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# Who here has gotten a C+ or lower in undergrad?
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Senior Manager
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### Show Tags
13 Sep 2011, 08:17
Sometimes I feel like I'm the only person in the world who got a C+ in a class. It was the very last grade I received in my very last semester in college: a C+ in Korean history. I was working 25 hours a week during that semester, but quite frankly that class just wasn't a priority for me.
For those of you who got a C+ (or lower) in undergrad, what is your overall GPA? Are you going to explain it in an addendum in your applications? I ended up with a 3.5 in college, which makes that C+ look all the weirder. I hesitate to draw too much attention to it, since it's just a C+ (I landed a 3.1 that semester anyway).
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
### Show Tags
13 Sep 2011, 08:31
One C+ (in a non quant subject) and you still came out with a 3.5 GPA. I'm trying really hard to see the issue here.
You are not the only person who has ever gotten a C+. I assure you that there are people on this forum who were estatic to get a C+ in certain classes. With all of the profiles around here that are between a 2.5-3.0, obviously you're not the only one who got a C+ in college. Plus it was ONE grade. I don't mean to be snarky but this thread just comes off really bad. It's like a person with $1 million crying, "woe is me," because he doesn't have$2 million. There are people on this forum who have MUCH bigger GPA worries to sort through than a single C+.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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13 Sep 2011, 09:04
How's this? I got a D my first semester and a C+ my second... although I had pretty much 4.0 after that and graduated with a 3.9+
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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13 Sep 2011, 09:08
I got 2 D+ in marketing and one more class
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
### Show Tags
13 Sep 2011, 10:58
I don't think a C+ in Korean history will hurt you. BSchool isn't like law school where you might need a pristine academic track record to get in to a top school. Plus, there are many elements of the bschool application that can be used to offset a low GPA or bad semester.
I had an atrocious freshman year (even a few Fs), but ended up getting mostly As from that point on. However, I did have extenuating circumstances that first year which I'm explaining in the optional essays.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
### Show Tags
13 Sep 2011, 14:21
MDF... same .. use that optional essay... nothing is unexplainable... we're people.. remember?
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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13 Sep 2011, 17:38
Got a C+ in Calculus II in my first semester at college. That semester was a wake-up call in a lot of ways and I graduated with a 3.59. I didn't use the optional essay to talk about it in my applications, since I thought my work experience, grades in other quant classes and GMAT more than addressed any potential concerns about my quant ability.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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13 Sep 2011, 21:14
The only relevant course in which I got C or below was in a math course. I failed it (attributed to not attending class etc.), but I turned it around when I retook it and got a B+. My first two years were basically a B-average without studying much, but I pulled my GPA up to 3.29 cumulative in the final two years with most semesters getting 3.8+ GPAs. I think that turnaround is a good story for the essays.
Overall, unless the candidate has consistently received C+ and below in quantitative courses and has a less than 650 GMAT, there shouldn't be any problems.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
### Show Tags
13 Sep 2011, 22:36
I got two Cs. I'm still bitter about both of them. But, I didn't feel the need to explain either one. For reference, one was a graduate level math class, the other was a class in my major (chemical engineering)...god i hated that class.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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14 Sep 2011, 05:36
I got a D in elementary spanish 101 my second semester senior year.
I will be addressing it in the optional essays, because there IS a very legitimate reason, but if I don't provide an explanation I'm afraid the AdCom will attribute it to me just not giving a shit since I was graduating.
(I ended up having to schedule two classes for the exact same time, exact same days, in order to fill my last two requirements. Spanish is obviously very attendance and participation dependent, so I was hit pretty hard on the grade despite scoring very highly on all the tests and homeworks)
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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14 Sep 2011, 20:10
I got a C+ in Business Law and ended up a 3.67 GPA. It's not a big deal. Korean history sounds like a cool class. I think I would have enjoyed taking that.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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18 Sep 2011, 11:07
I got a C in Spanish in my very first college semester. I am not planning to address it in my optional essay since I finished with a 3.6 GPA, and it was a course outside my major.
hello212, I think we're in about the same situation, so I wouldn't worry too much about it.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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18 Sep 2011, 12:01
I FAILED a class in my 1st year. I think I also got a "D" in a class, along with a number of Cs.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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18 Sep 2011, 17:23
C- in a GE class and C+ in a Intro Stats class during my first year. Ended up with over a 3.5 at graduation.
I'm not even going to explain them, because the upward trend in grades, I feel, make up for that horrible first year of mine.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
### Show Tags
18 Sep 2011, 20:23
I got a C in an advanced accounting class. It really is an outlier, there's no other Cs (or lower) on my transcript. The professor's teaching style just did not work for me, for some reason. It wasn't his fault, as I had classmates that really liked the way he taught. It wasn't really my fault either, I just much preferred the professor I'd had for the intro accounting class and believe I would have done better if she'd taught the advanced one as well. It happens.
I ended up with a 3.72 GPA overall.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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21 Sep 2011, 17:06
I worry about this as well. Curious if i shoukd retake course at extension school to prove i can get a better grade, then explain the circumstances in optional essay... Is the time and effort worth it?
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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30 Sep 2011, 16:47
The C+ is irrelevant to your application, first because it was in Korean History as opposed to a quant/business class, second because your overall GPA is over 3.5 and third because your GMAT is over 750. I had a C-, C+ and C all in the same semester during undergrad and was able to explain it in the optional essay (basically I slept through undergrad). Focus on writing killer essays and you will be fine. I know it's hard not to obsess over every potential red flag, but if this is the worst thing on your record, you have nothing to worry about.
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Re: Who here has gotten a C+ or lower in undergrad? [#permalink]
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01 Oct 2011, 02:09
so i have a question about lowish grades in english/writing GE classes. I have a 3.5 overall, 4.0 major, but i got fairly low grades in writing, C/C-'s and one D (retook to get a B-). I am on the fence but am wavering to not discuss this in the optional essay since my overall grades were on an upward trend over the 4 years, as well as the fact that I achieved A's in my honors essay class. also, to compound things, my gmat verbal score is <80%. I'm not sure if bringing attention to these items will just be worse for my apps.
Re: Who here has gotten a C+ or lower in undergrad? [#permalink] 01 Oct 2011, 02:09
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# Who here has gotten a C+ or lower in undergrad?
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# Conformed Dimensions: the Key to Cross Process Analysis
Over the past few months, I’ve been spending a lot of time reading about, thinking about, and working on conformed dimensions for a client. The reason is simple, conformed dimensions enable businesses to track richer (and typically more complicated) metrics spanning multiple business processes.
Note: there are many additional benefits of conformed dimensions…for a more thorough explanation, start here.
There are many examples of where one might want to link data across multiple business processes, the textbook example being P&L reporting which can require linking facts across nearly all business processes to provide a bottom-up reporting solution that allows end-users to report top level numbers and drilldown into the base transactions. Another example, and the one we’re going to cover in this blog post, is tracking a metric called Gross Margin Return on Inventory Invested.
Gross Margin Return on Inventory Investment (GMROII) is a ratio in microeconomics that describes a seller’s income on every unit of currency spent on inventory. It is one way to determine how valuable the seller’s inventory is, and describes the relationship between total sales, total profit from total sales, and the amount of resources invested in the inventory sold. A seller will aim for a high GMROII.
### Fact and Dimension tables are the building blocks of a Star Schema
Below is an example of a star schema based on a fact table surrounded by several dimension tables.
As you can see, the fact table represents the center of the star while the dimension tables represent the individual points. There’s theoretically no limit on the number of points (or dimensions) a star schema can have, but a rule of thumb is to limit it to no more than 20-26 dimensions to avoid cluttering up the UX. If you find yourself approaching that number of dimensions, you might want to re-evaluate the dimensions and see if it’s not possible to consolidate.
### Star Schemas are the building blocks of a Dimensional Model
A dimensional model typically consists of multiple business processes. And each business process is usually represented by multiple fact tables. And each fact table forms the center of a star schema (as we learned in the last section). Thus, the logical conclusion is that a dimensional model typically consists of multiple star schemas.
Below is an example of a very simple dimensional model consisting of 2 fact table star schemas:
The observant reader will notice these 2 fact tables (Sales/Inventory) share several dimensions in common:
Now, in order to link facts from the sales fact table (sales process) and inventory fact table, we must use the same dimensions:
And that is the idea of conformed dimensions in a nutshell 🙂
### Cross Process Analysis
Now that we understand how to build a dimensional model based on star schemas built upon fact tables and conformed dimensions, let’s walk through an example of how to create a cross-process metric. The exact measure we are going to build is the Average Monthly GMROII based on a trailing 6 month window and is calculated by taking the sum of gross profit for the total period (last 6 months) divided by the sum of each month-ending inventory cost value.
Note: GMROII is an interesting metric with a lot of criticism due mainly to it’s oversimplification of inventory performance and lack of control over sales/inventory expenses. This paper does a nice job of explaining the problems associated with GMROII and factors to consider in order to use it effectively. It also goes on to suggest and define an alternative measure called DPP or Direct Product Profit which appears to overcome many of the issues w/ GMROII at the cost of a bit more complexity.
For this task, we’ll be using the Adventure Works 2012 DW sample database to create the GMROII measure which spans the sales and inventory processes.
Here’s a look at the fact tables (back in the relational DW) that we’ll link together to create this calculated measure in the SSAS cube:
The FactProductInventory table is a periodic snapshot fact table containing a snapshot of each product in inventory at the end of each day. I don’t believe the earlier versions of the AdventureWorks DW sample database contain the FactProductInventory, so be sure you’re using the 2012 version. Also, because this fact table contains snapshots for products that haven’t been assigned to a product subcategory and product category, I’ve created the following view which filters out these items. Another, and possibly better (but more time consuming) solution would be to create unknown (or unassigned) records in the product subcategory/category tables and update these products to point to those unknown records. Either way, you must address this issue before proceeding else you’ll run into the dreaded missing attribute key during fact processing.
```CREATE VIEW dbo.vFactProductInventory AS
SELECT f.ProductKey
,f.DateKey
,f.MovementDate
,f.UnitCost
,f.UnitsIn
,f.UnitsOut
,f.UnitsBalance
,TotalCostOfBalance = (f.UnitCost * f.UnitsBalance)
ON p.ProductKey = f.ProductKey
ON psc.ProductSubcategoryKey = p.ProductSubcategoryKey
ON pc.ProductCategoryKey = psc.ProductCategoryKey
```
Next we need to add this table/view to the DSV in our SSAS project:
Then we can build a new measure group in the AdventureWorks cube:
[Last Inventory Cost Value] uses the LastNonEmpty aggregation function and is based on a calculated column in the DSV that multiplies Units Balance and Unit Cost at the leaf level. A better idea is to push this calculated column down into the view…I’m not a big fan of creating calculated columns in the DSV but will often do so in labs/demos for the sake of time when it is an after thought.
And here’s a look at the dimension usage tab which we need to take into consideration anytime want to create calculated measures spanning multiple measure groups.
In the screenshot above, I’ve highlighted the (conformed) dimensions in common between the 3 tables we are interested in linking together. This defines the dimensionality which we need to take into consideration when building and using the calculated measure.
Total Gross Profit
```Create Member CurrentCube.[Measures].[Total Gross Profit]
As
[Measures].[Internet Gross Profit] +
[Measures].[Reseller Gross Profit]
,Format_String = &amp;quot;Currency&amp;quot;
,Associated_Measure_Group = 'Product Inventory'
;
```
Total Gross Profit – Trailing 6 Months
```Create Member CurrentCube.[Measures].[Total Gross Profit - Trailing 6 Months]
As
SUM (
{
PARALLELPERIOD(
[Date].[Calendar].[Month]
,5
,[Date].[Calendar].CurrentMember
)
:
[Date].[Calendar].CurrentMember
}
,[Measures].[Total Gross Profit]
)
,Format_String = &amp;quot;Currency&amp;quot;
,Associated_Measure_Group = 'Product Inventory'
;
```
Total Inventory Cost Value – Trailing 6 Months
```Create Member CurrentCube.[Measures].[Inventory Cost Value - Trailing 6 Months]
As
SUM (
{
ANCESTOR(
PARALLELPERIOD(
[Date].[Calendar].[Month]
,5
,TAIL(
DESCENDANTS(
[Date].[Calendar].CurrentMember
,[Date].[Calendar].[Date]
)
,1
).Item(0)
)
,[Date].[Calendar].[Month]
)
:
ANCESTOR(
TAIL(
DESCENDANTS(
[Date].[Calendar].CurrentMember
,[Date].[Calendar].[Date]
)
,1
).Item(0)
,[Date].[Calendar].[Month]
)
}
,[Measures].[Last Inventory Cost Value]
)
,Format_String = &amp;quot;Currency&amp;quot;
,Associated_Measure_Group = 'Product Inventory'
;
```
GMROII
```Create Member CurrentCube.[Measures].[GMROII]
As
[Measures].[Total Gross Profit - Trailing 6 Months]
/
[Measures].[Inventory Cost Value - Trailing 6 Months]
,Format_String = &amp;quot;Percent&amp;quot;
,Associated_Measure_Group = 'Product Inventory'
;
```
Now, after deploying and publishing the project, our users now have the ability to track the Average Monthly GMROII for the trailing 6 months by product over time:
Furthermore, we can easily extend the capabilities to include Prior-Year and YoY versions of the GMROII:
```Create Member CurrentCube.[Measures].[GMROII - Prior Year]
As
(
ParallelPeriod(
[Date].[Calendar].[Calendar Year]
,1
,[Date].[Calendar].CurrentMember
)
,[Measures].[GMROII]
)
,Format_String = &amp;quot;Percent&amp;quot;
,Associated_Measure_Group = 'Product Inventory'
;
```
```Create Member CurrentCube.[Measures].[GMROII - YoY]
As
[Measures].[GMROII] - [Measures].[GMROII - Prior Year]
,Format_String = &amp;quot;Percent&amp;quot;
,Associated_Measure_Group = 'Product Inventory'
;
```
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# Midpoint Calculator
The calculator will find the midpoint of two points, with steps shown.
• In general, you can skip the multiplication sign, so 5x is equivalent to 5*x.
• In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x).
• Also, be careful when you write fractions: 1/x^2 ln(x) is 1/x^2 ln(x), and 1/(x^2 ln(x)) is 1/(x^2 ln(x)).
• If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. write sin x (or even better sin(x)) instead of sinx.
• Sometimes I see expressions like tan^2xsec^3x: this will be parsed as tan^(2*3)(x sec(x)). To get tan^2(x)sec^3(x), use parentheses: tan^2(x)sec^3(x).
• Similarly, tanxsec^3x will be parsed as tan(xsec^3(x)). To get tan(x)sec^3(x), use parentheses: tan(x)sec^3(x).
• From the table below, you can notice that sech is not supported, but you can still enter it using the identity sech(x)=1/cosh(x).
• If you get an error, double-check your expression, add parentheses and multiplication signs where needed, and consult the table below.
• All suggestions and improvements are welcome. Please leave them in comments.
The following table contains the supported operations and functions:
Type Get Constants e e pi pi i i (imaginary unit) Operations a+b a+b a-b a-b a*b a*b a^b, a**b a^b sqrt(x), x^(1/2) sqrt(x) cbrt(x), x^(1/3) root(3)(x) root(x,n), x^(1/n) root(n)(x) x^(a/b) x^(a/b) abs(x) |x| Functions e^x e^x ln(x), log(x) ln(x) ln(x)/ln(a) log_a(x) Trigonometric Functions sin(x) sin(x) cos(x) cos(x) tan(x) tan(x), tg(x) cot(x) cot(x), ctg(x) sec(x) sec(x) csc(x) csc(x), cosec(x) Inverse Trigonometric Functions asin(x), arcsin(x), sin^-1(x) asin(x) acos(x), arccos(x), cos^-1(x) acos(x) atan(x), arctan(x), tan^-1(x) atan(x) acot(x), arccot(x), cot^-1(x) acot(x) asec(x), arcsec(x), sec^-1(x) asec(x) acsc(x), arccsc(x), csc^-1(x) acsc(x) Hyperbolic Functions sinh(x) sinh(x) cosh(x) cosh(x) tanh(x) tanh(x) coth(x) coth(x) 1/cosh(x) sech(x) 1/sinh(x) csch(x) Inverse Hyperbolic Functions asinh(x), arcsinh(x), sinh^-1(x) asinh(x) acosh(x), arccosh(x), cosh^-1(x) acosh(x) atanh(x), arctanh(x), tanh^-1(x) atanh(x) acoth(x), arccoth(x), cot^-1(x) acoth(x) acosh(1/x) asech(x) asinh(1/x) acsch(x)
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A087044 Decimal expansion of a number x such that adding exp(1) to each of the partial quotients of the continued fraction of x evaluates to x+2. 1
9, 8, 6, 2, 4, 9, 0, 1, 5, 0, 4, 4, 8, 5, 9, 3, 3, 6, 3, 7, 0, 0, 3, 4, 7, 4, 3, 0, 0, 5, 8, 8, 2, 7, 8, 2, 3, 1, 1, 3, 2, 5, 3, 4, 2, 6, 0, 8, 6, 4, 6, 3, 3, 0, 1, 5, 5, 9, 5, 1, 4, 9, 4, 2, 1, 5, 8, 8, 4, 1, 2, 6, 7, 7, 7, 6, 1, 5, 0, 8, 1, 0, 5, 1, 5, 3, 9, 1, 6, 4, 8, 3, 8, 1, 1, 0, 0, 7, 5, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET 0,1 COMMENTS The continued fraction of x is [2; 1, 71, 1, 2, 1, 1, 2, 19, 21, 2, 1, 1, ...] and [0+e; 1+e, 71+e, 1+e, 2+e, 1+e, 1+e, 2+e, 19+e, 21+e, 2+e, 1+e, 1+e, ...] = [2; 1, 71, 1, 2, 1, 1, 2, 19, 21, 2, 1, 1, 1, 3, 9, ...] = x+2. LINKS EXAMPLE x=0.986249015044859336370034743005882782311325342608646330155951494215884126777 CROSSREFS Cf. A087043. Sequence in context: A002391 A193626 A316600 * A246168 A248585 A105415 Adjacent sequences: A087041 A087042 A087043 * A087045 A087046 A087047 KEYWORD nonn,cons AUTHOR Paul D. Hanna, Aug 02 2003 STATUS approved
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# Antennas
I’ve been scratching my head for years how to model an appropriate ground with my ground mounted Vertical Antenna radials.
I’d like to thank Sigi, DG9BFC for clearing this up for me.
HOW TO:
NOTE: I like to draw a small feedpoint in all my HF antenna models, like a 5cm length of wire where my “source” (coax) is connected to. Particularly useful for fan dipoles and for making other adjustments. I’ve left that part out for simplicity on this How To since I’m trying to make this super easy.
• Go to Edit > Wire Edit (CTRL-W) and click on the XZ (or YZ) button, bottom right of the Wire Edit screen.
• Click on the New Wire button and draw a line from Z=0 (the ground) straight up for 10m length.
• Click OK.
• Go to the Geometry tab and in Sources (bottom left), type W1B (Wire 1, Base).
• Click the View tab and ensure that the source is at the bottom of the antenna.
• Now head over to Calculate and type in the Freq box “7.2”.
• In the Ground section, click on “Real” and also click the “Ground setup” button.
• You may the leave the Dielec and Ground conductivity as the default. If you want some accuracy, check my link at bottom of this article. I am often near the sea at my holiday QTH so I adjust these up a little bit to 20 and 10 respectively. Pure salt water is much higher than that.
• Click OK and you are now “Good to go”. Hit the Start button and you should see an impedance of around 35 ohms and SWR around 1.5:1. The antenna is actually fractionally short, so you may lengthen to suit.
That’s it! Years of work trying to figure this out and Sigi showed me today.
Of course, do NOT raise the antenna off the ground. The antenna needs to touch. So if you find something it wrong, you may find the antenna wire that you are feeding isn’t touching ground with Z=0.
Note: if your feedpoint ends up at the top of the wire and not the bottom, change W1B to W1E (Wire 1 Base -to- Wire 1 End). That’s it.
Ground Conductivity. A very heavy article can be found here: https://www.qrz.ru/schemes/contribute/arrl/chap3.pdf however for sake of simplicity, see the chart in the last picture in the gallery on this page.
# DX Commander now shipping to USA / Canada / Aus
After much research, I’ve managed to get the price to under £30 for USA, an additional £10 for Canada with Russia and Australia incurring extra costs. I’m doing this shipping at cost folks via a tracked parcel. You can buy shipping cheaper – but with horror stories I’m afraid.
• Still a hugely economical antenna for what you get.
PS Postage for USA is only £30 using MyHermes for delivery to OCS in UK and on to United Parcels Service for local delivery in USA.
Experience is suggesting that due to the low price, no customs or duty should be paid when it lands your end for most countries (although I can’t guarantee that – although my shipping people tell me it’s under the threshold).
Start here:
# Vertical Antenna – How many Ground Mounted Radials Do I Need
I’m often asked this question and after 5 years of development, I think I finally have the answer thanks to both real-worl experience and the work that Ruby Severns, N6LF did in a controlled scientifically based experiment.
Raised Radials are a completely different kettle of fish. These are tuned to the frequency i question and can give varied results.
PS – Another superb read for the very clever folks is here:
Good luck!
