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# American Institute of Mathematical Sciences

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2012, 9(3): 539-552. doi: 10.3934/mbe.2012.9.539

## Impact of vaccine arrival on the optimal control of a newly emerging infectious disease: A theoretical study

 1 Department of Mathematics and Applications, University of Naples Federico II, via Cintia, I-80126 Naples, Italy

Received  April 2011 Revised  March 2012 Published  July 2012

When a newly emerging human infectious disease spreads through a host population, it may be that public health authorities must begin facing the outbreaks and planning an intervention campaign when not all intervention tools are readily available. In such cases, the problem of finding optimal intervention strategies to minimize both the disease burden and the intervention costs may be addressed by considering multiple intervention regimes. In this paper, we consider the scenario in which authorities may rely initially only on non-pharmaceutical interventions at the beginning of the campaign, knowing that a vaccine will later be available, at an exogenous and known switching time. We use a two-stage optimal control problem over a finite time horizon to analyze the optimal intervention strategies during the whole campaign, and to assess the effects of the new intervention tool on the preceding stage of the campaign. We obtain the optimality systems of two connected optimal control problems, and show the solution profiles through numerical simulations.
Citation: Bruno Buonomo, Eleonora Messina. Impact of vaccine arrival on the optimal control of a newly emerging infectious disease: A theoretical study. Mathematical Biosciences & Engineering, 2012, 9 (3) : 539-552. doi: 10.3934/mbe.2012.9.539
##### References:

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# Solution - State Ampere'S Circuital Law, Expressing It in the Integral Form - Ampere’s Circuital Law

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State Ampere’s circuital law

State Ampere's circuital law, expressing it in the integral form

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## Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole

# Chapter 8 - Section 8.2 - Perimeter and Area of Polygons - Exercises - Page 360: 11

#### Answer

1764$mm^{2}$

#### Work Step by Step

Given, AB = 39 mm, BC =52mm CD =25mm, DA = 60mm By Brahmaguptas formula, the area of cyclic quadrilateral with sides of length a,b,c and d is A = $\sqrt (s-a)(s-b)(s-c)(s-d)$ s = $\frac{a+b+c +d}{2}$ AB =a = 39 mm, BC = b = 52mm CD = c =25mm, DA = d = 60mm s = $\frac{39+52+25 +60}{2}$ =88mm A = $\sqrt (88-39)(88-52)(88-25)(88-60)$ =$\sqrt 49*36*63*28$ =1764$mm^{2}$ Therefore the area of the cyclic quadrilateral ABCD = 1764$mm^{2}$.

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# Math Help - iteration

1. ## iteration

im trying to write an iteration for x in x^x-y=0

i'v used x_(n+1)= x_n - (f(x_n)/f'(x_n))

maybe i'm doing somthing wrong...

so far the math i'v been doing latly has been all wrong..

dan

2. Originally Posted by dan
im trying to write an iteration for x in x^x-y=0

i'v used x_(n+1)= x_n - (f(x_n)/f'(x_n))
That looks correct to me.
---
Only problem is that you are might be breaking the theorem. Meaning some theorem do not work because your problem does not satisfy the conditions for the theorem. For example, the fundamental theorem of calculus sometimes fails, not because it is wrong but if you use a discountinous function it will not work (those were the assumed conditions). In this algorithm I know there is a special condition your function needs to satisfy (besides for being differenciable on some open interval) I forget what it is. So even if you did everything correct you might not satify a special condition.

3. Hello, Dan!

I'm trying to write an iteration for x in: . x^x - y = 0

I've used: .x_(n+1) .= .x_n - f(x_n)/f '(x_n)

It seems that you are seeking an x-intercept of: .y .= .x^x
. . (If I'm wrong, disregard this entire post.)

Is your derivative correct? . . . It's tricky.

Take logs: . ln y .= .ln(x^x) .= .x·ln(x)

Differentiate: . y'/y .= .x(1/x) + ln(x)

Then: . y' .= .y[(1 + ln(x)] .= .x^x[1 + ln(x)]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x^x
The interation becomes: . x_(n+1) . = . x_n - -----------------
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x^x[1 + ln(x)]
. . . . . . . . . . . . . . . . . . . . . . . . .1
Therefore: .x_(n+1) . = . x_n - ------------
. . . . . . . . . . . . . . . . . . . . . . 1 + ln(x)

4. Originally Posted by Soroban
Hello, Dan!

It seems that you are seeking an x-intercept of: .y .= .x^x
. . (If I'm wrong, disregard this entire post.)

Is your derivative correct? . . . It's tricky.

Take logs: . ln y .= .ln(x^x) .= .x·ln(x)

Differentiate: . y'/y .= .x(1/x) + ln(x)

Then: . y' .= .y[(1 + ln(x)] .= .x^x[1 + ln(x)]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x^x
The interation becomes: . x_(n+1) . = . x_n - -----------------
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x^x[1 + ln(x)]
. . . . . . . . . . . . . . . . . . . . . . . . .1
Therefore: .x_(n+1) . = . x_n - ------------
. . . . . . . . . . . . . . . . . . . . . . 1 + ln(x)

that looks good ...

thanks

dan

5. Originally Posted by dan
that looks good ...

thanks

dan
I am curious does the iteration work now? Because it still might not as I said in the first post.


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or
Jack DouglasCC0 1.0
TopAnswers has been designed to host multiple, independent[^1], communities of creators, curators, and beneficiaries of high quality Q&A.

If you'd like a particular community to be created here, please let us know in an answer below. We'll need:

1. A topic that lends itself to the focused Q&A format.
1. At least a few people who want to get involved, who are passionate about building that community here.

It would also be nice to know up front if you would hope to eventually get any particular customized tooling, things like:

* specialist markdown plugins like MathML or MathJax
* particular font support
* integration with external tools or services/apis (for example ['databases'](/databases) has [db<>fiddle integration](/databases?q=30))

[^1]: independent but perhaps sometimes overlapping, both in terms of topicality and participation.
We have 6 (5 + myself) LaTeX aficionados who would like to try getting a TeX based community going. All are current or formerly active members from the [TeX.SE](https://tex.stackexchange.com/). A private beta period would be preffered over public just to get things started on the right foot, experiment with how it will go, and make sure everything is copasetic before making a more public invite.

1. The scope would be TeX and all major direct dirivatives[^1] and the tooling dedicated to the TeX ecosystem.

1. :check:

As for special needs, we won't need anything major front.

* Adding the [LPPL license](https://www.latex-project.org/lppl/) commonly used in LaTeX derivative projects as an option for source code in answers would be lovely, possibly eeven making a CC0/LPPL combo the default.[^2] See [this SE post](https://tex.meta.stackexchange.com/q/1255/5100) for previous discussion of this issue.

* It is important **not** to enable MathML or MathJax support as this will inevitably conflict with users trying to covey what they want to do in LaTeX terms.

* I'm sure there will be font preferences, but I don't know what they are right now.

* Currently there are no "fiddle" style services that would be easy to integrate[^3], but eventually if there was such a thing that covered enough LaTeX engines it definitely be a nice thing to have.

Long term vector graphics suppport would also be nice. Vector or otherwise, expect this site to have a _lot_ of images (as screen shots will be used in almost every post). Making sure the image system scales well and handles the load will be an important  point when considering long term viability.

[^1]: Personally [I'd actually advocate for a bit wider scope](https://tex.meta.stackexchange.com/q/1255/5100) covering several similar digital typesetting systems, but I'm clearly in the minority right now as far as the potential community goes.

[^2]: The default would need some further discussion, at least having LPPL as an option would be very much appreciated as a minimum.

[^3]: Although [SwiftLaTeX](https://github.com/SwiftLaTeX/SwiftLaTeX/issues/12) released earlier today is a promising step towards being able to make one!
CC BY-SA 4.0Monica
**Update:** The community ended up trying [something that's more similar to SE](https://writing.codidact.com/questions/39219) at least for now.  (Basically, we needed a home Right Away because of SE, and a temporary site that looks more like SE made for an easier transition.  The long-term decision remains open.)

----

Representatives of [Writing.SE](https://writing.stackexchange.com/) request a community.

I'm writing (ha!) on behalf of a few of the "senior" users of our community.  We haven't had a meta discussion (which I suspect SE would shut down anyway) yet; we thought it better to set up something here and then invite people to check it out.  This means our community here would be provisional, but I have high hopes for it -- we're in crisis over on SE and our users would like a place to continue as a community.

