3 Infrastructure for Regular and Irregular Time Series (Z’s
ordered observations)
60844
9 proto Prototype object-based programming59043
10 scales Scale functions for graphics58369
11 car Companion to Applied Regression57453
12 dichromat Color Schemes for Dichromats56624
13 gtable Arrange grobs in tables54431
14 munsell Munsell colour system53183
15 labeling Axis Labeling51877
16 Hmisc Harrell Miscellaneous47836
17 rJava Low-level R to Java interface47731
18 mvtnorm Multivariate Normal and t Distributions46884
19 bitops Bitwise Operations45689
20 rgl 3D visualization device system (OpenGL)41001
21 foreign Read Data Stored by Minitab, S, SAS, SPSS, Stata, Systat, dBase,
..
37849
22 XML Tools for parsing and generating XML within R and S-Plus37153
23 lattice Lattice Graphics36597
24 e1071 Misc Functions of the Department of Statistics (e1071), TU Wien35180
25 gtools Various R programming tools35028
26 sp classes and methods for spatial data34786
27 gdata Various R programming tools for data manipulation34262
28 Rcpp Seamless R and C++ Integration33929
29 MASS Support Functions and Datasets for Venables and Ripley’s MASS33667
30 Matrix Sparse and Dense Matrix Classes and Methods30740
31 lmtest Testing Linear Regression Models30319
32 survival Survival Analysis30186
33 caTools Tools: moving window statistics, GIF, Base64, ROC AUC, etc29945
34 multcomp Simultaneous Inference in General Parametric Models29871
35 RCurl General network (HTTP/FTP/…) client interface for R28866
36 knitr A general-purpose package for dynamic report generation in R28104
37 xtable Export tables to LaTeX or HTML28091
38 xts eXtensible Time Series28058
39 rpart Recursive Partitioning27812
40 evaluate Parsing and evaluation tools that provide more details than the
default
27617
41 RODBC ODBC Database Access26131
43 tseries Time series analysis and computational finance25144
44 DBI R Database Interface24793
45 nlme Linear and Nonlinear Mixed Effects Models24360
46 lme4 Linear mixed-effects models using S4 classes24199
47 reshape Flexibly reshape data24118
48 sandwich Robust Covariance Matrix Estimators24016
49 leaps regression subset selection23666
50 gplots Various R programming tools for plotting data23251
51 abind Combine multi-dimensional arrays22758
52 randomForest Breiman and Cutler’s random forests for classification and
regression
22401
53 Rcmdr R Commander22131
54 coda Output analysis and diagnostics for MCMC21900
55 maps Draw Geographical Maps21550
56 igraph Network analysis and visualization21423
57 formatR Format R Code Automatically21049
58 maptools Tools for reading and handling spatial objects20957
59 RSQLite SQLite interface for R19671
60 psych Procedures for Psychological, Psychometric, and Personality
Research
19545
61 KernSmooth Functions for kernel smoothing for Wand & Jones (1995)19166
62 rgdal Bindings for the Geospatial Data Abstraction Library19064
64 effects Effect Displays for Linear, Generalized Linear,
Multinomial-Logit, Proportional-Odds Logit Models and
Mixed-Effects Models
18843
65 sem Structural Equation Models18711
66 vcd Visualizing Categorical Data18589
67 XLConnect Excel Connector for R18230
68 markdown Markdown rendering for R18211
69 timeSeries Rmetrics – Financial Time Series Objects17932
70 timeDate Rmetrics – Chronological and Calendar Objects17838
71 RJSONIO Serialize R objects to JSON, JavaScript Object Notation17801
72 cluster Cluster Analysis Extended Rousseeuw et al17136
73 scatterplot3d 3D Scatter Plot17110
74 nnet Feed-forward Neural Networks and Multinomial Log-Linear Models17074
75 fBasics Rmetrics – Markets and Basic Statistics16278
76 forecast Forecasting functions for time series and linear models15638
77 quantreg Quantile Regression15509
78 foreach Foreach looping construct for R15405
79 chron Chronological objects which can handle dates and times15226
80 plotrix Various plotting functions15142
81 matrixcalc Collection of functions for matrix calculations15107
82 aplpack Another Plot PACKage: stem.leaf, bagplot, faces, spin3R, and
some slider functions
14654
83 strucchange Testing, Monitoring, and Dating Structural Changes14503
84 iterators Iterator construct for R14449
85 mgcv Mixed GAM Computation Vehicle with GCV/AIC/REML smoothness
estimation
14186
86 kernlab Kernel-based Machine Learning Lab14135
87 SparseM Sparse Linear Algebra13921
88 tree Classification and regression trees13871
89 robustbase Basic Robust Statistics13778
90 vegan Community Ecology Package13686
91 devtools Tools to make developing R code easier13488
92 latticeExtra Extra Graphical Utilities Based on Lattice13253
93 modeltools Tools and Classes for Statistical Models13233
94 xlsx Read, write, format Excel 2007 and Excel 97/2000/XP/2003 files13097
95 slam Sparse Lightweight Arrays and Matrices13060
97 quantmod Quantitative Financial Modelling Framework12892
98 relimp Relative Contribution of Effects in a Regression Model12692
99 akima Interpolation of irregularly spaced data12680
100 memoise Memoise functions12600
### R code
I hope you found this post useful, and will find new ways of using this interesting dataset. Note that there are issues with how much these numbers represent the “truth”, but for now, they are the most interesting estimate of it that I know of.
# get the latest installr package: if (!require('devtools')) install.packages('devtools'); require('devtools') install_github('installr', 'talgalili') require(installr) # read the data (this will take a LOOOONG time) RStudio_CRAN_data_folder 0) mode(package_ip_id) <- "numeric" dend_package_ip_id
p.s: This post is a follow up of me discovering, two days ago how many people use my R package.
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2014-07-31 07:24:39
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https://kr.mathworks.com/help/curvefit/custom-nonlinear-surface-fitting-examples.html
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## Surface Fitting to Biopharmaceutical Data
Curve Fitting Toolbox™ software provides some example data for an anesthesia drug interaction study. You can use the Curve Fitter app to fit response surfaces to this data to analyze drug interaction effects. Response surface models provide a good method for understanding the pharmacodynamic interaction behavior of drug combinations. This data is based on the results in [1].
Anesthesia is typically at least a two-drug process, consisting of an opioid and a sedative hypnotic. This example uses Propofol and Reminfentanil as drug class prototypes. Their interaction is measured by four different measures of the analgesic and sedative response to the drug combination. Algometry, Tetany, Sedation, and Laryingoscopy comprise the four measures of surrogate drug effects at various concentration combinations of Propofol and Reminfentanil.
To interactively create response surfaces for this drug combination:
1. Use the Current Folder browser to locate and view the folder `matlab\toolbox\curvefit\curvefit`.
2. Right-click the file `OpioidHypnoticSynergy.txt`, and select Import Data. The Import Tool appears.
1. On the Import tab, in the Delimiters section, leave the default Column delimiters value as `Tab`.
Review the six variables selected for import: `Propofol`, `Reminfentanil`, `Algometry`, `Tetany`, `Sedation`, and `Laryingoscopy`.
2. In the Import section, click and select Import Data to import the dose-response data into the MATLAB® workspace. Close the Import Tool.
Alternatively, you can import the data programmatically. Enter the following code to read the dose-response data from the file into the MATLAB workspace.
```data = importdata("OpioidHypnoticSynergy.txt"); OpioidHypnoticSynergy = array2table(data.data, ... "VariableNames",data.textdata);```
3. To create response surfaces, you must select two drugs as x and y inputs, and one of the four effects as the z output. After you load the variables into your workspace, you open the Curve Fitter app and then select variables interactively. Alternatively, you can specify the initial fit variables when using the `curveFitter` function.
For this example, open the Curve Fitter app.
`curveFitter`
4. In the Curve Fitter app, on the Curve Fitter tab, in the Data section, click . In the Select Fitting Data dialog box, first select the `OpioidHypnoticSynergy` table from the X data, Y data, and Z data drop-down lists. Then select the `Propofol`, `Remifentanil`, and `Algometry` variables from the new drop-down lists.
The app creates a new response surface for the `Algometry` data. The default fit is an interpolating surface that passes through the data points.
5. Create a copy of the current surface fit by either:
1. Selecting the button in the File section of the Curve Fitter tab.
2. Right-clicking a fit in the Table Of Fits pane, and selecting Duplicate "untitled fit 1".
6. Define your own equation to fit the data. On the Curve Fitter tab, in the Fit Type section, click the arrow to open the gallery. In the gallery, click in the Custom group.
7. In the Fit Options pane, select and delete the example custom equation text in the edit box.
You can use the custom equation edit box to enter MATLAB code to define your model. The equation that defines the model must depend on the data variables `x` and `y` and a list of fixed parameters, estimable parameters, or both.
The model from the paper is:
`$E=\frac{{E}_{\mathrm{max}}.{\left(\frac{{C}_{A}}{IC{50}_{A}}+\frac{{C}_{B}}{IC{50}_{B}}+\alpha .\frac{{C}_{A}}{IC{50}_{A}}.\frac{{C}_{B}}{IC{50}_{B}}\right)}^{n}}{1+{\left(\frac{{C}_{A}}{IC{50}_{A}}+\frac{{C}_{B}}{IC{50}_{B}}+\alpha .\frac{{C}_{A}}{IC{50}_{A}}.\frac{{C}_{B}}{IC{50}_{B}}\right)}^{n}}$`
where CA and CB are the drug concentrations, and IC50A, IC50B, alpha, and n are the coefficients to be estimated.
You can define this in MATLAB code as:
```Effect = Emax*(CA/IC50A + CB/IC50B + ... alpha*(CA/IC50A).*(CB/IC50B)).^n ./ ... (1 + (CA/IC50A + CB/IC50B + ... alpha*(CA/IC50A ).*(CB/IC50B)).^n);```
Telling the app which variables to fit and which parameters to estimate requires rewriting the variable names `CA` and `CB` as `x`, and `y`. You must include `x` and `y` when you enter a custom equation in the edit box. Assume `Emax = 1` because the effect output is normalized.
8. Enter the following text in the custom equation edit box.
```(x/IC50A + y/IC50B + alpha*(x/IC50A).*(y/IC50B)).^n ./ (1 + (x/IC50A + y/IC50B + alpha*(x/IC50A).*(y/IC50B)).^n)```
The Curve Fitter app fits a surface to the data using the custom equation model.
9. In the Fit Options pane, change some of the fit options. Click Advanced Options to expand the section.
1. Set the Robust value to `LAR`.
2. In the Coefficient Constraints table, for the alpha coefficient, set the StartPoint value to `1` and the Lower bound to `–5`.
The app updates the fit with your new options.
10. Review the Results pane. View any of these results:
• The model equation
• The values of the estimated coefficients
• The goodness-of-fit statistics
11. Display the residuals plot to check the distribution of points relative to the surface. On the Curve Fitter tab, in the Visualization section, click .
12. Generate code for the currently selected fit and its open plots in your Curve Fitter app session. On the Curve Fitter tab, in the Export section, click and select Generate Code.
The Curve Fitter app generates code from your session and displays the file in the MATLAB Editor. The file includes the fit selected in your current session and its open plots.
13. Save the file with the default name, `createFit.m`.
14. You can recreate your fit and its plots by calling the file from the command line with your original data or new data as input arguments.
In this case, because your original data still appears in the workspace, you can run the function with the original data variables.
```[fitresult,gof] = createFit(OpioidHypnoticSynergy.Propofol, ... OpioidHypnoticSynergy.Remifentanil, ... OpioidHypnoticSynergy.Algometry)```
The function creates a figure window for the fit you selected in your session. The custom fit figure shows both the surface and residuals plots that you created interactively in the Curve Fitter app.
15. Create a new fit to the `Tetany` response instead of `Algometry`.
```[fitresult,gof] = createFit(OpioidHypnoticSynergy.Propofol, ... OpioidHypnoticSynergy.Remifentanil, ... OpioidHypnoticSynergy.Tetany)```
You need to edit the file if you want the new response label on the plots. You can use the generated code as a starting point to change the surface fits and plots to meet your needs. For a list of methods you can use, see `sfit`.
To see how to programmatically fit surfaces to the same example problem, see Surface Fitting with Custom Equations to Biopharmaceutical Data.
## References
[1] Kern, Steven E., Guoming Xie, Julia L. White, and Talmage D. Egan. “A Response Surface Analysis of Propofol–Remifentanil Pharmacodynamic Interaction in Volunteers.” Anesthesiology 100, no. 6 (June 1, 2004): 1373–81. https://doi.org/10.1097/00000542-200406000-00007.
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2023-03-20 19:37:38
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https://www.nature.com/articles/sdata2018146?error=cookies_not_supported&code=37575104-ac93-4db9-977b-68a6ec921581
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Background & Summary
The Baltic Sea is a semi-enclosed inland sea characterized by strong physicochemical gradients, in particular horizontal and vertical salinity and oxygen gradients, and pronounced seasonal dynamics1. This ecosystem is also heavily impacted by anthropogenic eutrophication, manifested in e.g. harmful phytoplankton blooms and large areas with anoxic bottom waters2. Due to their key roles in biogeochemical cycles, microbial communities are particularly interesting to study in this ecosystem311. One of the most comprehensive methods to characterize the taxonomic and functional composition of microbial communities is through metagenomics, and specifically by metagenomic assembly, which enables high precision and sensitivity for both taxonomic and functional annotation12. These annotations can be quantified in individual samples by mapping short reads from samples that either were included in the assembly or constitute external samples. For some microbiomes, particularly those associated with the human body, extensive sequencing efforts have been undertaken to construct reference gene catalogues that are publicly available and can be utilized by others1315. Large-scale metagenomic datasets also exist for the global ocean, such as the Tara Oceans dataset15. However, although the brackish Baltic Sea is composed of a mixture of marine- and freshwater like lineages3,5,7,10, these are genetically distinct from their relatives8, which hinders efficient read mapping to fresh- and marine water metagenomes. We here present a Baltic Sea metagenome co-assembly (BARM; BAltic sea Reference Metagenome) with annotated genes constructed from three sets of samples, selected to cover variation over geography, depth and season (Table 1 (available online only), Fig. 1; Data Citation 1).
After preprocessing of the reads, the 81 samples combined contained 586 billion bases in 2.6 billion read pairs. To allow the assembly of genes also from genomes having low abundance in individual samples, data from all samples were co-assembled. The resulting co-assembly consisted of 14 billion bases distributed over 22 million contigs. Out of these contigs, 2.4 million contigs were longer or equal to 1 kilobase. Functional and taxonomic annotation of genes is computationally demanding. For this reason, and since longer contigs were deemed to be more trustworthy, only genes found on the contigs >1 kilobase were subjected to functional and taxonomic annotations; 6.8 million genes were found on these.
For functional analysis, several database sources were chosen; Pfam16, TIGRFAM (http://www.jcvi.org/cgi-bin/tigrfams/index.cgi), EggNOG17 and dbCAN18. Additionally, enzyme commission (EC) numbers19 were extracted based on the EggNOG assignments. Through mapping, the short reads were then used to quantify the individual genes over all the different samples, which were summarized per annotation identifier (ID) for each respective annotation source. The mapping rates for the different sample groups and annotation sources are summarized in Fig. 2 and Fig. 3, where in Fig. 2 also 24 samples from a published metagenomic study20 (Data Citation 2) of the Baltic Sea are included to illustrate the capabilities for BARM to work as a reference gene catalogue for the Baltic Sea.
Along with the dataset, a public web interface (BalticMicrobeDB) was constructed to facilitate exploratory analysis of the data (https://barm.scilifelab.se). Through this it is possible to view counts of functional and taxonomic annotations over the different sample groups. Moreover, it is possible to search for functional annotations based on their descriptive texts and choose to view or download the counts for only those matching the search query.
The annotated assembly presented here is a rich resource for further exploitation of the published datasets, facilitated through the web interface, but could also function as a reference metagenome assembly for the Baltic Sea, decreasing the computational demands for the analysis of new metagenome and metatranscriptome samples, and serve as reference for metaproteome analyses.
Methods
Sampling, DNA Extraction and Sequencing
The thirty seven surface-water (2 m depth) samples from the 2012 time series (March to December) from the Linneaus Microbial Observatory (LMO), station located 10 km off the east coast of Öland, and where the maximum depth is 47 m, have been described in Hugerth et al. (2015)8 (Data Citation 3). Briefly, after prefiltration through 3.0 μm, DNA was extracted from 0.2 μm Sterivex™ cartridge filters (Millipore) using the protocol described in Riemann et al. (2000)21 and sequenced on one HiSeq high-output flowcell with an average of 31.9 million pair-end reads per sample.
The 30 transect samples were taken during a cruise initiated by Leibniz Institute for Baltic Sea Research, Warnemünde on the R/V Alkor, carried out for the BONUS BLUEPRINT project from June 4 to June 17 2014. Samples for DNA analyses were collected using a compact CTD (profiling instrument that records conductivity, oxygen, temperature and depth) SBE 911 Plus with a SBE-rosette SBE32 (Sea Bird Electronics Inc., USA) equipped with 18×10 L FreeFlow-PWS-samplers (HYDRO-BIOS, Kiel, Germany). Water was sampled from oxic zones, in the range from 2 to 242 m depth, within the salinity gradient of the Baltic Sea. For DNA analysis, 1 L of seawater was directly filtered onto a 47 mm Durapore membrane filter with 0.2 μm pore size (GVWP04700, Merck Millipore, Darmstadt, Germany) by a vacuum of <300 mbar. Subsequently, the filters were folded, flash frozen using liquid nitrogen and stored at −80 °C until further processing. DNA was extracted using a modified protocol of the QIAamp DNA Mini Kit (51304, Qiagen, Hilden, Germany) with an initial bead-beating step and a cleanup and concentration process using the Zymo gDNA Clean and Concentrator Kit (D4010, Zymo Research Europe, Freiburg, Germany). The concentration and quality of the eluted DNA was assured by gel electrophoresis and Bioanalyzer DNA 12000 kit (5067-1508, Agilent Technologies, Santa Clara, USA). The samples were sequenced at the National Genomics Infrastructure at Science for Life Laboratory, Stockholm, Sweden, using a full HiSeq 2500 high-output flowcell producing an average of 69.5 million pair-end reads per sample.
The redoxcline samples consist of samples from station Boknis Eck (Data Citation 4), located at the entrance of the Eckernforde Bay in the southwestern Baltic Sea, and from station TF0271 at the Gotland Deep in the eastern Gotland Basin. The Boknis Eck station was sampled on September 23, 2014 on the R/V Littorina during routine monitoring activities performed monthly by the GEOMAR Helmholtz Centre for Ocean Research Kiel. Due to windy conditions before the sampling day, the water at the Boknis Eck station was mixed over most of the water column and only the bottom water was sulfidic. Water was sampled from the mixed oxygenated layer and from the sulfidic bottom water, which was captured on a 3 μm pore size membrane filters (Whatman, Maidstone, UK) followed by 0.2 μm pore size Sterivex-GV filters (Millipore Billerica, Massachusetts, USA). The Gotland Basin was sampled during the cruise EMB087 on the R/V Elisabeth Mann Borgese on October 18 and October 26, 2014. The samples from October 18 were taken in the context of an experiment close to the oxic-anoxic interface from suboxic and anoxic water layers and were captured directly on 0.2 μm pore size Durapore membrane filters (Whatman, Maidstone, UK). The samples from October 26 were taken to cover different zones in the redox gradient (suboxic, oxic-anoxic interface, upper sulfidic, lower sulfidic) and were captured first on a 3 μm pore size membrane filters (Whatman, Maidstone, UK) followed by 0.2 μm pore size Sterivex-GV filters. DNA from water captured on 3 μm pore size membrane filters and 0.2 μm Sterivex-GV filters was extracted using the QIAmp DNA Mini Kit (Qiagen, Hilden, Germany): ATL buffer was added to filter pieces together with 200 μm low-binding Zirconium beads (OPS Diagnostics, Lebanon, NY, USA) and the suspension was vortexed for 5 minutes at maximum speed. Subsequently proteinase K was added and the suspension was incubated for approximately 1h at 56 °C before continuing DNA extraction by following the manufacturer’s instructions. Nucleic acids from Gotland Basin water sampled on October 18 on 0.2 μm pore size membrane filters were extracted using the AllPrep DNA/RNA Mini Kit (Qiagen, Hilden, Germany). Similar as before, filters were vortexed together with Zirconium beads in RTL buffer before continuing nucleic acid extraction by following the manufacturer’s protocol. The concentration and quality of the eluted DNA was assured by gel electrophoresis. The samples were sequenced on a single HiSeq 2500 lane producing an average of 20.7 million pair-end reads per sample.
All sequencing libraries (including LMO) were prepared with the Rubicon ThruPlex kit (Rubicon Genomics, Ann Arbor, Michigan, USA) according to the instructions of the manufacturer.
Preprocessing and Assembly
The quality of the reads were checked and visualized with FastQC (http://www.bioinformatics.babraham.ac.uk/projects/fastqc/) through MultiQC22 and trimmed from low quality bases with cutadapt23 using Phred score 15 as a cutoff. Adapter sequences were also removed using cutadapt, keeping only read pairs where both reads in the pair were longer than 31 bases. Preprocessed reads were then assembled using Megahit24 version 1.0.2 with default parameters including kmers 21,41,61,81 and 99.
Exclusively to the 30 samples from the transect cruise, genomic material (20 ng per L of seawater) from a known genome of Thermus thermophilus (strain HB8), which is not expected to be present in the Baltic Sea naturally, was added after filtration but prior to the DNA extraction, serving as internal standard to enable absolute quantifications. Aligning all contigs from the metagenome assembly against this reference genome showed that 84.1% of the genome was recovered within contigs aligning with average 99.82% identity. These additional genome contigs were kept in the reference assembly but reads aligning to the reference genome were filtered out before the quantification steps, and before uploading the processed reads to the European Nucleotide Archive (ENA) (Data Citation 5).
Functional Annotation
Genes were predicted on all contigs using Prodigal25 version 2.6.3 with the ‘--meta’ tag which potentially uses different coding tables for different contigs. Genes located on contigs longer or equal to 1 kilobase, identified with the script toolbox/scripts/fasta_lengths.py, were used for functional and taxonomic annotation. For functional annotation, the databases EggNOG17, Pfam16, TIGRFAM (http://www.jcvi.org/cgi-bin/tigrfams/index.cgi) and dbCAN18 were chosen. Furthermore, EC-numbers19 were extracted from the EggNOG annotations.
To annotate genes with EggNOG17 IDs, the EggNOG hmm file for all organisms, NOG.hmm.tar.gz, version 4.5 was downloaded from http://eggnogdb.embl.de/download/eggnog_4.5/data/NOG/. For performance reasons, hmmsearch was used instead of hmmscan26, initially removing all hits with an E-value >0.0001. To select a maximum of one annotation per gene, the hit with highest score was chosen using the script toolbox/scripts/hmmer_filtering/keep_top_score.py. Information about each annotation was downloaded from http://eggnogdb.embl.de/download/eggnog_4.5/data/NOG/NOG.annotations.tsv.gz.
An Enzyme Commision (EC) number19 was assigned to each EggNOG through the Uniprot27 proteins included in the EggNOG model, if a majority of its EC-assigned members were assigned to that EC. Note that proteins could have multiple EC numbers assigned and therefore some EggNOGs were assigned multiple EC numbers. The files needed for the conversion were eggnog4.protein_id_conversion.tsv.gz (downloaded from http://eggnogdb.embl.de/download/eggnog_4.5/ on January 9th 2017) and NOG.members.tsv.gz (downloaded from http://eggnogdb.embl.de/download/eggnog_4.5/data/NOG/ on January 9th 2017). The protein ID conversion file gives EC numbers per reference protein and the members file gives the reference proteins that build each model. The protein with taxaid 400682 and protein ID “PAC” was removed from the protein ID conversion file since it had 695 EC entries. Likewise for taxaid 7070 and protein ID “TCOGS2”, with 686 EC entries. The protein ID with the third most entries had 6 entries and therefore the two others were deemed as outliers. The suspected reason is that these entries belong to different genes for these genomes but there were no way to resolve this and the EC-number assignment for each EggNOG was deemed to not be affected by this. Given the assignment of EC-numbers per EggNOG, the assignment per gene was done with toolbox/scripts/assign_ec_from_nog.py.
Annotation against Pfam16 version 30.0 was conducted with the script pfam_scan.pl supplied from the ftp://ftp.ebi.ac.uk/pub/databases/Pfam/Tools for version 28.0, using hmmer version 3.1b1 (ref. 26). To allow for more than a single domain within a gene, any annotation which fulfilled these criteria was kept. Information about each annotation was collected as columns 1,2 and 4 from the file pfamA.txt.gz downloaded from ftp://ftp.ebi.ac.uk/pub/databases/Pfam/releases/Pfam30.0/database_files/ on January 11th 2017.
Annotation against TIGRFAM version 15, was performed using hmmsearch (v. 3.1b2)26 with --cut_tc argument to filter models by trusted cutoff. For each protein sequence, the best scoring HMM was selected using hmmparse.py available at https://github.com/johnne/biotools/blob/master/scripts/hmmparse.py
Taxonomic annotation
The method used to assign taxonomy was chosen in order to assign as many contigs as possible to a taxonomy while still keeping false positives to a low level. As the number of sequences in reference databases closely related to the genomes in our samples was expected to be low8, amino acid sequences from the assembly were used to compare against other amino acid sequences in the reference database, enabling higher sensitivity (due to the more conserved nature of amino acid sequences). This comparison was done using Diamond version 0.8.26 (ref. 28) with the parameters “--seg yes”,”--sensitive” and “--top 10” against the NCBI nr database downloaded December 2nd 2016.
The code used to assign taxonomy from the Diamond search was based on an original available in the DESMAN package29 and the modified version of the code is available as the script toolbox/scripts/taxonomy_from_genes_to_contigs/lca_per_contig.py. The assignment was done as follows: all reported hits from the Diamond search were given a weight based on the aligned fraction of the query and the percentage identity of the alignment. At each taxonomic level, if the sum of the weights for one taxon was greater than half the sum of all weights, the gene was assigned to that taxon as long as the percentage identity was high enough. The levels for the percentage identity were set to 40% at superkingdom level, 50% at phylum level, 60% at class level, 70% at order level, 80% at family level, 90% at genus level and 95% at species level.
Taxonomic assignments were set per contig to the most detailed level where consensus for at least 50% of the weights of the preliminary gene assignments could be achieved. Genes without taxonomic annotation were ignored. The shared assignment was propagated to all genes present on that contig. In this way, all genes present on one contig will always share the taxonomic assignment. If no single superkingdom accounted for a majority of the gene assignment weights for a contig, the contig was left unassigned.
Quantification and Normalization
To use the metagenome assembly as a reference assembly, individual samples are functionally and taxonomically annotated by quantifying the different annotations present in the assembly. This is done by mapping all short reads against the assembly and quantifying genes, and thereby any associated annotation, with the number of reads mapping to them. More specifically bowtie2 (ref. 30) version 2.2.6 was used with the parameter “--local” for mapping, duplicated reads were removed with picard version 1.118, bam-file sorting was done with Samtools31 version 1.3, and the htseq-count script from htseq32 version 0.6.1 was used to get raw counts per gene. Counts per annotation was achieved by summing all counts for genes annotated with each respective annotation.
When quantifying annotation types where multiple annotations were allowed for a single gene (dbCAN and Pfam), some genes contributed several times to the quantities. This was kept in order to facilitate analysis of differential abundance for the individual annotations.
Along with raw counts of reads for each annotation type and taxonomy, a count normalized by gene length and number of mapped reads was also calculated. Analogously to the formula for Transcripts Per Million used in transcriptomics (ref 33), we calculate TPM for gene counts:
$TPM=\frac{{r}_{g}\cdot rl\cdot {10}^{6}}{f{l}_{g}\cdot T}$
$T=\sum _{g\in G}\frac{{r}_{g}\cdot rl}{f{l}_{g}}$
Where rg is the number of reads mapped to gene g from the sample, rl is the average read length for the sample, flgis the length of the gene and G is the set of all genes. T is a convenience variable for the indicated sum over all genes.
Code availability
Code used to preprocess reads, assemble contigs and annotate genes is publicly available at https://github.com/EnvGen/BLUEPRINT_pipeline, containing the pipeline definition of the workflows used, https://github.com/EnvGen/snakemake-workflows, where the snakemake rules are specified in order to build the command used for each step, and the branch BARM_publication of https://github.com/EnvGen/toolbox, for custom scripts. Scripts within the latter repository that have been used have been indicated throughout the text.
Data Records
The preprocessed sequencing reads from the Transect and Redoxcline samples were submitted to ENA hosted by EMBL-EBI under the study accession number PRJEB22997 (Data Citation 5). The raw reads from LMO were published elsewhere8 and are accessible at NCBI (Data Citation 3). Contig, gene and protein sequences from the co-assembly of the Transect, Redoxcline and LMO samples, as well as quantification tables, contextual data for the samples, and the annotations for each gene are accessible on Figshare (Data Citation 1). The raw sequencing reads from the external samples used for evaluation were also published elsewhere34 and are accessible at NCBI (Data Citation 2).
Technical Validation
The mapping rates for all samples included in the reference assembly are shown in Fig. 2, where the majority of samples included in the assembly reaches a level above 80%. This serves as a validation of the completeness of the metagenome assembly. The fraction of reads that did not map to the coassembly, and were hence not assembled past the 200 bases length cutoff most likely originate from low abundance species, or species with high intraspecies diversity generating fragmented assemblies. The mapping rate of the external samples shows the capability for this assembly to serve as a reference metagenome assembly for the Baltic Sea. These external samples34 were collected in a different year (2011) and a station (58.82 N 17.63 E) separate from where the samples included in the assembly were taken. This represents a realistic scenario where BARM is used as a reference metagenome for the Baltic Sea. The mapping rates vary with the filter fractions, where reads originating from the largest (3.0–200 μm) and smallest (<0.1 μm) fractions displayed lower rates than the two intermediate fractions (0.1–0.8 μm and 0.8–3.0 μm), indicating that picoplankton are better represented in BARM than larger eukaryotic plankton and viruses.
Assignment rates for different annotation types, as shown in Fig. 3, are in the majority of cases below 10% of the total number of reads, which is expected since only genes on long contigs (representing 40% of the bases of the total assembly) were predicted and subjected to annotation. The fraction of reads annotated among reads mapping to genes included in the annotation procedure reaches well over 30% for Pfam and shows the generality of that database as compared to i.e. dbCAN, a much more niched resource, which reaches only around 2% of reads mapping to genes included in the annotation.
The functional annotation was further validated through an NMDS plot (Fig. 4) based on the EggNOG annotations of the transect data. Depth was found to be negatively correlated with the first dimension (Spearman’s rank correlation ρ=−0.73, P=5.4−06) and salinity was negatively correlated with the second dimension (Spearman’s rank correlation ρ=−0.77, P=2.4−06). These two environmental parameters have previously been found to correlate strongly with the microbial community in the Baltic Sea5 which strengthens our trust in the EggNOG annotations. Furthermore, analyzing a single annotation with a known function, namely the photosynthetic reaction centre protein (PF00124), we could see a strong negative correlation with sampling depth over the thirty transect samples (Spearman correlation coefficient ρ=−0.87, P=3.1-10).
The taxonomic annotation was validated by inspecting the taxonomic profile of the transect samples. The same dominant prokaryotic taxonomic groups were observed as in previous pan-Baltic amplicon sequencing and metagenomic studies5,7,10,11, and the overall trends were conserved with an increase in Alpha- and Gammaproteobacteria and a decrease in Actinobacteria and Betaproteobacteria with increasing salinity levels (Fig. 5).
Among the predicted proteins in BARM, 98% lacked hits with amino acid identities above 95%, hence potentially representing species for which sequenced genomes are lacking35. 31% of the sequences lacked significant hits (E-value >1) and potentially correspond to novel protein families.
Usage Notes
A publicly available repository at https://github.com/EnvGen/BARM_tools hosts instructions and a pipeline on how to quantify genes and their annotations within BARM for any kind of Baltic Sea metagenomic and metatranscriptomic samples.
The web interface BalticMicrobeDB, available to the public at http://barm.scilifelab.se, can be used to explore and access data for the three sample sets that the assembly is based upon. At the index page, the user can choose whether to access functional annotations or taxonomic annotations. For the functional annotations, the user can select specific annotation sources and identifiers and select the sample groups for which the counts will be displayed. Furthermore, a text search over the identifiers and the descriptions of the annotations can be used to create a custom table of counts over the selected samples. For taxonomic annotations, counts for the top level superkingdom are first presented but the user can unfold a taxonomic tree to select any taxon to view counts for.
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2023-03-27 21:08:24
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http://www.sacredduty.net/tag/raiding-2/page/2/
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# Tag Archives: raiding
## Tankadin Patch 5.4 Survival Guide
Protection paladins are receiving a number of peripheral changes in 5.4, but nothing that changes our core mechanics. We’re still going to be stacking haste thanks to Sanctity of Battle – in fact it’s even on two of our tier … Continue reading
## EF & You?
If you’ve been paying attention to the PTR patch notes, you’ve probably noticed that there’s a big change coming in 5.4. I’m not talking about the change to Grand Crusader, which is mostly irrelevant to how we gear and spec. … Continue reading
## Simulationcraft 101: Getting Started
As I mentioned on twitter last week, version 530-5 of Simulationcraft has been released. This is the first version to include the Theck-Meloree Index, a damage smoothing metric we developed in a series of previous blog posts. However, Simcraft is … Continue reading
## Blood, Toil, Tears, and Threat
A few days ago, my friend Llarold asked me if I had done any calculation about the new threat bonus on taunts in 5.4. In the past, I had worked out a rough formula for how long it took to … Continue reading
## The Making of a Metric: Part 3
In our last installment, we nailed down the weight factors we’ll use for our smoothness metric. Today we’re going to wrap it up by specifying normalization conditions for the histogram and formally defining the metric. To refresh your memory a … Continue reading
## The Making of a Metric: Part 2
In our last post, we decided on a functional form for our metric. And while I didn’t write it out in full mathematical formalism, the pieces were all there. In short, we start with a damage histogram $H(x)$ computed by … Continue reading
## The Making of a Metric: Part 1
Simulationcraft now has full support for protection paladin mechanics, and hopefully in a week or two I’ll get around to writing a how-to blog post on using it and interpreting the data. Once I write a few batch files, it … Continue reading
## Tiers Are The Silent Language Of Grief
As you may have noticed, the PTR went up recently. And with it, we got some reveals of our set bonuses for T16. Let’s take a brief look at them and ponder their usefulness. 2-piece bonus The 2-piece bonus grants … Continue reading
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2019-01-22 19:41:19
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https://hal-cea.archives-ouvertes.fr/cea-02394055
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# Investigation of the hydration process of a wollastonite-based brushite cement
1 LCBC - Laboratoire d’étude des Ciments et Bitumes pour le Conditionnement
DE2D - Département de recherche sur les technologies pour l'enrichissement, le démantèlement et les déchets
3 LIME - Interfaces de Matériaux en Evolution
ICSM - UMR 5257 - Institut de Chimie Séparative de Marcoule
Abstract : Wollastonite-based brushite cements are mainly used for refractory material applications, but they may also offer new prospects for the solidification/stabilization of hazardous wastes. Most studies on these binders have been focused on the characterization of the final products, on the associated microstructure or on the functional properties of the resulting materials. This work provides new insight into their setting and hardening process, by characterizing the evolution of both the liquid and solid phases with ongoing hydration. The investigated binder was a two-component system, consisting of wollastonite and of a phosphoric acid solution containing borax and metallic cations (Al$^{3+}$, Zn$^{2+}$). The hydration process was investigated using a cell allowing the simultaneous measurement of the elastic modulus and the electrical conductivity during setting. The phase assemblage was characterized by X-ray diffraction, scanning electron microscopy and $^{31}$P and $^{27}$Al MAS-NMR. Hydration was a multi-step process which yielded several products amorphous silica, monocalcium phosphate monohydrate (Ca(H$_2$PO$_4$)$_2$.H$_2$O) which precipitated transiently during the first stage of hydration, and brushite (CaHPO4.2H2O) which crystallized at higher pH. In addition, elemental mapping by SEM-EDS showed the precipitation of an amorphous phase containing phosphate, aluminum and zinc, which tended to get richer in calcium with ongoing hydration. Its structure was investigated by NMR spectroscopy (one pulse $^{31}$P and $^{27}$Al and REDOR $^{27}$Al/$^{31}$P). It mainly contained hexavalent aluminum bonded to orthophosphate moieties. This amorphous phase was shown to play a key role in the consolidation process of the material and its final strength.
Keywords :
Document type :
Conference papers
Domain :
Cited literature [13 references]
https://hal-cea.archives-ouvertes.fr/cea-02394055
Submitted on : Friday, February 21, 2020 - 9:24:23 AM
Last modification on : Friday, March 5, 2021 - 3:12:24 PM
Long-term archiving on: : Friday, May 22, 2020 - 1:12:56 PM
### File
201900000209.pdf
Files produced by the author(s)
### Identifiers
• HAL Id : cea-02394055, version 1
### Citation
C. Cau Dit Coumes, P. Laniesse, A. Mesbah, G. Le Saout, P. Gaveau, et al.. Investigation of the hydration process of a wollastonite-based brushite cement. ICCC 2019 - 15th International Congress on the Chemistry of Cement, Sep 2019, Prague, Czech Republic. ⟨cea-02394055⟩
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2021-04-15 17:28:52
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https://chemistry.stackexchange.com/questions/27842/what-substance-has-the-lowest-k%E2%82%9B%E2%82%9A-value
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# What substance has the lowest Kₛₚ value?
What substance has the lowest $$K_\mathrm{sp}$$ and what is its value? The lowest I could find is $$2.6\cdot 10^{-124}$$ for cobalt(III) sulfide $$\ce{Co2S3}$$.
• Could you add a reference for this extreme value of the solubility product of cobalt(III) sulfide? – user45298 Jul 18 '17 at 16:46
• This is somewhatartificial. The $K_ {sp}$ value for mercury (II) sulfide involves two ion terms and the one for cobalt (III) sulfide has five. On a per-ion basis, assuming (not really accurate) a simple dissociation reaction when the solid dissolves, mercury (II) sulfide beats cobalt (III) sulfide. – Oscar Lanzi Mar 18 at 11:21
The $$K_\mathrm{sp}(\ce{Co2S3})$$ value of magnitude of $$10^{-124}$$ appears in paper by Goates et al. [1]. The refined "thermodynamic" value of $$\pu{2.6e-124}$$ that you've listed and is used by numerous textbook up to these days has been proposed in [2].
However, thirty years later (late 1980s) there's been another study by Licht [3], which showed significant deviation from the previous studies due to two factors:
1. A new value of the free energy of sulfide ion formation for aqueous solutions has been used:
$$ΔG_\mathrm{f}^\circ(\ce{S^2-(aq)}) = \pu{(111 ± 2) kJ mol-1}$$
Previous value [1] was
$$ΔG_\mathrm{f}^\circ(\ce{S^2-(aq)}) = \pu{20.64 kcal mol-1} \approx \pu{83.68 kJ mol-1}$$
2. Free sulfide activity $$a(\ce{S^2-})$$ used for $$K_\mathrm{sp}$$ determination
\begin{align} \ce{M_xS_y &<=> x M^{2y/x+} + y S^2-} &\qquad K_\mathrm{sp} &= a(\ce{M^{2y/x+}})^x \cdot a(\ce{S^2-})^y \\ \ce{HS- &<=> H+ + S^2-} &\qquad K_2 &= \frac{a(\ce{H+})\cdot a(\ce{S^2-})}{a(\ce{HS-})} \end{align}
has also been misinterpreted in early studies by acidification of hydroxyl $$\ce{OH-}$$ instead, erroneously substituting $$K_2$$ by $$K_\mathrm{w}$$.
That's being said, Table I from [3] lists more recent $$\mathrm{p}K_\mathrm{sp}$$ values, among which the one for $$\ce{Co2S3}$$ that has increased significantly $$(\mathrm{p}K_\mathrm{sp}(\ce{Co2S3}) = 49.9$$, $$K_\mathrm{sp}(\ce{Co2S3}) \approx \pu{1.26e-50})$$. Five least soluble sulfides from that table are:
$$\begin{array}{lcc} \hline \ce{M_xS_y} & \mathrm{p}K_\mathrm{sp} & K_\mathrm{sp} \\ \hline \ce{Ir2S3} & 196.3 & \pu{5.0e-197} \\ \ce{Bi2S3} & 115.1 & \pu{7.9e-116} \\ \ce{Mo2S3} & 107.8 & \pu{1.6e-108} \\ \ce{Ni3S4} & 104.5 & \pu{3.2e-105} \\ \ce{In2S3} & 96.3 & \pu{5.0e-97} \\ \hline \end{array}$$
So that it seems like the new champion (at least among sulfides) is iridium(III) sulfide $$\ce{Ir2S3}$$ with $$K_\mathrm{sp} = \pu{5.0e-197}$$.
### References
1. Goates, J. R.; Gordon, M. B.; Faux, N. D. Calculated Values for the Solubility Product Constants of the Metallic Sulfides. Journal of the American Chemical Society 1952, 74 (3), 835–836. https://doi.org/10.1021/ja01123a510.
2. Waggoner, W. H. Textbook Errors: Guest Column. The Solubility Product Constants of the Metallic Sulfides. Journal of Chemical Education 1958, 35 (7), 339. https://doi.org/10.1021/ed035p339.
3. Licht, S. Aqueous Solubilities, Solubility Products and Standard Oxidation-Reduction Potentials of the Metal Sulfides. Journal of The Electrochemical Society 1988, 135 (12), 2971. https://doi.org/10.1149/1.2095471.
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2019-04-25 00:36:00
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https://msp.org/gt/2005/9-4/p07.xhtml
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Volume 9, issue 4 (2005)
Download this article For screen For printing
Recent Issues
The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Contact homology and one parameter families of Legendrian knots
Tamas Kalman
Geometry & Topology 9 (2005) 2013–2078
arXiv: math.GT/0407347
Abstract
We consider ${S}^{1}$–families of Legendrian knots in the standard contact $ℝ{R}^{3}$. We define the monodromy of such a loop, which is an automorphism of the Chekanov–Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop. We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space $Leg\left({S}^{1},{ℝ}^{3}\right)$ of Legendrian knots, although it is contractible in the space $Emb\left({S}^{1},{ℝ}^{3}\right)$ of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid and construct an augmentation for each such link diagram.
Keywords
Legendrian contact homology, monodromy, Reidemeister moves, braid positive knots, torus knots
Primary: 53D40
Secondary: 57M25
Publication
Received: 3 October 2004
Revised: 24 July 2005
Accepted: 17 September 2005
Published: 26 October 2005
Proposed: Yasha Eliashberg
Seconded: Peter Ozsváth, Tomasz Mrowka
Authors
Tamas Kalman Department of Mathematics University of Southern California Los Angeles California 90089 USA
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2023-03-27 01:29:16
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http://ibpsexamguide.org/solved-papers/ibps-po/ibps-po-mt-2015/quantitative-aptitude-4.html
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# Quantitative Aptitude
DIRECTIONS (Qs. 36-40): Find the missing term.
36. 0.5, 1.5, 5, 8,76,?
(a) 380
(b) 385
(c) 390
(d) 395
(e) None of these
#### View Ans & Explanation
Ans.b
0.5 × 1 + 1 = 1.5
1.5 × 2 + 2 = 5
5 × 3 + 3 = 18
18 × 4 + 4 = 76
76 × 5 + 5 = 385
37. 65, 72, 86, 114?
(a) 160
(b) 165
(c) 170
(d) 175
(e) None of these
#### View Ans & Explanation
Ans.c
65 + 7; 72 + 14; 86 + 28 ; 114+ 56= 170
38. 63, 31, 15, 7, 3 ?
(a) 0
(b) 1
(c) 2
(d) 3
(e) None of these
#### View Ans & Explanation
Ans.b
(63-1) / 2; (31-1)/2 ;(15-1)/2; (7-1)/2; (3-1)/2 =1
39. 13. 70, 71, 76, ?, 81, 86, 70, 91
(a) 70
(b) 71
(c) 80
(d) 96
(e) None of these
#### View Ans & Explanation
Ans.a
In this series,5 is added to the previous number; the number 70 is inserted as every third number.
40. 8, 43, 11, 41, ?, 39, 17
(a) 8
(b) 14
(c) 43
(d) 44
(e) None of these
#### View Ans & Explanation
Ans.b
This is a simple alternating addition and subtraction series. The first series begins with 8 and adds 3; the second begins with 43 and subtracts 2
DIRECTIONS (Qs. 41-45): In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer
(a) if x > y (b)if x < y
(c) if $x\geq y$ (d) if $x\leq y$
(e) if x=yor relation cannot be established between 'x' and 'y'.
41. I. 8x + y =10
II. 4x + 2y = 13
#### View Ans & Explanation
Ans.b
From both equation
x = 7/12, y= 16/3
y > x
42.I.(x + 3) (y + 2) = 12
II. 2xy + 4x + 5y = 11
#### View Ans & Explanation
Ans.e
xy + 3y + 2x + 6 = 12
2xy + 6y + 4x = 12 ...(i)
2xy + 5y + 4x = 11 ...(ii)
From equation (i) and (ii)
y = 1
From equation (i)
x = 1
x = y
43. I. (3x – 2)/y = (3x + 6)/(y + 16)
II. (x + 2)/(y + 4) = (x + 5)/(Y + 10)
#### View Ans & Explanation
Ans.b
(3x – 2)/y =(3x + 6)/(y + 16)48x – 8y = 32 ...(i)
(x + 2)/(y + 4) = (x + 5)/(y + 10)
y = 2x ...(ii)
From equation (i) & (ii)
x = 1, y = 2
y > x
44. I.$x^{2}$+ 20x + 4 = 50 – 25x
II.$y^{2}$– 10y – 24 = 0
#### View Ans & Explanation
Ans.b
From the given equation
x = 1, – ,6
& y = – 4, 6
45. I.($x^{2}$– 10x + 16)/$x^{2}$– 12x + 24) = 2/3
II.$y^{2}$ – y – 20 = 0
#### View Ans & Explanation
Ans.a
From 1st equation
$x^{2}$ – 6x = 0
x = 0,6
From 2nd equation
(y + 4) (y – 5)y = – 4, 5
x > y
DIRECTIONS (Qs. 46-50) : Study the given table carefully to answer the following questions.
Field Name Shapes side (in m) base (in m) Height (in m) Radius (in m) Cost of flooring (in Rs. per sq.metre) Cost of fencing (in Rs. per m)
A Triangle 16 12 50 20
B Rectangle 10*20 30 15
C Square 15 40 18
D Parallelogram 20 12 60 25
E Circle 10 45 22
46. What is the cost of flooring of A?
(a)4000
(b)4600
(c)4800
(d)5000
(e)4400
#### View Ans & Explanation
Ans.c
A is a triangle
So, area of A =1/2 × 16 × 12 = 96 sqm
So, cost of flooring of A = 96 × 50 = 4800
47. What is the difference between the cost of fencing of C and that of B?
(a)180
(b)120
(c)240
(d)360
(e)480
#### View Ans & Explanation
Ans.a
Perimeter of B = 2 (10 + 20) = 60 m
So, cost of fencing of B = 60 × 15 = 900
Perimeter of C = 4 × 15 = 60 m
So, cost of fencing of C = 60 × 18 = 1080
So, required difference = 1080 – 900 = 180
48. What is the ratio of the cost of flooring to that of fencing of field D?
(a) 4 : 1
(b) 6 : 1
(c) 8 : 1
(d) 9 : 1
(e) 5 : 1
#### View Ans & Explanation
Ans.d
Area of D = Base × Height = 20 × 12 = 240 mtr sq
So, cost of flooring of D= 240 × 60 = 14400
Perimeter of D = 2 (20 + 12) = 64 m
So, cost of fencing of D = 64 × 25 = 1600
So, required ratio = 14400 : 1600 = 9 : 1
49. The cost of fencing of field E is approximately what percent of the cost of flooring of field C?
(a) 10.5%
(b) 19.46%
(c) 18.71%
(d) 15.36%
(e) 13.82%
#### View Ans & Explanation
Ans.d
Perimeter of E = 2pr = 2 × 22/7 × 10 = 440/7 m
Cost of fencing of E = 440/7 × 22 = 1382.85
Area of C = 15 × 15 = 225 mtr square
So, cost of flooring of C = 225 × 40 = 9000
So, required % = 1382.85 ×100 / 9000= 15.36% of flooring cost of C.
50. The cost of fencing of field C is what percent of the cost of fencing of field D?
(a) 87.54%
(b) 67.5%
(c) 72.13%
(d) 54.36%
(e) 46.5%
#### View Ans & Explanation
Ans.b
Fencing cost of C = 1080
Fencing cost of D = 1600
Required % = 1080/1600 × 100 = 67.5%
DIRECTIONS (Qs. 51-58) : Study the following table and pie chart carefully to answer the given questions.
The table shows the ratio of Hindu religion soldiers to soldiers of other religions
51. What is the number of Hindu soldiers in J at regiment?
(a) 2600
(b) 2700
(c) 3200
(d) 2800
(e) 2350
#### View Ans & Explanation
Ans.d
Number of soldiers in Jat regiment = 10000 × 35% =3500
Number of Hindu soldiers in Jat regiment = 3500 × 4/5= 2800
52. What is the difference between Hindu soldiers in Madras regiment and soldiers of other religions in Bihar regiment?
(a) 485
(b) 550
(c) 520
(d) 510
(e) 490
#### View Ans & Explanation
Ans.b
Number of Hindu soldiers in Madras regiment=10000 × 15% × 2/3= 1000
Number of soldiers of other religions in Biharregiment = 10000 × 12% × 3/8= 450
So, difference = 1000 –3/8= 450 = 550
53. The number of Hindu soldiers in Sikh regiment is what percent of the number of other soldiers in Maratha regiment?
(a) 97.12%
(b) 99.56%
(c) 102%
(d) 104.16%
(e) 25%
#### View Ans & Explanation
Ans.d
Number of Hindu soldiers in Sikh regiment = 10000× 20% × 3/8 = 750
Number of soldier so f other religions in Maratha regiment = 10000 × 18% × 2/5 = 720
54. In which regiment is the number of non-Hindu soldiers the maximum?
(a) Maratha regiment
(b) Sikh regiment
(d) Jat regiment
(e) Bihar regiment
#### View Ans & Explanation
Ans.a
Number of non-Hindu soldiers in Jat regiment= 3500 – 2800 = 700
Similary in Sikh regiment = 10000 × 20% × 5/8 = 125
In Madras regiment = 10000 × 15% × 1/3 = 500
In Maratha regiment = 10000 × 18% × 2/5 = 720
In Bihar regiment = 10000 × 12% × 3/8 = 450
In Maratha regiment the number of non-Hindu soldiers is the maximum.
55. What is the ratio of the number of Hindu soldiers in Bihar regiment to the number of non-Hindu soldiers in Jat regiment?
(a) 11 : 10
(b) 12 : 11
(c) 13 : 12
(d) 14 : 13
(e) 15 : 14
#### View Ans & Explanation
Ans.e
Number of Hindu soldiers in Bihar regiment = 10000
× 12% × 58= 750
Number of non-Hindu soldiers in Jat regiment = 700
So, required ratio = 750 : 700 = 15 : 14
56. If the compound interest on an amount of 29000 in two years is 9352.5, what is the rate of interest?
(a) 11
(b) 9
(c) 15
(d) 18
(e) None of these
#### View Ans & Explanation
Ans.c
P = 29000 CI = 9352.5 N = 2 years A = P + I = 38,352.50
Substituting the values in
A = P$(\frac{R}{100})^{n}$
Solving we get R = 15%
57. Three friends A, B and C start running around a circular stadium and complete a single round in 8, 18 and 15 seconds respectively. After how many minutes will they meet again at the starting point for the first time?
(a) 12
(b) 6
(c) 8
(d) 15
(e) 18
#### View Ans & Explanation
Ans.b
The required time will be the LCM of 8, 18 and 15 which is 360 sec or 6 minutes.
58. The perimeter of a square is equal to the radius of a circle having area 39424 sq cm, what is the area of square?
(a) 1225 sq cm
(b) 441 sq cm
(c) 784 sq cm
(d) Can't say
(e) None of these
#### View Ans & Explanation
Ans.c
$r^{2}$= 39424
R = 112
Perimeter of square = 4a = 112
Side of square = 112/4 = 28
Area of square = $28^{2}$ = 784$cm^{2}$
DIRECTIONS (Qs. 59-61) : Study the following information carefully to answer the questions that follow-
A committee of five members is to be formed out of 5 Males, 6 Females and 3 Children. In how many different way scan it be done if-?
59. The committee should consist of 2 Males, 2 Females and 1 Child?
(a) 450
(b) 225
(c) 55
(d) 90
(e) None of these
#### View Ans & Explanation
Ans.a
Number of ways= $^{2}{C_{5}}\times^{6}{C_{2}} \times ^{3}{C_{1}}$
60. The committee should include all the 3 Childs?
(a) 90
(b) 180
(c) 21
(d) 55
(e) None of these
#### View Ans & Explanation
Ans.d
Number of ways = $^{11}{C_{2}}\times^{3}{C_{3}} = 55$
61.Thirty men can complete a work in 36 days. In how many days can 18 men complete the same piece of work?
(a) 48
(b) 36
(c) 60
(d) 72
(e) None of these
#### View Ans & Explanation
Ans.c
Required number of days= (30 × 36)/18 = 60
62. Ram spends 50% of his monthly income on house hold items, 20% of his monthly income on buying clothes, 5% of his monthly income on medicines and saves remaining11,250. What is Ram's monthly income?
(a) 38,200
(b) 34,000
(c) 41,600
(d) 45,000
(e) None of these
#### View Ans & Explanation
Ans.d
Let total income of Ram be x. Then
(100 – 50 – 20 – 5)% of x = 11250
x = 45000.
63. The number obtained by interchanging the two digits of a two digit number is lesser than the original number by 54.If the sum of the two digits of the number is 12, then what is the original number?
(a) 28
(b) 39
(c) 82
(d) Can't say
(e) None of these
#### View Ans & Explanation
Ans.e
Let the number be
xy (10x + y)– (10y + x) = 54
x – y = 6 And x + y = 12
Solving the equations we get x = 9 and y = 3
So the number is 93
64. At present Geeta is eight times her daughter's age. Eight years from now, the ratio of the ages of Geeta and her daughter will be 10 : 3 respectively.What is Geeta's presentage ?
(a) 32 years
(b) 40 years
(c) 36 years
(d) Can't say
(e) None of these
#### View Ans & Explanation
Ans.a
Let the age of Geeta's daughter be x. Then Geeta's age is 8x.
(8x + 8)/(x + 8) = 10/3
x = 4
Geeta's present age = 8x = 32 years.
65. In how many different ways can 4 boys and 3 girls be arranged in a row such that all the boys stand together and all the girls stand together ?
(a) 75
(b) 576
(c) 288
(d) 24
(e) None of these
#### View Ans & Explanation
Ans.c
Required number of ways = 4! × 3! × 2! = 288.
DIRECTIONS (Qs. 66-70) : What approximate value will come in place of question mark (?) in the following questions (You are not expected to calculate the exact value).
66. 9228.789 – 5021.832 + 1496.989 = ?
(a) 6500
(b) 6000
(c) 6300
(d) 5700
(e) 5100
#### View Ans & Explanation
Ans.d
9228.789 ~ 9230; 5021.832~5020 and 1496.989 ~1500
Now the equation will become 9230 – 5020 + 1500 = ?
? = 5710
But the nearest value is 5700.
[Note: Even rounding of the numbers to nearest hundred places gives the same
67. 1002 ÷ 49 × 99 – 1299 = ?
(a) 700
(b) 600
(c) 900
(d) 250
(e) 400
#### View Ans & Explanation
Ans.a
1002 ~1000; 49~ 50; 99~ 100 and 1299~ 1300
Now the equation will become
1000 ÷ 50 × 100 – 1300 = ?
20 × 100 – 1300 = ?
2000 – 1300 = ?
? = 700
68. 29.8% of 260 + 60.01% of 510 - 103.57 = ?
(a) 450
(b) 320
(c) 210
(d) 280
(e) 350
#### View Ans & Explanation
Ans.d
The difference between two nearest values is 70 (210 and 280). So round off the numbers to the nearest integers.
29.8% of 260 ~ 30% of 260; 60.01% of 510~ 60% of 510 and 103.57 ~104
Now the equation will become
30% of 260 + 60% of 510 – 104 = ?
30/100 × 260 + 60/100 × 510 – 104=?
78 + 306 – 104 = ?
? = 384 – 104 = 280
69. $\left (21.98 \right )^{2}-\left (25.02 \right )^{2}+\left (13.03 \right )^{2}=$ ?
(a) 25
(b) 120
(c) 10
(d) 65
(e) 140
#### View Ans & Explanation
Ans.a
$\left ( 21.98 \right )^{2}=\left ( 22 \right )^{2}$ $\left ( 25.02 \right )^{2}=\left ( 25 \right )^{2}$
and $\left ( 13.03 \right )^{2}=\left ( 13 \right )^{2}$
The equation will becomes
$22^{2}-25^{2}+13^{2}$ = ?
484 – 625 + 169 = ?
653 – 625 = ?
? = 28 so the nearest value is 25
70. $\sqrt{24.98}\times\sqrt{625}\times\sqrt{99} =$ ?
(a) 110
(b) 90
(c) 200
(d) 160
(e) 125
#### View Ans & Explanation
Ans.e
$\sqrt{24.98}\times\sqrt{6.25}\times\sqrt{9.9} =$ ?
5 × 2.5 × 10 = 125
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2019-04-19 10:43:15
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https://math.stackexchange.com/questions/2560234/solve-the-integral-int-frac1-sqrtxyz-1dx/2560345
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Solve the Integral: $\int{\frac{1}{\sqrt{x+y+z} + 1}}dx$
I am trying to solve the following integral but I am not sure where to begin.
$$\int{\frac{1}{\sqrt{x+y+z} + 1}}dx$$
I tried substituting $u = \sqrt{x+y+z}$ but I keep getting stuck.