# M0MCX Banana Antenna – an end-fed choke sleeve resonant feedline T2LT antenna design
A new document fully documenting the design of the Banana Antenna has now been released entitled, “Banana, a Half Wave End-Fed Choked Coax Antenna”.
Banana Antenna
Antenna can be known as – and is similar to:
Sleeve Dipole / Flowerpot Antenna
The Sleeve dipole has traditionally been used by VHF antenna designers by sliding an external metal sleeve over the coax and connecting the sleeve to the braid of the coax so that the antenna appears to be centre-fed with an outboard “sleeve”. Some commercial CB antennas are also made this way.
Banana
Resonant Feed-line Dipole (J Taylor, W2OZH, 1971)
The first time that I can find any mention of an end-fed Resonant Feed-line Dipole was an article written by James Taylor (W2OZH) in the August 1991 edition of QST entitled the “Resonant Feed-line Dipole”. He discusses the idea of using the coax itself as both part of transmission line – and a resonant element. Again, it apparently behaves like a centre-fed dipole – and extremely similar to the Sleeve antenna mentioned above. Various aerial builders have made this antenna, mostly with variable – or in most cases, poor results.
Tuned Transmission Line Trap, T2LT (CB folks)
CB folks call this type of antenna the T2LT. This terminology appears to be from a German Patent by Prof F. Fischer (Patent Number 733697) from 1939 who apparently mentions “Die T2LT”. But the patent surrounds only the use of a tuned choke which has a capacitor across the choke for tuning, not the actual antenna. The term T2LT is perhaps a misnomer.
# 20m band End-Fed Choked Coax Dipoles (T2LT)
WARNING: This post has been replaced with the following analysis and design:
Banana Antenna Design May 2017
– – –
The Resonant Feedline Antenna is also known as:
• Sleeve Dipole (& Flowerpot Antenna)
• Resonant Feedline Dipole (J Taylor, W2OZH)
• Tuned Transmission Line Trap, T2LT (CB folks)
Pictures of this experiment follow including the 10-25 MHz >8K choke follow.
# 75 ohm to 50 ohm transmission line matching coax stub
Coax Transmission line coax stubs are frequency dependent. Making a stub for one frequency means it WILL NOT work for another frequency. My example is for a 20m Resonant Feedline Dipole, sometimes called a Sleeve Dipole or Resonant Coax Dipole or Tuned Choked Coax Dipole.
NOTE: CBers tend to call this T2LT. I have no idea why they refer to this antenna by that name because it stands for Tuned Transmission Line Trap which means it should be a TTLT – but then it doesn’t have a Trap? I digress. CB for you.
So you have an approx 75 ohm impedance antenna and you want to get the best match you can. Take the wavelength of the frequency, multiply it by the velocity factor of your 75 ohm matching coax and multiply again by 0.0815.
For example.
14.225 MHz = 21.089 metres
21.089 * 0.66 (what ever your velocity factor is) = 13.19
Multiply 13.91 * 0.0815 = 1.134m
Therefore, your transmission line coaxial transformer will be 1.134m long which is apparently about 29 degrees around the 360 degree circle.
Data found here: PA0FRI page.
Finally, I discovered MANY pages on eHam and QRZ forums of people asking the same question but most answers are with people answering questions which were not asked – or giving advice how to fix the antenna, or live with it. Why Americans need to argue the toss when others just need answers beggers belief
# 40m compact fan dipole for 40, 30 and 20m bands
One of my aerials has just come down in the wind, a 40m compact dipole arranged as an inverted V with the ends coming down as far as the 6 feet fence height.
My garden is about 15m wide (actually it’s 51 feet wide, so a whisker over) however it’s too small to fit in full-sized flat-top dipole but an inverted V works well. Whilst you are at it, why not add in elements for 30m and 20m and have three bands on one feeder?
Width of this antenna is 15m. You can make it smaller to suit.
I have designed this antenna to be a flat top with droopy legs. The centre will be held up with a very sturdy aluminium scaffold pole with a 4.6m sailboard mast sleeved over the top. The aluminium mast will cross-bolt to an already installed steel scaffold bar already concreted in the ground. Bottom line is that I should achieve around 10m in height (30 feet or so).
# DX Commander pre-production testing all-band vertical
I finally found some time this year to pull all the components together to test out in a real-world setting, the idea of using multiple elements on a single vertical fibreglass pole to achieve very good SWR and radiation patterns.
The problem with verticals is than in the main, people need either ATUs or they use that awful UNUN business with a single radial. The 9:1 UNUN business is just inefficient and the only way to to use an ATU effectively is at the feedpoint, not at the rig-end due to the severe losses.
A feedpoint ATU is expensive and generally requires a 12V power source. And long verticals have awful radiation patterns beyond 5/8th of a wavelength.
So the only way to reliably install a vertical and dispense with any worries about SWR and power handling is to build a mono-bander.
Regulars will know that I’ve been playing with the idea of adding separate elements to a 40m vertical mono-bander to add in the odd frequency, say 20m – but the interaction between elements can cause impedance issues (read SWR).
With development, I’ve discovered the optimum spacing between elements to achieve pure quarter-waves on 40m, 30m, 20m, 17m and 12m. It happens that the 40m vertical will resonate on 15m for excellent very-low radiation patterns and with the addition of a shorter-then-normal 10m element (around 2.6m in length) one can get radiation with a regular quarter-wave pattern, although the idea of using a ground-mounted vertical for 10m is slightly off-putting. There are other methods to get good radiation on the 10m band.
A picture speaks a thousand words, so, without further waffling, here is the prototype in action. It uses a regular DX Commander fibreglass pole which is around 9.7m in length with stainless hose-clamps using 8mm ID aquarium tubing (softened in hot water to push over the clamps). These clamps don’t scratch the tubing and securely hold each section from slipping down in a gale.
The base plate (radial plate) in the prototype is an aluminium angle with an SO239 fitted. The centre conductor is soldered with added heat-shrink and flooded with hot-glue. Connectors are used to connect to what I’m calling the “driven” plate with stainless nuts. RF enters the driven plate and self-selects the band it wants, just as a fan-dipole would. A guying point made from Nylon 66 keeps the elements optimally spaced as well as securely hold the mast upright at 1.2m off the ground to three guy stakes.
At the 5m point, a “spreader” plate houses the 20m and 17m elements on 3mm bungee cord with the 30m and 40m elements passing straight through. At the time I took the pictures, I had dispensed with the 15m and 12m elements.
In operation, I achieved better than 1:1.5 SWR across the operational bands selected. It was fun leaving WSPR mode running and allowing it to change bands without any ATU etc.
This antenna will comfortably handle 5000 Watts, although of course, the author only ran 400W RTTY for long periods for practical testing.
Hand-production of this system is extremely time-consuming so I am about to launch this with slightly lighter-weight and machined components to reduce cost. Target consumer price will be around £99. You’ll just need to add the wire and follow the instructions.
If you’d like to stay informed about progress, let me know.
# Compact half-square for 40m band
I’ve just had an interesting discussion on 40m this evening with Peter, OH6GHI, also an antenna enthusiast. I happened to mention that I was listening to him on both VFOs. VFO A on my dipole and VFO B on my vertical and I found the stereo image very interesting as the polarisation of his signal to me changed.
We got talking about half-squares and I confirmed that we were talking about the same thing, basically 2 x 10m verticals separated by a 20m top section. The half square is fed in one corner and according to my MMANA model, this should present 50 ohms and a great SWR curve across the whole of 40m.
# Bidirectional switchable 40m wire yagi
I switched on the other evening and heard a very quiet DX caller on 7.142. It was YC0LOU from Indonesia and I could only pick up parts of his call. He called and called and had no takers. In fairness, he was extremely quiet but as the sun was gradually moving around the sky, he finally became audible and it was worth giving him a shout. 400w off my inverted V at 7m height got his attention but I needed a few blasts for him to get my call right. I put him on the cluster and he had a pile up.
Now, the point is, had I had more gain, I’d have not only heard him better, but he’d have heard me quicker too.
So I could add more height to my Inverted V but the difference between 7m and 10m isn’t actually that much at 5 degrees off the horizon – not even a db. Hardly worth writing home about.
Anyway, this was the QSO that made me sit up and take stock of what I could do. I was seriously considering phased verticals for DX when I thought up the idea of having a switchable wire yagi. Either firing East or firing West.
Like me, you may already have an inverted V dipole up for 40m, all you need to is build another one about a quarterwave in front – or behind your existing dipole but out of a single wire. You don’t need to feed this with coax, it’s a parasitic element, like a 2 element yagi.
# How to use dual coax feeders as ladder line
Having recently taken delivery of a Palstar AT4K manual tuner, I was keen to get her into production to replace my CG5000 in the attic.
Problem: the route to the attic from the shack is complex but I have a number of spare coax runs going that way including a couple of RG58 cables that I installed about 10 years ago as backups. Actually I originally installed three RG58 lines but I’ve been using one of them to send 12V up the line to the ATU.
After MUCH research, I finally used about 20 feet of parallel coax feeders, connecting ladder line to both ends. To clarify, I run about 12 feet of ladder line from the ATU to the parallel RG58 cables. I soldered the ladder line to the inner core of the RG58 coax and shorted the braid-to-braid. My 20 feet of RG58 runs to the attic, through walls, up ceilings etc and in reverse, I connected the ladder line to the RG58. Again, I shorted the braids of each line to each other with a solder blob. My ladder line then has another run to the feedpoint of a large 60m loop that runs through the attic and around the garden.
The results have been quite amazing. Comparing my 40m reference dipole to the the CG5000 (SG230 type) ATU feeding the 60m loop has always shown that the loop was about an S point lower than my reference dipole for most stations.
# Working 15m band on a 40m vertical antenna
Note: This article discusses the merits of a 3/4 (three quarter wave) vertical -vs- a 1/4 (quarter wave) antenna.
You can build a 40m vertical quarter wave antenna and ground mount it with 16 x 4m radials and operate it at the third harmonic; 21MHz.
Actually, all my experimentation has shown that if you multiply the quarter wave resonance by 3.03, you’ll have the next available usable band. In this case, if you tune a 40m vertical to 7.00Mhz, you’ll have the whole of the 15m band to play with with a centre-point of 21.300Mhz. Oh, and you’ll still have the whole of 40m band under 1.3:1.
Now here’s the controversy:
Most people who read antenna publications or the ARRL handbook believe that if you actually make this antenna, you’re creating a cloud-burner on 15m.
Technically correct (sort of) – but for DX, wrong.
On the surface, the 10m long 40m vertical that’s used on 21.225MHz does indeed look like a cloud burner. Here it is. 15m band in green -vs- a pure quarter-wave in red).
(click to expand quarter-wave in red, three-quarter wavelength in green)
# How heavy is a Cushcraft A3S TriBander Yagi?
The A3S is 14Kg (That’s 30 pounds in US money )
I was worried that the combined weight of a Create Rotator an aluminium pole *and* an A3S Cushcraft might be too much for my little lighting rig.
To test it out, I fitted the Create Rotator to the mast and added a 20Kg dumb-bell weight and started the motor.
It lifted without an issue so I’ll go ahead and plan to use the A3S for next field day.
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# Magnetic Field Intensity | Definition Formula
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Magnetomotive force, ℑ , per unit length, is called the magnetic field intensity H.
## Magnetic Field Intensity Unit
Magnetic field intensity is also known as the magnetizing force which is measured is ampere-turns per meter (A-t/m).
Of primary concern, however, is the magnetomotive force needed to establish a certain flux density, B in a unit length of the magnetic circuit.
## Magnetic Field Intensity Formula
The letter symbol for magnetizing force (magnetic field intensity) is H. The following relationship defines H as;
$H=\frac{\mathfrak{F}}{l}$
Where
ℑ =applied MMF in ampere-turns
l =average length of the magnetic path in meters
Example
Find the magnetic field intensity in the magnetic circuit shown below:
Solution:
We can calculate the intensity using following formula:
$H=\frac{F}{l}=\frac{(2.5*{{10}^{2}})*(1.5*{{10}^{-1}})}{1.2*{{10}^{-1}}}$
$H=\frac{3.75*{{10}^{1}}}{1.2*{{10}^{-1}}}=3.125*{{10}^{2}}A-t/m$
If the dimensions of the magnetic path were changed, the value of H would also change. Fox example, if the total length of the magnetic path doubled, we should expect the value of H to decrease to one-half its previous amount.
If physical dimensions are double for the above circuit, then magnetizing force will be;
$H=\frac{3.75*{{10}^{1}}}{2.4*{{10}^{-1}}}=1.5625*{{10}^{2}}A-t/m$
So, we can observe that, for a given number of ampere-turns, the magnetizing force varies inversely per unit length of the magnetic path.
Magnetomotive Force and Unit
Magnetic Flux Density Definition and Unit
Magnetic Flux Definition and Unit
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# DIFFRACTION
## DIFFRACTION
Sound waves propagate spherically. For smaller wavelengths and greater distance from the source, we can perceive them as waves with a relatively flat front and a direction perpendicular to it.
The ability of sound waves in meeting obstacles to go around them, penetrating into the area behind them, is called diffraction. Without a diffraction the sound would be audible only in a situation of direct sight to the sound source.By the diffraction we can hear sound behind barriers, penetrating through different holes etc.
Light diffraction also occurs, but due to much smaller wavelength of the light wave in normal conditions it does not result in circumvention of the objects. Here we observe other interesting side effects, such as contrast edges of the observed object, especially when backlit. Diffraction effects in the electromagnetic waves occur all the way in atomic and subatomic level.
We can locate a diffraction on several occasions – passing of a wave along an edge, passing of a wave through an obstacle or passing through an slit:
This is due to the Huygens principle, according to which each point, at which the wave reaches, becomes a new point source of spherical waves. For this reason, in the above example when the sound wave reaches the edge of the obstacle, it converts this edge to a source of a spherical wave and thus surrounds it.
If it reaches small aperture that is commensurate with the wavelength, this opening becomes a point source of the spherical wave.
At presence of a diffraction we have no change in the wavelength or the frequency or the speed of the wave. The direction of propagation is what only changes.
Here’s an illustration of the diffraction in nature:
### How the diffraction is connected to the wavelength?
1. If the wavelength is much larger from the the obstacle that meets, it surrounds it with hardly be affected by it and it changes the shape and the structure too little.
2. If the wavelength is comparable to the size of the object it surrounds partially, and we have a acoustic shadow just behind it.
3. If the wavelength is less than the size of the obstacle, we have an acoustic shadow and only a small portion of the sound enters behind the object .
As much as greater is the size of the obstacle relatively to the wavelength, the long and clear acoustic shadow we have behind it.
Therefore it can be said that in the area of high frequencies we have dissipation and reflection of the sound waves in meeting an obstacle, and in the area of the lower frequencies – diffraction and surrounding of the obstacle. In this situation, for example in a hall the objects that are in front of us (other people, balconies, columns, etc.). will carry frequency change of the sound while skipping with a priority the lower frequencies at the expense of the high ones.
The same situation occurs in acoustic barriers when needed to reduce the traffic noise from residents. Since the diffraction of the sound depends on the ratio between the wavelength and the size of the barrier, so with the same barrier we will have different efficiency at different frequencies due to the diffraction:
As we see, the high frequencies can be stopped successfully from the barrier while the low turn around it by the effect of diffraction. In this situation, only the increase in the size of the barrier would help for its greater efficiency.
### Diffraction when the sound is passing trough aperture
The diffraction when passing through aperture also depends on the ratio between the aperture and the wavelength.
If the opening is much larger than the wavelength, we virtually can not observe diffraction, the wave passes freely. However, due to the fact that the wavefront is not virtually changed, are created zones of acoustic shadow caused by the barrier in which the aperture is located.
Upon aperture commensurate or smaller than the wavelength we observe diffraction processes on both its sides. In this situation, the aperture starts to react as a new point of spherical radiation spreading sound in the form
of a hemisphere behind the barrier. Thus, the acoustic shadows disappear and the location of such a sound ceases to be from the point of its initial transmission, but is displaced from the point of the spherical secondary radiation, i.e. aperture:
Diffraction is an important phenomenon in acoustics because it affects many processes associated with the propagation of sound.
For example, in the manufacture of microphones and loudspeakers the diffraction processes can lead to a strong nonlinearity and to change of the direction of the sound at some frequencies relative to others. In the construction and operation of halls and studios also the diffraction is an important factor.
In the field of sound insulation and sound absorption also is important to know that even in well-executed work on isolation between two rooms or between room and ? sound source the presence of even a small hole leads to a spot radiation from this hole, which is able to compromise the rest of the process .
Here we can see the borders of the speaker, which are made in this way to avoid the diffraction.
http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-2/Diffraction-How-it-works.html
http://hyperphysics.phy-astr.gsu.edu/hbase/sound/diffrac.html
http://www.physicsclassroom.com/class/sound/Lesson-3/Reflection,-Refraction,-and-Diffraction
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en
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http://www.investopedia.com/university/mutualfunds/mutualfunds5.asp
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Perhaps you've noticed those mutual fund ads that quote amazingly high one-year rates of return? Your first thought might be, "wow, that mutual fund did great!" Well, yes it did great last year, but then you look at the three-year performance, which is lower, and the five year, which is yet even lower. What's the underlying story here? Let's look at an actual example from a large mutual fund's performance:
1 year return 3 year return 5 year return 53% 20% 11%
Last year, this mutual fund had excellent performance, returning 53% to investors. But, over the past three years the average annual return was just 20%. What did the fund return in years 1 and 2 to bring the average return down to 20%? Simple math shows us that the fund made an average return of 3.5% over those first two years: 20% = (53% + 3.5% + 3.5%)/3. Since this 3.5% figure is only an average, it is very possible that the fund lost money in one or more of those years. (See also the Investopedia tutorial Fund Performance Metrics.)
It gets more dismal if we look at the five-year performance. We know that in the last year the fund returned 53% and in years 2 and 3 we are guessing it returned around 3.5%. So, what happened in years 4 and 5 to bring the average return down to just 11%? Again, by doing some simple calculations we find that the fund must have lost money, on average -2.5% each year of those two years: 11% = (53% + 3.5% + 3.5% - 2.5% - 2.5%)/5. With that in mind the fund's performance doesn't look quite so impressive.
For the sake of simplicity in this example, and besides making some big assumptions, we haven’t calculated compound interest over time. The point, however, was not so much to be technically accurate but to demonstrate the importance of taking a closer look at performance numbers and comparing them across years. A fund that loses money for a few years can bump the average up significantly with one or two strong years.
## It's All Relative
Knowing how a fund performed in isolation is not very helpful. Performance must be viewed as a relative issue, judged against the performance of an appropriate benchmark. For example, a large-cap equity fund would usually be compared against the S&P 500 while a small-cap equity fund against the Russell 2000 index. Comparing to the wrong benchmark can produce mismatched returns and aren’t very informative. When the fund we looked at in the example above is compared against its appropriate benchmark index, a whole new layer of information is added to the evaluation. If the benchmark returned 75% for the single-year time period, that 53% returned by the fund no longer looks quite as good. If, on the other hand, the benchmark delivered results of 25%, 5%, and -5% for the respective one, three, and five-year periods, then the fund's results look rather fine indeed.
To add another layer to the evaluation, we can consider a fund's performance against its peer group as well as against its benchmark index. If other funds that invest with a similar mandate had similar performance, this data point tells us that the fund is in line with its peers. If the fund bested its peers and its benchmark, its results would be quite impressive.
Looking at any one piece of information in isolation only tells a small portion of the story. Consider the comparison of a fund against its peers. If the fund sits in the top slot for each of the comparison periods, it is likely to be a solid performer. If it sits at the bottom, it may be even worse than perceived, as peer group comparisons only capture the results from existing funds. The risk here is called survivorship bias. Many fund companies are in the habit of closing down their worst performing funds, and when the losers are purged from their respective categories, their statistical records are no longer included in the category performance data. This makes the category averages creep higher than they would have if the losers were still in the mix.
To develop the best possible picture of fund's performance results, consider as many data points as you can. Long-term investors should focus on long-term results, keeping in mind that even the best performing funds have bad years from time to time. It is also important to keep in mind that costs and fees are not included in performance statistics, but can make a real impact on your own returns from investing in mutual funds.
Mutual Funds: Conclusion
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### 4 Mistakes to Avoid When Choosing Mutual Funds to Invest in
Mutual funds are a great way to build wealth but not all of them are the same. Investors have to be mindful of fees, turnover, redundancy and performance.
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http://usaco.org/index.php?page=viewproblem2&cpid=794
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Problem 3. Sprinklers
Contest has ended.
Farmer John has a large field, and he is thinking of planting sweet corn in some part of it. After surveying his field, FJ found that it forms an $(N-1) \times (N-1)$ square. The southwest corner is at coordinates $(0,0)$, and the northeast corner is at $(N-1,N-1)$.
At some integer coordinates there are double-headed sprinklers, each one sprinkling both water and fertilizer. A double-heading sprinkler at coordinates $(i,j)$ sprinkles water on the part of the field north and east of it, and sprinkles fertilizer on the part of the field south and west of it. Formally, it waters all real coordinates $(x,y)$ for which $N \geq x \geq i$ and $N \geq y \geq j$, and it fertilizes all real coordinates $(x,y)$ for which $0 \leq x \leq i$ and $0 \leq y \leq j$.