Our site has about 8,000 questions that are not closed.  I have the impression you're not necessarily aiming for bulk imports; we could start with everything created by the users who come over (including questions that our users answered).  There might be some specific  tags we'd want to bring over wholesale.  (At least *I* do.)  How should we discuss import?

Our users will almost certainly want to see a [way to track their own content](https://topanswers.xyz/meta?q=233), and if we don't do something about [account recovery](https://topanswers.xyz/meta?q=234) or [passwords](https://topanswers.xyz/meta?q=237) we're likely to run into problems.  (Many of our users are not hardcore tech-geeks.)  These issues don't keep us from starting to use the new site but would probably be barriers to full-on adoption.  (Nobody wants to invest and risk losing access in a cookie accident.)  We know that other areas are under active development and are likely to be improved; we're ok with a site that's still under construction.  Do expect questions where we don't yet have help/FAQ posts, of course. :-)

We have no special tool or font needs.  We were using the generic beta theme on SE.

I don't know how you're handling moderation or even what moderators can do.  Our site had moderators on SE, so from a *community* perspective it's probably cleanest to appoint some of them as mods here.  If the community takes root here as I hope it will, and as TopAnswers works out how it wants to handle moderation, I would expect to then follow whatever process is developed for a community to choose moderators.  (In other words, I think of the initial mods as truly *pro-tem*, not serving for years and years without community buy-in.)

What other matters do we need to work out with TopAnswers so we can then invite our community in?

CC BY-SA 4.0Jonathan Mee
I'd like to start a C++ programming questions site. I ask and answer a lot of C++ questions over at StackOverflow, or at least I did. I'm looking for a new home: https://stackoverflow.com/users/2642059/jonathan-mee?tab=summary

Most C++ examples on Stack Overflow were externally hosted by ideone.com or similar, so I don't think any integration would be required. However, code markdown inside backticks or indentation to markdown a codeblock would be helpful.
I'd like to see a community for programming questions. I'm quite shocked to see it hasn't been created yet.

For a more focused community/topic, I'd suggest starting with Java programming questions.

I'm ready to say goodbye to a certain other popular site for programming questions, and to be active here. I don't have a whole group of people with me, but I hope others will join.
fenced code blocks are always the answer!
Jack Douglas replying to Jonathan Mee
I think [fenced code blocks](https://spec.commonmark.org/0.27/#fenced-code-blocks) are probably the answer? Assuming you are trying with indented code blocks
Jonathan Mee
Guys is there a way to place a code block immediately following a list? On StackOverflow I used the HTML comment between them to break the list (<!-- -->) but that doesn't seem to work here.
I used to be pretty good at it though
kinda rusty, and it has grown a lot since the last time I used it
My guess is that a .NET comunity would make sense but it is only a guess…
Lennart
Would a community ".NET" make sense, in contrast to a singluar C# or Ff# (or...) site? It would also encompass things like Visual Studio, NuGet, ASP, WPF etc. This would be something I can contribute to by importing from SO and general participation.
Jonathan Mee
Looks great I'll start populating it as I see questions I can link and we'll see what kind of traction we can get! Thanks for the help!
Jack Douglas
please let us know what tags you'd like to see and we are happy to add more at any time. And as soon user-tag-creation is a thing, we'll let you know.
Jack Douglas
OK @Jonathan, let's give it a go: https://topanswers.xyz/cplusplus — thanks for the vote of confidence and for your enthusiasm — we'll continue to try and encourage others to participate, but if there is reasonably regular activity in there I think it's likely that we'll gradually get people joining. Perhaps even a few from reddit :)
Jack Douglas
looks like the C++ logo uses shades of blue
Jack Douglas replying to Jonathan Mee
if we implement a dark theme it will be a profile settings, swapping black/white and the dark/light community shades, leaving the 'mid' shade alone. However we still need to have some sort of colour to build the theme around (the 'mid' shade) — grey on meta, blue on databases, green on code golf, etc
Jonathan Mee
I know that's a constant request even now of Stack Overflow.
Jonathan Mee
Oh, man could we do a dark theme? Is that legal?
Jack Douglas replying to Jonathan Mee
what about a colour scheme? any particular colour make sense to use as a base?
Jonathan Mee
creates less mayhem if I'm linking it in different contexts.
Jonathan Mee
probably cplusplus?
Jack Douglas replying to Jonathan Mee
any thoughts on what our url should be? I think https://topanswers.xyz/c++ would be legal, but perhaps https://topanswers.xyz/cplusplus would be more sensible?
Jonathan Mee
I'm down with that. I'd start asking my own core questions and answers if we had something available, and just link to the core answer externally.
Jack Douglas
are you prepared for us to set it up here on a trial basis here for  a few months, and then reassess whether it's going anywhere?
Jack Douglas
that sounds interesting
Jack Douglas replying to Jonathan Mee
oh I see, so you'd be producing canonical selfies here?
Jonathan Mee
Ideally putting content on our site which deals with underlying issues which could be linked to would be a good linkable foothold.
Jonathan Mee
Oh I wasn't saying that I'd reproduce anything. For example there are often questions of: "How do I set an array size to a variable?" The short answer is: "C++ doesn't currently allow this." I'd like to add a link to a question on our site which says: "What is the progress toward variable length arrays?"
Jack Douglas
does that make sense or am I missing something obvious?
Jack Douglas
it would be fine to pose the question here as a brief summary and a link to the reddit question but that would only be workable as a stepping stone. The final outcome (I think) woul have to be to either get the OPs permission to reproduce the full question here, or to delete the question here and post the full answer on reddit :(
Jack Douglas
reddit have a license from the copyright holder but we'd need permission from the OP to reproduce their question here
Jack Douglas replying to Jonathan Mee
there is a question mark over copyright in the question I think : https://www.reddit.com/r/TheoryOfReddit/comments/3uz0s1/who_owns_the_copyright_on_reddit_comments_and/
Jonathan Mee
I've been keeping myself busy in my leave from StackOverflow over here: https://www.reddit.com/r/cpp_questions/ but I realize I could even ask the real questions somewhere else and provide link answers to the content here.
Jonathan Mee
Lol, sounds like me in Java. I just keep trying to write C++code and having to go look up the statement analogs.
but I didn't code in C in quite a while now and every time I start a small function I feel like I've forgotten everything. So, I'd consider myself neither a C nor a C++ coder really. But if I have to decide one, I'd say C, as I never really used C++.
Jonathan Mee
That applies for the most part.
I feel comfortable in the subset of C++ that is C.
same question to you @Skillmon — are you a C++ coder as well as C?
Jack Douglas
@aditsu how is your C++?
Jack Douglas replying to Jonathan Mee
OK, we'll consider that as an option if we don't get any traction with other SO users. I've contacted a couple who I know are not completely happy with how things are going there.
Jonathan Mee
I do have a couple coworkers who are relatively advanced who I may be able to persuade to also be a part... If we had something running anyway.
Jonathan Mee replying to Jack Douglas
As much as I regret it, I don't know anyone on there personally. So I don't think I can pull any of them over.
If you look at a lot of questions, you'll see I answer my own pretty frequently, often because of their complexity or the unsatisfactory nature of answers. So even if it's just me talking to myself I'd like to get something going.
Jack Douglas
@Jonathan this is something we'd be very keen to get running here if we can get 2 or 3 more participants for a private beta to get things up and running — they don't need to have your level of expertise, just a decent level of understanding of the topic and enthusiasm for the project. I've [asked in an SO chat room](https://chat.stackoverflow.com/transcript/message/48554314#48554314) to see if we can get others on board but you may also know of other folk who'd be interested you can ask?
I don't have a really strong opinion, but I tend to agree with you
Then I have to oppose to @Jack's idea. I think it would be better to have a general landing point for now and split out languages once we have the manpower to dedicate a community to that language. Starting with a fragmented approach while being so small isn't the right thing to do, imho.
Andriy M replying to Jack Douglas
Oh, so that's another reason for me to be glad to have ended up in such a rational-minded company, then. I don't know if any of Monica's SE communities have suffered much from cross-posting but if they have and it was the way she described (similar questions with different focus posted on different sites still considered to be against the rules), then I'm sorry they didn't have enough consensus to agree to take a more reasonable view on the matter. Or maybe they just didn't have a chance to discuss the issue, who knows.
Jack Douglas replying to Andriy M
I've got a feeling dba.se probably handled that better than the average SE site — i.e. we did it the way @Monica is suggesting but others just rejected everything
That doesn't quite match my experience, to be honest. As far as I remember, the aversion has always been to posting *exactly the same* questions on different sites within the network. The two sites I've mainly participated in answering on are SO and DBA.SE, and there's a definite overlap in what both sites consider to be on topic. Certain types of questions could be posted on either site without the need to rephrase them to tailor to the requirements of that community, but we wouldn't tolerate word-for-word cross-postings. On the other hand, asking the same question under a different angle would be perfectly acceptable if not even encouraged.
Andriy M
-
oh, I just realized I can click a user from the site of the chat
btw, do these @ mentions actually work?
@Skillmon my original idea was a single community for programming questions (should be easier to get it started) and I still prefer it, but since @Jack wanted to separate languages, I guess I'm ok with that too
I've never understood the aversion on SE to *responsible* multi-posting.  If your question has different aspects that involve different communities, and you focus each question on the community where you're asking it, (and bonus points if you cross-link them), what's the problem?
there are a few differences already: 'labels' are implemented as distinct 'question types' here rather than with tags, and different question types have different voting rules (though that's not fully implemented yet)
Skillmon
(and if we don't have enough expertise to help people programming in A or B in the beginning, so be it, questions not getting answers happens on all Q&A)
see, and "now" there are enough people to create their own sub-community. So if @aditsu has enough supporters for a Java only community I think he should go on and get it going. If not, I don't see why he should limit the scope in the beginning if there are enough people interested in a general purpose programming community.
Code golf used to be on SO too,
As I already said, don't ask me, I don't have all the answers. I'd advocate for the communities to decide what's on-topic for them. If there are enough people confident to build a .NET community then they should go ahead. As long as there aren't it'd be good to have a community like SO which functions as a gathering place for those programming things which don't have a sub-community of their own. Similar to how LaTeX questions used to be asked on SO before TeX.SE came into existence.
How about things like C# vs .NET? They are hard to separate, but .NET also belongs with many other languages.
Definitely not. Neither would you want one community per implementation of the same language, even if there are small differences.
Don't ask me, I don't know. I think that many people who can help you with C can also help you with C++ and reducing the scope leads to a fragmented user base effectively reducing the number of people reached who can help you. So I'd say sticking together similar enough conceptual languages isn't a bad thing. Or would you want a Python2 and a Python3 community, because the languages are different enough?
There could also be one community for related languages, as certain principles could apply to the group as a whole. E.g. all APL-like array languages.
Is there anything *wrong* with lots of communities?
Would you say that C and C++ are like JavaScript or a related technology? I'd say no. And C# is different enough from C and C++ to not be in the same community. I have never looked at B, so can't say anything about it. But do we really want a sub-community for each and every programming language?
C, C#, C++? B too?
For things like that I'd suggest a single JavaScript community which also deals with the related technologies, similar to how the TeX community deals with TeX, LaTeX, ConTeXt, MetaFont, MetaFun, ...
Andriy M
Of course you may not even need to ask it twice if the gist of the problem defines its scope clearly enough
Andriy M
There's a large body of 'cross-discipline' problems where one component is client-side and the other server-side. I'm not sure if it's a good idea to create separate sites for combinations of languages/technologies, but for now, as a workaround, a question could be asked more than once in each relevant community, just with enough focus on the relevant aspect of the problem to make it on topic there.
I think one site per lang is much better. However, how do you deal with things variations on a language. E.g. JavaScript, JavaScript+jQuery, TypeScript, etc.
Skillmon
@aditsu do you think we should create sub-sites for every popular programming language or have a single community which deals with every popular languages?
Thanks, so far it looks a lot like PPCG (now called CGCC I guess)
you are in — we have some plans for customizing things so we can cater for Code Golf better — some of them are being hashed out on meta questions in there
oh yes I looked you up on SE
For code golfing, there is already https://code-golf.io/ which I like, but I guess I can check out your beta too. I think the generic question/answer format is not really suitable for that kind of challenges.
But you somehow knew I was into java and code golfing.. or did you get that from my SE profile? :)
do you mind updating your answer here to be a request for a Java community in particular? Then we can start to ask around to see if we can get other folk interested in kicking it off.
No I wasn't on PPCG, but we have [a private beta here](/meta?q=624) now and are adding people on request…
Oh sure, I'm all for Java :) Were you on PPCG too? Sorry your name doesn't ring a bell, maybe it's due to my sleep deprivation :p
Jack Douglas
@aditsu We are leaning towards having separate communities for e.g. C# and Java. SO is a whole big mess and we'd rather have things a bit more tightly defined here — along with adding a feature so we can get a question feed from multiple communities in a single place. If we can get a couple more Java experts interested we can start with that here if you'd like to get involved. In the meantime would you like access to the Code Golf beta?
*Programming Questions* would be appropriate. I don't like the clever names either. It requires in-knowledge to know that "Ask Different" is about Apple products, *Cross Validated* is about "statistics, machine learning, data analysis, data mining, and data visualization", etc.
Oh, ok. Whooosh :)
It was means as a joke building on your (tongue-in-cheek?) avoiding explicit mention of SO.
Does it need to have a "clever" name? The database [sub]site is simply called "Databases". And I see your suggestions as basically imitations of "Stack Overflow". I'd rather go with something different.