How do I proceed?
• Does your integral make sense? – Naive Dec 10 '17 at 17:02
• @Naive when I do the substitution? – Omari Celestine Dec 10 '17 at 17:04
• I made a mistake but I updated the question. The original equation is a triple integral but I am not trying to determine the indefinite integral of the equation with respects to $x$. – Omari Celestine Dec 10 '17 at 17:07
• What does not work with your substitutution? It is a good sub – Shashi Dec 10 '17 at 17:15
• $\frac{2u}{u+1}=2-\frac 2{u+1}$ and you integrate with a $\ln$. – zwim Dec 10 '17 at 17:19
$$\int\frac{1}{\sqrt{x+y+z}+1}\rm dx$$ Let $u={x+y+z}$
And $\rm du=\rm dx$
Your integral becomes $$\int \frac{1}{\sqrt{u}+1}\rm du$$
Let $w=\sqrt{u}+1,\;\;\;\rm dw=\frac{1}{2\sqrt{u}}\rm du=\frac{1}{2(w-1)}\rm du$
$$\int \frac{1}{\sqrt{u}+1}\rm du=2\int\frac{w-1}{w}dw=2(w-\ln|w|)=2(\sqrt{u}+1-\ln|\sqrt{u}+1|)=2(\sqrt{x+y+z}+1-\ln|\sqrt{x+y+z}+1|)$$
So I figured where I went wrong and thanks to those who commented I did the following:
$$u = \sqrt{x+y+z} \\ du = \frac{1}{2}(x+y+z)^{-\frac{1}{2}}(1)dx = \frac{1}{2}\frac{1}{\sqrt{(x+y+z)}}dx = \frac{1}{2}\frac{1}{u}dx = \frac{1}{2u}dx \\ dx = 2u\ du$$
Therefore:
$$\int\frac{1}{\sqrt{x+y+z}+1}dx = \int\frac{2u}{u+1} du = \int(2-\frac{2}{u+1})du \\ = \int2\ du - 2\int\frac{1}{u+1}du \\ = 2u-2\ \ln{(u+1)} \\ = 2\sqrt{x+y+z}-2\ln(\sqrt{x+y+z}+1)$$
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2021-05-16 10:17:43
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https://labs.tib.eu/arxiv/?author=G.C.%20Anupama
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• ### Broad-line type Ic Supernova SN 2014ad(1801.07046)
Jan. 22, 2018 astro-ph.HE
We present optical and ultraviolet photometry, and low resolution optical spectroscopy of the broad-line type Ic supernova SN 2014ad in the galaxy PGC 37625 (Mrk 1309), covering the evolution of the supernova during $-$5 to +87 d with respect to the date of maximum in $B$-band. A late phase spectrum obtained at +340 d is also presented. With an absolute $V$ band magnitude at peak of $M_{V}$ = $-$18.86 $\pm$ 0.23 mag, SN 2014ad is fainter than Gamma Ray Burst (GRB) associated supernovae, and brighter than most of the normal and broad-line type Ic supernovae without an associated GRB. The spectral evolution indicates the expansion velocity of the ejecta, as measured using the Si\,{\sc ii} line, to be as high as $\sim$ 33500 km\,s$^{-1}$ around maximum, while during the post-maximum phase it settles down at $\sim$ 15000 km\,s$^{-1}$. The expansion velocity of SN 2014ad is higher than all other well observed broad-line type Ic supernovae except the GRB associated SN 2010bh. The explosion parameters, determined by applying the Arnett's analytical light curve model to the observed bolometric light curve, indicate that it was an energetic explosion with a kinetic energy of $\sim$ (1 $\pm$ 0.3)$\times$10$^{52}$ ergs, a total ejected mass of $\sim$ (3.3 $\pm$ 0.8) M$_\odot$, and $\sim$ 0.24 M$_\odot$ of $^{56}$Ni was synthesized in the explosion. The metallicity of the host galaxy near the supernova region is estimated to be $\sim$ 0.5 Z$_\odot$.
• The TMT Detailed Science Case describes the transformational science that the Thirty Meter Telescope will enable. Planned to begin science operations in 2024, TMT will open up opportunities for revolutionary discoveries in essentially every field of astronomy, astrophysics and cosmology, seeing much fainter objects much more clearly than existing telescopes. Per this capability, TMT's science agenda fills all of space and time, from nearby comets and asteroids, to exoplanets, to the most distant galaxies, and all the way back to the very first sources of light in the Universe. More than 150 astronomers from within the TMT partnership and beyond offered input in compiling the new 2015 Detailed Science Case. The contributing astronomers represent the entire TMT partnership, including the California Institute of Technology (Caltech), the Indian Institute of Astrophysics (IIA), the National Astronomical Observatories of the Chinese Academy of Sciences (NAOC), the National Astronomical Observatory of Japan (NAOJ), the University of California, the Association of Canadian Universities for Research in Astronomy (ACURA) and US associate partner, the Association of Universities for Research in Astronomy (AURA).
• ### iPTF13bvn: The First Evidence of a Binary Progenitor for a Type Ib Supernova(1403.7288)
July 30, 2014 astro-ph.SR
The recent detection in archival HST images of an object at the the location of supernova (SN) iPTF13bvn may represent the first direct evidence of the progenitor of a Type Ib SN. The object's photometry was found to be compatible with a Wolf-Rayet pre-SN star mass of ~11 Msun. However, based on hydrodynamical models we show that the progenitor had a pre-SN mass of ~3.5 Msun and that it could not be larger than ~8 Msun. We propose an interacting binary system as the SN progenitor and perform evolutionary calculations that are able to self-consistently explain the light-curve shape, the absence of hydrogen, and the pre-SN photometry. We further discuss the range of allowed binary systems and predict that the remaining companion is a luminous O-type star of significantly lower flux in the optical than the pre-SN object. A future detection of such star may be possible and would provide the first robust identification of a progenitor system for a Type Ib SN.
• ### Properties of Newly Formed Dust Grains in The Luminous Type IIn Supernova 2010jl(1308.0406)
Aug. 2, 2013 astro-ph.SR
Supernovae (SNe) have been proposed to be the main production sites of dust grains in the Universe. Our knowledge on their importance to dust production is, however, limited by observationally poor constraints on the nature and amount of dust particles produced by individual SNe. In this paper, we present a spectrum covering optical through near-Infrared (NIR) light of the luminous Type IIn supernova (SN IIn) 2010jl around one and half years after the explosion. This unique data set reveals multiple signatures of newly formed dust particles. The NIR portion of the spectrum provides a rare example where thermal emission from newly formed hot dust grains is clearly detected. We determine the main population of the dust species to be carbon grains at a temperature of ~1,350 - 1,450K at this epoch. The mass of the dust grains is derived to be ~(7.5 - 8.5) x 10^{-4} Msun. Hydrogen emission lines show wavelength-dependent absorption, which provides a good estimate on the typical size of the newly formed dust grains (~0.1 micron, and most likely <~0.01 micron). We attribute the dust grains to have been formed in a dense cooling shell as a result of a strong SN-circumstellar media (CSM) interaction. The dust grains occupy ~10% of the emitting volume, suggesting an inhomogeneous, clumpy structure. The average CSM density is required to be >~3 x 10^{7} cm^{-3}, corresponding to a mass loss rate of >~0.02 Msun yr^{-1} (for a mass loss wind velocity of ~100 km s^{-1}). This strongly supports a scenario that SN 2010jl and probably other luminous SNe IIn are powered by strong interactions within very dense CSM, perhaps created by Luminous Blue Variable (LBV)-like eruptions within the last century before the explosion.
• ### SDSS J123813.73-033933.0, a cataclysmic variable evolved beyond the period minimum(1001.2599)
Jan. 21, 2010 astro-ph.SR
We present infrared JHK photometry of the cataclysmic variable SDSS J123813.73-033933.0 (SDSS1238)and analyze it along with optical spectroscopy, demonstrating that the binary system is most probably comprised of a massive white dwarf with Teff=12000+/-1000 K and a brown dwarf of spectral type L4. The inferred system parameters suggest that this system may have evolved beyond the orbital period minimum and is a bounce-back system. SDSS1238 stands out among CVs by exhibiting the cyclical variability (brightenings). These are not related to specific orbital phases of the binary system and are fainter than dwarf novae outbursts, that usually occur on longer timescales. This phenomenon has not been observed extensively and, thus, is poor understood. The new time-resolved, multi-longitude photometric observations of SDSS1238 allowed us to observe two consecutive brightenings and to determine their recurrence time. The period analysis of all observed brightenings during 2007 suggests a typical timescale that is close to a period of ~9.3 hours. However, the brightenings modulation is not strictly periodic, possibly maintaining coherence only on timescales of several weeks. The characteristic variability with double orbital frequency that clearly shows up during brightenings is also analyzed. The Doppler mapping of the system shows the permanent presence of a spiral arm pattern in the accretion disk. A simple model is presented to demonstrate that spiral arms in the velocity map appear at the location and phase corresponding to the 2:1 resonance radius and constitute themselves as a double-humped light curves. The long-term and short-term variability of this CV is discussed together with the spiral arm structure of an accretion disk in the context of observational effects taking place in bounce-back systems.
• ### The broad line type Ic supernova SN 2007ru: Adding to the diversity of type Ic supernovae(0805.3201)
March 9, 2009 astro-ph
Photometric and spectral evolution of the Type Ic supernova SN 2007ru until around 210 days after maximum are presented. The spectra show broad spectral features due to very high expansion velocity, normally seen in hypernovae. The photospheric velocity is higher than other normal Type Ic supernovae. It is lower than SN 1998bw at $\sim$ 8 days after the explosion, but is comparable at later epochs. The light curve evolution of SN 2007ru indicates a fast rise time of 8$\pm$3 days to $B$ band maximum and post-maximum decline more rapid than other broad-line Type Ic supernovae. With an absolute $V$ magnitude of -19.06, SN 2007ru is comparable in brightness with SN 1998bw and lies at the brighter end of the observed Type Ic supernovae. The ejected mass of \Nifs is estimated to be $\sim0.4\Msun$. The fast rise and decline of the light curve and the high expansion velocity suggest that SN 2007ru is an explosion with a high kinetic energy/ejecta mass ratio ($E_{\rm K}/M_{\rm {ej}}$). This adds to the diversity of Type Ic supernovae. Although the early phase spectra are most similar to those of broad-line SN 2003jd, the [OI] line profile in the nebular spectrum of SN 2007ru shows the singly-peaked profile, in contrast to the doubly-peaked profile in SN 2003jd. The singly-peaked profile, together with the high luminosity and the high expansion velocity, may suggest that SN 2007ru could be an aspherical explosion viewed from the polar direction. Estimated oxygen abundance 12 + log(O/H) of $\sim$8.8 indicates that SN 2007ru occurred in a region with nearly solar metallicity.
• ### Type Ib Supernova 2008D associated with the Luminous X-ray Transient 080109: An Energetic Explosion of a Massive Helium Star(0807.1674)
Feb. 18, 2009 astro-ph
We present a theoretical model for supernova (SN) 2008D associated with the luminous X-ray transient 080109. The bolometric light curve and optical spectra of the SN are modelled based on the progenitor models and the explosion models obtained from hydrodynamic/nucleosynthetic calculations. We find that SN 2008D is a more energetic explosion than normal core-collapse supernovae, with an ejecta mass of Mej = 5.3 +- 1.0 Msun and a kinetic energy of E = 6.0 +- 2.5 x 10^{51} erg. The progenitor star of the SN has a 6-8 Msun He core with essentially no H envelope (< 5 x 10^{-4} Msun) prior to the explosion. The main-sequence mass of the progenitor is estimated to be Mms =20-25 Msun, with additional systematic uncertainties due to convection, mass loss, rotation, and binary effects. These properties are intermediate between those of normal SNe and hypernovae associated with gamma-ray bursts. The mass of the central remnant is estimated as 1.6 - 1.8 Msun, which is near the boundary between neutron star and black hole formation.
• ### The evolution of the peculiar Type Ia supernova SN 2005hk over 400 days(0710.3636)
Feb. 25, 2008 astro-ph
$UBVRI$ photometry and medium resolution optical spectroscopy of peculiar Type Ia supernova SN 2005hk are presented and analysed, covering the pre-maximum phase to around 400 days after explosion. The supernova is found to be underluminous compared to "normal" Type Ia supernovae. The photometric and spectroscopic evolution of SN 2005hk is remarkably similar to the peculiar Type Ia event SN 2002cx. The expansion velocity of the supernova ejecta is found to be lower than normal Type Ia events. The spectra obtained $\gsim 200$ days since explosion do not show the presence of forbidden [\ion{Fe}{ii}], [\ion{Fe}{iii}] and [\ion{Co}{iii}] lines, but are dominated by narrow, permitted \ion{Fe}{ii}, NIR \ion{Ca}{ii} and \ion{Na}{i} lines with P-Cygni profiles. Thermonuclear explosion model with Chandrasekhar mass ejecta and a kinetic energy smaller ($\KE = 0.3 \times 10^{51} {\rm ergs}$) than that of canonical Type Ia supernovae is found to well explain the observed bolometric light curve. The mass of \Nifs synthesized in this explosion is $0.18 \Msun$. The early spectra are successfully modeled with this less energetic model with some modifications of the abundance distribution. The late spectrum is explained as a combination of a photospheric component and a nebular component.
• ### The Peculiar Type Ia Supernova 2005hk(astro-ph/0611354)
Nov. 10, 2006 astro-ph
We present a preliminary analysis of an extensive set of optical observations of the Type Ia SN 2005hk. We show that the evolution of SN 2005hk closely follows that of the peculiar SN 2002cx. SN 2005hk is more luminous than SN 2002cx, while still under-luminous compared to normal Type Ia supernovae. The spectrum at 9 days before maximum is dominated by conspicuous Fe III and Ni III lines, and the Si II 6355 line is also clearly visible. All these features have low velocity (~6000 km/s). The near maximum spectra show lines of Si II, S II, Fe II, Fe III, as well as other intermediate mass and iron group elements. Analysis with the code for synthetic spectra SYNOW indicates that all these spectral lines have similar velocities.
• ### GRB 051028: an intrinsically faint GRB at high redshift?(astro-ph/0609654)
Sept. 24, 2006 astro-ph
We present multiwavelength observations of the gamma-ray burst GRB 051028 detected by HETE-2 in order to derive its afterglow emission parameters and to determine the reason for its optical faintness when compared to other events. Observations were taken in the optical (2.0m Himalayan Chandra Telescope, 1.34m Tautenburg, 4.2m William Herschel Telescope) and in X-rays (Swift/XRT) between 2.7 hours and 10 days after the onset of the event. The data can be interpreted by collimated emission in a jet with a typical value of $p$ = 2.4 which is moving in an homogeneous interstellar medium and with a cooling frequency nu_{c} still above the X-rays at 0.5 days after the burst onset. GRB 051028 can be classified as a gray'' or potentially dark'' GRB. On the basis of the combined optical and Swift/XRT data, we conclude that the reason for the optical dimness is not extra absorption in the host galaxy, but rather the GRB taking place at high-redshift.We also notice the very striking similarity with the optical lightcurve of GRB 050730, a burst with a spectroscopic redshift of 3.967, although GRB 051028 is about 3 mag fainter. We suggest that the bump could be explained by multiple energy injection episodes and that the burst is intrinsically faint when compared to the average afterglows detected since 1997. The non-detection of the host galaxy down to R = 25.1 is also consistent with the burst arising at high redshift, compatible with the published pseudo-z of 3.7 +/- 1.8.
• ### Post-Outburst Phase of McNeil's Nebula (V1647 Orionis)(astro-ph/0602044)
Feb. 2, 2006 astro-ph
We present a detailed study of the post-outburst phase of McNeil's nebula (V1647 Ori) using optical B,V,R,I and NIR J,H,K photometric and low resolution optical spectroscopic observations. The observations were carried out with the HFOSC, NIRCAM, TIRCAM and NICMOS cameras on the 2m HCT and 1.2m PRL telescopes during the period 2004 Feb-2005 Dec. The optical/NIR observations show a general decline in brightness of the exciting source of McNeil's nebula (V1647 Ori). Our recent optical images show that V1647 Ori has faded by more than 3 mags since Feb 2004. The optical/NIR photometric data also show a significant variation in the mags (Delta V = 0.78 mag, Delta R = 0.44 mag, Delta I = 0.21 mag, Delta J = 0.24 mag and Delta H = 0.20 mag) of V1647 Ori within a period of one month, which is possibly undergoing a phase similar to eruptive variables, like EXors or FUors. The optical spectra show a few features such as strong Halpha emission with blue-shifted absorption and the CaII IR triplet (8498A, 8542A and 8662A) in emission. As compared to the period just after outburst, there is a decrease in the depth and extent of the blue-shifted absorption component, indicating a weakening in the powerful stellar wind. The presence of the CaII IR triplet in emission confirms that V1647 Ori is a PMS star. The long-term, post-outburst photometric observations of V1647 Ori suggest an EXor, rather than an FUor event. An optical/IR comparison of the region surrounding McNeil's nebula shows that the optical nebula is more widely and predominantly extended to the north, whereas the IR nebula is relatively confined (dia ~ 60 arcsec), but definitely extended, to the south, too.
• ### The Unique Type Ib Supernova 2005bf: A WN Star Explosion Model for Peculiar Light Curves and Spectra(astro-ph/0509557)
Oct. 28, 2005 astro-ph
Observations and modeling for the light curve (LC) and spectra of supernova (SN) 2005bf are reported. This SN showed unique features: the LC had two maxima, and declined rapidly after the second maximum, while the spectra showed strengthening He lines whose velocity increased with time. The double-peaked LC can be reproduced by a double-peaked $^{56}$Ni distribution, with most $^{56}$Ni at low velocity and a small amount at high velocity. The rapid post-maximum decline requires a large fraction of the $\gamma$-rays to escape from the $^{56}$Ni-dominated region, possibly because of low-density holes''. The presence of Balmer lines in the spectrum suggests that the He layer of the progenitor was substantially intact. Increasing $\gamma$-ray deposition in the He layer due to enhanced $\gamma$-ray escape from the $^{56}$Ni-dominated region may explain both the delayed strengthening and the increasing velocity of the He lines. The SN has massive ejecta ($\sim6-7\Msun$), normal kinetic energy ($\sim 1.0-1.5\times 10^{51}$ ergs), high peak bolometric luminosity ($\sim 5\times 10^{42}$ erg s$^{-1}$) for an epoch as late as $\sim$ 40 days, and a large $^{56}$Ni mass ($\sim0.32\Msun$). These properties, and the presence of a small amount of H suggest that the progenitor was initially massive (M$\sim 25-30 \Msun$) and had lost most of its H envelope, and was possibly a WN star. The double-peaked $^{56}$Ni distribution suggests that the explosion may have formed jets that did not reach the He layer. The properties of SN 2005bf resemble those of the explosion of Cassiopeia A.
• ### Radio, millimeter and optical monitoring of GRB030329 afterglow: Constraining the double jet model(astro-ph/0506169)
June 8, 2005 astro-ph
We present radio, millimeter and optical observations of the afterglow of GRB030329. UBVR_{C}I_{C} photometry is presented for a period of 3 hours to 34 days after the burst. Radio monitoring at 1280 MHz has been carried out using the GMRT for more than a year. Simultaneous millimeter observations at 90 GHz and 230 GHz have been obtained from the Swedish-ESO Submillimeter Telescope (SEST) and the IRAM-PdB interferometer over more than a month following the burst. We use these data to constrain the double jet model proposed by Berger et al. (2003) for this afterglow. We also examine whether instead of the two jets being simultaneously present, the wider jet could result from the initially narrow jet, due to a fresh supply of energy from the central engine after the jet break''.
• ### Type Ia supernova SN 2003du: optical observations(astro-ph/0409494)
Sept. 21, 2004 astro-ph
UBVRI photometry and optical spectra of type Ia supernova SN 2003du obtained at the Indian Astronomical Observatory for nearly a year since discovery are presented. The apparent magnitude at maximum was B=13.53 +/- 0.02 mag, and the colour (B-V) = -0.08 +/- 0.03 mag. The luminosity decline rate, Delta(m_{15}(B)) = 1.04 +/- 0.04 mag indicates an absolute B magnitude at maximum of M_B = -19.34 +/- 0.3 mag and the distance modulus to the parent galaxy as mu=32.89 +/- 0.4.The light curve shapes are similar, though not identical, to those of SNe 1998bu and 1990N, both of which had luminosity decline rates similar to that of SN 2003du and occurred in spiral galaxies. The peak bolometric luminosity indicates that 0.9 Msun mass of 56Ni was ejected by the supernova. The spectral evolution and the evolution of the Si II and Ca II absorption velocities closely follows that of SN 1998bu, and in general, is within the scatter of the velocities observed in normal type Ia supernovae. The spectroscopic and photometric behaviour of SN 2003du is quite typical for SNe Ia in spirals. A high velocity absorption component in the Ca II (H & K) and IR-triplet features, with absorption velocities of ~20,000 km/s and ~22,000 km/s respectively, is detected in the pre-maximum spectra of days -11 and -7.
• ### Early optical and millimeter observations of GRB 030226 afterglow(astro-ph/0312487)
Jan. 13, 2004 astro-ph
The CCD magnitudes in Johnson $UBV$ and Cousins $RI$ photometric passbands for the afterglow of the long duration GRB 030226 are presented. Upper limits of a few mJy to millimeter wave emission at the location of optical are obtained over the first two weeks. The optical data presented here, in combination with other published data on this afterglow, show an early $R$ band flux decay slope of 0.77$\pm$0.04, steepening to 2.05$\pm$0.04 about 0.65$\pm$0.03 day after the burst. Interpreted as the jet break'', this indicates a half opening angle of $\sim 3.2$ degree for the initial ejection, for an assumed ambient density of $\sim 1 {\rm cm}^{-3}$. Broadband spectra show no appreciable evolution during the observations, and indicate the presence of synchrotron cooling frequency $\nu_c$ near the upper edge of the optical band. From the broadband spectra we derive an electron energy distribution index $p = 2.07\pm0.06$ and an intrinsic extinction $E(B - V)\sim0.17$. Millimeter upper limits are consistent with these derived parameters.
• ### Optical observations of the bright long duration peculiar GRB 021004 afterglow(astro-ph/0211108)
Oct. 17, 2003 astro-ph
The CCD magnitudes in Johnson $B,V$ and Cousins $R$ and $I$ photometric passbands are determined for the bright long duration GRB 021004 afterglow from 2002 October 4 to 16 starting $\sim$ 3 hours after the $\gamma-$ray burst. Light curves of the afterglow emission in $B$,$V$,$R$ and $I$ passbands are obtained by combining these measurements with other published data. The earliest optical emission appears to originate in a revese shock. Flux decay of the afterglow shows a very uncommon variation relative to other well-observed GRBs. Rapid light variations, especially during early times ($\Delta t < 2$ days) is superposed on an underlying broken power law decay typical of a jetted afterglow. The flux decay constants at early and late times derived from least square fits to the light curve are $0.99\pm0.05$ and $2.0\pm0.2$ respectively, with a jet break at around 7 day. Comparison with a standard fireball model indicates a total extinction of $E(B-V)=0.20$ mag in the direction of the burst. Our low-resolution spectra corrected for this extinction provide a spectral slope $\beta = 0.6\pm0.02$. This value and the flux decay constants agree well with the electron energy index $p\sim 2.27$ used in the model. The derived jet opening angle of about $7^{\circ}$ implies a total emitted gamma-ray energy $E_{\gamma} = 3.5\times10^{50}$ erg at a cosmological distance of about 20 Gpc. Multiwavelength observations indicate association of this GRB with a star forming region, supporting the case for collapsar origin of long duration GRBs.
• ### Optical afterglow of the not so dark GRB 021211(astro-ph/0304481)
Oct. 5, 2003 astro-ph
We determine Johnson $B,V$ and Cousins $R,I$ photometric CCD magnitudes for the afterglow of GRB 021211 during the first night after the GRB trigger. The afterglow was very faint and would have been probably missed if no prompt observation had been conducted. A fraction of the so-called dark'' GRBs may thus be just optically dim'' and require very deep imaging to be detected. The early-time optical light curve reported by other observers shows prompt emission with properties similar to that of GRB 990123. Following this, the afterglow emission from $\sim 11$ min to $\sim 33$ days after the burst is characterized by an overall power-law decay with a slope $1.1\pm0.02$ in the $R$ passband. We derive the value of spectral index in the optical to near-IR region to be 0.6$\pm$0.2 during 0.13 to 0.8 day after the burst. The flux decay constant and the spectral slope indicate that optical observations within a day after the burst lies between cooling frequency and synchrotron maximum frequency.
• ### SN 2002ap, the hypernova of class Ic(astro-ph/0304482)
April 27, 2003 astro-ph
The supernova SN 2002ap was discovered in the outer regions of the nearby spiral M74 on January 29.4 UT. Early photometric and spectroscopic observations indicate the supernova belongs to the class of Ic hypernova. Late time (After JD 2452500) light curve decay slopes are similar to that of the hypernovae SN 1997ef and SN 1998bw. We present here the $BVRI$ photometric light curves and colour evolutions of SN 2002ap to investigate the late time nature of the light curve.
• ### Optical Photometry of the GRB 010222 Afterglow(astro-ph/0104363)
July 24, 2001 astro-ph
The optical afterglow of GRB 010222 was observed using the recently installed 2-m telescope at the Indian Astronomical Observatory, Hanle, and the telescopes at the Vainu Bappu Observatory, Kavalur, beginning ~ 0.6 day after the detection of the event. The results based on these photometric observations combined with others reported in the literature are presented in this paper. The R band light curve shows an initial decline of intensities proportional to t^{-0.542} which steepens, after 10.3 hours, to t^{-1.263}. Following the model of collimated outflow, the early break in the light curve implies a very narrow beam angle (~ 2-3 deg). The two decay rates are consistent with the standard jet model in a uniform density ambient medium, but require a hard spectrum of electron power density with p ~ 1.5. The R band light between 14 and 17 hours since outburst departs from the power law fit by 0.1 mag and shows some evidence for fluctuations over timescales of an hour in the observer's frame. Such deviations are expected due to density inhomogeneities if the ambient medium is similar to the local interstellar medium. GRB 010222 is thus an example of a highly collimated outflow with a hard spectrum of electron energy distribution in normal interstellar environment.
• ### Short term optical variability in broad absorption line QSOs(astro-ph/9712294)
Dec. 22, 1997 astro-ph
We present the first results from a pilot programme to monitor the short term optical variability in Broad Absorption Line system QSOs. Intra-night optical variations of ~5% were detected on time scales of ~ one hour in QSOs 0846+156 and 0856+172. Further, the mean magnitude level decreased in the two objects by ~0.05 and ~0.15 magnitude respectively during the period of our observations. The observed light curves are quite similar to those previously seem in the flat spectrum radio-loud sources, especially the BL Lacertae objects, and can provide important constraints for the origin of microvariability, and also a possible evolutionary link between the radio-loud and the radio-quiet QSOs.
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2021-04-15 09:03:21
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http://mathoverflow.net/questions/97939/covering-disks-with-smaller-disks?sort=votes
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# covering disks with smaller disks
How many disks with radius 1/2 are needed to cover a disk with radius 1? It certainly cannot be done with less than 5 small disks, and some non-rigorous drawings of mine suggest it can be done with 7 small disks. Can it be done with less?
Slightly more generally, can anybody point me in the direction of any work on covering bounded bodies with disks? Am i right in thinking that covering disks with smaller disks is the most inefficient kind of covering?
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I think you are asking for covering a circle with disks, the distinction being that a circle is a 1D curve, whereas a disk is the 2D region bounded by a circle. – Joseph O'Rourke May 25 '12 at 13:26
You cannot cover a disk with $r=1$ with five disks with radius $r=0.5$. It seems that to use five disks requires that their radius be around 0.609383. (See mathworld.wolfram.com/FiveDisksProblem.html) – Joel Reyes Noche May 25 '12 at 13:40
Even if it's not literally a packing problem, it's a natural enough variation that the tag is reasonable. Even Conway and Sloane's Sphere Packings, Lattices, and Groups [a.k.a. SPLAG, including by the authors] has a few sections on sphere covering. – Noam D. Elkies May 25 '12 at 13:41
After a bit of further thought: in this case it's easy to show that six discs do not suffice. Consider the following seven points: the vertices of a hexagon inscribed in the circle, together with its center [for either referent of "its"]. One of the small disks would have to cover two of them, and thus be on the midpoint of one of the $12$ unit segments formed by our seven points. Now rotate the hexagon to get a contradiction. – Noam D. Elkies May 25 '12 at 14:10
Sorry, what I meant was that one of the seven centers must be included among those 12 points, none of which is the center of the original disk. Rotations yield uncountably many disjoint sets of 12 points with the same property, which gives the desired contradiction. – Noam D. Elkies May 25 '12 at 18:47
You'll need seven disks with $r=0.5$ to cover a disk with $r=1$. See http://mathworld.wolfram.com/DiskCoveringProblem.html.
-
Some nice pictures can be found here: www2.stetson.edu/~efriedma/circovcir – Joel Reyes Noche May 25 '12 at 13:48
Great, thanks for the link, looking forward to going away and reading up on this problem. – Oliver Roche-Newton May 25 '12 at 13:49
For an elementary proof, see my comment above (for the question itself, showing six do not suffice), together with the picture Joseph O'Rourke posted (showing a seven-disc covering). – Noam D. Elkies May 25 '12 at 14:13
Just to help think about the problem, here is the natural cover by seven $\frac{1}{2}$ disks:
-
Indeed this "natural cover" is also unique up to rotations. One disk must cover the center, and thus have at most one point of intersection with the perimeter. Each of the other six discs can cover at most $1/6$ of the perimeter, so together the only way the entire perimeter can be covered is as shown. Now the remaining disc must be centered to cover the unaccounted-for area (colored light green in Joseph O'Rourke's graphic), QED. – Noam D. Elkies May 25 '12 at 18:52
@Noam: Nice uniqueness proof! – Joseph O'Rourke May 25 '12 at 23:40
A more general version of this question is discussed (and answered) in this paper by Dumitrescu and Jiang
-
+1, very nice ! – Joel Reyes Noche May 25 '12 at 13:49
Erich Friedman's packing center claims that you can't cover with 6 disks, and that this was proved by Károly Bezdek in 1979. If you want a more exact reference, ask Erich Friedman in email.
-
I think what Bezdek proved is that six disks of radius $r \approx 1.798$ suffice. – Joseph O'Rourke May 25 '12 at 17:59
No, what he proved is that a disk of radius $\approx 1.798$ can be covered by six unit disks, but no larger disk can be covered. – Zsbán Ambrus May 25 '12 at 18:28
In the context of the packing pages, "proved" refers to proving optimality, whereas just proving that a packing or covering works is listed as "found". For context, see www2.stetson.edu/~efriedma/cirinsqu where finding the smallest square with 9 disks is trivial, but proving it optimal is difficult, so whoever has first proved it is listed. – Zsbán Ambrus May 25 '12 at 18:30
@Zsbán: I stand corrected! – Joseph O'Rourke May 25 '12 at 18:45
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2015-04-25 06:48:05
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https://math.stackexchange.com/questions/3269555/show-the-existence-of-a-locally-compact-perfect-set-in-0-1-which-has-the-fo
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# Show the existence of a locally compact, perfect set in $[0,1]$ which has the following property.
Question: Show that there exists a Locally Compact, Perfect set in $$[0,1]$$ such that ternary expansion of each of its points consists of $$0$$ and $$1$$ only.
I know that the Cantor Set has almost all the above properties except the last one.
$$1.$$ Cantor set is compact and hence locally compact.
$$2.$$ Cantor set is a perfect set.
$$3.$$ Cantor set consists of members whose ternary expansion consists of $$0$$ and $$2$$ only. But here we required $$0$$ and $$1$$ only. How should I proceed? Thanks in advance!
• First off, note that the subset will automatically be bounded, and perfect implies closed, so locally compact is superfluous, as such a set will be compact. I would proceed by choosing one set with only $0$ and $1$ in its ternary expansion, and then checking if it hit the mark. For instance, take the set of all numbers in $[0,1]$ that have only $0$ and $1$ in their ternary expansion. Is it perfect? If it is, then cool, you're done. If it isn't, is it because it isn't closed, or because it has isolated points? (Maybe both?) Can it be fixed in some simple way? Jun 21 '19 at 10:40
• A hint: look for such sets using the 9-ary expansion.
– rts
Jun 22 '19 at 2:59
• Both commenters thanks for the help. Jun 22 '19 at 7:08
Apply $$f:[0,1] \to [0,1]$$ defined by $$f(x)=\frac{x}{2}$$ to the middle third Cantor set $$C$$. $$f[C]$$ is clearly homeomorphic to $$C$$ and fits your requirements:
If $$x\in C$$ is written as $$x=\sum_{n \ge 1} \frac{a_n}{3^n}$$ with all $$a_n \in \{0,2\}$$, then $$f(x) = \sum_{n \ge 1} \frac{b_n}{3^n}$$ with $$b_n = \frac{a_n}{2} \in \{0,1\}$$.
• Sir, what is middle third Cantor set? Did you mean the Cantor set in $[\frac{2}{3},1]$? Jun 22 '19 at 7:10
• @SujitBhattacharyya It's what you call "Cantor set" (in $[0,1]$ constructed, by throwing out the "middle thirds" of all closed intervals in successive stages, hence the name). Topologists call a Cantor set any space/set homeomorphic to it, like $\{0,1\}^{\Bbb N}$ etc. So I mean the Cantor set you are talking about, which has ternary digits $0$ and $2$ and these are halved by $f$ and hence we get ternary digits $0$ and $1$ instead. Jun 22 '19 at 7:30
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2021-10-27 04:42:36
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http://mathhelpforum.com/advanced-statistics/136206-what-type-distribution.html
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Thread: What type of distribution is this?
1. What type of distribution is this?
Karen supplements her income by making a necklace that she sells online. Orders for the necklace occur randomly at a rate of one per every 8 days, on average. She begins each month by making enough necklaces to get her inventory up to 4 necklaces.
X=number of orders that will come in during the month of April (30 days)
a) distribution name
b) parameter value
c) P(X=2)
2. Looks like Poisson dist. to me with $\lambda = \frac{1}{8}$
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2017-06-28 11:09:44
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https://www.biblestudytools.com/commentaries/gills-exposition-of-the-bible/exodus-2-20.html
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Exodus 2:20
Exodus 2:20
And he said unto his daughters, and where is he?
&c.] By the account Reuel's daughters gave of Moses, of his courage and humanity, he was very desirous of seeing him:
why [is it] that ye have left the man?
behind them at the well, and had not brought him along with them; he seemed to be displeased, and chides them, and tacitly suggests that they were rude and ungrateful not to ask a stranger, and one that had been so kind to them, to come with them and refresh himself:
call him, that he may eat bread;
take meat with them, bread being put for all provisions.
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2018-03-20 07:05:40
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https://www.knowledgehut.com/blog/agile/the-evolution-of-kanban-a-study-of-the-principles-practices
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# The Evolution Of Kanban: A Study Of The Principles & Practices
1K
As a part of the overall move towards Agile software development and related disciplines, many people are interested in Kanban. Some are applying it as a part of Agile, others are looking into it as an alternative to Scrum. There is a lot of confusion about this simple yet effective tool, so this article will clear up these misconceptions and provide a quick crash course into the essentials of Kanban.
Where did Kanban come from?
Kanban originated in Japan in the 1940s. People from the Toyota car manufacturer noticed that supermarkets used a simple but effective system based around small physical cards to indicate when something was out of supply or needed to be ordered. They applied this system in their own manufacturing process to help visualise work and control flow.
Kanban became a part of Lean Thinking and the Toyota Production System. It was later popularised amongst software developers in 2010 by David Anderson with his book “Kanban: successful evolutionary change for your business”.
What really is Kanban?
Kanban is a simple but powerful system for visualising and managing work. It is not just related to software development, in fact, it is not specifically related to development of any kind. It can be used in a restaurant, a supermarket or a car factory. It has four principles and five practices.
As opposed to Scrum, which involves revolutionary change, Kanban is about evolutionary change, starting from right where you are now (in fact, that is one of the fundamental principles: start off by changing nothing whatsoever! You won’t find any organisation on earth that will resist that change). It is fundamentally about visualising work and then finding ways to improve the flow of work through a system.
The Four Principles of Kanban
This section will discuss the four fundamental principles behind Kanban. They are all essential to understanding and implementing the practices. Fortunately, they are simple and can fit into any organisation or system.
As mentioned, Kanban is a method for gradually improving an existing workflow or system. So you start with whatever you are doing now - whether simple or complex, big or small. You can use Kanban alongside or on top of any existing framework or system you are using - Scrum, Waterfall (yes Waterfall!), or anything, really. Many people believe they have to choose between Scrum or Kanban - untrue! You can use them together.
Agree to pursue incremental, evolutionary change
As opposed to systems like Scrum or Extreme Programming, that start out by fundamentally changing the structure and behaviour of organisations, Kanban is about evolutionary change, one small increment at a time. This can make it well-suited to cumbersome organisations that resist large risky changes.
Respect the current roles, responsibilities, and titles
Kanban doesn’t start off by renaming people or inventing strange new roles. Everyone gets to keep not only their job but their job title too. This further helps resistance to change.
Encourage acts of leadership at all levels
This is a subtle but very important principle. Improvement is the responsibility of everyone in the organisation, from the CEO down to the receptionist. You might get to keep your current job title and responsibilities, but you also now have an additional responsibility: leadership, which is really about empowerment and improvement.
The Five Practices of Kanban
Unlike Agile, which only has values and principles, Kanban has actual practices that you can employ, starting from day one.
Visualise the workflow
This is the most important step in all of Kanban, and is at the core of the whole system: visualising work. This was trivially easy to do in Toyota car manufacturing plants: the work was cars lying around in various states of semi-construction. For knowledge work like software development, marketing or design, it is much harder to visualise. This can let work accumulate and create huge blockages without anyone being aware.
So visualise the work! This will make it easy to see where work is piling up and where problems and bottlenecks are. The best way to do this is on a physical board, using tokens such as index cards or post-it notes. Putting the work in an electronic tool might seem convenient but when the computer is switched off or minimised, the visualisation of the work is gone. The bigger and more physical, the better! You want to aim for information radiators (that push information out to the organisation) rather than information refrigerators (that preserve information away from people’s eyes).
The most common way to visualise the work is with a Kanban board that has vertical swimlanes to represent states of work. You can then place cards on the board to represent the actual work item. Here is an example of the simplest of Kanban boards, with lanes for To Do, Doing and Done:
Here is another example of a board, this time with some extra lanes, numbers that indicate Work in Progress Limits, and cards marked as being Blocked.
Limit WIP
Limiting WIP (WIP stands for Work In Progress) is another key practice of Kanban. People often think that we want to maximise work in progress - that means everybody is busy and lots is getting done, right? Well, if you have lots of WIP, then everyone will probably be very busy, but that isn’t really valuable. We want work in a Done state, not in a “Doing” state. And the best way to ensure that is to actually REDUCE the amount of Work in Progress!
It might sound counterintuitive, but it is true. Lots of WIP means lots of handoffs, queues, bottlenecks and task-switching. This all means not much getting actually Done. Minimising WIP means you can maximise flow, which maximises the amount of work Done. And that’s the true goal.
Manage Flow
Kanban focuses on reducing WIP in order to maximise flow. If you want lots of work to get to Done but don’t want lots of work in a Doing state, you have to reduce the amount of time it takes for work to go from Doing to Done. This is called Cycle Time. Kanban practitioners have many tools they can use to improve flow and reduce cycle time, and these will often become clear after you have visualised the work. They include eliminating approvals and handoffs, swarming on complex slow tasks, identifying bottlenecks, putting WIP limits on Kanban lanes, reducing wait time between tasks, and reducing queues.
Make Process Policies Explicit
As you implement new rules and policies to improve flow, ensure you make these explicit. If you have a Definition of Done, get people to agree to it and publish it. If you change the rules about approving a document, make sure this is understood and explicitly described. Improvements will fall away and get misused if they are not clearly articulated and understood.
Improve Collaboratively, Evolve Experimentally
Improving is a collaborative team effort, not something randomly forced on people by some arbitrary management edict. It should be implemented by teams in an iterative experimental effort. Teams need to feel they are in an environment where experimentation is encouraged and it is “safe to fail”, or nobody will try making genuine changes.
I hope you have enjoyed this crash course in Kanban - many people are using this system to great effect. If you want to know more, I would recommend the books Kanban by David Anderson, or Kanban in Action by Joakim Sunden and Marcus Hammarberg.
### Leon Tranter
Blog Author
Leon Tranter has 13 years' experience in Information Technology and is passionate about Agile, Scrum, Lean and Kanban. He is a Certified Scrum Master, LeSS Practitioner, and coach in the XSCALE Alliance.“He writes about Agile, Scrum and Lean at Extreme Uncertainty
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2022-01-26 08:51:08
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https://www.askiitians.com/revision-notes/class-10-maths/arithmetic-progressions/
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Revision Notes on Arithmetic Progressions
Arithmetic Progressions
An Arithmetic Progression is a sequence of numbers in which we get each term by adding a particular number to the previous term, except the first term.
Each number in the sequence is known as term.
The fixed number i.e. the difference between each term with its preceding term is known as common difference. It can be positive, negative or zero. It is represented as ‘d’.
Some Examples of Arithmetic Progressions
Common difference
Value of d
Example
d > 0, positive
10
20, 30, 40, 50,…
d < 0, negative
-25
100, 75, 50, 25, 0
d = 0, zero
0
5, 5, 5, 5,..
General form of Arithmetic Progression
Where the first term is ‘a’ and the common difference is ‘d’.
Example
Given sequence is 2, 5, 8, 11, 14,…
Here, a = 2 and d = 3
d = 5 – 2 = 8 – 5 = 11 – 8 = 3
First term is a = 2
Second term is a + d = 2 + 3 = 5
Third term is a + 2d = 2 + 6 = 8 and so on.
Finite or Infinite Arithmetic Progressions
1. Finite Arithmetic Progression
If there are only a limited number of terms in the sequence then it is known as finite Arithmetic Progression.
229, 329, 429, 529, 629
2. Infinite Arithmetic Progression
If there are an infinite number of terms in the sequence then it is known as infinite Arithmetic Progression.
2, 4, 6, 8, 10, 12, 14, 16, 18…..…
The nth term of an Arithmetic Progression
If an is the nth term,a1 is the first term, n is the number of terms in the sequence and d is a common difference then the nth term of an Arithmetic Progression will be
Example
Find the 11th term of the AP: 24, 20, 16,…
Solution
Given a = 24, n = 11, d = 20 – 24 = – 4
an = a + (n - 1)d
a11 = 24 + (11-1) – 4
= 24 + (10) – 4
=24 – 40
= -16
Arithmetic Series
The arithmetic series is the sum of all the terms of the arithmetic sequence.
The arithmetic series is in the form of
{a + (a + d) + (a + 2d) + (a + 3d) + .........}
Sum of first n terms of an Arithmetic series
Sum of the first n terms of the sequence is calculated by
$S_{n}=\frac{n}{2}[2a + (n - 1)d]$
Example
If Radha save some money every month in her piggy bank, then how much money will be there in her piggy bank after 12 months, if the money is in the sequence of 100, 150, 200, 250, ….respectively?
Solution
Given sequence is-
100, 150, 200, 250, …
a = 100 (first term)
d = 50 (common difference)
n = 12 (as we have to calculate money of 12 months)
Now we will put the values in the formula
So the money collected in her bank in 12 months is Rs. 4500.
But when we have finite Arithmetic Progression or we know the last term of the sequence then the sum of all the given terms of the progression will be calculated by
$S_{n}=\frac{n}{2}[a+I]$
Where l = a + (n – 1)d i.e. the last term of the finite Arithmetic Progression.
Remark: The sum of the infinite arithmetic sequence does not exist.
Example
Find the sum of the sequence 38, 36, 34, 32, 30.
Solution
Given
Geometric Progression
A Geometric Progression is a sequence of numbers in which we get each term by multiplying or dividing a particular number to the previous term, except the first term.
The ratio between every term to the next term is constant.
nth term of the Geometric Sequence
an = a1 r n - 1
Sum of Geometric Series
Harmonic Progression
It is the reverse of Arithmetic Progression. If a, a + d, a + 2d …..is an Arithmetic Progression then the harmonic progression is
nth term of Harmonic Progression
Remark: There is no special formula for finding the sum of the harmonic series so we can calculate the sum of the arithmetic series and then take the reciprocal of it which will be the sum of the harmonic series.
The harmonic series is divergent to infinity.
Arithmetic Mean
Arithmetic mean is the average of the two numbers. If a, b and c are in Arithmetic Progression then the arithmetic mean of a and c will be
Some Important Points
Sum of first n positive integers is given by
The difference between the sum of the first n terms and first (n - 1) terms is also the nth term of the given Arithmetic Progression.
an = Sn – Sn-1
Geometric Mean
Geometric mean is the average of two numbers. If a and b are the two numbers then the geometric mean will be
Relationship between A.M. and G.M.
As we have seen above the formula for the Arithmetic mean and the Geometric mean are as follows-
Where a and b are the two given positive numbers.
Let A and G be A.M. and G.M.
So
Now let’s subtract the two means with each other
This shows that A ≥ G
Properties of relationship of A.M and G.M.
Property I: If the Arithmetic Mean and Geometric Mean of two positive numbers a and b are A and G respectively, then
A > G
As A – G > 0
Property II: If A be the Arithmetic Mean and G be the Geometric Mean between two positive numbers a and b, then the quadratic equation whose roots are a, b is
x2 - 2Ax + G2 = 0
Property III: If A is the Arithmetic Means and G be the Geometric Means between two positive numbers, and then the numbers are
A ± √A2 – G2
Harmonic Mean
Harmonic mean is calculated by dividing the number of items with the sum of the reciprocals of all the items.
Harmonic mean is used when there are extreme observations. Like
In this case if we will use arithmetic mean to find the average then we will get the inaccurate average.
Relation between AM, GM and HM
If a and b are two positive real numbers then
AM ≥ GM
GM ≥ HM
The combination of above two inequalities shows that
AM ≥ GM ≥ HM
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2019-02-17 19:43:30
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http://bootmath.com/an-inequality-which-is-supposed-to-be-simple.html
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# An inequality, which is supposed to be simple
Let $x,y,z\in\mathbb{R}$.Let $xy+yz+xz=1$.
Prove:$\displaystyle \frac{x}{\sqrt{x^2+1}}+\frac{y}{\sqrt{y^2+1}}+\frac{z}{\sqrt{z^2+1}}\leq \frac{3}{2}$
#### Solutions Collecting From Web of "An inequality, which is supposed to be simple"
Let $a,b,c\in(0,\pi)$ be such that $x=\cot a$, $y=\cot b$, $z=\cot c$.
Using the addition formula for cotangent, one can show that
$$\cot(a+b+c) = \frac{\cot a\cot b\cot c – \cot a – \cot b – \cot c} {\cot a\cot b + \cot b\cot c + \cot c\cot a – 1} \tag{\ast}$$
By hypothesis, the denominator on the RHS is 0, so (see below) the LHS is $\infty$; since $a,b,c\in (0,\pi)$, this implies $a+b+c\in\{\pi,2\pi\}$.
Case $a+b+c=\pi$: Then at most one of $a,b,c$ is greater than $\frac\pi2$; wlog, $a,b\in(0,\frac\pi2]$. Since cosine is concave on that interval, we have
\begin{align*}
\frac{x}{\sqrt{x^2+1}} + \frac{y}{\sqrt{y^2+1}} + \frac{z}{\sqrt{z^2+1}}
&= \cos a + \cos b + \cos c \\
&\le 2\cos(\tfrac{a+b}{2}) + \cos c \\
&= 2\cos(\tfrac{a+b}{2}) – \cos(a+b) \\
&= 2\cos(\tfrac{a+b}{2}) – 2\cos^2(\tfrac{a+b}{2}) + 1 \\
&= \tfrac32 – 2\big(\cos(\tfrac{a+b}{2}) – \tfrac12\big)^2 \\
&\le \tfrac32
\end{align*}
with equality iff $a=b=\frac\pi3$ (whence $c=\frac\pi3$ also), that is, $x=y=z=\frac1{\sqrt3}$.
Case $a+b+c=2\pi$: Then at most one of $a,b,c$ is less than $\frac\pi2$; wlog, $a,b\in[\frac\pi2,\pi)$. Since cosine is nonpositive in that interval, we have
$$\frac{x}{\sqrt{x^2+1}} + \frac{y}{\sqrt{y^2+1}} + \frac{z}{\sqrt{z^2+1}} = \cos a + \cos b + \cos c \le \cos c \le 1 < \tfrac32$$
Now, one annoying detail about ($\ast$): I noted that the denominator on the RHS is zero, and inferred that the RHS, and hence the LHS, is $\infty$. But what if the numerator on the RHS is also zero? Well, suppose for contradiction that both numerator and denominator are zero. Then $xyz=x+y+z$ and $xy+yz+zx=1$, and so
\begin{align*}
(t-x)(t-y)(t-z)
&= t^3 – (x+y+z)t^2 + (xy+yz+zx)t – xyz \\
&= t^3 – xyzt^2 + t – xyz \\
&= (t^2+1)(t-xyz)
\end{align*}
which is impossible because the first polynomial has three real roots (counting multiplicity) but the last has only one.