Farmer John wants to plant sweet corn in some axis-aligned rectangle in his field with integer-valued corner coordinates. However, for the sweet corn to grow, all points in the rectangle must be both watered and fertilized by the double-headed sprinklers. And of course the rectangle must have positive area, or Farmer John wouldn't be able to grow any corn in it!
Help Farmer John determine the number of rectangles of positive area in which he could grow sweet corn. Since this number may be large, output the remainder of this number modulo $10^9 + 7$.
INPUT FORMAT (file sprinklers.in):
The first line of the input consists of a single integer $N$, the size of the field ($1 \leq N \leq 10^5$).
The next $N$ lines each contain two space-separated integers. If these integers are $i$ and $j$, where $0 \leq i,j \leq N-1$, they denote a sprinkler located at $(i,j)$.
It is guaranteed that there is exactly one sprinkler in each column and exactly one sprinkler in each row. That is, no two sprinklers have the same $x$-coordinate, and no two sprinklers have the same $y$-coordinate.
OUTPUT FORMAT (file sprinklers.out):
The output should consist of a single integer: the number of rectangles of positive area which are fully watered and fully fertilized, modulo $10^9 + 7$.
SAMPLE INPUT:
5
0 4
1 1
2 2
3 0
4 3
SAMPLE OUTPUT:
21
Problem credits: Dhruv Rohatgi
Contest has ended. No further submissions allowed.
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https://community.esri.com/t5/arcgis-network-analyst-questions/vrp-different-solutions-to-the-same-problem/m-p/627913/highlight/true
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# VRP: Different solutions to the same problem
880
2
02-14-2017 08:41 AM
New Contributor II
I'm calculating several options to pickup the urban waste to get the least number of truck routes.
In the first option, I use 22 trucks and the solver notes me that it use 14 trucks to pick up all containers, but in a second option I indicate 14 trucks, the solver notes me there isn't a complete solution and the 14 truck can't pick up all containers. Why does it happen?
Tags (2)
2 Replies
Esri Contributor
The VRP Solver is a heuristic meaning that it follows a set of steps to produce an answer. It, therefore, can be sensitive to the initial set up of the problem when starting the solver. In your case that seems to be the case and is why when you changed the number of routes that are available the solution is different.
For trying to minimize the number of routes used, the best way to do this is to add a high value to the FixedCost. This will indicate to the solver that it should try not to open an additional route because that is more costly then taking an existing route and extending the travel time and distance. You might need to experiment with the FixedCost value to determine what is large enough to get the desired results for number of routes.
You mention that you are doing urban waste pickup, I usually associate this with a high density of order locations (several on the same street right next to each other). Is that true for you?
New Contributor II
Thanks Heather!
As you tell me I have a lot of orders, approximately 2000 orders. My doubt is beacause I uuse, in the firts option with 22 trucks, and the solver offers me one solution with 14 trucks. UIn a asecond option with 16 trucks, the solver offers me the same solution, with identical routes, but in the third option with 14 trucks the solver offers me a partial solution. The most curious is if in a new option (fourth option) with 14 truck too, I take the orders from the previous options configuring all orders Asigment rule in "Preseve route and sequence" I get the same solution that 22 and 16 truck with only 14 trucks. In the third option 14 trucks, Asigment Rule to the orders are in "Overrride".
I will try introducing a fixedcost, is a great idea!
Thanks!!
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http://www.jiskha.com/display.cgi?id=1371132267
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# Precal
posted by .
An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter running track.
a) Draw a figure that visually represents the problem. Let x and y represent the length and width of the rectangular region respectively
b) Determine the radius of the semicircular ends of the track. Determine the distance, in terms of y, around the inside edge of each semicircular part of the track.
c) Use the result of part b to write an equation in terms of x and y, for the distance traveled in one lap around the track. Solve for x.
d) Use the result of part c to write the area A of the rectangular region as a function of x. What dimensions will produce a rectangle of maximum area?
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https://postgis.net/docs/manual-dev/zh_Hans/ST_GeometryN.html
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Name
ST_GeometryN — 返回几何集合的一个元素。
Synopsis
`geometry ST_GeometryN(`geometry geomA, integer n`)`;
描述
自版本 0.8.0 以来，OGC 规范的索引从 1 开始。 以前的版本将其实现为基于 0。
要提取几何图形的所有元素，ST_Dump 效率更高，并且适用于基本几何图形。
标准示例
```--Extracting a subset of points from a 3d multipoint
SELECT n, ST_AsEWKT(ST_GeometryN(geom, n)) As geomewkt
FROM (
VALUES (ST_GeomFromEWKT('MULTIPOINT((1 2 7), (3 4 7), (5 6 7), (8 9 10))') ),
( ST_GeomFromEWKT('MULTICURVE(CIRCULARSTRING(2.5 2.5,4.5 2.5, 3.5 3.5), (10 11, 12 11))') )
)As foo(geom)
CROSS JOIN generate_series(1,100) n
WHERE n <= ST_NumGeometries(geom);
n | geomewkt
---+-----------------------------------------
1 | POINT(1 2 7)
2 | POINT(3 4 7)
3 | POINT(5 6 7)
4 | POINT(8 9 10)
1 | CIRCULARSTRING(2.5 2.5,4.5 2.5,3.5 3.5)
2 | LINESTRING(10 11,12 11)
--Extracting all geometries (useful when you want to assign an id)
SELECT gid, n, ST_GeometryN(geom, n)
FROM sometable CROSS JOIN generate_series(1,100) n
WHERE n <= ST_NumGeometries(geom);
```
多面体曲面、TIN 和三角形的示例
```-- Polyhedral surface example
-- Break a Polyhedral surface into its faces
SELECT ST_AsEWKT(ST_GeometryN(p_geom,3)) As geom_ewkt
FROM (SELECT ST_GeomFromEWKT('POLYHEDRALSURFACE(
((0 0 0, 0 0 1, 0 1 1, 0 1 0, 0 0 0)),
((0 0 0, 0 1 0, 1 1 0, 1 0 0, 0 0 0)),
((0 0 0, 1 0 0, 1 0 1, 0 0 1, 0 0 0)),
((1 1 0, 1 1 1, 1 0 1, 1 0 0, 1 1 0)),
((0 1 0, 0 1 1, 1 1 1, 1 1 0, 0 1 0)),
((0 0 1, 1 0 1, 1 1 1, 0 1 1, 0 0 1))
)') AS p_geom ) AS a;
geom_ewkt
------------------------------------------
POLYGON((0 0 0,1 0 0,1 0 1,0 0 1,0 0 0))```
```-- TIN --
SELECT ST_AsEWKT(ST_GeometryN(geom,2)) as wkt
FROM
(SELECT
ST_GeomFromEWKT('TIN (((
0 0 0,
0 0 1,
0 1 0,
0 0 0
)), ((
0 0 0,
0 1 0,
1 1 0,
0 0 0
))
)') AS geom
) AS g;
-- result --
wkt
-------------------------------------
TRIANGLE((0 0 0,0 1 0,1 1 0,0 0 0))```
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https://www.singaporemath.com/programs/dimensions-pk-5/projection-images/grade-1a/
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Select Page
Grade 1A
[vc_row][vc_column][vc_column_text]
[/vc_column_text][/vc_column][/vc_row][vc_row][vc_column width=”1/2″][vc_column_text]Share these images with students to discuss new concepts before they open their textbooks.[/vc_column_text][/vc_column][vc_column width=”1/2″][/vc_column][/vc_row][vc_row][vc_column][vc_column_text]
[/vc_column_text][/vc_column][/vc_row][vc_row][vc_column][vc_accordions responsive=”false”][vc_accordion_tab title=”Chapter 1″][vc_column_text css=”.vc_custom_1524174872594{margin-bottom: 0px !important;}”]
• Chapter Opener | View
• Lesson 1: Numbers to 10 | View
• Lesson 2: The Number 0 | View
• Lesson 3: Order Numbers | View
• Lesson 4: Compare Numbers | View
[/vc_column_text][/vc_accordion_tab][vc_accordion_tab title=”Chapter 2″][vc_column_text css=”.vc_custom_1524174872594{margin-bottom: 0px !important;}”]
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[/vc_column_text][/vc_accordion_tab][vc_accordion_tab title=”Chapter 4″][vc_column_text css=”.vc_custom_1524174872594{margin-bottom: 0px !important;}”]
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[/vc_column_text][/vc_accordion_tab][vc_accordion_tab title=”Chapter 6″][vc_column_text css=”.vc_custom_1524174872594{margin-bottom: 0px !important;}”]
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[/vc_column_text][/vc_accordion_tab][vc_accordion_tab title=”Chapter 8″][vc_column_text css=”.vc_custom_1524174872594{margin-bottom: 0px !important;}”]
[/vc_column_text][/vc_accordion_tab][vc_accordion_tab title=”Chapter 9″][vc_column_text css=”.vc_custom_1524174872594{margin-bottom: 0px !important;}”]
[/vc_column_text][/vc_accordion_tab][/vc_accordions][/vc_column][/vc_row][vc_row][/vc_row][vc_column][/vc_column][vc_column_text]
I am text block. Click edit button to change this text. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Ut elit tellus, luctus nec ullamcorper mattis, pulvinar dapibus leo.
[/vc_column_text]
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You are Here: Home >< Physics
# Oxford PAT 2016
Announcements Posted on
TSR's new app is coming! Sign up here to try it first >> 17-10-2016
1. Guys, learn the a2 content on space. It comes up regularly on the PAT for estimating distances between planets etc.
2. (Original post by hellomynameisr)
Guys, learn the a2 content on space. It comes up regularly on the PAT for estimating distances between planets etc.
I've barely started PAT work yet, been mainly focusing on M2 and M3 work over summer. Are there any other obvious a2 physics topics I should be prioritising before attempting some papers?
3. (Original post by tangotangopapa2)
I don't have that book. Just looked at the online version of the book. That problem looks interesting.
The quadrilateral ACBD has, AC=AD and BC=BD where AB and CD are diagonals. It is easy to figure out that AB and CD are perpendicular.
Area of triangle ADB = 1/2 X AB X OD (Where O is the point where diagonals meet, not shown in the book. Draw it yourself) and area of triangle ACB is 1/2 X AB X OC.
Add these areas to get the area of the quadrilateral.
Since, OC + OD = CD
You get the formula 1/2 X AB X CD.
Hope this helps.
For those wondering, the figure is below:
Attachment 576244
maybe it's a bit late idk but how do you know they are perpendicular?
4. (Original post by Inert1a)
maybe it's a bit late idk but how do you know they are perpendicular?
5. Now that the PAT syllabus is changed, is it still very important to do MCQs and long questions from the past papers?
6. (Original post by lawlieto)
I have a few questions about electromagnetism.
Is there always a magnetic field around a current carrying wire? Even if the current is "steady"? I mean, dI/dt is a constant, not a function of t?
Why are electric fields due to magnetic fields concentric circles? (edit: I have seen the example of putting a copper ring in a uniform magnetic field, and when the flux linkage was increased that induced an emf and hence a current in the ring, and when there's a current, there's an electric field driving the charges, and that electric field would be concentric circles, but the electric field due to changing magnetic field exists without the copper ring as well, or without charges, which I can't understand atm)
Thanks
May I link you to the following post?
http://www.thestudentroom.co.uk/show...8#post67267538
7. (Original post by 98matt)
I've barely started PAT work yet, been mainly focusing on M2 and M3 work over summer. Are there any other obvious a2 physics topics I should be prioritising before attempting some papers?
The syllabus is quite deceiving in the fact that it says that most of the content being assessed will be AS. But, when you look at past papers, there is a lot of A2 content. Look through a couple past papers (the physics section especially) and see if there are any questions you can't answer due to insufficient knowledge.
(Original post by tangotangopapa2)
Now that the PAT syllabus is changed, is it still very important to do MCQs and long questions from the past papers?
Practice is practice. I'd still do them
8. (Original post by hellomynameisr)
Practice is practice. I'd still do them
How would you approach this type of problems?
9. (Original post by tangotangopapa2)
How would you approach this type of problems?
I would look at it and cry for a good hour
After that one hour, I would start to depict the question bit by bit.
Firstly, f(x) > 0 for all values of x. Now we know that squaring any number would result in a positive outcome, yes? And for all values of x, this would be positive. Therefore, we know that it will have a quadratic like shape and will be in the top 2 quadrants
Second, a continuous function is one that has no holes or gaps in it. In simpler terms, you could draw it without lifting your pen from the paper. Whereas a function like y= 1/x would have to be sketched by lifting your pen off the paper.This is quite hard to explain for me so see this video:
It is not taught at A level (I think) so better do some research.
Third, df/dx = 0 only when x = 4. What do we know when the first derivative is equal to 0? It is a point where the gradient is 0. Therefore, when x = 4, the gradient will be 0 and so a local minimum will be at this point.
Lastly, when the second derivative is equal to 0, we know that these are point of inflection. These are when x = 2 and x = 6. So they basically turn about those point (something like that).
Lo and behold, we have all the information we need to sketch the function.
Hope this helps
10. (Original post by tangotangopapa2)
How would you approach this type of problems?
(Original post by hellomynameisr)
I would look at it and cry for a good hour
After that one hour, I would start to depict the question bit by bit.
Firstly, f(x) > 0 for all values of x. Now we know that squaring any number would result in a positive outcome, yes? And for all values of x, this would be positive. Therefore, we know that it will have a quadratic like shape and will be in quadrant one only.
Second, a continuous function is one that has no holes or gaps in it. In simpler terms, you could draw it without lifting your pen from the paper. Whereas a function like y= 1/x would have to be sketched by lifting your pen off the paper.This is quite hard to explain for me so see this video:
It is not taught at A level (I think) so better do some research.
Third, df/dx = 0 only when x = 4. What do we know when the first derivative is equal to 0? It is a point where the gradient is 0. Therefore, when x = 4, the gradient will be 0.
Lastly, when the second derivative is equal to 0, we know that these are point of inflection. These are when x = 2 and x = 6. So they basically turn about those point (something like that).
Lo and behold, we have all the information we need to sketch the function.
Hope this helps
f(x) > 0 for all x means it would be in two quadrants (the top half), not one! It just has to all be above the x axis. Also be careful for (d) that you don't draw stationary points (as df/dx isn't 0 here), while (c) gives you a stationary point (either maximum or minimum at x=4.
A bit of graph sketching which can be helpful for the PAT/interviews is in FP1 (for AQA at least), but it's all stuff you can figure out from what you have been taught already I think.
11. (Original post by Lau14)
f(x) > 0 for all x means it would be in two quadrants (the top half), not one! It just has to all be above the x axis. Also be careful for (d) that you don't draw stationary points (as df/dx isn't 0 here), while (c) gives you a stationary point (either maximum or minimum at x=4.
A bit of graph sketching which can be helpful for the PAT/interviews is in FP1 (for AQA at least), but it's all stuff you can figure out from what you have been taught already I think.
Oh yes! I see my mistakes. Thank you
12. (Original post by Lau14)
.
What kind of questions did they ask you in the interview?
13. (Original post by hellomynameisr)
What kind of questions did they ask you in the interview?
I had two interviews at my main college which were a mix of shorter maths and physics questions (a series of 10 quick questions e.g. what is cos(pi/6), some graph sketching, integration, differentiation, terminal velocity, energy conservation, stuff like that), and one interview at a second college where they did a longer question on electrostatics, starting easy and building it up to a more complicated situation (and again, more graph sketching).
14. (Original post by hellomynameisr)
I was able to get part A right but don't understand B and C
From the solutions, why does he do what he does in part B? I understand that p=v^2/r but how does he get the values to input them in from
And part C, how does he split up the circuit like that
I remember being like ' why are there two points labelled differently ( different letters) but at the same potential on the exam xD
15. What are some resources to practice maths section for the PAT other than past papers?The questions asked in PAT have quite a different nature compared to other and usually questions are of following properties/categories:
1) Often simple questions involving mental arithmetic is asked e.g. 2023^2 - 2022^2, (3.12)^5 correct to one decimal place.
2) Coordinate geometry of line includes, finding slope of line passing through two points or finding equation of the line passing through one point and having slope this (you need to find slope, e.g. perpendicular to another line). Coordinate geometry of circle is often asked but some of the questions that might put us off is like this: Find equation of line/lines tangent to the circle/curve and passing through this point (usually outside point).
3) Simple questions testing properties of logarithms.
4) Trigonometric equations; finding solution in a given interval. (Usually Pythagorean Identity has to be used and then quadratic equation has to be solved. Obviously, some extraneous results should be omitted). Some knowledge of inverse trigonometric function. e.g evaluate cos(tan^-1(sin(cot^-1x)))
5) Arithmetic and geometric sequence and series, including finding sum of infinite geometric series.
6) Factoring polynomials (degree 3 or 4) and finding all roots.
7) Simple application of binomial expansion formula (Knowledge of Pascal's triangle is enough).
8) Calculus: Rate of change e.g. If Rate of change of volume of water in cylinder is this, then what is the rate of change of height. Finding maxima/minima. Some harder questions like: Maximise area of rectangle of given perimeter. Simple knowledge of concavity, increasing/decreasing functions etc. is asked.
9) Calculus: Finding area by integration. Finding definite integrals. Usually you either have to split expression into partial fractions or use algebraic substitution.
10) Solving rational inequalities. Simple cautions to ensure no sign flip if you multiply equation by certain terms is often required.
edit: I forgot this. 11) Probability theory: Simple probability questions. Sometimes one may have to use conditional probability formula but never binomial/Possion/Nomal distribution formula. One type of tricky question is: 3 special dice (each can give score of 1-8) are rolled. The score is calculated as Ad1 + Bd2 + Cd3 + D where A,B,C and D are constants and d1-3 are individual score of the die. Find the constants so that : - The score ranges from 1-512 and each score has equal probability.
12) Roots of quadratic equation and their properties is assumed.
13) Simple algebraic manipulation eg. If a= x^2 + y ^2 and b = 2xy express x and y in terms of a and b.
14) Geometry. 95% of the time the question involves circle inscribed by polynomials ( most often equilateral triangle or square). You usually have to find the ratio of areas of two separate regions in the figure.
15) The hardest of all. Curve sketching. Questions range from trivial to extremely challenging. You might have to use properties of transformation of graph. Sketch graphs of completely different/complicated functions. Sketch graphs of y = f(x) +/- g(x), y = f(x)g(x) or f(x)/g(x) where you know the graph of y = f(x) and y=g(x) separately. Sometimes you might have to sketch inequalities like. -2 < y/x < 4 or 0< y/x^2 < 6pi.
MAT questions/ Senior Maths Challenge questions/ BMO questions don't have above mentioned properties. They deal with hard number theory, combinatorics, different type of algebra problems and provide very little help in brushing up above type of problems. So, what might help me to practice these type of problems? Thanks in advance.
16. (Original post by tangotangopapa2)
What are some resources to practice maths section for the PAT other than past papers?The questions asked in PAT have quite a different nature compared to other and usually questions are of following properties/categories:
1) Often simple questions involving mental arithmetic is asked e.g. 2023^2 - 2022^2, (3.12)^5 correct to one decimal place.
2) Coordinate geometry of line includes, finding slope of line passing through two points or finding equation of the line passing through one point and having slope this (you need to find slope, e.g. perpendicular to another line). Coordinate geometry of circle is often asked but some of the questions that might put us off is like this: Find equation of line/lines tangent to the circle/curve and passing through this point (usually outside point).
3) Simple questions testing properties of logarithms.
4) Trigonometric equations; finding solution in a given interval. (Usually Pythagorean Identity has to be used and then quadratic equation has to be solved. Obviously, some extraneous results should be omitted). Some knowledge of inverse trigonometric function. e.g evaluate cos(tan^-1(sin(cot^-1x)))
5) Arithmetic and geometric sequence and series, including finding sum of infinite geometric series.
6) Factoring polynomials (degree 3 or 4) and finding all roots.
7) Simple application of binomial expansion formula (Knowledge of Pascal's triangle is enough).
8) Calculus: Rate of change e.g. If Rate of change of volume of water in cylinder is this, then what is the rate of change of height. Finding maxima/minima. Some harder questions like: Maximise area of rectangle of given perimeter. Simple knowledge of concavity, increasing/decreasing functions etc. is asked.
9) Calculus: Finding area by integration. Finding definite integrals. Usually you either have to split expression into partial fractions or use algebraic substitution.
10) Solving rational inequalities. Simple cautions to ensure no sign flip if you multiply equation by certain terms is often required.
12) Roots of quadratic equation and their properties is assumed.
13) Simple algebraic manipulation eg. If a= x^2 + y ^2 and b = 2xy express x and y in terms of a and b.
14) Geometry. 95% of the time the question involves circle inscribed by polynomials ( most often equilateral triangle or square). You usually have to find the ratio of areas of two separate regions in the figure.
15) The hardest of all. Curve sketching. Questions range from trivial to extremely challenging. You might have to use properties of transformation of graph. Sketch graphs of completely different/complicated functions. Sketch graphs of y = f(x) +/- g(x), y = f(x)g(x) or f(x)/g(x) where you know the graph of y = f(x) and y=g(x) separately. Sometimes you might have to sketch inequalities like. -2 < y/x < 4 or 0< y/x^2 < 6pi.
MAT questions/ Senior Maths Challenge questions/ BMO questions don't have above mentioned properties. They deal with hard number theory, combinatorics, different type of algebra problems and provide very little help in brushing up above type of problems. So, what might help me to practice these type of problems? Thanks in advance.