@aditsu Ideas to for the name of the site you're suggesting:
* Segmentation Fault
* Buffer Overrun
* Memory Full
Both of our projects are interesting and have very different setups and constraints.  It'll be interesting to see how they both develop.  I'm glad we're talking with each other and working together where we can.
Thanks @Monica :) Having lots of chefs has it's advantages but can also be harder work — I appreciate the relative freedom we have here, while nonetheless hoping that the dev team will grow over time!
Monica
I appreciate all of your offers of help.  Large projects involve more cat-herding, it seems. :-)
Please don't worry about it!  It seemed like an obvious idea to me -- start with your chat, replace the business logic because of the different tech stack, but that seems easier than starting from scratch.  The people who'd actually be doing it told me no, so oh well.  Maybe others who are technically proficient will say otherwise, and if so they can duke it out.  I figured the SQL at least would be something we could reuse, and maybe when we get to chat that'll still happen.
sorry about the chat code-sharing thing, I wasn't taking into account internal factors like 'Codidact want a clean consistent code base' when I said I thought it would be fairly easy (I meant from a purely technical point of view)
Also (and this was my motivation), if you're trying to get users to come here from there, being able to say "and you can keep all your stuff together here" is valuable.  Users don't want to fork their stuff.
that sounds like a good balance. I've been wondering about having a tool for importing everything for certain users — some people produce almost exclusively high quality content.
Monica
We wanted to bring users over with their stuff, which gets more complicated with selective importing.  Had I the chance to do it over again, I would have done a more-restrictive bulk import coupled with a way for any user to import specific questions.
I regret importing everything.  I wouldn't have wanted to go as restrictive as TopAnswers did (import specific questions only), but we should have written a query to exclude stuff of lower value.  Of course, "lower value" is in the eye of the beholder, but at the very least, old closed questions with no answers aren't going to be valuable.  Old closed questions *with* answers that didn't garner many votes are highly questionable.
Do you mind me asking how it is going on writing.codidact? I'm particularly interested to hear whether you think importing the entire database of Q&A is a definite win or whether you would have preferred a selective import in hindsight.
I was thinking of you when I added the footnote plugin ;) hello, welcome and Happy New Year (Decade)!
David
@Jack Footnotes! :) Best wishes for 2020, while I'm at it.
ArtOfCode did that from the most recent data dump, yes.  It was a one-time operation; we're not doing any syncing.
oh, I see there is info on that in the answers!
I'd be interested to know how you imported all the Q&A, was that from an SE data dump?
thanks for providing the links, it's very interesting to see.
This is older Q&A software that he built as a side project a while back, and has been recently updating for us so that the SE community has someplace to go while we wait for a more durable solution.
It's not a Codidact alpha; Codidact isn't that far along.  It's run by ArtOfCode, who's a lead on Codidact, and he'll keep it up for us until either Codidact is ready and we move there, or we decide to go elsewhere (like here).
Writing is temporarily living at https://writing.codidact.com/ .  See this meta post for an introduction: https://writing.codidact.com/questions/39219`


text:
American Institute of Mathematical Sciences

November  2013, 33(11&12): 5441-5455. doi: 10.3934/dcds.2013.33.5441

Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements

 1 Department of Mathematics, Tulane University, New Orleans, LA 70118, United States 2 Mathematics Department, Tulane University, New Orleans, LA 70118 3 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, United States