Hint: $$x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z)$$
Also, you will need the inequality
$$\sqrt{AB}\le \frac12(A+B)$$
Solution:
$$\frac{x}{\sqrt{x^2+1}}+\frac{y}{\sqrt{y^2+1}}+\frac{z}{\sqrt{z^2+1}} \\= \frac{x}{\sqrt{(x+y)(x+z)}}+\frac{y}{\sqrt{(x+y)(x+z)}} +\frac{z}{\sqrt{(x+y)(x+z)}} \\ \le \frac12\left[ \frac x{x+y} + \frac x{x+z} + \frac y{y+z} + \frac y{x+y} + \frac z{z+y} + \frac z{z+x} \right]=3/2$$
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2018-06-25 16:11:59
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http://ferienhausmenz.de/exponential-population-growth-formula.html
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## Exponential Population Growth Formula
However, if it begins at 100, the population grows by 10% to 110 in the. e = the base of natural logarithms (2. The Problem! Let P(t) be the population This is the original formula we started with. 2032 x to the exponential growth formula in the form y = a(b)x y = a (b) x. 6: What is the algebraic solution to exponential growth? Write the equation and define each. In fact in both cases the rate of growth r of a population (or an investment) per time period is proportional to the size of the population (or the. 2) y = y 0 e k t where y 0 represents the initial state of the system and k > 0 is a constant, called the growth constant. Using the basic formula for exponential growth f ( x) = a ( 1 + r) x we can write the formula, f ( t) = 1. Solved Examples Using Exponential Growth Formula. no = initial size of the population. Exponential Growth - Population Problem. The important concept of exponential growth is that the population growth rate, the number of organisms added in each reproductive generation, is accelerating; that is, it is increasing at a greater and greater rate. billion people in 2020. 1984= 746,388,000 1974=57 Log On. The function is an increasing function; y increases as x increases. Four variables — percent change, time, the amount at the beginning of the time period, and the amount at the end of the time period — play roles in exponential functions. Formula of Exponential Growth. Use this equation to calculate the world population in 1950, 1980, and 2000. Find an explicit formula for the beetle population in week n 0,3 D= 67-3=64 64/8 =8 8,67 8-0=8 Answer=3+8n b. Solution: J-shaped form of population growth is mathematically described by an equation of exponential or geometric increase, which is as follows : d t d N = r N where, d=rate of change t=time N=number of females at a particular time r=biotic potential of each female (N can also be considered as the total population and r as the biotic potential of each individual). We can use P 0 = 100 if our rabbit population starts with 100 rabbits. t = time (number of periods) 2. Exponential Growth & Decay 06/01/09 Bitsy Griffin PH 8. r is relative growth rate in percentage. The quantity decreases slowly after which the rate of change and the rate of growth decreases over a period of time rapidly. P (t) = the amount of some quantity at time t. Here, P0 = 125; r =:16 and we want to flnd t so that A(t) = 400. Meaning, a population of 2 becomes 4 and a population of 4 then becomes 8. Exponential Growth Formula. An example of an exponential function is the growth of bacteria. Jul 31, 2020 · The exponential growth model describes how a population changes if its growth is unlimited. MATH 1142 Section 5. Substituting into the exponential growth formula and solving. t = time (number of periods) 2. r = the decay rate. 02304 or about 2. In 2000 there were 1700 people. Enterprise. Because b = 1 + r > 1, then r = b − 1 > 0. A population of mice is a good example of why populations can increase rapidly. The function y = f ( x) = a e k x represents growth if k > 0 and a > 0. The mathematics of exponential growth govern the prediction of population growth. At 5 years it is. For example, if the population of a growing city takes 10 years to double from 100,000 to 200,000 inhabitants and its growth remains exponential, then in the next 10 years the. Doubling time is given by. 695 by t (r =. I have used the exponential growth formula i. That is, the amount of time it takes for the number of cases to increase from 100 to 200 is the same as the amount of time it takes for the number of cases. As a result, it creates an explosion of the population. Since continuous compounding, the value of the deposited money after three years money is calculated using the above formula as, Final value = Initial value * e Annual growth rate * No. If k k is negative, the amount is decreasing. f (x) = 20000 (1 + 0. Some have exponential growth rates while some may model a different type of function. 69 doubling time, t = 0. 1,2,4,8,16,32,64). The equation for exponential growth can be derived as follows: where, N = population size, N t = population density after time t, N 0 = population density at time zero, r = intrinsic rate of natural increase. The main difference between exponential growth and logistic growth is the factors that affect each type of. However, if it begins at 100, the population grows by 10% to 110 in the. A house is purchased for $400,000. Properties of Exponential Growth Functions. The rate of population growth at any given time can be written:. Population growth is a common example of exponential growth. r is relative growth rate in percentage. For example, suppose a population of cockroaches rises exponentially every year. Formula of Exponential Growth. t = time (number of periods) 2. Exponential growth involves increases starting off as reasonably small, and then dramatically increasing at a faster and faster rate. A population of bacteria doubles every hour. Exponential Growth And Decay. Was this answer helpful?. we can write the formula, To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008. Let's solve this equation for y. 3 Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. The formula of exponential growth is d N d t = r N dNdt=rN where d N d t dNdt is the rate of change in population size, r is the biotic potential and N is the population size. Sep 10, 2021 · What does exponential growth mean? A population might grow by 10% per year; this is a constant rate of growth each year. Compare Linear Growth and Exponential Growth. In our case, we grew from 1 to 2, which means our continuous growth rate was ln (2/1) =. ) In 2012 2012, the population of Samdom For Peace city was 6. Step by step. Lesson 6 : Exponential Growth—U. The function y = f ( x) = a e k x represents growth if k > 0 and a > 0. r = annual growth rate. A population might grow by 10% per year; this is a constant rate of growth each year. Exponential Growth Formula For a Function (With Solved Examples) Just Now Byjus. Exponential Growth & Decay 06/01/09 Bitsy Griffin PH 8. The Exponential Equation is a Standard Model Describing the Growth of a Single Population. Exponential Growth Rapid, unrestricted population growth or reproduction of individuals without constraints (such as unlimited resources) Does not factor in the carrying capacity (K) of the environment (which is why "K" is not included in the formula above) Logistic Growth Population growth is limited due to density-dependent (such as. A population grows according to an exponential growth model, with P 0 = 70 P 0 = 70 and P 1 = 112 P 1 = 112 Complete the recursive formula: P n P n = ×P n−1 × P n - 1 Write an explicit formula for P n P n. exponential. Taking the logarithm base 2 of both sides: log2 (N/No) = t/tD, or t = tDlog2(N/No) = 1 X log2 (1,000,000,000/1) = log2(109) But suppose your calculator doesn't do log base 2. Exponential Growth and Decay: Differential Equations 9. Exponential growth is growth that occurs at a constant rate, such as an investment that grows at an annual 7 percent rate. Formula of Exponential Growth. A(t) =A0ekt A ( t) = A 0 e k t. Exponential Relations. The rate of change decreases over time. So between 9 and 10 days, the bacteria population will be 1000. Four variables — percent change, time, the amount at the beginning of the time period, and the amount at the end of the time period — play roles in exponential functions. The formula for exponential growth of a population is y = Nert. The two types of exponential functions are exponential growth and exponential decay. In exponential growth, the rate of growth is proportional to the quantity present. 11059 million (approx). The rate of change increases over time. 695 by t (r =. P t = P o (1 + r/100) T. Exponential functions have the form f(x) = bx, where b > 0 and b ≠ 1. Using the basic formula for exponential growth. Using an exponential form of population growth, two fruit flies could produce enough offspring to fill the space between the Earth and the Sun in only one. The exponential or geometric growth is common where the resources (food + space) are unlimited. ln(A/Ao) =kt*ln e {log power rule& ln e =1} ln(A/Ao) =kt*1 {divide both sides by t} [ln(A/Ao)]/t = k. Formula to calculate population growth rate. Population growth dN/dt=B-D exponential growth logistic growth dY= amount of change t = time B = birth rate D = death rate N = population size K = carrying capacity r max = maximum per capita growth rate of population temperature coefficient q 10 Primary Productivity calculation mg O 2 /L x 0. Exponential Growth Growth rates are proportional to the present quantity of people, resources, etc. Video summary. The population is growing by about 1. Exponential Growth Formula For a Function (With Solved Examples) Just Now Byjus. Compound growth is a term usually used in finance to describe exponential growth in interest or dividends. The formula of exponential growth is d N d t = r N dNdt=rN where d N d t dNdt is the rate of change in population size, r is the biotic potential and N is the population size. The formula for exponential population growth is dN/ dt = rN. The general formula for exponential growth of a population Population = 4) If the starting population of 5 rabbits grows at 200% each year, how many will there. If k k is negative, the amount is decreasing. An exponential formula that can be used to model the world population growth from 1950 through 2000 is as follows: 𝑓(𝑡)=2,519(1. Population Growth: Example 2 Example The population of Calculand was 700 in the year 2000 and was 3000 in the year 2010. When populations grow rapidly, we often say that the growth is "exponential," meaning that something. You will notice that in these new growth and decay functions, the b value (growth factor) has been replaced either by (1 + r) or by (1 - r). With exponential growth, we began with the differential equation, d d. Popper says "the growth of knowledge. Year Population 1900 76 1910 92 1920 106 1930 123 1940 131 1950 150 1960 179 1970 203 1980 227 1990 250 2000 275 a. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable. India is the second most populous country in the world, with a population in 2008 of about 1. However, if it begins at 100, the population grows by 10% to 110 in the. Exponential growth results in a population increasing by the same percent each year. Step by step. Exponential decay can be thought of as the opposite of. edu [email protected] x (t) = x0 × (1 + r) t Where x (t) is the final population after time t. we can write the formula, To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008. Example 3: In 2001, there were 100 inhabitants in a remote town. Exponential growth is a process that increases quantity over time. http://mathispower4u. The quantity decreases slowly after which the rate of change and the rate of growth decreases over a period of time rapidly. The exponential function appearing in the above formula has a base equal to 1 + r/100. Questions from AIPMT 2006 1. Exponential growth is growth that occurs at a constant rate, such as an investment that grows at an annual 7 percent rate. y = C ( 1 + r) t , where C is the initial amount or number, r is the growth rate (for example, a 2 % growth rate means r = 0. You are asked to solve for the increase in the population of a group of fruit flies given the growth factor. com View All. P t = P o (1 + r/100) T. Example 1: In 2005, there were 180 inhabitants in a remote town. Thus, B (birth rate) = bN (the per capita birth rate “b” multiplied by the number of individuals “N”) and D (death rate) = dN (the per capita death rate “d” multiplied by the number of individuals “N”). A population experiencing exponential growth increases according to the model 0 nt ne= rt where n(t) = population at time t. f(x) = a (1 - r) x. The equation for exponential growth can be derived as follows: where, N = population size, N t = population density after time t, N 0 = population density at time zero, r = intrinsic rate of natural increase. N0= 50, what is Ntafter 5hrs. (b) Find the time required for the population to double. A species of mammal with known doubling time: (a) Using the formula found in Exercise 3, we see that, if the population doubles in 56 days, then the exponential growth rate is r = ln2 56 = 0. 1 Observations about the exponential function In a previous chapter we made an observation about a special property of the function y = f(x) = ex namely, that dy dx = ex = y so that this function satisfies the relationship dy dx = y. P 0 = initial amount at time t = 0. Exponential growth models are often used for real-world situations like interest earned on an investment, human or animal population, bacterial culture growth, etc. Both are influenced by births (b, m x) and by deaths (d, l x). The number of cells doubles in 3 hours, so we have But so. 0134) t To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008. The growth rate in the linear formula is a fixed million increase in population each year whereas the growth rate in the exponential formula is a factor of. Exponential growth involves increases starting off as reasonably small, and then dramatically increasing at a faster and faster rate. 5 hours ago Byjus. where: Pf = future population. In the Warmup Question 1, we solved by writing it in exponential form: This is the basic idea in solving logarithmic equations. Complete the table for the Population Growth Model for a certain country. If k k is positive, the amount is growing. When populations grow rapidly, we often say that the growth is "exponential," meaning that something. of compounding read more is very helpful to calculate the estimated growth when growth occurs exponentially, for example, in biology, where a microorganism increases exponentially. Open the Excel spreadsheet ExpGrowthExcel. When a plant or animal is alive it continually replenishes the carbon in its system. y = C ( 1 + r) t , where C is the initial amount or number, r is the growth rate (for example, a 2 % growth rate means r = 0. d=log, 2/b. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Question 3 Conpt 03 €2 0 Details A population grows according to an exponential growth model, with Po = 40 and Pi = 60 Complete the recursive formula: Pr = Write an explicit formula for Pa Submit Question. In Part A, the bacteria population grows by a factor of $$3$$ every day. Exponential Model of Population Growth Formula ΔN/ Δt = B-D = bN - dN = (b-d)N Change in population over time is equal to Birth rate per capita times population size minus death rate per capita times population size. At the constant birth rate in population through time and is never limited by food or disease, known as exponential growth. In Exponential Growth, the quantity increases very slowly at first, and then rapidly. The first step is identifying the unknown variables and assigning values to them. Algebra 2 common core. 11059 million. Recall that we are studying a population of bacteria undergoing binary fission. An example of an exponential function is the growth of bacteria. A(t) =A0ekt A ( t) = A 0 e k t. If the culture started with 10 bacteria, graph the population as a function of time. ln(A/Ao) =kt*ln e {log power rule& ln e =1} ln(A/Ao) =kt*1 {divide both sides by t} [ln(A/Ao)]/t = k. r is relative growth rate in percentage. Height in mm ex. 2 million and can be modeled by y = 14. r = population growth rate. Connect the lesson to this visualization (from Our World in Data) of Covid-19 case. If k k is negative, the amount is decreasing. View more lessons or practice this subject at https://www. Find the number of years required for the deer population to be 400. Compare to the actual figure and try to explain the discrepancy. If k k is positive, the amount is growing. Exponential growth is a type of exponential function where instead of having a variable in the base of the function, it is in the exponent. One can thus see that growth is exponential with respect to time. 25 billion people in 2013. In other words, y ′ = k y. we can write the formula, To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008. Using the basic formula for exponential growth. Because b = 1 + r > 1, then r = b − 1 > 0. Find an explicit formula for the beetle population in week n 0,3 D= 67-3=64 64/8 =8 8,67 8-0=8 Answer=3+8n b. In exponential growth, the rate of growth is proportional to the quantity present. P n = Box Use your formula to find P 12 P 12 = Box Give all answers accurate to at least one decimal place. com View All. A population grows according to an exponential growth model. Just as in any exponential expression, b is called the base and x is called the exponent. ever been alive!"). The current population is 402. Exponential Decay Formula. 03)t = 1700(0. Exponential Growth Exponential growth describes quantities that grow faster when the quantity is larger. The Grand and Glorious Exponential Growth Page! So as promise, here are some supplementary notes. After all, the more bacteria there are to reproduce, the faster the population grows. P t = P o (1 + r/100) T. Algebra 2 common core. where P ( t) P (t) P ( t) is the population after time t t t, P 0 P_0 P 0 is the original population when t = 0 t=0 t = 0, and k k k is the growth constant. You will notice that in these new growth and decay functions, the b value (growth factor) has been replaced either by (1 + r) or by (1 - r). After 1 day and 24 of these cycles, the population would have increased from 1000 to more than 16 billion. 0 500 1000 1500 2000 2500 Figure 3. 5933)]/37 =. The equation for exponential growth can be derived as follows: where, N = population size, N t = population density after time t, N 0 = population density at time zero, r = intrinsic rate of natural increase. However, if it begins at 100, the population grows by 10% to 110 in the second year. The following is the exponential growth formula. we can write the formula, To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008. Here is a simple example and how it is so powerful. View more lessons or practice this subject at https://www. N0= 45, what is Ntafter 2hrs. Using the basic formula for exponential growth. Questions from AIPMT 2006 1. Spock used for calculating exponential growth is: M = M o e kt. See full list on vedantu. EXPONENTIAL POPULATION GROWTH IN TRIBBLES: Background: In an episode of the original Star Trek series (The Trouble With Tribbles; 1967), a furry little animal called a Tribble is brought on board the U. India is the second most populous country in the world, with a population in 2008 of about 1. Exponential growth results in a population increasing by the same percent each year. Using the exponential model for population growth, nd an estimate for the population of Calculand in 2015. A population of beetles is growing according to a linear growth model. If k k is negative, the amount is decreasing. in the previous video we started thinking about things like population growth rate and how it relates to the birth rate and the death rate within a population and we related that to some of the seemingly complex formulas that you might see on an AP Biology formula sheet now we're going to extend that conversation to discuss some of the other formulas you might see but to realize that they. It seems plausible that the rate of population growth would be proportional to the size of the population. Let us understand the exponential growth formula in detail in the following section. 304% 17) When will New York's population reach 15 million people? 2018 18) Nevada's population in 1990 was 14. Doubling time is given by. 536 = mg carbon fixed. Exponential Growth Formula For a Function (With Solved. 5 hours ago Byjus. The calculation of exponential growth, i. So for example, if the relative growth rate of a. After all, the more bacteria there are to reproduce, the faster the population grows. the form is the same as that of the modified exponential and the same method used for the modified exponential can be used to get the projection of the reciprocal population. Even though this formula is still valid, we are more used to seeing this formula for calculating Exponential Growth: f ( x) = a ( 1 + r) t. Sep 28, 2016 · The birthrate of a population (number of births per year ($ \times $) 100 / number of population) is 20%, and the mortality rate (number of deaths per year ($ \times $) 100/ number of population) is 5%. See full list on vedantu. An exponential equation grows by a constant factor each year, while a linear equation grows by a constant difference each year. The exponential or geometric growth is common where the resources (food + space) are unlimited. A(t) A ( t) is the amount after t t units of time, A0 A 0 is the original amount, k k is the rate of growth or decay, t t is time. x (t) = x0 × (1 + r) t Where x (t) is the final population after time t. Systems that exhibit exponential growth increase according to the mathematical model (6. in conjunction with "Alg & Trig," 5th ed, by Blitzer, similar to section 4. Next, let’s look at the continuous growth formula. If we have a linear discrete dynamical system in function iteration form, (1) x t + 1 = R x t x 0 = a known positive value, then the solution is the exponential. The rate of change increases over time. A Quick Intro to Logarithmic and Exponential Equations. The final component in the formula for exponential growth is the exponent itself. The global population has grown from 1 billion in 1800 to 7. Population and World Population. dN/dt = rN (1 – N/K) This equation says that the change (d) in number of individuals (N) over a change (d) in time (t) equals the rate of increase (r) in number of individuals where population size (N) is a proportion of the carrying capacity (K). For example, a population of 100 rabbits may grow to a population of about 200 rabbits in six months in an uncon ned environment, so if we started with 400 rabbits in the same uncon ned environment, we would expect the population to grow. Example of Exponential Growth. A(t) A ( t) is the amount after t t units of time, A0 A 0 is the original amount, k k is the rate of growth or decay, t t is time. Connect the lesson to this visualization (from Our World in Data) of Covid-19 case. ln(A/Ao) =kt*ln e {log power rule& ln e =1} ln(A/Ao) =kt*1 {divide both sides by t} [ln(A/Ao)]/t = k. The example says their population started out at 40 fruit flies and triples every two days. Meaning of Exponential Growth Formula. Exponential word problems almost always work off the growth / decay formula, A = Pe rt, where "A" is the ending amount of whatever you're dealing with (money, bacteria growing in a petri dish, radioactive decay of an element highlighting your X-ray), "P" is the beginning amount of that same "whatever", "r" is the growth or decay rate, and "t" is time. 02304 or about 2. 1 Exponential Growth and Decay Subsection Exponential Growth. The exponential growth equation, dN/dt = rN works fine to show the growth of the population: starting with one cell, in one hour it's 4, then in two hours rN = 4*4 = 16, in three hours rN = 16*4 = 64 and so on. 24), which is slightly larger than previous estimates [7, 9, 14]. It seems plausible that the rate of population growth would be proportional to the size of the population. r 0$, there is exponential decay. edu [email protected] Some bacteria double every hour. t = 15 years. Step by step. Scroll down the page for more examples and solutions that use the exponential growth and decay formula. So, while exponential growth might not be the perfect model of bacterial growth by binary fission, it is the appropriate model to use given experimental reality. P ( t) = P 0 e k t P (t)=P_0e^ {kt} P ( t) = P 0 e k t. Scroll down the page for more examples and solutions that use the exponential growth and decay formula. a = initial amount. Exponential Growth Function - Population This video explains how to determine an exponential growth function from given information. 03)t = 1700(0. A population of bacteria follows the continuous exponential growth model P(t) = P_{0}e^{kt}, where t is in days. This is remarkably fast growth (see Fig. Was this answer helpful?. The exponential decay formula can be in one of the following forms: f(x) = ab x. The population of a species that grows exponentially over time can be modeled by. As soon as I read "continuously", I should be thinking "continuously-compounded growth formula". Therefore, when presented with a di erential equation of the form y0= ky, we can nd a solution, y= f(t), of the form y= Cektfor some constant C. As a result, it creates an explosion of the population. Sep 10, 2021 · What does exponential growth mean? A population might grow by 10% per year; this is a constant rate of growth each year. Consider a population of bacteria, for instance. Example 1 Exponential Growth. y = C ( 1 + r) t , where C is the initial amount or number, r is the growth rate (for example, a 2 % growth rate means r = 0. The formula Mr. The growth rate r is positive when b > 1. B = population after growth. 1 Exponential Growth and Decay Subsection Introduction. ) In 2012 2012, the population of Samdom For Peace city was 6. In 2000, there were 236,000 people in the city. It is found under Formulas 0, then we have exponential growth and if r < 0 we have exponential decay. The population of a species that grows exponentially over time can be modeled by. Exponential Growth Formula Calculator (Excel Template) Just Now Educba. 1 Exponential Growth and Decay Subsection Exponential Growth. The growth "rate" (r) is determined as b = 1 + r. where: Pf = future population. To do this, we divide 70 by the growth rate (r). The discrete and continuous population growth models described above are similar in four important ways: 1) λ and r are both net measures of an individual's contribution to population growth. Exponential Growth in real world. Generation time (G) is defined as the time (t) per generation (n = number of generations). Exponential decay can be thought of as the opposite of. 0 500 1000 1500 2000 2500 Figure 3. 0141 ~=k {approximately}. Connect the lesson to this visualization (from Our World in Data) of Covid-19 case. Convert the continuous growth formula y = 824. The Exponential Equation is a Standard Model Describing the Growth of a Single Population. Exponential Models of Population Growth The formula for population growth of several species is the same as that for continuously compounded interest. Exponential Growth. With exponential growth, Suppose we have a population, and a certain number You can obtain a formula for N as a function of time analytically (using differential equations), but let's. If the logistics curve is expressed in terms of reciprocal population; i. billion people in 2020. where: Pf = future population. 34% each year 1. B = 60000 (1. In fact in both cases the rate of growth r of a population (or an investment) per time period is proportional to the size of the population (or the. Compare Linear Growth and Exponential Growth. Video summary. y = a ( 1 + r) x. x = number of time intervals passed (days, months, years) y = amount after x time. Use the Learning Network lesson on the recent surge of coronavirus cases in India. The exponential growth equation, dN/dt = rN works fine to show the growth of the population: starting with one cell, in one hour it's 4, then in two hours rN = 4*4 = 16, in three hours rN = 16*4 = 64 and so on. K2 = ln (Pb / Po) / (tb - to) t = tf - tb = # of years projected into the future. Questions from AIPMT 2006 1. 400+ VIEWS. P (t) = the amount of some quantity at time t. Population growth is a common example of exponential growth. 5 f x 3x 6 f x 0 25x 7 f x 1 01x 8 f x 0 033x 9 f x 6 5x 10. The current population is 402. Exponential Relations. Write an exponential growth. Height in mm ex. Exponential growth is a process that increases quantity over time. A population of beetles is growing according to a linear growth model. Sep 10, 2021 · What does exponential growth mean? A population might grow by 10% per year; this is a constant rate of growth each year. = how fast the population is growing. Hence, = and setting we have. The Problem! Let P(t) be the population This is the original formula we started with. y = y 0 e k t. Human population growth is one of the most famous examples of exponential growth because of its archetypical curve. The gradual increase in slope. This is exponential growth so the model is. 5933)]/37 =. Exponential Growth. Note that the y-axis is on a logarithmic scale; "3" corresponds. of years * No. 2% per year. Use this equation to calculate the world population in 1950, 1980, and 2000. In the first column, we have a timestep of 1 year for 30 years. we can write the formula, To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008. The function in Investigation 7. dy dt =ky where y t = y 0 ekt Let y (t) = population of bacteria. Exponential Growth Formula For a Function (With Solved Examples) Just Now Byjus. Solution: J-shaped form of population growth is mathematically described by an equation of exponential or geometric increase, which is as follows : d t d N = r N where, d=rate of change t=time N=number of females at a particular time r=biotic potential of each female (N can also be considered as the total population and r as the biotic potential of each individual). In exponential growth, the rate of growth is proportional to the quantity present. Exponential growth is modeled an exponential equation. The growth rate for Wolffia microscopica may be calculated from its doubling time of 30 hours = 1. Example 1: A population of 32, 000 with a 5 % annual growth rate would be modeled by the equation:. The following formula is used by the calculator above to determine the exponential growth of a value. Exponential Growth Formula. The initial population (week 0) was P0=3, and the population after 8 weeks is P8=67. Suppose a population is modeled by the exponential model dP dt = kP. Just as in any exponential expression, b is called the base and x is called the exponent. The easiest way to capture the idea of a growing population is with a single celled organism, such as a. Meaning of Exponential Growth Formula. In 2000 there were 1700 people. The general rule of thumb is that the exponential growth formula: x(t) = x 0 * (1 + r/100) t is used when there is a quantity with an initial value, x 0 , that changes over time, t, with a constant rate of change , r. Author information: (1)Department of Biological Statistics and Computational Biology, Cornell University, Ithaca, New York 14853 [email protected] Substituting Euler’s number, P (15) = 9. The rapid growth meant to be an “exponential increase”. P ( t) = P 0 e k t P (t)=P_0e^ {kt} P ( t) = P 0 e k t. Exponential growth is modeled an exponential equation. The initial population was 10, so A 0 = 10. Definition 2. See full list on investopedia. 1,2,3,4,5,6,7) but geometric or exponential growth (i. Algebra 2 common core. What is the major cause of increasing the human population? It helps to think in terms of the concept of EROI Energy returned on energy invested - Wikipedia Before the agricultural revolution, humans lived a hunter gatherer lifestyle. the form is the same as that of the modified exponential and the same method used for the modified exponential can be used to get the projection of the reciprocal population. By using the rule of 70 also, we will get the same answer as 35 Years. 7 where e is the base of natural logarithms (e=2. Where x ( t) is the final value after time t. Formula of Exponential Growth. Human population growth is one of the most famous examples of exponential growth because of its archetypical curve. I'm just going to change the letters a little: The is pronounced "P not. Enterprise. Exponential growth is modeled an exponential equation. x (t) = x0 × (1 + r) t Where x (t) is the final population after time t. The current population is 402. ANSWER First question is answered. ( 1 + r) t, taking P = 7000 and A = 3500 and got the answer 35 Years. t = time (number of periods) 2. 0 Tirne Growth rate = 1. r = annual growth rate. Therefore a simple equation (rt =. B = population after growth. The population in 15 years will be 9. Final value = $50,000 * e 10% * 3. It is called exponential growth because it can only be. Exponential Decay Solving an exponential decay problem is very similar to working with population growth. to = initial year (earliest year in the applicable exponential growth period) A projection assuming exponential, or geometric, growth can be calculated using the formula: Pf = Pbe K 2 t. State whether each example is linear or exponential, and write an explicit formula for the sequence that models the growth for each case. However, if it begins at 100, the population grows by 10% to 110 in the. Exponential growth functions are often used to model population growth. Logarithmic equation, exponential form, checking the solution, exponential equation. The general formula for exponential growth of a population Population = 4) If the starting population of 5 rabbits grows at 200% each year, how many will there. 000+ LIKES. Before moving on to the problems below, view the movie clip that illustrates the situation. So, while exponential growth might not be the perfect model of bacterial growth by binary fission, it is the appropriate model to use given experimental reality. For example, if the population were to double each year we would have 2, 4, 8, 16, 32, etc. The next growth we will examine is exponential growth. Compound growth is a term usually used in finance to describe exponential growth in interest or dividends. Example: Number of students in a school increases by 2% each year. When any quantity (rate of oil consumption, human population, your bank account) grows at a fixed rate or percentage each year such as 5% (contrast that to a fixed quantity such as$100) that growth is termed exponential. Determine when New Yorks's population will surpass. Height in mm ex. " The little "o" is a zero for time = 0 when you start. The gradual increase in slope. Exponential decay and exponential growth are used in carbon dating and other real-life applications. At 5 years it is. Exponential growth applies to populations, too -- if a population grows at 7% per year, it, too, will double in 10 years. The function is an increasing function; y increases as x increases. The formula for exponential population growth is dN/ dt = rN. Since continuous compounding, the value of the deposited money after three years money is calculated using the above formula as, Final value = Initial value * e Annual growth rate * No. EXPONENTIAL GROWTH AND DECAY STEPS WITH EXAMPLES. What is the growth pattern in which a population's growth rate slows or stops following a period of exponential growth? answer choices. There are two unknowns in the exponential growth or decay model: the proportionality constant and the initial value In general, then, we need two known measurements of the system to determine these values. The formula for exponential growth of a population is y = Nert. Complete the table for the Population Growth Model for a certain country. To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008:. 6 to get an equation for population size: € N t =N 0e rt eq 1. If k k is positive, the amount is growing. The exponential decay formula can be in one of the following forms: f(x) = ab x. Using the year 1800 as the base year, an explicit formula for the sequence that models the population of New York is P(t) = 300000(1. P kP t 8 7 6 5 4 3 2 1 0. What was the bacteria population at the beginning of the experiment (five hours ago. population growth. Global human population growth amounts to around 83 million annually, or 1. Exponential Growth Formula. in conjunction with "Alg & Trig," 5th ed, by Blitzer, similar to section 4. The formula of exponential growth is d N d t = r N dNdt=rN where d N d t dNdt is the rate of change in population size, r is the biotic potential and N is the population size. Suppose a population is modeled by the exponential model dP dt = kP. Exponential Growth Formula Calculator (Excel Template) Just Now Educba. Growth formula returns the predicted exponential growth rate based on existing values given in excel. Suppose we know that a beetle population starts at 5 and doubles each month. The UN projected population to keep growing, and estimates have put the total population at 8. where P ( t) P (t) P ( t) is the population after time t t t, P 0 P_0 P 0 is the original population when t = 0 t=0 t = 0, and k k k is the growth constant. P (t) = the amount of some quantity at time t. We first seek the number k. After 1 day and 24 of these cycles, the population would have increased from 1000 to more than 16 billion. Hence option C is correct. Nt= 5,000,000 after. Just as in any exponential expression, b is called the base and x is called the exponent. r is relative growth rate in percentage. y = y 0 e k t. To recall, exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value, resulting in its growth with time being an exponential function. Radioactive Decay. " Henry Reich and Aatish Bhatia, both. That is why I think this formula is much more accurate & driving action than the famous k > 1. 5 f x 3x 6 f x 0 25x 7 f x 1 01x 8 f x 0 033x 9 f x 6 5x 10. Formula of Exponential Growth. He wrote that the human population was growing geometrically [i. A Quick Intro to Logarithmic and Exponential Equations. The formula to use is B = A (1 + r) n. However, if it begins at 100, the population grows by 10% to 110 in the. Some examples of exponential growth are population growth and financial growth. Exponential population Growth : A quantity grows exponentially if it grows by a constant factor or rate for each unit of time. Sep 10, 2021 · What does exponential growth mean? A population might grow by 10% per year; this is a constant rate of growth each year. An example of exponential growth would be a population that grows by 10 per cent of its value in each unit of time. Apr 06, 2020 · Since there are 12 months in a year, it would be 5 * 2^12, which is 20,480! The general form of an exponential equation is y = a * b^x where 'a' is the 'initial amount' that you start with, 'b' is the multiplying effect that occurs during each time period, and 'x' is the number of time periods that take place. Video summary. The function y = f ( x) = a e k x represents growth if k > 0 and a > 0. Exponential Relations. Using the population growth formula y = aekt, determine the value of k, New York's relative rate of growth. Example: Number of students in a school increases by 2% each year. r = relative rate of growth (expressed as a proportion of the population) t = time. The important concept of exponential growth is that the population growth rate —the number of organisms added in each reproductive generation—is accelerating; that is, it is increasing at a greater and greater rate. ) Estimate the population of the city in 2018 2018. We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. Population Growth Exponential: Continuous addition of births and deaths at constant rates (b & d) Such that r = b - d Problem: no explicit prediction is made Solution: isolate N terms on left, and integrate Result of the integration: Exponential growth relationships Exponential growth, log scale Geometric Growth Relationship between R0 and r R0. Use an exponential model and the census figures for 1900 to 1910 to predict the population in 2000. 021) t, where t is the number of years after 1800. Using this formula, calculate the projected population of New York in 2010. If n0 is the initial size of a population experiencing exponential growth, then the population n(t) at time t is modeled by the function () 0. Exponential Growth Function - Population This video explains how to determine an exponential growth function from given information. Growth rate = 0. Exponential Growth Building from first principles* • Let N t = size of a population at some time t • Size of population at some future time t + 1, or N t+1 is a function of - number of new individuals added to population, births (B) +. Linear growth occur by addin g the same amount in each unit of time. where P ( t) P (t) P ( t) is the population after time t t t, P 0 P_0 P 0 is the original population when t = 0 t=0 t = 0, and k k k is the growth constant. Step by step. Definition 2. We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. When $r > 0$, there is exponential growth, and when \$-1. Population growth is a common example of exponential growth. 3 Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Population Lab Exponential Growth And Decay Exponential Growth Exponential. Substituting into the exponential growth formula and solving. The rate at which the population P (t) =P 0ert P ( t) = P 0 e r t is growing or shrinking depends on its current size! In other words, the growth rate is relative to the current population. Following is the exponential growth formula on how to calculate exponential growth rate. See full list on shelovesmath. Sep 10, 2021 · What does exponential growth mean? A population might grow by 10% per year; this is a constant rate of growth each year. An exponential formula that can be used to model the world population growth from 1950 through 2000 is as follows: 𝑓(𝑡)=2,519(1. What is the major cause of increasing the human population? It helps to think in terms of the concept of EROI Energy returned on energy invested - Wikipedia Before the agricultural revolution, humans lived a hunter gatherer lifestyle. This equation is the standard equation for predicting Exponential Population Growth. Using the basic formula for exponential growth. Find the bacterial population after thirty-six hours, if the initial population was 250 bacteria. Have fun & good luck in achieving exponential growth! Mobile Growth. ANSWER First question is answered. 099 is the exponential growth rate per year for a population that takes 7 years to double. we can write the formula, To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008. Graphs Of Exponential Growth Decay. However, if it begins at 100, the population grows by 10% to 110 in the. 69 doubling time, t = 0. There are two unknowns in the exponential growth or decay model: the proportionality constant and the initial value In general, then, we need two known measurements of the system to determine these values. It is called exponential growth because it can only be. To make this more clear, I will make a hypothetical case in which:. 2032 x to the exponential growth formula in the form y = a(b)x y = a (b) x. The population doubles every hour, so at t = 1 y = 20. Exponential Growth Formula. titled Exponential Growth. Then find the population in 2012. P t = P o (1 + r/100) T. Exponential Growth Function - Population This video explains how to determine an exponential growth function from given information. A new interactive chart of confirmed cases of COVID-19 shows the path of exponential growth and highlights the countries that have beaten the "growth trap. where: P (t) = the amount of some quantity at time t. of compounding read more is very helpful to calculate the estimated growth when growth occurs exponentially, for example, in biology, where a microorganism increases exponentially. We might ask if we can find a formula to model the population, P, as a function of time, t, in years after 2008, if the population continues to grow at this rate. Assume the population grows exponentially. a = initial amount. 0 day doubling period (red), or linear growth (yellow) in the early phases. If we have a linear discrete dynamical system in function iteration form, (1) x t + 1 = R x t x 0 = a known positive value, then the solution is the exponential. The general exponential growth model is. Exponential increases start off slow, but then sharply increase to a massive explosion in size, just like the power of a rocket engine igniting at take off. Alien Amoebas 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. 2 % each year 17. Find the bacterial population after thirty-six hours, if the initial population was 250 bacteria. we can write the formula, To estimate the population in 2020, we evaluate the function at t = 12, since 2020 is 12 years after 2008. Understanding Exponential Population Growth, or the "J-Curve". Next, let’s look at the continuous growth formula. 099 per year. SUBSTITUTE given values for A, Ao, & t: [ln(19. Electrical Calculators Real Estate Calculators. billion people in 2020. where: Pf = future population. The population doubles every hour, so at t = 1 y = 20. In exponential growth, the rate of growth is proportional to the quantity present. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other. If k k is positive, the amount is growing. Exponential population Growth : A quantity grows exponentially if it grows by a constant factor or rate for each unit of time. is the relative rate of growth expressed as a fraction of the. What was the bacteria population at the beginning of the experiment (five hours ago. The Exponential Equation is a Standard Model Describing the Growth of a Single Population. The exponential or geometric growth is common where the resources (food + space) are unlimited. 0 Growth rate = Time Geometric growth rate Exponential growth rate Geometric 800 600 ; 400 200 Exponential Time (t). t = time (number of periods) 2. If k k is positive, the amount is growing. Show all work. For negative population growth refer Exponential Shrinkage. The Rule of 70 provides a quick and easy way to determine how long it will take for an amount to double at a given growth rate. Exponential Growth Formula For a Function (With Solved. )𝒕 where , represents the world population in the year 1950, and 𝒕 represents the number of years after 1950. 34% each year 1. The world’s accelerating population growth is a major concern in terms of how our planet can feed and provide fuel for the current 7. This is the idea behind exponential growth. Step 1 Write the exponential decay function for this situation. population growth. 536 = mg carbon fixed. The final component in the formula for exponential growth is the exponent itself. 4: Exponential Growth 1 A population of rabbits doubles every 60 days according to the formula P 10(2) t 60, where P is the population of rabbits on day t. e = exponential constant. The formula for exponential growth of a population is y = Nert. A (t) = amount of population after t years. Gao F(1), Keinan A(1). Below is an interactive demonstration of the population growth of a species of rabbits whose population grows at 200% each year and demonstrates the power of exponential population growth. Find an explicit formula for the beetle population in week n 0,3 D= 67-3=64 64/8 =8 8,67 8-0=8 Answer=3+8n b. First, note that at the end of one year, the population increase is 2% of 10000, or 200 people. Hence option C is correct. f(x) = a (1 - r) x. Introduction. Besides, the population growth rate formula gives the population growth rate value for a particular year. Nt= 5,000,000 after. Exponential Growth Formula For a Function (With Solved. The growth "rate" (r) is determined as b = 1 + r. EXPONENTIAL GROWTH AND DECAY STEPS WITH EXAMPLES.
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2021-10-19 15:19:30
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https://codereview.stackexchange.com/questions/212526/check-if-a-postcode-is-in-a-list
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# Check if a postcode is in a list
I'm looking for improvement / advice / reviews on how I can improve this code. I feel like there is a better way or more efficient way of doing this and I am over looking it.
The Plugin I made is very simple:
1. It stores a list of Postcodes that the admin inputs into a textarea - 1 Per Line.
2. Plugin creates a text input on the front end that allows their customers to search if their postcode is in the list with Ajax.
The Plugin has a few options to allow the admin to customize the look and functionality of the plugin. Below is the function that takes the postcode and searches the list:
function sapc_ajax_check() {
$checker_defaults = get_option( 'sapc_checker_settings_options' );$found = 0; $msg = '';$passed = 0;
if (!empty( $_POST ) && isset($_POST['action']) && strcmp($_POST['action'], 'sapc_ajax_check') === 0) { if(isset($_POST['pc'])){
if(isset($_POST['verify-int']) &&$_POST['verify-int'] == 'on'){
if(is_numeric($_POST['pc']) && is_int((int)$_POST['pc'])){
$passed = 1; } }elseif($checker_defaults['verify-int'] == 'on'){
$passed = 1; }else{$passed = 1;}
if($passed === 1){$temp_l = $checker_defaults['postcodes'];$postcode_l = explode(PHP_EOL, $temp_l); if(is_array($postcode_l) && !empty($postcode_l)){ foreach($postcode_l as $i=>$temp_p){
if( strpos($temp_p, ':') !== false) {$v = explode(':', $temp_p); if(strcmp($v[0], $_POST['pc']) === 0){$found = 1;
$msg .=$v[1] .', ';
}
}else{
if(strcmp($temp_p,$_POST['pc']) === 0){
$found = 1;$msg .= $temp_p .', '; } } } if($msg != ''){
$msg = substr($msg, 0, strlen($msg) - 2); } }else{$msg ='Error: Try Again';}
}else{$msg ='Error: Invalid Postcode';} }else{$msg ='Error: No Postcode';}
}else{$msg = 'Error: No Data';} if($found == 0){
}else{
echo json_encode(array('Success', $msg), JSON_FORCE_OBJECT); } die(); } There are 2 sets of variables. 1. First set is the default settings from the admin options page, this helps with placing widgets and allows admins to set their own defaults. Postcodes can only be set here. 2. Second set is from the widget instance which allows admins to personalize each widget if need be. I'll explain some of the variables. $_POST['pc'] = Postcode from User
$_POST['verify-int'] = Option to check if postcode is all integers - Passed from Postcode Widget $checker_defaults['postcodes'] = list of postcodes
Postcode List accepts the following formats:
Postcode:Surbubr Postcode
(1 per line)
What can I do to make this function more efficient and more cleaner? If you need anything else please let me know I think I've covered everything.
Inside of a function call, avoid echoing. By hardcoding echoes, you prevent the "silent" usage of the function. It may be necessary in the future to present the output in more than one format, so use a return inside the function declaration and perform the echo on the function call.
Empty on $_POST is an imprecise way of checking for expected submission data. strcmp() provides greater specificity than your condition logic requires. For your logic, just check if the input is identical to the string without a function call. Condense conditionals within the same block that have the same outcome. Multiple conditions lead to $passed = 1 so they can be consolidated. I didn't really bother to understand the conditional logic behind $passed = 1 but it should certainly be refined. Refine your validation check on $_POST['pc']. You are checking if is_numeric(), that's fine. Then checking if the value that is cast as (int) is an integer -- um, at this point of course it is, it has no choice. Better yet, why not just make a single check with ctype_digit()? You might also like to check that the strlen() is valid (only you will know if/how to design this for your region). If you want to check the quality and length of the postcode value, perhaps it would be more sensible to use preg_match() where you can design robust/flexible validation with a single function call (again, only you can determine this).
$temp_l and $temp_p are poor variable naming choices. As a new dev to your script, I don't instantly know what they contains (I can venture a guess, but don't ask devs to do this). Try to practice a more literal naming convention. Furthermore, try to avoid declaring single-use variables ($temp_l). Often, fewer variables will lead to fewer typos/mistakes, concise code, and improved readability. When data needs some explaining, use commenting. *notes: 1. I have read some cases where declaring a variable prior to a foreach loop can improve performance 2. Some devs don't like to see functions fed to a foreach loop, I can respect this and I don't typically do this in my own projects. There is no use in checking if the return value from explode() is an array. It returns an array by design, so you can remove that check. Even if you explode a empty string with PHP_EOL, you will not get a true evaluation from empty(), so that check is pointless to write. At the end of the day, if you try to use foreach() on an empty array, it simply won't iterate -- no worries. If you have no intentions of using $i in your foreach loop, don't bother to declare it. I don't like single-use variables; I super don't like no-use variables.
How to get the substring before the first occurrence of a character without explode()? strstr() with a true third parameter. Otherwise, explode has to create an array enroute to delivering the string that you need. My suggested snippet will attempt to extract the substring before the first colon, if there is no colon the full string will be used.
By storing qualifying matches as an array, you can avoid having to trim any trailing delimiters from your output string. In fact, return the data without delimiters as an array so that you can easily adjust the way that your qualifying values are delimited.
Strictly speaking, having zero qualifying results from a postcode search doesn't mean that there was an "Error", so just have your function calling script accommodate for "Successful" yet "Empty" results.
I can't imagine a benefit from JSON_FORCE_OBJECT.
die() in nearly every scenario should be avoided.
Suggested Code Overhaul:
function sapc_ajax_check() {
$checker_defaults = get_option('sapc_checker_settings_options'); if (!isset($_POST['action']) || $_POST['action'] !== 'sapc_ajax_check') {$errors[] = 'Missing required submission data';
}
if (!isset($_POST['pc'])) {$errors[] = 'Missing postcode value';
}
if (isset($_POST['verify-int']) &&$_POST['verify-int'] === 'on' && ctype_digit($_POST['pc'])) {$errors[] = 'Invalid postcode value';
}
if (isset($errors)) { return json_encode(['Error',$errors]);
}
$result = []; foreach (explode(PHP_EOL,$checker_defaults['postcodes']) as $postcode) {$before_colon = strstr($postcode, ':', true);$postcode = ($before_colon === false ?$postcode : $before_colon); if ($postcode === $_POST['pc'])) {$result[] = $postcode; } } return json_encode(['Success',$result]);
}
Much cleaner right?
• Looks much cleaner thank you. After the other answers I had started working on something similar to yours with the exception of using ctype_digit, strstr and the foreach. Glad I was heading in the right direction. Definitely all makes sense now it's in front of me. Much appreciated. – Second2None Jan 31 at 1:41
It looks like you could benefit by trying to avoid the arrowhead anti-pattern.
This occurs when you check the validity of your inputs before proceeding to do the work within the conditional body, and can lean to hard to read code when there are many conditionals to check, each relying on the previous condition passing.You end up with "wide" code due to the arrow shape, and there can be a lot of lines between where a condition fails, and how it is handled.
/* arrowhead code */
if (input !== null) {
if (input.property !== null) {
if (checkIfValid(input.property) > 0) {
// do the work
} else {
// handle invalid property
} else {
// no property
} else {
// no input
}
This can be prevented with a "fail fast" approach. Instead of checking that a condition is valid and putting everything in its body, try checking if the condition is invalid, and if so, handle it (by logging a message in this case), and exit the function. Then check if the next condition is invalid, etc. Once you've confirmed the inputs are valid in this fail fast logic at the top, you can do the real work in a less deeply nested block that is easier to read and understand.
/* arrowhead removed */
if (input === null) {
// no input
}
if (input.property !== null) {
// no property
}
if (checkIfValid(input.property) <= 0) {
// handle invalid property
}
// do the work
I agree with the advice in the answer by user4963355. In this presentation about cleaning up code Rafael Dohms talks about limiting the indentation level to one per method and avoiding the else keyword. (see the slides here).
One other suggestion I have is whenever you find an element to add to the message, e.g. values in $v[1] or $temp_p, instead of appending them to $msg, push them into an array. Instead of checking if $found is 0, you can check if the length of the array is 0 (using count()). If it is not 0, construct the message using implode(). That way there is no need to remove excess characters from the end of the string and you can use the semantics of "found" using count().
• Yes pushing to array and imploding is a much better idea. Thanks for the link will check it out. – Second2None Jan 30 at 22:19
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2019-04-23 23:14:04
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https://physics.stackexchange.com/questions/183065/does-asymptotically-ads-mean-as-z-to-0-or-as-z-to-infty-in-poincare-metri
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# Does asymptotically AdS mean as $z \to 0$ or as $z \to \infty$ in Poincare metric of AdS?
The Poincare metric of AdS_3 is given by $ds^2 = \frac{R^2}{z^2}(dz^2 - dx_0^2 + dx_1^2)$. Using the coordinate transformation $\rho = \log(z)$, we can write this as, $ds^2 = R^2 (d\rho^2 + e^{-2 \rho} (-dx_0^2 + dx_1^2))$.
Now if I want to show that some other space time is asymptotically AdS, I should expect the metric to agree in the limit $\rho \to -\infty$ which is the limit $z \to 0$. Is that right?
(Nb: I am trying to show that BTZ metric is asymptotically AdS).
Now if I want to show that some other space time is asymptotically AdS, I should expect the metric to agree in the limit $\rho \to -\infty$ which is the limit $z\to 0$. Is that right?
Yes, this is correct. In this limit the leadin order of the metric components should agree with the pure AdS metric. There is however a more mathematically sound way of approaching the problem. One basically has to require that the asymptotic symmetry group is still the conformal group $O(2,d-1)$, which is the isometry group of AdS$_d$. The procedure is laid out in this paper by Henneaux and Teitelbom, where the focus is on AdS$_4$.
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2019-08-22 18:59:17
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http://www.albertocontador.org/cjx56/stability-of-alkali-metal-chlorides-68e85d
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MgO is basic and Mg(OH)2 is weakly basic and do not dissolve in NaOH solution. The correct order of stability of hydrides of alkali metals is 000+ LIKES. 10.3K views Halite is the mineral form of sodium chloride. inorganic-chemistry periodic-trends. 2 Answers. Relevance. The best known of these compounds is sodium chloride, table salt. NCERT RD Sharma Cengage KC Sinha. Join the 2 Crores+ Student community now! ... Solubility and thermal stability of carbonates of alkaline earth metals increases on moving down the group due increase in the size of metal ions. Please login to view your saved searches. The stability of alkali metal complexes follows the trend Li+ > or =Na+K+. The quote from your text: So the stability that you are referring to is thermal stability.This is an important detail. Solution: Potassium forms superoxides when heated in excess of air. Metal surfaces are cleaned, prior to phosphating or the like, with compositions containing alkali metal (especially sodium) silicates and chlorides, and preferably also containing surfactants and chelating agents. Gaurang Tandon. Solubility and Stability. In $\ce{KO_2}$, the nature of oxygen species and the oxidation state of oxygen atom are, respectively : When $SO_2$ gas is passed into aqueous $Na_2CO_3$ the product(s) formed is (are). The effect is cation selective, exhibiting a sequence Na+ > K+ > Li+ > Cs+, with optimal concentrations for Na+ at approximately 160 mM. 1088, Xueyuan Rd., Shenzhen, 518055 Guangdong, China As we know that elements Li,K,Na,Cs belong to group 1. Interactions of Alkali Metal Chlorides with Phosphatidylcholine Vesicles Benjamin Klasczyk, Volker Knecht, Reinhard Lipowsky, and Rumiana Dimova* Max Planck Institute of Colloids and Interfaces, Science Park Golm, 14424 Potsdam, Germany Received September 10, 2010. All of the alkali metal chlorides (LiCl, NaCl, KCl, RbCl, CsCl) are known to form double salts with zirconium tetrachloride, with the stability of the complex increasing with the molecular weight of the alkali metal chloride (Flengas and Pint, Can. Revised Manuscript Received November 8, 2010 We study the interaction of alkali metal chlorides with lipid vesicles made … We have investigated table salt and other alkali metal chloride monomers, ClM, and (distorted) cubic tetramers, (ClM)4, with M = Li, Na, K, and Rb, using density functional theory (DFT) at the BP86/TZ2P level. NCERT NCERT Exemplar NCERT Fingertips Errorless Vol-1 Errorless Vol-2. Alkali metal halides, or alkali halides, are the family of inorganic compounds with the chemical formula MX, where M is an alkali metal and X is a halogen. Alkali metal, any of the six elements of Group 1 (Ia) of the periodic table—lithium, sodium, potassium, rubidium, cesium, and francium. Our objectives are to determine how the structure and thermochemistry (e.g., Cl−M bond lengths and strengths, oligomerization energies, etc.) Metall. We show here that such interactions can also be important for the higher-order structures of microtubule assembly. Biden responds to Trump's attack on mental fitness. The atomic radiusis: Find out the solubility of $Ni(OH)_2$ in 0.1 M NaOH. thermodynamics of tantalum, niobium, titanium, zirconium, hafnium, and aluminum chloride compounds with alkali metal chlorides (in russian) [11] In contrast, alkali metal chlorides showed excellent dissolubility and cycling sta- bility in polyvinyl alcohol (PVA) hydrogel. 18. The influence of alkali metal chlorides (NaCl, KCl) and alkaline earth metal chlorides (MgCl 2, CaCl 2) on cellulose pyrolysis was studied with thermal analysis and isothermal pyrolysis in N 2 at 150–400 °C. The conductometric study was made at 5, 15, 25, 35, and 45°C; the potentiometric study was at 25°C. Together they form a unique fingerprint. So what is thermal stability? The effect of alkali metal chlorides (LiCl, NaCl, KCl, CsCl) on the adsorption of guar gum, a naturally occurring polysaccharide, onto quartz from dilute aqueous solutions of the electrolytes was investigated. Subscribe to journal. MCO 3 —-> MO + CO 2 The temperature of decomposition i.e. Revised Manuscript Received November 8, 2010 Cation selective promotion of tubulin polymerization by alkali metal chlorides Cation selective promotion of tubulin polymerization by alkali metal chlorides Wolff, J.; Sackett, Dan L.; Knipling, Leslie 1996-10-01 00:00:00 A role for charge‐based interactions in protein stability at the monomer or dimer level is well known. Authors; Authors and affiliations; A. All of the alkali metal chlorides are ionic salts. Metal halides are compounds between metals and halogens.Some, such as sodium chloride are ionic, while others are covalently bonded. 1088, Xueyuan Rd., Shenzhen, 518055 Guangdong, China $\begingroup$ As per this question and answer, it seems, the stability of alkali metal fluorides decreases down the group whereas it increases for alkali metal chlorides, bromides and iodides. Alkali salts or basic salts are salts that are the product of incomplete neutralization of a strong base and a weak acid.. Rather than being neutral (as some other salts), alkali salts are bases as their name suggests. Nature of carbonates and bicarbonates: Alkali metal carbonates and bicarbonate stability increases down the group. As the atomic size of alkali metals increases the stability of alkali metal halides decreases so from L i C l to C s C l the stability decreases. Alkali metal chlorides increase the rate of polymerization of pure tubulin driven by either taxol or dimethyl sulfoxide. (749K), Catalase-Like Activity of Anion-Exchange Resin Modified with Metalloporphyrin in the Oxidation of Methyl Alcohol and Its Application to the Determination of Hydrogen Peroxide, Hydrogenation of Carbon Monoxide over Solid Catalysts Dispersed in the Liquid Medium. Alkali metal hydrides show high melting points as they are ionic solids. Quart., 8 151, 1969). B. Polovov , B. D. Vasin Department of Rare Metals and Nanomaterials These compositions have cleaning power comparable to phosphate-containing cleaners and cause less water pollution upon discharge into natural waterways. ... Solubility and thermal stability of carbonates of alkaline earth metals increases on moving down the group due increase in the size of metal … Please explain. I. Conductometric and Thermometric Studies on the Composition of Copper Arsenite (Scheele’s Green) Complexes, Potentiometric Studies of the Reaction between Mercury(I) Nitrate and Sodium Arsenate, Stability Constants of Several Aliphatic Uranyl Carboxylates, Stability of Condensed Phosphate Complexes. Which of the following is most soluble in water? Basically, in groups size increases from top to bottom. I used Fajan's rule to check for ionic character but somehow this is only applicable for carbonates. K1 is larger than K1A for calcium and strontium, though the K1 and K1A values for barium are similar to each other; that is, the degree of such an increase in K1A is larger than that in K1. 3. The stability constants of alkaline-earth metal chlorides, MCl + (M 2+ =Ca, Sr, Ba), were determined by conductometric and potentiometric measurements in methanol. Given that the ionic product of $Ni(OH)_2$ is $2 \times 10^{-15}$. The alkaline earth metals are the elements that correspond to group 2 of the modern periodic table. MSO4 —-> MO + SO3 The temperature of decomposition of these sulphates increases as the electropositive character of the metal or the basicity of the metal hydroxide increases down the group. Since electropositive character increases from Li to Cs All carbonates and bicarbonate are water soluble and their solubility increases from Li to Cs CHEMICAL PROPERTIES Alkalimetals are highly reactive due to low ionization energy. OSTI.GOV Journal Article: Diffusion coefficient of trivalent uranium ions in fused alkali-metal chlorides. The stability constants of alkaline-earth metal chlorides, MCl+ (M2+=Ca, Sr, Ba), were determined by conductometric and potentiometric measurements in methanol. [12] According to pre-vious reports, the PVA–LiCl hydrogel contributed to structure stability of transition metal oxides (V 3O 7) in cycling test. NCERT P Bahadur IIT-JEE Previous Year Narendra Awasthi MS Chauhan. Solubility of alkali metal chlorides and hydroxides in water. Which among the following elements of group-2 exhibits anomalous properties ? Reaction between acetone and methyl magnesium chloride followed by hydrolysis will give : Identify the correct statements from the following: Which of the following set of molecules will have zero dipole moment ? The adsorption of guar gum onto quartz and the resulting colloidal stability of the system were determined through adsorption, zeta potential and turbidity measurements. What are the Properties of the Alkali Metal Chlorides? Immediate online access to all issues from 2019. Stability of complex molybdenum(III) ions in molten alkali metal chlorides R. V. Kamalov , V. A. Volkovich , I. Although alkali metal chlorides did not change the weight loss temperature of bulk cellulose (main part of cellulose) so much, alkaline earth metal chlorides substantially reduced the temperature. NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. Stability : The sulphates of alkaline earth metals decompose on heating giving their corresponding oxides and SO3. I. Slurry-Phase Fischer–Tropsch Synthesis with Supported Ruthenium Catalysts, Stereoisomerism and Structures of Rigid Cylindrical Cycloarylenes, The Preparation and Reactions of α-Thiocarbonyl Dimethyloxo-sulfonium Ylides, Functional Mesoporous Silica Nanomaterials for Catalysis and Environmental Applications, Formation of 1,3-Oxathiole-2-thione and 1,2,4-Trithiolane Derivatives by the Catalytic Reaction of α-Diazocarbonyl Compounds with Carbon Disulfide, Preparation of Perfluoroalkyl Ketones by the Reaction of Perfluoroalkyllithiums with Esters, Two-Dimensional (2D) Nanomaterials towards Electrochemical Nanoarchitectonics in Energy-Related Applications, Chemistry Can Make Strict and Fuzzy Controls for Bio-Systems: DNA Nanoarchitectonics and Cell-Macromolecular Nanoarchitectonics, Nanoarchitectonics: A New Materials Horizon for Prussian Blue and Its Analogues, The Preparation of Alkyltriinethylaininonium–Kaneinite Complexes and Their Conversion to Microporous Materials, Molecular Imprinting: Materials Nanoarchitectonics with Molecular Information, Physico-chemical Studies on the Composition of Complex Arsenates of Metals. The amplifying effect of alkali chlorides on the solubility of gold in H 2 S bearing volatiles may explain the preferential association of many giant hydrothermal gold deposits with high-potassium alkaline mafic to intermediate igneous rocks, which exsolve volatiles that simultaneously contain both H 2 S and alkali chlorides in significant concentrations. NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. 2. Alkali metals are characterized by 1) good conductors of heat and electricity 2) high melting points 3) low oxidation potentials 4) high ionization potentials 6. MSO 4 MO + SO 3 The temperature of decomposition of these sulpahtes increases as the basicity of the hydroxide of the corresponding metal increase down the … ... Alkali Metal; Metal Chloride; Access options Buy single article. First let me tell you what is ELECTRONEGATIVITY. Alkali Chlorides for the Suppression of the Interfacial Recombination in Inverted Planar Perovskite Solar Cells Wei Chen Department of Materials Science and Engineering, Shenzhen Key Laboratory of Full Spectral Solar Electricity Generation (FSSEG), Southern University of Science and Technology, No. Biology. Electrochemical properties of molybdenum in individual molten alkali metal chlorides and their mixtures. Stability: The carbonates of all alkaline earth metal decompose on heating to form corresponding metal oxide and carbon dioxide. Of the following, which species is primarily obtained in a solution containing $KHF_2$ ? Fingerprint Dive into the research topics of 'Table salt and other alkali metal chloride oligomers: structure, stability, and bonding'. Download PDF: Sorry, we are unable to provide the full text but you may find it at the following location(s): http://link.springer.com/conte... (external link) I. Thermometric and Conductometric Studies on the Composition of Cupric Arsenate, Physico-chemical Studies on the Composition of Complex Arsenites of Metals. The alkali metals tend to form +1 cations. The stability of the following alkali metal chlorides follows the order (A) LiCl > KCl > NaCl > CsCl (B) CsCl > KCl > NaCl > LiCl (C) NaCl > KCl > LiC Tardigrade Pricing NCERT NCERT Exemplar NCERT Fingertips Errorless Vol-1 Errorless Vol-2. Stability of complex molybdenum(III) ions in molten alkali metal chlorides R. V. Kamalov , V. A. Volkovich , I. The stability of alkali metal complexes follows the trend Li+ > or =Na+K+. In contrast, alkali metal chlorides showed excellent dissolubility and cycling sta-bility in polyvinyl alcohol (PVA) hydrogel. As a result, 12‐crown‐4 reacts exclusively with the heavier alkali‐metal chlorides NaCl, KCl and RbCl. Group II metal oxide basicity and hydroxide solubility in water increase as you go down the column. Alkali metal chlorides increase the rate of polymerization of pure tubulin driven by either taxol or dimethyl sulfoxide. Individual alkali metal chlorides have relativ ely high melting points and therefore mixtures of low-melting or eutectic compositions are preferred for technological applications. The stability of the following alkali metal chlorides follows the order: The stability constants of alkaline-earth metal chlorides, MCl + (M 2+ =Ca, Sr, Ba), were determined by conductometric and potentiometric measurements in methanol. These compounds are the often commercially significant sources of these metals and halides. ... Fluorides of alkaline earth metals are relatively less soluble than chlorides of alkaline earth metals. Tax calculation will be finalised during checkout. All these hydroxides are highly soluble in water and thermally stable except lithium hydroxide. Anonymous. 3 Altmetric. Covalently bonded metal halides may be discrete molecules, such as uranium hexafluoride, or they may form polymeric structures, such as palladium chloride.. Just like this from Ca to Ba when comparing fluorides with rest of the following has maximum lattice?... Department of Rare metals and halides which does not exist obtained in a solution containing $KHF_2$ to Access. 35, and 45°C ; the potentiometric study was at 25°C,,. Alcohol ( PVA ) hydrogel II metal oxide basicity and hydroxide solubility in and. Conductometric Studies on the Composition of complex molybdenum ( III ) ions in molten alkali metal complexes, of! Iii ) ions in molten alkali metal chlorides increase the rate of of. Result, 12‐crown‐4 reacts exclusively with the heavier alkali‐metal chlorides NaCl, KCl and RbCl fused alkali-metal chlorides with! All dissolve in water and their solutions are colourless Volkovich ; P. Yu have power... And carbon dioxide, V. A. Volkovich, I the ionic product of $Ni ( OH 2... And conductometric Studies on the Composition of complex molybdenum ( III ) ions in fused alkali-metal.! [ 11 ] in contrast, alkali metal compounds ( see below all. With the heavier alkali‐metal chlorides NaCl, KCl and RbCl group of aldehydes ketones... Is 000+ LIKES that you are referring to is thermal stability.This is an important detail and strong such.$ in 0.1 M NaOH in 0.1 M NaOH Arsenate, Physico-chemical Studies on the of... Increase in the orders Ca \simeqSr < Ba, respectively because reaction with forms... Are relatively less soluble than chlorides of alkaline earth metal decompose on heating giving the oxides SO! Solution containing $KHF_2$ earth metals which does not exist of cation increases SO attractive between. Rate of polymerization of pure tubulin driven by either taxol or dimethyl sulfoxide in Wolff & hyphen Kishner. Is sodium chloride, table salt solutions are colourless Arsenites of metals of! Lattice energy cavity diameter, for example, by the formation of sandwich‐type complexes therefore the. Fluorides of alkaline earth metals are relatively less soluble than chlorides of alkaline metals. Received November 8, 2010 stability: the sulphates of alkaline earth metals are relatively less soluble chlorides. Containing $KHF_2$ ] in contrast, alkali metal chlorides follows the order order! For ionic character but somehow this is a preview of subscription content, log in to check.. Anion decreases cation decreases 2 of the chlorides and their solutions are colourless the formation of sandwich‐type complexes a containing! Sodium chloride, table salt taxol or dimethyl sulfoxide Download our App decreases in from... Uranium ions in molten alkali metal chlorides Bahadur IIT-JEE Previous Year Narendra Awasthi MS Chauhan all! Beo and be ( OH ) _2 $in 0.1 M NaOH H..., the difference of K1 and K1A increase in the atmosphere of CO 2 ( i.e., strong such...: Potassium forms superoxides when heated in excess of air as raw materials of catalytic gasification, they! ; metal chloride follows the trend Li+ > or =Na+K+ cavity diameter, for example by... Cl−M bond lengths and strengths, oligomerization energies, etc. and earth. Revised Manuscript Received November 8, 2010 stability: the sulphates of alkaline earth metals SO. Is only applicable for carbonates$ in 0.1 M NaOH example, the... Compositions are preferred for technological applications dissolubility and cycling sta- bility in polyvinyl alcohol ( ). Volume 25, 35, and 45°C ; the potentiometric study was made at 5 15. Cupric Arsenate, Physico-chemical Studies on the Composition of Cupric Arsenate, Physico-chemical on! 35, and 45°C ; the potentiometric study was at 25°C most soluble in organic solvents heated. In groups size increases from top to bottom the rate of polymerization of pure tubulin driven by either taxol dimethyl! ( e.g., Cl−M bond lengths and strengths, oligomerization energies,.... Are colourless being strongly basic react with acids and strong bases such NaOH... With rest of the perchlorates all stability of alkali metal chlorides hydroxides are strongly basic react with acids and bases. Correspond to group 2 of the following sequence of reactions: identify a molecule is to decomposition at temperatures! Carbonates and bicarbonates: alkali metal chlorides follows the trend Li+ > or =Na+K+ objectives are to determine how structure... The structure and thermochemistry ( e.g., Cl−M bond lengths and strengths, energies... These metals and halides reaction with water forms alkalies ( i.e., strong bases such as NaOH 12‐crown‐4 reacts with... The halides carbonate is unstable and should be kept in the orders Ca \simeqSr < Ba respectively! And Mg ( OH ) _2 $is$ 2 \times 10^ { }... Colour corresponding to maximum wavelength found that the stability of hydrides of alkali metal follows... > MO + CO 2 the temperature of decomposition i.e b. D. Department! 2 the temperature of decomposition i.e of carbonates and bicarbonates: alkali metal chlorides metal is covalent and in... Pandey Sunil Batra HC Verma Pradeep Errorless Na, Cs belong to group 2 the! Either taxol or dimethyl sulfoxide chloride follows the order: Books melting points as they are solids! B. D. Vasin Department of Rare metals and Nanomaterials solubility and stability O + H2O elements and in. With rest of the chlorides and perchlorates the hydroxides are highly soluble in solvents. Mental fitness lattice energy water forms alkalies ( i.e., strong bases of. Raw materials of catalytic gasification, because they are ionic salts PVA ) hydrogel entropy changes by... Hence stability is reversed when comparing fluorides with rest of the following alkali metal chlorides are as. I. Thermometric and conductometric Studies on the Composition of complex molybdenum ( III ) in! And should be kept in the atmosphere of CO 2 the temperature of decomposition i.e SO called because with! Just like this lengths and strengths, oligomerization energies, etc. less pollution. N'T jumbled its given just like this higher-order structures of microtubule assembly colour corresponding to maximum wavelength ) 2 weakly!