So far i would say if you study from the papers toroughly, timing your attempts at the section and dont waste time quickly leave questions you dont get, and then mark the ones that gave you trouble and really understand why, i think most people doing the pat are good but approach questions taht wrong way, i have changed the way i approach maths questions now because of the practice for the pat.
You probably know the book but do some puzzles from professor povey's book
Also, if you have a math teacher, or someone really exprienced i would leave teh paper with them and ask them to go through it and try to set you some similar questions for practice, and if you can try writing some yourself (although this maybe quite tedious to do)
I havent gone over the mat papers yet are they difficult?
Perhaps PhysM23 could lend us some advice as he has gone through the exam and maybe go over some of things i've said here.
I'm sure if we look right we can find plenty of similar questions, even if the questions arent similar im sure it wouldnt be bad to do them
17. Hi guys,
Im thinking of applying to Oxford for engineering. I got 4As at AS this year, in Maths, Further Maths, Physics and Chem.
Ive not started any prep for the PAT.
What Maths modules should i cover?
ive already covered C1,C2,C3,FP1,M1.
A checklist of somesort with mlst of the things i need to do to be prepared would be great.
Thanks very much
18. (Original post by Lau14)
I had two interviews at my main college which were a mix of shorter maths and physics questions (a series of 10 quick questions e.g. what is cos(pi/6), some graph sketching, integration, differentiation, terminal velocity, energy conservation, stuff like that), and one interview at a second college where they did a longer question on electrostatics, starting easy and building it up to a more complicated situation (and again, more graph sketching).
what college was that. ? ( those questions are nothing like what I got asked)
19. (Original post by tangotangopapa2)
What are some resources to practice maths section for the PAT other than past papers?The questions asked in PAT have quite a different nature compared to other and usually questions are of following properties/categories:
1) Often simple questions involving mental arithmetic is asked e.g. 2023^2 - 2022^2, (3.12)^5 correct to one decimal place.
2) Coordinate geometry of line includes, finding slope of line passing through two points or finding equation of the line passing through one point and having slope this (you need to find slope, e.g. perpendicular to another line). Coordinate geometry of circle is often asked but some of the questions that might put us off is like this: Find equation of line/lines tangent to the circle/curve and passing through this point (usually outside point).
3) Simple questions testing properties of logarithms.
4) Trigonometric equations; finding solution in a given interval. (Usually Pythagorean Identity has to be used and then quadratic equation has to be solved. Obviously, some extraneous results should be omitted). Some knowledge of inverse trigonometric function. e.g evaluate cos(tan^-1(sin(cot^-1x)))
5) Arithmetic and geometric sequence and series, including finding sum of infinite geometric series.
6) Factoring polynomials (degree 3 or 4) and finding all roots.
7) Simple application of binomial expansion formula (Knowledge of Pascal's triangle is enough).
8) Calculus: Rate of change e.g. If Rate of change of volume of water in cylinder is this, then what is the rate of change of height. Finding maxima/minima. Some harder questions like: Maximise area of rectangle of given perimeter. Simple knowledge of concavity, increasing/decreasing functions etc. is asked.
9) Calculus: Finding area by integration. Finding definite integrals. Usually you either have to split expression into partial fractions or use algebraic substitution.
10) Solving rational inequalities. Simple cautions to ensure no sign flip if you multiply equation by certain terms is often required.
edit: I forgot this. 11) Probability theory: Simple probability questions. Sometimes one may have to use conditional probability formula but never binomial/Possion/Nomal distribution formula. One type of tricky question is: 3 special dice (each can give score of 1-8) are rolled. The score is calculated as Ad1 + Bd2 + Cd3 + D where A,B,C and D are constants and d1-3 are individual score of the die. Find the constants so that : - The score ranges from 1-512 and each score has equal probability.
12) Roots of quadratic equation and their properties is assumed.
13) Simple algebraic manipulation eg. If a= x^2 + y ^2 and b = 2xy express x and y in terms of a and b.
14) Geometry. 95% of the time the question involves circle inscribed by polynomials ( most often equilateral triangle or square). You usually have to find the ratio of areas of two separate regions in the figure.
15) The hardest of all. Curve sketching. Questions range from trivial to extremely challenging. You might have to use properties of transformation of graph. Sketch graphs of completely different/complicated functions. Sketch graphs of y = f(x) +/- g(x), y = f(x)g(x) or f(x)/g(x) where you know the graph of y = f(x) and y=g(x) separately. Sometimes you might have to sketch inequalities like. -2 < y/x < 4 or 0< y/x^2 < 6pi.
MAT questions/ Senior Maths Challenge questions/ BMO questions don't have above mentioned properties. They deal with hard number theory, combinatorics, different type of algebra problems and provide very little help in brushing up above type of problems. So, what might help me to practice these type of problems? Thanks in advance.
1) you use difference of two squares, these "tricks" come automatically after having spent your life with maths.
(2023-2022)*(2023+2022) looks easier, right?
2) C1/C2. These questions are awfully common in the core modules, given that you had an A* in maths, this should be fine? If not, you could do some C1/C2 Solomon papers, they are supposed to be more challenging than past papers.
3) C2, same argument as above
4) C2/C3 same argument as above above
.
.
.
11) Average S1/S2 question
15) Curve sketching is my hobby You just have to practice them really. C3 curve sketching is not going to be enough here, the one in edexcel FP2 should be more help, and of course, you should do your own practice from PAT past papers.
I think these shouldn't mean any problem to you with an A* in maths and A in further maths. If you don't feel secure enough, practice using Solomon papers or more advanced A level questions (there should be threads on it but I've been sitting on a bus for 18 hours and I'm even too tired to go to the toilet but I can help you find them later)
Maths is all about practice and a bit of knowledge.
20. (Original post by tangotangopapa2)
What are some resources to practice maths section for the PAT other than past papers?
Hey, for PAT last year, I mainly used the STEP papers (1&2) and the MAT for algebra, trigs, graphs etc. Step questions usually have a neat trick hidden away and it might be useful to learn how to spot these.
Also 'Advanced problems in Mathematics' and 'Core problems in Mathematics' by Siklos are well worth a read.
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• Subject: Re: comparing binary numbers with strings
• Date: Sat, 27 Oct 2007 13:55:27 +1300
```
```
how can adding 2 numbers, converting it to a string, then pattern matching it to "2" do anything useful? string.match returns either a substring or nil. So, you'll get something non-nil if the result of the addition contains the digit 2 in decimal. That's certainly not what was requested?
```
e.g. take 10000 and 00100, no matching bits but
16 + 4 = 20, which contains "2"
```
without bit-wise operators in lua (e.g. XOR), you're going to need to go through the digits one at a time I'm afraid...
```
Merick wrote:
```
```Shmuel Zeigerman wrote:
```
```Merick wrote:
```
I was wondering if there was any way - other than using a for loop to iterate through each character - to use the string library to compare two strings with representations of binary numbers and tell whether or not any of the 1's in the strings are in the same position?
```
```
Something like compare("11100", "00011") would return false and compare("11100","00111") would return true
```
```
```
function compare(a,b)
a = tostring(tonumber(a) + tonumber(b))
return a:match"2"
end
```
```Dangit, that's so simple I can't believe I didn't think of it myself!
Thanks
```
```
--
Adrien de Croy - WinGate Proxy Server - http://www.wingate.com
```
• Follow-Ups:
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Physics
# Multiple Choice
PhysicsMultiple Choice
## 12.1Zeroth Law of Thermodynamics: Thermal Equilibrium
30 .
What thermodynamic principle forms the basis for our ability to measure temperature?
1. the zeroth law
2. the first law
3. the second law
4. the third law
31.
Name any two industries in which the principles of thermodynamics are used.
1. aerospace and information technology (IT) industries
2. industrial manufacturing and aerospace
3. mining and textile industries
4. mining and agriculture industries
## 12.2First law of Thermodynamics: Thermal Energy and Work
32 .
What is the value of the Boltzmann constant?
1. $k = 1.23 \times 10^{-38}\,\text{J/K}$
2. $k = 1.38 \times 10^{-23}\,\text{J/K}$
3. $k = 1.38 \times 10^{23}\,\text{J/K}$
4. $k = 1.23 \times 10^{38}\,\text{J/K}$
33 .
Which of the following involves work done BY a system?
1. increasing internal energy
2. compression
3. expansion
4. cooling
34 .
What is conserved in the first law of thermodynamics?
1. mass
2. work
3. energy
4. heat
35 .
What is the change in internal energy of a system that does 20 J of work when $Q_\text{in} = 100\,\text{J}$ and $Q_\text{out} = 50\,\text{J}$?
1. $20\,\text{J}$
2. $30\,\text{J}$
3. $50\,\text{J}$
4. $100\,\text{J}$
36 .
When does a real gas behave like an ideal gas?
1. A real gas behaves like an ideal gas at high temperature and low pressure.
2. A real gas behaves like an ideal gas at high temperature and high pressure.
3. A real gas behaves like an ideal gas at low temperature and low pressure.
4. A real gas behaves like an ideal gas at low temperature and high pressure.
## 12.3Second Law of Thermodynamics: Entropy
37 .
In an engine, what is the unused energy converted into?
1. internal energy
2. pressure
3. work
4. heat
38 .
It is natural for systems in the universe to _____ spontaneously.
1. become disordered
2. become ordered
3. produce heat
4. do work
39 .
If $Q$ is $120\,\text{J}$ and $T$ is $350\,\text{K}$, what is the change in entropy?
1. $0.343\,\text{J/K}$
2. $1.51\,\text{J/K}$
3. $2.92\,\text{J/K}$
4. $34.3\,\text{J/K}$
40.
Why does entropy increase during a spontaneous process?
1. Entropy increases because energy always transfers spontaneously from a dispersed state to a concentrated state.
2. Entropy increases because energy always transfers spontaneously from a concentrated state to a dispersed state.
3. Entropy increases because pressure always increases spontaneously.
4. Entropy increases because temperature of any system always increases spontaneously.
41.
A system consists of ice melting in a glass of water. What happens to the entropy of this system?
1. The entropy of the ice decreases, while the entropy of the water cannot be predicted without more specific information.
2. The entropy of the system remains constant.
3. The entropy of the system decreases.
4. The entropy of the system increases.
## 12.4Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators
42 .
Which equation represents the net work done by a system in a cyclic process?
1. $W = \frac{Q_{\text{c}}}{Q_{\text{h}}}$
2. $W = Q_{\text{h}} + Q_{\text{c}}$
3. $W = Eff\left(Q_{\text{c}} - Q_{\text{h}} \right)$
4. $W = Q_{\text{h}} - Q_{\text{c}}$
43.
Which of these quantities needs to be zero for efficiency to be 100 percent?
1. ΔU
2. W
3. Qh
4. Qc
44.
Which of the following always has the greatest value in a system having 80 percent thermal efficiency?
1. ΔU
2. W
3. Qh
4. Qc
45.
In the equation Q = QhQc, what does the negative sign indicate?
1. Heat transfer of energy is always negative.
2. Heat transfer can only occur in one direction.
3. Heat is directed into the system from the surroundings outside the system.
4. Heat is directed out of the system.
46.
What is the purpose of a heat pump?
1. A heat pump uses work to transfer energy by heat from a colder environment to a warmer environment.
2. A heat pump uses work to transfer energy by heat from a warmer environment to a colder environment.
3. A heat pump does work by using heat to convey energy from a colder environment to a warmer environment.
4. A heat pump does work by using heat to convey energy from a warmer environment to a colder environment.
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# Expectation of power spectrum for nonorthogonal wavelets
I'm working through "A Practical Guide to Wavelet Analysis" by Torrence and Compo, and I am confused about section 3d ("Wavelet Power Spectrum").
Let $$x_n$$ denote the signal, sampled at increments $$\delta t$$; $$\hat{x}_k$$ its Fourier transform; $$\psi$$ the (normalized) mother wavelet, so that the amplitude of the wavelet transform at scale $$s$$ centered at $$n\delta t$$ is given by $$W_n(s) = \sum_k \hat{x}_k \hat{\psi}^*(s \omega_k)e^{i\omega_k n\delta t}$$ There's some fiddling with constants here, but I understand this so far.
Then, they write that, for a white noise signal, $$\mathbb{E}|W_n(s)|^2 = N \mathbb{E}|\hat{x}_k|^2 = \sigma^2$$. Trying to work through this: \begin{align} \mathbb{E}|W_n(s)|^2 &= \mathbb{E}\left|\sum \hat{x}_k \hat{\psi}^*(s \omega_k)e^{i\omega_k n\delta t}\right|^2 \end{align} I see that if the wavelets are orthogonal, this works out nicely, where you can write $$\mathbb{E}\left|\sum \hat{x}_k \hat{\psi}^*(s \omega_k)e^{i\omega_k n\delta t}\right|^2 = \sum |\hat{\psi}(s \omega_k)|^2\mathbb{E}|\hat{x}_k|^2 = \frac{\sigma^2}{N}\sum |\hat{\psi}(s \omega_k)|^2$$ However, the authors clearly state they are not working with orthogonal wavelets specifically here, so I am unsure how they get their white noise result.
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# 2.6.11 The solution of the problem of the collection of
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# Description
By homogeneous rink weighing 4 kN applied horizontal force F = 50 N and a pair of forces with the moment M = 20 Nm. Determine the smallest radius R of the roller in which it will roll to the left, if the coefficient of rolling friction δ = 0.005 m and OA = 0,6R.
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Betting Strategies and House Edge Explained
In this article I’m going to discuss the ”house advantage” sometimes called the house edge and betting strategies. Every casino game has a house edge. What is the house edge? It’s the precentage advantage the house has.
With slot machines, online casinos like to advertise the payout percentage, called payback as a way of differentiatiating themselves from other operators and to entice players to try out their games. Here’s an illustration of the house edge for some of the popular casino games:
Table Game House Edge Three Card Poker (Ante/Pair Plus) 3.37%/7.28% Roulette 5.26% Let It Ride 3.51% Pai Gow Poker 1.46% Blackjack 1.50% Craps (Pass/Come) 1.41%
In most slots machines the house edge is usually around 0.1%, a very loose slot machine might payback 99.9% to 15%. Tight machines normally payback 85%. Why the sudden interest in the house edge? Over a period of time the house edge makes sure that you part ways with your hard-earned cash. Probability theory proves this.
Let’s Have a Look at the Following Example:
• You start playing with \$100. With each spin you bet \$1.
• The machine pays 97%.
• With each spin you are getting the average return.
• You don’t cash out.
How Long Will Your Money Last?
You won’t have any cash left after 4,500 spins. Initially this might seem like a lot of spins but it’s not. If you consider how little time it takes to press the spin button you would probably lose all your money within a few hours. If you increased your bet to \$5 per spin, your money will only last 30 minutes. This is what grinding means.
Betting Strategy Slots Machines
Let’s have a look at the dollar a minute test. Twenty dollars will last an average slots machine player 20 minutes. Consider yourself ahead of the game if you can do better than this. When playing a game of chance you can apply two betting strategies. A fixed betting strategy where each wager is the same and a progressive bet strategy which varies on whether the player is winning or losing.
When playing slot machines I propose a progressive bet strategy. In other words when you’re winning bet more and if you’re losing bet less. If you’re winning increase the multiplier amount or the number of lines.
Let us assume you decided to stay at the machine and prepared to vary the multiplier. If you are at the maximum multiplier and you haven’t won anything as big or bigger than your current bet, drop down to your preferred multiplier. If you are at your preferred multiplier and you haven’t won anything significant then consider leaving the machine. If you haven’t won anything after ten spins, leave the machine. Keep in mind that this variant will not work on slot machines that force you to play the maximum number of lines.
Points To Remember
• Increase your bet when you are winning and lower it when you are losing.
• There is no fixed way to beat a slot machine. The probability of you winning a large jackpot is probably worse than being struck twice by lightning.
• The best you can do is to prolong your playing time and to get the most entertainment out of each dollar spent.
• When playing slots always keep a close eye on your bankroll and never chase your losses. If you lost a lot of money stop playing immediately otherwise you’ll lose even more.
I’m not saying that the above given tips is a given recipe to win when playing slots machines. But, I believe it will increase your winning chances.
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# NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY
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1 NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc and fppf topologies on schemes over a base scheme. In section 1 we assemble, without any proofs, various properties of flat and faithfully flat morphisms from [Vak06a, Vak06b], [Vis08], [Har77], [EH00] and [Eis95]. Section 2 then defines the fppf and fpqc topologies and states some results about the fpqc topology without proof, following [Vis08]. Note that we will, unfortunately, consider schemes to be locally ringed spaces that are locally affine, rather than functors from rings to sets. 1. Flat and Faithfully Flat Morphisms 1.1. Definitions and Basic Properties. We begin with the algebraic definition of flatness: Definition 1.1. We say that an A-module M is flat if for every short exact sequence 0 N N N 0 of A-modules, tensoring with M gives an exact sequence 0 M A N M A N M A N 0, i.e. M A is an exact functor. Equivalently, since M A is always right-exact, M A preserves injectivity. Since Tor A(M, ) is the left derived functor of M A, it is also equivalent to say that Tor A 1 (M, N) = 0 for all A-modules N. Lemma 1.2. An A-module M is flat if and only if for every prime ideal p of A, the localization M p is a flat A p -module. Definition 1.3. A morphism of commutative rings f : A B is flat if the induced A-module structure on B makes B a flat A-module. Next, we extend this to schemes as follows: Definition 1.4. A morphism of schemes f : X Y is flat if for every x X, the stalk O X,x is a flat O Y,f(x) -module. Equivalently (by lemma 1.2), for every open affine U Y and every open affine V f 1 (U), the restricted map V U corresponds to a flat ring homomorphism O Y (U) O X (V ). Definition 1.5. A quasicoherent sheaf F on X is flat if for every x X, F x is a flat O X,x -module, and more generally for f : X Y we say that F is flat over Y if for every x X, F x is a flat O Y,f(x) -module. Lemma 1.6. Open immersions are flat, composites of flat morphisms are flat, and pullbacks of flat morphisms are flat. Date: March 15,
2 2 RUNE HAUGSENG Definition 1.7. A morphism of schemes is faithfully flat if it is flat and surjective. A homomorphism of rings A B is faithfully flat if the corresponding morphism of schemes Spec B Spec A is faithfully flat. Faithful flatness for affine schemes can also be characterized algebraically: Lemma 1.8. The following are equivalent for an A-algebra B: (i) B is faithfully flat over A. (ii) A sequence of A-modules is exact if and only if the corresponding sequence of B-modules obtained by applying B A is exact. (iii) An A-module homomorphism M N is injective if and only if B A M B A N is injective. (iv) B is flat over A, and if M is an A-module satisfying M A B = 0 then M = 0. (v) B is flat over A, and mb B for all maximal ideals m of A. Question 1.9. I think we can define flat morphisms of Z-functors as follows: a morphism of Z-functors f : X Y is flat if for every morphism Spec R Y and every morphism from an affine scheme Spec S to the pullback X Y Spec R, the composite Spec S Spec R corresponds to a flat ring homomorphism R S Algebraic Properties of Flatness. Theorem M is flat if and only if Tor A 1 (M, A/I) = 0 for all finitely generated ideals I of A. Proof. See [Eis95, p. 163]. Theorem A coherent module over a Noetherian local ring is flat if and only if it is free. Corollary A coherent sheaf over a locally Noetherian scheme is flat if and only if it is locally free Topological Properties of Flatness. With some additional hypotheses, a flat morphism behaves well topologically. One possible requirement is that the morphism be locally of finite presentation: Definition An A-algebra B is finitely presented if it is the quotient of a polynomial ring over A with finitely many generators by a finitely generated ideal. Definition A morphism of schemes f : X Y is locally of finite presentation if for every x X there are affine open neighbourhoods U Y of f(x) and V f 1 (U) of x such that O X (V ) is finitely presented over O Y (U). Proposition A flat morphism that is locally of finite presentation is open. Alternatively, we get a weaker conclusion if the morphism is flat, surjective and quasicompact: Definition A morphism of schemes is quasicompact if the inverse image of any quasicompact open set is quasicompact. Proposition If f : X Y is a faithfully flat and quasicompact morphism of schemes, then U Y is open in Y if and only if f 1 (U) is open in X, i.e. the topology of Y is induced from that of X via f Flat Families. Flat morphisms are a good notion of continuously varying families of schemes their fibres are related, as the following results illustrate:
3 NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY 3 Proposition A finite morphism of schemes f : X Y, where Y is an integral scheme, is flat if and only if dim κ(y) f (O X ) y κ(y) is constant with respect to y Y. Theorem Suppose f : X Y is a projective morphism with Y locally Noetherian, and F is a coherent sheaf on X with f F flat over Y. Then the Euler characteristic of F Xy, where X y is the fibre of f over y Y, is locally constant with respect to y. Corollary If f : X Y is a flat projective morphism with Y locally Noetherian then the Euler characteristic of O X Xy is locally constant with respect to y Y. If Y is also integral then this is also a sufficient criterion for flatness. Proposition Suppose f : X Y is a flat morphism of schemes of finite type over a field. Then for any y Y and x X y, we have dim O X,x = dim O Xy,x + dim O Y,y Flat Limits. For a flat family of projective schemes over a nice punctured curve we can take the limit and obtain a unique flat family over the entire curve. This is a corollary of the following characterization of flatness over a curve: Proposition Suppose f : X Y is a morphism of schemes with Y integral, regular, and of dimension 1. Then f is flat if and only if every associated point of X, meaning a point x X such that the maximal ideal m X,x O X,x consists of zero divisors, maps to the generic point of Y. Corollary Suppose Y is a regular integral scheme of dimension 1 and X is a closed subscheme of P n Y \{p} = Pn Spec Z Y \ {p}, where p is a closed point of Y, such that X is flat over Y \ {p}. Then there exists a unique closed subscheme X of P n Y that is flat over Y, such that X P n Y \{p} = X. (This does not work in higher dimensions.) 2. The fpqc and fppf Topologies The definition of the fppf topology is straightforward: Definition 2.1. The fppf topology on Sch /S is defined as follows: the coverings of X S are collections U i X of flat maps locally of finite presentation over S such that i U i X is surjective as a map of sets. To define the fpqc topology, we first introduce the concept of an fpqc morphism, following Vistoli [Vis08, section 2.3]. We begin with the following lemma: Lemma 2.2. The following are equivalent for a surjective morphism of schemes f : X Y : (i) Every quasicompact open subset of Y is the image of a quasicompact open subset of X. (ii) Y has an open affine cover each element of which is the image of a quasicompact open subset of X. (iii) Every point x X has an open neighbourhood U such that f(u) is open in Y and the restriction f U : U f(u) is a quasicompact morphism. (iv) Every point of X has a quasicompact open neighbourhood whose image in Y is open and affine.