Received  August 2011 Revised  March 2012 Published  May 2013

We propose a new method for estimating the eigenvalues of the thermal tensor of an anisotropically heat-conducting material, from transient thermal probe measurements of a heated thin cylinder.
We assume the principal axes of the thermal tensor to have been identified, and that the cylinder is oriented parallel to one of these axes (but we outline what is needed to overcome this limitation). The method involves estimating the first two Dirichlet eigenvalues (exponential decay rates) from transient thermal probe data. These implicitly determine the thermal diffusion coefficients (thermal tensor eigenvalues) in the directions of the other two axes. The process is repeated two more times with cylinders parallel to each of the remaining axes.
The method is tested by simulating a temperature probe time-series (obtained by solving the anisotropic heat equation numerically) and comparing the computed thermal tensor eigenvalues with their true values. The results are generally accurate to less than $1\%$ error.
Citation: Steve Rosencrans, Xuefeng Wang, Shan Zhao. Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5441-5455. doi: 10.3934/dcds.2013.33.5441
References:
 [1] F. M. Arscott, "Periodic Differential Equations," Pergamon Press, 1964. Google Scholar [2] G. Backstrom and J. Chaussy, Determination of thermal conductivity tensor and heat capacity of insulating solids, J. Phys. E: Sci. Instrum., 10 (1977), 767-769. doi: 10.1088/0022-3735/10/8/003.  Google Scholar [3] U. Brydsten and G. Backstrom, Hot strip determination of the thermal conductivity tensor and heat capacity of crystals, Int'l. J. Thermophysics, 4 (1983), 369-387. doi: 10.1007/BF01178787.  Google Scholar [4] H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids," $2^\text{nd}$ edition, Oxford University Press, 1959.  Google Scholar [5] V. Isakov, "Inverse Problems for Partial Differential Equation," Springer, 1998.  Google Scholar [6] J. Li, S. Rosencrans, X. Wang and K. Zhang, Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body, Proceedings of the American Math. Society, 137 (2009), 1711-1721. doi: 10.1090/S0002-9939-08-09766-9.  Google Scholar [7] J. M. Macleod, Instructions for operating the divided bar apparatus for thermal conductivity measurement at the GSC, calgary, Open File 3444, Geological Survey of Canada, (1997). doi: 10.4095/209020.  Google Scholar [8] V. Pereyra and G. Scherer, Exponential data fitting, in "Exponential Data Fitting and Its Applications," 1-26, Bentham, (2010). doi: 10.2174/97816080504821100101.  Google Scholar [9] Per Sundqvist, Exponential curve-fitting without start-guess,, \url{http://www.mathworks.com/matlabcentral/fileexchange/21959}., ().   Google Scholar

show all references

References:
 [1] F. M. Arscott, "Periodic Differential Equations," Pergamon Press, 1964. Google Scholar [2] G. Backstrom and J. Chaussy, Determination of thermal conductivity tensor and heat capacity of insulating solids, J. Phys. E: Sci. Instrum., 10 (1977), 767-769. doi: 10.1088/0022-3735/10/8/003.  Google Scholar [3] U. Brydsten and G. Backstrom, Hot strip determination of the thermal conductivity tensor and heat capacity of crystals, Int'l. J. Thermophysics, 4 (1983), 369-387. doi: 10.1007/BF01178787.  Google Scholar [4] H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids," $2^\text{nd}$ edition, Oxford University Press, 1959.  Google Scholar [5] V. Isakov, "Inverse Problems for Partial Differential Equation," Springer, 1998.  Google Scholar [6] J. Li, S. Rosencrans, X. Wang and K. Zhang, Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body, Proceedings of the American Math. Society, 137 (2009), 1711-1721. doi: 10.1090/S0002-9939-08-09766-9.  Google Scholar [7] J. M. Macleod, Instructions for operating the divided bar apparatus for thermal conductivity measurement at the GSC, calgary, Open File 3444, Geological Survey of Canada, (1997). doi: 10.4095/209020.  Google Scholar [8] V. Pereyra and G. Scherer, Exponential data fitting, in "Exponential Data Fitting and Its Applications," 1-26, Bentham, (2010). doi: 10.2174/97816080504821100101.  Google Scholar [9] Per Sundqvist, Exponential curve-fitting without start-guess,, \url{http://www.mathworks.com/matlabcentral/fileexchange/21959}., ().   Google Scholar
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# 12.2 The hyperbola  (Page 4/13)

 Page 6 / 13

Graph the hyperbola given by the equation $\text{\hspace{0.17em}}\frac{{x}^{2}}{144}-\frac{{y}^{2}}{81}=1.\text{\hspace{0.17em}}$ Identify and label the vertices, co-vertices, foci, and asymptotes.

vertices: $\text{\hspace{0.17em}}\left(±12,0\right);\text{\hspace{0.17em}}$ co-vertices: $\text{\hspace{0.17em}}\left(0,±9\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(±15,0\right);\text{\hspace{0.17em}}$ asymptotes: $\text{\hspace{0.17em}}y=±\frac{3}{4}x;$

## Graphing hyperbolas not centered at the origin

Graphing hyperbolas centered at a point $\text{\hspace{0.17em}}\left(h,k\right)$ other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms $\text{\hspace{0.17em}}\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1\text{\hspace{0.17em}}$ for horizontal hyperbolas, and $\text{\hspace{0.17em}}\frac{{\left(y-k\right)}^{2}}{{a}^{2}}-\frac{{\left(x-h\right)}^{2}}{{b}^{2}}=1\text{\hspace{0.17em}}$ for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.

Given a general form for a hyperbola centered at $\text{\hspace{0.17em}}\left(h,k\right),$ sketch the graph.

1. Convert the general form to that standard form. Determine which of the standard forms applies to the given equation.
2. Use the standard form identified in Step 1 to determine the position of the transverse axis; coordinates for the center, vertices, co-vertices, foci; and equations for the asymptotes.
1. If the equation is in the form $\text{\hspace{0.17em}}\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{\hspace{0.17em}}$ then
• the transverse axis is parallel to the x -axis
• the center is $\text{\hspace{0.17em}}\left(h,k\right)$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(h±a,k\right)$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(h,k±b\right)$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(h±c,k\right)$
• the equations of the asymptotes are $\text{\hspace{0.17em}}y=±\frac{b}{a}\left(x-h\right)+k$
2. If the equation is in the form $\text{\hspace{0.17em}}\frac{{\left(y-k\right)}^{2}}{{a}^{2}}-\frac{{\left(x-h\right)}^{2}}{{b}^{2}}=1,\text{\hspace{0.17em}}$ then
• the transverse axis is parallel to the y -axis
• the center is $\text{\hspace{0.17em}}\left(h,k\right)$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(h,k±a\right)$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(h±b,k\right)$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(h,k±c\right)$
• the equations of the asymptotes are $\text{\hspace{0.17em}}y=±\frac{a}{b}\left(x-h\right)+k$
3. Solve for the coordinates of the foci using the equation $\text{\hspace{0.17em}}c=±\sqrt{{a}^{2}+{b}^{2}}.$
4. Plot the center, vertices, co-vertices, foci, and asymptotes in the coordinate plane and draw a smooth curve to form the hyperbola.

## Graphing a hyperbola centered at ( h , k ) given an equation in general form

Graph the hyperbola    given by the equation $\text{\hspace{0.17em}}9{x}^{2}-4{y}^{2}-36x-40y-388=0.\text{\hspace{0.17em}}$ Identify and label the center, vertices, co-vertices, foci, and asymptotes.

Start by expressing the equation in standard form. Group terms that contain the same variable, and move the constant to the opposite side of the equation.

$\left(9{x}^{2}-36x\right)-\left(4{y}^{2}+40y\right)=388$

Factor the leading coefficient of each expression.