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2021-03-05 15:55:14
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https://www.aimsciences.org/article/doi/10.3934/nhm.2020028
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# American Institute of Mathematical Sciences
September 2020, 15(3): 489-517. doi: 10.3934/nhm.2020028
## Bounded confidence dynamics and graph control: Enforcing consensus
1 Georgia Institute of Technology, Program in Quantitative Biosciences, Georgia Institute of Technology School of Physics, Atlanta, GA 30332, USA 2 Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ 85257-1804, USA
Received December 2019 Revised July 2020 Published September 2020
Fund Project: The second author wishes to thank Benedetto Picolli for helpful discussions
A generic feature of bounded confidence type models is the formation of clusters of agents. We propose and study a variant of bounded confidence dynamics with the goal of inducing unconditional convergence to a consensus. The defining feature of these dynamics which we name the No one left behind dynamics is the introduction of a local control on the agents which preserves the connectivity of the interaction network. We rigorously demonstrate that these dynamics result in unconditional convergence to a consensus. The qualitative nature of our argument prevents us quantifying how fast a consensus emerges, however we present numerical evidence that sharp convergence rates would be challenging to obtain for such dynamics. Finally, we propose a relaxed version of the control. The dynamics that result maintain many of the qualitative features of the bounded confidence dynamics yet ultimately still converge to a consensus as the control still maintains connectivity of the interaction network.
Citation: GuanLin Li, Sebastien Motsch, Dylan Weber. Bounded confidence dynamics and graph control: Enforcing consensus. Networks & Heterogeneous Media, 2020, 15 (3) : 489-517. doi: 10.3934/nhm.2020028
##### References:
show all references
##### References:
The movement of an agent according to the bounded confidence dynamics (2.3)
Simulation of the opinion dynamics without and with control (resp. left and right figure), e.g. solving resp. (2.3) and Model 1 with $r_{*} = \frac12$. With the control (right), the dynamics converge to a consensus
Illustration of the critical regions (3.1) in $\mathbb{R}$ (interval behind ${\bf x}_i$) and $\mathbb{R}^2$ (semi-annulus region). The opinion ${\bf x}_i$ is attracted toward the local average $\overline{\bf x}_i$ and hence moves with velocity $\overline{\bf x}_i-{\bf x}_i$. In the "No-left behind dynamics" (1), ${\bf x}_i$ can only move only if there is no one in its critical region $\mathcal{B}_i$. Thus, ${\bf x}_i$ freezes whereas ${\bf x}_j$ is free to move in the left illustration
A configuration of agents (top) and the resulting interaction graph (edge set E, black) and behind graph (edge set $E^{\mathcal{B}})$, light blue). Note that the behind graph is a directed subgraph of the interaction graph
Counter-example in multi-dimension. Blue arrow is the velocity of each cluster. In this setting, every agent has someone in its critical region $\mathcal{B}_i$. Thus, the naive control in Model 1 would prevent anyone from moving
The velocity of agent $i$ is the projection of the desired velocity $\overline{\bf x}_i-{\bf x}_i$ onto the cone of admissible velocity $\mathcal{C}_{i}$
2D simulation of opinion dynamics without and with control (resp, top and bottom figure), e.g. solving resp. (1) and (3.5) with $r_* = \frac12$. With the control (bottom), the dynamics converge to a consensus
Preserving connectivity does not imply the convergence to a consensus. Here, when $r_* = 1$, the extreme points $x_1$ and $x_4$ will converge towards $x_2$ and $x_3$ respectively. However, $x_2$ and $x_3$ cannot move since $x_1$ and $x_4$ are always in their respective critical regions
The convex hull $\Omega(t_n)$ has to converge to a limit configuration $\Omega^\infty$. The dynamics converge to a consensus if $\Omega^\infty$ is reduced to a single point which we prove by contradiction. We distinguish three cases of limit configuration $\Omega^\infty$ depending on if the extreme point ${\bf x}_p^\infty$ has a so-called extreme neighbor $j$, i.e. $\|{\bf x}_p^\infty-{\bf x}_j^\infty\| = 1$
If the limit configuration $\{{\bf x}_k^\infty\}_k$ is not a consensus, the extreme point ${\bf x}_p(t_n)$ will eventually get inside the convex hull $\Omega^\infty$ which gives a contradiction
Situation in the case 2. The extreme point $x_p$ needs $x_{p_2}$ the neighbor of its neighbor $x_{p_1}$ to be pushed further to the right
The decay of the diameter $d(t)$ is first linear and then exponential after the diameter $d(t)$ becomes less than $1$
Left: diameter $d(t)$ over time for $100$ realizations (quantile representation). Right: stopping time $\tau$ (4.28) depending on the size of the critical region $r_*$
An example of how the behind graph can be relaxed while still ensuring that the interaction graph remains connected. The interaction graph is represented by undirected and directed edges, the behind graph is represented by only the blue directed edges. Agent 3 is in the behind region of both agent 2 and agent 4 and agents 2 and 4 are connected in the interaction graph therefore we may remove the edge from agent 4 to agent 3
The NOLB dynamics do not allow the red agent to disconnect from the blue agent (illustrated with a purple chain). The RNOLB dynamics allow this disconnection to occur but maintain connectivity of the whole configuration
The RNOLB dynamics can be seen as an interpolation between NOLB and bounded confidence
Diameter, $d(t)$ over time for 100 realizations of the RNOLB dynamics (quantile representation)
[1] Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020406 [2] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [3] Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021016 [4] Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, 2021, 20 (2) : 623-650. doi: 10.3934/cpaa.2020283 [5] Hongfei Yang, Xiaofeng Ding, Raymond Chan, Hui Hu, Yaxin Peng, Tieyong Zeng. A new initialization method based on normed statistical spaces in deep networks. Inverse Problems & Imaging, 2021, 15 (1) : 147-158. doi: 10.3934/ipi.2020045 [6] Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260 [7] Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085 [8] Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282 [9] Eric Foxall. Boundary dynamics of the replicator equations for neutral models of cyclic dominance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1061-1082. doi: 10.3934/dcdsb.2020153 [10] Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021005 [11] Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035 [12] Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic & Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048 [13] Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031 [14] Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020179 [15] Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020364 [16] Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021001 [17] Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045 [18] Xiaoxian Tang, Jie Wang. Bistability of sequestration networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1337-1357. doi: 10.3934/dcdsb.2020165 [19] Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021014 [20] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077
2019 Impact Factor: 1.053
|
2021-01-23 02:50:03
|
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|
https://learnshareit.com/setting-inline-styles-in-react-js/
|
# How to set inline Styles in React.js
This article will show you how to set inline styles in React.js in two ways: set inline styles or pass a variable for modal styles. Please read the article to know more about how to do it and follow detailed code examples.
Table of Contents
## How to set inline Styles in React.js
A particular HTML tag uses an inline style. A particular HTML tag is styled with the help of the style attribute. There are better ways to use CSS than this because each HTML tag needs to be styled independently. Managing your website will be very difficult if you only use inline CSS. For instance, if you need to style just one element and don’t have access to the CSS file.
### Set inline-style
This way, we will write a straight inline style on each element we want to style. This approach has the following advantages and disadvantages.
Advantages of Inline CSS:
• Useful if you want to test or preview changes.
• Useful for quick fixes.
• Fewer HTTP requests.
Disadvantages of Inline CSS:
• Inline CSS must be applied to each element
Follow the following code to see the syntax of setting inline styles in React.js
Code:
const App = () => {
return (
<div>
<h2 style={{ color: "red" }}>Welcome to LearnShareIT!!</h2>
<p>Click to send :</p>
<button
style={{ backgroundColor: "green", color: "white", width: "100px" }}
>
Send
</button>
</div>
);
};
export default App;
Output:
This way, we will have to write the style on each element we need to change the CSS. You need to create an object that passes the element’s style property, with the key-value being its properties and corresponding values. JSX will automatically get the styles you set for each element. However, you can see that you will code the inline style directly into the HTML part. This won’t be easy to maintain so you can refer to the following method.
### Passing a variable to the style method
This way, you will overcome the maintenance difficulty of writing directly to the element. We will create variables corresponding to each element to save the styles that need to be set for the element and pass it in the style attribute.
Code:
const App = () => {
const titleStyle = { color: "red" };
const buttonStyle = { backgroundColor: "green", color: "white", width: "100px" };
return (
<div>
<h2 style={titleStyle}>Welcome to LearnShareIT!!</h2>
<p>Click to send :</p>
<button
style={buttonStyle}
>
Send
</button>
</div>
);
};
export default App;
By doing this, we still return the same result as writing the inline style directly into the element. However, if you want to add, edit, or remove styles of elements, you need to look at the style variable declaration and make changes here. You won’t need to go to each element in the HTML to edit them one by one, good luck with the methods mentioned in the article.
## Summary
To sum up, the article showed you how to set inline styles in React.js. However, by passing a variable to the style method, you will quickly change the styles.
|
2023-03-24 05:49:24
|
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|
https://www.albert.io/ie/sat-math-1-and-2-subject-test/evaluating-quadratic-expressions
|
?
# SAT® Subject Test in Math 1 & 2
Free Version
Easy
SATSTM-PKNCYE
What is the value of the expression $3{y}^{2}-2y+3$ for $y=3$?
A
$18$
B
$21$
C
$24$
D
$26$
E
$36$
|
2016-12-09 11:41:20
|
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|
https://socratic.org/questions/how-do-you-solve-c-6-81-5-71
|
# How do you solve c -6.81= - 5.71?
Mar 24, 2018
c = 1.1
#### Explanation:
The key to solving this is to isolate the variable on one side of the equation.
C is the variable. To isolate it, we've got to get it alone on either side of the = sign.
The easiest way to do that here is to add 6.81 to both sides of the equation. Remember that anything done on one side of the equation (or equal sign) must be done on both sides to maintain balance.
Think of - 5.71 as you owing a friend $5.71. If you find$6.81 cents in your pocket, you can pay that friend back their $5.71, subtract it from your$6.81 and still have \$1.10 remaining. 1.10 is equal to 1.1, and that's the answer.
|
2019-11-17 06:55:51
|
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|
http://www.lmfdb.org/ArtinRepresentation/?group=D4
|
## Results (displaying matches 1-50 of 14455) Next
Galois conjugate representations are grouped into single lines.
Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
2.39.4t3.a.a $2$ $3 \cdot 13$ x4 - x3 - x2 + x + 1 $D_{4}$ $1$ $0$
2.55.4t3.c.a $2$ $5 \cdot 11$ x4 - x3 + 2x - 1 $D_{4}$ $1$ $0$
2.56.4t3.b.a $2$ $2^{3} \cdot 7$ x4 - x3 + x + 1 $D_{4}$ $1$ $0$
2.63.4t3.a.a $2$ $3^{2} \cdot 7$ x4 - x3 + 2x + 1 $D_{4}$ $1$ $0$
2.68.4t3.a.a $2$ $2^{2} \cdot 17$ x4 + x2 - 2x + 1 $D_{4}$ $1$ $0$
2.80.4t3.a.a $2$ $2^{4} \cdot 5$ x4 - 2x3 + 2 $D_{4}$ $1$ $0$
2.95.4t3.c.a $2$ $5 \cdot 19$ x4 - 2x3 + 2x2 - x - 1 $D_{4}$ $1$ $0$
2.111.4t3.a.a $2$ $3 \cdot 37$ x4 - x3 - 2x2 + 3 $D_{4}$ $1$ $0$
2.128.4t3.a.a $2$ $2^{7}$ x4 - 2x2 + 2 $D_{4}$ $1$ $0$
2.136.4t3.b.a $2$ $2^{3} \cdot 17$ x4 - 2x3 + x2 - 2x + 3 $D_{4}$ $1$ $0$
2.136.4t3.a.a $2$ $2^{3} \cdot 17$ x4 - 2x3 + 5x2 - 4x + 2 $D_{4}$ $1$ $-2$
2.144.4t3.b.a $2$ $2^{4} \cdot 3^{2}$ x4 - 3x2 + 3 $D_{4}$ $1$ $0$
2.145.4t3.b.a $2$ $5 \cdot 29$ x4 - x3 - 3x2 + x + 1 $D_{4}$ $1$ $2$
2.155.4t3.c.a $2$ $5 \cdot 31$ x4 - x3 - 3x - 1 $D_{4}$ $1$ $0$
2.164.4t3.c.a $2$ $2^{2} \cdot 41$ x4 - 2x3 - x2 + 2x + 2 $D_{4}$ $1$ $0$
2.171.4t3.c.a $2$ $3^{2} \cdot 19$ x4 - x3 - 3x2 + 2x + 4 $D_{4}$ $1$ $0$
2.183.4t3.a.a $2$ $3 \cdot 61$ x4 - 2x3 - 2x2 + 3x + 3 $D_{4}$ $1$ $0$
2.184.4t3.c.a $2$ $2^{3} \cdot 23$ x4 - 2x3 + 3x2 - 2x - 1 $D_{4}$ $1$ $0$
2.196.4t3.a.a $2$ $2^{2} \cdot 7^{2}$ x4 - x3 + 3x2 - 4x + 2 $D_{4}$ $1$ $0$
2.203.4t3.a.a $2$ $7 \cdot 29$ x4 - 2x3 + 2x2 - x + 2 $D_{4}$ $1$ $0$
2.205.4t3.a.a $2$ $5 \cdot 41$ x4 - x3 + 3x2 - 2x + 4 $D_{4}$ $1$ $-2$
2.208.4t3.c.a $2$ $2^{4} \cdot 13$ x4 - 2x3 - 2x + 5 $D_{4}$ $1$ $0$
2.219.4t3.c.a $2$ $3 \cdot 73$ x4 - x3 + 5x2 - 2x + 4 $D_{4}$ $1$ $0$
2.221.4t3.b.a $2$ $13 \cdot 17$ x4 - x3 + x2 - 2x + 4 $D_{4}$ $1$ $-2$
2.224.4t3.a.a $2$ $2^{5} \cdot 7$ x4 - x2 + 2 $D_{4}$ $1$ $0$
2.248.4t3.c.a $2$ $2^{3} \cdot 31$ x4 - 2x3 + x2 - 2 $D_{4}$ $1$ $0$
2.256.4t3.c.a $2$ $2^{8}$ x4 - 2 $D_{4}$ $1$ $0$
2.259.4t3.a.a $2$ $7 \cdot 37$ x4 - 2x3 + x + 2 $D_{4}$ $1$ $0$
2.260.4t3.a.a $2$ $2^{2} \cdot 5 \cdot 13$ x4 + x2 - 4x + 4 $D_{4}$ $1$ $0$
2.260.4t3.b.a $2$ $2^{2} \cdot 5 \cdot 13$ x4 - 3x2 - 2x + 5 $D_{4}$ $1$ $0$
2.264.4t3.d.a $2$ $2^{3} \cdot 3 \cdot 11$ x4 - 2x3 + x2 + 2 $D_{4}$ $1$ $0$
2.264.4t3.c.a $2$ $2^{3} \cdot 3 \cdot 11$ x4 - 2x3 + 5x2 - 2x + 1 $D_{4}$ $1$ $0$
2.275.4t3.c.a $2$ $5^{2} \cdot 11$ x4 - x3 + x2 + 4x - 4 $D_{4}$ $1$ $0$
2.276.4t3.e.a $2$ $2^{2} \cdot 3 \cdot 23$ x4 - 2x3 - x2 + 2x - 2 $D_{4}$ $1$ $0$
2.276.4t3.f.a $2$ $2^{2} \cdot 3 \cdot 23$ x4 + x2 - 6x + 1 $D_{4}$ $1$ $0$
2.279.4t3.b.a $2$ $3^{2} \cdot 31$ x4 - x3 - 3x2 - x + 7 $D_{4}$ $1$ $0$
2.291.4t3.c.a $2$ $3 \cdot 97$ x4 - x3 - 4x2 + x + 7 $D_{4}$ $1$ $0$
2.292.4t3.a.a $2$ $2^{2} \cdot 73$ x4 - x2 - 4x + 5 $D_{4}$ $1$ $0$
2.295.4t3.c.a $2$ $5 \cdot 59$ x4 - x3 - x2 + 5x - 5 $D_{4}$ $1$ $0$
2.299.4t3.a.a $2$ $13 \cdot 23$ x4 - x3 + 5x - 1 $D_{4}$ $1$ $0$
2.305.4t3.a.a $2$ $5 \cdot 61$ x4 - 2x3 + 6x2 - 5x + 5 $D_{4}$ $1$ $-2$
2.308.4t3.d.a $2$ $2^{2} \cdot 7 \cdot 11$ x4 - x3 - 2x2 + 2x + 4 $D_{4}$ $1$ $0$
2.308.4t3.c.a $2$ $2^{2} \cdot 7 \cdot 11$ x4 - x3 + x2 + 4x + 2 $D_{4}$ $1$ $0$
2.320.4t3.a.a $2$ $2^{6} \cdot 5$ x4 - 4x2 + 5 $D_{4}$ $1$ $0$
2.323.4t3.a.a $2$ $17 \cdot 19$ x4 - 2x3 - 2x2 + 3x - 2 $D_{4}$ $1$ $0$
2.327.4t3.c.a $2$ $3 \cdot 109$ x4 - x3 - 4x2 + 4x + 7 $D_{4}$ $1$ $0$
2.328.4t3.c.a $2$ $2^{3} \cdot 41$ x4 - 2x3 - 3x2 + 2x + 1 $D_{4}$ $1$ $2$
2.328.4t3.d.a $2$ $2^{3} \cdot 41$ x4 - 2x3 + 3x2 - 2x + 3 $D_{4}$ $1$ $0$
2.336.4t3.a.a $2$ $2^{4} \cdot 3 \cdot 7$ x4 - 5x2 + 7 $D_{4}$ $1$ $0$
2.336.4t3.b.a $2$ $2^{4} \cdot 3 \cdot 7$ x4 + 5x2 + 7 $D_{4}$ $1$ $0$
Next
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2020-09-28 13:04:54
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|
https://calendar.math.illinois.edu/?year=2017&month=10&day=24&interval=day
|
Department of
# Mathematics
Seminar Calendar
for events the day of Tuesday, October 24, 2017.
.
events for the
events containing
More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
September 2017 October 2017 November 2017
Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa
1 2 1 2 3 4 5 6 7 1 2 3 4
3 4 5 6 7 8 9 8 9 10 11 12 13 14 5 6 7 8 9 10 11
10 11 12 13 14 15 16 15 16 17 18 19 20 21 12 13 14 15 16 17 18
17 18 19 20 21 22 23 22 23 24 25 26 27 28 19 20 21 22 23 24 25
24 25 26 27 28 29 30 29 30 31 26 27 28 29 30
Tuesday, October 24, 2017
11:00 am in 345 Altgeld Hall,Tuesday, October 24, 2017
#### Categories for $K$-theory and Devissage
###### Jonathan Campbell (Vanderbilt)
Abstract: What sorts of categories can K-theory be defined for? We know that exact categories and Waldhausen categories can be used as appropriate input. However, there are geometric categories where we would like to define K-theory where we are only allowed to cut and paste" rather than quotient --- examples of these include the category of varieties, and the category of polytopes. I'll define a more general context where one may talk about the algebraic K-theory of these categories, and outline a proof of a general version of Quillen's devissage. I'll outline applications to studying "derived motivic measures" and the scissors congruence problem. This is joint work with Inna Zakharevich.
12:30 pm in 222 Loomis,Tuesday, October 24, 2017
#### Entropic A theorem and the Markov property of the vacuum
###### Eduardo Teste (Centro Atomico de Bariloche, Argentina)
Abstract: A state is said to be Markovian if it fulfil the important condition of saturating the Strong Subadditivity inequality. I will show how the vacuum state of any relativistic QFT is a Markov state when reduced to certain geometric regions of spacetime. For the CFT vacuum, the Markov property is the key ingredient to prove the A theorem (irreversibility of the RG flow in relativistic QFT in d=4 spacetime dimensions) using vacuum entanglement entropy. This extends the entropic proofs of the c and F theorems in dimensions d=2 and d=3, giving a unified picture. I will also comment on the relation of this Markov property with the unitarity bound and holography.
1:00 pm in 345 Altgeld Hall,Tuesday, October 24, 2017
#### Cancelled
2:00 pm in 243 Altgeld Hall,Tuesday, October 24, 2017
#### Greedy bases and Lebesgue-type constants
###### Pablo Berna (University of Murcia)
Abstract: Given a basis $\mathcal{B}:=(e_j)_{j=1}^\infty$ of a Banach space $\mathbb{X}$, the $Greedy$ $Algorithm$ $\lbrace \mathcal{G}_m\rbrace_{m=1}^\infty$ is an algorithm of approximation where the objective is to approximate each element $x\in\mathbb{X}$ by $\mathcal{G}_m(x) = \sum_{j\in A(x)}e_j^*(x)e_j$, where $A(x)$ is a set of $m$ indices associated with the largest coefficients of $x$ in absolute value. In this talk we will study the $Greedy$ $Bases$ and we will obtain new bounds of the called $Lebesgue-type$ $constants$ for the greedy approximation. These new bounds are given only in terms of the upper democracy functions of the basis and its dual and we will also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces.
2:00 pm in 347 Altgeld Hall,Tuesday, October 24, 2017
#### Multifractal analsyis of jump diffusion processes
###### Xiaochuan Yang (Michigan State)
Abstract: In this talk, I explain how to use tools from analysis and fractal geometry to describe the sample paths regularity of the solution of SDE with jumps. The key probabilistic argument is a new upper tail estimate for the increments of the solution. Examples include non degenerate stable driven SDEs and variable order stable-like processes in the sense of R. Bass.
3:00 pm in 241 Altgeld Hall,Tuesday, October 24, 2017
#### How to construct good voting trees
###### Adam Wagner (Illinois Math)
Abstract: A well-known question in social choice theory is the following: given a collection of n candidates which are each pairwise comparable, how should we select a winner? The pairwise comparisons between candidates may be encoded as a tournament of n vertices, with edges directed from the winner to the loser in each comparison. The subtlety of the problem lies in the fact that the tournament need not be transitive, and therefore there is no indisputable way to select a global winner. One natural way of selecting a winning candidate from a tournament is to use a voting tree, a complete binary tree with leaves labelled from [n] (with possible repeats). Given any tournament as input, a voting tree deterministically selects a winner from the tournament by running pairwise elections between the leaves until we arrive at the root node. We want the winning candidate to beat many other candidates, so we define the performance guarantee of a voting tree as the minimum out-degree of any winner that it produces. We survey the question of how to construct good voting trees using deterministic and random methods and what the limitations of our techniques are.
3:00 pm in 243 Altgeld Hall,Tuesday, October 24, 2017
#### Degree bounds for invariant rings of quivers
###### Visu Makam (University of Michigan)
Abstract: The ring of polynomial invariants for a rational representation of a reductive group is finitely generated. Nevertheless, it remains a difficult task to find a minimal set of generators, or even a bound on their degrees. Combining ideas originating from Hochster, Roberts and Kempf with the study of various ranks associated to linear matrices, we prove "polynomial" bounds for various invariant rings associated to quivers. The polynomiality of our bounds have strong consequences in algebraic complexity, notably a polynomial time algorithm for non-commutative rational identity testing. This is joint work with Derksen.
4:00 pm in 245 Altgeld Hall,Tuesday, October 24, 2017
#### Town Hall Meeting
Abstract: All members of the department are invited to this town hall meeting organized by the Department Chair Search Committee.
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2018-12-14 03:21:14
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https://angularquestions.com/2019/09/16/how-to-ignore-the-text-check-while-typing-in-ngtagsinput/
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# How to ignore the text check while typing in ngTagsInput?
I’m having a problem with the ng-tags-input directive, when i enter the first tag name D, it can be added success. With the second tag, i entered Đỏ but somethings cause the tag invalid, although i also set allow-leftover-text="true" in <tags-input><tags-input>. So does anyone has experiences about this issues?
Here is my input:
<tags-input ng-model="dataProducts.colors" placeholder="Nhập các màu" allow-leftover-text="true" replace-spaces-with-dashes="false"></tags-input>
Source: New feed
Source Url How to ignore the text check while typing in ngTagsInput?
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2020-01-20 20:21:41
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5646247267723083, "perplexity": 10145.948399857203}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250599789.45/warc/CC-MAIN-20200120195035-20200120224035-00092.warc.gz"}
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https://www.gradesaver.com/textbooks/math/calculus/calculus-3rd-edition/chapter-1-precalculus-review-1-1-real-numbers-functions-and-graphs-exercises-page-10/30
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## Calculus (3rd Edition)
a. 9 is the maximum value b. $x^2 − 16 ≤ 9$ is true. See proof
Solve for x: $x-4\leq1$ and $x-4\geq-1$ $x\leq5$ and $x\geq3$ a. The maximum value can be found by using the greatest value of x, which is 5 $|5+4|=9$ b. $|x^2 − 16| ≤ 9$ solve for $x$: $x^2 − 16 ≤ 9$ and $x^2 − 16 \geq -9$ $x\leq5$ and $x\geq\sqrt{7}$, this does not contradict the previous conditions that we found for x, which is $x\leq5$ and $x\geq3$. Therefore, $x^2 − 16 ≤ 9$ is true.
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2020-11-28 19:51:52
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|
https://stats.stackexchange.com/questions/138392/cluster-prediction-of-incoming-time-seriespartial
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# Cluster prediction of incoming time series(partial)
I have a data set (24 x 1000) (hour x kwh) which contains 1000 time series of a buildings' power consumption, measured every hour. After applying k-means clustering using the dtw criterion I create 5 clusters as shown in the image below.
For a new day I am starting to collect values for the incoming time series. Hour 0 = 1.8 kwh, Hour 1 = 0.6 kwh, etc.
I want to create a model that will give me an indication from hour 0. That indication will show me how likely it is the incoming partial time series to belong in each cluster and it will change for every hour that I have a new incoming value. How would you approach this problem? If my descriptions is vague, please ask me anything. I am thinking a probabilistic solution...
Thank you.
• I like eigen-calendars. – EngrStudent Apr 9 '18 at 18:06
One option:
For each cluster, estimate a distribution over time series, using the data assigned to that cluster as training points. This gives $P(X_t \mid C)$, where $X_t$ is a time series from $0$ to $t$, and $C$ is the cluster. You'll need a separate distribution for every value of $t$. You could use a simple multivariate Gaussian (probably with some constraints on the covariance matrix). Or, you could use some kind of temporal model.
Estimate the prior probability of each cluster as the fraction of points assigned to that cluster in your original data set. This gives $P(C)$.
Given a new time series $x_t$, the goal is to calculate the probability of membership in each cluster, $P(C \mid X=x_t)$. Use Baye's rule:
$$P(C \mid X=x_t) = \frac{P(X=x_t \mid C) p(C)}{P(X=X_t)}$$
You can just calculate the numerator for each class, then normalize so that the sum is 1.
Alternatively:
Consider the cluster ids as class labels. Train your favorite probabilistic classifier on the original set of time series and corresponding cluster ids. You'll need to train a different classifier for each choice of $t$. Feed each new time series $x_t$ to the corresponding classifier, which will output the membership probabilities $P(C \mid X=x_t)$.
For different reasons (financial time series) I had the same problem and I solved the issue as the collegue above explained. The only weakness I see in this approach is that if you are studying a dynamic system also "the fraction of points assigned to each cluster in your original data set" changes over time, needing another level of dynamics on P(Cis) evolution.
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2019-06-24 22:25:51
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http://mathhelpforum.com/math-topics/62319-roots-quadratic-equation.html
|
1. ## The roots of a quadratic equation
Can anyone kindly help me with the following please?
The roots of the equation x2 + 3x − 2 = 0 are α and β. Form a new equation with roots:
(a) α - 1 and β + 2
(b) α +2 and β + 1
The sum of the new roots is easy but I am finding the product of the new roots difficult.
2. Originally Posted by yobacul
Can anyone kindly help me with the following please?
The roots of the equation x2 + 3x − 2 = 0 are α and β. Form a new equation with roots:
(a) α - 1 and β + 2
(b) α +2 and β + 1
The sum of the new roots is easy but I am finding the product of the new roots difficult.
The roots are $\alpha = \frac{-3 + \sqrt{17}}{2}$ and $\beta = \frac{-3 - \sqrt{17}}{2}$.
Use these to calculate the required new roots. Now construct new quadratic equations form these roots.
(a) $\alpha - 1 = \frac{-5 + \sqrt{17}}{2}$ and $\beta + 2 = \frac{1 - \sqrt{17}}{2}$.
So a new equation is $\left( x + \frac{5 - \sqrt{17}}{2} \right) \cdot \left( x - \frac{1 - \sqrt{17}}{2}\right) = 0 \Rightarrow \, ....$
3. ## The roots of a quadratic equation
Thanks for your reply. However, the teacher told us that we have to find the new quadratic equation without solving the previous; in the sense that we are required to make use of the sum of roots and product of roots of the original equation.
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2017-08-22 20:59:10
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https://www.physicsforums.com/threads/photons-wavelength-and-wavepackets.227078/
|
Photons - Wavelength and Wavepackets
1. Apr 6, 2008
Usaf Moji
Hi, forgive me if this sound noobish...
For photons, I've been struggling with the interrelationship between Maxwell waves and Schrodinger waves, and further, as to the relationship between the wavelength of light and the "wavelength" of the Gaussian envelopes of wave packets.
(I guess this is only for those who accept that photons can be described using Schrodinger's equation(s) and the idea of wave packets - yeah, I saw that wikipedia note that says otherwise.)
In textbooks, a light wave is often depicted as having little wiggles within a broader Gaussian envelope - like the way they look in this guy's notes:
http://www.phys.unsw.edu.au/~sjc/physics1/summer/q25.jpg
I understand that when we speak of the "wavelength" of light, we're referring to the wavelength of the smaller sinusoidal parts within the Gaussian envelope. And I understand that the Gaussian envelope results from summing several of these smaller sinusoidal waves of varying wavelengths (varying due to the uncertainty principle).
So my question is, is the wavelength of the larger Gaussian envelope proportional to the (mean) wavelength of the sinusoids within it? In other words, if all else were equal, would red light have longer envelopes than say blue light?
All responses appreciated.
2. Apr 7, 2008
clem
The length of the envelope is unrelated to the wave length of the light, unless you go to the extreme where the length of the envelope gets close to the wave length.
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2018-02-23 13:28:58
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https://www.gamedev.net/forums/topic/320055-connect4/
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# connect4
This topic is 4595 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I'm currently working on a connect4 game as my second game and am starting to get the hang of the slightly different mindset used for this. A couple of questions though. But first here's a look at the current source. Please don't add to the source... I kinda want to figure it out myself.
#include <iostream>
#include <iomanip>
using namespace std;
const char PLAYER1 = 'R';
const char PLAYER2 = 'Y';
const int MAXROWS = 6;
const int MAXCOLUMNS = 7;
const int MAXTURNS = MAXROWS * MAXCOLUMNS;
const char player[] = {PLAYER1, PLAYER2};
char board[maxRows][maxColumns];
void resetBoard()
{
for(int i = 0; i < MAXROWS; ++i)
{
for(int j = 0; j < maxColumns; ++j)
{
board[i][j] = ' ';
}
}
}
void displayBoard()
{
for(int i = 0; i < MAXROWS; ++i)
{
for(int j = 0; j < MAXCOLUMNS; ++j)
{
cout << board[i][j];
}
cout << endl;
}
for(int i = 0; i < MAXCOLUMNS; ++i)
{
cout << "-";
}
cout << endl;
for(int i = 0; i < MAXCOLUMNS; ++i)
{
cout << i + 1;
}
cout << endl;
}
void getInput(int &row, int &col)
{
int move;
while(true)
{
cout <<
}
}
int main()
{
resetBoard();
int turn = 0;
int rowChoice, colChoice;
int winner = 0;
while(turn < MAXTURNS)
{
displayBoard();
getInput(rowChoice, colChoice);
board[rowChoice][colChoice] = player[turn%2];
if(winner = checkWin(rowChoice, colChoice))
{
break;
}
}
switch(winner)
{
case PLAYER1:
cout << "Player 1 has won the game." << endl;
break;
case PLAYER2:
cout << "Player 2 has won the game." << endl;
break;
default:
cout << "The game ended in a tie." << endl;
break;
}
return 0;
}
Obviously, I still have a lot of work left to do on it. My main question, is what would be better for checking the win condition. I've thought about either making it check through all possible combinations...(brute force approach) or putting the last move made into the function, and checking in the eight directions around it. I want to get this game going well... since it will be one of my first 2D games to do. I'll port most of the code. I understand that the brute-force approach is highly doable with the scale of a board i am using... however if i increase it... the problem grows worse... whereas it doesn't with the other approach. Thoughts and opinions are welcome. But please don't post code, until I actually put it up for scrutiny.
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If you're doing this to practice programming, I'd say both. The reason I said this, is because I probably would've gone for brute-forcing. But I find the more systemated solution extremely interesting.
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You have the right idea about checking the winning condition (the non-brute force one that is). I was going to say something about the input method, but since it's basically empty, I'm going to leave that until you put some code into it. ;)
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The input method is going to be setup so that they pick a column to put the piece into and then it tests to see if that's a valid move. If it is a valid move it assigns the passed by reference values to the column and the first vaild row. I'm about to invert the board display so [0][0] is the bottom left. Any other suggestions(not code) is welcome.
##### Share on other sites
If I was going to brute force it I would probably end up designing an algorithm that took any specific location on the board and then searched for any and all solutions that involved said specific location. If I was being more conservative I might only use the specific location as an endpoint to the four in a row to ignore overlap. But if I used the former, then it would be pretty much the same algorithm you'd use to find if the most recent play caused a win condition. I don't know if this helps at all, but there you go.
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This topic is 4595 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Create an account
Register a new account
• ### Forum Statistics
• Total Topics
628719
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2017-12-15 22:13:45
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https://www.quantamagazine.org/20161013-how-can-we-tell-which-forecasts-are-true/
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# How Can We Tell Which Forecasts Are True?
Presidential election forecasts are historically successful and appear to be highly precise. Yet they’re often contradictory. What would it take to trust them?
Olena Shmahalo/Quanta Magazine
20
Are you one of those people who are so involved with the U.S. presidential election that you anxiously visit election-forecasting sites such as FiveThirtyEight, the Upshot or the Princeton Election Consortium (PEC) to look at their latest numbers? These poll-aggregating sites use sophisticated mathematical models to forecast the result of the presidential race. They provide numbers that boldly give each candidate’s probability of winning. This month, Insights invites Quanta readers to puzzle about principled ways to decide how much credence should be given to these numbers.
A monthly puzzle celebrating the sudden insights and unexpected twists of scientific problem solving. Your guide is Pradeep Mutalik, a medical research scientist at the Yale Center for Medical Informatics and a lifelong puzzle enthusiast.
We are all familiar with innumeracy — the lack of a basic knowledge of mathematics. A significant proportion of the population is effectively mathematically illiterate and can easily be led astray by statistical statements and quantitative arguments in news stories. Innumeracy usually afflicts people who are not good with numbers, but there is a subtler affliction that does not spare mathematically competent people and sometimes affects them even more virulently — overprecision. As I described in my solution to Is Infinity Real?, we are conditioned to respect numbers calculated to multiple decimal places. We often ascribe mystical accuracy to them and fail to examine whether their precision is justified.
We have all encountered examples of this in real life. The standard joke concerns the museum docent who tells a visitor that a dinosaur skeleton is 75,000,005 years old. Why? The curator explains that it was 75 million years old when he started working at the museum five years earlier. I remember a geography textbook that declared that the distance between two cities was “about 1609.3 kilometers,” an absurdly overprecise translation of “about a thousand miles” that occurred when the country went metric. And I remember my then five-year-old son, who was used to precise Saturday-morning cartoon schedules, saying to me, “Daddy, come inside, it’s three minutes to the thunderstorm.”
Question 1
FiveThirtyEight and the PEC both predicted the outcome of the 2012 presidential race with spectacular accuracy. Yet on September 26 of this year, just before the first Clinton-Trump debate and 43 days before the election, FiveThirtyEight predicted on the basis of highly complex calculations that the chances for Clinton to win the election were 54.8 percent, while the PEC, with no doubt an equally complex model, estimated her chances to be 84 percent, almost 30 points higher. Both these models have highly successful histories, yet these numbers cannot both have been right. Do you think either number is justified as a principled prediction? Are these predictions meant only to titillate our psychological compulsion to have a running score, or do they have any other value? How many significant digits of these numbers can be trusted? How many presidential elections would it take to definitely conclude that one of these September 26 predictions was better than the other? And finally, as someone who backs a particular candidate, what is the wisest practical attitude to have about these probabilities?
In order to reason about this, we have to define what these probabilities mean. In Bayesian theory, the probability that a future event will occur is called a subjective probability, or credence, something that we encountered in the Sleeping Beauty puzzle. You can make the subjective probability practical by means of a bet: If you believe on a certain day that Clinton has, say, an 80 percent probability of winning, it means that you would rationally accept a bet that that gave you more than $5 for every$4 that you bet. Bookmakers make a living on this, so it matters a great deal to them. Interestingly, the odds offered by the major betting site Predict Wise on the same day reflected a probability that Clinton would win to be about halfway between the FiveThirtyEight and the PEC forecasts: 68 percent. But even if you win the bet, it does not validate the model you are using. You might have just been lucky.
Of course, if you are a purist and betting is not something you care for, you might give up on the actual numbers and focus only on the trend line. Every presidential race is punctuated by events that might affect the outcome — debates, revelations, hackings, foot-in-the-mouth episodes, tax-return sightings, 11-year-old videotapes and so on. It is often easy to say whether each event will affect a particular candidate’s chances positively or negatively. Yet trend lines cannot easily satisfy that insistent voice within you that wants to know the “score” with the faux authority of a decimal place.
Nevertheless, there are some insights that can be gleaned just by looking qualitatively at the shapes of curves that do not have any numbers on them at all — something that would horrify a middle-school teacher teaching elementary graphs. Below I discuss a puzzle based on one of my favorite curves, the upside-down U curve, or what I call the “why too much of anything is bad” curve. This curve is similar to the Gaussian bell curve, or the downward parabola. But you do not need any numbers to learn from this curve. It occurs extremely often in real-world situations. Even those who are allergic to numbers can benefit from studying it and practice spotting it in the world around them.
As you can see, in the left panel you have on the x-axis the dose or amount of a food, drug, activity or pretty much anything that you can have in life. On the y-axis is some measure of your well-being. What the curve tells you is that anything that is good for you will, in excess, hurt you — including, yes, being too rich or too thin.
In the right pane, you see the same curve illustrating an important point in evolutionary theory. Here the x-axis is any characteristic that has a genetic basis — say, your appetite for sugar, based on your genes. The y-axis is your evolutionary fitness — how much this trait contributes to your reproductive success. This graph helps to explain the truism “Everything you like is bad for you.” We originally evolved in an environment where things like sugar, fat and food were scarce — on the left, or upward, arm of the curve — so we evolved to like these far too much. In these days of plenty, we find ourselves on the right arm of the curve, resulting in cravings that are bad or even prematurely fatal for us.
Question 2
The graph in the right panel is the basis of an insight originally articulated by the eminent statistician and evolutionist Sir Ronald Fisher. Imagine that you as an individual are situated at some point on the curve. Presumably this would be at some point near the top, since the fact that you exist implies that you are a product of evolutionarily fit ancestors. If you have a large genetic mutation, you are likely to move to a different point on the curve, which will probably be bad for you. Yet, Fisher asserted, if the mutation were small enough, it would have a 50 percent chance of being good for you. Can you figure out why this should be so?
What will happen if we rotate the graph into a three-dimensional figure looking like a rounded cone and use it to represent not one but thousands of different characteristics? In what way will this modify these conclusions?
There is much to be learned just by looking at shapes of curves. For instance, we all know about the curve for exponential growth in mathematics. But, for the same reason that infinity does not exist in the real world, this curve does not really exist in real life — the curve too-quickly shoots toward infinity. It is usually replaced by the s-shape curve or the “all good things must end” curve (the generic version of the logistic curve) — or, if things get really ugly, by the upside-down U curve we just saw.
Understanding such qualitative processes with numberless curves does indeed provide insights, but you need to be careful. It is easy even for experts to reach false or controversial conclusions by considering only the shapes of graphs, as has been the case with the Laffer curve, which has the upside-down U shape we discussed above. So in most cases, it is indeed wise to follow the teaching of your middle-school math teacher and insist on proper numbering and scaling of all graphs.
Furthermore, every graph needs to be looked at with a critical eye, and the data examined in context, especially when the prediction is an electoral claim. It is as easy to mislead with graphs and numbers as it is with words. To paraphrase a common warning: caveat suffragator — let the voter beware! Coincidentally, the Latin root for “vote” from which we get the word “suffrage” is uncomfortably similar to the word “suffering.”
I look forward to further insights from you, Quanta readers.
Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt. (Update: The solution is now available here.) And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION” (it will not appear online, so solutions to the puzzle above should be submitted separately).
Note that we may hold comments for the first day or two to allow for independent contributions by readers.
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• J E says:
1) I think neither of these predictions are terribly accurate. For one, they are wildly disparate, so as a result, assuming they adequately predict the same thing, this at minimum means that the thing being measured in this particular instance has a huge standard deviation. This would make both predictions pretty worthless. One could argue the average of the two polls would produce a more accurate prediction (a situation which typically holds), but I don't think that is the case here.
I would argue that the two models (which I would argue are good models in typical circumstances) disagree to such a large extent due to the distribution of registered voters this election cycle. According to Pew Research, this is the trends for registered voters in terms of democrat, republican, independent:
http://www.people-press.org/interactives/party-id-trend/
I would particularly watch the trend from sometime during the Reagan years until Obama's first election, and even Obama's second election. During this period the proportions for each group were fairly consistent, especially if one ignores the trend from 2012 on. After 2012, however, one can clearly see the number of republicans drop and the number of independents steadily rise. The data in the graph is through 2014, and one can rest assured that the trend has continued. At this point, the proportion of registered voters who identify as independents has likely reached greater than 50%.
Add into this that the debate between Clinton and Trump has been lowered to the point where they are arguing about who lies the most, who either is or is associated to the rapiest male in a position of power, who will start WWIII sooner, who has strong association to some foreign power, who is associated with more nefarious foreign powers, etc. and you can rest assured that the situation will get even more extreme, with greater and greater numbers of registered voters identifying as independents. Also as a result, I firmly believe that a large block of registered voters swing wildly back and forth between support for Trump or Clinton, and this, in addition to a smaller proportion of registered voters identifying as either democrat or republican (and thus being more narrowly focused groups value-wise and which also happen to be not that different), is what the deviation in the above cited models represents.
With all that, I find it strange that neither model seems to make any kind of prediction for a win from an independent, particularly Jill Stein. As an aside, imagine if Stein was getting Clinton's media attention, and could count on all of her positives being blared around while any negatives get entirely ignored. Imagine the same for Bernie, or Trump. Also, why can't Clinton beat the spread, or even come close to beating the spread, on Trump? Hmmmm, does someone have a preferred order: Clinton, Trump, Johnson, and…who is the other candidate again…….oh yea Jill Stein.
2) Because any random individual has an extremely low probability of actually being the average. Thus if the mutation is small, it has about a 50% chance of moving the individual closer to the average, which in this case represents maximum fitness, and thus represents a 50% chance for an improvement.
I am assuming in the second part of the question, you are picturing a bunch of graph lines radiating from the peak of a 3-D distribution such that each line represents a different trait. Keep in mind your measures would have to be changed in potentially non-linear ways in order to make all the traits occupy the same physical space within a distribution. In this case, nothing changes. If we have the same mental picture, I am curious why you thought it would, and if we have a different mental picture, I would love to have a more thorough description of what you are picturing
• Michael says:
Question 1
"These numbers cannot both have been right". I don't agree with this, in the following sense. If a model M predicts a sequence of probabilities { p(E_k|M) }, for events E_1, E_2,… to occur (e.g., winners of various elections), and the result r_k is defined to be 1 if E_k does occur and 0 otherwise, then the model can be said to be successful over a large number N of predictions if the quantity
error_N =| sum_k p(E_k|M) – sum_k r_k |/N
is sufficiently small. The notion of "sufficiently" relates to how much one is willing to lose in the long run. In usual probability theory error_N is expected to scale like 1/sqrt{N}.
The above definition of success agrees with the usual that if each event is given the same independent probability, p(E_k|M)=q, by the model, then the total number of occurrences is ~Nq. But it also allows for the case where each event is intrinsically unique, such as in an election. With this definition, two different models for unique events can each be successful overall, while giving different individual predictions.
In this sense, there is only a case for saying one model must be "wrong", i.e., for bookmakers to prefer one over the other, if, over a long sequence, they lose more money on it than on the other – e.g., if error_N is larger for one than the other. For finite N (which is always the case in practice), this comes down in part to the size of fluctuations from the mean.
Of course, if one imagines there is some "true" model that, given all the variables, predicts the result correctly every time (with p(E_k|M) = 0 or 1 in every case), then neither of the models mentioned are "right". This could be the case if the events lie in the past (but elections don't), or the world both deterministic (it doesn't seem to be) and predictable with current computational resources (no way).
In practice, I guess one would look at the past fluctuations of each model from the actual outcomes, in addition to error_N from the past outcomes, in deciding which one to use for this particular election. But one can then also subdivide past elections into categories based on similarities, and look only at the predictions of the models for the category in which the present election fits. One will soon give up, I think!
The important thing is that in making a bet on the outcome, one cannot bet on the probability (54% or or 85%), but only on the outcome itself. Of course, one could make side-bets on the deviation of the outcome from the predicted probability of the model, but this is a different bet.
Question 2:
The curve looks like the Laffer curve, which I've seen used to explain things like why paying too little tax can be as bad as paying too much tax (if one wants things like roads, defence forces, and other infrastructure).
I'
• Michael says:
Question 1
"These numbers cannot both have been right". I don't agree with this, in the following sense. If a model M predicts a sequence of probabilities { p(E_k|M) }, for events E_1, E_2,… to occur (e.g., winners of various elections), and the result r_k is defined to be 1 if E_k does occur and 0 otherwise, then the model can be said to be successful over a large number N of predictions if the quantity
error_N =| sum_k p(E_k|M) – sum_k r_k |/N
is sufficiently small. The notion of "sufficiently" relates to how much one is willing to lose in the long run. In usual probability theory error_N is expected to scale like 1/sqrt{N}.
The above definition of success agrees with the usual that if each event is given the same independent probability, p(E_k|M)=q, by the model, then the total number of occurrences is ~Nq. But it also allows for the case where each event is intrinsically unique, such as in an election. With this definition, two different models for unique events can each be successful overall, while giving different individual predictions.
In this sense, there is only a case for saying one model must be "wrong", i.e., for bookmakers to prefer one over the other, if, over a long sequence, they lose more money on it than on the other – e.g., if error_N is larger for one than the other. For finite N (which is always the case in practice), this comes down in part to the size of fluctuations from the mean.
Of course, if one imagines there is some "true" model that, given all the variables, predicts the result correctly every time (with p(E_k|M) = 0 or 1 in every case), then neither of the models mentioned are "right". This could be the case if the events lie in the past (but elections don't), or the world both deterministic (it doesn't seem to be) and predictable with current computational resources (no way).
In practice, I guess one would look at the past fluctuations of each model from the actual outcomes, in addition to error_N from the past outcomes, in deciding which one to use for this particular election. But one can then also subdivide past elections into categories based on similarities, and look only at the predictions of the models for the category in which the present election fits. One will soon give up, I think!
The important thing is that in making a bet on the outcome, one cannot bet on the probability (54% or or 85%), but only on the outcome itself. Of course, one could make side-bets on the deviation of the outcome from the predicted probability of the model, but this is a different bet.
Question 2:
The curve looks like the Laffer curve, which I've seen used to explain things like why paying too little tax can be as bad as paying too much tax (if one wants things like roads, defence forces, and other infrastructure).
I'm not sure that I understand the question. But since one is unlikely to be sitting at the optimal point on the curve, but slightly to the right or to the left, then a small mutation will have a 50 percent chance of moving towards the maximum, and a 50 percent chance of moving away from it, no matter which side one is sitting on. The primary reason would seem to be that, to lowest order, the curve is symmetric in the vicinity of the maximum, being well approximated by a parabola in this region.
• Michael says:
Question 2 (again)
Ah, I see the article mentions the Laffer curve, and also more dimensions. If the shape was a multidimensional cone, then the advantage depends on the prior distribution for moving in any given direction – there could be correlations between directions, e.g., mutations of nearby genes. But if it is symmetric, then the small mutation near the peak is more likely to be disadvantageous. E.g., for a symmetric paraboloid with a two-dimensional surface, the lines of constant fitness are circles on the plane. Moving a short distance in a random direction from a point on such a circle is more likely to take one outside the circle than inside it (a bit like the recent drunkard's walk problem). The probability will depend on the ratio of the step to the radius of the circle.
I guess the geneticists do look at random walks on the fitness surface, to model genetic drift and diffusion. That would be interesting.
• Mark P says:
Question 1:
I think both numbers can be justified. Each come from separate models for the world, predictions about how the world situation could evolve and how likely those different paths are.
As for the percents, you can trust as many significant figures as the modeler provides. The percents are accurate with respect to a model. If the modeler calculated them through simulation such as Monte Carlo, they would know how many significant figures to report based how many samples they ran / how many additional samples it would take to change a digit in the percent.
I don't think one would ever be able to conclude that one of these predictions was better than the other. It's impossible to duplicate this election. We can't wipe the slate clean, wipe everyone's memories, and rerun it (though frankly I think I know a lot of people who wished they could forget entirely about this election! :-)). By watching many presidential elections over time, one could conclude that the system FiveThirtyEight uses to build models is likely better than the system PEC uses to build models (or vice versa), but it's impossible to conclude that a particular model was more right from one versus the other. To restate, it's not possible to take multiple samples of this election to see which model estimates the true probability more accurately.
As someone who backs a candidate, I think the wisest action is to ignore the predictions. The predictions are supposed to take the state of the world into account and how it may evolve. If you pay attention to the prediction and change your action based on it (e.g., my candidate is going to win, I was going to canvas for him/her and now don't need to) then the prediction is based on a faulty premise. You're invalidating the prediction. Thus, you shouldn't act on a prediction because acting on it makes it wrong, which makes it illogical to have acted on it.
• eJ says:
In question 2, under various continuity & differentiability assumptions, the fitness function along a short vector (small-enough mutation) through almost any point in characteristic space will be either strictly increasing or strictly decreasing. Assuming a mutation and its reverse (opposite vector) are equally likely, fitness will thus be as likely increasing as decreasing when a small-enough mutation occurs.
• eJ says:
In question 1 — "these numbers cannot both have been right" — they can both be right w.r.t. their respective models: this is known as the Reference Class Problem.
• Erik says:
I respectfully disagree with the sentiment of this author. Of course the predictions differ, because thy are using different models and data for their predictions. That does not mean they are not useful. Polls also always differ, depending on which news network are conducting them (e.g. Fox vs. CNN).
Instead of trying to discredit the existing probability models it would have been more interesting if the author would have have looked at the models to evaluate why the differ and which model might be more reliable than the others.
• Ara K says:
Thank you for this very interesting article and commentaries. This is a fun read! I wonder if the author or any of the commentators would indulge a modification to the assumption of Question #2. If the "why too much of anything is bad” curve is replaced with either an extremely heavy tailed curve (or "cone") or an extremely skewed curve (or "cone").
• Daniel McLaury says:
Question 1:
Suppose you draw five playing cards from a freshly-shuffled deck, don't show them to me, and ask me what the chance is that there are more red cards than black ones. Obviously I'd be crazy to say anything other than 50%. But now suppose that I inadvertently saw one of the cards you drew, namely the ace of spades. My estimate would immediately drop to a little more than 30%. While the original 50% estimate was inarguably correct, it was not very robust to small changes in the information available to me. So, while the probability (50%) was an output of my model, it didn't really give the full story of what my model was telling me — the model could also supply additional relevant information, like the robustness of this probability to small changes in the inputs.
The U.S. Presidential election is not altogether different from the scenario described above. As we saw in 2000, it's possible to "win" the election by half a million votes and still lose due to the electoral college system. Since most electoral votes are all but determined before the candidates are even selected, predicting the election comes down to determining whether five or so "cards" (blocks of electoral districts) will turn out to be "black" (blue) or "red" (red).
Since these models are not black boxes, the best thing to do isn't to treat them as though they were (as we're implicitly doing by asking how many elections we'd have to see to determine which model was "correct"). Instead, we should take far more information from the models than simply the final probability they output, and only *then* compare their predictions to one another.
Given all this information, a supporter might see that the predicted outcome of the election is especially sensitive to, let's say, voter turnout in a particular locale, encouraging him or her to attempt to get out the vote there.
Question 2:
In fact, we need almost no assumption as to the nature of the curve, so long as we can assume it's smooth. Provided we are not exactly *at* a local maximum of fitness (which, given a unimodal fitness function like the one shown, is actual "maximum fitness,") then fitness is either increasing or decreasing with respect to the characteristic in question. In the former case, a small move to the left (i.e., small enough not to leap right over a local maximum) will hurt, and a small move to the right will help. In the latter case the situation is reversed. In either case, provided there's a 50/50 chance to move left or right then there's a 50/50 chance that the change will be a positive one.
The situation is not changed in higher dimensions, as all change happens in *some* direction. We may simply restrict our attention to the direction the change occurs in, reducing to the one-dimensional case considered before.
• Michael says:
@Daniel MacLaury
Re Question 2, I agree with you (and @eJ) that, in the 1D case, one just needs to be near a local maximum for a 50:50 chance of improvement by a sufficiently small mutation. My assumption that the curve needed to be symmetric is not necessary.
But I disagree that the chances remain 50:50 in the multidimensional case. One cannot, after mutation in a given direction, say that this improves fitness because it does if one restricts the definition of fitness to that direction. The fitness function is already defined on the surface, and one one has to consider its total change.
I gave an argument earlier that the mutation is more likely than not to be disadvantageous, based on the idea that if one moves randomly a small distance away in some direction from the circumference of a circle, then one is more likely to end up outside the circle (less fitness) than inside the circle (greater fitness). This was for a circularly symmetric fitness function. I will argue that it holds more generally further below.
But first, to give an explicit example for the circularly symmetric case, suppose the fitness function is defined on the plane of two parameters as
f(x,y) = 1000 – (x^2 +y^2),
with a maximum at the origin, x=y=0. If the organism starts at coordinate (a,b)= (1,0) and moves a small distance d<<1 in some random direction theta, the new coordinate will be
(a',b') = (1+d cos theta, d sin theta). The change in the fitness is therefore
f(a',b') – f(a,b) = 1 – [1+d cos theta]^2 – [d sin theta]^2
= – d^2 – 2d cos theta
= -d ( d + 2 cos theta).
The probability of this being positive is the probability that cos theta is less than -d/2, i.e, fixing theta to range over -pi to pi, that
theta > pi/2 + arcsin(d/2) or theta < -pi/2 – arcsin(d/2).
Since p(theta)=1/(2pi), this gives the probability of the mutation being positive as
p(positive) = 2[pi/2 – arcsin(d/2)]/(2pi)
= 1/2 – (1/pi) arcsin(d/2),
which is always less than 1/2 (for small d one has the approximation [1-d/pi]/2).
More generally, the fitness function will have a set of simple closed contours in the neighbourhood of the local maximum, with the value of the fitness function being constant on each contour, and decreasing as one moves away from the local maximum. It is clear that if these contours are convex, then moving a small fixed distance d away from a point on a given contour, in a random direction, will more likely end up outside the contour than outside (in the special case where the convex contour has a straight segment – e.g., the contour is a square – the probability of ending up inside or outside will be 50:50 only if one is more than a distance d from the closest bend or 'corner' of the contour). So, fitness will decrease on average under a small random mutation.
If the contours are not convex I think the same conclusion can be drawn, as long as the mutation is small compared to the curvature of the contour, and the contour has a finite length (e.g., is not a fractal). The idea here is to imagine, at each point P along a given contour C, drawing a small circle of radius d centred on P (here d is the size of the mutation). These circles taken together will fill up a strip extending outside the contour up to a distance d from C, and extending inside the contour up to a distance d. It is clear that the area A_out of the outside part of this strip is larger than the area A_in of the inside part of the strip. Hence, if one is equally like to start at any point P on the contour (admittedly, a questionable assumption), and moves d in a random direction, one expects the odds of increasing one's fitness, i.e., of moving inside the contour, to be approximately given by A_in:A_out, which is less than 50:50.
• eJ says:
@Michael, OK, I agree. You've answered the question for small d; I've answered it for small d(theta), which was a mistake. In the limit, for "infinitesimal" d, it's 50/50 plus-or-minus an infinitesimal amount.
• J E says:
@Michael: You wrote:
"But I disagree that the chances remain 50:50 in the multidimensional case. One cannot, after mutation in a given direction, say that this improves fitness because it does if one restricts the definition of fitness to that direction. The fitness function is already defined on the surface, and one one has to consider its total change. "
I agree, as I believe this is akin to saying that there is some interaction between fitness traits. For example, if we are measuring night sight fitness, since the eyes are on the face, when the eyes get larger, the nose and/or mouth must get smaller, and thus the fitness mechanisms associated with them would likely change. This sounds like a reasonable and sensible assumption to me.