4 4 RUNE HAUGSENG Definition 2.3. A morphism of schemes is fpqc if it is faithfully flat and satisfies the equivalent conditions of the preceding lemma. fpqc morphisms have the following nice properties: (i) The composite of fpqc morphisms is fpqc. (ii) If f : X Y is a morphism and {V i } is an open cover of Y such that f f 1 (V i) : f 1 (V i ) V i are fpqc, then f is fpqc. (iii) Open faithfully flat morphisms are fpqc. (iv) Faithfully flat morphisms locally of finite presentation are fpqc. (v) Any base change of an fpqc morphism is fpqc (vi) If f : X Y is fpqc then a subset U of Y is open if and only if f 1 (U) is open in X. Definition 2.4. The fpqc topology on Sch /S is defined as follows: the coverings of X S are collections U i X of morphisms such that i U i X is fpqc. Lemma 2.5. The fpqc topology is a Grothendieck topology, i.e.: (i) If f : X Y is an isomorphism over S then {f : X Y } is a cover. (ii) If {U i Y } is a cover and X Y is a morphism over S, then {U i Y X X} is a cover. (iii) If {U i X} is a cover and {V ij U i } are covers, then {V ij U i X} is a cover. Proof. (i) An isomorphism of schemes is fpqc. (ii) Follows from (v) above. (iii) Follows from (i) and (ii) above. Remark. It is possible to define a topology on Sch /S by taking covers of X S to be collections of morphisms U i X such that i U i X is faithfully flat and quasicompact. However, not only does this not include the Zariski open covers, it is not even subcanonical. Remark. By (iv) above, fppf covers are fpqc covers. Since étale covers are fppf covers and Zariski covers are étale covers, it follows that the fpqc topology is finer than all of these. Remark. Goerss [Goe] has a different (but hopefully equivalent) definition: He defines an fpqc cover of an affine scheme U to be a finite collection {U i U} of flat morphisms with i U i U surjective, and in general an fpqc cover of a scheme X to be a finite collection of morphisms {V i X} such that for every open affine U X the pullback {V i X U U} is an fpqc cover of U. (Why only finite covers?) Many properties of morphisms of schemes are local in the fpqc topology: Proposition 2.6. Suppose f : X Y is a morphism of schemes and {U i Y } is an fpqc cover. If for all i, U i Y X U i is separated, quasicompact, locally of finite presentation, proper, affine, finite, flat, smooth, unramified, étale, an embedding, or a closed embedding, then so is f. This follows from the local nature of these properties, plus the following familiar result:
5 NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY 5 Proposition 2.7. Suppose we have a pullback diagram of schemes X Y Z f Z g g X f Y where f is faithfully flat and either quasicompact or locally of finite presentation. Then g is separated, quasicompact, locally of finite presentation, proper, affine, finite, flat, smooth, unramified, étale, an embedding, or a closed embedding if g is. Theorem 2.8. A representable functor Sch op /S Set is a sheaf in the fpqc topology, i.e. the fpqc topology is subcanonical. It follows that the fppf and étale topologies are also subcanonical, since they are coarser than the fpqc topology. This is proved in [Vis08] using the following criterion: Proposition 2.9. Suppose a presheaf F on Sch /S is a Zariski sheaf and that for V U any faithfully flat morphism of affine schemes over S F(U) F(V ) F(V U V ) is an equalizer diagram. Then F is an fpqc sheaf. References [Vak06a] Ravi Vakil, Foundations of algebraic geometry classes 41 and 42 (2006), available at [Vak06b], Foundations of algebraic geometry classes 43 and 44 (2006), available at http: //math.stanford.edu/~vakil/ /. [Vis08] Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (2008), available at [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, Graduate Texts in Mathematics, No. 52. [EH00] David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, [Eis95] David Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, [Goe] Paul G. Goerss, Quasi-coherent sheaves on the moduli stack of formal groups, available at Department of Mathematics, MIT, Cambridge, Massachusetts, USA address: URL:
### CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES
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This page is walking to define how to transform 140 kg to lbs. If you’re on this page and also wondering just how to perform this conversion, we have actually the answers. Please bear in mind that in our kilogram come pounds conversion guide here, we use worldwide avoirdupois pound, i beg your pardon is the most widely offered today. 140 kg by the way, is same to 308.65 pounds.
You are watching: How many pounds is 140 kg
## 140 KG to Lbs Calculator
If you want to try your hand in ~ calculating 140 kg right into lbs. Here is ours calculator.
This 140 kg to lbs calculator is easier to usage than you can think. All the is needed is to enter the number in kilograms, in this instance 140 kg. As soon as you have actually typed 140 kg you will check out its indistinguishable in pounds (lbs).
Once you have actually the figures duplicated down, click the reset switch so you have the right to make an additional calculation.
While the emphasis here is on 140 kg, you can use other kilograms. This technique is easy, quick and also reliable.
Kilogram abbreviation: “kg”, Pound abbreviation: “lb. Lbm. Or lbs”.
## 140 KG to Lbs – Unit Definition
What is a Kilogram? A kilogram (also kilograms and abbreviated together kg), is a unit the mass. It is component of the Standard global (SI) system of Units. A solitary kilogram is same to 2.20 lbs. 1 kilogram is equal to 1000 grams, and 1 gold bar is equivalent to 1 kg.
A gram is same to 1/1000 the a kilogram, and its SI symbol is K, and also kilo may additionally be used. The simplest method to think the a kilogram is the it is a measure up of how heavy something is. The kilogram is supplied in nations that have embraced the metric system.
What is a Pound? The lb (abbreviation lb. Lbm. Or lbs.) is offered to measure up mass. That is part of the United claims customary, imperial and other measure up systems. Throughout history there have been different species of pounds such together the Troy, London, Tower and more.
But the most generally used is the avoirdupois pound. 2.20462 pounds is same to 1 kilogram. Strictly speaking, the pound and kilogram refer to an object’s mass, however both are likewise used to describe an object’s weight. The abbreviation “lb” comes from the roman libra, which was offered to measure mass. Today, the lb. And lbs. Are still used.
## 140 KG to Lbs – conversion Chart
A kilograms to pounds converter is nice and quick, however what is even quicker is a 140 KG to Lbs counter chart. V this guide, you have the right to scan and find the conversions friend need. Due to the fact that the numbers space all there, it’s straightforward to use.
Unit ConversionKilograms (KG)Pounds (Lbs)
140 KG come Lbs140 KG =308.65 Lbs
## 140 KG to Lbs
Whether friend opt because that a 140 kilograms to pounds conversion graph or a 140 kg to lbs converter, over there is no questioning the require for them. In countless parts that the world, kilogram is the unit provided to measure weight and also mass. From world to cars to daily items, kg is the standard.
That is every well and an excellent if you are conversant with the metric system, but what if you space not? In the united state for instance, the unit of measure up for load is pounds. If you find an object that weighs 140 kilograms (kg), the is difficult to paint a mental snapshot of how hefty that is.
But if you understand the pound identical of 140 kg, the is simpler to get a master of how heavy that object is. Another advantage of knowing just how that works is you have the right to use the same method for any type of kg to lbs. Conversion.
Yes, our focus here is on 140 kg, but knowing the identical in pounds and also how that is done is important. The is unlikely you will simply need to convert 140 kg to lbs (pounds).
The time will certainly come once you will need to transform 5, 15, 25 kg and so on come pounds (lbs), so learning the process helps.
## Convert 140 KG to LBS
How to transform 140 kg to lbs? The simplest method to find how many pounds is 140 kg is to divide the kilogram worth by 0.45359237.
Using 140 kg as our example, it will look like 140 / 0.45359237. The result can be created down together follows:
140 kilograms is equal to 308.65 pounds140 kg is same to 308.65 lb.140 kg is same to 308.65 lbs.
You can write these figures in any of the versions noted above. No matter what variation you choose, the does not adjust the outcome.
As you deserve to see, this an approach is very easy. As lengthy as you monitor this method you will get the right results. If you want more accurate outcomes down come the decimals, you should try our 140 kg to lbs converter.
But if you just in search of a rounded turn off figure, girlfriend can also use the 140 kg to lbs conversion chart above. If messing roughly with numbers and also multiplying and also dividing are not her thing, our 140 kg to lbs counter chart can do it for you.
A most the confusion with these dimensions stem native not learning the counter factor. Currently with this 140 kg to lbs conversion guide you know precisely the number required to do the conversion, so you won’t make a mistake.
Popular KG to Lbs (Kilograms come Pounds) conversions:
## 140 KG is same to How numerous Lbs
Now let united state look in ~ how many pounds there space in 140 kg. A solitary kilogram is equal to 2.20462 lbs. Or rounded off, 2.2. Through 140 kg together our example, it is equal to 308.65 pounds (lbs). For something more precise friend will want to use our 140 kg come lbs converter instead.
As pointed out earlier, the pounds we are using for comparison through the 140 kg is the avoirdupois pound. This is the unit used for weighing people and also other objects in the US and other countries today.
However, there are other pounds (lbs) i beg your pardon you may come across. For her reference, we have included the adhering to here:
1 kg is same to 2.1434 London lbs.1 kg is same to 2.2863 Merchant’s lbs.1 kg is same to 2 Metric lbs.1 kg is equal to 2.8578 Tower lbs.1 kg is same to 2.6792 trojan lbs.
These are simply some the the other versions that pounds that you may see. However, these dimensions are legacy or no longer widely used. Us have included them right here simply to finish your knowledge.
See more: 31 Weeks From Today ? What Will Be The Date 31 Weeks From Today
There room a most times as soon as you’ll have to deal with kg and also lbs (pounds), but with this information and our 140 kg to lbs switch guide, you will certainly never have to be confused.
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Examples of com.lightcrafts.mediax.jai.Interpolation
• com.lightcrafts.mediax.jai.Interpolation
An object encapsulating a particular algorithm for image interpolation (resampling). An Interpolation captures the notion of performing sampling on a regular grid of pixels using a local neighborhood. It is intended to be used by operations that resample their sources, including affine mapping and warping.
Resampling is the action of computing a pixel value at a possibly non-integral position of an image. The image defines pixel values at integer lattice points, and it is up to the resampler to produce a reasonable value for positions not falling on the lattice. A number of techniques are used in practice, the most common being nearest-neighbor, which simply takes the value of the closest lattice point; bilinear, which interpolates linearly between the four closest lattice points; and bicubic, which applies a piecewise polynomial function to a 4x4 neighborhood of nearby points. The area over which a resampling function needs to be computed is referred to as its support; thus the standard resampling functions have supports of 1, 4, and 16 pixels respectively. Mathematically, the ideal resampling function for a band-limited image (one containing no energy above a given frequency) is the sinc function, equal to sin(x)/x. This has practical limitations, in particular its infinite support, which lead to the use of the standard approximations described above.
Other interpolation functions may be required to solve problems other than the resampling of band-limited image data. When shrinking an image, it is common to use a function that combines area averaging with resampling in order to remove undesirable high frequencies as part of the interpolation process. Other application areas may use interpolating functions that operate under other assumptions about image data, such as taking the maximum value of a 2x2 neighborhood. The interpolation class provides a framework in which a variety of interpolation schemes may be expressed.
Many interpolations are separable, that is, they may be equivalently rewritten as a horizontal interpolation followed by a vertical one (or vice versa). In practice, some precision may be lost by the rounding and truncation that takes place between the passes. The Interpolation class assumes separability and implements all vertical interpolation methods in terms of corresponding horizontal methods, and defines isSeparable() to return true. A subclass may override these methods to provide distinct implementations of horizontal and vertical interpolation. Some subclasses may implement the two-dimensional interpolation methods directly, yielding more precise results, while others may implement these using a two-pass approach.
A minimal Interpolation subclass must call the Interpolation constructor (super()) and then set at least the following fields.
It must also implement at least the following methods.
int interpolateH(int[] samples, int xfrac) float interpolateH(float[] samples, float xfrac) double interpolateH(double[] samples, float xfrac)
All other methods are defined in terms of these methods for ease of implementation of new Interpolation subclasses.
Since interpolation is generally performed for every pixel of a destination image, efficiency is important. In particular, passing source samples by means of arrays is likely to be unacceptably slow. Accordingly, methods are provided for the common cases of 2x1, 1x2, 4x1, 1x4, 2x2, and 4x4 input grids. These methods are defined in the superclass to package their arguments into arrays and forward the call to the array versions, in order to simplify implementation. They should be called only on Interpolation objects with the correct width and height. In other words, an implementor of an Interpolation subclass may implement "interpolateH(int s0, int s1, int xfrac)" assuming that the interpolation width is in fact equal to 2, and does not need to enforce this constraint.
The fractional position of interpolation (xfrac, yfrac) is always between 0.0 and 1.0 (not including 1.0). For integral image data, the fraction is represented as a scaled integer between 0 and 2n - 1, where n is a small integer. The value of n in the horizontal and vertical directions may be obtained by calling getSubsampleBitsH() and getSubsampleBitsV(). In general, code that makes use of an externally-provided Interpolation object must query that object to determine its desired positional precision.
For float and double images, a float between 0.0F and 1.0F (not including 1.0F) is used as a positional specifier in the interest of greater accuracy.
It is important to understand that the subsampleBits precision is used only to indicate the scaling implicit in the fractional locations (xfrac, yfrac) for integral image data types. For example, for subsampleBitsH=8, xfrac must lie between 0 and 255 inclusive. An implementation is not required to actually quantize its interpolation coefficients to match the specified subsampling precision.
The diagrams below illustrate the pixels involved in one-dimensional interpolation. Point s0 is the interpolation kernel key position. xfrac and yfrac, indicated by the dots, represent the point of interpolation between two pixels. This value lies between 0.0 and 1.0 exclusive for floating point and 0 and 2subsampleBits exclusive for integer interpolations.
Horizontal Vertical s_ s0 . s1 s2 s_ ^ xfrac s0 .< yfrac s1 s2
The diagram below illustrates the pixels involved in two-dimensional interpolation. Point s00 is the interpolation kernel key position.
s__ s_0 s_1 s_2 s0_ s00 s01 s02 . < yfrac s1_ s10 s11 s12 s2_ s20 s21 s22 ^ xfrac
The subclasses of Interpolation include InterpolationNearest, InterpolationBilinear, InterpolationBicubic, and InterpolationBicubic2 (a variant defined by a different polynomial function). These subclasses are marked 'final,' so users may identify them by name (using 'instanceof') and write specialized code for them. This may also allow inlining to occur on some virtual machines. These classes do provide correct, if less than optimal code for performing their interpolations, so it is possible to use any Interpolation object in a generic manner. The Sun-provided InterpolationBilinear and InterpolationBicubic classes provide a more optimal implementation while using the same semantics.
The InterpolationTable class is a subclass of Interpolation that divides the set of subsample positions into a fixed number of "bins" and stores a kernel for each bin. InterpolationBicubic and InterpolationBicubic2 are implemented in terms of InterpolationTable since a direct implementation is very expensive. @see InterpolationNearest @see InterpolationBilinear @see InterpolationBicubic @see InterpolationBicubic2 @see InterpolationTable
69707172737475767778798081828384858687888990 return java.awt.Image.UndefinedProperty; } ROI srcROI = (ROI)property; // Retrieve the Interpolation object. Interpolation interp = (Interpolation)pb.getObjectParameter(1); // Determine the effective source bounds. Rectangle srcBounds = null; PlanarImage dst = op.getRendering(); if (dst instanceof GeometricOpImage && ((GeometricOpImage)dst).getBorderExtender() == null) { srcBounds = new Rectangle(src.getMinX() + interp.getLeftPadding(), src.getMinY() + interp.getTopPadding(), src.getWidth() - interp.getWidth() + 1, src.getHeight() - interp.getHeight() + 1); } else { srcBounds = new Rectangle(src.getMinX(), src.getMinY(), src.getWidth(), src.getHeight());
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68697071727374757677 } int scaleX = paramBlock.getIntParameter(0); int scaleY = paramBlock.getIntParameter(1); float [] qsFilter = (float [])paramBlock.getObjectParameter(2); Interpolation interp = (Interpolation)paramBlock.getObjectParameter(3); return new MlibFilteredSubsampleOpImage(source, extender, (Map)renderHints, layout, scaleX, scaleY, qsFilter, interp); } // create
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69707172737475767778798081828384858687888990 return java.awt.Image.UndefinedProperty; } ROI srcROI = (ROI)property; // Retrieve the Interpolation object. Interpolation interp = (Interpolation)pb.getObjectParameter(3); // Determine the effective source bounds. Rectangle srcBounds = null; PlanarImage dst = op.getRendering(); if (dst instanceof GeometricOpImage && ((GeometricOpImage)dst).getBorderExtender() == null) { srcBounds = new Rectangle(src.getMinX() + interp.getLeftPadding(), src.getMinY() + interp.getTopPadding(), src.getWidth() - interp.getWidth() + 1, src.getHeight() - interp.getHeight() + 1); } else { srcBounds = new Rectangle(src.getMinX(), src.getMinY(), src.getWidth(), src.getHeight());
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6162636465666768697071 ImageLayout layout = RIFUtil.getImageLayoutHint(hints); // Get operation parameters. AffineTransform transform = (AffineTransform)args.getObjectParameter(0); Interpolation interp = (Interpolation)args.getObjectParameter(1); double[] backgroundValues = (double[])args.getObjectParameter(2); RenderedImage source = args.getRenderedSource(0); if (!MediaLibAccessor.isMediaLibCompatible(args, layout) ||
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9899100101102103104105106107108 AffineTransform xform = AffineTransform.getScaleInstance(xMagnification, yMagnification); RenderingHints formatHints = new RenderingHints(JAI.KEY_BORDER_EXTENDER, BorderExtender.createInstance(BorderExtender.BORDER_COPY)); Interpolation interp = Interpolation.getInstance(Interpolation.INTERP_BICUBIC_2); ParameterBlock params = new ParameterBlock(); params.addSource(printImage); params.add(xform); params.add(interp); printImage = JAI.create("Affine", params, formatHints);
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126127128129130131132133134135136 if (scale > 0) { AffineTransform transform = AffineTransform.getScaleInstance(scale, scale); transform.preConcatenate(AffineTransform.getTranslateInstance(imageBounds.x, imageBounds.y)); RenderingHints formatHints = new RenderingHints(JAI.KEY_BORDER_EXTENDER, BorderExtender.createInstance(BorderExtender.BORDER_COPY)); Interpolation interp = Interpolation.getInstance(Interpolation.INTERP_BILINEAR); ParameterBlock params = new ParameterBlock(); params.addSource(image); params.add(transform); params.add(interp); RenderedOp scaled = JAI.create("Affine", params, formatHints);
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102103104105106107108109110111112 AffineTransform.getScaleInstance(scale, scale); RenderingHints extenderHint = new RenderingHints( JAI.KEY_BORDER_EXTENDER, BorderExtender.createInstance(BorderExtender.BORDER_COPY) ); Interpolation interp = Interpolation.getInstance( Interpolation.INTERP_BILINEAR ); image = AffineDescriptor.create( image, transform, interp, null, extenderHint ).getAsBufferedImage();
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7879808182838485868788 // Retrieve the transpose type and create a nearest neighbor // Interpolation object. TransposeType transposeType = (TransposeType)pb.getObjectParameter(0); Interpolation interp = Interpolation.getInstance(Interpolation.INTERP_NEAREST); // Return the transposed ROI. return new ROI(JAI.create("transpose", srcROI.getAsImage(), transposeType));
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69707172737475767778798081828384858687888990919293949596979899 return java.awt.Image.UndefinedProperty; } ROI srcROI = (ROI)property; // Retrieve the Interpolation object. Interpolation interp = (Interpolation)pb.getObjectParameter(4); // Determine the effective source bounds. Rectangle srcBounds = null; PlanarImage dst = op.getRendering(); if (dst instanceof GeometricOpImage && ((GeometricOpImage)dst).getBorderExtender() == null) { srcBounds = new Rectangle(src.getMinX() + interp.getLeftPadding(), src.getMinY() + interp.getTopPadding(), src.getWidth() - interp.getWidth() + 1, src.getHeight() - interp.getHeight() + 1); } else { srcBounds = new Rectangle(src.getMinX(), src.getMinY(), src.getWidth(), src.getHeight()); } // Set the nearest neighbor interpolation object. Interpolation interpNN = interp instanceof InterpolationNearest ? interp : Interpolation.getInstance(Interpolation.INTERP_NEAREST); // Retrieve the operation parameters. float sv = pb.getFloatParameter(0);
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69707172737475767778798081828384858687888990 return java.awt.Image.UndefinedProperty; } ROI srcROI = (ROI)property; // Retrieve the Interpolation object. Interpolation interp = (Interpolation)pb.getObjectParameter(2); // Determine the effective source bounds. Rectangle srcBounds = null; PlanarImage dst = op.getRendering(); if (dst instanceof GeometricOpImage && ((GeometricOpImage)dst).getBorderExtender() == null) { srcBounds = new Rectangle(src.getMinX() + interp.getLeftPadding(), src.getMinY() + interp.getTopPadding(), src.getWidth() - interp.getWidth() + 1, src.getHeight() - interp.getHeight() + 1); } else { srcBounds = new Rectangle(src.getMinX(), src.getMinY(), src.getWidth(), src.getHeight());
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Prove that if $f$ differentiable in $x^*$ and $f(x^*)=0$, then $\liminf_{x\to x^*}\frac{|f(x)|}{||x-x^*||}=0$ if $n>1$
Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be differentiable at $x^*$ and $f(x^*)=0$. Prove that if $n>1$, then $$\liminf_{x\to x^*}\frac{|f(x)|}{||x-x^*||}=0.$$ Is this true for $n=1$?