$9\left({x}^{2}-4x\right)-4\left({y}^{2}+10y\right)=388$

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

$9\left({x}^{2}-4x+4\right)-4\left({y}^{2}+10y+25\right)=388+36-100$

Rewrite as perfect squares.

$9{\left(x-2\right)}^{2}-4{\left(y+5\right)}^{2}=324$

Divide both sides by the constant term to place the equation in standard form.

$\frac{{\left(x-2\right)}^{2}}{36}-\frac{{\left(y+5\right)}^{2}}{81}=1$

The standard form that applies to the given equation is $\text{\hspace{0.17em}}\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,$ where $\text{\hspace{0.17em}}{a}^{2}=36\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{b}^{2}=81,$ or $\text{\hspace{0.17em}}a=6\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b=9.\text{\hspace{0.17em}}$ Thus, the transverse axis is parallel to the x -axis. It follows that:

• the center of the ellipse is $\text{\hspace{0.17em}}\left(h,k\right)=\left(2,-5\right)$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(h±a,k\right)=\left(2±6,-5\right),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-4,-5\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,-5\right)$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(h,k±b\right)=\left(2,-5±9\right),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(2,-14\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(2,4\right)$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(h±c,k\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}c=±\sqrt{{a}^{2}+{b}^{2}}.\text{\hspace{0.17em}}$ Solving for $\text{\hspace{0.17em}}c,$ we have

$c=±\sqrt{36+81}=±\sqrt{117}=±3\sqrt{13}$

Therefore, the coordinates of the foci are $\text{\hspace{0.17em}}\left(2-3\sqrt{13},-5\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(2+3\sqrt{13},-5\right).$

The equations of the asymptotes are $\text{\hspace{0.17em}}y=±\frac{b}{a}\left(x-h\right)+k=±\frac{3}{2}\left(x-2\right)-5.$

Next, we plot and label the center, vertices, co-vertices, foci, and asymptotes and draw smooth curves to form the hyperbola, as shown in [link] .

#### Questions & Answers

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function


text:
Stef's Coding Community

Text in Neighboring Div's too close.

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Hello,

I have two adjacent divs that are placed throughout a few pages. The text is too close to the neighboring div. What is the best way to put a small buffer b/w these two?

Thanks

Edited by 803andy

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In most cases, adding a bit of margins or padding is the way to go about this. Sorta depends on how your site is currently coded.... You have a link to one of the pages (and specify which are the two divs that are "too close")?

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Thanks Ben,

I've been trying to use padding and margin, but the side text then flys off the page.

Below is my file (bare minimum):

http://www.sendspace.com/file/4kv863

Please let me know where I'm wrong.

Thnx

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//The right div will contain text and sometimes contain an image.

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Can you clarify -- in the HTML file, which are the two divs that you are talking about, and how exactly are you hoping to change what is there? Where are you wanting to add spacing?

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The solo_text & solo_left divs are too close. I would like some more space b/w the two. Depending on the words or the sentence the text and the two divs nearly touch in some instances.

I've tried padding and margin, but the results have always pushed one div completely out of focus.

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In general, I try to stay away from setting both a height and top/bottom margins/padding on the same element (or similarly, setting a width and a left/right margins or padding). This avoids some potential browser compatibility issues.

Your best bet here is to add a wrapper div within one of the two that will give it the padding you need. For example:

```
content here...

```

And then the css:

``` .left_wrapper { padding-right: 40px; }
```

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Thanks a lot, that idea is very slick..........

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Hmmmm... now the only problem is IE or IE6,

I'm using:

.left_wrapper {

}

I'm using the left_wrapper div right under the main_left div.

It works well in Firefox but it's going off the page in IE6

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I took a look at your code again... Check the "images/right_line.jpg" images which is within the .solo_text div. You have given them a width of 540px, but the .solo_text div is only 532px wide. That may be causing the issue you are experiencing.

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Creative Web Designs

Left left left left left left left left left left Left left left left

left left left left left left Left left left left left left left left left left

Affordable Pricing

ALeft left left left left left left left left left Left left left left

left left left left left left Left left left left left left left left left left

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Could it be the doctype??

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here is one way that I have tested and know that it works...

-- I've added a wrapper div surrounding the two columns, giving that the specific width and margin: 0 auto (margin top and bottom 0, margin left and right auto). I also added "overflow:auto" so it would correctly wrap around the two text columns

-- I also added a background color on both the body and the wrapper div to show where the columns were

Changing the doctype shouldn't affect the positioning of the columns, though I always use the strict doctype rather than transitional.

css:

```body { margin:0; background: #ffff00; }

.wrapper {
width: 780px;
margin:0 auto;
background: #fff;
overflow: auto;
}

.solo_text {
width: 540px;
float: right;
color: #999999;
font-size: 14px;
font-family: Verdana, Arial, Helvetica, sans-serif;

}
.solo_left {
width: 240px;
float: left;
font-family: Verdana, Arial, Helvetica, sans-serif;
font-size: 14px;

}
.left_wrapper {
}```

html:

```br />
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">

Text, text, text, text,

text, text Text, text, text, text, text, text Text, text, text, text, text, text Text, text,

text, text, text, text Text, text, text, text, text, text Text, text, text, text, text, text

Text, text, text, text, text, text Text, text, text, text, text, text:

- Text Text

- Text Text

- Text Text

- Text Text

- Text Text

- Text Text

- Text Text

- Text Text

- Text Text

- Text Text

Text, text, text, text, text, text Text, text, text, text, text, text Text, text, text, text,

text, text Text, text, text, text, text, text Text, text, text, text, text, text Text, text,

text, text, text, text Text, text, text, text, text, text Text, text, text, text, text, text