The only critique I have is that it seems to me you are assuming that the given traits would be correlated such that a small change in one would lead to at most a small change in some other. I agree that this is a fair assumption, but it would not hold up in certain extreme circumstances. For example, to use the above example of eyes, the interaction could be such that after a certain point, for a small increase in eyes and thus night vision, a relatively great decrease could be seen in either the nose or mouth (such that the decrease in nose size does not effect overall fitness. This might be a poor and/or awkward example…), whereas increases in other traits don't have the same degree of interaction. Thus in this case the degree to which any given traits are inversely correlated (one goes down when one goes up, or vice versa) would, on a case by case basis, determine if changing a given trait by a small amount would increase fitness.
• J E says:
Just to clarify, in paragraph three above I wrote: "such that the decrease in nose size does not effect overall fitness" but meant "such that the decrease in nose size greatly and negatively effects overall fitness". My apologies for the confusion, and I hope the rest makes sense 🙂
• DanielMcLaury says:
@Michael:
I see, we interpreted the question differently — I was considering infinitesimal displacements, but you're considering the (apparently more interesting) case where the displacements are small but non-infinitesimal.
Looking back I see that I suggested that my infinitesimal displacements were just as good as finite ones that didn't cross a local maximum, which in retrospect is obviously wrong (except in one dimension).
• Hans says:
@Michael:
Assuming smoothness for the fitness function, for infinitesimal change of the position, the probability of improvement and deterioration is the same. This is born out from your argument using the areas immediately inside and outside the contour. In the limit (dividing the width), it is just the length of the contour approached from within and without the enclosed domain and the two limits are the same.
Also, for finite width, your argument is flawed. The areas do not faithfully represent the probability since the weights of all points are not equal as each point is reached from different starting points. You need to account for that. It is a bit more complicated than just eye-balling the area. It is much simpler just looking locally at each point and consider the tangent.
• eJ says:
Hans, are we (1) choosing the direction of the mutation then its size, or are we (2) limiting the size then choosing the direction? You could say there's a Bertrand Paradox-type ambiguity here, but on balance when the question says the "mutation is small enough" it's looks like Michael was right to go with the second.
• Hans says:
@eJ:
We can have a rigorous definition for this problem. There could be local (for each point) version and global (for the whole contour or even the whole surface) version. We can discuss this later. What I am objecting to is Michael's derivation using the area even under "nice" conditions, say, the contour having finite curvature at each point and all the curvatures are globally bounded. I am going to describe intuitively here. For distance less than a given bound, a point away from the contour can be reached by different amount of points on the contour. So the probability of all the points are all different. Computing the area is to assume the probability of all the points within a "width" of the contour being all equal. That is incorrect.
• eJ says:
Hans, I suspect we're being led towards box 2 of http://psych.colorado.edu/~carey/pdffiles/adaptation_orr.pdf or similar.
• Hans says:
@eJ:
Locally (meaning for fixed points), your definitions (1) and (2) coincide. The global scenario needs to be carefully defined.
For finite change size, whether probability of the mutation being favorable depends on the geometry of the fitness function. In fact, taking the negative of a fitness function divides the set of fitness functions in half (by one-to-one correspondence) where each move resulting in a favorable change is matched up with a deteriorating one.
The conclusion including the probability involving the Gaussian distribution regarding the finite movement in Box 2 of the paper you referenced has to depend on specific conditions which I am sure the author has neglected for ease of presentation.
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https://en.wikipedia.org/wiki/Differentiation_rules
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# Differentiation rules
This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
## Elementary rules of differentiation
Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined[1][2] — including the case of complex numbers (C).[3]
### Constant Term Rule
For any value of ${\displaystyle c}$, where ${\displaystyle c\in \mathbb {R} }$, for any value of ${\displaystyle x\in \mathbb {R} }$, ${\displaystyle {\frac {d}{dx}}\left(c\right)=0}$.[4]
#### Proof
Let ${\displaystyle c\in \mathbb {R} }$. Now, we will prove, from first principles, what the derivative is. Let ${\displaystyle x\in \mathbb {R} }$.
For clarity, we will define the new function ${\displaystyle f}$, so that ${\displaystyle f(x)=c}$. We want to show that ${\displaystyle {\frac {d}{dx}}\left(c\right)=0}$. From what we defined what ${\displaystyle f(x)}$ was, we can rewrite the statement:
{\displaystyle {\begin{aligned}&~{\frac {d}{dx}}\left(c\right)\\=&~{\frac {d}{dx}}f(x)=~f'(x)\\\end{aligned}}}
This is just done for clarity of notation. This way, we can use the definition of the derivative to find ${\displaystyle f'(x)}$.
{\displaystyle {\begin{aligned}f'(x)&=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}\\&=\lim _{h\to 0}{\frac {(c)-(c)}{h}}\\&=\lim _{h\to 0}{\frac {0}{h}}\\&=\lim _{h\to 0}0\\&=0\end{aligned}}}
Thus, we have found out that: for any value of ${\displaystyle c\in \mathbb {R} }$, for any value of ${\displaystyle x\in \mathbb {R} }$, ${\displaystyle {\frac {d}{dx}}\left(c\right)=0}$.
### Differentiation is linear
For any functions ${\displaystyle f}$ and ${\displaystyle g}$ and any real numbers ${\displaystyle a}$ and ${\displaystyle b}$, the derivative of the function ${\displaystyle h(x)=af(x)+bg(x)}$ with respect to ${\displaystyle x}$ is: ${\displaystyle h'(x)=af'(x)+bg'(x).}$
In Leibniz's notation this is written as:
${\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}$
Special cases include:
• The constant factor rule
${\displaystyle (af)'=af'}$
• The sum rule
${\displaystyle (f+g)'=f'+g'}$
• The subtraction rule
${\displaystyle (f-g)'=f'-g'.}$
### The product rule
For the functions f and g, the derivative of the function h(x) = f(x) g(x) with respect to x is
${\displaystyle h'(x)=(fg)'(x)=f'(x)g(x)+f(x)g'(x).}$
In Leibniz's notation this is written
${\displaystyle {\frac {d(fg)}{dx}}={\frac {df}{dx}}g+f{\frac {dg}{dx}}.}$
### The chain rule
The derivative of the function ${\displaystyle h(x)=f(g(x))}$ is
${\displaystyle h'(x)=f'(g(x))\cdot g'(x).}$
In Leibniz's notation, this is written as:
${\displaystyle {\frac {d}{dx}}h(x)=\left.{\frac {d}{dz}}f(z)\right|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),}$
often abridged to
${\displaystyle {\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.}$
Focusing on the notion of maps, and the differential being a map ${\displaystyle {\text{D}}}$, this is written in a more concise way as:
${\displaystyle [{\text{D}}(f\circ g)]_{x}=[{\text{D}}f]_{g(x)}\cdot [{\text{D}}g]_{x}\,.}$
### The inverse function rule
If the function f has an inverse function g, meaning that ${\displaystyle g(f(x))=x}$ and ${\displaystyle f(g(y))=y,}$ then
${\displaystyle g'={\frac {1}{f'\circ g}}.}$
In Leibniz notation, this is written as
${\displaystyle {\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}}.}$
## Power laws, polynomials, quotients, and reciprocals
### The polynomial or elementary power rule
If ${\displaystyle f(x)=x^{r}}$, for any real number ${\displaystyle r\neq 0,}$ then
${\displaystyle f'(x)=rx^{r-1}.}$
When ${\displaystyle r=1,}$ this becomes the special case that if ${\displaystyle f(x)=x,}$ then ${\displaystyle f'(x)=1.}$
Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.
### The reciprocal rule
The derivative of ${\displaystyle h(x)={\frac {1}{f(x)}}}$for any (nonvanishing) function f is:
${\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}}}$ wherever f is non-zero.
In Leibniz's notation, this is written
${\displaystyle {\frac {d(1/f)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.}$
The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule.
### The quotient rule
If f and g are functions, then:
${\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}}\quad }$ wherever g is nonzero.
This can be derived from the product rule and the reciprocal rule.
### Generalized power rule
The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g,
${\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\quad }$
wherever both sides are well defined.
Special cases
• If ${\textstyle f(x)=x^{a}\!}$, then ${\textstyle f'(x)=ax^{a-1}}$when a is any non-zero real number and x is positive.
• The reciprocal rule may be derived as the special case where ${\textstyle g(x)=-1\!}$.
## Derivatives of exponential and logarithmic functions
${\displaystyle {\frac {d}{dx}}\left(c^{ax}\right)={ac^{ax}\ln c},\qquad c>0}$
the equation above is true for all c, but the derivative for ${\textstyle c<0}$ yields a complex number.
${\displaystyle {\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}}$
${\displaystyle {\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>1}$
the equation above is also true for all c, but yields a complex number if ${\textstyle c<0\!}$.
${\displaystyle {\frac {d}{dx}}\left(\ln x\right)={1 \over x},\qquad x>0.}$
${\displaystyle {\frac {d}{dx}}\left(\ln |x|\right)={1 \over x},\qquad x\neq 0.}$
${\displaystyle {\frac {d}{dx}}\left(W(x)\right)={1 \over {x+e^{W(x)}}},\qquad x>-{1 \over e}.\qquad }$where ${\displaystyle W(x)}$ is the Lambert W function
${\displaystyle {\frac {d}{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).}$
${\displaystyle {\frac {d}{dx}}\left(f(x)^{g(x)}\right)=g(x)f(x)^{g(x)-1}{\frac {df}{dx}}+f(x)^{g(x)}\ln {(f(x))}{\frac {dg}{dx}},\qquad {\text{if }}f(x)>0,{\text{ and if }}{\frac {df}{dx}}{\text{ and }}{\frac {dg}{dx}}{\text{ exist.}}}$
${\displaystyle {\frac {d}{dx}}\left(f_{1}(x)^{f_{2}(x)^{\left(...\right)^{f_{n}(x)}}}\right)=\left[\sum \limits _{k=1}^{n}{\frac {\partial }{\partial x_{k}}}\left(f_{1}(x_{1})^{f_{2}(x_{2})^{\left(...\right)^{f_{n}(x_{n})}}}\right)\right]{\biggr \vert }_{x_{1}=x_{2}=...=x_{n}=x},{\text{ if }}f_{i0{\text{ and }}}$ ${\displaystyle {\frac {df_{i}}{dx}}{\text{ exists. }}}$
### Logarithmic derivatives
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):
${\displaystyle (\ln f)'={\frac {f'}{f}}\quad }$ wherever f is positive.
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.[citation needed]
Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for taking derivatives.
## Derivatives of trigonometric functions
${\displaystyle (\sin x)'=\cos x={\frac {e^{ix}+e^{-ix}}{2}}}$ ${\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}}$ ${\displaystyle (\cos x)'=-\sin x={\frac {e^{-ix}-e^{ix}}{2i}}}$ ${\displaystyle (\arccos x)'=-{1 \over {\sqrt {1-x^{2}}}}}$ ${\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x}$ ${\displaystyle (\arctan x)'={1 \over 1+x^{2}}}$ ${\displaystyle (\cot x)'=-\csc ^{2}x=-{1 \over \sin ^{2}x}=-1-\cot ^{2}x}$ ${\displaystyle (\operatorname {arccot} x)'={1 \over -1-x^{2}}}$ ${\displaystyle (\sec x)'=\sec {x}\tan {x}}$ ${\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}}$ ${\displaystyle (\csc x)'=-\csc {x}\cot {x}}$ ${\displaystyle (\operatorname {arccsc} x)'=-{1 \over |x|{\sqrt {x^{2}-1}}}}$
The derivatives in the table above is for when the range of the inverse secant is ${\displaystyle [0,\pi ]\!}$ and when the range of the inverse cosecant is ${\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]\!}$.
It is common to additionally define an inverse tangent function with two arguments, ${\displaystyle \arctan(y,x)\!}$. Its value lies in the range ${\displaystyle [-\pi ,\pi ]\!}$ and reflects the quadrant of the point ${\displaystyle (x,y)\!}$. For the first and fourth quadrant (i.e. ${\displaystyle x>0\!}$) one has ${\displaystyle \arctan(y,x>0)=\arctan(y/x)\!}$. Its partial derivatives are
${\displaystyle {\frac {\partial \arctan(y,x)}{\partial y}}={\frac {x}{x^{2}+y^{2}}}}$, and ${\displaystyle {\frac {\partial \arctan(y,x)}{\partial x}}={\frac {-y}{x^{2}+y^{2}}}.}$
## Derivatives of hyperbolic functions
${\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}$ ${\displaystyle (\operatorname {arsinh} x)'={1 \over {\sqrt {1+x^{2}}}}}$ ${\displaystyle (\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}}$ ${\displaystyle (\operatorname {arcosh} x)'={\frac {1}{\sqrt {x^{2}-1}}}}$ ${\displaystyle (\tanh x)'={\operatorname {sech} ^{2}x}={1 \over \cosh ^{2}x}=1-\tanh ^{2}x}$ ${\displaystyle (\operatorname {artanh} x)'={1 \over 1-x^{2}}}$ ${\displaystyle (\coth x)'=-\operatorname {csch} ^{2}x=-{1 \over \sinh ^{2}x}=1-\coth ^{2}x}$ ${\displaystyle (\operatorname {arcoth} x)'={1 \over 1-x^{2}}}$ ${\displaystyle (\operatorname {sech} x)'=-\operatorname {sech} {x}\tanh {x}}$ ${\displaystyle (\operatorname {arsech} x)'=-{1 \over x{\sqrt {1-x^{2}}}}}$ ${\displaystyle (\operatorname {csch} x)'=-\operatorname {csch} {x}\coth {x}}$ ${\displaystyle (\operatorname {arcsch} x)'=-{1 \over |x|{\sqrt {1+x^{2}}}}}$
See Hyperbolic functions for restrictions on these derivatives.
## Derivatives of special functions
Gamma function
${\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,dt}$
{\displaystyle {\begin{aligned}\Gamma '(x)&=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt\\&=\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)\\&=\Gamma (x)\psi (x)\end{aligned}}}
with ${\displaystyle \psi (x)}$ being the digamma function, expressed by the parenthesized expression to the right of ${\displaystyle \Gamma (x)}$ in the line above.
Riemann Zeta function
${\displaystyle \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}}$
{\displaystyle {\begin{aligned}\zeta '(x)&=-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots \\&=-\sum _{p{\text{ prime}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ prime}},q\neq p}{\frac {1}{1-q^{-x}}}\end{aligned}}}
## Derivatives of integrals
Suppose that it is required to differentiate with respect to x the function
${\displaystyle F(x)=\int _{a(x)}^{b(x)}f(x,t)\,dt,}$
where the functions ${\displaystyle f(x,t)}$ and ${\displaystyle {\frac {\partial }{\partial x}}\,f(x,t)}$ are both continuous in both ${\displaystyle t}$ and ${\displaystyle x}$ in some region of the ${\displaystyle (t,x)}$ plane, including ${\displaystyle a(x)\leq t\leq b(x),}$ ${\displaystyle x_{0}\leq x\leq x_{1}}$, and the functions ${\displaystyle a(x)}$ and ${\displaystyle b(x)}$ are both continuous and both have continuous derivatives for ${\displaystyle x_{0}\leq x\leq x_{1}}$. Then for ${\displaystyle \,x_{0}\leq x\leq x_{1}}$:
${\displaystyle F'(x)=f(x,b(x))\,b'(x)-f(x,a(x))\,a'(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}\,f(x,t)\;dt\,.}$
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.
## Derivatives to nth order
Some rules exist for computing the n-th derivative of functions, where n is a positive integer. These include:
### Faà di Bruno's formula
If f and g are n-times differentiable, then
${\displaystyle {\frac {d^{n}}{dx^{n}}}[f(g(x))]=n!\sum _{\{k_{m}\}}f^{(r)}(g(x))\prod _{m=1}^{n}{\frac {1}{k_{m}!}}\left(g^{(m)}(x)\right)^{k_{m}}}$
where ${\textstyle r=\sum _{m=1}^{n-1}k_{m}}$ and the set ${\displaystyle \{k_{m}\}}$ consists of all non-negative integer solutions of the Diophantine equation ${\textstyle \sum _{m=1}^{n}mk_{m}=n}$.
### General Leibniz rule
If f and g are n-times differentiable, then
${\displaystyle {\frac {d^{n}}{dx^{n}}}[f(x)g(x)]=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {d^{n-k}}{dx^{n-k}}}f(x){\frac {d^{k}}{dx^{k}}}g(x)}$
## References
1. ^ Calculus (5th edition), F. Ayres, E. Mendelson, Schaum's Outline Series, 2009, ISBN 978-0-07-150861-2.
2. ^ Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, ISBN 978-0-07-162366-7.
3. ^ Complex Variables, M.R. Speigel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 978-0-07-161569-3
4. ^ "Differentiation Rules". University of Waterloo - CEMC Open Courseware. Retrieved 3 May 2022.
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2022-09-27 09:45:21
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https://math.stackexchange.com/tags/diophantine-equations/hot
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# Tag Info
## Hot answers tagged diophantine-equations
3
I could manage to find many almost solutions (with the aim to derive full solutions) by a systematic search: My first trials, code and ideas are posted here, where I received very helpful input that directed me to a promising approach, proposed by Peter: Search for 6-tuples $(s,t,u,t+u,t+u−s,t−s)$ consisting of perfect squares. I implemented a Python script ...
1
I am going from here: instead of dividing two, I am going to multiple two. $$4044=2((x-y)^2+(y-z)^2+(z-x)^2)=(2x-y-z)^2+3(y-z)^2$$ WLOG $x\ge y\ge z$ Therefore set $2x-y-z=a$ and $y-z=b$ we are finding the solution of $a^2+3b^2=4044$. After trying, we have three solutions: $(63,5),(39,29),(24,34)$. $(63,5)$ yields $x=34+z,y=5+z$, $(39,29)$ yields $x=34+z,y=... 1 No,there are no other positive integral solutions except the one you have already guessed. For this solution ,I would be replacing (n,m) with (x,y).(as I comfortable in solving with those variables)$\implies$$x=y^{x} \implies y=x^{1/x} By differentiating the expression,you will realise that the maximum value of the expression is at e which is e^{1/e}.... 1 For the case M=5, K=6, all solutions have x divisible by 6. Modulo 2, you'd get x^2\equiv 0\pmod{2}, so x is even. Modulo 3, you get x^2+y^2\equiv 0\pmod 3. A little work shows that this implies x must be divisible by 3. In fact, this shows that if x^2-20y^2 is divisible by 6 then x is divisible by 6. It really has nothing to do ... 1 x^4 + 2x^3 -3x^2-4x-y^2-2y-6=0\iff(x^2+x)^2-4(x^2+x)=(y+1)^2+5 Putting X=x^2+x we get (X-2)^2=(y+1)^2+9 whose only solutions are clearly$$(X-2,y+1)=(\pm3,0),(\pm5,\pm4)$$in both cases, since X=x^2+x, we have the equations for the unknown x$$x^2+x-5=0\text{ and } x^2+x+1=0\\x^2+x-7=0\text{ and } x^2+x+3=0 these four equations have not integral ...
1
Since this is a Diophantine equation, we only seek for integer solutions. Notice that $x^4 + 2x^3 -3x^2-4x-y^2-2y-6=0$ implies $(y-x^2-x+3)(y+x^2+x-1)=-9$, we only have $6$ cases: $((y-x^2-x+3),(y+x^2+x-1))$ must be one of $(1,-9),(9,-1),(3,-3),(-1,9),(-9,1),(-3,3)$. However, after checking all these cases there are no solutions. A simpler way to do this: ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2022-01-27 00:52:02
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https://cungthi.vn/cau-hoi/find-a-mistake-in-the-four-underlined-parts-of-the-sentence--19691-261.html
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# Find a mistake in the four underlined parts of the sentence and correct it: Population density in the Red River Delta is by now more densely populated region in Vietnam with 1,136 people per square kilometer.
A.
more
B.
densely populated
C.
with 1,136 people
D.
per square kilometer
Đáp án:A
Lời giải:
more → the most
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2019-12-07 16:54:39
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https://www.degruyter.com/view/j/phys.2018.16.issue-1/phys-2018-0048/phys-2018-0048.xml
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Show Summary Details
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# Open Physics
### formerly Central European Journal of Physics
Editor-in-Chief: Seidel, Sally
Managing Editor: Lesna-Szreter, Paulina
1 Issue per year
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Volume 16, Issue 1
# An optimal solution for software testing case generation based on particle swarm optimization
Shi Jianqi
• National Trusted Embedded Software Engineering Technology Research Center, East China Normal University, China; Shanghai, 200062, China
• Other articles by this author:
/ Huang Yanhong
• Corresponding author
• National Trusted Embedded Software Engineering Technology Research Center, East China Normal University, China; Shanghai, 200062, China
• Email
• Other articles by this author:
/ Li Ang
• National Trusted Embedded Software Engineering Technology Research Center, East China Normal University, China; Shanghai, 200062, China
• Other articles by this author:
/ Cai Fangda
• National Trusted Embedded Software Engineering Technology Research Center, East China Normal University, China; Shanghai, 200062, China
• Other articles by this author:
Published Online: 2018-06-21 | DOI: https://doi.org/10.1515/phys-2018-0048
## Abstract
Searching based testing case generation technology converts the problem of testing case generation to function optimizations, through a fitness function, which is usually optimized using heuristic search algorithms. The particle swarm optimization (PSO) optimized testing case generation algorithm tends to lose population diversity of locally optimal solutions with low accuracy of local search. To overcome the above defects, a self-adaptive PSO based software testing case optimization algorithm is proposed. It adjusts the inertia weight dynamically according to the current iteration and average relative speed, to improve the performance of standard PSO. An improved alternating variable method is put forward to accelerate local search speed, which can coordinate both global and local search ability thereby improving the overall generation efficiency of testing cases. The experimental results demonstrate that the approach outlined here keeps higher testing case generation efficiency, and it shows certain advantages in coverage, evolution generation amount and running time when compared to standard PSO and GA-PSO.
Keywords: PSO; testing case; local search; inertia; instrumentation
## 1 Introduction
Software engineering is a subject of Engineering Research on software development and software testing is the key component of software engineering, which directly affects the development prospects of software engineering. Currently, with the extension of the software scale, traditional software testing methods cannot satisfy actual demand, so automated testing of software becomes the emphasis of research today [1, 2, 3, 4]. The key of automated testing lies in automated generation. Test case means a scientific organization induction for the activities of software testing, aiming at converting the activities of software testing to manageable modes. Testing case is also verified to be an effective process of quantifying the detailed tests [5, 6, 7, 8, 9]. Therefore, the research of testing case generation has great significance on software tests, and on the development of software engineering.
In the field of testing data autogeneration, there exists representative methods such as the symbolic execution method and random number method based on requirement specification, as well as the Korel [10] method etc. Symbolic execution method is often used to generate static testing data and it is not suitable for object-oriented programming. Random number method belongs to dynamic approaches. It generates random number according to the demand. When performing sample testing using a testing case, it can generate testing data with high effectiveness. The defect of such methods is that it is hard to acquire suitable solution sets for the high constraint test method such as path coverage or branch coverage. Korel is designed for the solution of path coverage of high constraint demand. It can adjust input variables continuously to achieve branch function minimization based on the information of branch functions in running programs. But it will also cause local optimal solution and is able to execute needed path when the initial data is bad. Jones et al. [11] adopts branch coverage criterion to propose a GA-based algorithm for testing case generation. They prove that GA-based testing data has a higher speed than random testing method in triangle classification programs. Anand [12] proposes simulated annealing to be implemented on the basic structure of testing generation data and provides empirical evidence for his research. However the algorithm lacks corresponding experiments and analysis. Sadiq [13] adopts variants of integrated learning PSO in structural software testing. It is used in testing for multiple artificial generation programs that works as testing objects, and is compared to GA, thus verifying the effectiveness of PSO in software testing case generation. Jee [14] proposes an approximation algorithm like Tracy and he integrates global search to local search in PSO-based testing case generation algorithm. These algorithms also show the defects caused by the disadvantage of PSO, that is, bad local search accuracy and solutions with low quality.
The rest of this paper is organized as follows: The standard PSO algorithm is briefly presented in the next section. Section 3 fully introduces SPSOLS and the whole process of the improved scheme. Simulation and comparison results are provided in section 4. Finally, a conclusion is provided in section 5.
## 2.1 PSO model
PSO [15, 16] is an optimization search technique based on population. It is a new branch of evolutionary computation and the particle swarm can be seen as a simple social system. In PSO, each individual is called a particle and each particle denotes a potential solution, which forms the population set. The particles in searching space influence each other, exchanging information, to adjust the location and speed of itself and approach to the optimal solution. PSO initialize a group of particles, finding he optimal solution by iteration. During each iteration, the particle updates its speed and location by tracing individual extremum and global extremum.
$vidt+1=vidt+r1c1(pid−xid′)+r2c2(pgd−xid′)$(1)
$xidt+1=xidt+vidt+1$(2)
vi = {vi1, vi2, …, vid} denotes the speed of the ith particle; Xi = {xi1, xi2, …, xid}, i = 1, 2, …, N denotes a vector point of ith particle in d-dimension solution space, t is the iteration number; $\begin{array}{}{x}_{id}^{t}\end{array}$ and $\begin{array}{}{{v}_{id}}^{\prime }\end{array}$ denote the location after tth iteration and dth dimensional component of speed vector of ith particle, r1 and r2 are random numbers obeying the distribution of U(0, 1); c1 and c2 are accelerate factors and usually c1 = c2 = 2. We use Pi = {pi1, pi2, …, pid} to record the optimal point that is searched by the ith particle, as pbest. Thus there must exist an optimal point in the population, whose number is g. Then Pg = {pg1, pg2, …, pgd} is the optimal value of the population search.
During the process of particle optimization, it is crucial to balance the local developing ability and the global detecting ability. For different problems the balance of these two abilities are not the same. Therefore, Shi [17] introduces inertia weight in equation 1 and it is revised as
$vidt+1=ωvid′+r1c1(pid−xid′)+r2c2(pgd−xid′)$(3)
ω, called inertia factor, is a normal number or it may be a linear or nonlinear positive number, with time as a variable. When ω = 1, equation 3 is changed to equation 1, so PSO with inertia factor is an extended form of standard PSO. ω keeps motion inertia of particle and the ability such as trend of extending search space and explore novel regions. It balances global detecting and local development. Therefore, a suitable value of ω implies the improvement of convergence accuracy and speed, which increases the probability of PSO to find the global optimal solution.
When ω ≥ 1, the speed is increased with time and the particle will not change direction towards a good region, leading to population divergence; when 0 < ω < 1, the particle decelerates and the convergence of the whole population is decided by c1 and c2. To get better global search ability at an early stage, and better local development ability at a later stage, ω is set to undergo linear reduction with the evolution.
## 2.2 Implementation of PSO algorithm on automatic testing data generation
The problem of automatic testing data generation can be expressed as a function minimization problem. The branches of the program are taken as a function. When we execute the function and approach to a expected location in the code, the function will be expected to reach the desired location of one or more variable values as a parameter of function [18]. As long as a set of input data can be found to achieve the minimum value, they are the requiring input data in need.
Assuming the 324th line in a program contains the condition if (cps≥ 20).
The object of use is to ensure the execution of this branch whose condition is true. Therefore we must find an input to ensure that when the 112th line arrives at a value of variable cps that is greater than or equal to 20. A simple way is to record the value of cps when program comes to 112th line. We use cps121 to record this value with input data x. Then the branch problem is converted to a function minimization problem, when 112th execution is true:
$f(x)=20−cps121(x),cps121(x)<200,otherwise$(4)
To find the expected input data, it is needed to find a value for x such that f(x) achieves the minimized value, so f(x) indicates the degree that input data approaches the expected result. Generally it is believed that the smaller of f(x) is, the closer of input data approaching expected result is. Thus, the initial input data can be adjusted and f(x) can be used to evaluate the method used to adjust the data for a closer result to the expected object. Test data generator is instructed by the value of f(x) to approach the final object by a series of adjustments. In PSO, f(x) often works as the fitness value to compute the testing case, determining the speed and direction of particles.
## 3.1 System model of testing case generation
The core problem of PSO-based testing data generation is ensuring the cooperation of PSO search algorithm and testing process. The system is decomposed into test modules and core algorithm modules, as depicted in figure 1. From the test point of view, it can be divided into the initial preparation of the search algorithm and the interaction part with search algorithm during the execution. The preparation part mainly includes the following work:
Figure 1
Testing case generation system model
1. Establish fitness function of coverage criterion according to the internal structure of program;
2. Perform instrumentation on the program structure elements corresponding to the coverage criterion to acquire coverage information;
3. Extract interface information of program to prepare for the work that describes the input parameter codes for the location vector of the particles.
Algorithm module is a key part of the system. It creates an initialized particle swarm according to previous procedures, and instructs the particles to approach the object, finding final solution or arriving at the expected number of iterations, by repeated evaluation and operation of the particles in the population. The testing running environment provides testing programs that can be called in real-time and instrumented in running phase. It evaluates the status of each particle in population and return a fitness value for the core algorithm.
The generation process will observe the following steps:
• Step 1
Analyze the source codes of tested program to draw its control flowchart.
• Step 2
Choose a needed testing path and specify fitness function according to the conditions in path. Make instrumentation on source programs.
• Step 3
Generate testing case set in the definition domain of program randomly in demand.
• Step 4
Acquire corresponding fitness with initial testing case executing instrumented programs and check whether it executes the expected path. If it does, turn to step 6; otherwise, turn to step 5.
• Step 5
Run improved PSO algorithm package according to the fitness of particle to generate new testing cases, and turn to step 4.
• Step 6
End of run, output appropriate testing cases.
Inertia weight factor ω directly influences the convergence of the algorithm and it is crucial to the performance. The adjustment has been performed in literature [19], but the diversity of particles is still an important factor. During the search for a solution, the diversity is gradually decreased, which shows homoplasy. It tends to get to the local extremum before the global solution is found. The approach of chasing diversity of particles will cause bad convergence speed. Therefore, this article proposes a self-adaptive scheme, which integrates the fitness and aggregation degree of particles to set ω with different methods. Then ω can be dynamically adjusted to ensure the global search ability of algorithm without affecting the convergence speed.
The novel adjusting method is based on fuzzy logic and it can improve the performance of the PSO algorithm, First we set two decision variables of ω, the number of current iteration Nt and the average relative speed ΔVcs(t).
$ΔVcs(t)=|Vcs(t)−Vcs(t−1)|Vcs(t)=∑i∑jvidij$(5)
The input variables are t and ΔVcs(t), and the output variable is ω in the fuzzy structure. During the process of fuzzification we choose fuzzy word set (S, M, L): S →small, M →middle, L →large. The diagram of membership is depicted as figure 2.
Figure 2
Membership degree change in fuzzy inference system
We establish 9 dynamic fuzzy adjustment rules for ω, as described in table 1. It is obvious to satisfy different demand with adaptive mechanism for the inertia weight at different search stages. It also means a large inertia expectation will change the value of x at the initial stage to a certain extent, whilst a smaller inertia expectation can be taken as the global optimal solution in certain interval in search. The fuzzy rules are based on the following three facts:
Table 1
Fuzzy rules to compute ω
• When the algorithm starts, a bigger inertia weight is adopted to increase the diversity of searching space, causing a strong exploration in population. In addition, the last procedure in inertia weight decreasing can lead to a faster convergence of population to find a global solution. Thus it has better performance to search the solution in a small region.
• When the particle approaches the object and pidpgd, the value of [pidxid(t)] and [pgdxid(t)] approaches zero. It means that with the decreasing speed, the decreasing speed of small inertia weight will affect the location of the next particle, so the particle may be converged to a solution up to now. There exists a risk that if a population converges too fast it may not be the global solution for the particle to explore the next attempt. Therefore, larger inertia weight should be chosen to avoid getting into local optimization.
• In PSO, a suitable control of global exploration and local development is to find the optimal solution effectively. To balance exploration and development we can adjust the value of ω dynamically to adapt the change of ΔVcs(t) for relative speed V. In other words, we expect to maintain the diversity of population by choosing a larger inertia weight value. Though the average relative speed is small, local optimization can be avoided, while smaller inertia weight value also maintain a better convergence ability to get a smaller ΔVcs finally. A larger ΔVcs is suggested to avoid the swarm lies on the boundary of space, with setting of a small ω.
The detailed steps for fuzzy adjustment are:
• Step 1
Initialize the particles with asymmetric method. Repeat step 2 to step 6 until the demand is satisfied.
• Step 2
Evaluate the fitness of each particle
• Step 3
For each particle i and pgd = xid, if fi < fbest, ∀iN
• Step 4
For each particle i and pgd = xid, if fi < fbest[i], ∀iN
• Step 5
Adjust the value of ω based on the normalization of input variables
• Step 6
update the speed and location of each particle according to equations 1 and 2.
## 3.3 Local search strategy based on improved alternating variable method
The major concept of alternating variable method (AVM) is to adjust each dimension of input variable in turn until the fitness cannot be raised any further. Then the value of the next dimensional value will be revised until the fitness value of all the variables cannot be improved any more. The adjusting method adopts a hill-climbing algorithm, that is, finding an optimal solution in the neighbour range of the variable. If a new solution is found and it is better than the current solution, alternate current solution and continue to search the neighbour domain of the new solution. Then the search is iterated to find the optimal solutions so the basis of AVM is a hill-climbing algorithm. The result in literature [20] indicates that for most simple programs AVM is better than GA, but AVM is only suitable for unimodal function. When the test programs are multiple peaks, it will get into local optimization and lose its ability to find the optimal solution.
We propose an improved local search based on AVM to make a more effective search in optimal peak function with geometric hierarchical progression, which is a variant of binary search. This idea executes pattern search and uses binary search until the object in found. It can be seen that the optimization of any peak function depends on the discovery of the logarithm of the initial time distance, which is called geometric research since the pattern search adopts a number of geometric sequences. The following parts offer the procedure of research process:
Geometrical hierarchical progression differentiates the same geometry to perform a search as IPS, while it provides two optimal solutions by variant binary search. If we adopt pattern search and the search points xj–1, xj, xj+1, and f(xj–1) > f(xj) ≤ f(xj+1), it indicates that if f is a unimodal function, the global optimization will be produced in set {xj–1}, xj, …, xj+1}.
Input: The location of x[] of the bet particle Output: The location of x[] of the bet particle in Pt 1 iff (x – 1) ≥ f (x) and f (x + 1) ≥ f (x) then 2 | return x 3 end 4 iff(x – 1) ≤ f(x + 1) then 5 | k = –1 6 else 7 | k =1 8 end 9 whilef (x + k) < f (x) do 10 | Let l = min(x – k/2, x + k), r = max(x – k/2, x + k) 11 end 12 whilel ≠ rdo 13 | iff([(l + r)/2]) < f([(l + r)/2] + 1) then 14 | | r = [(l + r)/2] 15 | else 16 | | l = [(l + r)/2] +1 17 | | return l 18 | end 19 end
## 3.4 Construction and instrumentation of fitness function
The fitness function is constructed by the Branch distance method [10]. Branch distance denotes the distance between the input variable and a given Branch predicate, which offers corresponding methods to compute distance among predicates for different relations. Tracey et al [21] improved this algorithm, so we adopt their improved scheme to construct the fitness function of each branch in the programs under test. In addition, we introduce parameter weight w to assign different weights according to difficulty level of branch coverage, which is determined by two factors: boundary condition and branch hierarchy. This is illustrated by the program code “Is Valid Date” [22] described below. Branch 1-3 is more difficult to cover than branch 4-6, in view of the boundary condition; In view of the branch hierarchy, branch 5-6 lies in the third layer and they can be covered only after branch 3 and 4.
Program Is Valid Date … // branch 1 If (month==l||)|| month==3|| month==5|| month==7 || month==8 || month==10 || month==12) { …; } // branch 2 …; } else if (month==4||)|| month==6|| month==9|| month==7 || month==11) { …; } // branch 3 else if (month==2||) { …; } // branch 4 If ((year%4==0 && year%100!=0) || year%400==0) // leap year judgement { // branch5 If (date>29 || date <1) { …; } // branch6 Else { …; } } … }} …
The fitness value computation is shown as follows:
$f=1/[0.01+∑i=1xwif(brai)],$(6)
where f(brai) is the branch distance function of ith branch; wi is corresponding weight and $\begin{array}{}{\sum }_{i=1}^{x}{w}_{i}=1.\end{array}$ The fitness value is decided by the number of branch coverage. The fitness function increases the difference between a good candidate solution and a bad one. Therefore, it fastens the convergence speed of search algorithm to a certain extent.
## 4.1 Testing case to generate common triangular paths
In this experiment we choose classic Triangular decision procedures without polymorphic information to test our algorithm with other evolutionary algorithms. The problem is: input three integers as the sides of a triangle. Compare the sides and output the type of triangle, including ordinary triangle, equilateral triangle, isosceles triangle, and non triangle, output by program modules. Since the decision procedure has many judgements and there is no loop structure or polymorphic, we use branch function overlaps to construct the instrument function.
First we focus on three indicators of the algorithms, average iteration, coverage rate and time consumption. Table 2 depicts the related test results about three algorithms in triangle generation test. SPSOLS has the minimum iteration and average time consumption of the other algorithms. The iteration number is reduced to 1/10 of standard PSO and 1/6 of GAPSO, while the time cost is about 1/11 of standard PSO and 1/8 of GAPSO. All the algorithms have higher coverage rate and can correctly output the testing paths of testing cases. SPSOLS keeps relative stable coverage rate with the increase of convergence speed. The increase of population size brings few effects on the algorithm performance. So our scheme shows better comprehensive performance improvement with little cost of price.
Table 2
Testing results of three algorithms in iteration number, time consumption and coverage rate
Table 3
Program under test
## 4.2 Performance comparisons under standard testing programs
In this experiment we choose 8 programs for testing. 5 of them are of independent patterns and the other 3 are not. The detailed information of these programs is described in table 2.
From the experimental results as shown in figure 3 which provides the performance comparison distribution in three indexes, SPSOLS-based testing data generation are superior than the other two algorithms in all programs. Towards the average coverage rage of 8 testing programs, its results are more than 98%, meaning it arrives at complete branch coverage in most tests. For standard PSO and GAPSO, they can achieve part of complete branch coverage. In the programs, the worst coverage of PSO is 88.79% and that of GAPSO is 95.47%.
Figure 3
Testing results of three algorithms about their performance indexes
The convergence speed of SPSOLS is 2 times of PSO and GAPSO. Illustrated by program “Equals”, PSO needs 16 generation to get the optimal solution, while GAPSO only needs 5 generation to get convergence. For other programs with simple independent pattern, the average speed of SPSOLS increased by 70 percent or so compared to the other two algorithms.
For the time consumption, GAPSO-based testing case generation has obvious advantage over the other two algorithms. Standard PSO has nearly ten times average time consumption than SPSOLS and GAPSO. Illustrated by program “Density”, SPSOLS saves about 100 times of seconds of PSO.
Specially, we choose 4 of representative programs to analyze the relation between evolutionary generation number and average coverage and the comparison curves are shown in figure 4. As summarized above, SPSOLS always obtain the highest branch coverage rate. For most of the programs, GAPSO has faster increasing speed of coverage rate than standard PSO. For program “line” and “Printcalendar”, the convergence speed relation of PSO and GAPSO are contrary. PSO shows frequent fluctuation in local range and the coverage rate has a stable increase while GAPSO has local stability. For a few programs the coverage rate shows a little degeneration, that is, the coverage rate is inversely proportional to the evolutionary generation number. The algorithm proposed in this article has comprehensive performance improvement and it is more stable in the convergence process.
Figure 4
The relation between branch coverage rate and evolution generation
## 5 Conclusion
PSO algorithm has deficiency in premature convergence, and the local search accuracy is low. It also brings factors of program structure in testing case generation. Therefore the PSO-based testing case generation technology is discussed intensively in this article. To balance the ability of exploring and self-improvement of algorithms, and to achieve better convergence speed in global search, we provide adaptive particle inertia weight factor adjusting approach, integrated with fitness and particle aggregation degree. During evolution, a local searching strategy is performed on the optimal individual of each generation to further improve the efficiency of testing case generation. The simulations indicate that the SPSOLS algorithm proposed in this article shows better testing case generation performance and it achieves coverage rate optimization of tested cases. It also has certain advantages in testing case generation compared to homogeneous algorithm under approximate environment.
## Acknowledgement
This work is partially supported by the Science and Technology Commission of Shanghai Municipality Projects (No. 15511104700, No. 16DZ1100600 and No. 15ZR1410400), National Natural Science Foundation of China Projects (No. 61602178), The Integration of Industry, Education and Research Project of SHEITC, Shanghai. (No. CXY-2015-015)
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Accepted: 2017-04-26
Published Online: 2018-06-21
Citation Information: Open Physics, Volume 16, Issue 1, Pages 355–363, ISSN (Online) 2391-5471,
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2018-11-14 01:07:04
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https://neuralpde.sciml.ai/dev/pinn/wave/
|
1D Wave Equation with Dirichlet boundary conditions
Let's solve this 1-dimensional wave equation:
\begin{align*} ∂^2_t u(x, t) = c^2 ∂^2_x u(x, t) \quad & \textsf{for all } 0 < x < 1 \text{ and } t > 0 \, , \\ u(0, t) = u(1, t) = 0 \quad & \textsf{for all } t > 0 \, , \\ u(x, 0) = x (1-x) \quad & \textsf{for all } 0 < x < 1 \, , \\ ∂_t u(x, 0) = 0 \quad & \textsf{for all } 0 < x < 1 \, , \\ \end{align*}
with grid discretization dx = 0.1 and physics-informed neural networks.
Further, the solution of this equation with the given boundary conditions is presented.
using NeuralPDE, Flux, ModelingToolkit, GalacticOptim, Optim, DiffEqFlux
import ModelingToolkit: Interval, infimum, supremum
@parameters t, x
@variables u(..)
Dxx = Differential(x)^2
Dtt = Differential(t)^2
Dt = Differential(t)
#2D PDE
C=1
eq = Dtt(u(t,x)) ~ C^2*Dxx(u(t,x))
# Initial and boundary conditions
bcs = [u(t,0) ~ 0.,# for all t > 0
u(t,1) ~ 0.,# for all t > 0
u(0,x) ~ x*(1. - x), #for all 0 < x < 1
Dt(u(0,x)) ~ 0. ] #for all 0 < x < 1]
# Space and time domains
domains = [t ∈ Interval(0.0,1.0),
x ∈ Interval(0.0,1.0)]
# Discretization
dx = 0.1
# Neural network
chain = FastChain(FastDense(2,16,Flux.σ),FastDense(16,16,Flux.σ),FastDense(16,1))
initθ = Float64.(DiffEqFlux.initial_params(chain))
discretization = PhysicsInformedNN(chain, GridTraining(dx); init_params = initθ)
@named pde_system = PDESystem(eq,bcs,domains,[t,x],[u(t,x)])
prob = discretize(pde_system,discretization)
cb = function (p,l)
println("Current loss is: l") return false end # optimizer opt = Optim.BFGS() res = GalacticOptim.solve(prob,opt; cb = cb, maxiters=1200) phi = discretization.phi We can plot the predicted solution of the PDE and compare it with the analytical solution in order to plot the relative error. using Plots ts,xs = [infimum(d.domain):dx:supremum(d.domain) for d in domains] analytic_sol_func(t,x) = sum([(8/(k^3*pi^3)) * sin(k*pi*x)*cos(C*k*pi*t) for k in 1:2:50000]) u_predict = reshape([first(phi([t,x],res.minimizer)) for t in ts for x in xs],(length(ts),length(xs))) u_real = reshape([analytic_sol_func(t,x) for t in ts for x in xs], (length(ts),length(xs))) diff_u = abs.(u_predict .- u_real) p1 = plot(ts, xs, u_real, linetype=:contourf,title = "analytic"); p2 =plot(ts, xs, u_predict, linetype=:contourf,title = "predict"); p3 = plot(ts, xs, diff_u,linetype=:contourf,title = "error"); plot(p1,p2,p3) 1D Damped Wave Equation with Dirichlet boundary conditions Now let's solve the 1-dimensional wave equation with damping. \begin{aligned} \frac{\partial^2 u(t,x)}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 u(t,x)}{\partial t^2} + v \frac{\partial u(t,x)}{\partial t} \\ u(t, 0) = u(t, L) = 0 \\ u(0, x) = x(1-x) \\ u_t(0, x) = 1 - 2x \\ \end{aligned} with grid discretization dx = 0.05 and physics-informed neural networks. Here we take advantage of adaptive derivative to increase accuracy. using NeuralPDE, Flux, ModelingToolkit, GalacticOptim, Optim, DiffEqFlux using Plots, Printf using Quadrature,Cubature, Cuba import ModelingToolkit: Interval, infimum, supremum @parameters t, x @variables u(..) Dxu(..) Dtu(..) O1(..) O2(..) Dxx = Differential(x)^2 Dtt = Differential(t)^2 Dt = Differential(t) # Constants v = 3 b = 2 L = 1.0 @assert b > 0 && b < 2π / (L * v) # 1D damped wave eq = Dx(Dxu(t, x)) ~ 1 / v^2 * Dt(Dtu(t, x)) + b * Dtu(t, x) # Initial and boundary conditions bcs_ = [u(t, 0) ~ 0.,# for all t > 0 u(t, L) ~ 0.,# for all t > 0 u(0, x) ~ x * (1. - x), # for all 0 < x < 1 Dtu(0, x) ~ 1 - 2x # for all 0 < x < 1 ] ep = (cbrt(eps(eltype(Float64))))^2 / 6 der = [Dxu(t, x) ~ Dx(u(t, x)) + ep * O1(t, x), Dtu(t, x) ~ Dt(u(t, x)) + ep * O2(t, x)] bcs = [bcs_;der] # Space and time domains domains = [t ∈ Interval(0.0, L), x ∈ Interval(0.0, L)] # Neural network chain = [[FastChain(FastDense(2, 16, Flux.tanh), FastDense(16, 16, Flux.tanh), FastDense(16, 16, Flux.tanh), FastDense(16, 1)) for _ in 1:3]; [FastChain(FastDense(2, 6, Flux.tanh), FastDense(6, 1)) for _ in 1:2];] initθ = map(c -> Float64.(c), DiffEqFlux.initial_params.(chain)) dx = 0.05 strategy = GridTraining(dx) discretization = PhysicsInformedNN(chain, strategy; init_params=initθ) @named pde_system = PDESystem(eq, bcs, domains, [t, x], [u(t, x), Dxu(t, x), Dtu(t, x), O1(t, x), O2(t, x)]) prob = discretize(pde_system, discretization) pde_inner_loss_functions = prob.f.f.loss_function.pde_loss_function.pde_loss_functions.contents inner_loss_functions = prob.f.f.loss_function.bcs_loss_function.bc_loss_functions.contents bcs_inner_loss_functions = inner_loss_functions cb = function (p, l) println("Current loss is:l")
println("pde_losses: ", map(l_ -> l_(p), pde_inner_loss_functions))
println("bcs_losses: ", map(l_ -> l_(p), bcs_inner_loss_functions))
return false
end
# Optimizer
res = GalacticOptim.solve(prob, BFGS(); cb=cb, maxiters=1000)
prob = remake(prob, u0=res.minimizer)
res = GalacticOptim.solve(prob, BFGS(); cb=cb, maxiters=3000)
phi = discretization.phi[1]
# Analysis
ts, xs = [infimum(d.domain):dx:supremum(d.domain) for d in domains]
μ_n(k) = (v * sqrt(4 * k^2 * π^2 - b^2 * L^2 * v^2)) / (2 * L)
b_n(k) = 2 / L * -(L^2 * ((2 * π * L - π) * k * sin(π * k) + ((π^2 - π^2 * L) * k^2 + 2 * L) * cos(π * k) - 2 * L)) / (π^3 * k^3) # vegas((x, ϕ) -> ϕ[1] = sin(k * π * x[1]) * f(x[1])).integral[1]
a_n(k) = 2 / -(L * μ_n(k)) * (L * (((2 * π * L^2 - π * L) * b * k * sin(π * k) + ((π^2 * L - π^2 * L^2) * b * k^2 + 2 * L^2 * b) * cos(π * k) - 2 * L^2 * b) * v^2 + 4 * π * L * k * sin(π * k) + (2 * π^2 - 4 * π^2 * L) * k^2 * cos(π * k) - 2 * π^2 * k^2)) / (2 * π^3 * k^3)
# Plot
analytic_sol_func(t,x) = sum([sin((k * π * x) / L) * exp(-v^2 * b * t / 2) * (a_n(k) * sin(μ_n(k) * t) + b_n(k) * cos(μ_n(k) * t)) for k in 1:2:100]) # TODO replace 10 with 500
anim = @animate for t ∈ ts
@info "Time \$t..."
sol = [analytic_sol_func(t, x) for x in xs]
sol_p = [first(phi([t,x], res.minimizer)) for x in xs]
plot(sol, label="analytic", ylims=[0, 0.1])
title = @sprintf("t = %.3f", t)
plot!(sol_p, label="predict", ylims=[0, 0.1], title=title)
end
# Surface plot
u_predict = reshape([first(phi([t,x], res.minimizer)) for t in ts for x in xs], (length(ts), length(xs)))
u_real = reshape([analytic_sol_func(t, x) for t in ts for x in xs], (length(ts), length(xs)))
diff_u = abs.(u_predict .- u_real)
p1 = plot(ts, xs, u_real, linetype=:contourf, title="analytic");
p2 = plot(ts, xs, u_predict, linetype=:contourf, title="predict");
p3 = plot(ts, xs, diff_u, linetype=:contourf, title="error");
plot(p1,p2,p3)
We can see the results here:
Plotted as a line one can see the analytical solution and the prediction here:
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2021-09-18 16:09:55
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https://enbilgi.ir/post/if-nominal-gdp-increases-from.p19972
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if you want to remove an article from website contact us from top.
if nominal gdp increases from year 1 to year 2 then we must have produced more goods and services in year 2 than we did in year 1.
Category :
James
Guys, does anyone know the answer?
get if nominal gdp increases from year 1 to year 2 then we must have produced more goods and services in year 2 than we did in year 1. from EN Bilgi.
Lesson summary: Real vs. nominal GDP (article)
In this lesson summary review and remind yourself of the key terms and calculations used in calculating real and nominal GDP. Topics include the distinction between real and nominal GDP and how to calculate and use the GDP deflator.
Real vs. nominal GDP
Lesson summary: Real vs. nominal GDP
In this lesson summary review and remind yourself of the key terms and calculations used in calculating real and nominal GDP. Topics include the distinction between real and nominal GDP and how to calculate and use the GDP deflator.
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Lesson Overview
Even GDP needs to keep it real. When we calculate GDP using today’s prices, we are creating a measure called nominal GDP. However, prices can change even if output doesn’t change. Because of that, our measure of output might get distorted by something like inflation.
We account for this using real GDP, which is a measure of GDP that has been adjusted for the price level. In this way, real GDP is a truer measure of output in an economy. There are two approaches to adjusting nominal GDP to get real GDP: 1) using the same prices every year or 2) using the GDP deflator.
Key Terms
Term Definition
nominal GDP the market value of the final production of goods and services within a country in a given period using that year’s prices (also called “current prices”)real GDP nominal GDP adjusted for changes in the price level, using prices from a base year (constant prices) instead of “current prices” used in nominal GDP; real GDP adjusts the level of output for any price changes that may have occurred over timeGDP deflator a price index used to adjust nominal GDP to find real GDP; the GDP deflator measures the average prices of all finished goods and services produced within a nation’s borders over time.base year the year used for comparison in the determination of price changes using the GDP deflator price index; the deflator in a base year is always equal to
=100 =100 equals, 100 .
current prices the prices at which goods are sold in a nation in a particular year; current prices are used when calculating nominal GDP.constant prices the prices from a base year that are used to calculate real GDP in other years; this allows for a more accurate measure of how a country’s actual output changes over time, because using constant prices cancels out any changes in the price level between years.
Key takeaways
Definitions of nominal v. real GDP
Nominal GDP is a measure of how much is spent on output. For example, in Canada during 2015,
\text{CAN }\$1{,}994.9\text{ billion} CAN$1,994.9 billion
start text, C, A, N, space, end text, dollar sign, 1, comma, 994, point, 9, start text, space, b, i, l, l, i, o, n, end text
was spent on the goods and services produced in Canada. Nominal GDP measures aggregate output (meaning the value of all of the final goods and services produced) using current prices. In other words, these figures reflect the amount spent on Canada’s output in the country’s prices in 2015.
Real GDP weighs output using prices from a base year
Real GDP is a measure of how much is actually produced. Real GDP measures aggregate output using constant prices, thus removing the effect of changes in the overall price level. For example, in 2015 the value of Canada’s output expressed in constant 2010 prices was
\text{CAN }\$1{,}857\text{ bilion} CAN$1,857 bilion
start text, C, A, N, space, end text, dollar sign, 1, comma, 857, start text, space, b, i, l, i, o, n, end text
.
Here’s another way to think about Real GDP: if we add up all of the output that was produced in Canada during 2015 by using the prices that these goods sold for in 2010, the value of GDP in Canada is
\$1{,}857\text{ billion}$1,857 billion
dollar sign, 1, comma, 857, start text, space, b, i, l, l, i, o, n, end text
. But if we add up all of the output that was produced in Canada during 2015, using the prices that they sold for in 2015, the value of GDP in Canada is
\$1{,}995\text{ billion}$1,995 billion
dollar sign, 1, comma, 995, start text, space, b, i, l, l, i, o, n, end text
. This means prices must have increased between 2010 and 2015.
However, there is a slight problem with the method above. Calculating real GDP by weighting final goods and services by their prices in a base year can lead to an overstatement of real GDP growth because the prices of some goods decrease over time. Therefore, this method overstates growth in real GDP because it makes it seem like goods make up a bigger share of spending than they really do.
[Can you explain that further?]
The GDP deflator and real GDP
Another method of calculating real GDP involves converting nominal GDP to real GDP by using the GDP deflator, which tracks price changes of a nation’s output over time. Canada’s GDP deflator for its base year of 2010 was
100 100 100
since this is the year against which prices are compared. By 2015 the deflator had increased to
107.4 107.4 107, point, 4
, indicating that the average prices of Canada’s output had increased by
7.4\% 7.4%
7, point, 4, percent
.
By expressing 2015’s output in 2015 prices, therefore, Canada’s output would appear to have increased by
7.4% 7.4 7, point, 4
more than it actually did. Canada’s nominal GDP, which has been “inflated” by higher prices, can be “deflated” by dividing the country’s nominal GDP of
Nominal Gross Domestic Product
Nominal gross domestic product measures the value of all finished goods and services produced by a country at their current market prices.
ECONOMICS MACROECONOMICS
Nominal Gross Domestic Product
By THE INVESTOPEDIA TEAM Updated January 30, 2022
Reviewed by THOMAS BROCK
Fact checked by YARILET PEREZ
What Is Nominal Gross Domestic Product?
Nominal gross domestic product is gross domestic product (GDP) evaluated at current market prices. GDP is the monetary value of all the goods and services produced in a country. Nominal differs from real GDP in that it includes changes in prices due to inflation, which reflects the rate of price increases in an economy.
KEY TAKEAWAYS
Nominal GDP is an assessment of economic production in an economy but includes the current prices of goods and services in its calculation.
GDP is typically measured as the monetary value of goods and services produced.
Since nominal GDP doesn't remove the pace of rising prices when comparing one period to another, it can inflate the growth figure.
0 seconds of 0 secondsVolume 75%
2:37
Understanding Nominal Gross Domestic Product
Nominal GDP is an assessment of economic production in an economy that includes current prices in its calculation. In other words, it doesn't strip out inflation or the pace of rising prices, which can inflate the growth figure. All goods and services counted in nominal GDP are valued at the prices that are actually sold for in that year.
Effects of Inflation on Nominal GDP
Because it is measured in current prices, growing nominal GDP from year to year might reflect a rise in prices as opposed to growth in the amount of goods and services produced. If all prices rise more or less together, known as inflation, then this will make nominal GDP appear greater. Inflation is a negative force for economic participants because it diminishes the purchasing power of income and savings, both for consumers and investors.
Inflation is most commonly measured using the Consumer Price Index (CPI) or the Producer Price Index (PPI). The CPI measures price changes from the buyer's perspective or how they impact the consumer.1 The PPI, on the other hand, measures the average change of selling prices that are paid to producers in the economy.2
When the overall price level of the economy rises, consumers have to spend more to purchase the same amount of goods. If an individual’s income rises by 10% in a given period but inflation rises 10% as well, then the individual’s real income (or purchasing power) is unchanged. The term real in real income merely reflects the income after inflation has been subtracted from the figure.
Nominal GDP vs. Real GDP
Likewise, if we were comparing the GDP growth between two periods, the nominal GDP growth might overstate the growth if inflation is present. Economists use the prices of goods from a base year to act as a reference point when comparing GDP from one year to another. The difference in prices from the base year to the current year is called the GDP price deflator.
For example, if prices rose by 1% since the base year, the GDP deflator would be 1.01. Overall, real GDP is a better measure any time the comparison is over multiple years.
Real GDP starts with nominal GDP but factors in any change in prices from one period to the other. Real GDP is calculated by taking the total output for GDP and dividing it by the GDP deflator.
For example, let's say the current year's nominal GDP output was $2,000,000, while the GDP deflator showed a 1% increase in prices since the base year. Real GDP would be calculated as$2,000,000/1.01 or $1,980,198 for the year. One of the limitations of using nominal GDP is when an economy is mired in recession or a period of negative GDP growth. Negative nominal GDP growth could be due to a decrease in prices, called deflation. If prices declined at a greater rate than production growth, nominal GDP might reflect an overall negative growth rate in the economy. A negative nominal GDP would be signaling a recession when, in reality, production growth was positive. Compete Risk Free with$100,000 in Virtual Cash
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quiz 5,6&7 Flashcards
Start studying quiz 5,6&7. Learn vocabulary, terms, and more with flashcards, games, and other study tools.
quiz 5,6&7
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A period during which real GDP declines is called a recession.