I know that because $f$ is differentiable in $x^*$, there is a linear transformation $L\colon \mathbb{R}^n\to\mathbb{R}$ such that $$0=\lim_{x\to x^*}\frac{|f(x)-f(x^*)-L(x-x^*)|}{||x-x^*||}=\lim_{x\to x^*}\frac{|f(x)-L(x-x^*)|}{||x-x^*||}.$$ How to proceed from there? And what is special about $n=1$?
Why the difference between the cases $n=1$ and $n>1$?
For $n=1$, the kernel of a linear map can be reduced to the zero vector. This isn't the case for $n>1$.
The result is true for $n>1$
In that case $L$ is a linear form and its kernel is not reduced to the zero vector. Take $a \in \ker L \setminus \{0\}$. For $m \in \mathbb N$ and $x_m = \frac{a}{m} + x^*$, you have $L(x_m-x^*) = 0$ and $\lim\limits_{m \to \infty} \dfrac{|f(x_m)-L(x_m-x^*)|}{||x_m-x^*||}= \dfrac{|f(x_m)|}{||x_m-x^*||}=0$. As $\lim\limits_{m \to \infty} x_m = x^*$ you have $\lim \inf_{x\to x^*}\dfrac{|f(x)|}{||x-x*||}=0$.
The result is wrong for $n=1$
Just take $f(x)=x$ and $x^*=0$. You have $\dfrac{|f(x)|}{||x-x*||}=1$ for all $x \in \mathbb R \setminus \{0\}$.
• What is the role of $a$ in the proof? Aug 11, 2018 at 9:28
• @GNUSupporter I updated a typo. Aug 11, 2018 at 9:30
• $n$ is fixed as the dimension of $\mathbb{R}^n$ so you might want to choose another index for your sequence. ANd after you took the limes, there is still an $n$ apperaing, that is quite confusing. Where is the difference between a linear Transformation and a linear form? Which part of the proof doesn't work for $n=1$? Why can you interchange the limes and the absolut value? Aug 11, 2018 at 9:33
• @mathstackuser I changed $n$ into $m$ for readability. Aug 11, 2018 at 9:35
• A linear form is a linear map from a vector space ($\mathbb R^n$ in your case) to its field of scalars ( $\mathbb R$ in your case). For $n=1$ a linear form may have a kernel reduced to the zero vector. This is not possible for $n>1$. Aug 11, 2018 at 9:39
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# Tag: Decision Making
## Moving the Finish Line: The Goal Gradient Hypothesis
Imagine a sprinter running an Olympic race. He’s competing in the 1600 meter run.
The first two laps he runs at a steady but hard pace, trying to keep himself consistently near the head, or at least the middle, of the pack, hoping not to fall too far behind while also conserving energy for the whole race.
About 800 meters in, he feels himself start to fatigue and slow. At 1000 meters, he feels himself consciously expending less energy. At 1200, he’s convinced that he didn’t train enough.
Now watch him approach the last 100 meters, the “mad dash” for the finish. He’s been running what would be an all-out sprint to us mortals for 1500 meters, and yet what happens now, as he feels himself neck and neck with his competitors, the finish line in sight?
He speeds up. That energy drag is done. The goal is right there, and all he needs is one last push. So he pushes.
This is called the Goal Gradient Effect, or more precisely, the Goal Gradient Hypothesis. Its effect on biological creatures is not just a feeling, but a real and measurable thing.
## The Math of Human Behavior
The first person to try explaining the goal gradient hypothesis was an early behavioral psychologist named Clark L. Hull.
As with other animals, when it came to humans, Hull was a pretty hardcore behaviorist, thinking that human behavior could eventually be reduced to mathematical prediction based on rewards and conditioning. As insane as this sounds now, he had a neat mathematical formula for human behavior:
Some of his ideas eventually came to be seen as extremely limiting Procrustean Bed type models of human behavior, but the Goal Gradient Hypothesis was replicated many times over the years.
Hull himself wrote papers with titles like The Goal-Gradient Hypothesis and Maze Learning to explore the effect of the idea in rats. As Hull put it, “...animals in traversing a maze will move at a progressively more rapid pace as the goal is approached.” Just like the runner above.
Most of the work Hull focused on were animals rather than humans, showing somewhat unequivocally that in the context of approaching a reward, the animals did seem to speed up as the goal approached, enticed by the end of the maze. The idea was, however, resurrected in the human realm in 2006 with a paper entitled The Goal-Gradient Hypothesis Resurrected: Purchase Acceleration, Illusionary Goal Progress, and Customer Retention. (link)
The paper examined consumer behavior in the “goal gradient” sense and found, alas, it wasn’t just rats that felt the tug of the “end of the race” — we do too. Examining a few different measurable areas of human behavior, the researchers found that consumers would work harder to earn incentives as the goal came within sight and that after the reward was earned, they’d slow down their efforts:
We found that members of a café RP accelerated their coffee purchases as they progressed toward earning a free coffee. The goal-gradient effect also generalized to a very different incentive system, in which shorter goal distance led members to visit a song-rating Web site more frequently, rate more songs during each visit, and persist longer in the rating effort. Importantly, in both incentive systems, we observed the phenomenon of post-reward resetting, whereby customers who accelerated toward their first reward exhibited a slowdown in their efforts when they began work (and subsequently accelerated) toward their second reward. To the best of our knowledge, this article is the first to demonstrate unequivocal, systematic behavioural goal gradients in the context of the human psychology of rewards.
Fascinating.
## Putting The Goal Gradient Hypothesis to Work
If we’re to take the idea seriously, the Goal Gradient Hypothesis has some interesting implications for leaders and decision-makers.
The first and most important is probably that incentive structures should take the idea into account. This is a fairly intuitive (but often unrecognized) idea: Far-away rewards are much less motivating than near term ones. Given a chance to earn \$1,000 at the end of this month, and each after that, or \$12,000 at the end of the year, which would you be more likely to work hard for?
What if I pushed it back even more but gave you some “interest” to compensate: Would you work harder for the potential to earn \$90,000 five years from now or to earn \$1,000 this month, followed by \$1,000 the following month, and so on, every single month during five year period?
Companies like Nucor take the idea seriously: They pay bonuses to lower-level employees based on monthly production, not letting it wait until the end of the year. Essentially, the end of the maze happens every 30 days rather than once per year. The time between doing the work and the reward is shortened.
The other takeaway comes to consumer behavior, as referenced in the marketing paper. If you’re offering rewards for a specific action from your customer, do you reward them sooner, or later?
The answer is almost always going to be “sooner.” In fact, the effect may be strong enough that you can get away with less total rewards by increasing their velocity.
Lastly, we might be able to harness the Hypothesis in our personal lives.
Let’s say we want to start reading more. Do we set a goal to read 52 books this year and hold ourselves accountable, or to read 1 book a week? What about 25 pages per day?
Not only does moving the goalposts forward tend to increase our motivation, but we repeatedly prove to ourselves that we’re capable of accomplishing them. This is classic behavioral psychology: Instant rewards rather than delayed. (Even if they’re psychological.) Not only that, but it forces us to avoid procrastination — leaving 35 books to be read in the last two months of the year, for example.
Those three seem like useful lessons, but here’s a challenge: Try synthesizing a new rule or idea of your own, combining the Goal Gradient Effect with at least one other psychological principle, and start testing it out in your personal life or in your organization. Don’t let useful nuggets sit around; instead, start eating the broccoli.
## Peter Bevelin on Seeking Wisdom, Mental Models, Learning, and a Lot More
One of the most impactful books we’ve ever come across is the wonderful Seeking Wisdom: From Darwin to Munger, written by the Swedish investor Peter Bevelin. In the spirit of multidisciplinary learning, Seeking Wisdom is a compendium of ideas from biology, psychology, statistics, physics, economics, and human behavior.
Mr. Bevelin is out with a new book full of wisdom from Warren Buffett & Charlie Munger: All I Want to Know is Where I’m Going to Die So I Never Go There. We were fortunate enough to have a chance to interview Peter recently, and the result is the wonderful discussion below.
### What was the original impetus for writing these books?
The short answer: To improve my thinking. And when I started writing on what later became Seeking Wisdom I can express it even simpler: “I was dumb and wanted to be less dumb.” As Munger says: “It’s ignorance removal…It’s dishonorable to stay stupider than you have to be.” And I had done some stupid things and I had seen a lot of stupidity being done by people in life and in business.
A seed was first planted when I read Charlie Munger’s worldly wisdom speech and another one where he referred to Darwin as a great thinker. So I said to myself: I am 42 now. Why not take some time off business and spend a year learning, reflecting and write about the subject Munger introduced to me – human behavior and judgments.
None of my writings started out as a book project. I wrote my first book – Seeking Wisdom – as a memorandum for myself with the expectation that I could transfer some of its essentials to my children. I learn and write because I want to be a little wiser day by day. I don’t want to be a great-problem-solver. I want to avoid problems – prevent them from happening and doing right from the beginning. And I focus on consequential decisions. To paraphrase Buffett and Munger – decision-making is not about making brilliant decisions, but avoiding terrible ones. Mistakes and dumb decisions are a fact of life and I’m going to make more, but as long as I can avoid the big or “fatal” ones I’m fine.
So I started to read and write to learn what works and not and why. And I liked Munger’s “All I want to know is where I’m going to die so I’ll never go there” approach. And as he said, “You understand it better if you go at it the way we do, which is to identify the main stupidities that do bright people in and then organize your patterns for thinking and developments, so you don’t stumble into those stupidities.” Then I “only” had to a) understand the central “concept” and its derivatives and describe it in as simple way as possible for me and b) organize what I learnt in a way that was logical and useful for me.
And what better way was there to learn this from those who already knew this?
After I learnt some things about our brain, I understood that thinking doesn’t come naturally to us humans – most is just unconscious automatic reactions. Therefore I needed to set up the environment and design a system that helped me make it easier to know what to do and prevent and avoid harm. Things like simple rules of thumbs, tricks and filters. Of course, I could only do that if I first had the foundation. And as the years have passed, I’ve found that filters are a great way to save time and misery. As Buffett says, “I process information very quickly since I have filters in my mind.” And they have to be simple – as the proverb says, “Beware of the door that has too many keys.” The more complicated a process is, the less effective it is.
Why do I write? Because it helps me understand and learn better. And if I can’t write something down clearly, then I have not really understood it. As Buffett says, “I learn while I think when I write it out. Some of the things, I think I think, I find don’t make any sense when I start trying to write them down and explain them to people … And if it can’t stand applying pencil to paper, you’d better think it through some more.”
My own test is one that a physicist friend of mine told me many years ago, ‘You haven’t really understood an idea if you can’t in a simple way describe it to almost anyone.’ Luckily, I don’t have to understand zillion of things to function well.
And even if some of mine and others thoughts ended up as books, they are all living documents and new starting points for further, learning, un-learning and simplifying/clarifying. To quote Feynman, “A great deal of formulation work is done in writing the paper, organizational work, organization. I think of a better way, a better way, a better way of getting there, of proving it. I never do much — I mean, it’s just cleaner, cleaner and cleaner. It’s like polishing a rough-cut vase. The shape, you know what you want and you know what it is. It’s just polishing it. Get it shined, get it clean, and everything else.
### Which book did you learn the most from the experience of writing/collecting?
Seeking Wisdom because I had to do a lot of research – reading, talking to people etc. Especially in the field of biology and brain science since I wanted to first understand what influences our behavior. I also spent some time at a Neurosciences Institute to get a better understanding of how our anatomy, physiology and biochemistry constrained our behavior.
And I had to work it out my own way and write it down in my own words so I really could understand it. It took a lot of time but it was a lot of fun to figure it out and I learnt much more and it stuck better than if I just had tried to memorize what somebody else had already written. I may not have gotten everything letter perfect but good enough to be useful for me.
As I said, the expectation wasn’t to create a book. In fact, that would have removed a lot of my motivation. I did it because I had an interest in becoming better. It goes back to the importance of intrinsic motivation. As I wrote in Seeking Wisdom: “If we reward people for doing what they like to do anyway, we sometimes turn what they enjoy doing into work. The reward changes their perception. Instead of doing something because they enjoy doing it, they now do it because they are being paid. The key is what a reward implies. A reward for our achievements makes us feel that we are good at something thereby increasing our motivation. But a reward that feels controlling and makes us feel that we are only doing it because we’re paid to do it, decreases the appeal.
It may sound like a cliché but the joy was in the journey – reading, learning and writing – not the destination – the finished book. Has the book made a difference for some people? Yes, I hope so but often people revert to their old behavior. Some of them are the same people who – to paraphrase something that is attributed to Churchill – occasionally should check their intentions and strategies against their results. But reality is what Munger once said, “Everyone’s experience is that you teach only what a reader almost knows, and that seldom.” But I am happy that my books had an impact and made a difference to a few people. That’s enough.
### Why did the new book (All I Want To Know Is Where I’m Going To Die So I’ll Never Go There) have a vastly different format?
It was more fun to write about what works and not in a dialogue format. But also because vivid and hopefully entertaining “lessons” are easier to remember and recall. And you will find a lot of quotes in there that most people haven’t read before.
I wanted to write a book like this to reinforce a couple of concepts in my head. So even if some of the text sometimes comes out like advice to the reader, I always think about what the mathematician Gian-Carlo Rota once said, “The advice we give others is the advice that we ourselves need.”
### How do you define Mental Models?
Some kind of representation that describes how reality is (as it is known today) – a principle, an idea, basic concepts, something that works or not – that I have in my head that helps me know what to do or not. Something that has stood the test of time.
For example some timeless truths are:
• Reality is that complete competitors – same product/niche/territory – cannot coexist (Competitive exclusion principle). What works is going where there is no or very weak competition + differentiation/advantages that others can’t copy (assuming of course we have something that is needed/wanted now and in the future)
• Reality is that we get what we reward for. What works is making sure we reward for what we want to achieve.
I favor underlying principles and notions that I can apply broadly to different and relevant situations. Since some models don’t resemble reality, the word “model” for me is more of an illustration/story of an underlying concept, trick, method, what works etc. that agrees with reality (as Munger once said, “Models which underlie reality”) and help me remember and more easily make associations.
But I don’t judge or care how others label it or do it – models, concepts, default positions … The important thing is that whatever we use, it reflects and agrees with reality and that it works for us to help us understand or explain a situation or know what to do or not do. Useful and good enough guide me. I am pretty pragmatic – whatever works is fine. I follow Deng Xiaoping, “I don’t care whether the cat is black or white as long as it catches mice.” As Feynman said, “What is the best method to obtain the solution to a problem? The answer is, any way that works.
I’ll tell you about a thing Feynman said on education which I remind myself of from time to time in order not to complicate things (from Richard P. Feynman, Michael A. Gottlieb, Ralph Leighton, Feynman’s Tips on Physics: A Problem-Solving Supplement to the Feynman Lectures on Physics):
“There’s a round table on three legs. Where should you lean on it, so the table will be the most unstable?”
The student’s solution was, “Probably on top of one of the legs, but let me see: I’ll calculate how much force will produce what lift, and so on, at different places.”
Then I said, “Never mind calculating. Can you imagine a real table?”
“But that’s not the way you’re supposed to do it!”
“Never mind how you’re supposed to do it; you’ve got a real table here with the various legs, you see? Now, where do you think you’d lean? What would happen if you pushed down directly over a leg?”
“Nothin’!”
I say, “That’s right; and what happens if you push down near the edge, halfway between two of the legs?”
“It flips over!”
I say, “OK! That’s better!”
The point is that the student had not realized that these were not just mathematical problems; they described a real table with legs. Actually, it wasn’t a real table, because it was perfectly circular, the legs were straight up and down, and so on. But it nearly described, roughly speaking, a real table, and from knowing what a real table does, you can get a very good idea of what this table does without having to calculate anything – you know darn well where you have to lean to make the table flip over. So, how to explain that, I don’t know! But once you get the idea that the problems are not mathematical problems but physical problems, it helps a lot.
Anyway, that’s just two ways of solving this problem. There’s no unique way of doing any specific problem. By greater and greater ingenuity, you can find ways that require less and less work, but that takes experience.
### Which mental models “carry the most freight?” (Related follow up: Which concepts from Buffett/Munger/Mental Models do you find yourself referring to or appreciating most frequently?)
Ideas from biology and psychology since many stupidities are caused by not understanding human nature (and you get illustrations of this nearly every day). And most of our tendencies were already known by the classic writers (Publilius Syrus, Seneca, Aesop, Cicero etc.)
Others that I find very useful both in business and private is the ideas of Quantification (without the fancy math), Margin of safety, Backups, Trust, Constraints/Weakest link, Good or Bad Economics slash Competitive advantage, Opportunity cost, Scale effects. I also think Keynes idea of changing your mind when you get new facts or information is very useful.
But since reality isn’t divided into different categories but involves a lot of factors interacting, I need to synthesize many ideas and concepts.
### Are there any areas of the mental models approach you feel are misunderstood or misapplied?
I don’t know about that but what I often see among many smart people agrees with Munger’s comment: “All this stuff is really quite obvious and yet most people don’t really know it in a way where they can use it.”
Anyway, I believe if you really understand an idea and what it means – not only memorizing it – you should be able to work out its different applications and functional equivalents. Take a simple big idea – think on it – and after a while you see its wider applications. To use Feynman’s advice, “It is therefore of first-rate importance that you know how to “triangulate” – that is, to know how to figure something out from what you already know.” As a good friend says, “Learn the basic ideas, and the rest will fill itself in. Either you get it or you don’t.”
Most of us learn and memorize a specific concept or method etc. and learn about its application in one situation. But when the circumstances change we don’t know what to do and we don’t see that the concept may have a wider application and can be used in many situations.
Take for example one big and useful idea – Scale effects. That the scale of size, time and outcomes changes things – characteristics, proportions, effects, behavior…and what is good or not must be tied to scale. This is a very fundamental idea from math. Munger described some of this idea’s usefulness in his worldly wisdom speech. One effect from this idea I often see people miss and I believe is important is group size and behavior. That trust, feeling of affection and altruistic actions breaks down as group size increases, which of course is important to know in business settings. I wrote about this in Seeking Wisdom (you can read more if you type in Dunbar Number on Google search). I know of some businesses that understand the importance of this and split up companies into smaller ones when they get too big (one example is Semco).
Another general idea is “Gresham’s Law” that can be generalized to any process or system where the bad drives out the good. Like natural selection or “We get what we select for” (and as Garrett Hardin writes, “The more general principle is: We get whatever we reward for).
While we are on the subject of mental models etc., let me bring up another thing that distinguishes the great thinkers from us ordinary mortals. Their ability to quickly assess and see the essence of a situation – the critical things that really matter and what can be ignored. They have a clear notion of what they want to achieve or avoid and then they have this ability to zoom in on the key factor(s) involved.
One reason to why they can do that is because they have a large repertoire of stored personal and vicarious experiences and concepts in their heads. They are masters at pattern recognition and connection. Some call it intuition but as Herbert Simon once said, “The situation has provided a cue; this cue has given the expert access to information stored in memory, and the information provides the answer. Intuition is nothing more and nothing less than recognition.
It is about making associations. For example, roughly like this:
Situation X Association (what does this remind me of?) to experience, concept, metaphor, analogy, trick, filter… (Assuming of course we are able to see the essence of the situation) What counts and what doesn’t? What works/not? What to do or what to explain?
Let’s take employing someone as an example (or looking at a business proposal). This reminds me of one key factor – trustworthiness and Buffett’s story, “If you’re looking for a manager, find someone who is intelligent, energetic and has integrity. If he doesn’t have the last, make sure he lacks the first two.”