Creative Web Designs

Left left left left left left left left left left Left left left left

left left left left left left Left left left left left left left left left left

Affordable Pricing

ALeft left left left left left left left left left Left left left left

left left left left left left Left left left left left left left left left left

```

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I hate to sound greedy....

The above code does work; however, when I try and implement it into the entire page, I am still getting an issue in IE6. I believe there is something small I'm missing:

I have attached the "full-webpage" with the wrapper class implemented. Its fine in fireworks but still acts up IE6.

http://www.sendspace.com/file/ac6u38

I appreciate your time in helping me troubleshoot this matter greatly!!

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text:
# Create a Genome Topology for your Agent

In order to create a agent you first need to create its genome. An agent’s genome is very much like a genome in biology where it is an organism’s complete set of genetic instructions. In this case, the genome is for an AI agent. A genome can be created manually, algorithmically, or using the Intelligent Artifacts Lab GUI (Lab). For now let’s a genome that is located in the config folder named, ‘demo-all.’ All you need to do is upload the Genome in the Lab’s Topology Creator. Moreover, a genome can be downloaded and uploaded for later use or sharing after spawning an agent (covered in the next section below). The genome file is a JavaScript Object Notation (JSON) string that can be imported as below:

[10]:

from ia.gaius.genome_info import Genome
import json

genome_topology = '''{"elements":{"nodes":[{"data":{"name":"P1","id":"p7ae0a0612","type":"primitive","sources":["observables"],"manipulatives":[],"always_update_frequencies":false,"auto_act_method":"none","auto_act_threshold":0.8,"auto_learn_method":"none","classifier":"CVC","datastore":"mongodb","dynamic_sequence_length":true,"max_sequence_length":0,"sort":true,"process_predictions":true,"quiescence":3,"search_depth":10,"recall_threshold":0.1,"smoothness":3,"max_predictions":100,"persistence":5},"position":{"x":587,"y":272.5975},"group":"nodes","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":false,"classes":""},{"data":{"name":"P2","id":"p5b55a8399","type":"primitive","sources":["m44ee0fd21"],"manipulatives":["m44ee0fd21"],"always_update_frequencies":false,"auto_act_method":"none","auto_act_threshold":0.8,"auto_learn_method":"none","classifier":"CVC","datastore":"mongodb","dynamic_sequence_length":true,"max_sequence_length":0,"sort":true,"process_predictions":true,"quiescence":3,"search_depth":10,"recall_threshold":0.1,"smoothness":3,"max_predictions":100,"persistence":5},"position":{"x":687,"y":272.5975},"group":"nodes","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":false,"classes":""},{"data":{"name":"P3","id":"p85643c122","type":"primitive","sources":["observables"],"manipulatives":["mb94faab95"],"always_update_frequencies":false,"auto_act_method":"none","auto_act_threshold":0.8,"auto_learn_method":"none","classifier":"CVC","datastore":"mongodb","dynamic_sequence_length":true,"max_sequence_length":0,"sort":true,"process_predictions":true,"quiescence":3,"search_depth":10,"recall_threshold":0.1,"smoothness":3,"max_predictions":100,"persistence":5},"position":{"x":789,"y":271.5975},"group":"nodes","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":false,"classes":""},{"data":{"displayName":"Multiplier","mtype":"normal","name":"emotiveMultiplier","inputs":["strings","emotives","vectors"],"outputs":["strings","emotives","vectors","predictions"],"description":"Multiplies emotive values by a configurable value.","tags":["emotives","multiply","multiplication","multi","value","factor"],"category":"emotives","genes":{"sources":{"alleles":[],"mutability":0,"value":["observables"],"volatility":0},"multiplier":{"alleles":[1,2,3,4,5,6,7,8,9,10],"mutability":0,"value":5,"volatility":0}},"id":"m44ee0fd21","type":"manipulative","primitive":"p5b55a8399"},"position":{"x":687,"y":372.5975},"group":"nodes","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":false,"classes":""},{"data":{"displayName":"abstraction","name":"abstraction","category":"standard","mtype":"output","description":"Output manipulative to forward abstracted predictions to other Primitives. The chosen field of the predictions are forwarded as. The are also forwarded. No are forwarded.","tags":["abstraction","name","present","past","future","missing","matches","extras","output","forward","abstract","emotives","strings","emotive","classification"],"genes":{"sources":{"alleles":[],"mutability":0,"value":["cognition-data"],"volatility":0},"primitives":{"alleles":[],"mutability":0,"value":["pc4dff12d6"],"volatility":0},"field":{"alleles":["name","past","present","future","missing","extras","matches","classification"],"mutability":0,"value":["name"],"volatility":0}},"id":"mb94faab95","type":"manipulative","primitive":"p85643c122"},"position":{"x":789,"y":171.59750000000003},"group":"nodes","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":false,"classes":""},{"data":{"name":"P4","id":"pc4dff12d6","type":"primitive","sources":["observables"],"manipulatives":[],"always_update_frequencies":false,"auto_act_method":"none","auto_act_threshold":0.8,"auto_learn_method":"none","classifier":"CVC","datastore":"mongodb","dynamic_sequence_length":true,"max_sequence_length":0,"sort":true,"process_predictions":true,"quiescence":3,"search_depth":10,"recall_threshold":0.1,"smoothness":3,"max_predictions":100,"persistence":5},"position":{"x":788,"y":81.59750000000003},"group":"nodes","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":false,"classes":""}],"edges":[{"data":{"id":"eb7965422f","source":"m44ee0fd21","target":"p5b55a8399","type":"edge"},"position":{"x":0,"y":0},"group":"edges","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":true,"classes":""},{"data":{"id":"ec26380337","source":"p85643c122","target":"mb94faab95","type":"edge"},"position":{"x":0,"y":0},"group":"edges","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":true,"classes":"abstractionEdge"},{"data":{"id":"e5dd3a5044","source":"mb94faab95","target":"pc4dff12d6","type":"edge"},"position":{"x":0,"y":0},"group":"edges","removed":false,"selected":false,"selectable":false,"locked":false,"grabbable":true,"pannable":true,"classes":"abstractionEdge"}]},"style":[{"selector":"edge","style":{"target-arrow-shape":"triangle","curve-style":"straight","control-point-distances":10,"control-point-weights":0.1,"width":1,"overlay-color":"#fff"}},{"selector":"edge.curved","style":{"curve-style":"bezier"}}],"data":{},"zoomingEnabled":true,"userZoomingEnabled":true,"zoom":1,"minZoom":1e-50,"maxZoom":1,"panningEnabled":true,"userPanningEnabled":true,"pan":{"x":0,"y":0},"boxSelectionEnabled":false,"renderer":{"name":"canvas"},"wheelSensitivity":0.1,"motionBlur":false,"agent":"demo-all","description":"tutorial"}'''

Parameter values in genomes are intended to be modified. Since the genome file is a JSON object, modifying values is easy. You can also use the Genome object class to easily find nodes and values for modification.

The following example will change the 'recall_threshold' and 'max_predictions' of the primitive ‘P1’:

[20]:

for p in genome.primitives.values():
if p['name'] == 'P1':
p['recall_threshold'] = 0.1
p['max_predictions'] = 10

print("%s: \
\n ID               : %s \
\n Recall Threshold : %s \
\n Max Predictions  : %s" %(p['name'], p['id'], p['recall_threshold'], p['max_predictions']))

P4:
ID               : pc4dff12d6
Recall Threshold : 0.1
Max Predictions  : 100

[ ]:


text:
APRG Seminar

Venue: Microsoft Teams (online)

In this talk, we discuss Carleman estimates for Laplacian, which implies strong unique continuation for $-\Delta u+Vu$ with potential $V \in L^{\infty}.$ We briefly discuss unique continuation in certain critical situations such as when the potential is in $L^{n/2}_{loc}$ assuming Fourier restriction theorems. Then $L^{n/2}_{loc}$ case is a well-known result of Kenig-Jerison. Our proof of unique continuation is based on the Carleman estimate where it is a consequence of the spectral gap of Laplace Beltrami on the sphere.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 24 Mar 2023


text:
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# Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.$$y=-2-2 \cos 3 \pi x$$

## $x=1 / 3$

### Discussion

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### Video Transcript

so we have. Why is equal to negative two minus two course site. Three. High X So far, things was theon executes of the antic you just to and the senior is true pie divided by three pipe, which is equal to to over three, and then we need to find the X intercept. So there is wanted X intercept, which is at X is equal to one or 30 and then the graph is increasing over the interval where X is greater than zero, but it is less than 1/3.

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text:
## Loren on the Art of MATLABTurn ideas into MATLAB

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English version of the page.

# Convenient nargout Behavior8

Posted by Loren Shure,

Here's a question that was recently posed at MathWorks.

I'm trying to invoke a function handle. It either returns 0 or 1 output arguments. If it returns an output argument, I want to get it; if not I would like to just invoke the function and be on my way. I have no way of knowing ahead of time whether the function returns something or not.

This situation can arise when you are building an application in which you expect to work with user-supplied functions. You can solve this using try construct, but this situation arises frequently enough that we decided to introduce the nargout bump. So what happens?

### nargout Bump from 0 to 1

Many MATLAB functions return 1 result even when called with an nargout is 0. For these, here's a technique that may be of interest. It allows for a bump in nargout on the left-hand side.

Let's assume the variable c is unitialized and then call a function f.

   [c{1:0}] = f(...);

f is called with nargout of 0, because 1:0 is an empty vector. If f returns no arguments, c is an empty cell. However, if f returns a result, c is a 1x1 cell containing that result.