True False
Click card to see definition 👆
true
Click again to see term 👆
A farmer grows wheat and sells it to the miller for $87. The miller turns the wheat into flour and sells it to the baker for$125. The baker turns the flour into bread and sells it to consumers for $194. What is the value added by the baker? Enter a whole number with no other characters. Click card to see definition 👆 69 Click again to see term 👆 1/68 Created by erinrussell1999 Terms in this set (68) A period during which real GDP declines is called a recession. True False true A farmer grows wheat and sells it to the miller for$87. The miller turns the wheat into flour and sells it to the baker for $125. The baker turns the flour into bread and sells it to consumers for$194. What is the value added by the baker? Enter a whole number with no other characters.
69
Capital goods are goods that will be used in the future to produce more goods and services.
True False true
A firm produces 100 units of some good in the year 2017 but only manages to sell 90 of the units. When we calculate GDP for the year 2017 we will count the value of the 90 units sold and ignore the value of the 10 units not sold.
True False false
Net export spending will have a negative value when the country exports more goods than it imports from other countries.
True False false
When a business such as CarMax sells a used car to an individual for $9,000 then no part of this transaction counts in GDP because the car is used. True False false When a private individual sells a used car to another individual, this does not count in GDP. True False true Nominal GDP increases from year 1 to year 2. Therefore: - We produced more goods and services in year 1. - It is not possible to determine in which year we produced a greater quantity of goods and services. - We produced more goods and services in year 2. It is not possible to determine in which year we produced a greater quantity of goods and services. In 1988 the value of RGDP was$8,474.492 billion, the population was 244.499 million, and the number of people employed was 115,060 measured in thousands. What was the value of real GDP per capita? Round to the nearest whole number. Do not enter a dollar sign.
34661
In a given year, nominal GDP is $442 and real GDP is$381. Find the GDP deflator. Enter a number rounded to two decimal places.
116.01
When your parents pay your tuition bill to UCF this counts as a consumption expenditure.
True False true
A farmer grows wheat and sells it to the miller for $47. The miller turns the wheat into flour and sells it to the baker for$79. The baker turns the flour into bread and sells it to consumers for $132. As a result of these transactions, GDP will increase by how much? Enter a whole number with no other characters. 132 When the government pays social security to elderly people, this counts as: a transfer payment. investment spending. government purchases. consumption spending. transfer payment We can use the GDP deflator to calculate the inflation rate in the economy. True False true The four categories of expenditure (spending) in the economy are wages, rent, interest, and profit. True False false Company A produces paper and sells it to Company B for a total of$159. Company B uses the paper to make books and then sells the books to consumers for a total of $321. What is the value of GDP in this simple economy? Enter a whole number with no other characters. 321 When Toyota produces cars in the United States this does count in GDP even though Toyota is not an American company. True False true The government pays social security benefits to your grandfather every month. When your grandfather spends his entire social security check on groceries at the supermarket this counts as: investment spending. consumption spending. government spending. a transfer payment. consumption The largest component of GDP is investment spending. True False false If the price of a banana is twice the price of an apple, then the sale of a banana contributes twice as much to GDP as the sale of an apple. True False true In the economy, income earned when producing new final goods and services must equal expenditure on the new final goods and services. True False true GDP increased from$16.35 trillion to \$17.1 trillion. What was the growth rate of GDP? Enter a number rounded to two decimal places. Do not enter any other characters.
4.59
There are 82 unemployed people and 1121 employed people. What is the unemployment rate? Enter a number (measured in percentage terms) rounded to two decimal places. Do not enter a % sign.
6.82
According to the Bureau of Labor Statistics, you are considered to be unemployed if you have no job.
True False false
An increase in the minimum wage is likely to _____ the quantity of labor demanded and at the same time is likely to _____ the quantity of labor supplied.
Source : quizlet.com
Do you want to see answer or more ?
James 11 month ago
Guys, does anyone know the answer?
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2023-02-04 18:05:52
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http://www.perlmonks.org/index.pl?node_id=1042738
|
by kennethk (Abbot)
on Jul 05, 2013 at 16:02 UTC ( #1042738=note: print w/replies, xml ) Need Help??
my \$cookie_jar = HTTP::Cookies->new(file => 'C:\path\to\cookies.dat',
autosave=> 1,
[download]
#11929 First ask yourself How would I do this without a computer?' Then have the computer do it the same way.
Replies are listed 'Best First'.
by PerlSufi (Friar) on Jul 05, 2013 at 16:45 UTC
Thanks, kennethk. That worked for me. :)
The cookie is different from what my browser gives me so I'm working on how to get that now
Firebug, the free Firefox plug-in, makes that easy.
#11929 First ask yourself How would I do this without a computer?' Then have the computer do it the same way.
Ty, kennethk. I have been using HTTP live headers. I forgot firebug did that. I actually want my script to get them automatically. I will be using WWW::Mechanize::Firefox to do crawl the site. I need the JSESSIONID to be a part of an HTTP get request.
Oh. It turns out that WWW::Mechanize::Firefox doesn't save cookies to the jar? :(
Thanks again for recommending the Data::Dumper to get those cookies. I am another step closer to automating the page. Now I think I need to use a regex to match for the JSESSIONID and concatenate it into my get request.
Create A New User
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2017-04-26 18:46:42
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3653981685638428, "perplexity": 5431.872456167589}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121528.59/warc/CC-MAIN-20170423031201-00277-ip-10-145-167-34.ec2.internal.warc.gz"}
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https://golem.ph.utexas.edu/category/2014/12/a_categorical_understanding_of.html
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## December 7, 2014
### A Categorical Understanding of the Proof of Cantor-Schröder-Bernstein?
#### Posted by Emily Riehl
Over the past year I have become increasingly fascinated by set theory and logic. So this morning when I was meant to be preparing a talk, I instead found myself thinking about the Cantor–Schröder–Bernstein theorem.
Theorem (Cantor–Schröder–Bernstein). Let $A$ and $B$ be sets. If there exist injections $f \colon A \to B$ and $g \colon B \to A$, then $|A|=|B|$.
This is an incredibly powerful tool for proving that two sets have the same cardinality. (Exercise: use CSB to prove that $|\mathbb{N}|=|\mathbb{Q}|$ and that $|P(\mathbb{N})|=|\mathbb{R}|$.) Earlier this fall, I taught a proof of this result that I learned from Peter Johnstone’s Notes on logic and set theory. The question that’s distracting me is this: how categorical is this argument?
## Proof of Cantor–Schröder–Bernstein
Let me start by describing the proof I have in mind. The hope is that $f$ and $g$ might be used to construct a bijection $A \cong B$ in the following manner: by finding subsets $A^\ast \subset A$ and $B^\ast \subset B$ so that $f$ defines a bijection between $A^\ast$ and its image, the complement of $B^\ast$, and $g$ defines a bijection between $B^\ast$ and its image, the complement of $A^\ast$.
The first trick — and this is the part that I do not understood categorically — is that a pair of subsets with the above property is encoded by a fixed point to the function
$h \colon P(A) \to P(A)$ defined by $h(X) = A \backslash g(B \backslash f(X)).$
Here I’m thinking of the powerset $P(A)$ as a poset, ordered by inclusion, and indeed $h$ is a functor: $X \subset Y \subset A$ implies that $h(X) \subset h(Y)$. Now the second trick is to remember that a terminal coalgebra for an endofunctor, if it exists, defines a fixed point.
Here the coalgebras are just those subsets $X$ with the property that $X \subset h(X)$. Let $\mathcal{C} \subset P(A)$ be the subposet of coalgebras (if you will, defined by the forming the inserter of the identity and $h$). I don’t know whether it is a priori clear that $\mathcal{C}$ has a terminal object, but if it does, then it is given by forming the union
$C = \cup \{X \subset A \mid X \subset h(X)\}.$
And indeed this works: it’s easy to check that $C \subset h(C)$ and moreover that this inclusion is an equality. The bijection $A \cong B$ is defined by $f$ applied to $C$ and $g$ applied to the complement of $f(C)$.
Alternatively, we could apply a theorem of Adámek to form the terminal coalgebra: it is the limit of the inverse sequence
\cdots \subset h^3(A) \subset h^2(A) \subset h(A) \subset A]
defined by repeatedly applying the endofunctor $h$ to the terminal object $A \in P(A)$. Because $P(A)$ is complete, this limit must exist.
This construction seems to be related to the other standard proof of Cantor–Schröder–Bernstein, which Wikipedia tells me is due to Julius König. The injections $f$ and $g$ and their partially-defined inverses define a partition of $A \sqcup B$ into disjoint infinite (possibly repeating) chains of elements contained alternately in $A$ and in $B$
$\cdots a , f(a), g(f(a)), f(g(f(a)), \ldots$
that terminate at the left whenever there is an element that is not in the image of $f$ or $g$. For those elements in chains that terminate at the left at an element of $A$ (resp. $B$), $f$ (resp. $g$) is used to define the bijection. For the remaining chains, either $f$ or $g$ may be used.
Arguing inductively by cases, you can see that the limit of the inverse sequence, i.e., the terminal coalgebra of $h$ is the union of those elements $a \in A$ that appear in chains that either terminate at an element of $A$ or continue forever (possibly repeating) in both directions. (Side question: is there a slick way to demonstrate this?) This argument tells us that the terminal coalgebra is the maximal fixed point of $h$, but in general it isn’t the only one. The minimal fixed point consists only of those elements in chains that terminate at an element of $A$ on the left.
So, how categorical is this argument? Am I seeing terminal coalgebras just because I learned this proof from a category theorist? Or is this interpretation less superficial than the way I am presenting it?
Concluding remarks:
• The $n$Lab explains that a dual version of this proof holds in any boolean topos, but not in all toposes, because the argument given above requires the law of the excluded middle.
• A category theorist might ask whether Cantor–Schröder–Bernstein holds in other categories. For those wishing to dive into that rabbit hole, I recommend starting here.
Posted at December 7, 2014 10:45 PM UTC
TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2788
### Re: A categorical understanding of the proof of Cantor-Schröder-Bernstein?
No comments so far, so let me start.
First of all, I do not think that the original Cantor-Schröder-Bernstein theorem is about sets. It is, secretly, about Boolean algebras. In fact, it is about direct products of $\aleph_0$-complete Boolean algebras:
Theorem (Tarski) Let $A_1,A_2,B,C$ be $\aleph_0$-complete Boolean algebras such that $A_1\simeq A_2\times B$ and $A_2\simeq A_1\times C$. Then $A_1\simeq A_2$.
To see that Tarski’s theorem implies CSBT observe that the existence of a monomorphism $X_1\to X_2$ in $Set$ is the same thing as existence of a set $Y$ such that $X_1\simeq X_2\oplus Y$ and this is the same thing as $P(X_1)\simeq P(X_2)\times P(Y)$ in the category of $\aleph_0$-complete Boolean algebras.
The proof of the more general theorem is, basically, the same as the original one. However, the surroundings are different now: instead of dealing with injections in $Set$ we are dealing with intervals $[0,a]$ in $A_1$ and instead of the “same cardinality” equivalence we are dealing with isomorphism of $\aleph_0$-complete Boolean algebras.
Let me say that Tarski’s theorem is not true if we drop “$\aleph_0$-complete” from the assumptions. The counterexample is a little bit complicated, as far as I remember. It can be found in the “Handbook of Boolean Algebras”, the proof is based on the theorem that every countable commutative semigroup embeds into the semigroup of isomorphism classes of countable Boolean algebras, equipped with the operation arising from product. It is somewhere in the Volume 3, I think.
However, looking at the proof of Tarski’s theorem carefully, one realizes that we do not use all properties of the Boolean algebras. In fact, we use only disjoint unions. Moreover, we do not use all properties of the isomorphism of intervals $[0,a]$, only some of them. Years ago, as an unexperienced pup, I have tried to distill out as general version of the proof as I could. I replaced Boolean algebras with effect algebras and isomorphism of intervals with an equivalence $\sim$ satisfying countable additivity and the axiom
If $a\sim b+c$, then there are $a_1,a_2$ such that $a=a_1+a_2$, $a_1\sim b$ and $a_2\sim c$.
The paper appeared in Algebra Universalis here, the preprint (in an obscure *.ps.zip form) can be found here . The paper reflects the priorities I have had at that time. From todays perspective, the main result should probably be Proposition 3.
Finally, let me mention that there are non-Boolean examples of dimensional equivalences like this appearing in the wild: for example Murray-von Neumann equivalence of projections in a von Neumann algebra.
Posted by: Gejza Jenča on December 10, 2014 8:08 AM | Permalink | Reply to this
### Re: A categorical understanding of the proof of Cantor-Schröder-Bernstein?
Gejza,
Thanks for weighing in. This is very interesting.
I’m traveling at the moment so will have to respond slowly, when I find myself graced with a sporadic wifi connection.
Could you explain this:
instead of dealing with injections in $Set$ we are dealing with intervals $[0,a]$ in $A_1$
Here I’m guessing $0$ is the initial object and $a$ is a generic interval in the boolean algebra $A_1$ and the interval is the set of elements $0 \leq x \leq a$. I guess if we identify an injection in $Set$ with the subset of its image $X$, that corresponds to an element $a \in P(X)$. Is that the idea?
Posted by: Emily Riehl on December 10, 2014 4:56 PM | Permalink | Reply to this
### Re: A categorical understanding of the proof of Cantor-Schröder-Bernstein?
Yes, that is the idea. It is really that simple.
I think that the CSBT is about coproducts in $Set$ rather than about monos in $Set$. And that the proper generalization of CSBT is some form of Tarski’s theorem for monoidal categories.
Posted by: Gejza Jenča on December 11, 2014 9:12 AM | Permalink | Reply to this
### Re: A categorical understanding of the proof of Cantor-Schröder-Bernstein?
As you may know, your last question (together with a commented link to the SB seminar post) appeared a while ago on MO.
Posted by: L Spice on December 10, 2014 11:41 PM | Permalink | Reply to this
### Re: A Categorical Understanding of the Proof of Cantor-Schröder-Bernstein?
Small correction: I think your definition of $h$ should be $h(X) = A\setminus g(B\setminus f(X))$. Note that this also can be written as $g_\ast(f_!(X))$, where $f_!$ is the left adjoint to $f^{-1}$ and $g_\ast$ is the right adjoint to $g^{-1}$. This version at least makes sense in a non-Boolean topos (or more generally a Heyting category), and perhaps we can still find its terminal coalgebra, but I guess we won’t then be able to define a bijection by cases on the fixed point and its complement.
Posted by: Mike Shulman on December 20, 2014 7:02 PM | Permalink | Reply to this
### Re: A Categorical Understanding of the Proof of Cantor-Schröder-Bernstein?
Thanks Mike. I’ve corrected the typo.
Curiously this is the second time that Heyting categories have come up today (a term I hadn’t heard before this afternoon).
Posted by: Emily Riehl on December 20, 2014 10:57 PM | Permalink | Reply to this
Post a New Comment
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2020-01-29 05:27:08
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https://living.medicareful.com/how-to-brighten-your-smile
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Whether you’ve smoked for years or enjoyed the occasional cup of coffee in the morning, having to deal with yellow practically an eventuality. There are many causes of teeth staining, a lot of which we encounter daily. As a result, many people’s teeth yellow with age. If you fall into this camp, there are many ways you can whiten your teeth.
First, it’s important to understand why our teeth may not always stay pearly whites. What determines the color of our teeth is their enamel — their hard, outer surface — and the dentin underneath that layer. Light reflects off tooth enamel, helping teeth appear whiter. As we age, our enamel wears away, revealing dentin, which is darker. Hence, our teeth become less bright. Certain foods, drinks, and medications can also stain enamel. This is where the problem lies.
Reversing the ravages of time can be difficult, but you’re not in it alone. Let’s look at some of the ways you can get rid of yellow teeth.
## Getting Your Teeth Professionally Cleaned and Whitened
The most effective methods of teeth whitening can be done by a dentist. There are two main types of whitening services, vital whitening and non-vital whitening. If your teeth have live nerves, you can get vital whitening. If a tooth has a dead nerve, like after a root canal, you can get non-vital whitening.
Non-vital whitening is different because the dentist whitens the tooth from the inside out. It’s also less common than vital whitening. Vital whitening uses a hydrogen peroxide gel that lightens and removes stains on the outside of the tooth. The dentist will apply the gel directly to the teeth while guarding the gums. They will then activate the gel with a laser or special light, which can help the gel work faster.
Vital whitening uses a hydrogen peroxide gel that lightens and removes stains on the outside of the tooth.
All said, a vital whitening office visit should take 30 to 90 minutes. Since the session is in-office, the gel the dentist uses may be stronger than an at-home option. However, to reach your desired whiteness, you may still need to attend several bleaching sessions.
The major downside of using professional teeth whitening services is the price. Since this is a purely cosmetic procedure, it’s very rarely covered by any form of insurance. This can lead you to foot the bill yourself. Depending on how many sessions you need, the service could cost you anywhere from $300 to$1,000. Many dentists offer whitening services, so it may be worth shopping around. Tooth bleaching can also leave your teeth sensitive for a little while as well.
## At-Home Options
If you decide that an in-office method isn’t your cup of tea, an at-home remedy could work for you. Your dentist may be able to provide gel trays that are custom-fitted to your teeth. Much like the in-office option, this can be expensive, though not as expensive as going that route. You can also try an over-the-counter option.
For example, teeth whitening strips are available in most grocery stores or pharmacies. These work similarly to the bleaching process at a dentist’s office. Simply apply the strips after brushing your teeth and keep them on your teeth for a set amount of time before removing them. You can also buy whitening toothpaste in many stores or online. Unlike bleaching, whitening toothpaste essentially scrubs the teeth clean. To achieve this, it’s more abrasive than regular toothpaste.
Your at-home whitening options include strips and toothpaste. Both are available at most grocery stores.
The downside of whitening strips and toothpaste is that they take longer to produce results than in-office methods. Whitening strips can take days or weeks of continued use, and toothpaste can take longer, before you see significant improvement. Both can also cause teeth sensitivity.
## Natural Options?
Some people choose to eschew chemical options and look for more natural solutions. The belief is that some natural substances either eliminate staining bacteria or brighten the enamel.
Coconut oil has an antibacterial effect, which studies show can reduce gingivitis and plaque.
One of the popular natural teeth whitening options is oil pulling. Coconut oil has an antibacterial effect, which studies show can reduce gingivitis and plaque. This can reduce staining, and some believe it whitens your smile. Other natural remedies include apple cider vinegar and activated charcoal. The effectiveness of these methods is disputed. They may not work for everyone, but they may be worth a try if the previous options don’t interest you.
● ● ●
Ultimately, the best way to keep your smile pearly is to avoid foods and drinks that stain your teeth, quit smoking, and have good dental hygiene. Eventually, though, time can take its toll, and teeth will age. When this happens, if you’d like to keep your teeth shining bright, it helps to know your options. Luckily, there are plenty out there, and you should be able to find something that works best for you.
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2018-08-20 08:31:17
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http://physics.stackexchange.com/questions/35034/where-does-the-electricity-generated-by-a-solar-panel-go-if-you-dont-use-the/35039
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# Where does the electricity, generated by a solar panel, go if you don't use the electricity?
I'm sorry if this question is too trivial for this Q&A forum. I am a layman when it comes to physics (though I did cover the high-school physics courses).
I was wondering what happens to the electricity, that is generated by your own solar panel, in case you don't use any electricity in your home. Does it turn from electrical energy into heat? Or does something else happen? Is the energy "lost" for practical purposes?
A follow-up question: In case the energy is "lost" for practical purposes, wouldn't that be an enormous waste of energy? Is this what the "Smart Grid" network is intended for? To distribute the energy to other places where it is needed, thereby minimizing the energy loss and CO_2 emissions?
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There are several ways to design the circuit.
If it's a Smart system, then when there's surplus power, additional devices will get turned on, to use it: dishwashers, washing machines, or immersion heaters in hot-water storage tanks. If there's still surplus after that, then it's as below.
If it's grid-connected, with an inverter, then it's usually designed to export surplus power to the grid. There's typically either an export meter, or the property's main electricity meter runs backwards during those times, to ensure that payment is made for the exports.
If it's designed as standalone, then it might have a battery where surplus power is initially sent. When the battery's full, then it's as below.
In the absence of other uses of the surplus, there's a resistance heater and heat sink where the generated power is dumped.
And in general, the economics tends to ensure that enormous amounts of energy are not wasted. If there's going to be a large surplus, that will be enough to justify expenditure on one or more of: a Smart system, a grid export connection, local storage.
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What if there is no heat think? Can you understand that the question is about the case when there are no sinks? – Val Apr 22 '14 at 9:10
Hmm, I think it would depend on the panel circuit. If the panel is not connected, for example, the charge potential would still be created at the leads, but since it's not being drained into a storage device (or otherwise used), the solar medium would saturate at some measurable voltage boundary. Whether it turns to heat at that point depends on whether the solar medium leaks the charge (variant on light intensity) or the other possibility is that the solar cell's reflectivity changes and the sunlight no longer gets absorbed (not the case in any solar technology that I've heard of).
In general, to have heat (in/from a solar panel), you have to have current flow. That flow can happen from leaky charges (at the battery bank or the solar panel itself) or intentional due to your own usage with the inefficiencies in your electrical equipment.
As far as wasted energy, it's been vegetation that has trapped this solar energy in earth's history, creating biomass -- ancient sunlight which we now use in the form of coal and oil. Consider that the energy lost cannot be greater than the amount of sunlight now denied in the square footage that lies in the shadows beneath your panels.
Otherwise, generally, when generated electricity isn't used, it can be sold to the power company. If that's not an option (because you're off the grid), you should get more storage (batteries, flywheel...)
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I come across this cause I am working on this problem. I see long talk and they don't know the answer.
Yes, the energy is lost, when all systems are fully charged and outside the sun is still shining. No meter PWM or MPPT charging system. You just cant put more into batteries when its full. Full means full.
Solution is ultracapcitors. Assemble a capacitor module, for $12v$ solar system, rated $23$ volts $3000$ Farads, if you got money make it $6000$ Farads, and connect it directly to solar panel. No need controllers for this one cause caps handle $23v$ and $12v$ solar panel give $17v$ to $22v$.
I built it, cant wait to buy more caps and attach another cap module to double or triple or more the power.
Warning, this work only with solar charger controllers that are rated $80A$ that's a minimum. But if you play with small $10F$ caps $7amp$ charger fit.
Result. When the sun is gone solar charger controller is indicating Charging, cause my ultracapacitors still contain power that is left over for you.
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2015-08-30 09:53:59
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https://covtestr.bearstatistics.com/reference/structurestatistics
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Performs Tests for the structure of covariance matrices.
Ahmad2015(x, Sigma = "identity", ...)
Chen2010(x, Sigma = "identity", ...)
Fisher2012(x, Sigma = "identity", ...)
LedoitWolf2002(x, Sigma = "identity", ...)
Nagao1973(x, Sigma = "identity", ...)
Srivastava2005(x, Sigma = "identity", ...)
Srivastava2011(x, Sigma = "identity", ...)
## Arguments
x data as a list of matrices Population covariance matrix as a matrix other options passed to covTest method
## Value
A list with class "htest" containing the following components:
statistic the value of equality of covariance test statistic parameter the degrees of freedom for the chi-squared statistic p.value the p=value for the test estimate the estimated covariances if less than 5 dimensions null.value the specified hypothesized value of the covariance difference alternative a character string describing the alternative hyposthesis method a character string indicating what type of equality of covariance test was performed data.name a character string giving the names of the data
## References
Ahmad, M. R. and Rosen, D. von. (2015). Tests for High-Dimensional Covariance Matrices Using the Theory of U-statistics. Journal of Statistical Computation and Simulation, 85(13), 2619-2631. 10.1080/00949655.2014.948441
Chen, S., et al. (2010). Tests for High-Dimensional Covariance Matrices. Journal of the American Statistical Association, 105(490):810-819. 10.1198/jasa.2010.tm09560
Fisher, T. J. (2012). On Testing for an Identity Covariance Matrix when the Dimensionality Equals or Exceeds the Sample Size. Journal of Statistical Planning and Infernece, 142(1), 312-326. 10.1016/j.jspi.2011.07.019
Ledoit, O., and Wolf, M. (2002). Some Hypothesis Tests for the Covariance Matrix When the Dimension Is Large Compared to the Sample Size. The Annals of Statistics, 30(4), 1081-1102. 10.1214/aos/1031689018
Nagao, H. (1973). On Some Test Criteria for Covariance Matrix. The Annals of Statistics, 1(4), 700-709
Srivastava, M. S. (2005). Some Tests Concerning the Covariance Matrix in High Dimensional Data. Journal of the Japan Statistical Society, 35(2), 251-272. 10.14490/jjss.35.251
Srivastava, M. S., Kollo, T., and Rosen, D. von. (2011). Some Tests for the Covariance Matrix with Fewer Observations then the Dimension Under Non-normality. Journal of Multivariate Analysis, 102(6), 1090-1103. 10.1016/j.jmva.2011.03.003
Other Testing for Structure of Covariance Matrices: structureCovariances
## Examples
Chen2010(as.matrix(iris[1:50, 1:3]))#>
#> Chen et al. 2010 Test of Covariance Matrix Structure
#>
#> data:
#> Standard Normal = -180.68, Mean = 0, Variance = 1, p-value < 2.2e-16
#> alternative hypothesis: true difference between the Sample Covariance Matrix and the Null Covariance Matrix Structure is not equal to 0
#> sample estimates:
#> Sepal.Length Sepal.Width Petal.Length
#> Sepal.Length 0.12424898 0.09921633 0.01635510
#> Sepal.Width 0.09921633 0.14368980 0.01169796
#> Petal.Length 0.01635510 0.01169796 0.03015918
#>
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2018-12-15 18:38:39
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https://mdtpmodules.org/linr-1-lesson-3-explore-vertical-lines-solutions/
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# LINR 1 | Lesson 3 | Explore (Vertical Lines) Solutions
1. Answer may vary.
2. $$x$$ values are all the same
3. Change in $$\frac{Δy}{Δx}= \frac{1}{0}$$
4. The slope is undefined.
5. There is no $$y$$-intercept
6. Equation is $$x = 2$$
7. Not in slope-intercept form
Table made in Desmos.com
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2022-05-27 22:28:42
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http://mathhelpforum.com/advanced-algebra/145165-solved-b-c-right-ideals-b-c-ac-bc.html
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Thread: [SOLVED] A,B,C right ideals. (A+B)C = AC+BC?
1. [SOLVED] A,B,C right ideals. (A+B)C = AC+BC?
Hi:
In a textbook I find: if A, B, C are right ideals in a ring R, then A(B+C) = AB+AC. This I have prove. But I could also prove (A+B)C = AC+BC (1)
where again A, B, C are right ideals in R. Can (1) be true? Any hint will be welcome.
2. Originally Posted by ENRIQUESTEFANINI
Hi:
In a textbook I find: if A, B, C are right ideals in a ring R, then A(B+C) = AB+AC. This I have prove. But I could also prove (A+B)C = AC+BC (1)
where again A, B, C are right ideals in R. Can (1) be true? Any hint will be welcome.
yes it is true even for left ideals. since the product of (left, right or two-sided) ideals $I,J$ is defined to be $IJ= \{\sum_{i=1}^n x_iy_i: \ x_i \in I, y_i \in J, \ n \in \mathbb{N} \},$ we only need to prove that
$(a+b)c_1 \in AC+BC$ and $ac_1 + b c_2 \in (A+B)C,$ for all $a \in A, \ b \in B, \ c_1,c_2 \in C.$ these are very easy to prove:
$(a+b)c_1=ac_1 + bc_1 \in AC + BC$ and $ac_1 + bc_2 =(a+0)c_1 + (0+b)c_2 \in (A+B)C.$
3. Ideal products.
Thanks a lot, NonCommAlg.
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2017-05-28 01:21:36
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https://www.serwer1494638.home.pl/proximal-and-otrbap/marginal-revenue-function-calculator-40572d
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You can use the marginal revenue calculator below to quickly calculate a firm’s Marginal revenue concerning its total revenues and quantities used or units sold by entering the required numbers. Marginal Revenue (MR) of the firm at any quantity of output sold is the increment in its total revenue (TR) that is obtained when the firm sells the marginal (or the additional) unit of that quantity. This brings marginal revenue to $25 as the total revenue ($25) divided by the quantity sold (1) is $25. Assume that a … University. For example, the first 10 units could sell for$100. If x is the number of units produced and sold and p is its unit price, then the total revenue function R(x) is defined as R(x) =px, where x and p are positive. And there's other similar ideas. Besides marginal revenue, another thing that is no less important is calculating … The marginal revenue function is the first derivative of the total revenue function or MR = 120 - Q. It states that a firm should select the level of output where marginal revenue is equal to the marginal cost to maximize its profits. This explains why firms in a perfectly competitive market maximize profits when the marginal costs, market price and, marginal revenues are equal as illustrated in example 1 above, This is not the same case for monopolies because the firms are in a market with limited competitors accompanied by a sets of demands. James is producing corn in a one-acre piece of land using fertilizer as the variable input. He can sell 20 units in a day at $180 each, but for him to sell 22 units in a day, he must reduce the per-unit price to$170. This formula needs only three variables; units of inputs used, units of output obtained, and the price for the outputs. C = $50 + 0.10 x lemonade +$0.50 x cookie. MRPL = marginal product of labour x marginal revenue. Cost Revenue and Profit Functions (English) Study guide. In other words, marginal revenue at any output (q) is the (extra) revenue that the firm obtains from selling the marginal (or the additional) unit of that output. If we modeled our profit as a function of quantity, if we took the derivative, that would be our marginal profit. To sell the next 10 units (#11 – 20) they would have to sell for $90. Marginal Product – The ratio of change between an input and an output. change in quantity). Now we need to calculate the Marginal Revenue for Anand Group of companies. Geben Sie Ihren Namen und Ihre E-Mail-Adresse in das unten stehende Formular ein und laden Sie jetzt die kostenlose Vorlage herunter! Marginal Revenue Calculator How to Calculate Producer Surplus GDP per Capita Calculator GDP Deflator Calculator Money ... Demand Function Calculator helps drawing the Demand Function. Cost function of a company is 6 2 5 Calculate the marginal cost function. a. marginal profit . The profit function is just the revenue function minus the cost function. We can write And increase the selling price per unit by 40%. You can use the following Marginal Revenue Calculator. To calculate change in quantity, subtract the new quantity of products sold by the previous quantity sold. Indicates an incremental cost change. Thus, the denominator is typically one. cost, revenue and profit functions cost functions cost is the total cost of producing output. [i.e. At Rs. We can calculate marginal revenue for a company by dividing the change in total revenue by change in total output quantity produced by the company. Financial Ratios. You can easily calculate the Marginal Revenue using Formula in the template provided. For instance, using the demand function above, total revenue for production … the marginal revenue equates the marginal cost. Company adjust their output and restructure its pricing to optimize their profitability. ALL RIGHTS RESERVED. Step 1: First we need to calculate the change in revenue. We need to find the Marginal revenue of Anand Machine works Pvt Ltd. Total Revenue from Production = 400 * 100 =$40,000, Total Revenue from Production = 800 * 150 = $1,20,000, We can calculate Marginal Revenue by using the below formula, Marginal Revenue (MR)= Change in Revenue / Change in Quantity. Anand Machine Works Pvt Ltd. is a manufacturer of office printing & Stationery items. In this case, companies/sellers use specific prices as the market dictates the optical price levels. Because some production costs are fixed and some are variable, marginal revenue usually changes as a business sells more product. In this example, we are assuming here that the firm employing labour is operating in a perfectly competitive market so that each unit of output sold generates a revenue of$20. Marginal Revenue Product Calculation. Marginal revenue is the derivative of the revenue function, so take the derivative of R (x) and evaluate it at x = 100: Thus, the approximate revenue from selling the 101st widget is $50. So, to get the change in total revenue first, we get the total revenues by multiplying the output by the price. Next, we can see that at$500 per ton, James’ revenue went from $2500 to$3250. In a line of the same, we need to calculate the Marginal Revenue for Anand & Son’s Shops. It is very easy and simple. These points are found most easily on a graphing calculator. The marginal revenue for a new calculator is given by MR = 60,000 − 20,000 (10 + x)2 where x represents hundreds of calculators and revenue is in dollars. For this example, the marginal revenue would be -$5. In words: To perform marginal analysis on either profit, revenue or cost, find the derivative function for the one quantity out of these three that you are estimating for. For example, suppose, when the quantities sold (q) are 9, 10 and 11 units, the firm’s TRs are, 50, 55 and 58 (Rs) respectively. We need to find the Marginal revenue of Anand Machine works Pvt Ltd.Here we have, 1. (ii) The marginal revenue [MR] is approximately equal to the additional revenue made on selling of (x+1) th unit, whenx the sales level is x units. It is a financial ratio that is used to compute the overall change in income obtained from the sales of one additional product or unit. 23. Marginal revenue, or MR, is the incremental revenue from selling an additional unit. Marginal revenue is the additional revenue that a producer receives from selling one more unit of the good that he produces. Marginal revenue is expressed as a financial ratio that is used to compute the overall change in income obtained from the sales of one additional product or unit. Grenzerlösformel. Let us say Mr. X is selling boxes of candy. The derivatives of these quantities are called marginal profit function, marginal revenue function and marginal cost function, respectively. Use the graph input tool to help you answer the following questions. There are limited alternatives available for the products, the selling price is affected by the production level of the product. for this financial year on the basis of the below data. Total revenue equals price, P, times quantity, Q, or TR = P×Q. Learning Outcomes At the end of this section you will be able to: † Understand the difierence between the total revenue and the marginal revenue, † Calculate the marginal revenue from the total revenue. Of course, you can simply do things by hand to get a sense of marginal revenue from a change in quantity. The marginal revenue formula is calculated by dividing the change in total revenue by the change in quantity sold. But don’t worry, we can always use the following formula to calculate it: Marginal revenue = change in revenue / change in quantity. To calculate the change in revenue, we simply subtract the revenue figure before the last unit was sold from the total revenue after the last unit was sold. Find the total revenue function, R(x), for these calculators. Marginal revenue (MR) is the additional revenue that will be generated increasing product sales one unit. 42. Sie müssen sich nur daran erinnern, dass der Grenzerlös der Umsatz ist, der aus … R(x) = In our widget example, dTotalCost(X)/dX = 2X+ 3. Now that we understand what these curves are and what their function is, let us discuss marginal revenue in the context of marginal cost. Marginal revenue equal to the sale price of a single additional item sold. Course. If we modeled revenue, that would be our marginal revenue. TC = Q(AVC) + TFC TC = Q*(Q 2 - 10Q + 60) + 1000 = Q 3-10Q 2 +60Q + 1000 MC = 3Q 2-20Q + 60 b. In this example, we are assuming here that the firm employing labour is operating in a perfectly competitive market so that each unit of output sold generates a revenue of$20. In other words, revenue is the income that an organization receives from normal business activities. Marginal revenue can be defined as the increase in revenue, as a result of the one additional unit sold. Article Shared by Nitisha. ADVERTISEMENTS: Revenue can be defined as receipts or returns from the sale of products of an organization. Let’s take the example of a farmer who sells barley. Marginal revenue measures the change in the revenue when one additional unit of a product is sold. GDP per Capita Calculator. 3 d. What profit or loss will One and Only earn? He sells 25 boxes every day for $2 each and makes a profit of$0.50 on every box that he sells. The marginal revenue function can be derived by taking the first derivative of the TR function: $$\text{MR}=\frac{\text{dTR}}{\text{dQ}}=\text{500}\ -\ \text{20Q}$$ A marginal revenue curve is a graphical representation of the relationship between marginal revenue and quantity. Get more help from Chegg. In this video we explore one of the most fundamental rules in microeconomics: a rational producer produces the quantity where marginal revenue equals marginal costs. Let’s look at it another way through a different example. You can use the marginal revenue calculator below to quickly calculate a firm’s Marginal revenue concerning its total revenues and quantities used or units sold by entering the required numbers. Marginal revenue refers to the increase in revenue realized from the sale of an additional one unit of output. The formula above breaks this calculation into two parts: one, change in revenue (total revenue – old revenue) and two, change in quantity (total quantity – old quantity). Marginal revenue is expressed as a financial ratio that is used to compute the overall change in income obtained from the sales of one additional product or unit. This means if the demand will be increased by supply and consumer will be ready to pay higher prices. Given the cost of producing a good, what is the best quantity to produce? Let’s take an example, the Market price is equal to Marginal revenue in a truly competitive market where the manufacturers are selling homogenous products, mass-produced products. What is the Marginal Revenue for James? Marginal revenue is the revenue a business receives from selling one more unit of a product. Anand is currently planning to introduce the production of a new category of pens. Midpoint Elasticity – An alternate way of calculating … Because marginal revenue is the derivative of total revenue, we can construct the marginal revenue curve by calculating total revenue as a function of quantity and then taking the derivative. So he had an increase of $750. Depending on the context, you can divide the change in revenue by the quantity/units of input used or units of output sold. In other words, every time a producer increases one unit of input holding other factors constant, the resulting marginal gain will be increased up to a certain point. If fixed costs are equal to$1,000, derive the firm's total cost function and marginal cost function. Demand Function Calculator helps drawing the Demand Function. The consumer demand arrived at by the management further helps in the planning of production schedules. Calculating the Profit Function. Let’s take an example to understand the calculation of the Marginal Revenue formula in a better manner. P'(x)=0 Enter your answer in the answer box and then click Check Answer . Marginal revenue follows the law of diminishing returns, which states that holding other factors constant, if a production process, as one factor of production (input) is varied, there will be a point at which the marginal per unit output will start to decrease. Marginal Revenue (MR) function is expressed as the first derivative of the Revenue function (TR) with respect to quantity Examples 1. of Units to be Produced: (2000 + 50% of 2000) = (2000 + 1000) = 3000, Forecasted Selling Price: $(50 + 30% of 50) =$(50 + 15 ) = $65, Marginal Revenue =$ (1,95,000 – 1,00,000) / (3000 – 2000), Marginal Revenue = (2,50,000 – 2,00,000) / (3,000 – 1,500). Let’s take a look at how to calculate marginal revenue and some other uses for this metric. Here we will do the example of the Marginal Revenue formula in Excel. The revenue concepts are concerned with Total Revenue, Average Revenue and Marginal Revenue. Because profit maximization happens at the quantity where marginal revenue equals marginal cost, it's important not only to understand how to calculate marginal revenue but also how to represent it graphically: The marginal revenue gained by producing that second hockey stick is $10 because the change in total revenue ($25-$15) divided by the change in quantity sold (1) is$10. Marginal revenue (or marginal benefit) is a central concept in microeconomics that describes the additional total revenue generated by increasing product sales by 1 unit. For any linear demand function with an inverse demand equation of the form P = a - bQ, the marginal revenue function has the form MR = a - 2bQ. Marginal revenue varies across firms: In a perfectly competitive market MR is constant while in a monopolistic. Marginal Revenue Calculator. We need to calculate the Marginal Revenue for Jagriti & Sons if they introduce the new production line. Demand Function Calculator helps drawing the Demand Function. This instructional aid was prepared by the TCC Learning Commons. If every cookie cost 50 cents to make, our revenue function becomes . Fortunately, it is easy to calculuate the revenue function. Anand Machine Works Pvt Ltd. is a manufacturer of office printing & Stationery items. GDP Deflator Calculator. From the examples above, the concept of marginal revenue is straightforward, and computing it takes the shortest time possible so long as you have a good mastery of the revenues generated in the activity and the respective costs attached to each. Currently, they are producing 2000 units and selling them at $50 each. Calculating Marginal Revenue . Marginal Revenue formula also plays a vital role in the invention of the Profit Maximization Rule. Calculating marginal revenue is one of the most important thing to do in running a business. Share. You’ll need to find the first derivative of the total cost function to find the marginal cost function. Determining marginal revenue helps a business set production levels to maximize revenue. So the company sells a second unit for$15. The cost of barley is set by the market every year. In the case of Ice Cream Wonderland we can calculate marginal revenue as follows. For every marginal revenue gained, there is a marginal cost attached to it of which the marginal revenue has to cover. Again, let’s break down the variables in this problem. 6 3. e. If fixed costs were $1,200, how would your answer to (c) change? He has forecasted to produce 800 pens and will be selling them at$150. The formula for calculating Marginal Revenue: Start Your Free Investment Banking Course, Download Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others. To calculate a change in revenue is a difference in total revenue and revenue figure before the additional unit was sold. Anand & Sons are planning to increase the current production by 50% and increase the selling price per unit by 30% to optimize their profitability. We can plug these variables into the equation again. Firms in the perfect competition market continue producing output until marginal revenue equates marginal cost. © 1999-2021 Study Finance. They want to increase their profitability and find out the sale price of a single additional item sold. Juan sells hospital equipment. of Units Produced: 400 2. By using this website, you agree to our Cookie Policy. Some standard results. For instance, in a perfectly competitive firm where there is full information about prices, and the products are homogenous, the marginal revenue remains typically constant. Diagrammatical explanation of Marginal Revenue [MR] Marginal revenue is the change in aggregate revenue when the volume of selling unit is increased by one unit. C(x) = 7x 2; R(x) = x 3 + 11x + 16. marginal cost. In this article we will discuss about the formula and equation for calculating the marginal revenue that the seller acquires by selling the good. Total Revenue from Production = 5000 * 140 = $7,00,000, Total Revenue from Production = 7500 * 196 =$14,70,000. But let's say the firm must lower its price to increase sales. By understanding these three key concepts, the manufacturer will be better placed in the market because there will be no product shortage resulting from customer demand misjudgment. You will not be graded on any changes you make to this graph. Marginal revenue of Anand Machine works Pvt Ltd is $200. In a perfectly competitive market, or one in which no firm is large enough to hold the market power to set price of a good, if a business were to sell a mass-produced good and sells all of its goods at market price, then the marginal revenue would simply be equivalent to the market price. Hence, perfectly competitive firms are only comfortable producing until the time when marginal revenue equals marginal cost. Marginal revenue increase in revenue realized from the sale of an additional one unit of output. Remember that revenue is simply the number of units times the price. Here, you can see that you’re putting so much work into your business, but are losing money instead. This formula requires two variables: Change in Total Revenue and Change in Quantities sold. Now let’s break it down and identify the meaning and values of different variables in the problem. If you try to sell at a higher price, consumers will buy from other competitors in the market because the products are homogeneous. In the words of Dooley, ‘the revenue of a firm is its sales, receipts or income’. marginal revenue. To calculate marginal cost, try some marginal cost example problems. The total is earned from sales of products, and these products can be from farm or firm. Then, subtract the original revenue from the alternate revenue. The inverse demand function is useful in deriving the total and marginal revenue functions. Since marginal revenue is defined to be the change in total revenue resulting from a one unit change in output, this means that marginal revenue will be less than the price. Let’s assume Anand Group of Companies Financial has shown the following details. The total revenue (TR) received from the sale of Q goods at price P is given by TR = PQ. Over a certain level of output, Marginal revenue can remain constant as it follows the law of diminishing returns and Marginal revenue can eventually decelerate as the output level increases. Calculate the net profit margin, net profit and profit percentage of sales from the cost and revenue. Currently, they are producing 400 pens and sell them at$100 each. However, we were not given a revenue function in the problem. Profit function. the cost function consists of two different types of cost: Sign in Register; Hide. In microeconomics, supply and demand is an economic model of price determination in a market. All rights reserved. You may wish to use a derivative calculator for this math. The net profit margin is net profit divided by revenue (or net income divided by net sales). This is the point we conclude that the diminishing returns take effect. MRPL calculation. Calculator Use. Laut grundlegender ökonomischer Prinzipien wird eine Firma, wenn sie den Preis eines ihrer Produkte senkt, auch eine größere Stückzahl dieses Produkts verkaufen. © 2020 - EDUCBA. All you need to remember is that marginal revenue is the revenue obtained from the additional units sold. Study Finance is an educational platform to help you learn fundamental finance, accounting, and business concepts. In microeconomics, supply and demand is an economic model of price determination in a market. It identifies the exact unit when the business entity should stop its production of further products so that it is still in the green. Marginal revenues vary from one firm to another. Revenue Types : Total, Average and Marginal Revenue! Application 1 - Marginal Revenue (MR) Aim To demonstrate an application of difierentiation. He has forecasted to produce 800 pens and will be selling them at $150. MR changes depending on how many units sell. Calculating Marginal Revenue. Find all values of x for which the marginal profit is zero. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 -.5Q) × Q = 120Q - 0.5Q². Essentially, you’re paying your customer$2.50 per candle to take them off of your hands. Trader's how to calculate marginal revenue function calculator p0172 fuel trim system rich Mason Cash Mixing Bowl Uk For example, let's calculate the margin requirements for buying one lot of EURUSD, while the size of one contract is 100 000 and the leverage is 1:100. For the companies in the Manufacturing Industries, marginal revenue plays a huge role in determining the production levels and product pricing. Money Multiplier Calculator. 1. IncFile Review: The Best Company Formation Service? The profit maximization formula: Marginal Revenue = Marginal cost. By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy, Download Marginal Revenue Formula Excel Template, Special Offer - All in One Financial Analyst Bundle (250+ Courses, 40+ Projects) Learn More, You can download this Marginal Revenue Formula Excel Template here –, 250+ Online Courses | 1000+ Hours | Verifiable Certificates | Lifetime Access, Marginal Revenue Formula in Excel (With Excel Template), Finance for Non Finance Managers Course (7 Courses), Investment Banking Course(117 Courses, 25+ Projects), Financial Modeling Course (3 Courses, 14 Projects), Marginal Benefit Formula | Calculator with Examples, Finance for Non Finance Managers Training Course, Current Revenue from Production: $100 each, Marginal Revenue = ($1,20,000 – $40,000) / (800 – 400), Forecasted No. Despite being a microeconomics concept, it still incorporates some managerial accounting and financial applications. Current No. The marginal revenue formula is calculated by dividing the change in total revenue by the change in quantity sold. Derive a total revenue function and marginal revenue function for the firm. Calculating marginal revenue from a linear demand curve The blue curve on the following graph represents the demand curve facing a firm that can set its own prices. The calculation of the most important thing to do in running a sells. Marginal revenue the equation again let us say Mr. x is selling boxes candy... By hand to get the change in total revenue when one additional unit of the below data that. Production line companies become successful by guiding them on how to manage their growth and development$... Price per ton of corn in the planning of production schedules a difference in total by! Demands, which is just the revenue margin from his 22nd sale each and makes profit. Increasing product sales one unit of output where marginal revenue for jagriti & Sons are planning introduce... Where, marginal revenue formula shall consider What happens to total revenue from the sale price of below. Be $750 s take the example of a company is 6 2 5 calculate marginal. Discuss how to manage their growth and development: Both functions are linear good that he.... The variable input 50 =$ 50 + 0.10 x lemonade + $0.50 x Cookie box that produces... Platform to help you learn fundamental Finance, accounting, and these products can defined... 11 – 20 ) they would have to sell for$ 90 but if fertilizer is increased to bags! Sons are indulged in production and selling them at $150 cost, revenue and change in revenue the... Below data … marginal revenue has to cover producing 5000 units and selling of commodities be marginal! Shown the following characteristics: Both functions are linear its production of a farmer who sells.. Of their RESPECTIVE OWNERS revenue refers to the change in revenue of cost Sign! Is marginal revenue function calculator corn in a certain period of time ) is the increase in revenue from! Where the production levels and product pricing and profit marginal revenue function calculator ( English study... ( # 11 – 20 ) they would have to sell the next 10 units ( # 11 20! Functions cost functions cost is the revenue margin from his 22nd sale of Rs price levels increase.. He was able to sell the next 10 units ( # 11 – 20 ) would! Different prices manufacturer of office printing & Stationery items revenue of Anand Works... Our profit as a business set production levels to maximize revenue this,! Calculator with a downloadable Excel template c ) change sell the next 10 units could sell marginal revenue function calculator 15! Laut grundlegender ökonomischer Prinzipien wird eine Firma, wenn Sie den kostenlosen Vorlagenrechner herunter set production.. Kostenlose Vorlage herunter: ( i ) Average revenue [ AR ] and price [ p ] marginal revenue function calculator... Maximization formula: marginal revenue, as we did for the same marginal revenue function calculator we will need to the! Affecting the revenue when the business entity should stop its production of line and expecting to their! For which the marginal revenue 100 each the old amount, consumers will buy from other competitors in market. 40 % learn fundamental Finance, accounting, and marginal cost to maximize revenue 7,00,000, revenue. Company adjust their output and restructure its pricing to optimize their profitability and find out the sale of goods price...$ 3250 along with practical examples we will do the example of single! 0.50 x Cookie way of calculating … den Grenzerlös berechnen ) =0 your... Dooley, ‘ the revenue in a better manner difference in total revenue from production = 3000 * =... And financial applications Vorlage herunter output until marginal revenue – incremental revenue from selling one unit. Downloadable Excel template say the firm changes output from 3 to 4 units = 3000 * 65 = $,! Just the revenue of Anand Machine Works Pvt Ltd is$ 200 a producer receives from selling one unit! Consumers will buy from other competitors in the number of units times the price manufacturer raises price... S look at it another way through a different example may wish to use a Calculator. Linear cost function consists of two different Types of cost: Sign Register! Revenue varies across firms: in a perfectly competitive firms are only comfortable producing until time... More unit of a product is sold of $0.50 x Cookie the products, the is. Producing 2000 units and selling them at$ 150 calculate from the cost and revenue figure before the additional sold! Ar ] and price [ p ] are the same, we do. See this, we will do the example of a product formula marginal! Revenue, subtract the new production line how would your answer in Manufacturing. Running a business generated a total revenue from selling one more unit of output obtained, and these products be. Managerial accounting and financial applications Laden Sie jetzt die kostenlose Vorlage herunter revenue from! ( # 11 – 20 ) marginal revenue function calculator would have to sell 5 additional boxes of for! Find the marginal revenue can be defined as receipts or returns from the problem cost... 2 5 calculate the change in quantity, if we modeled revenue, as a result of the revenue! Equals price, p, times quantity, if we modeled revenue, Average revenue [ AR ] and [... Use specific prices as the increase or decrease in the Manufacturing Industries, marginal revenue along with practical.! Same, we can obtain another key concept: marginal revenue is easy to calculate marginal revenue = marginal.. By an additional unit 2 bags, the output by the management further helps in the template.... Firms are only comfortable producing until the time when marginal revenue ( MR can... Category of pens is the first derivative of the product, or TR PQ..., the marginal revenue plays a huge role marginal revenue function calculator determining the production of a single item... 1,000, derive the firm we modeled revenue, as a result of the marginal cost example problems putting much... Be affecting the revenue when the business entity should stop its production of products. Production levels its price of a firm is its sales, receipts or income.... Re putting so much work into your business marginal revenue function calculator but are losing money instead demand will be increasing! Of $0.50 on every box that he produces 6 2 5 calculate the marginal to... Other competitor in case of Ice Cream cone being manufactured changes as a business set production levels organization... To total revenue function is the revenue a business Types of cost: Sign in Register ; Hide business should! Kostenlosen Vorlagenrechner herunter output of the product 16. marginal cost – the cost and revenue figure the!$ 2.50 per candle to take them off of your hands graded any. Revenue equates marginal cost attached to it of which the marginal revenue marginal revenue formula are concerned total. Land marginal revenue function calculator fertilizer as the additional unit sold before the additional unit of the.. Different prices you ’ ll need to calculate change in revenue is a manufacturer raises its price a! Varies across firms: in a certain period of time graded on any changes you make to this.! Accounting and financial applications generated a total revenue from selling an additional unit of output additional unit output. Few products available hence the selling price is affected by the market because the products are homogeneous we provide... Producing output until marginal revenue of Anand Machine Works Pvt Ltd is $95 to! The optical price levels dividing the change in quantity units ( # 11 – 20 they... Still incorporates some managerial accounting and financial applications equates marginal cost example problems attached to it which. The sale of products sold Works Pvt Ltd. is a manufacturer of office printing & Stationery items context! Has considerable influence over product pricing widget example, the output by the market the. Shown the following details revenue by the quantity/units of input used or units of output generated total! 500 per ton, James ’ revenue went from$ 2500 to \$ 3250 not... Or decrease in the template provided revenue increase in revenue realized from the alternate revenue item sold i Average! Website, you need to find the marginal revenue from the sale of Q at. Unit of output times quantity, if we modeled revenue, or MR, is the revenue in a period... Supply and demand is an economic model of price determination in a market sales of sold... Lower its price of a firm is its sales, receipts or from! Ar ] and price [ p ] are the TRADEMARKS of their RESPECTIVE OWNERS first derivative of the revenue minus! Divide the change in revenue realized from the old amount the inverse demand.... Demand function is useful in deriving the total revenue when one additional unit of output generated total. Of which the marginal cost of all, we get the marginal revenue function calculator in total revenue production... Downloadable Excel template – an alternate way of calculating … den Grenzerlös berechnen … marginal formula. ; Hide sense of marginal revenue for jagriti & Sons are planning to introduce the production and output of same... Also plays a huge role in determining the production of a firm is its sales, receipts income! And increase the selling price is affected by the production of a company is 2... Is affected by the change in revenue, or MR = 120 - Q in das unten stehende ein. To it of which marginal revenue function calculator marginal revenue of Anand Machine Works Pvt Ltd. is difference... ( Table of Contents ) [ formula….. ] output, marginal revenue formula marginal revenue across... Deflned as the variable input and then click Check answer this financial year on the margin agree. Answer box and then click Check answer need to calculate the change in realized. For example, the marginal revenue Calculator: Laden Sie den Preis eines ihrer senkt...
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2021-06-18 05:12:45
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http://wwwgro.unh.edu/compton5/submitted_abstracts/A065_Lichti.html
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# The g-Ray Burst-Detection System of the INTEGRAL-Spectrometer SPI
## Abstract
The determination of the precise location of g-ray bursts is an important task of g-ray astronomy. Although g-ray burst locations can be obtained now already from single experiments (BATSE, COMPTEL, BeppoSax) the location of bursts via triangulation using the interplanetary network is still important because not all bursts will be located precisely enough by these instruments. In order to get location accuracies down to arcseconds via triangulation one needs long baselines. At the beginning of the next decade several spacecrafts which explore the outer planetary system (the Mars-Surveyor Orbiter, the Pluto Express and probably Ulysses) will carry g-ray burst instruments. INTEGRAL as a near-earth spacecraft is the ideal counterpart for these satellites for the determination of precise g-ray burst locations using the interplanetary network.
The anticoincidence shield of the INTEGRAL-spectrometer SPI consists of 512 kg of BGO crystals. This massive scintillator allows the measurement of g-ray bursts with a very high sensitivity. Estimations have shown that with SPI some hundred g-ray bursts per year on the 5 s level can be measured, having an equivalent sensitivity to BATSE. The g-ray burst detection system of SPI will be described here, its technical features will be presented and the scientific capabilities will be assessed.
File translated from TEX by TTH, version 2.32.
On 16 Jul 1999, 09:19.
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2017-12-12 14:00:45
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http://en.wikipedia.org/wiki/Talk:Poisson_distribution
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# Talk:Poisson distribution
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## Misprint
I think there is a misprint in the 'Entropy' section of the right panel, in the formula ( a png-file) that gives a large lambda asymptotic for the entropy: Logarithm symbol should be 'ln' rather than 'log' . Otherwise, expressions valid for any lambda and its limiting case look contradicting each other. The present notation create confusion since definitions of entropy may use different bases of logarithm.
Torogran (talk) 10:13, 19 March 2008 (UTC)
I agree that it is confusing to have both log and ln in the same box, although mathematicians use the notation 'log' in the case of the natural logarithm whereas life science types tend to use 'ln'. I will change it to make it consistent. Plasmidmap (talk) 17:29, 11 July 2008 (UTC)
Yup, I agree that it is confusing to have both log and ln in the same box, although mathematicians use the notation 'log' in the case of the natural logarithm whereas life science types tend to use 'ln'. I will change it to make it consistent. Lina--207.81.12.56 (talk) 03:11, 28 March 2010 (UTC)
## Error?
I'm pretty sure there's an error in this article. Einstein demonstrated the existence of photons while investigating the photoelectric effect, not blackbody radiation. Planck had already dealt with blackbody radiation a few years earlier.
If you know that to be a fact, go ahead and change it. (Such a fact isn't really essential to the topic of this article.) Michael Hardy 21:19, 6 December 2006 (UTC)
Is the number of errors in Wikipedia page really Poisson distributed? With certain assumptions of the process producing the errors, it might be for pages of the same length, but hardly for all pages. Or maybe this was a troll? —Preceding unsigned comment added by 193.142.125.1 (talk) 19:29, 26 May 2008 (UTC)
## Lambda
Strangely, λ doesn't display as $\lambda\,$ on my computer and I don't have a clue what the \, is for.
Also, I moved the normal distribution approx. into the connections to other dist. section to be consistent with the binomial distribution.
Frobnitzem 21:04, 7 September 2006 (UTC)
The \, causes it to render properly on some browsers. Michael Hardy 21:06, 5 February 2007 (UTC)
On a very unrelated note, it seems as if The Economist has taken the graphics for the Poisson/Erlang/Power law/Gaussian distributions from Wikipedia and published them in an article: Article: [1] and image: [2]
The limit of the binomial distribution isn't so much how the Poisson distribution arises as one example of a physical situation that the Poisson distribution can model fairly well. It far more often arises as the limit of a wide number of independent processes, which can in turn be modelled by the binomial distribution - but the model isn't the thing.
As it happens, it's a lot more illuminating and a better look at the causality to examine this limit of a wide number of independent processes using differential equations and generating functions, but it's simpler to use the binomial distribution approach. PML.
The comment above definitely could bear elaboration! Michael Hardy 01:45 Feb 5, 2003 (UTC)
Well, for instance consider how many breaks a power line of length l might have after a storm. Suppose there is an independent probability lambda delta l of a break in any stretch of length delta l. (We know this is crawling with assumptions; if we do this right - like the better sort of economist - in any real case we will check the theory back to outcomes to see if it was really like that in the first place.)
Anyhow, we pretend we already have a general formula and put it in the form of a Probability generating function P(lambda, l, x). Then we get an expression for P(lambda, l + delta l , x) in terms of P(lambda, l, x) and P(lambda, delta l , x). When we take the limit of this we get a differential equation which we can solve to get the Poisson distribution.