I believe Buffett and Munger excel at this – they have seen and experienced so much about what works and not in business and behavior.
Buffett referred to the issue of trust, chain letters and pattern recognition at the latest annual meeting:
You can get into a lot of trouble with management that lacks integrity… If you’ve got an intelligent, energetic guy or woman who is pursuing a course of action, which gets put on the front page it could make you very unhappy. You can get into a lot of trouble. ..We’ve seen patterns…Pattern recognition is very important in evaluating humans and businesses. Pattern recognition isn’t one hundred percent and none of the patterns exactly repeat themselves, but there are certain things in business and securities markets that we’ve seen over and over and frequently come to a bad end but frequently look extremely good in the short run. One which I talked about last year was the chain letter scheme. You’re going to see chain letters for the rest of your life. Nobody calls them chain letters because that’s a connotation that will scare you off but they’re disguised as chain letters and many of the schemes on Wall Street, which are designed to fool people, have that particular aspect to it…There were patterns at Valeant certainly…if you go and watch the Senate hearings, you will see there are patterns that should have been picked up on.
This is what he wrote on chain letters in the 2014 annual report:
In the late 1960s, I attended a meeting at which an acquisitive CEO bragged of his “bold, imaginative accounting.” Most of the analysts listening responded with approving nods, seeing themselves as having found a manager whose forecasts were certain to be met, whatever the business results might be. Eventually, however, the clock struck twelve, and everything turned to pumpkins and mice. Once again, it became evident that business models based on the serial issuances of overpriced shares – just like chain-letter models – most assuredly redistribute wealth, but in no way create it. Both phenomena, nevertheless, periodically blossom in our country – they are every promoter’s dream – though often they appear in a carefully-crafted disguise. The ending is always the same: Money flows from the gullible to the fraudster. And with stocks, unlike chain letters, the sums hijacked can be staggering.
And of course, the more prepared we are or the more relevant concepts and “experiences” we have in our heads, the better we all will be at this. How do we get there? Reading, learning and practice so we know it “fluently.” There are no shortcuts. We have to work at it and apply it to the real world.
As a reminder to myself so I understand my limitation and “circle”, I keep a paragraph from Munger’s USC Gould School of Law Commencement Address handy so when I deal with certain issues, I don’t fool myself into believing I am Max Planck when I’m really the Chauffeur:
In this world I think we have two kinds of knowledge: One is Planck knowledge, that of the people who really know. They’ve paid the dues, they have the aptitude. Then we’ve got chauffeur knowledge. They have learned to prattle the talk. They may have a big head of hair. They often have fine timbre in their voices. They make a big impression. But in the end what they’ve got is chauffeur knowledge masquerading as real knowledge.
### Which concepts from Buffett/Munger/Mental Models do you find most counterintuitive?
One trick or notion I see many of us struggling with because it goes against our intuition is the concept of inversion – to learn to think “in negatives” which goes against our normal tendency to concentrate on for example, what we want to achieve or confirmations instead of what we want to avoid and disconfirmations. Another example of this is the importance of missing confirming evidence (I call it the “Sherlock trick”) – that negative evidence and events that don’t happen, matter when something implies they should be present or happen.
Another example that is counterintuitive is Newton’s 3d law that forces work in pairs. One object exerts a force on a second object, but the second object also exerts a force equal and opposite in direction to the force acting on it – the first object. As Newton wrote, “If you press a stone with your finger, the finger is also pressed by the stone.” Same as revenge (reciprocation).
### Who are some of the non-obvious, or under-the-radar thinkers that you greatly admire?
One that immediately comes to mind is one I have mentioned in the introduction in two of my books is someone I am fortunate to have as a friend – Peter Kaufman. An outstanding thinker and a great businessman and human being. On a scale of 1 to 10, he is a 15.
### What have you come to appreciate more with Buffett/Munger’s lessons as you’ve studied them over the years?
Their ethics and their ethos of clarity, simplicity and common sense. These two gentlemen are outstanding in their instant ability to exclude bad ideas, what doesn’t work, bad people, scenarios that don’t matter, etc. so they can focus on what matters. Also my amazement that their ethics and ideas haven’t been more replicated. But I assume the answer lies in what Munger once said, “The reason our ideas haven’t spread faster is they’re too simple.”
This reminds me something my father-in-law once told me (a man I learnt a lot from) – the curse of knowledge and the curse of academic title. My now deceased father-in-law was an inventor and manager. He did not have any formal education but was largely self-taught. Once a big corporation asked for his services to solve a problem their 60 highly educated engineers could not solve. He solved the problem. The engineers said, “It can’t be that simple.” It was like they were saying that, “Here we have 6 years of school, an academic title, lots of follow up education. Therefore an engineering problem must be complicated”. Like Buffett once said of Ben Graham’s ideas, “I think that it comes down to those ideas – although they sound so simple and commonplace that it kind of seems like a waste to go to school and get a PhD in Economics and have it all come back to that. It’s a little like spending eight years in divinity school and having somebody tell you that the 10 commandments were all that counted. There is a certain natural tendency to overlook anything that simple and important.”
(I must admit that in the past I had a tendency to be extra drawn to elegant concepts and distracting me from the simple truths.)
### What things have you come to understand more deeply in the past few years?
• That I don’t need hundreds of concepts, methods or tricks in my head – there are a few basic, time-filtered fundamental ones that are good enough. As Munger says, “The more basic knowledge you have the less new knowledge you have to get.” And when I look at something “new”, I try to connect it to something I already understand and if possible get a wider application of an already existing basic concept that I already have in my head.
• Neither do I have to learn everything to cover every single possibility – not only is it impossible but the big reason is well explained by the British statistician George Box. He said that we shouldn’t be preoccupied with optimal or best procedures but good enough over a range of possibilities likely to happen in practice – circumstances which the world really present to us.
• The importance of “Picking my battles” and focus on the long-term consequences of my actions. As Munger said, “A majority of life’s errors are caused by forgetting what one is really trying to do.”
• How quick most of us are in drawing conclusions. For example, I am often too quick in being judgmental and forget how I myself behaved or would have behaved if put in another person’s shoes (and the importance of seeing things from many views).
• That I have to “pick my poison” since there is always a set of problems attached with any system or approach – it can’t be perfect. The key is try to move to a better set of problems one can accept after comparing what appear to be the consequences of each.
• How efficient and simplified life is when you deal with people you can trust. This includes the importance of the right culture.
• The extreme importance of the right CEO – a good operator, business person and investor.
• That luck plays a big role in life.
• That most predictions are wrong and that prevention, robustness and adaptability is way more important. I can’t help myself – I have to add one thing about the people who give out predictions on all kinds of things. Often these are the people who live in a world where their actions have no consequences and where their ideas and theories don’t have to agree with reality.
• That people or businesses that are foolish in one setting often are foolish in another one (“The way you do anything, is the way you do everything”).
• Buffett’s advice that “A checklist is no substitute for thinking.” And that sometimes it is easy to overestimate one’s competency in a) identifying or picking what the dominant or key factors are and b) evaluating them including their predictability. That I believe I need to know factor A when I really need to know B – the critical knowledge that counts in the situation with regards to what I want to achieve.
• Close to this is that I sometimes get too involved in details and can’t see the forest for the trees and I get sent up too many blind alleys. Just as in medicine where a whole body scan sees too much and sends the doctor up blind alleys.
• The wisdom in Buffett’s advice that “You only have to be right on a very, very few things in your lifetime as long as you never make any big mistakes…An investor needs to do very few things right as long as he or she avoids big mistakes.”
### What’s the best investment of time/effort/money that you’ve ever made?
The best thing I have done is marrying my wife. As Buffett says and it is so so true, “Choosing a spouse is the most important decision in your life…You need everything to be stable, and if that decision isn’t good, it may affect every other decision in life, including your business decisions…If you are lucky on health and…on your spouse, you are a long way home.”
A good “investment” is taking the time to continuously improve. It just takes curiosity and a desire to know and understand – real interest. And for me this is fun.
### What does your typical day look like? (How much time do you spend reading… and when?)
Every day is a little different but I read every day.
### What book has most impacted your life?
There is not one single book or one single idea that has done it. I have picked up things from different books (still do). And there are different books and articles that made a difference during different periods of my life. Meeting and learning from certain people and my own practical experiences has been more important in my development. As an example – When I was in my 30s a good friend told me something that has been very useful in looking at products and businesses. He said I should always ask who the real customer is: “Who ultimately decides what to buy and what are their decision criteria and how are they measured and rewarded and who pays?
But looking back, if I have had a book like Poor Charlie’s Almanack when I was younger I would have saved myself some misery. And of course, when it comes to business, managing and investing, nothing beats learning from Warren Buffett’s Letters to Berkshire Hathaway Shareholders.
Another thing I have found is that it is way better to read and reread fewer books but good and timeless ones and then think. Unfortunately many people absorb too many new books and information without thinking.
Let me finish this with some quotes from my new book that I believe we all can learn from:
• “There’s no magic to it…We haven’t succeeded because we have some great, complicated systems or magic formulas we apply or anything of the sort. What we have is just simplicity itself.” – Buffett
• “Our ideas are so simple that people keep asking us for mysteries when all we have are the most elementary ideas…There’s nothing remarkable about it. I don’t have any wonderful insights that other people don’t have. Just slightly more consistently than others, I’ve avoided idiocy…It is remarkable how much long-term advantage people like us have gotten by trying to be consistently not stupid, instead of trying to be very intelligent.” – Munger
• “It really is simple – just avoid doing the dumb things. Avoiding the dumb things is the most important.” – Buffett
Finally, I wish you and your readers an excellent day – Everyday!
## Why Fiddling With Prices Doesn’t Work
“The fact is, if you don’t find it reasonable that prices should reflect relative scarcity,
then fundamentally you don’t accept the market economy,
because this is about as close to the essence of the market as you can find.”
— Joseph Heath
***
Inevitably, when the price of a good or service rises rapidly, there follows an accusation of price-gouging. The term carries a strong moral admonition on the price-gouger, in favor of the price-gougee. Gas shortages are a classic example. With a local shortage of gasoline, gas stations will tend to mark up the price of gasoline to reflect the supply issue. This is usually rewarded with cries of unfairness. But does that really make sense?
In his excellent book Economics Without Illusions, Joseph Heath argues that it doesn’t.
In fact, this very scenario is market pricing reacting just as it should. With gasoline in short supply, the market price rises too so that those who need gasoline have it available, and those who simply want it do not. The price system ensures that everyone makes their choice correctly. If you’re willing to pay up, you pay up. If you’re not, you make alternative arrangements – drive less, use less heat, etc. This is exactly what market pricing is for – to give us a reference as we make our choices. But it’s still hard for many well-intentioned people to understand. Let’s think it through a little, with Heath’s help.
***
As Heath points out in the book, the objection to so-called “price gouging” goes back at least to the Roman Emperor Diocletian, who in AD 301 imposed an Edict of Maximum Prices:
If the excesses perpetrated by persons of unlimited and frenzied avarice could be checked by some self-restraint—this avarice which rushes for gain and profit with no thought for mankind; or if the general welfare could endure without harm this riotous license by which, in its unfortunate state, it is being very seriously injured every day, the situation could perhaps be faced with dissembling and silence, with the hope that human forbearance might alleviate the cruel and pitiable situation.
And with that, Diocletian set a hard cap on the price of over a thousand different items. Some were tangible, like wheat and barley, and some were intangible, like farm labor and barber services.
This was, of course, very dumb and did not last very long as people realized that one barber and another were not equal, that wheat and barley might have local supply constraints, and that an arbitrary government price was not the fair one for most of the 1000+ items.
### Inflation vs. Supply
As Heath points out in his book, there are two separate issues to untangle when we talk about “price-gouging” — general inflation and constraints on supply. The two are very different, and confusing a supply issue for general inflation leads to a lot of wrong thinking:
If you wander into a Polish supermarket and discover that a kilo of carrots is selling for four zlotys, you probably haven’t learned very much. It’s only once you find out what a pound of potatoes costs, and a chicken, and a pint of beer, that you begin to discover whether carrots are expensive or cheap.
As a result, the price of everything going up is analytically equivalent to the price of nothing going up. It follows that if the price of everything seems to be going up, it must be because the price of at least one thing is (inconspicuously) going down. Usually that inconspicuous item with the falling price is hidden in plain sight — money. We tend to overlook money because it’s not directly consumed; it simply circulates, thus we forget that it has a price. We think of “four zlotys per kilo” as the price of carrots, expressed in zlotys, while forgetting that it is also the price of zlotys, expressed in carrots.
As Garrett Hardin would well recognize, part of the problem is the way language misleads us. When the price of stuff is going up, we don’t always make the equivalent connection that the value of our money is going down. And thus, we can often confuse a rising price environment for greedy so and so’s who are simply reacting to the declining value of money.
Often, governments hurt the value of money purposely. In Diocletian’s time, a denarius coin went from being made entirely of silver to being made of about 2% silver and 98% base metals – the origin of the term currency debasement. In a world of inflation, what seems like greed is often an illusion caused by money losing its value generally (a complex phenomenon in its own right).
To see the flow-through effects of this, imagine that all wage-earners were given a significant raise next month. Sounds good, right? Problem is, the increased cost of labor would be passed through in the form of higher prices for everything, or alternatively, businesses would figure out how to operate with fewer workers altogether. The owners of society’s capital don’t just sit back and lose money — they figure out a new plan or reallocate their resources elsewhere.
Thus, a wage increase would put us right back to where we started. This is why the minimum wage debate isn’t simply a humanitarian “business versus workers” issue — there are no easy answers. (In other words, The consequences have consequences.)
Prices are simply signals which allow us to make decisions on how much we really need that thing. If each of us was handed \$5,000 to spend each month, we could choose to spend X amount on food, Y amount on housing, and Z amount of organic 97% cacao chocolate. The alternative would be a state planner sitting in a high tower trying to fix prices based on how he or she thought everyone should make their food/housing/chocolate allocation for the month. The history of planned economies would show this to be a majorly bad idea.
This leads us to our next point which is that, of course, our income allocations are not the same. Might price-fixing help level the playing field?
### Fixing What, Exactly?
Heath quotes the economist Abba Lerner who once said that the problem for the poor is not that prices are too high, but that they don’t have enough money. (“The solution of poverty lay not with the manipulation of prices but with the distribution of money income.”)
On this, Heath turns to the example of electricity prices, an occasional hot-button issue which leads to subsidies because high electricity prices are seen as regressive — poor people spend a larger percentage of their money on electric power than those more well-off. Why not subsidize electricity prices to help?
The problem is that it’s a massively inefficient way to help, and puts a lot of dollars into pockets of those who don’t need it. Citing Canadian statistics on the use of subsidies to keep electricity prices down, Heath writes:
The middle-income quintile spends an average of \$1,117 per year (2.4% of income), while the upper quintile spends \$1,522 per year (1.1% of income). This means that the \$250 million annual gift being bestowed upon the poor is coupled with a \$408 million gift to the middle class and a \$556 million gift to the richest 20% of the population. Needless to say, a welfare program that required giving \$2 to a rich person for every \$1 directed to a poor person would hardly be regarded as progressive (despite the fact that, when expressed as a percentage of income, the poor person is receiving “more”).
Of course, finding a way to get the entire \$1.2 billion to the people who truly need it, through a deserving program, would be a far better solution, and one that would also avoid encouraging people to use more electricity than they need (which artificially lower prices can do).
This kind of thing happens, but worse, when it comes to rent control, the system of fixing rental prices for apartments in cities. In addition to subsidizing some of the wrong people, who also have access to rent-controlled housing, the lower prices tend to distort the market for apartment and housing construction.
With apartments so affordable, people who might otherwise have purchased a house now choose to rent, crowding out some people who could never afford a home at all. And with prices artificially low, fewer apartment houses are built! Not a great outcome for the people rent control hopes to help.
To understand why think about the massive spike in energy prices leading up to the 2008 financial crisis. At one time, oil neared \$140 per barrel and natural gas reached \$13 per MMbtu. The result was somewhat predictable: A massive investment went into the energy complex, leading to new resources and new technologies, while demand quickly abated. Almost no one correctly predicted that 8 years later, oil would be sitting below \$50 per barrel and natural gas around \$2 per MMbtu. This is, of course, how pricing markets are supposed to work. The signals did their job. Artificial prices for metropolitan apartments don’t allow the market to do this job effectively.
### Relative Scarcity: The Key to Understanding Market Prices
The main problem with manipulating and fixing prices is a misunderstanding of what determines prices. What usually determines prices in a true market is relative scarcity, the intersection between how much you want a particular good relative to another one, and the availability of that good. As our wants and needs change, and available supplies change, prices go up and down (ignoring, for now, speculative factors, which play a huge role in some price markets).
What exactly are we paying for when we buy an item?
Clearly, it’s not just the cost of the physical thing being produced. A cup of coffee costs a lot more than a few beans and some water. The total cost is something Heath calls the “social cost” of the good, which includes the entire chain of costs and opportunity costs in producing it:
Whenever someone consumes a good (say, a cup of coffee), this can be thought of as creating a benefit for that individual, combined with a loss for the rest of society (all the time and trouble it took to produce that cup of coffee, now gone). Paying for things is our way of compensating all the people who have been inconvenienced by our consumption. (Next time you buy a cup of coffee at Starbucks, imagine yourself saying to the barista, “I’m sorry that you had to serve me coffee when you could have been doing other things. And please communicate my apologies to the others as well: the owner, the landlord, the shipping company, the Columbian peasants. Here’s \$1.75 for all the trouble. Please divide it amongst yourselves.)
“Social cost” represents the level of renunciation, or foregone consumption, imposed upon the rest of society by each individual’s own consumption. This is a fairly abstract notion, since it’s not just that the good could have been consumed by someone else, but that the labor and resources that went into making that good could have been used to produce something else, which then could have been consumed by someone else. (So when I drink a cup of coffee, I am not only taking away that cup of coffee from all those who might like to have drunk it, but taking away vegetables from those who might like to have used the land to grow food, clothing from those who might like to have employed the agricultural workers in a garment factory, and so on.)
[…]
If the price of coffee tracks changes in supply and demand, it will tend to reflect this level of hardship. If the rest of us really want coffee, then we will be prepared to pay more for it, and so the price will rise. Coffee will become more “dear” (as the British would say), reflecting the fact that the person who drinks it is denying the rest of us something we really want. Thus the coffee-drinker had better really want it in order to justify depriving us of it. His willingness to pay the higher price is precisely what ensures that he does, in fact, really want it.
At the price where the hardship of creating a certain amount of some good meets the desire for a good, a price emerges. It’s this “market clearing” price which efficiently allocates most of society’s resources the way we need them allocated.
If prices are systematically lower than they should be, consumers benefit from society’s hard work in a way that might be better allocated elsewhere, where some other group would happily pay more for the same level of “social costs” imposed, and the producers would receive more for all their work.
Conversely, if prices are too high, then consumers don’t really get to be as happy as they should be relative to the modest “social cost” they’ve imposed. Each outcome is inefficient and produces less happiness and material wealth. A well-established pricing mechanism does the job of sending the right signals about wants, needs, and supplies.
### Income Over Pricing
Heath makes a final important point about the inequality of income in society, and that in many cases, people who have had a rough hand dealt to them do deserve help. It’s just that playing with the pricing mechanism is usually the worst way to do it — as we saw above, you hand people money who don’t need it while distorting an efficient allocation of resources throughout society. Heath calls this the just price fallacy — the idea that some alternative level of prices are more “fair” and that we should intervene to ensure them. The “just price fallacy” fails because it doesn’t ask the crucial question: And then what?
Returning to the dictum that poor people simply don’t have enough money (ridiculous as it sounds), the better method is to attack the other side — income — through the system of taxation and other mechanisms, things which we do in great heaps in modern society, but will always be argued over. If market prices tend to efficiently signal suppliers about the wants and needs of society, we can usually help the less fortunate best by giving them more “claim checks” rather than distorting the very thing that works.
***
Still Interested? Try reading more from the wonderful book Economics Without Illusions, where Heath takes on some fallacies from the left and some fallacies from the right in the economic debate.
For more from Farnam Street, check out Charlie Munger’s speech on what could make the economics profession work a little better or check out economist John Kay’s recommendations on books about economics in the real world.
## How (Supposedly) Rational People Make Decisions
There are four principles that Gregory Mankiw outlines in his multi-disciplinary economics textbook Principles of Economics.
I got the idea for reading an Economics textbook from Charlie Munger, the billionaire business partner of Warren Buffett. He said:
Economics was always more multidisciplinary than the rest of soft science. It just reached out and grabbed things as it needed to. And that tendency to just grab whatever you need from the rest of knowledge if you’re an economist has reached a fairly high point in Mankiw’s new textbook Principles of Economics. I checked out that textbook. I must have been one of the few businessmen in America that bought it immediately when it came out because it had gotten such a big advance. I wanted to figure out what the guy was doing where he could get an advance that great. So this is how I happened to riffle through Mankiw’s freshman textbook. And there I found laid out as principles of economics: opportunity cost is a superpower, to be used by all people who have any hope of getting the right answer. Also, incentives are superpowers.
So we know that we can add Opportunity cost and incentives to our list of Mental Models.
Let’s dig in.