### Why does nargout Behave this Way?

Since this is not a widely known technique, I'll explain one more use of it. Suppose you are writing a function foo, and after some preparation you want foo to return all of the results, if any, of a call on bar, passing along to bar the number of arguments given to foo. The template looks like this.

   function varargout = foo(...)
... whatever ...
[varargout{1:nargout}] = bar(...);

Anonymous functions use exactly this technique to pass along nargout to their body and to return that body's results as the result of the anonymous function. This requirement on anonymous functions, and the observation that a significant number of other functions work in the same way, is why we introduced the bump.

### Will this Simplify Some Code?

I wonder if this information allows you to simplify some of your code. Please share your results here.

Get the MATLAB code

Published with MATLAB® 7.8

### Note

Oliver Woodford replied on : 1 of 8

Useful. It allowed me to change:

str = sprintf('"%s" %s', xpdf_path, cmd);
if nargout
[varargout{:}] = system(str);
else
system(str);
end

to:

[varargout{1:nargout}] = system(sprintf('"%s" %s', xpdf_path, cmd));

Oliver

Sung Soo replied on : 2 of 8

I use similar technique to access subfunctions because I don’t want to have many small functions in different m-files though ‘class’ can solve this too.

Kieran replied on : 3 of 8

Thanks Loren. This code really helped me in using subsref in my classes. I wanted to only change some subsref functionality and call the builtin version for the rest, but ran into output argument issues. I ended up doing something like:

function varargout = subsref(this, s)
use_built_in = true;
% Add custom subsref here and set use_built_in to false.

% Use default subsref for others.
if use_built_in
[varargout{1:nargout}] = builtin(‘subsref’, this, s);
end
end

Loren replied on : 4 of 8

I’m delighted that several of you find good use in this nargout behavior.

–Loren

OysterEngineer replied on : 5 of 8

I understand that 1:0 is an empty matrix.

But, by what is happening with [c{1:0}] = f(…)

Is it really doing something like: [c{1:0}] = [] ?

Also, what do you mean when you say, “when called with an nargout is zero . . . ?” The FRP is nicely clear in explaining that it can be called without an input argument to return the number of output arguments a user has supplied. Or, it can be called with a function handle or a function name as an input argument to return the number of output arguments the function was written to provide. So, which use are you referring to?

Loren replied on : 6 of 8

OysterEngineer-

When MATLAB goes to calcualate [c{1:0}] = f(…),
it is finding 0 elements on the left side (numel([c{1:0}]) IS 0) which IS nargout(f), and therefore f gets called as if nargout is 0.

–Loren

OysterEngineer replied on : 7 of 8

That helps some. I see how the numel([c{1:0}0] is 0.

But, I haven’t been able to invent a function, f, with no outputs such that nargout(‘f’) or nargout(@f)is also 0. I keep getting -1.

Clemens Buchacher replied on : 8 of 8

Thanks. This trick makes the following function almost trivial.

function v = mdgrid(A, N)
% MDGRID multi-dimensional grid
%
% Returns all possible combinations of N elements of A
%


text:
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Cardinals allow us to distinguish between finite and infinite sets very easy. But to really understand how this is accomplished, we have to make a connection between two facts:

• Classic mathematics allows some atomic objects which are not sets themselves, like numbers, points, etc. In modern mathematics, almost all mathematical objects are sets. We will see later in number systems, how numbers, including natural numbers, can be defined from a purely set-theoretical perspective. For the time being, just please keep in mind that not only the set of natural numbers $\mathbb N$ is a set, but also every natural number $n\in\mathbb N$ is a set.
• For this reason, each natural number $n$ is a representative of its own cardinal equivalence class $|n|.$

Now, we are able to define exactly what it means for a set to be finite or to be infinite.

## Definition: Finite Set, Infinite Set

A set $X$ is called finite, if both, $X$ and a natural number $n\in\mathbb N$ belong to the same cardinal, formally $|X|=|n|$ (we could even write $X\in |n|$ to express this1). If such a natural number $n$ does not exist, then the set $X$ is called infinite.2

1 Please note that the cardinal $|n|$ is an equivalence class, by definition, and must not to be mixed up with the absolute value of $|n|$, with the same notation! This fact justifies the notation $X\in |n|.$

2 Please note that the existence of infinite sets is ensured by the axiom of infinity and therefore, there are also infinite cardinal numbers.

| | | | | created: 2014-10-10 20:41:12 | modified: 2016-12-22 23:04:03 | by: bookofproofs | references: [656], [983]

(none)


text:
Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

Calculate and Interpret a Forward Discount or Premium

Forward Discount

A forward discount is when the domestic current spot exchange rate is traded at a higher level than the current domestic future spot rates. The analysis of the expectations from the market depends mostly on discounts and premiums. Also, they enable one to know the currencies that should appreciate and those that will depreciate in the near future.

A forward premium is a situation when the forward exchange rate is higher than the spot exchange rate. Conversely, a forward discount is when the forward exchange rate is lower than the spot exchange rate.

Irrespective of the quoting convention, the currency with the higher (lower) interest rate will always trade at a discount (premium) in the forward market.

Calculation

Interest parity states that both the spot and forward exchange rates between two currencies must be at equilibrium with the two nation’s interest rates. The formula includes four variables:

$$F_{f/d} = S_{f/d} (\frac{1 + i_f}{1 + i_d})$$

Where

$$S_{f/d}$$ = current spot exchange rate

$$F_{f/d}$$ = current forward exchange rate

$$i_d$$ = domestic interest rate

$$i_f$$ = foreign interest rate

We can rewrite the above equation to show the forward rate as a percentage of the spot rate:

$$\frac{F_{f/d}}{S_{f/d}} = \frac{1 + i_f}{1 + i_d}$$

It is easy to see that, given the foreign/domestic convention (f/d), the forward rate will be at a premium to the spot rate if the foreign interest rates are higher than the domestic interest rate.

Let’s say the current Kenyan Shilling to Ugandan Shilling exchange rate is Sh 100. The domestic interest rate in Kenya is 5%, and the foreign interest rate is 4.75%, causing the resulting equation to be:

$$F = \text{Ksh} 100 (\frac{1.0475}{1.05}) = \text 99.7619$$

The forward rate relates to the spot rate by a premium or discount, which is proved in the following relationship:

$$F = S (1 + x)$$

Where F is the current premium or discount.

Spot exchange rates differ from the forward currency exchange rates. When the forward currency exchange rate happens to be higher than the spot rate, then the currency is said to be at a premium. Conversely, discounts occur when the spot rates are higher than the forward exchange rates. Hence, a negative premium is equal to a discount.

A simpler way to look at it is when $$F/S – 1 > 0$$, the denominator is at a discount. When $$F/S – 1 < 0$$, the denominator is at a premium, and the numerator is at a discount. For instance, if we want to ascertain the premium of a forward trading rate, we can also do that using the formula below:

$$F_{f/d}=S_{f/d} (\frac{1 + i_f τ}{1+i_d τ})$$

Where (1 + i_f τ) is the interest earned in the foreign market and (1 + i_d τ) is the interest earned in the domestic market. Where τ is the fraction of the investment horizon at which the interest rates are quoted.

If we want to know the 31-day forward exchange rate from a 31-day domestic risk-free interest rate of 2.5% per year, given that the foreign 31-day risk-free interest rate is 3.5% with a spot exchange rate, $$S_{f/d}$$, of 1.5630, then we simply have to substitute these values into the forward rate equation:

$$F_{f/d}=S_{f/d} (\frac{1 + i_f τ}{1+i_d τ})$$

$$F_{f/d}=1.5630 (\frac{1+0.035×\frac{31}{360}}{1+0.025×\frac{31}{360}})=1.5643$$

$$F_{f/d} – S_{f/d} = 1.5643 – 1.5630 = 0.0013$$

Since forward premiums or discounts are usually quoted in pips or points (1/100 of 1%), multiplying the result by 10,000 will give us $$0.0013 \times 10,000 = 13$$ pips, which is the forward trading premium quoted in pips or points.

Question

When is a foreign currency most likely trading at a forward premium?

A. When the forward rate expressed in the domestic currency is below the spot rate.

B. When the forward rate expressed in the domestic currency is above the spot rate.

C. When the forward rate expressed in the foreign/domestic currency is at equilibrium.

Solution

A foreign currency is at a forward premium if the forward rate expressed in domestic currency is above the spot rate.

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text:
• # question_answer Study the following passage carefully and answer the questions given below: Corporate managerial efficacy is a key driver of success within firms across all industries and markets. Parsing the skill sets that separate high performing managerial teams from underperformers yields insight into underserved and underestimated value providers within this arena and reveals that significant financial impact stems directly from training managers on strategies for talent retention and layoff protocol. Companies in general undervalue the financial effect of reduced turnover through an insufficient training programs to handle situations where underestimated employees are laid off, which leads to higher than necessary turnover in stable growth periods and further exacerbates the negative financial impact to a firm. What is the strategy necessarily to be inculcated by the companies to avoid exacerbation in a firms financial condition? A) Parsing the necessary sets of skills that shows off the mediocre and underestimated work force B) Cultivating devised training programs that equips the stable employees to stabilize the turnover C) Promoting and training the managers to have efficacy for the success within the industries D) Maintaining a standard lay off protocols that will equip the company to detect and eliminate the underserved