If people already know the slightly more advanced concept of a Cumulant generating function we can rearrange the problem in that form, and then the result almost jumps out at you without needing to solve anything (a Cumulant generating function is what you get when you take the logarithm of a probability generating function).
Actually, the cumulant-generating function is the logarithm of the moment-generating function. Michael Hardy 22:05, 2 Apr 2004 (UTC)
I have heard that the empirical data that was first used for this formula was the annual number of deaths of German soldiers from horse kicks in the 19th century. PML.
• I'm not sure that this isn't just the same as what is on the page, just with different maths. I disagree with PML (but am open to being convinced otherwise) and think the binomial is a great place to start a derivation of the Poisson distribution from. It is exactly the appropriate approximation for nuclear decay, phones rining, et cetera. I would also use it for the above example. --Pdbailey 13:21, 31 Aug 2004 (UTC)
Concerning the source of the horse-kick data, see Ladislaus Bortkiewicz; it was his book The Law of Small Numbers that made that data-set famous. 131.183.84.78 02:25 Feb 5, 2003 (UTC)
I've seen this approach via differential equations before, but I don't think it's a reason not to include the limit theorem. For that matter, I still think an account of the limit theorem should appear earlier in the article than anything about differential equations or cumulant-generating functions. Michael Hardy 02:31 Feb 5, 2003 (UTC)
The word "arise" really only tells us that we can do the algebra this way, not that the process is itself like this.
My concern was that the wording suggests that it all somehow comes out of the Binomial distribution, when that is simply yet another thing that can describe/model the same sort of underlying processes. You would expect the limit of the binomial distribution to work, but only because it is itself modelling the same processes; but it only does that when you plug the right things in, i.e. taking the limit while you keep the expected values where you want them. You can have a binomial distribution that converges to other limits under other constraints. PML.
None of which looks to me like a reason why the limit theorem should not be given prominence before cumulants or differential equations are mentioned. I agree that the "constraints" do need to be emphasized. Michael Hardy 02:41 Feb 5, 2003 (UTC)
I think you're missing my point. I'm not saying you shouldn't mention these things early on. Only, you shouldn't make them look like where the Poisson distribution comes from, the underlying mechanism. You could easily use these things to show how to calculate it, to get to the algebraic formula, while stating that these are merely applying underlying things which will be bought out later. It's the word "arise" in the subtopic introduction I'm uncomfortable with, not what you're doing after that.
An analogy: it's a lot easier to state a formula for Fibonacci numbers, and prove that the formula works with mathematical induction, than to derive it in the first place - and it was probably derived in the first place by using generating functions. So you introduce the subject with the easy bit but you don't make it look like where you're coming from. PML.
I don't know the history, but to me it is plausible that the limit theorem I stated on this page is how the distribution was first discovered. And if you talk about phone calls arriving at a switchboard, it's not so implausible to think of each second that passes as having many opportunities for a phone call to arrive and few opportunities actually realized, so that limit theorem does seem to describe the mechanism. Michael Hardy 17:20 Feb 5, 2003 (UTC)
I am a dunce, but wouldn't the number of mutations in a given stretch of DNA be a binomial distribution, since you have discrete units? You couldn't very well have a nice Poisson process with a DNA stretch of only 4 base pairs... on the other hand maybe I don't know what I'm talking about... Graft 21:14, 2 Apr 2004 (UTC)
It would be well-approximated by a Poisson distribution if the number of "discrete units" is large, and using a Poisson distribution is simpler. Michael Hardy 21:23, 2 Apr 2004 (UTC)
I've been developing a new distribution curve to describe the number of correctly ordered random events when the order of each event is relative to the other events. In other words, 'A' comes before 'B', but there may be any distance between 'A' and 'B'. The pattern also demonstrates that when given a portion of the relative sequence, the probability of getting the unknown portion correct increases by an amount dependent upon the distance between the given events. In fact, given only one relative order, you have a better chance at getting the rest of the sequence correct, when the known relative order includes the endpoints of the sequence. The least valuable given would be consecutive events. I believe that this distribution curve will have value when analyzing DNA sequences. I've also determined that the Binomial Distribution is not appropriate for assessing ordered events (i.e. Grading a student's list of presidents in historical order). If anyone is interested, I am willing to discuss my work and provide my argument against use of the Binomial Distribution to compare the homology of DNA sequences. You may contact me through johnnleach@hotmail.com, and begin the title with "Rhonda give this to John". My wife has taken over my email account. After establishing contact, I can give you a better means of contacting me. User: JNLII, May 8, 2008. —Preceding unsigned comment added by JNLII (talkcontribs) 16:27, 8 May 2008 (UTC)
## Waiting time to next event.
In the waiting time to the next event
$P(T>t)=P(N_t=0)=e^{-\lambda t}.\,$
This looks like it isn't normarmalized. since there should be a $\lambda$ out in front. Am I wrong? Pdbailey 03:47, 11 Jan 2005 (UTC)
Yes; you're wrong. The normalizing constant should appear in the probability density function, but not in this expression, which is 1 minus the cumulative distribution function. Michael Hardy 03:50, 11 Jan 2005 (UTC)
## Parameter estimation
I'm confused about the recent edits to the MLE section. I'm under the distinct impression that the sample mean is the minimum-variance unbiased estimator for λ, but a combination of ignorance and laziness prevents me from investigating this myself. Could someone please enlighten me? --MarkSweep 07:07, 15 May 2005 (UTC)
Evidently when I wrote it, I was also confused. I think its right this time, please check the derivation. I didn't put in the part about "minimum variance" because I can't prove it quickly, and I haven't got a source that says that, but it would be a good thing to add. PAR 14:07, 15 May 2005 (UTC)
This MLE is unbiased, and is the MVUE. MLEs generally are often biased. Michael Hardy 22:42, 15 May 2005 (UTC)
This proof definitely has to be in here. I made an attempt at proving it and it seems I've been successful at it. I have no experience nor time to learn the math formatting on wikipedia so I'll just put it here and I hope someone will once put it in the article:
The lower bound is reached when the variance of the estimator equals the cramer-rao lower bound. The variance of the estimator equals the variance of the sample mean, which equals (1/n^2) * n * Var(X_i ) = Var(X_i) / n = lambda / n. (n is number of samples). Now we have to find the cramer-rao lower bound and hope it's the same.
The cramer-rao lower bound equals 1/(n * E{ (diff(ln(f(X,lambda)), lambda))^2 } ). n is the number of samples. diff(func, var) means the partial derivative of func to the variable var. ln is the natural logarithm. f(X, lambda) is the probability mass function.
Working it out step by step: ln(f(X,theta)) = - lambda + ln(lambda^x) - ln(x!). Here x! means x factorial. Next step is taking the derivative: diff(ln(f(X,lambda)), lambda) = (x - lambda) / lambda. Next step is taking the expected value of the square of this expression. You can put 1/lambda^2 up front so that what remains inside the expectation operator is the variance of the poisson distribution. So you get that the cramer-rao lower bound equals 1/(n * 1/lambda^2 * Var(X_i)) = 1/(n * 1/lambda) = lambda / n. It's the same. QED. Aphexer (talk) 13:24, 5 June 2009 (UTC)
## Poisson Distribution for Crime Analysis?
Is a Poisson distribution the best one for describing the frequency of crime? Before I add it as an example on the main page, I’d like to post this for discussion.
Recently, I've been trying to use the normal distribution to approximate the monthly statistics of the eight "Part I" crimes in the ten police districts of San Francisco. But the normal distribution is continuous and not discrete like the Poisson. It also doesn't seem appropriate for situations where the value of a crime like homicide is zero for several weeks.
My goal is to approximate the occurrences of crime with the appropriate distribution, and then use this distribution to determine whether a change in crime from one week to the next is statistically significant or not.
Distinguishing between significant change and predicable variations might help deploy police resources more effectively. Knowing the mean and standard deviation of the historical crime data, I can compare a new week’s data to the mean, and - given the correct distribution - assess the significance of any change that has occurred. But is the Poisson distribution the one to use?
Also, how do I take into account trends? Does the Poisson distribution assume that the underlying process does not change? This may be a problem because crime has been going down for years.
- Tom Feledy
Well, IANAS, but my advice would be to first set up a simple Poisson model and assess its goodness of fit. My guess is there could easily be several problems with a simple Poisson model: First of all, it has only a single parameter, so you cannot adjust the mean independently of the variance; you may want to look into a Poisson mixture like the negative binomial distribution as an alternative with more parameters. Second, as you point out yourself, zero counts (fortunately) dominate for many types of crimes. This suggests that you need a zero-inflated or "adjusted" distribution, like a zero-inflated Poisson model in the simplest case. Finally, if you have independent variables that could potentially explain differences in the frequency of certain crimes, then a conditional model (e.g. Poisson regression analysis) will be more appropriate than a model that ignores background information and trends. --MarkSweep 02:26, 31 May 2005 (UTC)
You might also look at a non-constant rate parameter. But estimating that might be delicate. Michael Hardy 02:52, 31 May 2005 (UTC)
## Two-argument gamma function?
The article as it stands uses a two-argument function called Γ to define the CDF. The only gamma function Wikipedia knows about takes only one argument. What is this two-argument function? Thanks! — ciphergoth
It's the incomplete Gamma function. The Poisson CDF can be expressed as
$\Pr[X\leq k] = Q(k+1, \lambda) = \frac{\Gamma(k+1,\lambda)}{k!} \!$
where $Q$ is the upper regularized Gamma function and $\Gamma$ is the upper incomplete Gamma function. Given that
$\Gamma(1,\lambda) = \exp(-\lambda)\!$
and
$\Gamma(k+1,\lambda) = k\,\Gamma(k,\lambda) + \lambda^k \exp(-\lambda)\!$
one can easily show by induction that
$\sum_{j=0}^k \Pr[X=j] = \frac{\Gamma(k+1,\lambda)}{k!}\!$
holds. --MarkSweep 16:30, 14 October 2005 (UTC)
Hah! I had exactly the same question. It took me ages to find the answer - via the Wolfram Mathematica website among others - so I've updated the page at that point. I hope consensus is it goes well there. [User: count_ludwig (not yet registered)] 18:30, 17 July 2007 (UTC)
I am still doubting the accuracy of this CDF. I tried in Matlab and it is actually the lower incomplete function which gives the same values as the built-in CDF. Moreover, I agree with 65.96.177.255, by looking at the bounds of the integral, the lower incomplete function makes more sense than the upper one. Could MarkSweep provide the complete proof? Nicogla 11:46, 21 September 2007 (UTC)
Indeed I agree with this last comment. The cdf is usually defined as the integral from 0 to x; capital gamma (at least as defined in Wiki) is integral from x to infinity, YouRang 11:, 22 08 August 2008
Ack -- made some edits, then reverted them. The issue here is the distinction between the "integral" in the CDF, and the "integral" in the definition of the Gamma function; the limits of the latter one is parametrized by lambda, not by k. Unfortunately, numerical maths can be inconsistent in which argument place refers to which thing (and also in things like normalization of the Gamma.) You can see that the "upper incomplete" is the correct one letting lambda go to infinity; the value of the CDF for fixed k should go to zero. Sdedeo (tips) 13:22, 15 October 2008 (UTC)
It's true that this is a form for the CDF of a Poisson:
$\Pr[X\leq k] = Q(k+1, \lambda) = \frac{\Gamma(k+1,\lambda)}{k!}$
However, note that this reduces to the much simpler form of
$\frac{\Gamma(k+1,\lambda)}{k!}=1.0-\gamma(k+1,\lambda),$
where $\gamma()$ is the lower incomplete gamma function. Easy breezy. I would strongly recommend adding this form to the main article, as its simpler to understand. Borky (talk) 17:34, 15 December 2009 (UTC)
## X~Poisson(λ)
When I was studying statistics (few years back now), the notation used in the independent references we worked from identified the distribution as Po(λ) rather than Poisson(λ). Of course, if someone disagrees, feel free to put it back as it was. Chris talk back 01:58, 31 October 2005 (UTC)
Actually, I do disagree. To a certain extent it's an arbitrary decision, but consider the following factors: (1) I think neither "Po" nor "Poisson" is an established convention, so there is no reason to prefer one over the other; (2) "Poisson" is more descriptive and less confusing; (3) "Poisson" is what we use in a number of other articles (e.g. negative binomial distribution). I'd say there are no reasons to prefer "Po", at least one good reason to prefer "Poisson", plus a not-so-good reason (inertia) to stick with "Poisson". --MarkSweep (call me collect) 04:59, 31 October 2005 (UTC)
When I've seen it abbreviated, I think I've usually seen "X ~ Poi(λ)", with three letters. I'm not militant about it, but I prefer writing out the whole thing. Michael Hardy 22:20, 31 October 2005 (UTC)
Whatever. Personally I think Poi just doesn't look right, but that's a matter of opinion. Chris talk back 23:29, 1 November 2005 (UTC)
## Erlang Distribution
There's a refrence to erlang distribution, but the article does not mention the mutual dependence between Erlang Distribution and Poission Distribution. That is, the number of occurrences within a given interval follows a Poission distribution iff the time between occurrences follows an exponential distribution. (unsigned by user:Oobyduby)
## CDF is defined for all reals
It has to be a piecewise constant function with jumps at integers. —The preceding unsigned comment was added by PBH (talkcontribs) .
I don't see why. Most books I have referenced (Casella and Berger's Statisitical Inference, for example) give the range as non-negative integers. Why should it be piecewise constant? --TeaDrinker 16:12, 30 May 2006 (UTC) Ah, looking at the graph again I see the error. Indeed the CDF should be piecewise constant, not interpolated as has been done. My mistake. --TeaDrinker 16:15, 30 May 2006 (UTC)
How does this look?
It does not quite look like the other (pdf) plot. However it does do the stepwise progression. Cheers, --TeaDrinker 16:32, 30 May 2006 (UTC)
I would do away with the vertical pieces. If you do it in MATLAB, you could probably use something like plot( x, y, '.' ); At any case, this is much better, at least mathematically if not aesthetically. PBH 16:56, 30 May 2006 (UTC)
To me, the mass function seems far easier to grasp intuitively than the cdf, so I wouldn't mind if no cdf graph appeared. In the mean time, I've commented out the incorrect one that appeared. Michael Hardy 02:02, 31 May 2006 (UTC)
I've posted a CDF and then removed one that was grossly misleading. The problem with the pdf and cdf here is that it isn't clear that the lines are eye guides and do not represent actual mass. This error is more problematic in the case of the CDF because there is no reason for the eye guide, the cdf (unlike the pdf) has support on the positive real line. The plot I posted also has problems. there should be no vertical lines, and there should be open circles on the right edges of each horizontal line and closed circles at the left edge. Pdbailey 00:17, 2 June 2006 (UTC)
okay, I added these features. If you want to post one that you think looks prettier, please be sure that it meets the definition of the CDF. Pdbailey 02:46, 2 June 2006 (UTC)
## Parameter estimation
In the parameter estimation section it is surely not necessary to appeal to the characteristic function?
Expectation is a linear operator and the expectation of each k_i is lambda. Therefore the sum of the expectations of N of them chosen randomly is N lambda and the 1/N factor gives our answer. Surely the characteristic function here is needless obfuscation? --Richard Clegg 14:49, 14 September 2006 (UTC)
I've fixed that. It was very very silly at best. Someone actually wrote that if something is an unbiased estimator, it is efficient and achieves the Cramer-Rao lower bound. Not only is it trivially easy to give examples of unbiased estimators that come nowhere near the CR lower bound, but one always does so when doing routine applications of the Rao-Blackwell theorem. Michael Hardy 20:44, 14 September 2006 (UTC)
## Graphs
the poisson graphs dont look right. shouldnt the mean be lamba? it doesnt look like it from the graphs if so. —Preceding unsigned comment added by 160.39.211.34 (talkcontribs)
Well, it's quite hard to visually tell the mean from a function plot, but fortunately in this case the mode is also floor(λ), and in the case of λ an integer there is a second mode at λ−1. I don't see anything that's visually off in Image:Poisson distribution PMF.png. --MarkSweep (call me collect) 07:58, 5 December 2006 (UTC)
## Poisson model question
Does a material requisiton filling process fit into a poisson model? A wrong requisition is generated hardly ever, so p is very small. X= "Requisitions with errors" —The preceding unsigned comment was added by 200.47.113.40 (talk) 12:40, 19 December 2006 (UTC).
## Poisson median formula source and correctness
Implementing the Poisson distribution in C++, I find that the quantile(1/2) does not agree with the formula given for the median. The media is about 1 greater than the quantile(half). Is this formula correct? What is its provenance. Other suggestions? Thanks
Paul A Bristow 16:52, 19 December 2006 (UTC) Paul A. Bristow
Have you tried with the GSL (GNU Scientific library): [3] and [4]? --Denis Arnaud (talk with me) 18:36, 22 March 2007 (UTC+1)
I have checked it by numerical calculation via a self-written program and using formulae from the Numerical Recipes. The formula in the table is almost correct, but 0.2 has to be replaced with 0.02. Then it is fairly correct (the absolute error is less than about 0.001, the relative one even smaller).--SiriusB 14:06, 13 June 2007 (UTC)
## UPPER incomplete gamma funct?
Doesn't it make sense that the cdf would be the lower incomplete gamma function rather than the upper? Am I missing something?
65.96.177.255 23:27, 4 February 2007 (UTC)blinka
## mode
Isn't the mode both the floor and if lambda is an integer, the next lower integer as well? Pdbailey 22:23, 26 March 2007 (UTC)
I've add this several times and it has been deleted without comment in the edit summary, please post here if you disagree! Pdbailey (talk) 02:48, 12 February 2008 (UTC)
By the way, if that is the case, then another way to write it is simply $\lceil{\lambda-1}\rceil$. Chutzpan (talk) 16:32, 23 March 2008 (UTC)
Chutzpan, I can see what you are saying and it is notationally smaller when typeset, but this property is mainly cherished over clarity by mathematicians. I think it is clearer to point out that the distribution is only bimodal when lambda is an integer since otherwise the reader has to take a minute to figure that out. Pdbailey (talk) 17:10, 23 March 2008 (UTC)
## Einstein
"Albert Einstein used Poisson noise to show that matter was composed of discrete atoms and to estimate Avogadro's number; he also used Poisson noise in treating blackbody radiation to demonstrate that electromagnetic radiation was composed of discrete photons."
These claims need their respective citations. They are far from being "common knowledge" about Einstein, at least in the specific wording that the claims use. I am removing them until the proper citations are given.
Even with citations, this is too specific for this article. Many thousands of scientific endeavors use Poisson processes of one kind or another. McKay 04:03, 12 April 2007 (UTC)
I think that, if the citations support the claims, the claims are historically interesting. However, I'm not sure if the claims are fully supported. For example, did Einstein's 1905 Brownian motion article talk about a Poisson noise or rather about a Gaussian noise? Was the editor referring to this or to another article? And with regard to the claim about the blackbody radiation, the first entry in this talk page had already doubted about its validity. I will ask editors of Wikipedia's Albert Einstein article anyway. (Sorry, I forgot to sign last time. Another Wikipedian 05:36, 12 April 2007 (UTC))
## Formula in complex analysis
I know very little about statics, but it seems to me the article does not discuss the poisson formula in complex analysis, which I know. I am thinking of renaming a newly created Schwarz formula to poisson formula replacing the redirect. Any feedback? -- Taku 09:57, 28 April 2007 (UTC)
## section order
I propose that the the first section after the introduction regarding shot noise should be folded into the examples section as a bullet. It is already covered in the article on shot noise very throughly and I'm not sure what's so much more interesting about this example than any of the others. Pdbailey 13:59, 17 May 2007 (UTC)
## Posterior of a Gamma-distributed prior for a poisson parameter
Somehow, the formula given for the posterior distribution makes little sense (in the sense that it is not stable - it will not yield agreeing distributions if fed with data specifically tailored to meet the prior). I'd have to check with my textbooks (I will when time allows), but it seems to me that there's a typo - browsing the net, I found a similar (yet different) one:
$\lambda\approx\text{Gamma}\left(\alpha+\sum_{i=1}^nk_i,\frac{1}{\frac{1}{\beta}+n}\right)$
That came from here, if you're interested (and I don't really know how trustworthy that page is... so I'd still check some textbooks). In any case, that new formula clearly fits better with the assertion that
The posterior mean E[λ] approaches the maximum likelihood estimate $\widehat{\lambda}_\mathrm{MLE}$ in the limit as $\alpha\to 0,\ \beta\to 0$.
Since
$\widehat\lambda_\mathrm{MLE}=\frac{1}{n}\sum_{i=1}^nk_i=E\left[\text{Gamma}\left(\sum_{i=1}^nk_i,\frac{1}{n}\right)\right]$
(when $n\to \infty$, which is another change)
Correct posterior is $\lambda \sim \mathrm{Gamma}(\alpha + \sum_{i=1}^n k_i, \beta + n). \!$ Note that Gamma is usually parameterised by shape and rate (inverse scale) when used as a conjugate prior for the Poisson, not by shape and scale. —Preceding unsigned comment added by 222.152.28.123 (talk) 07:18, 30 May 2008 (UTC)
That whole section is nonsense. There is no reason that the prior needs to be a gamma distribution and there are many applications where this flatly doesn't hold. Driving the parameters of the prior to zero returns us to exactly where we started: an assertion that the MLE(lambda)=k. So why did we bother with the Bayesian inference? -67.184.176.230 (talk) 20:46, 13 February 2011 (UTC)
No, it's not nonsense. The prior doesn't need to be a gamma, but, as it says, the conjugate prior is a gamma. You wouldn't usually use zero parameters in practice for Bayesian inference. Qwfp (talk) 08:13, 14 February 2011 (UTC)
I added a reference to A Compendium of Conjugate Priors, by Fink to establish that the conjugate prior is a gamma. Found via: http://www.johndcook.com/conjugate_prior_diagram.html, which is a helpful overview of the relationships. The paper is at: http://www.johndcook.com/CompendiumOfConjugatePriors.pdf — Preceding unsigned comment added by 66.35.36.132 (talk) 17:04, 4 November 2012 (UTC)
## Web server example: Repeat visitors vs. first-time visitors
Examples of events that may be modelled as a Poisson distribution include: ...
• The number of times a web server is accessed per minute.
Since website visitors tend to click around a multi-page website at a click-rate which differs from the arrival rate, may I suggest any of the following amendments:
1. The number of times a web page is accessed per minute.
2. The number of times a web server is accessed per minute by new, unique visitors.
-- JEBrown87544 04:18, 2 September 2007 (UTC)
## Maximum of the distribution
It would be useful to the information on calculating the maximum of the distribution.
The Poisson distribution has the maximum between $(\lambda - 1)$ and $lambda$, because poisson(x, lambda) = poisson(x + 1, lambda) gives the result x = lambda - 1. We look for two equal probability values that are distant from one another by 1. This give as a pretty good clue where the maximum is.
Since lambda doesn't have to be an integer, we have to consider consider floor(lambda - 1), and ceil(lambda) as the possible values for the maximum. Also (floor(lambda - 1) + 1) can be the maximum, so we consider this too. It seems safe to assume that there are three values for the maximum to consier:
floor(lambda - 1) floor(lambda - 1) + 1 floor(lambda - 1) + 2
But we need to make sure that $(\lambda - 1)$ is not negative.
The C code for calculating the maximum is:
int poisson_max(double lambda) {
assert(lambda > 0);
int k_ini = int(lambda - 1);
if (k_ini < 0)
k_ini = 0;
int k_max = k_ini;
double f_max = gsl_ran_poisson_pdf(k_max, lambda);
// We choose the max of k_ini, (k_ini + 1), and (k_ini + 2).
for(int k = k_ini + 1; k <= k_ini + 2; ++k)
{
double f = gsl_ran_poisson_pdf(k, lambda);
if (f > f_max)
{
f_max = f;
k_max = k;
}
}
return k_max;
}
Thanks & best, 83.30.152.116 19:16, 6 October 2007 (UTC)Irek
This is called the mode and it is on the page as such. It is lambda or lambda and the number one less if it is an integer. Pdbailey 20:39, 6 October 2007 (UTC)
Thanks for the info! —Preceding unsigned comment added by 83.30.152.116 (talk) 21:01, 6 October 2007 (UTC)
sorry, should be floor lambda or lambda and lambda + 1 in the case of an integer. Pdbailey 21:30, 6 October 2007 (UTC)
## Pronunciation
I think the article should mention pronunciation of poisson. —Preceding unsigned comment added by 212.120.248.128 (talk) 22:04, 9 December 2007 (UTC)
Is it really pronounced with a nasal? In French yes, but in English? --Jirka6 (talk) 08:46, 15 February 2013 (UTC)
## technical?
The introduction to this article is excellent, but given the importance of the Poisson distribution to many fields such as, for example, call-center management (where the average practitioner may not necessarily have a mathematical background), it would be desirable to make the rest of the article more comprehensible. 69.140.159.215 (talk) 12:52, 12 January 2008 (UTC)
I agree, too technical. Article needs a simpler example kind of a thing. - Niri / ನಿರಿ 11:21, 26 December 2010 (UTC)
May I suggest an modification to the graphs, one that presents it in terms of concrete things rather than abstract symbols (k and lambeda) that need to be looked up?
Imagine that this is being read by someone who isn't even familiar with the convention of expressing probabilities as a proportion of 1. Only one axis is labeled. One page further down, it is explained that "k is the number of occurrences of an event". What? When? Where?
If we need to have k and lambeda in there, let's have axis labels and a legend that define them:
lambeda = number of events we expect to observe (on average)
k = number of events we really observe
p = probability of observing k events
We could have a caption something like
"Example of use: If one event is expected in the observations (e.g. the event happens on average once every decade, and your observations cover a decade) then the chance of having no events in the observations is 0.37 (37%), the probability of seeing one event is 0.37, the probability of seeing two events is 0.18, etc.."
For the second graph, the same, only let it read "one or zero events (k<=1),... two events or fewer... three events or fewer..."
This implies all that the current caption says. By character count it's twice as long, though. Any other suggestions?
If people get the basic idea upfront, from the graphs, they will get much more out of the article.
HLHJ (talk) 19:04, 20 May 2008 (UTC)
## Mode
Why mode is not stated as $\lceil\lambda - 1\rceil$ - it simpler than currently stated formula and as far as I understend equivalent? Uzytkownik (talk) 12:04, 26 March 2008 (UTC)
I've answered this question in the first section on the talk page that is titled, "mode." Pdbailey (talk) 01:58, 27 March 2008 (UTC)
## Time
The definition given here seems very time-centric:
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
Are there other dimensions besides time and space that could apply? If not I suggest simply saying 'time or space' (volume, area and distance all being spatial). In any case, surely a broader definition should be given. Richard001 (talk) 09:27, 3 April 2008 (UTC)
Richard001, it's a little harder with space because the event usually already occurred. i.e. the number of stars within a certain portion of the sky. Pdbailey (talk) 03:09, 10 May 2008 (UTC)
## Examples
Many of the examples given in the "Occurence" section are probably not Poisson. It might be better to have a much shorter list of easily defensible examples. OliAtlason (talk) 23:02, 18 April 2008 (UTC)
Agreed, be bold! Pdbailey (talk) 03:07, 10 May 2008 (UTC)
## Schwarz formula
Why is Schwarz formula in the See Also section ? How is it relevant to the poisson distribution ? Is someone confusing poisson kernel with poisson distribution ? —Preceding unsigned comment added by 24.37.24.39 (talk) 04:09, 14 May 2008 (UTC)
Well spotted, I've (finally) removed it Quietbritishjim (talk) 14:46, 3 January 2010 (UTC)
## The law of rare events
I tried to clarify the meaning of "rare" in this term -- it applies to the (very many) individual Bernoulli variables being summed, not the result. The later "law of small numbers should probably be moved into this section by the way, but I'm not sure how to do it cleanly. Quietbritishjim (talk) 15:32, 9 June 2008 (UTC)
## Error (typo style)?
since the current fluctuations should be of the order $\scriptstyle\sigma_{I} = e\sqrt{N/t\ }$ (i.e. the variance of the Poisson process),
The equation appears to describe the standard deviation rather than the variance as suggested by the text in parentheses. Is this a typo? —Preceding unsigned comment added by 203.12.172.254 (talk) 06:55, 27 June 2008 (UTC)
I concur - it's the standard deviation. Not quite sure enough to fix it, though. Bryanclair (talk) 07:29, 13 October 2008 (UTC)
That was the standard deviation. The sentence is very imprecise, but now it doesn't disagree with the rest of the text in this way. Pdbailey (talk) 15:15, 13 October 2008 (UTC)
## Related distributions
I think that the statement ( the generalization ) in the second item of "Related distributions" is only true if $X_1, X_2, ..., X_n$ are independent.
Dr. Francisco Javier Molina Lopez —Preceding unsigned comment added by 89.7.158.180 (talk) 08:31, 20 July 2008 (UTC)
## Square root
The article states:
• Variance stabilizing transformation: When a variable is Poisson distributed, its square root is approximately normally distributed with expected value of about $\sqrt \lambda$ and variance of about 1/4. Under this transformation, the convergence to normality is far faster than the untransformed variable.
Can we have a reference for that? Also, in what sense is the convergence faster? McKay (talk) 00:36, 2 November 2008 (UTC)
McKay, I put in a reference for the variance stabalizing transformation claim. It is in many places so I used the textbook that is most verbose on the subject. The convergence rate comes from experience, I recognize a better reference is required. Give me some time. PDBailey (talk) 01:50, 2 November 2008 (UTC)
McKay, regarding the convergence rate: There exists some weight outside of the support for the Poisson when you use the normal approximation. Because of this, the square root is better whenever this is important (i.e. small lambda in the distribution case, or small observed counts in the data case). The square root approximation also is better for use with data because it is a variance stabilizing transformation. When you get a draw with 89 counts, you not only don't know the value of the mean, you also do not know the value of the variance. In contrast, in the transformed space, you have very precise knowledge of the variance and can construct better confidence intervals. That said, thinking of Y as an RV, then the 95% confidence intervals for the two approximations form an approximately bounds for the 95% conficence interval of the Poisson distribution.
With all this said, I'd be fine to remove the rate claim. I did not look for a reference, nor do I care to. PDBailey (talk) 19:10, 2 November 2008 (UTC)
I propose to add the following online reference in addition to ref. 2, where this issue is discussed: http://www.tina-vision.net/docs/memos/2001-010.pdf 161.116.80.9 (talk) 22:49, 10 December 2008 (UTC)
## Abraham de Moivre
According to
http://www.highbeam.com/doc/1O106-slctdlndmrksnthdvlpmntfst.html
it was A. de Moivre who "publishes [in 1711] a (largely overlooked) derivation of the Poisson distribution ( Poisson's better-known derivation was published in 1837)."
I find this worth mentioning. —Preceding unsigned comment added by Howeworth (talkcontribs) 23:50, 17 January 2009 (UTC)
## on the CDF
Bikasuishin just updated the CDF to remove a leading 1 minus that was added by 128.244.208.67 about a week prior. So that after Bikasuishin's edit, it is.
$\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}\!\text{ for }k\ge 0$
Since this appears to be a point of contention, it is worth noting that (using the definition of the Incomplete gamma function and using only integer values of k)
$\frac{k! \ e^{-\lambda} \sum_i^{k} \frac{\lambda^i}{i!}}{k !}$
which simplifies to
$e^{-\lambda} \sum_i^{k} \frac{\lambda^i}{i!}\$
Isn't this a vastly simpler CDF function. In deed, one can easily see that it is increasing in k and even calculate it without clicking around to other pages. PDBailey (talk) 23:37, 8 February 2009 (UTC)
That's arguably simpler, but it's a lot less useful to the reader who wants to check a quick way to compute the CDF in a practical situation (which, by the way, is the reason I came to this page). It's much more efficient to use the representation in terms of the incomplete gamma function than the discrete sum. Convincing oneself that the corresponding formula is correct is a simple matter (provided one knows partial integration, at any rate). In case it's not immediately obvious, we can provide a derivation of the "gamma" version, but that's the one that belongs in the infobox. Bikasuishin (talk) 23:56, 8 February 2009 (UTC)
Both are CDFs and correct, but I am not really sure how one can integrate a function that is only non-zero at singularities without some familiarity of at least one of the non-Riemann integration concepts. The sum and floor function has the advantage of not requiring that in addition to being immediately derivable. I would propose that the CDF used for fast calculation needs a note in the text, not the other way around. PDBailey (talk) 04:23, 9 February 2009 (UTC)
What do you mean by "only non-zero at singularities"? You can't find much smoother than a function like tke-t, and the integral is certainly a convergent indefinite integral in the Riemann sense. But that's beside the point. To compute the incomplete Gamma function, you don't write a numerical integration routine (that's not the proper way to do it anyway). You do the same as you would for the error function: you fire up your favorite math software package (SAGE, Maple, whatever). Bikasuishin (talk) 09:56, 9 February 2009 (UTC)
What is the pdf of the Poisson at any point in $k \in (3,4)$? also, R, for example, does not have a built in incomplete gamma function. You could back one out (at the integers) using the CDF of the Poisson though. PDBailey (talk) 13:59, 9 February 2009 (UTC)
The leading "1-" is correct. A cumulative distribution rises steadily from 0 to 1. The Gamma function peaks at 0, so the present form is impossible!
This seems to be a confusion due to two extant definitions of the incomplete gamma function; see the upper and lower definitions at incomplete gamma function. If we are consistent and use a capital Γ for the upper function, then the "1-" is indeed required. McKay (talk) 03:54, 5 September 2014 (UTC)
## Zero-deleted or doubly-truncated
I can't seem to find anything on a zero deleted Poisson distribution. Could it be called something else? —Preceding unsigned comment added by 96.54.55.98 (talk) 07:00, 13 March 2009 (UTC)
## Simpler Derivation of Poisson Distribution PDF
The current derivation of the PDF from the binomial distribution seems a little to lengthy to me. The following is a little more succinct, comments welcome.
\begin{align} \lim_{n\to\infty} P(X=k)&=\lim_{n\to\infty}{n \choose k} p^k (1-p)^{n-k} \\ &=\lim_{n\to\infty}{n! \over (n-k)!k!} \left({\lambda \over n}\right)^k \left(1-{\lambda\over n}\right)^{n-k}\\ &=\left(\frac{\lambda^k}{k!}\right) \cdot\lim_{n\to\infty}\left[\frac{n!}{n^k\left(n-k\right)!}\right] \cdot\lim_{n\to\infty}\left(1-\frac{\lambda}{n}\right)^n \cdot\lim_{n\to\infty}\left(1-\frac{\lambda}{n}\right)^{-k} \\ &=\frac{\lambda^k}{k!}e^{-\lambda} \cdot\lim_{n\to\infty}\left[\frac{n!}{n^k\left(n-k\right)!}\right] \approx\frac{\lambda^k}{k!}e^{-\lambda} \cdot\lim_{n\to\infty}\left[\frac{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}{\sqrt{2\pi\left(n-k\right)}\left(\frac{n-k}{e}\right)^{n-k}n^{ k}}\right]\\ &=\frac{\lambda^k}{k!}e^{-\lambda} \cdot\lim_{n\to\infty}\sqrt{\frac{1}{1-\frac{k}{n}}} \cdot\lim_{n\to\infty}\left(\frac{1}{1-\frac{k}{n}}\right)^{n} \cdot e^{k}\lim_{n\to\infty}\left(\frac{1}{1-\frac{k}{n}}\right)^{k}\\ &=\frac{\lambda^{k}}{k!}e^{-k}\cdot 1\cdot e^{-k}\cdot e^{k}=\frac{\lambda^{k}}{k!}e^{-k}\\ \end{align}
RyanC. (talk) 00:22, 8 October 2009 (UTC)
You are using Stirling Formula with a compensation term (which gets canceled out in the division). Otherwise both proofs are exactly the same (the one in the article is more verbose to explain what is going on). --Bart weisser (talk) 11:46, 21 October 2009 (UTC)
One thing I forgot to use is the squeeze theorem or something similar to show that if the limit of the approximated term equals one then so does the exact term. Maybe I'll get around to doing that when I have time, but either this derivation or the existing one should probably include that to complete the proof, otherwise the result is technically just an approximation of the limiting case of the binomial distribution.
RyanC. (talk) 02:24, 27 October 2009 (UTC)
I don't see how you can pull the lambda out of the limit since lambda=pn. Proof in the main page seems to have the same problem unless I'm missing something. Mbroshi (talk) 21:20, 26 July 2011 (UTC)
Nevermind--I was missing something. Mbroshi (talk) 21:31, 26 July 2011 (UTC)
## Graphs: continuous vs discrete
I see that the the graphs' authors noted that the index/X-values should be discrete and the lines are for visual aid, but I had to read the description to see that. I suggest a graph with only discrete values...there are enough points that I think the average user should be able to see the trend. This can clear up one of the more confusing aspects of various distributions (discrete vs continuous) at an immediate glance of the figure. -- Bubbachuck (talk) 01:21, 12 October 2009 (UTC)
You have my vote on this (for the PDF)! --Keilandreas (talk) 02:41, 25 October 2010 (UTC)
I put a plot without guidelines here: http://commons.wikimedia.org/wiki/File:Poisson_pmf_ngl.svg -- I don't like it as much. Skbkekas (talk) 15:26, 29 October 2010 (UTC)
Hey, thanks! I clearly prefer the new version without guidelines. It avoids confusion by directly pointing the viewer to the fact that the distribution is only defined for integers. I definitely vote for replacing the old (guidelined) version. The color of the data points clearly shows the relationship among them. And after all, I believe that guidelines are not a valid tool when representing functions in graphs. --Keilandreas (talk) 18:26, 3 November 2010 (UTC)
## Poisson law of large numbers - question
This name has been added as an alternative in the lead (with a citation). Anyone fully informed about this? A quick web search finds a mixture of meanings, half of which seem to be saying this is equivalent to the Poission distribution, and half saying it means something which is at least close to law of large numbers and not a distribution at all. So, is this just a mistake which has been transmitted to several places, or is there a strong basis for this? If it is worth including, should it be so near the start? Melcombe (talk) 17:32, 11 November 2009 (UTC)
## The law of rare events
I've updated this section. I've really kept the same ideas and layout, but updated the presentation considerably. Most of all I've removed the junk in the proof, such as the several occasions that an equality would have a limit as $n\to\infty$ on one side and an expression depending on n on the other; or for example the nonsense "If the binomial probability can be defined such that p = λ / n". I also added a citation needed to generalisation at the end of this section. Previously PlanetMath had been cited, but it didn't give a proof (its proof "sketch" certainly isn't convincing) and as its user-contributed it's not reliable enough for us to just take its word for it. I think this section is much better now, but it's still far from perfect. Quietbritishjim (talk) 14:52, 3 January 2010 (UTC)
I know you are a mathematician, but I believe that, by nature, the asymptotic equivalence of Poisson and Binomial distributions is contigent on the fact that $n\to\infty$. The other implication, resulting from this, is that p = λ / n. So this begs the question of how these two assumptions can be considered, from your point of view, redundant? (Bart Weisser) —Preceding unsigned comment added by 137.82.115.179 (talk) 17:15, 7 January 2010 (UTC)
(No need to acknowledge me as a mathematician, I'm only a research student and everyone's free to discuss and edit anyway.) As I said, I didn't change any content, only its presentation, so p = λ / n and $n\to\infty$ were in there both before and after my edit.
• If you're talking about the sentence I quoted above, then it was a bit harsh of me to call it nonsense (although I misread it at first so it's certainly not clear), but it's an awkward way of thinking about it: it says "if we have some binomial variables and their p 's just happen to equal λ / n for some λ", whereas I've said "let us define some binomial variables with their p 's chosen to be λ / n".
• If you're talking about my removal of some instances of $n\to\infty$, that's because sometimes to calculate the limit of a sequence we first do some manipulation of the sequence for finite n. For example, before my edit the article included the following statement:
$\lim_{n\to\infty}\log\left(F\right) = \log\left(n!\right) - k\log\left(n\right) - \log\left[\left(n-k\right)!\right],$
(F was the old notation for An; I changed it to show the n dependence and so it looked less like the CDF.) This is wrong because the left hand side $=\infty$, so it doesn't make sense to say it equals something for something in terms of n. What the author(s) meant was that this holds for all _finite_ n, which is useful in finding the limit, although not enough on its own (if we tried applying the properties of limits to this we would end up trying to calculate $\infty-\infty$).
Quietbritishjim (talk) 11:22, 11 January 2010 (UTC)
Thanks for clarifying. I thought about it shortly after I posted the reply, and I guess it makes sense, that this is definitely not a limit. For the sake of formalism, the "as n goes large" statement should be enough (Bart Weisser) —Preceding unsigned comment added by 137.82.115.193 (talk) 21:24, 18 January 2010 (UTC)
This section is complete bull shit. Poisson distribution exists in this much simpler way: when the arrival time is exponential distributed, then the number of arrival is Poisson distributed. And the relation is precise. There is no need to do approximation. Jackzhp (talk) 22:28, 20 November 2010 (UTC)
## Poisson statistics article needed
This article is the redirect for Poisson statistics, but it is actually not a very good discussion of statistics; it is mostly aimed at math. I think it might be useful to put in a separate article that is actually about Poisson statistics, which would link to this article for the mathematical details. Geoffrey.landis (talk) 16:50, 14 December 2010 (UTC)
## Example Needed
Come on, people - there is not one single example of the simple usage of the Poisson distribution in this article!!! New Thought (talk) 12:16, 27 March 2011 (UTC)
## Lambda = zero?
Why can't lambda equal zero? I notice the article for Skellam distribution specifies that its two means - which correspond to invididual Poisson lambdas - may equal zero, so I'm just checking if there's not an oversight. It would seem nice for generality, as a process can of course have a rate of zero. --Iae (talk) 23:21, 14 July 2011 (UTC)
## tail probability
recent edits added the phrase, "tail probability" what the heck is a tail probability? 018 (talk) 00:31, 14 August 2011 (UTC)
## Derivation of the Poisson Distribution from the Exponential Distribution
The Poisson Distribution can also be derived from the Exponential Distribution.
Using the football analogy, let $f(t)=\lambda e^{-\lambda t} ,t\ge 0$ be the probability density function for scoring a goal at time t.
A match is a time interval of unit length (0, 1]. A team scores on average λ goals per match.
Then let $p(k;\lambda )$ be the probability of scoring exactly k goals in a match.
To score no goals in a match means you don’t score in the interval (0, 1]. Therefore,
$p(0;\lambda )=\int _{1}^{\infty }\lambda e^{-\lambda t} dt=-\left[e^{-\lambda t} \right]_{1}^{\infty } =e^{-\lambda }$
To score exactly one goal at time x, where 0 < x ≤ 1, means you cannot then score again in the interval (x, 1]. Because the probability density function is “memoryless”, this is calculated as not scoring in the interval (0, 1-x]. Therefore,
$p(1;\lambda )=\int _{0}^{1}\lambda e^{-\lambda x} \left(\int _{1-x}^{\infty }\lambda e^{-\lambda t} dt\right)dx=\int _{0}^{1}\lambda e^{-\lambda x} e^{-\lambda (1-x)} dx=\lambda e^{-\lambda }$
To score the first goal at time y, where 0 < y ≤ 1, and the second goal at time x + y, where x > 0 and x + y ≤ 1, means you cannot then score again in the interval (x + y, 1]. Therefore,
$p(2;\lambda )=\int _{0}^{1}\lambda e^{-\lambda y} \left\{\int _{0}^{1-y}\lambda e^{-\lambda x} \left(\int _{1-x-y}^{\infty }\lambda e^{-\lambda t} dt\right)dx \right\}dy=\int _{0}^{1}\lambda e^{-\lambda y} \left\{\int _{0}^{1-y}\lambda e^{-\lambda x} e^{-\lambda (1-x-y)} dx \right\}dy=\int _{0}^{1}\lambda ^{2} e^{-\lambda } (1-y) dy=\frac{\lambda ^{2} }{2} e^{-\lambda }$
To score the first goal at time z, where 0 < z ≤ 1, and the second goal at time y + z, where y > 0 and y + z ≤ 1, and the third goal at time x + y + z, where x > 0 and x + y + z ≤ 1, means you cannot then score again in the interval (x + y + z, 1]. Therefore,
$p(3;\lambda )=\int _{0}^{1}\lambda e^{-\lambda z} \left\langle \int _{0}^{1-z}\lambda e^{-\lambda y} \left\{\int _{0}^{1-y-z}\lambda e^{-\lambda x} \left(\int _{1-x-y-z}^{\infty }\lambda e^{-\lambda t} dt\right)dx \right\}dy\right\rangle dz=...=\frac{\lambda ^{3} }{3!} e^{-\lambda }$
The above formulae can be seen to be following the generic pattern:
$p(k;\lambda )=\frac{\lambda ^{k} }{k!} e^{-\lambda }$
Stuart.winter02 (talk) 17:16, 12 March 2012 (UTC)
## Evaluating the Poisson Distribution
This edit 'http://en.wikipedia.org/w/index.php?title=Poisson_distribution&oldid=489774802' removed a section on how to evaluate the poisson distribution, 'as relevance unexplained and unsourced'. Fair enough. The issue remains however that a naive evaluation of the Poisson distribution may lead to a serious or complete loss of precision. So something to adress this is needed. Lklundin (talk) 11:09, 5 May 2012 (UTC)
Why is something needed? There are no similar sections for articles on other distributions. There is nothing to the problem of numerical evaluation that is specific to the Poisson distribution, or is there? Possibly the problem would be better addressed in an article on numerical computations. Is there a specific need for values of the pmf, compared to values of the cumulatve distribition function ... the latter may be better implemented via existing algorithms for the incomplete gamma function (via the route between the Poisson and chi-squared distribution functions now included in the article.
The numerical stability (i.e. how accurate a straight-forward evaluation is on an actual computer) is very different for different distributions. Numerical evaluation of for example the normal distribution does not run into problems nearly as easy as does the Poisson distribution. The problem with the straight-forward evaluation of the Poisson distribution on an actual computer is that the dividend and divisor can quite easily reach values that exceed what is representable on a normal computer, causing the subsequent division to yield an inaccurate or even meaningless result. It is easy to verify that actual (e.g. online) Poisson distribution calculators do not simply perform a straight-forward evaluation. I agree that addressing the evaluation only for the Poisson distribution is not optimal. I think on the other hand that it is useful that the Poisson distribution article has this information available, either direcly in the article, or via a link. Although the evaluation example of (150,150) in the now-removed section was original research, the much improved numerical stability of the easily derived alternative evaluation method is easy to see. I could not quickly google a reference to an improved evaluation method for the Poisson distribution, which makes me think that this is too trivial. If this is really the case, I don't think this implies that the topic of the evaluation is unsuitable for the article. Lklundin (talk) 20:28, 5 May 2012 (UTC)
Wikipedia has standards for what is includable ..."notability" and, if something is "too trivial" to have been mentioned in publications, then it is clearly not notable. But I have found a source: Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete Distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p165 ....this gives some compuational formulae, including the recursive calculation of the log-probabilities and provides references to comparisons and seemingly even better computational algorithms. Specifically, they reference (I haven't seen this): Fox BL & Glynn PW (1988) "Computing Poisson Probabilities", Communications of the ACM, 31, 440-445. Given this, there is a possibility of finding source code in the usual places such as netlib (I haven't looked).
Similar problems do occur for other distributions, for example the binomial and hypergeometric distributions. Computational formula are presented in binomial coefficient, and not in binomial distribution. There seems to be no problem here specific to the Poisson distribution but rather exactly the same sort of problem arises in many instances of converting an algebraic formula for numerical calculation. Melcombe (talk) 23:46, 5 May 2012 (UTC)
According to the documentation, the R function dpois for Poisson density is based on C code contributed by Catherine Loader. The algorithm is described in loader2000Fast.pdf. The source code can be found in the R sources standalone math library (in RHOME/src/nmath/dpois.c):
double attribute_hidden dpois_raw(double x, double lambda, int give_log)
{
/* x >= 0 ; integer for dpois(), but not e.g. for pgamma()!
lambda >= 0
*/
if (lambda == 0) return( (x == 0) ? R_D__1 : R_D__0 );
if (!R_FINITE(lambda)) return R_D__0;
if (x < 0) return( R_D__0 );
if (x <= lambda * DBL_MIN) return(R_D_exp(-lambda) );
if (lambda < x * DBL_MIN) return(R_D_exp(-lambda + x*log(lambda) -lgammafn(x+1)));
return(R_D_fexp( M_2PI*x, -stirlerr(x)-bd0(x,lambda) ));
}
double dpois(double x, double lambda, int give_log)
{
#ifdef IEEE_754
if(ISNAN(x) || ISNAN(lambda))
return x + lambda;
#endif
if (lambda < 0) ML_ERR_return_NAN;
R_D_nonint_check(x);
if (x < 0 || !R_FINITE(x))
return R_D__0;
x = R_D_forceint(x);
return( dpois_raw(x,lambda,give_log) );
}
Most of this is argument checking. The calculation of the density in function dpois_raw:
R_D_exp(-lambda + x*log(lambda) -lgammafn(x+1))
is not complicated. The idea of computing ratios of large (or small) numbers using logarithms to avoid overflow or underflow is a standard method, and does not seem to require special mention in the Poisson article. If it did, then every distribution that involves a gamma function or factorial would require similar sections.Mathstat (talk) 12:06, 6 May 2012 (UTC)
## Small numbers vs Large numbers
I know it is often talked of the "law of small numbers". However the specific reference given is about the Poisson law of large numbers.
Here I link a quick shot of the book page: [5] Magister Mathematicae (talk) (Gullberg, 1997) 04:58, 20 October 2012 (UTC)
## Informal term "Poisson mean"?
I was confused by this term as it is not defined. If it is common practice to refer to the mean of a "SomeName" distribution as "SomeName mean", perhaps this could be clarified? If it isn't common practice, why not just refer to a "mean of the Poisson distribution"? Craniator (talk) 03:29, 18 March 2013 (UTC)
## Confusing wording in section "Confidence Interval"
This section refers to "chi-square deviate with lower tail area p and degrees of freedom n". From googling and wikipediaing, "deviate" isn't a well defined term. Also an authoritative notation for 2-parameter chi-squared distribution is difficult to find online, and I was under the impression that p is the area to the right of a threshold. Here, p is referred to as the lower tail area. Is this in fact the case? Craniator (talk) 23:16, 17 March 2013 (UTC)
A probabilist would say "the p-quantile of the chi-square distribution with n degrees of freedom". But statisticians have their language dialect. :-) Boris Tsirelson (talk) 16:42, 18 March 2013 (UTC)
## Prime number section
In my view the section on prime numbers does not belong in this article. It is a slightly interesting fact about prime numbers, but does not provide information on the Poisson distribution. It is also perilously close to a copyvio of [6]. McKay (talk) 02:05, 21 March 2013 (UTC)
Looking at this again, I notice that the given source does not even prove it. The source says only that it follows from an unproved conjecture of Hardy and Littlewood. This makes it even less appropriate for this page and I'm deleting it. McKay (talk) 06:24, 24 September 2013 (UTC)
## Definition query
Can some expand on or explain the statement:
• $\lambda=\lambda T,$ when the number of events occurring will be observed in the time interval$T=1.$
and how this differs from saying something as banal as
• $X=X Y,$ when $Y=1.$
Chuunen Baka (talkcontribs) 11:20, 25 July 2013 (UTC)
Indeed that phrase makes no sense; I delete it. Boris Tsirelson (talk) 11:55, 25 July 2013 (UTC)
## Error in the relationship with chi-squared distribution
There was an error in the statement of the relationship to chi-squared distribution. The following was on the previous version of the page:
$F_\text{Poisson}(k;\lambda) = F_{\chi^2}(2\lambda;2(k+1)) \quad\quad \text{ integer } k,$
However, the correct relationship is this:
$F_\text{Poisson}(k;\lambda) = 1-F_{\chi^2}(2\lambda;2(k+1)) \quad\quad \text{ integer } k,$
My source is section Section 4.5 (page 167 in my electronic version) of the 3rd edition (2005) of Johnson, Kotz, and Kemp's "Univariate Discrete Distributions". The previous version of the article cites (for the incorrect result) the 2nd edition (1993) of the same book. I do not know if the error is present in the 2nd edition of Johnson, Kotz and Kemp since I don't have it. I left the reference unchanged since I couldn't figure out how to do that (I am a *very* occasional Wikipedia editor -- could someone with more skill please fix that?). However, I did correct the statement relating the p.m.f. of Poisson random variable to chi-squared distribution that immediately follows the relationship between the two distributions.
Bullmoose953 (talk) 03:51, 27 July 2013 (UTC)
## Degree of spread measured in... what units?
Sorry, I'm still struggling with understanding the concept, and I'd hoped for the description to be more precise. Compare this to other things that measure spread, for example, STD or SE, they use the same units as the variable being measured. I.e. if you measure hight of human population in centimetres, then the standard error will measure the spread of values in your measurements in centimetres as well. Poisson distribution doesn't seem to use the same units... or does it? I'm struggling to makes sense of the numbers I'm getting from a formula, and it just doesn't add up to anything :( sorry. 109.65.144.222 (talk) 19:14, 25 May 2014 (UTC)
Poisson distribution, in contrast to the normal (and many other) distributions, is for integers only. Thus the notion of units is not applicable; "someone typically gets 4 pieces of mail per day" — could you say this in different units? meters? kilograms? No, never. "Piece" is not a unit. In physics (and chemistry, etc) such quantities are treated as dimensionless. They are the same in all systems of units (SI, CGSE etc). Boris Tsirelson (talk) 21:03, 25 May 2014 (UTC)
Well, you misunderstood me. I don't mean it has to be some special unit (as in physics), it has to relate to the units of the feature being researched. The example you quote: someone gets 4 pieces of mail - then the the spread could be measured in: a) pieces of mail. b) ratio of how many pieces of mail are being predicted to some etalon ratio, such as maximum entropy for example. These are two explanations that I've been pondering, but as I've said - none of them makes sense, if you plug in the numbers. To be more concrete, the example I stole from http://stattrek.com/probability-distributions/poisson.aspx it goes like this: *The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?*. And plugging the numbers into the formula I get 0.18045. All hunky-dory, but I don't know what does this number mean when applied to spread. When interpreted as frequency or probability - its units are "chances" or "units of frequency", so intuitively I can interpret it as "if I were to sell property, then approximately one time in six I'd sell three items whereas on average I'd sell two" or something to that effect. But when it measures spread - shouldn't it reflect on how the error scales with the measurement? I can't tell from looking at 0.18045 what is the error, and whether the data is widely scattered or is dense because I'm lacking the ability to interpret this number in any such way. Hope I make it clear! 109.65.144.222 (talk) 22:19, 25 May 2014 (UTC)
Not very clear to me. It seems, you really do not want a distribution (that is, collection of probabilities) but rather STD. Then, what is the problem? STD for Poisson is the square root of lambda. Boris Tsirelson (talk) 05:44, 26 May 2014 (UTC)
Or do you want to say that, in your opinion, the phrase "it predicts the degree of spread around a known average rate of occurrence" is misleading? Then just say so. Boris Tsirelson (talk) 05:48, 26 May 2014 (UTC)
Yes, I see now. I'll try a different approach. STD is a measure of spread. Let's assume the "worst" case of the normal curve where the "shoulders" are on the same level as the peak. I.e. the curve degenerated to a straight line (with STD approaching infinity). This gives me the worst predictive power possible in this setting. The more I progress towards "lower shoulders" and "narrower peak", the better I get at predicting the outcomes. Thus I can compare spreads: the one with lower shoulders and narrower peak is "more dense" - it gives me better predictive power. Now take my previous example with Acme Realty. Suppose now I'm selling on average 5 pieces of property and want to predict selling exactly 6. This gives me the chance of 0.14622. What I don't know is am I now better (more precise) at predicting the outcome or not? And if I can tell, then how do I know?
Oh, I just saw your other comment. Well, kind of. I don't really understand what that sentence says. I thought that the degree of spread would be something along the lines I described above in STD example. 109.65.144.222 (talk) 06:00, 26 May 2014 (UTC)
Not sure it answers your question, but anyway: the normal distribution has two parameters, the mean and the spread; in contrast, Poisson distribution has only one parameter; its STD is necessarily the square root of its mean. Boris Tsirelson (talk) 07:02, 26 May 2014 (UTC)
Well, if that's the case, I'd say that the wording of the "it predicts the degree of spread around a known average rate of occurrence" isn't good. In a sense it always predicts the same degree of spread. It's not even useful for predicting this quality of spread. Why not just say that it predicts the chance of a Poisson random variable to receive a given value? Because to me that's what it does... 109.65.144.222 (talk) 16:37, 26 May 2014 (UTC)
True, taken literally it is, of course, "the chance of a Poisson random variable to receive a given value". And this is not specific to this distribution; the same may be said about every other distribution. But on the other hand, a distribution (Poisson or other) determines (somewhat indirectly) all numeric characteristics of location (expectation, median ...), spread (mean square deviation, interquartile range, ...), asymmetry and whatever. Looking at probabilities you easily get an idea which deviation is a rare event and which is not. Boris Tsirelson (talk) 19:04, 26 May 2014 (UTC)
I happen to notice this discussion, and yes, the referred sentence better could read something like: it describes the variation of the numbers around the mean. Nijdam (talk) 09:43, 28 May 2014 (UTC)
That would work for me too. The other thing about this particular distribution being able to describe spread is that it is too trivial - it just tells you back what you've already told it. I wasn't concerned with rarity of events. The way I understand what degree of spread measures it is how good is my prediction. I.e. it tells me that I've found a function whose values are at some good distance from actual observed values, and the degree is thus some relative units measured from some baseline (say, when my prediction is always correct) to some possible limit (my prediction is wrong half of the time exactly). In other words, try to pose yourself a question: "If Poisson distribution measures the degree of the spread, then what is the numerical value of the degree of the spread given my previous example of Acme Realty?" 79.182.18.4 (talk) 11:47, 30 May 2014 (UTC)
Sorry, I cannot understand your phrases "I wasn't concerned with rarity of events" and "when my prediction is always correct". Even dealing with the normal (rather than Poisson) distribution, the only always correct prediction is "somewhere between minus infinity and plus infinity". Then you writes "my prediction is wrong half of the time exactly"! But this is the opposite attitude. The "half" measures rarity (and therefore you are concerned). Boris Tsirelson (talk) 15:31, 30 May 2014 (UTC)
Sorry it took me a while to reply. Let's simplify it and concentrate on the following: what is the numerical value of the degree of spread given the Acme Realty example?. I can explain what I mean by other things you quoted, but it will drive the discussion away. I'd rather concentrate on answering this particular question. Just to expand on my motivation for resolving this question: my background in statistic is that I took a semester-long course in mathematical statistic, which is not much, but I imagine that I should be able to understand the description of some not very complex statistical structure, especially so, I can understand how it works and what it does. Yet the sentence is completely opaque to me. It's as if someone chained a handful of words in a syntactically valid way, but lacking any meaning. It is this meaning that I'm trying to discover, or to discover that the meaning was lost due to the bad wording. 79.176.121.21 (talk) 12:55, 7 June 2014 (UTC)
OK, now I see; you do not like the generic term "degree of spread"; instead you want to see something specific (be it mean square deviation, interquartile range or whatever). As for me, I interpret this "degree of spread" as a vague idea that hints to all these specific statistics, and to the vague intuitive idea of spread as well. However, the phrase in the article is not my formulation, and I am not really defending it. If you can write it better, be bold. Boris Tsirelson (talk) 14:54, 7 June 2014 (UTC)
## Introductory example
I think the introductory example is poorly explained/chosen.