Rationale: (b) Reasoning type: Deductive reasoning$\to$arriving reasoning from top down approach observed to solve or prove a theory Focus of the question: Conclusion and inference$\to$Observation and application Relationship analogy: Equanimity$\to$finding the similarity. The passage concludes with the message that the negative financial impact to a firm occurs when the underestimated employees are laid off due to unrevised training programs that undervalues the financial results of reduced turnover directly with underestimated employees. Therefore, the necessary strategy should be focused on the conclusion of the passage that requires a DEVISED TRAINING PROGRAM, which will avoid lay off and stabilize the turnover to reduce the exacerbate negative financial impact.


text:
# corbivgeo: Correlation Coefficient for the Basu-Dhar Bivariate Geometric... In BivGeo: Basu-Dhar Bivariate Geometric Distribution

## Description

This function computes the correlation coefficient analogous of the Pearson correlation coefficient for the Basu-Dhar bivariate geometric distribution assuming arbitrary parameter values.

## Usage

 1 corbivgeo(theta)

## Arguments

 theta vector (of length 3) containing values of the parameters θ_1, θ_2 and θ_{3} of the Basu-Dhar bivariate Geometric distribution. For real data applications, use the maximum likelihood estimates or Bayesian estimates to get the correlation coefficient.

## Details

The correlation coefficient between X and Y, assuming the Basu-Dhar bivariate geometric distribution, is given by,

ρ = \frac{(1 - θ_{3})(θ_1 θ_2)^{1/2}}{1 - θ_1 θ_2 θ_{3}}

Note that the correlation coefficient is always positive which implies that the Basu-Dhar bivariate geometric distribution is useful for bivariate lifetimes with positive correlation.

## Value

corbivgeo computes the correlation coefficient analogous to the Pearson correlation coefficient for the Basu-Dhar bivariate geometric distribution for arbitrary parameter values.

Invalid arguments will return an error message.

## Author(s)

Ricardo P. Oliveira rpuziol.oliveira@gmail.com

Jorge Alberto Achcar achcar@fmrp.usp.br

## Source

corbivgeo is calculated directly from the definition.

## References

Basu, A. P., & Dhar, S. K. (1995). Bivariate geometric distribution. Journal of Applied Statistical Science, 2, 1, 33-44.

Li, J., & Dhar, S. K. (2013). Modeling with bivariate geometric distributions. Communications in Statistics-Theory and Methods, 42, 2, 252-266.

Achcar, J. A., Davarzani, N., & Souza, R. M. (2016). Basu<e2><80><93>Dhar bivariate geometric distribution in the presence of covariates and censored data: a Bayesian approach. Journal of Applied Statistics, 43, 9, 1636-1648.

de Oliveira, R. P., & Achcar, J. A. (2018). Basu-Dhar's bivariate geometric distribution in presence of censored data and covariates: some computational aspects. Electronic Journal of Applied Statistical Analysis, 11, 1, 108-136.

de Oliveira, R. P., Achcar, J. A., Peralta, D., & Mazucheli, J. (2018). Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach. Journal of Applied Statistics, 1-19.

## Examples

 1 2 3 4 5 6 7 8 corbivgeo(theta = c(0.5, 0.5, 0.7)) # [1] 0.1818182 corbivgeo(theta = c(0.2, 0.5, 0.7)) # [1] 0.102009 corbivgeo(theta = c(0.8, 0.9, 0.1)) # [1] 0.822926 corbivgeo(theta = c(0.9, 0.9, 0.9)) # [1] 0.3321033

BivGeo documentation built on May 2, 2019, 6:12 a.m.


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# Understanding Recursion

## Introduction

In computer science, when a function (or method or subroutine) calls itself, we call it recursion. Most of the programming languages out there support recursion and its one of the fundamental concepts you need to master while learning data structures and algorithms. Recursion is the key to divide and conquer paradigm where we divide the bigger problem into smaller pieces, solve the smaller pieces individually and combine the results. Recursions are heavily used in Graphs and Trees and almost all the data structures that have a parent-child relationship.

## Why is recursion so useful?

Many problems we encounter in computer science are of recursive nature. Consider an example of finding a factorial of a number $n$. The formula for calculating the factorial of $n$ is given below.

$$n! = n \times (n - 1) \times (n - 2) \times … \times 1$$

If you observe the above equation carefully, you will notice that everything after first $n$ is the factorial of $n - 1$. So the equation can be written as
$$n! = n \times (n - 1)!$$
The equation above says that to find the value of $n!$, we need to first find the value of $(n-1)!$. If we find the value of $(n-1)!$, we can simply multiply it by $n$ and we are done!. Now, how do we calculate the value of $(n-1)?$ We repeat the same process i.e.
$$(n - 1)! = (n - 1) \times (n - 2)!$$
We go on like this. Now the important question is: how far do we go? or when do we stop? We stop the process when we can trivially find the factorial. We know the factorial of 0 is 1. That means we keep the process going until we reach 0. This is called the base case (or stopping criteria) and is mandatory for all the recursive programs. If we do not have a base case, the process goes on forever.

So, to find the value of $n$ factorial, we find the value of $0!$ trivially. Then we find the value of $1!$ by multiplying the result of $0!$ by 1. We find the value of $2!$ by multiplying the value of $1!$ by 2 and so on. Basically, we divide the bigger problem (i.e. finding the factorial of $n$) into smaller problems whose factorial can be found trivially (i.e. $0! = 1$) and solve the bigger problem by gluing together the result of smaller problems.

Using the very same concept discussed above, we can solve so many problems in computer science. Some of them are: quick sort, merge sort, FFT, Closest Pair of Points, Convex Hull and much more.

## How does it work?

We learned what is recursion and why it is useful. Now we learn how does it work? i.e. how is it implemented in programming languages? Knowing the working of it helps you to better understand the recursive programs, helps you to write the recursive programs and most importantly help you to think a recursive solution of a problem. Before understanding the working of recursion, we need to first understand how function calls work.

### Activation Record (Frame) and Call Stack

When a compiler detects a function call, it creates a data structure called Activation Record (also called Frame) and pushes this record into a call stack. The simplified structure of the Activation Record is shown in the figure below.

It contains three major parts - (i) Local Variables: All the local variables declared inside the body of the function, (ii) Pointer to the caller: The location in the memory from where it was called, and (iii) Function parameters: The formal parameters of the function. When a function calls another function the activation record of the called function lies on the top of the activation record of the caller. When a function returns, the activation record is popped from the stack, local variables and parameters are destroyed and the control goes to the location of the caller.

Consider a simple C program with two functions - main and sum.

When the above program compiles, the compiler detects a main function and creates an activation record for it. It adds a, b and result to the local variable section. It then detects that there is a function call to function sum. Now it creates the activation record for sum and places the activation record on the top of the stack. The call stack looks as follows

The sum function calculates the result and returns. As soon as the sum function returns, it is popped from the stack and all the variables are removed. The main function begins its execution and assigns the result to the result variable. Finally, the main function also returns and the stack becomes empty.

### Back to the recursion

We now know what does compiler handles the function call from one function to another. In the case of recursion, the exact same thing happens. The only difference, in this case, is that both the functions (caller and callee) are same. Let us try to understand how the factorial program described above works. The C code for the factorial function is given below (I have changed the base condition to 2 to reduce the area of call stack).

When you run the program, the main function calls the fact functions with n = 5. The fact function calls itself each time reducing the value of $n$ by 1 until the base condition is satisfied. The call stack is shown in the figure below.

When the control is inside the base condition, it returns value 2 and pops the record from the stack. The result from the stack popped is multiplied with 3 and the result is returned and so on. Finally, all the records for fact functions are popped off from the stack and we get the final result in the main function.

## Drawbacks of recursion

Although recursions are intuitive and feel more natural to many problems, there are many drawbacks. Some includes

• It is much slower than the iterative approach. For each recursive call, the compiler creates activation record, pushes it to the stack and pops it when the function returns. This adds extra overhead to the program.
• It uses more stack space than the iterative approach.
• It is hard to understand especially when the number of recursive calls (sequential) is more than one.
• For some programs, it takes really long to run. For example, the recursive solution of Fibonacci number is way slower than its iterative counterpart.

So, try to avoid recursion as much as possible when you are doing simple programs.

## When should we use recursion?

• When you are dealing with graphs or solving a problem like sorting, the recursive solution captures the idea more intuitively.
• The recursive solutions, most of the time, require less code and the code looks more elegant and clean.
• It helps to calculate the complexity of many problems (at least theoretically) using recurrence solution.
• We can use mathematical induction to prove the correctness of a recursive program.

Please note that every recursive program can be replaced by iterative solution and vice versa.