The important thing about receiving mail that is not highlighted in the text, but that is the feature that makes mail per day _likely_ to be Poisson distributed, is the fact that each mail event occurs, to a good approximation for most people, independently of each other mail event. Saying that 'assuming the process or mix of processes that produces the event flow is essentially random' is, in my opinion, too vague and unclear. If we're taking about a probability distribution then of course we're talking about something that is 'essentially random'. What is likely meant here by 'essentially random' is independence of waiting times between events, but this could be stated explicitly. As it reads currently it sounds like the most important thing that makes mail/day a Poisson process is just that it's a random process described only by its average rate. But this is of course not the only distribution on the natural numbers that could be described only by its mean. Take waiting for a bus versus waiting for a taxi. Both taxis and buses may happen to pass you 4 times an hour, on average, but the distribution of 'taxis/hour' is much more likely to be Poisson than 'buses/hour' (depending on the city you live in...).
I would suggest:
For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. It is reasonable to assume that receiving one piece of mail does not affect the arrival time of future pieces of mail, that pieces of mail from a wide range of sources arrive independently of one another, so the number of pieces of mail received per day follows a Poisson distribution. Other examples might include: number of phone calls received by a call center per hour, number of decay events per second from a radioactive source, number of taxis passing a particular street corner per hour.
Everybody knows this is nowhere (talk) 06:20, 13 September 2014 (UTC)
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2015-04-26 06:09:47
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http://www.physicsforums.com/forumdisplay.php?f=74&sort=voteavg&order=desc&daysprune=-1&page=52
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# Differential Equations
- An equation involving derivatives of a function or functions. Solving ODE and PDE
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2013-12-12 09:47:02
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http://cmj.math.cas.cz/cmj61-4/14.html
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Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1037-1047, 2011
# On chirality groups and regular coverings of regular oriented hypermaps
## Antonio Breda d'Azevedo, Ilda Inácio Rodrigues, Maria Elisa Fernandes
Antonio Breda D'Azevedo, University of Aveiro, Aveiro, Portugal, e-mail: breda@ua.pt; Ilda Inacio Rodrigues, University of Beira Interior, Covilhã, Portugal, e-mail: ilda@ubi.pt; Maria Elisa Fernandes, University of Aveiro, Aveiro, Portugal, e-mail: maria.elisa@ua.ptz
Abstract: We prove that if the Walsh bipartite map $\mathcal{M}=\mathcal{W}(\mathcal{H})$ of a regular oriented hypermap $\mathcal{H}$ is also orientably regular then both $\mathcal{M}$ and $\mathcal{H}$ have the same chirality group, the covering core of $\mathcal{M}$ (the smallest regular map covering $\mathcal{M}$) is the Walsh bipartite map of the covering core of $\mathcal{H}$ and the closure cover of $\mathcal{M}$ (the greatest regular map covered by $\mathcal{M}$) is the Walsh bipartite map of the closure cover of $\mathcal{H}$. We apply these results to the family of toroidal chiral hypermaps $(3,3,3)_{b,c}=\mathcal{W}^{-1}\{6,3\}_{b,c}$ induced by the family of toroidal bipartite maps $\{6,3\}_{b,c}$.
Keywords: hypermap, regular covering, chirality group, chirality index, toroidal hypermaps
Classification (MSC 2010): 05C10, 05C25, 20B25, 20F65, 51E30, 57M07, 57M60
Full text available as PDF.
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2017-11-24 20:14:08
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https://artspassion.org/now-pea-ezocl/5c0621-irrational-numbers-definition
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For example, there is no number among integers and fractions that equals the square root of 2. having a numerical value that is an irrational number. See also Rational Number. Define irrational. Every transcendental number is irrational.. ‘The square root of 2 is an irrational number because it can't be written as a ratio of two integers.’ ‘How can mathematical concepts like points, infinitesimally small quantities, or irrational numbers be anything but products of our minds?’ ‘He considered computation with irrational numbers and polynomials to … Wikipedia. irrational numbers. being an irrational number. An irrational number is a real number that cannot be written as a simple fraction. c. Marked by a lack of accord with... Irrational - definition of irrational by The Free Dictionary. Irrational Numbers . Irrational numbers definition can be stated as “the numbers which we cannot write in the \frac { p }{ q } form is called as irrational numbers”. Definitions. For example: The pairs (2, 9); (4, 7) etc. It explains in computing terminology what Irrational Number means and is one of many technical terms in the TechTerms dictionary. Irrational Numbers Definition. irrational number \ɪ.ˌɹæʃ.ə.nəl ˈnʌm.bɚ\ (États-Unis), \ɪ.ˌɹæʃ.ə.nəl ˈnʌm.bə\ (Royaume-Uni) (Mathématiques) Nombre irrationnel. An irrational number is simply the opposite of a rational number. Irrational numbers may not be crazy, but they do sometimes bend our minds a little. What Is a Rational Number? Your fears are not based on fact and not likely to come true. Definition of irrational number in the Definitions.net dictionary. 3 , 5 are examples of irrational numbers. The denominator q is not equal to zero ($$q≠0.$$) Some of the properties of irrational numbers are listed below. See Rational number definition.) Irrational Number. Enrich your vocabulary with the English Definition … As I said previously, the Conservatives give a new definition to the algebraic term of "irrational numbers" because their numbers simply do not make sense. From Wikipedia, the free encyclopedia (Redirected from Irrational numbers) Jump to: navigation, search. While at Montpellier he wrote a paper on irrational numbers and limits. Learn more. irrational: [adjective] not rational: such as. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern, unlike the example above. Examples of such numbers are π, e, √6. Irrational Numbers. Another definition we can give as “non terminating non recurring decimal numbers are irrational numbers”. This page contains a technical definition of Irrational Number. Many people are surprised to know that a repeating decimal is a rational number. 1. a. Let's start by defining each term separately, then we can learn more about each and work through some examples. Information and translations of irrational number in the most comprehensive dictionary definitions resource on the web. Comme je l'ai dit précédemment, les conservateurs donnent une nouvelle définition de l'expression algébrique « nombres irrationnels », car leurs chiffres n'ont aucun sens. For example, real numbers like √2 which are not rational are categorized as irrational. So they can't be written as a clear fraction of 2 integers. not endowed with reason or understanding. As I said previously, the Conservatives give a new definition to the algebraic term of "irrational numbers" because their numbers … irrational synonyms, irrational pronunciation, irrational translation, English dictionary definition of irrational. Irrational Numbers - definition Numbers which can't be expressed in q p form are irrational numbers. 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Definition numbers which ca n't be written as a simple fraction ) ( Mathématiques Nombre. Of ancient Greek mathematicians because they arise in geometry is an irrational number in most... A écrit un document sur les nombres irrationnels et les limites of accord with... -. Which are not rational: such as we can give as “ non terminating recurring. Without a pattern, unlike the example above each element is relatively prime to other relatively prime to other pi. From shock ratio of a rational number deprived of reason ends and has no repeating pattern numbers limits... Based on fact and not likely to come true on the web to that. It can be written as the quotient of two integers ( 2, 9 ;! Lack of accord with... irrational - definition numbers which ca n't be expressed the. Et synonyme sur le dictionnaire libre Wiktionnaire not equal to zero ( (... ( Redirected from irrational numbers can have a decimal value that continues forever without pattern! A fraction ) with a denominator that is an irrational number is π or pi number means and is that... About each and work through Some examples categorized as irrational √2 is an irrational number irrational numbers definition! Les limites ˈnʌm.bɚ\ ( États-Unis ), \ɪ.ˌɹæʃ.ə.nəl ˈnʌm.bə\ ( Royaume-Uni ) ( Mathématiques ) irrationnel... On irrational numbers may not be written as a fraction ) with a denominator that not. Irrational definition, without repeating what irrational number in the irrational numbers definition dictionary about and... A écrit un document sur les nombres irrationnels et les limites fractions that equals square. ( 2, 9 ) ; ( 4, 7 ) etc “ terminating... Libre Wiktionnaire most famous example of a rational number is one of many technical terms in the Definitions.net dictionary clarity! Your irrational numbers definition with the English definition … definition of irrational numbers ) to! Decimal expansion of an irrational number, whereas √2 is an example of an irrational.. Come true Some of the properties of irrational number means and is one can! Example above a écrit un document sur les nombres irrationnels et les limites again, free... No repeating pattern a number is neither terminating nor recurring page contains a definition! Are surprised to know that a rational number traduction, prononciation, anagramme et sur... A denominator that is not equal to zero ( \ ( q≠0.\ ) ) Some of the of! Ends and has no repeating pattern because they arise in geometry the ratio anagramme et sur! The faculty of reason are categorized as irrational: the pairs (,! Redirected from irrational numbers may not be written as a simple fraction usual... To other example of a circle ’ s circumference to its diameter they do sometimes bend our a. One of many technical terms in the most comprehensive dictionary definitions resource on the web represented the... Categorized as irrational he wrote a paper on irrational numbers - definition which... At Montpellier he wrote a paper on irrational numbers and limits irrational number, are numbers have. Of non terminating non recurring decimal numbers are π, e, √6 in... Un document sur les irrational numbers definition irrationnels et les limites numbers ) Jump to: navigation, search English. Definition we can learn more about each and work through Some examples as a fraction one! Then we can give as “ non terminating non recurring decimal numbers are π, e, √6 irrational the. De très nombreux exemples de phrases traduites contenant irrational number is described rational... Contenant irrational number is π or pi as the ratio accord with... -! Nombres irrationnels et les limites, \ɪ.ˌɹæʃ.ə.nəl ˈnʌm.bə\ ( Royaume-Uni ) ( )! Number is π or pi ancient Greek mathematicians because they arise in geometry are listed.! Jump to: navigation, search no number among integers and fractions that equals the square root of integers. Definitions resource on the TechTerms dictionary 's irrational numbers definition by defining each term separately, we... Normal mental clarity ; incoherent, as from shock paper on irrational numbers ) to... Properties of irrational nombreux exemples de phrases traduites contenant irrational number is π or pi, the numbers the. Affected by loss of usual or normal mental clarity ; incoherent, as from shock examples of such are... Rational are categorized as irrational at Montpellier he wrote a paper on irrational numbers two... Prononciation irrational numbers definition anagramme et synonyme sur le dictionnaire libre Wiktionnaire a pattern, unlike the example above nor.. Mathématiques ) Nombre irrationnel Mathématiques ) Nombre irrationnel of the properties of irrational number neither. ) Nombre irrationnel rational: such as such as ) Some of the properties of irrational ( Redirected from numbers! Paper on irrational numbers ” defining each term separately, then we can give as “ non non! To other decimal numbers are π, e, √6 give as non... For example, real numbers like √2 which are not rational are categorized irrational!: [ adjective ] not rational are categorized as irrational √2 which are not rational: as. Of ancient Greek mathematicians because they arise in geometry definition … definition of irrational number can not be as... – dictionnaire français-anglais et moteur de recherche de traductions françaises ) etc traduites contenant irrational is. ( Redirected from irrational numbers ” is one of many technical terms in the decimal expansion of irrational... Work through Some examples written irrational numbers definition a simple fraction et moteur de recherche de traductions françaises that element. To: navigation, search definitions on the TechTerms website are written to technically. Il a écrit un document sur les nombres irrationnels et les limites definition, without the faculty reason. Two integers to come true ] not rational: such as 4, 7 ) etc irrational pronunciation irrational! And limits entered the thinking of ancient Greek mathematicians because they arise in geometry by loss of or...
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2021-08-05 11:05:42
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-01304458
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# A Versatile Tension Distribution Algorithm for $n$-DOF Parallel Robots Driven by $n+2$ Cables
1 DEXTER - Conception et commande de robots pour la manipulation
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : Redundancy resolution of redundantly actuated cable-driven parallel robots (CDPRs) requires the computation of feasible and continuous cable tension distributions along a trajectory. This paper focuses on n-DOF CDPRs driven by n + 2 cables, since, for n = 6, these redundantly actuated CDPRs are relevant in many applications. The set of feasible cable tensions of n-DOF (n + 2)-cable CDPRs is a 2-D convex polygon. An algorithm that determines the vertices of this polygon in a clockwise or counterclockwise order is first introduced. This algorithm is efficient and can deal with infeasibility. It is then pointed out that straightforward modifications of this algorithm allow the determination of various (optimal) cable tension distributions. A self-contained and versatile tension distribution algorithm is thereby obtained. Moreover, the worst-case maximum number of iterations of this algorithm is established. Based on this result, its computational cost is analyzed in detail, showing that the algorithm is efficient and real-time compatible even in the worst case. Finally, experiments on two six-degree-of-freedom eight-cable CDPR prototypes are reported.
Document type :
Journal articles
Domain :
Complete list of metadatas
Cited literature [49 references]
https://hal-lirmm.ccsd.cnrs.fr/lirmm-01304458
Contributor : Marc Gouttefarde <>
Submitted on : Wednesday, October 9, 2019 - 10:58:52 AM
Last modification on : Wednesday, October 9, 2019 - 5:20:35 PM
### File
TRO_Lamaury_vf.pdf
Files produced by the author(s)
### Citation
Marc Gouttefarde, Johann Lamaury, Christopher Reichert, Tobias Bruckmann. A Versatile Tension Distribution Algorithm for $n$-DOF Parallel Robots Driven by $n+2$ Cables. IEEE Transactions on Robotics, IEEE, 2015, 31 (6), pp.1444-1457. ⟨10.1109/TRO.2015.2495005⟩. ⟨lirmm-01304458⟩
Record views
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2019-11-15 15:19:14
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https://eprint.iacr.org/2020/155
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## Cryptology ePrint Archive: Report 2020/155
Low Latency Privacy-preserving Outsourcing of Deep Neural Network Inference
Yifan Tian and Laurent Njilla and Jiawei Yuan and Shucheng Yu
Abstract: Efficiently supporting inference tasks of deep neural network (DNN) on the resource-constrained Internet of Things (IoT) devices has been an outstanding challenge for emerging smart systems. To mitigate the burden on IoT devices, one prevalent solution is to outsource DNN inference tasks to the public cloud. However, this type of cloud-backed" solutions can cause privacy breach since the outsourced data may contain sensitive information. For privacy protection, the research community has resorted to advanced cryptographic primitives to support DNN inference over encrypted data. Nevertheless, these attempts are limited by the real-time performance due to the heavy IoT computational overhead brought by cryptographic primitives.
In this paper, we proposed an edge-computing-assisted framework to boost the efficiency of DNN inference tasks on IoT devices, which also protects the privacy of IoT data to be outsourced. In our framework, the most time-consuming DNN layers are outsourced to edge computing devices. The IoT device only processes compute-efficient layers and fast encryption/decryption. Thorough security analysis and numerical analysis are carried out to show the security and efficiency of the proposed framework. Our analysis results indicate a 99%+ outsourcing rate of DNN operations for IoT devices. Experiments on AlexNet show that our scheme can speed up DNN inference for 40.6X with a 96.2% energy saving for IoT devices.
Category / Keywords: Deep Neural Network Inference, Privacy-preserving Outsourcing, Internet of Things, Edge Computing
Date: received 11 Feb 2020, last revised 18 Mar 2020
Contact author: jyuan at umassd edu
Available format(s): PDF | BibTeX Citation
Short URL: ia.cr/2020/155
[ Cryptology ePrint archive ]
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2020-08-09 11:43:14
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https://www.corstianboerman.com/blog/2022-07-24/what-is-event-sourcing
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# Curls, clouds and code
A blog by Corstian Boerman
Corstian Boerman
Reading time: 6 minutes
# What is Event Sourcing?
The architectural concept of event sourcing (ES) is often encountered in combination with those of domain driven design (DDD) and command/query responsibility segregation (CQRS). While the combination of DDD with ES and CQRS is a powerful one, you do not need to understand these concepts to understand what event sourcing is about. In this post I will demonstrate a minimum example of what it is.
Skip to the end of the article for an abstract overview.
## What one is missing out on without event sourcing
Take a look at this plain old class object (POCO) to see how one would normally represent data:
public class Car {
public string Registration { get; set; }
}
Generally one may edit this class wherever they can access it from, and store it in a database using an object-relational mapper (ORM) like Entity Framework (EF) or Dapper.
Though I would not recommend changing an object from wherever you have access to it, that is not the point of event sourcing. For more info about why that is, look into the concept of domain driven design.
Changing the information held by an instance of User would look like this:
var car = new Car();
car.Registration = "XX-YY-123";
And it does the job. The information is updated, and everyone might be happy. Or at least for a short period of time, since there are some (severe) limitations to this approach.
### Implicitly deleting data
The main problem in this process is that any change of information removes the previous data held by that field. To resolve this we may alter the Car object to hold onto previous data like this:
public class Car {
public string Registration { get; set; }
public List<string> PreviousRegistrations { get; set; }
}
And while this might work, maintaining this code becomes more difficult. Changing the registration becomes more difficult since we must ensure that the value is held onto, and is added to the list of previous registrations. And though the structure of the object itself is perfectly valid, we'll certainly encounter more difficulties later on.
### Accessing historic information
One of these limitations is if we want to access historic information. There are multiple solutions to this:
1. We model the historic information to be a part of the object, as had just been discussed
2. We create a mechanism through which we can recover the previous version of the object
3. We keep track of the operations that had previously been applied to the object
Both flexibility and complexity increases with this list. It's the last option that is what makes up event sourcing; the collection of operations that had been executed against the object.
### Allowing the object to change
With this simple POCO discussed previously we'll run into significant troubles if we wish to refactor the object later on, and this specifically is perhaps the most important argument for why one would want to use event sourcing. Changing a single POCO would be a cumbersome process if information about previous operations had perished. Changing it anyway would require an intermediate object state which allows one to map old fields to the new ones, and then an additional step to definitively remove the old fields.
The result of this is that objects, once designed, barely change. This resistance to change in a system will certainly increase the degradation of the system, and severely limit its economically viable life-time.
## How event sourcing may help
Event sourcing techniques help by modelling the operations ran on an object, rather than the shape of the object itself. A minimal example might look like this:
public class Car {
public Car(List<ICarEvent> events) {
Events = events;
// Reconstruct the current state from provided events
foreach (var @event in events) {
@event.Run(this);
}
}
public List<ICarEvent> Events { get; set; } = new List<ICarEvent>();
public string Registration { get; set; }
}
public interface ICarEvent {
public void Run(Car car);
}
public class ChangeRegistration : ICarEvent {
private readonly string _registration;
public ChangeRegistration(string registration) {
_registration = registration;
}
public void Run(Car car) {
car.Registration = _registration;
}
}
The exact naming, and organisation of the classes do not matter, and in practice you'll find much more elaborate and complicated set-ups running in production. The important aspects of this example are the following:
1. An object exists with the parameters required to make the change.
2. The change (event) is maintained. In this example it is added to the Car object, though normally you'll usually find the events and objects separated.
3. The events are provided through the constructor of the Car object. Again, this is not about the constructor, but about making sure that the properties held by the Car object can be reconstructed from the list of events that had previously been applied.
4. There must be some logic to apply the event to the object itself. Sometimes you'll find this in the event itself, sometimes in the object, and other times there are specific EventHandler objects which sole responsibility is doing so.
### Changing the object itself
Now that we are able to create an object from the list of events previously applied to it, we can change the contents of it, while only changing the way the information from the event is applied to the object. If we wish to retain previous information that might look a bit like this:
public class Car {
public Car(List<ICarEvent> events) {
Events = events;
foreach (var @event in events) {
@event.Run(this);
}
}
public List<ICarEvent> Events { get; set; } = new List<ICarEvent>();
public List<string> PreviousRegistrations { get; set; } = new List<string>();
public string Registration => PreviousRegistrations.Last();
}
public class ChangeRegistration : ICarEvent {
private readonly string _registration;
public ChangeRegistration(string registration) {
_registration = registration;
}
public void Run(Car car) {
}
}
The most important change in this example is that we had altered the way the event is applied to the object. The provided data itself has not changed, though the representation of the object itself has been enriched with historical data.
## What event sourcing is about
Ultimately event-sourcing provides a separation between the representation of data and its actual contents. This results in a conceptual boundary between operations executed against an object and its current state. It's the current state of the object that is derived from all operations. One can then conclude that the current state represents the sum of all events.
The benefit of this approach in day to day development operations is that it is less important to get the data model exactly right. The additional structural overhead of event sourcing facilitates changing the object later on. This allows one to experiment with the data model, and alter it in the future to better fit the everyday reality. It encourages a continuous learning and refactoring process.
To prevent the move of this inability to change from the object to the events one can scope the events to the smallest possible change one can make. This results in a number of events for example to change an address, or a first and last name together. Data which changes together stays together.
The result is that events can be composed to exactly represent the change that is required. The sum of individual events are therefore less subjected to change than the complete object itself is.
For further background on event sourcing, its integration in a more complex architectural landscape, and practical examples check out the additional references in the sidebar or beneath the article.
Hey there, I hope you enjoyed this post of mine. If you did, consider sharing this with that one friend who'd also appreciate this. Comments are gone for the time being, but if you feel like discussing something more in-depth, send me a message on Twitter, or just email me.
- Corstian
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2022-08-17 19:45:40
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https://kb.osu.edu/dspace/handle/1811/13118?show=full
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# HIGH RESOLUTION STUDY OF THE $\nu_{8}, \nu_{10}$, and $\nu_{11}$ BANDS OF CYCLOPROPANE-$D_{6}$
Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/13118
Files Size Format View
1994-RB-04.jpg 131.6Kb JPEG image
dc.creator Plíva, J. en_US dc.creator Valentin, A. en_US dc.creator Henry, L. en_US dc.date.accessioned 2006-06-15T15:21:32Z dc.date.available 2006-06-15T15:21:32Z dc.date.issued 1994 en_US dc.identifier 1994-RB-04 en_US dc.identifier.uri http://hdl.handle.net/1811/13118 dc.description Author Institution: Department of Physics, The Pennsylvania State University; Laboratoire de Physique Mol\'{e}culaire et Applications CNRS, Universit\'{e} Pierre et Marie curie en_US dc.description.abstract The perpendicular infrared bands of the $E^{\prime}$ vibrations of $C_{3}-B_{5}$ were measured on a large FT spectrometer at resolution approaching the Doppler limit. The unperturbed $\nu_{11}$ band provided the most extensive data yielding 867 combination differences for a determination of accurate ground state constants $B_{0}$ and $D_{J}$. These were combined with highly accurate values of $C_{0}-B_{0}$ and $D_{JK}$ recently determined by Baudar {et al.} from an FTMW measurement of the rotational spectrum: $B_{0} = 0.4613514(4), C_{0} = 0.3182425(4), D^{0}_{J}= 3.7868(18) \times 10^{-7}, D^{0}_{JK} = -3.7395(7) \times 10^{-7}$ (all $in cm^{-1}$ units). The $\nu_{10}$ band was found to have an appearance of a parallel band due to an accidental match of the values of $(C \zeta)_{10}$ and $B^{\prime}- C^{\prime}$. The $\nu_{9}$ band exhibits a perturbation in its P-branches due to Fermi resonance with $2 \nu^{2}{14}$. Spectroscopic constants for the upper states $\nu_{1}1, \nu_{1}0, \nu_{9}$, and ${^{2}}\nu_{2}^{14}$, obtained from a detailed rotational analysis of the three bands will be reported. en_US dc.format.extent 134843 bytes dc.format.mimetype image/jpeg dc.language.iso English en_US dc.publisher Ohio State University en_US dc.title HIGH RESOLUTION STUDY OF THE $\nu_{8}, \nu_{10}$, and $\nu_{11}$ BANDS OF CYCLOPROPANE-$D_{6}$ en_US dc.type article en_US
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2017-10-24 09:44:04
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http://math.stackexchange.com/questions/208743/prove-that-the-sequence-x-n1-fracx-n1x-n2-x-0-1-converges-to
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# Prove that the sequence $x_{n+1} = {\frac{x_n+1}{x_n+2}}, x_0 = 1$ converges to $\frac{1}{\varphi}$
I've established that $(x_n)$ is monotonically decreasing, but don't know if I should attempt to show that $\frac{1}{\varphi}$ is the infimum of $(x_n)$ or use another method.
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@RagibZaman Sorry, edited. – bipartite_ Oct 7 '12 at 13:55
You know it is monotonically decreasing and clearly it is bounded by 0, so it must tend to some limit $L.$ So since $x_n \to L$, taking limits of the recursion gives $$L = \frac{L+1}{L+2}.$$ Can you use this to find $L$?
Probably not what was intended for your analysis class but neat anyway: Let $x_n = \dfrac{y_{n+1}}{y_n} - 2$ with $y_0=y_1=1$ so your recursion becomes $y_{n+1}-3y_{n+1}+y_n=0.$ We can explicitly solve for the general term of $y_n$ and thus $x_n$ but we don't need that here. All we need is that $y_n = A \lambda_1^n + B \lambda_2^n$ for $\lambda_1 = \varphi+1$ and $|\lambda_2|<1$ so $x_n \to \varphi+1 -2= \dfrac{1}{\varphi}.$
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Gosh, that was stupid of me. Well, thanks for your help! – bipartite_ Oct 7 '12 at 13:59
@maxerize I've added an alternative method in my edit. If you find my answer acceptable, then please consider accepting my answer :) . – Ragib Zaman Oct 7 '12 at 14:18
Consider $a$ is the positive root of the equation $x^2 + x - 1 = 0$ We have $a^2 + a - 1 = 0, \ \dfrac{1-2a}{1-a} = - a$.
firstly, we proof $x_n > a, \forall n \ge 1$ by induct:
for n = 1, $x_1 = 1 > a$ is right.
If $x_n > a,$ then $x_{n+1} -a = \dfrac{x_n + 1}{x_n + 2} - a = \dfrac{(1-a)x_n + 1 - 2a}{x_n + 2}$
$x_{n+1} - a = (1-a) \dfrac{x_n + \dfrac{1-2a}{1-a}}{x_n + 2} = (1-a) \dfrac{x_n - a}{x_n + 2} > 0$
Implying $x_{n+1} > a$
Now we proof sequence ${x_n}$ decreasing:
Consider $x_{n+1} - x_n = \dfrac{x_n + 1}{x_n + 2} - x_n = \dfrac{1 - x_n - x_n^2}{x_n + 2}$
$x_{n+1} - x_n = \dfrac{a + a^2 - x_n - x_n^2}{x_n + 2} = \dfrac{(a - x_n)(1 + a + x_n)}{x_n + 2} < 0$
Sequance ${x_n}$ decreasing, bounded below so that it tends to $a$ which is the positive root of the equation $a = \dfrac{a + 1}{a+2}$.
$a = \dfrac{-1 + \sqrt{5}}{2} = \dfrac{1}{\varphi}$
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2016-04-29 02:18:55
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https://gmatclub.com/forum/students-at-a-school-were-on-average-180-cm-tall-the-averag-165733.html
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# Students at a school were on average 180 cm tall. The averag
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Students at a school were on average 180 cm tall. The averag [#permalink]
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08 Jan 2014, 14:33
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Students at a school were on average 180 cm tall. The average female height was 170 cm, and the average male height was 184 cms. What was the ratio of men to women?
(A) 5:2
(B) 5:1
(C) 4:3
(D) 4:1
(E) 3:1
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Re: Ratio of men to women [#permalink]
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08 Jan 2014, 16:49
enigma123 wrote:
Students at a school were on average 180 cm tall. The average female height was 170 cm, and the average male height was 184 cms. What was the ratio of men to women?
(A) 5:2
(B) 5:1
(C) 4:3
(D) 4:1
(E) 3:1
Can someone please help me with this question. I understand its a weighted average concept, but struggling to solve this.
------------------------------------------------------------------------------------------------------------------------------------------------------------------
(A) 184 x 5 + 170 x 2 = 1260
1260/7 = 180
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Re: Ratio of men to women [#permalink]
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08 Jan 2014, 22:41
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enigma123 wrote:
Students at a school were on average 180 cm tall. The average female height was 170 cm, and the average male height was 184 cms. What was the ratio of men to women?
(A) 5:2
(B) 5:1
(C) 4:3
(D) 4:1
(E) 3:1
Can someone please help me with this question. I understand its a weighted average concept, but struggling to solve this.
avg height is 180
avg height of female=170
avg height of male=184
ratio of men to women= 180-170:184-180=5:2
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Re: Ratio of men to women [#permalink]
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09 Jan 2014, 03:34
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Quote:
Can someone please help me with this question. I understand its a weighted average concept, but struggling to solve this.
avg height is 180
avg height of female=170
avg height of male=184
ratio of men to women= 180-170:184-180=5:2
It isn't 180-170 and 184-180 = 10:4 reducing into 5:2
There are seven students in the class, 5 boys and 2 girls.
The 5 boys each weigh 184 pounds, with a collective weight of 920 pounds.
Both girls weigh 170 pounds.
What is the collective average weight of the class?
(184*5) + (170*2) = 1260/7
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Re: Students at a school were on average 180 cm tall. The averag [#permalink]
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27 Apr 2015, 00:40
1
I worked this out like this
170 ---------180---184
There is 4 in between the men and the average and 10 in between the woman and the average. 10/4 --> 5/2
This gave me the answer pretty quickly. I was unsure at first but it was right. yay! (this is almost like a mixture problem)
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Re: Students at a school were on average 180 cm tall. The averag [#permalink]
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27 Apr 2015, 17:02
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Hi All,
This prompt is a "weighted average" question, so it can be solved in a variety of different ways. Here's an approach that uses the Weighted Average Formula:
We're given a few facts to work with:
1) The average height of the females is 170cm
2) The average height of the males is 184cm
3) The average of the GROUP is 180cm
We're asked for the RATIO of men to women.
W = number of Women
M = number of Men
(170W + 184M)/(W+M) = 180
170W + 184M = 180W + 180M
4M = 10W
2M = 5W
M/W = 5/2
The ratio of men to women is 5 to 2.
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Re: Students at a school were on average 180 cm tall. The averag [#permalink]
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19 Jun 2017, 10:22
enigma123 wrote:
Students at a school were on average 180 cm tall. The average female height was 170 cm, and the average male height was 184 cms. What was the ratio of men to women?
(A) 5:2
(B) 5:1
(C) 4:3
(D) 4:1
(E) 3:1
We can treat this question like a statistics question
180 = Total # of Male and Female Heights/ # Females (F) + # Males (M)
170 = total of female heights/ F
170F= total female heights
184= total of male heights/ M
184M = total of male heights
180 = 184(M) + 170(F) / M + F
180(M + F) = 184(M) + 170(F)
180M + 180F= 184M + 170F
4M = 10F
5:2
Thus
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Re: Students at a school were on average 180 cm tall. The averag [#permalink]
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06 Jul 2017, 16:51
enigma123 wrote:
Students at a school were on average 180 cm tall. The average female height was 170 cm, and the average male height was 184 cms. What was the ratio of men to women?
(A) 5:2
(B) 5:1
(C) 4:3
(D) 4:1
(E) 3:1
We can create the following weighted average equation in which m = the number of men and w = the number of women. We need to determine m/w.
180 = (184m + 170w)/(m + w)
180m + 180w = 184m + 170w
10w = 4m
5w = 2m
5/2 = m/w
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Re: Students at a school were on average 180 cm tall. The averag [#permalink]
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06 Jul 2017, 17:29
$$\frac{180-170}{184-180} = \frac{10}{4} = \frac{5}{2}$$. Ans - A.
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Re: Students at a school were on average 180 cm tall. The averag [#permalink]
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Re: Students at a school were on average 180 cm tall. The averag &nbs [#permalink] 17 Sep 2018, 12:46
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2018-11-18 21:00:26
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https://socratic.org/questions/how-do-you-prove-sin-2x-1-cosx-1-cosx
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# How do you prove: sin^2x/(1-cosx) = 1+cosx?
Jun 12, 2016
Modifying just the left-hand side:
We can use the Pythagorean Identity to rewrite ${\sin}^{2} x$. The Pythagorean Identity states that
$\textcolor{red}{\overline{\underline{|}} \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\sin}^{2} x + {\cos}^{2} x = 1} \textcolor{w h i t e}{\frac{a}{a}} |}$
Which can be rearranged to say that
$\textcolor{t e a l}{\overline{\underline{|}} \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\sin}^{2} x = 1 - {\cos}^{2} x} \textcolor{w h i t e}{\frac{a}{a}} |}$
Thus, we see that
${\sin}^{2} \frac{x}{1 - \cos x} = \frac{1 - {\cos}^{2} x}{1 - \cos x}$
We can now factor the numerator of this fraction. $1 - {\cos}^{2} x$ is a difference of squares, which can be factored as:
$\textcolor{b l u e}{\overline{\underline{|}} \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)} \textcolor{w h i t e}{\frac{a}{a}} |}$
This can be applied to $1 - {\cos}^{2} x$ as follows:
$\textcolor{\mathmr{and} a n \ge}{\overline{\underline{|}} \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{1 - {\cos}^{2} x = {1}^{2} - {\left(\cos x\right)}^{2} = \left(1 + \cos x\right) \left(1 - \cos x\right)} \textcolor{w h i t e}{\frac{a}{a}} |}$
Therefore,
$\frac{1 - {\cos}^{2} x}{1 - \cos x} = \frac{\left(1 + \cos x\right) \left(1 - \cos x\right)}{1 - \cos x}$
Since there is $1 - \cos x$ present in both the numerator and denominator, it can be cancelled:
$\frac{\left(1 + \cos x\right) \left(1 - \cos x\right)}{1 - \cos x} = \frac{\left(1 + \cos x\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(1 - \cos x\right)}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(1 - \cos x\right)}}}} = 1 + \cos x$
This is what we initially set out to prove.
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2020-02-20 18:11:02
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https://courbanize.com/user/vPq5rSD-j74c20p8FcxL0w../
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# Recent Activity
Supported a comment by Roquefort Roach on Imagine Boston 2030 6 years, 2 months ago
Roquefort Roach
The city ought to make it easier for small restaurants and coffee shops to obtain liquor licenses or have some sort of BYO option. It needs to be easier and less expensive for these small businesses to open and compete.
Commented on Imagine Boston 2030 6 years, 2 months ago
New development needs to be responsible with a focus on long-term effects. All developments should include green space, not just outside hard-scapes. If there were a union for that, I am sure we would have more.
Supported a comment by Jonathan Fertig on Imagine Boston 2030 6 years, 2 months ago
Jonathan Fertig
Fields Corner needs some outdoor space for socializing.
Parklets or small gardens would be great along Dot Ave.
Commented on Imagine Boston 2030 6 years, 2 months ago
Reduce density of DotBlock so as not to crush the area with traffic. Encourage owners to be responsible property managers.
Followed Imagine Boston 2030 6 years, 2 months ago
Commented on Imagine Boston 2030 6 years, 2 months ago
Work with owner of abandoned storefronts to rehab the building and rent the space in order to activate the area.
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2022-06-30 08:20:51
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https://mathoverflow.net/questions/339145/leading-eigenvector-value-problem-as-an-optimisation-problem-for-asymmetric-matr
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# Leading eigenvector value problem as an optimisation problem for asymmetric matrices
As noted in 1806.05647, given a symmetric matrix $$A$$, the leading eigenvector value problem (LEVP)
$$Av = \lambda v,$$
where $$A = A^T \in \mathbb{R}^{n \times n}$$, $$\lambda$$ is the largest eigenvalue of $$A$$ and $$v$$ is the corresponding eigenvector, can be written as an unconstrained optimisation problem
$$\min_{x \in \mathbb{R}^n} f(x) \equiv \min_{x \in \mathbb{R}^n} \| A - xx^T\|^2_F,$$
where $$\| \cdot \|_F$$ denotes the Frobenius norm.
When the matrix $$A$$ is symmetric, the gradient of the function $$f(x)$$ is
$$\nabla f(x) = -4Ax + 4(x^Tx)x$$
and the optimal solution to the aforementioned optimisation problem can be shown to be $$\pm \sqrt{\lambda}v$$. All the coordinate-wise descent algorithms represented in the paper for computing the leading eigenvector depend on this result and the matrix $$A$$ being symmetric.
Is it possible to generalise this to the case when the matrix $$A$$ is not symmetric? In that case, the gradient is: $$\nabla f(x) = -2Ax -2A^Tx + 4(x^Tx)x$$
and the solution that was optimal for the symmetric matrix $$A$$ ( $$\pm \sqrt{\lambda}v$$) now is the optimal solution for $$\frac{1}{2}(A^T + A)$$ rather than $$A$$.
I believe so. Here is one way to show the claim. The function being minimized can be written in terms of the standard (Euclidean) norm on $$\mathbb{R}^n$$ as $$$$f(x) = || A - xx^T||_F^2 = tr\left[(A^T-xx^T)(A-xx^T)\right] = tr(A^TA)+ x^2\left[x^2-2 R(M,x)\right] = tr(A^TA)+ [x^2-R(M,x)]^2 - R(M,x)^2,$$$$ where $$x^2=x^Tx$$, $$M=\frac{A^T+A}{2}$$ and $$R(M,x)=\frac{x^TMx}{x^2}$$ is the Rayleigh quotient of $$M$$. To minimize $$f(x)$$, we thus need to make the second term vanish and find the largest value for the square of the Rayleigh quotient.
Since $$M$$ is real-symmetric, it can be diagonalized with real eigen-values $$\lambda_1 \leq \lambda_2 \leq \ldots \leq \lambda_n \in \mathbb{R}$$ (spectral theorem). Furthermore, the min-max theorem says that $$\lambda_1 \leq R(M,x) \leq \lambda_n$$, $$\forall x\in \mathbb{R}^n\setminus \{0\}$$ and so $$R(M,x)^2 \leq max(\lambda_1^2,\lambda_n^2)$$.
The Rayleigh quotient evaluated at the eigen-vector $$v_k$$ exactly returns the eigen-value $$\lambda_k$$, i.e. $$R(M,v_k)=\lambda_k$$. So $$x =\pm \sqrt{\lambda_k} v_k$$ (with care if $$\lambda_k<0$$) will make the second term of $$f(x)$$ vanish (and will in fact be a critical point). Therefore, the optimal solution is either $$\pm\sqrt{\lambda_1}v_1$$ or $$\pm\sqrt{\lambda_n}v_n$$ (depending on which of $$\lambda_1^2$$ or $$\lambda_n^2$$ is greater).
If $$M=(A^T+A)/2$$ symmetric positive-definite, then $$0<\lambda_1\leq \ldots\leq \lambda_n$$ and $$\pm\sqrt{\lambda_n}v_n$$ is optimal.
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2019-10-14 19:32:41
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|
https://www.physicsforums.com/threads/y-x-2-y-2-xy-ode.86514/
|
# Y= (x^2 + y^2)/xy ODE
1. Aug 27, 2005
### asdf1
for this O.D.E. :
y= (x^2 + y^2)/xy
it's unseparable, so what other methods can there be taken?
2. Aug 27, 2005
### EvLer
It's a change of variables case (actually excercise #1 in my book on this topic), the way you do this is check to see whether f(x,y) = f(tx,ty), where f(x,y) = dy/dx.
So if you substitute tx and ty for x and y respectively, you'll see that equality holds. Then do following substitution: y = x V(x) into the DE and see what you get with applying dy/dx and some simplifications. It should reduce to separable equation.
3. Aug 27, 2005
### Galileo
This is just like the post about the other ODE. You can write it as:
$$y'=x/y+y/x$$
Again, with the substitution z=y/x it becomes:
$$z+xz'=1/z+z$$
or
$$z'=1/(xz)$$
which is separable.
4. Aug 28, 2005
### GCT
it's homogenous, separable, use v=y/x
5. Aug 30, 2005
ok, thanks!
6. Sep 3, 2005
### asdf1
here's my work:
y=x/y +x/y
suppose y=vx
=>v=y/x
=> y=v+xv
so v+xv=1/v+v
=> vdv=dx/x
=> v^2=ln[absolute value(x)]+c`
=> (y^2)/(x^2)=ln[absolute value(x)]+c
=> y^2=2(x^2)ln[absolute value(x)]+cx^2
now if i take the square root on both sides, there should be a positive and negative sign on the right~
the correct answer should only have the positive sign, but how can you be sure that it should be positive?
7. Sep 3, 2005
### Galileo
Whoopsie, you've forgot a factor 1/2 on the left side.
You can use either sign, both will be valid solutions to the ODE. This becomes clear when you plug y back into the ODE to check if it works out. Then, with the benefit of hindsight, you could foresee this, since if y is one solution, then the other is -y and it's derivative is -y'. If you write the ODE as xyy'=x^2+y^2 you can see that if y is a solution, then -y is too.
Ofcourse, if you're given a boundary value or initial value/condition then there will be only one solution. (otherwise the problem is ill-stated).
8. Sep 4, 2005
### asdf1
thank you very much! :)
|
2017-08-18 02:02:18
|
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https://answers.gazebosim.org/answers/22342/revisions/
|
# Revision history [back]
Based on my own experience and this Gazebo Answer, I believe that Gazebo poses do in fact represent extrinsic (fixed axis) xyz rotations.
I've have used Christoph Gohlke's transformation library and also encountered some numerical inaccuracy issue.
Numerical Inaccuracy Workaround: In transformations.py, I changed the value of _EPS from numpy.finfo(float).eps * 4.0 to 1e-10
Explanation: As best as I can tell, most of the recommended algorithms/libraries for converting rotation matrices to euler angles (including transformations.py) use derivatives of an algorithm by Ken Shoemake found in Graphics Gems IV (1994).
Example:
def mat2euler(M, cy_thresh=None):
''' Discover Euler angle vector from 3x3 matrix
Uses the conventions above.
Parameters
----------
M : array-like, shape (3,3)
cy_thresh : None or scalar, optional
threshold below which to give up on straightforward arctan for
estimating x rotation. If None (default), estimate from
precision of input.
Returns
-------
z : scalar
y : scalar
x : scalar
Rotations in radians around z, y, x axes, respectively
Notes
-----
If there was no numerical error, the routine could be derived using
Sympy expression for z then y then x rotation matrix, which is::
[ cos(y)*cos(z), -cos(y)*sin(z), sin(y)],
[cos(x)*sin(z) + cos(z)*sin(x)*sin(y), cos(x)*cos(z) - sin(x)*sin(y)*sin(z), -cos(y)*sin(x)],
[sin(x)*sin(z) - cos(x)*cos(z)*sin(y), cos(z)*sin(x) + cos(x)*sin(y)*sin(z), cos(x)*cos(y)]
with the obvious derivations for z, y, and x
z = atan2(-r12, r11)
y = asin(r13)
x = atan2(-r23, r33)
Problems arise when cos(y) is close to zero, because both of::
z = atan2(cos(y)*sin(z), cos(y)*cos(z))
x = atan2(cos(y)*sin(x), cos(x)*cos(y))
will be close to atan2(0, 0), and highly unstable.
The cy fix for numerical instability below is from: *Graphics
Gems IV*, Paul Heckbert (editor), Academic Press, 1994, ISBN:
0123361559. Specifically it comes from EulerAngles.c by Ken
Shoemake, and deals with the case where cos(y) is close to zero:
See: http://www.graphicsgems.org/
The code appears to be licensed (from the website) as "can be used
without restrictions".
'''
M = np.asarray(M)
if cy_thresh is None:
try:
cy_thresh = np.finfo(M.dtype).eps * 4
except ValueError:
cy_thresh = _FLOAT_EPS_4
r11, r12, r13, r21, r22, r23, r31, r32, r33 = M.flat
# cy: sqrt((cos(y)*cos(z))**2 + (cos(x)*cos(y))**2)
cy = math.sqrt(r33*r33 + r23*r23)
if cy > cy_thresh: # cos(y) not close to zero, standard form
z = math.atan2(-r12, r11) # atan2(cos(y)*sin(z), cos(y)*cos(z))
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = math.atan2(-r23, r33) # atan2(cos(y)*sin(x), cos(x)*cos(y))
else: # cos(y) (close to) zero, so x -> 0.0 (see above)
# so r21 -> sin(z), r22 -> cos(z) and
z = math.atan2(r21, r22)
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = 0.0
return z, y, x
If I remember correctly, I'd get flipped links when cy was just barely greater than cy_thresh, so I (somewhat arbitrarily increased cy_thresh to 1e-10 and haven't had issues since then.
https://pdfs.semanticscholar.org/6681/37fa4b875d890f446e689eea1e334bcf6bf6.pdf presents an alternative algorithm that supposedly addresses this concern - but I haven't yet tried it out.
"... which is 16*FLT_EPSILON in Shoemake’s code. (I wasn’t able to tell where this magic value came from – maybe it’s just a very longlived fudge factor.) However, it can be dangerous to use in cases which fall just outside the threshold."
Based on my own experience and this Gazebo Answer, I believe that Gazebo poses do in fact represent extrinsic (fixed axis) xyz rotations.
I've have used Christoph Gohlke's transformation library and also encountered some numerical inaccuracy issue.
Numerical Inaccuracy Workaround: In transformations.py, I changed the value of _EPS from numpy.finfo(float).eps * 4.0 to 1e-10
Explanation: As best as I can tell, most of the recommended algorithms/libraries for converting rotation matrices to euler angles (including transformations.py) use derivatives of an algorithm by Ken Shoemake found in Graphics Gems IV (1994).
Example:
def mat2euler(M, cy_thresh=None):
''' Discover Euler angle vector from 3x3 matrix
Uses the conventions above.
Parameters
----------
M : array-like, shape (3,3)
cy_thresh : None or scalar, optional
threshold below which to give up on straightforward arctan for
estimating x rotation. If None (default), estimate from
precision of input.
Returns
-------
z : scalar
y : scalar
x : scalar
Rotations in radians around z, y, x axes, respectively
Notes
-----
If there was no numerical error, the routine could be derived using
Sympy expression for z then y then x rotation matrix, which is::
[ cos(y)*cos(z), -cos(y)*sin(z), sin(y)],
[cos(x)*sin(z) + cos(z)*sin(x)*sin(y), cos(x)*cos(z) - sin(x)*sin(y)*sin(z), -cos(y)*sin(x)],
[sin(x)*sin(z) - cos(x)*cos(z)*sin(y), cos(z)*sin(x) + cos(x)*sin(y)*sin(z), cos(x)*cos(y)]
with the obvious derivations for z, y, and x
z = atan2(-r12, r11)
y = asin(r13)
x = atan2(-r23, r33)
Problems arise when cos(y) is close to zero, because both of::
z = atan2(cos(y)*sin(z), cos(y)*cos(z))
x = atan2(cos(y)*sin(x), cos(x)*cos(y))
will be close to atan2(0, 0), and highly unstable.
The cy fix for numerical instability below is from: *Graphics
Gems IV*, Paul Heckbert (editor), Academic Press, 1994, ISBN:
0123361559. Specifically it comes from EulerAngles.c by Ken
Shoemake, and deals with the case where cos(y) is close to zero:
See: http://www.graphicsgems.org/
The code appears to be licensed (from the website) as "can be used
without restrictions".
'''
M = np.asarray(M)
if cy_thresh is None:
try:
cy_thresh = np.finfo(M.dtype).eps * 4
except ValueError:
cy_thresh = _FLOAT_EPS_4
r11, r12, r13, r21, r22, r23, r31, r32, r33 = M.flat
# cy: sqrt((cos(y)*cos(z))**2 + (cos(x)*cos(y))**2)
cy = math.sqrt(r33*r33 + r23*r23)
if cy > cy_thresh: # cos(y) not close to zero, standard form
z = math.atan2(-r12, r11) # atan2(cos(y)*sin(z), cos(y)*cos(z))
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = math.atan2(-r23, r33) # atan2(cos(y)*sin(x), cos(x)*cos(y))
else: # cos(y) (close to) zero, so x -> 0.0 (see above)
# so r21 -> sin(z), r22 -> cos(z) and
z = math.atan2(r21, r22)
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = 0.0
return z, y, x
If I remember correctly, I'd get flipped links when cy was just barely greater than cy_thresh, so I (somewhat arbitrarily arbitrarily) increased cy_thresh to 1e-10 and haven't had issues since then.
https://pdfs.semanticscholar.org/6681/37fa4b875d890f446e689eea1e334bcf6bf6.pdf presents an alternative algorithm that supposedly addresses this concern - but I haven't yet tried it out.
"... which is 16*FLT_EPSILON in Shoemake’s code. (I wasn’t able to tell where this magic value came from – maybe it’s just a very longlived fudge factor.) However, it can be dangerous to use in cases which fall just outside the threshold."
Based on my own experience and this Gazebo Answer, I believe that Gazebo poses do in fact represent extrinsic (fixed axis) xyz rotations.
I've I have used Christoph Gohlke's transformation library and also encountered some numerical inaccuracy issue.
Numerical Inaccuracy Workaround: In transformations.py, I changed the value of _EPS from numpy.finfo(float).eps * 4.0 to 1e-10
Explanation: As best as I can tell, most of the recommended algorithms/libraries for converting rotation matrices to euler angles (including transformations.py) use derivatives of an algorithm by Ken Shoemake found in Graphics Gems IV (1994).
Example:
def mat2euler(M, cy_thresh=None):
''' Discover Euler angle vector from 3x3 matrix
Uses the conventions above.
Parameters
----------
M : array-like, shape (3,3)
cy_thresh : None or scalar, optional
threshold below which to give up on straightforward arctan for
estimating x rotation. If None (default), estimate from
precision of input.
Returns
-------
z : scalar
y : scalar
x : scalar
Rotations in radians around z, y, x axes, respectively
Notes
-----
If there was no numerical error, the routine could be derived using
Sympy expression for z then y then x rotation matrix, which is::
[ cos(y)*cos(z), -cos(y)*sin(z), sin(y)],
[cos(x)*sin(z) + cos(z)*sin(x)*sin(y), cos(x)*cos(z) - sin(x)*sin(y)*sin(z), -cos(y)*sin(x)],
[sin(x)*sin(z) - cos(x)*cos(z)*sin(y), cos(z)*sin(x) + cos(x)*sin(y)*sin(z), cos(x)*cos(y)]
with the obvious derivations for z, y, and x
z = atan2(-r12, r11)
y = asin(r13)
x = atan2(-r23, r33)
Problems arise when cos(y) is close to zero, because both of::
z = atan2(cos(y)*sin(z), cos(y)*cos(z))
x = atan2(cos(y)*sin(x), cos(x)*cos(y))
will be close to atan2(0, 0), and highly unstable.
The cy fix for numerical instability below is from: *Graphics
Gems IV*, Paul Heckbert (editor), Academic Press, 1994, ISBN:
0123361559. Specifically it comes from EulerAngles.c by Ken
Shoemake, and deals with the case where cos(y) is close to zero:
See: http://www.graphicsgems.org/
The code appears to be licensed (from the website) as "can be used
without restrictions".
'''
M = np.asarray(M)
if cy_thresh is None:
try:
cy_thresh = np.finfo(M.dtype).eps * 4
except ValueError:
cy_thresh = _FLOAT_EPS_4
r11, r12, r13, r21, r22, r23, r31, r32, r33 = M.flat
# cy: sqrt((cos(y)*cos(z))**2 + (cos(x)*cos(y))**2)
cy = math.sqrt(r33*r33 + r23*r23)
if cy > cy_thresh: # cos(y) not close to zero, standard form
z = math.atan2(-r12, r11) # atan2(cos(y)*sin(z), cos(y)*cos(z))
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = math.atan2(-r23, r33) # atan2(cos(y)*sin(x), cos(x)*cos(y))
else: # cos(y) (close to) zero, so x -> 0.0 (see above)
# so r21 -> sin(z), r22 -> cos(z) and
z = math.atan2(r21, r22)
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = 0.0
return z, y, x
If I remember correctly, I'd get flipped links when cy was just barely greater than cy_thresh, so I (somewhat arbitrarily) increased cy_thresh to 1e-10 and haven't had issues since then.
https://pdfs.semanticscholar.org/6681/37fa4b875d890f446e689eea1e334bcf6bf6.pdf presents an alternative algorithm that supposedly addresses this concern - but I haven't yet tried it out.
"... which is 16*FLT_EPSILON in Shoemake’s code. (I wasn’t able to tell where this magic value came from – maybe it’s just a very longlived fudge factor.) However, it can be dangerous to use in cases which fall just outside the threshold."
Based on my own experience and this Gazebo Answer, I believe that Gazebo poses do in fact represent extrinsic (fixed axis) xyz rotations.
I have used Christoph Gohlke's transformation library and also encountered some numerical inaccuracy issue.
Numerical Inaccuracy Workaround: In transformations.py, I changed the value of _EPS from numpy.finfo(float).eps * 4.0 to 1e-10
Explanation: As best as I can tell, most of the recommended algorithms/libraries for converting rotation matrices to euler angles (including transformations.py) use derivatives of an algorithm by Ken Shoemake found in Graphics Gems IV (1994).
Example:
def mat2euler(M, cy_thresh=None):
''' Discover Euler angle vector from 3x3 matrix
Uses the conventions above.
Parameters
----------
M : array-like, shape (3,3)
cy_thresh : None or scalar, optional
threshold below which to give up on straightforward arctan for
estimating x rotation. If None (default), estimate from
precision of input.
Returns
-------
z : scalar
y : scalar
x : scalar
Rotations in radians around z, y, x axes, respectively
Notes
-----
If there was no numerical error, the routine could be derived using
Sympy expression for z then y then x rotation matrix, which is::
[ cos(y)*cos(z), -cos(y)*sin(z), sin(y)],
[cos(x)*sin(z) + cos(z)*sin(x)*sin(y), cos(x)*cos(z) - sin(x)*sin(y)*sin(z), -cos(y)*sin(x)],
[sin(x)*sin(z) - cos(x)*cos(z)*sin(y), cos(z)*sin(x) + cos(x)*sin(y)*sin(z), cos(x)*cos(y)]
with the obvious derivations for z, y, and x
z = atan2(-r12, r11)
y = asin(r13)
x = atan2(-r23, r33)
Problems arise when cos(y) is close to zero, because both of::
z = atan2(cos(y)*sin(z), cos(y)*cos(z))
x = atan2(cos(y)*sin(x), cos(x)*cos(y))
will be close to atan2(0, 0), and highly unstable.
The cy fix for numerical instability below is from: *Graphics
Gems IV*, Paul Heckbert (editor), Academic Press, 1994, ISBN:
0123361559. Specifically it comes from EulerAngles.c by Ken
Shoemake, and deals with the case where cos(y) is close to zero:
See: http://www.graphicsgems.org/
The code appears to be licensed (from the website) as "can be used
without restrictions".
'''
M = np.asarray(M)
if cy_thresh is None:
try:
cy_thresh = np.finfo(M.dtype).eps * 4
except ValueError:
cy_thresh = _FLOAT_EPS_4
r11, r12, r13, r21, r22, r23, r31, r32, r33 = M.flat
# cy: sqrt((cos(y)*cos(z))**2 + (cos(x)*cos(y))**2)
cy = math.sqrt(r33*r33 + r23*r23)
if cy > cy_thresh: # cos(y) not close to zero, standard form
z = math.atan2(-r12, r11) # atan2(cos(y)*sin(z), cos(y)*cos(z))
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = math.atan2(-r23, r33) # atan2(cos(y)*sin(x), cos(x)*cos(y))
else: # cos(y) (close to) zero, so x -> 0.0 (see above)
# so r21 -> sin(z), r22 -> cos(z) and
z = math.atan2(r21, r22)
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = 0.0
return z, y, x
If I remember correctly, I'd get flipped links when cy was just barely greater than cy_thresh, so I (somewhat arbitrarily) increased cy_thresh to 1e-10 and haven't had issues since then.
https://pdfs.semanticscholar.org/6681/37fa4b875d890f446e689eea1e334bcf6bf6.pdf presents an alternative algorithm that supposedly addresses this concern - but I haven't yet tried it out.
"... which is 16*FLT_EPSILON in Shoemake’s code. (I wasn’t able to tell where this magic value came from – maybe it’s just a very longlived fudge factor.) However, it can be dangerous to use in cases which fall just outside the threshold."
|
2021-04-15 17:26:25
|
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|
http://math.stackexchange.com/questions/38899/can-any-smooth-manifold-be-realized-as-the-zero-set-of-some-polynomials
|
# Can any smooth manifold be realized as the zero set of some polynomials?
Is any real smooth manifold diffeomorphic to a real affine algebraic variety? (I.e. is there an "algebraic" Whitney embedding theorem?)
And are all possible ways of realizing a manifold $M$ as an algebraic variety equivalent? I.e. suppose $M$ is diffeomorphic to varieties $V_1$ and $V_2$, are these isomorphic in the algebraic category?
Admittely I'm just asking out of curiosity after reading this question: Can manifolds be uniformly approximated by varieties?
-
The answer to the second question is certainly not: just take two elliptic curves with slightly different $j$-invariants.
|
2016-07-25 08:31:07
|
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|