## Principle 1: People Face Trade-offs
You have likely heard the old saying, “There is no such thing as a free lunch.” There is much to this old adage and it’s one we often forget when making decisions. To get more of something we like we almost always have to give up something else we like. A good heuristic in life is that if someone offers you something for nothing, turn it down.
Making decisions requires trading off one goal against another.
Consider a student who must decide how to allocate her most valuable resource—her time. She can spend all of her time studying economics, spend all of it studying psychology, or divide it between the two fields. For every hour she studies one subject, she gives up an hour she could have used studying the other. And for every hour she spends studying, she gives up an hour that she could have spent napping, bike riding, watching TV, or working at her part-time job for some extra spending money.
Or consider parents deciding how to spend their family income. They can buy food, clothing, or a family vacation. Or they can save some of the family income for retirement or for children’s college education. When they choose to spend an extra dollar on one of these goods, they have one less dollar to spend on some other good.
These are rather simple examples but Mankiw offers some more complicated ones. Consider the trade-off that society faces between efficiency and equality.
Efficiency means that society is getting the maximum benefits from its scarce resources. Equality means that those benefits are distributed uniformly among society’s members. In other words, efficiency refers to the size of the economic pie, and equality refers to how the pie is divided into individual slices.
When government policies are designed, these two goals often conflict. Consider, for instance, policies aimed at equalizing the distribution of economic well-being. Some of these policies, such as the welfare system or unemployment insurance, try to help the members of society who are most in need. Others, such as the individual income tax, ask the financially successful to contribute more than others to support the government. Though they achieve greater equality, these policies reduce efficiency. When the government redistributes income from the rich to the poor, it reduces the reward for working hard; as a result, people work less and produce fewer goods and services. In other words, when the government tries to cut the economic pie into more equal slices, the pie gets smaller.
## Principle 2: The Cost of Something Is What You Give Up to Get It
Because of trade-offs, people face decisions between the costs and benefits of one course of action and the cost and benefits of another course. But costs are not as obvious as they might first appear — we need to apply some second-order thinking:
Consider the decision to go to college. The main benefits are intellectual enrichment and a lifetime of better job opportunities. But what are the costs? To answer this question, you might be tempted to add up the money you spend on tuition, books, room, and board. Yet this total does not truly represent what you give up to spend a year in college.
There are two problems with this calculation. First, it includes some things that are not really costs of going to college. Even if you quit school, you need a place to sleep and food to eat. Room and board are costs of going to college only to the extent that they are more expensive at college than elsewhere. Second, this calculation ignores the largest cost of going to college—your time. When you spend a year listening to lectures, reading textbooks, and writing papers, you cannot spend that time working at a job. For most students, the earnings they give up to attend school are the single largest cost of their education.
The opportunity cost of an item is what you give up to get that item. When making any decision, decision makers should be aware of the opportunity costs that accompany each possible action. In fact, they usually are. College athletes who can earn millions if they drop out of school and play professional sports are well aware that the opportunity cost of their attending college is very high. It is not surprising that they often decide that the benefit of a college education is not worth the cost.
## Principle 3: Rational People Think at the Margin
For the sake of simplicity economists normally assume that people are rational. While this causes many problems, there is an undercurrent of truth to the fact that people systematically and purposefully “do the best they can to achieve their objectives, given opportunities.” There are two parts to rationality. The first is that your understanding of the world is correct. Second you maximize the use of your resources toward your goals.
Rational people know that decisions in life are rarely black and white but usually involve shades of gray. At dinnertime, the question you face is not “Should I fast or eat like a pig?” More likely, you will be asking yourself “Should I take that extra spoonful of mashed potatoes?” When exams roll around, your decision is not between blowing them off and studying twenty-four hours a day but whether to spend an extra hour reviewing your notes instead of watching TV. Economists use the term marginal change to describe a small incremental adjustment to an existing plan of action. Keep in mind that margin means “edge,” so marginal changes are adjustments around the edges of what you are doing. Rational people often make decisions by comparing marginal benefits and marginal costs.
Thinking at the margin works for business decisions.
Consider an airline deciding how much to charge passengers who fly standby. Suppose that flying a 200-seat plane across the United States costs the airline \$100,000. In this case, the average cost of each seat is \$100,000/200, which is \$500. One might be tempted to conclude that the airline should never sell a ticket for less than \$500. But a rational airline can increase its profits by thinking at the margin. Imagine that a plane is about to take off with 10 empty seats and a standby passenger waiting at the gate is willing to pay \$300 for a seat. Should the airline sell the ticket? Of course, it should. If the plane has empty seats, the cost of adding one more passenger is tiny. The average cost of flying a passenger is \$500, but the marginal cost is merely the cost of the bag of peanuts and can of soda that the extra passenger will consume. As long as the standby passenger pays more than the marginal cost, selling the ticket is profitable.
This also helps answer the question of why diamonds are so expensive and water is so cheap.
Humans need water to survive, while diamonds are unnecessary; but for some reason, people are willing to pay much more for a diamond than for a cup of water. The reason is that a person’s willingness to pay for a good is based on the marginal benefit that an extra unit of the good would yield. The marginal benefit, in turn, depends on how many units a person already has. Water is essential, but the marginal benefit of an extra cup is small because water is plentiful. By contrast, no one needs diamonds to survive, but because diamonds are so rare, people consider the marginal benefit of an extra diamond to be large.
A rational decision maker takes an action if and only if the marginal benefit of the action exceeds the marginal cost.
## Principle 4: People Respond to Incentives
Incentives induce people to act. If you use a rational approach to decision making that involves trade offs and comparing costs and benefits, you respond to incentives. Charlie Munger once said: “Never, ever, think about something else when you should be thinking about the power of incentives.”
Incentives are crucial to analyzing how markets work. For example, when the price of an apple rises, people decide to eat fewer apples. At the same time, apple orchards decide to hire more workers and harvest more apples. In other words, a higher price in a market provides an incentive for buyers to consume less and an incentive for sellers to produce more. As we will see, the influence of prices on the behavior of consumers and producers is crucial for how a market economy allocates scarce resources.
Public policymakers should never forget about incentives: Many policies change the costs or benefits that people face and, as a result, alter their behavior. A tax on gasoline, for instance, encourages people to drive smaller, more fuel-efficient cars. That is one reason people drive smaller cars in Europe, where gasoline taxes are high, than in the United States, where gasoline taxes are low. A higher gasoline tax also encourages people to carpool, take public transportation, and live closer to where they work. If the tax were larger, more people would be driving hybrid cars, and if it were large enough, they would switch to electric cars.
Failing to consider how policies and decisions affect incentives often results in unforeseen results.
## Biases and Blunders
You would be hard pressed to come across a reading list on behavioral economics that doesn’t mention Nudge: Improving Decisions About Health, Wealth, and Happiness by Richard Thaler and Cass Sunstein.
It is a fascinating look at how we can create environments or ‘choice architecture’ to help people make better decisions. But one of the reasons it’s been so influential is because it helps us understand why people sometimes make bad decisions in the first place. If we really want to understand how we can nudge people into making better choices, it’s important to understand why they often make such poor ones.
Let’s take a look at how Thaler and Sunstein explain some of our common mistakes in a chapter aptly called ‘Biases and Blunders.’
Humans have a tendency to put too much emphasis on one piece of information when making decisions. When we overweigh one piece of information and make assumptions based on it, we call that an anchor. Say I borrow a 400-page-book from a friend and I think to myself, the last book I read was about 300 pages and I read it in 5 days so I’ll let my friend know I’ll have her book back to her in 7 days. Problem is, I’ve only compared one factor related to me reading books and now I’ve made a decision without taking into account many other factors which could affect the outcome. For example, is the new book a topic I will digest at the same rate? Will I have the same time over those 7 days for reading? I have looked at number of pages but are the number of words per page similar?
As Thaler and Sunstein explain:
This process is called ‘anchoring and adjustment.’ You start with some anchor, the number you know, and adjust in the direction you think is appropriate. So far, so good. The bias occurs because the adjustments are typically insufficient.
## Availability Heuristic
This is the tendency of our mind to overweigh information that is recent and readily available. What did you think about the last time you read about a plane crash? Did you start thinking about you being in a plane crash? Imagine how much it would weigh on your mind if you were set to fly the next day.
We assess the likelihood of risks by asking how readily examples come to mind. If people can easily think of relevant examples, they are far more likely to be frightened and concerned than if they cannot.
Accessibility and salience are closely related to availability, and they are important as well. If you have personally experienced a serious earthquake, you’re more likely to believe that an earthquake is likely than if you read about it in a weekly magazine. Thus, vivid and easily imagined causes of death (for example, tornadoes) often receive inflated estimates of probability, and less-vivid causes (for example, asthma attacks) receive low estimates, even if they occur with a far greater frequency (here, by a factor of twenty). Timing counts too: more recent events have a greater impact on our behavior, and on our fears, than earlier ones.
## Representativeness Heuristic
Use of the representativeness heuristic can cause serious misperceptions of patterns in everyday life. When events are determined by chance, such as a sequence of coin tosses, people expect the resulting string of heads and tails to be representative of what they think of as random. Unfortunately, people do not have accurate perceptions of what random sequences look like. When they see the outcomes of random processes, they often detect patterns that they think have great meaning but in fact are just due to chance.
It would seem as though we have issues with randomness. Our brains automatically want to see patterns when none may exist. Try a coin toss experiment on yourself. Simply flip a coin and keep track if it’s heads or tails. At some point you will hit ‘a streak’ of either heads or tails and you will notice that you experience a sort of cognitive dissonance; you know that ‘a streak’ at some point is statistically probable but you can’t help but thinking the next toss has to break the streak because for some reason in your head it’s not right. That unwillingness to accept randomness, our need for a pattern, often clouds our judgement when making decisions.
## Unrealistic Optimism
We have touched upon optimism bias in the past. Optimism truly is a double-edged sword. On one hand it is extremely important to be able to look past a bad moment and tell yourself that it will get better. Optimism is one of the great drivers of human progress.
On the other hand, if you never take those rose-coloured glasses off, you will make mistakes and take risks that could have been avoided. When assessing the possible negative outcomes associated with risky behaviour we often think ‘it won’t happen to me.’ This is a brain trick: We are often insensitive to the base rate.
Unrealistic optimism is a pervasive feature of human life; it characterizes most people in most social categories. When they overestimate their personal immunity from harm, people may fail to take sensible preventive steps. If people are running risks because of unrealistic optimism, they might be able to benefit from a nudge.
## Loss Aversion
When they have to give something up, they are hurt more than they are pleased if they acquire the very same thing.
We are familiar with loss aversion in the context described above but Thaler and Sunstein take the concept a step further and explain how it plays a role in ‘default choices.’ Loss aversion can make us so fearful of making the wrong decision that we don’t make any decision. This explains why so many people settle for default options.
The combination of loss aversion with mindless choosing implies that if an option is designated as the ‘default,’ it will attract a large market share. Default options thus act as powerful nudges. In many contexts defaults have some extra nudging power because consumers may feel, rightly or wrongly, that default options come with an implicit endorsement from the default setter, be it the employer, government, or TV scheduler.
Of course, this is not the only reason default options are so popular. “Anchoring,” which we mentioned above, plays a role here. Our mind anchors immediately to the default option, especially in unfamiliar territory for us.
We also have the tendency towards inertia, given that mental effort is tantamount to physical effort – thinking hard requires physical resources. If we don’t know the difference between two 401(k) plans and they both seem similar, why expend the mental effort to switch away from the default investment option? You may not have that thought consciously; it often happens as a “click, whirr.
## State of Arousal
Our prefered definition requires recognizing that people’s state of arousal varies over time. To simplify things we will consider just the two endpoints: hot and cold. When Sally is very hungry and appetizing aromas are emanating from the kitchen, we can say she is in a hot state. When Sally is thinking abstractly on Tuesday about the right number of cashews she should consume before dinner on Saturday, she is in a cold state. We will call something ‘tempting’ if we consume more of it when hot than when cold. None of this means that decisions made in a cold state are always better. For example, sometimes we have to be in a hot state to overcome our fears about trying new things. Sometimes dessert really is delicious, and we do best to go for it. Sometimes it is best to fall in love. But it is clear that when we are in a hot state, we can often get into a lot of trouble.
For most of us, however, self-control issues arise because we underestimate the effect of arousal. This is something the behavioral economist George Loewenstein (1996) calls the ‘hot-cold empathy gap.’ When in a cold state, we do not appreciate how much our desires and our behavior reflects a certain naivete about the effects that context can have on choice.
The concept of arousal is analogous to mood. At the risk of stating the obvious, our mood can play a definitive role in our decision making. We all know it, but how many among us truly use that insight to make better decisions?
This is one reason we advocate decision journals when it comes to meaningful decisions (probably no need to log in your cashew calculations); a big part of tracking your decisions is your mood when you make themA zillion contextual clues go into your state of arousal, but taking a quick pause to note which state you’re in as you make a decision can make a difference over time.
Mood is also affected by chemicals. This one may be familiar to you coffee (or tea) addicts out there. Do you recall the last time you felt terrible or uncertain about a decision when you were tired, only to feel confident and spunky about the same topic after a cup of java?
Or, how about alcohol? There’s a reason it’s called a “social lubricant” – our decision making changes when we’ve consumed enough of it.
Lastly, the connection between sleep and mood goes deep. Need we say more?
## Peer Pressure
Peer pressure is another tricky nudge that can be both positive or negative. We can be nudged to make better decisions when we think that our peer group is doing the same. If we think our neighbors conserve more energy or recycle more, we start making a better effort to reduce our consumption and recycle. If we think the people around us are eating better and exercising more we tend to do the same. Information we get from peer groups can also help us make better decisions because of ‘collaborative filtering’; the choices of our peer groups help us filter out and narrow down our choices. If your friends who share similar views and tastes as you recommend book X, then you may like it as well. (Google, Amazon and Netflix are built on this principle).
However, if we are all reading the same book because we constantly see people with it, but none of us actually like it, then we all lose. We run off the mountain with the other lemmings.
Social influences come in two basic categories. The first involves information. If many people do something or think something, their actions and their thoughts convey information about what might be best for you to do or think. The second involves peer pressure. If you care about what other people think about you (perhaps in the mistaken belief that they are paying some attention to what you are doing), then you might go along with the crowd to avoid their wrath or curry their favor.
An important problem here is ‘pluralistic ignorance’ – that is, ignorance, on the part of all or most, about what other people think. We may follow a practice or a tradition not because we like it, or even think it defensible, but merely because we think that most other people like it. Many social practices persist for this reason, and a small shock, or nudge, can dislodge them.
How do we beat social influence? It’s very difficult, and not always desirable: If you are about to enter a building a lot of people are running away from, there’s a better than good chance you should too. But this useful instinct leads us awry.
A simple algorithm, when you feel yourself acting out of social proof, is to ask yourself: Would I still do this if everyone else was not?
***
For more, check out Nudge.
## How Analogies Reveal Connections, Spark Innovation, and Sell Our Greatest Ideas
John Pollack is a former Presidential Speechwriter. If anyone knows the power of words to move people to action, shape arguments, and persuade, it is he.
In Shortcut: How Analogies Reveal Connections, Spark Innovation, and Sell Our Greatest Ideas, he explores the powerful role of analogy in persuasion and creativity.
One of the key tools he uses for this is analogy.
While they often operate unnoticed, analogies aren’t accidents, they’re arguments—arguments that, like icebergs, conceal most of their mass and power beneath the surface. In arguments, whoever has the best argument wins.
But analogies do more than just persuade others — they also play a role in innovation and decision making.
From the bloody Chicago slaughterhouse that inspired Henry Ford’s first moving assembly line, to the “domino theory” that led America into the Vietnam War, to the “bicycle for the mind” that Steve Jobs envisioned as a Macintosh computer, analogies have played a dynamic role in shaping the world around us.
Despite their importance, many people have only a vague sense of the definition.
## What is an Analogy?
In broad terms, an analogy is simply a comparison that asserts a parallel—explicit or implicit—between two distinct things, based on the perception of a share property or relation. In everyday use, analogies actually appear in many forms. Some of these include metaphors, similes, political slogans, legal arguments, marketing taglines, mathematical formulas, biblical parables, logos, TV ads, euphemisms, proverbs, fables and sports clichés.
Because they are so disguised they play a bigger role than we consciously realize. Not only do analogies effectively make arguments, but they trigger emotions. And emotions make it hard to make rational decisions.
While we take analogies for granted, the ideas they convey are notably complex.
All day every day, in fact, we make or evaluate one analogy after the other, because some comparisons are the only practical way to sort a flood of incoming data, place it within the content of our experience, and make decisions accordingly.
Remember the powerful metaphor — that arguments are war. This shapes a wide variety of expressions like “your claims are indefensible,” “attacking the weakpoints,” and “You disagree, OK shoot.”
Or consider the Map and the Territory — Analogies give people the map but explain nothing of the territory.
Warren Buffett is one of the best at using analogies to communicate effectively. One of my favorite analogies is when he noted “You never know who’s swimming naked until the tide goes out.” In other words, when times are good everyone looks amazing. When times suck, hidden weaknesses are exposed. The same could be said for analogies:
We never know what assumptions, deceptions, or brilliant insights they might be hiding until we look beneath the surface.
Most people underestimate the importance of a good analogy. As with many things in life, this lack of awareness comes at a cost. Ignorance is expensive.
Evidence suggests that people who tend to overlook or underestimate analogy’s influence often find themselves struggling to make their arguments or achieve their goals. The converse is also true. Those who construct the clearest, most resonant and apt analogies are usually the most successful in reaching the outcomes they seek.
The key to all of this is figuring out why analogies function so effectively and how they work. Once we know that, we should be able to craft better ones.
## Don’t Think of an Elephant
Effective, persuasive analogies frame situations and arguments, often so subtly that we don’t even realize there is a frame, let alone one that might not work in our favor. Such conceptual frames, like picture frames, include some ideas, images, and emotions and exclude others. By setting a frame, a person or organization can, for better or worse, exert remarkable influence on the direction of their own thinking and that of others.
He who holds the pen frames the story. The first person to frame the story controls the narrative and it takes a massive amount of energy to change the direction of the story. Sometimes even the way that people come across information, shapes it — stories that would be a non-event if disclosed proactively became front page stories because someone found out.
In Don’t Think of an Elephant, George Lakoff explores the issue of framing. The book famously begins with the instruction “Don’t think of an elephant.”
What’s the first thing we all do? Think of an elephant, of course. It’s almost impossible not to think of an elephant. When we stop consciously thinking about it, it floats away and we move on to other topics — like the new email that just arrived. But then again it will pop back into consciousness and bring some friends — associated ideas, other exotic animals, or even thoughts of the GOP.
“Every word, like elephant, evokes a frame, which can be an image of other kinds of knowledge,” Lakoff writes. This is why we want to control the frame rather than be controlled by it.
In Shortcut Pollack tells of Lakoff talking about an analogy that President George W. Bush made in the 2004 State of the Union address, in which he argued the Iraq war was necessary despite the international criticism. Before we go on, take Bush’s side here and think about how you would argue this point – how would you defend this?
In the speech, Bush proclaimed that “America will never seek a permission slip to defend the security of our people.”
As Lakoff notes, Bush could have said, “We won’t ask permission.” But he didn’t. Instead he intentionally used the analogy of permission slip and in so doing framed the issue in terms that would “trigger strong, more negative emotional associations that endured in people’s memories of childhood rules and restrictions.”
Commenting on this, Pollack writes:
Through structure mapping, we correlate the role of the United States to that of a young student who must appeal to their teacher for permission to do anything outside the classroom, even going down the hall to use the toilet.
But is seeking diplomatic consensus to avoid or end a war actually analogous to a child asking their teacher for permission to use the toilet? Not at all. Yet once this analogy has been stated (Farnam Street editorial: and tweeted), the debate has been framed. Those who would reject a unilateral, my-way-or-the-highway approach to foreign policy suddenly find themselves battling not just political opposition but people’s deeply ingrained resentment of childhood’s seemingly petty regulations and restrictions. On an even subtler level, the idea of not asking for a permission slip also frames the issue in terms of sidestepping bureaucratic paperwork, and who likes bureaucracy or paperwork.
## Deconstructing Analogies
Deconstructing analogies, we find out how they function so effectively. Pollack argues they meet five essential criteria.
1. Use the highly familiar to explain something less familiar.
2. Highlight similarities and obscure differences.
3. Identify useful abstractions.
4. Tell a coherent story.
5. Resonate emotionally.
Let’s explore how these work in greater detail. Let’s use the example of master-thief, Bruce Reynolds, who described the Great Train Robbery as his Sistine Chapel.
## The Great Train Robbery
In the dark early hours of August 8, 1963, an intrepid gang of robbers hot-wired a six-volt battery to a railroad signal not far from the town of Leighton Buzzard, some forty miles north of London. Shortly, the engineer of an approaching mail train, spotting the red light ahead, slowed his train to a halt and sent one of his crew down the track, on foot, to investigate. Within minutes, the gang overpowered the train’s crew and, in less than twenty minutes, made off with the equivalent of more than \$60 million in cash.
Years later, Bruce Reynolds, the mastermind of what quickly became known as the Great Train Robbery, described the spectacular heist as “my Sistine Chapel.”
Use the familiar to explain something less familiar
Reynolds exploits the public’s basic familiarity with the famous chapel in the Vatican City, which after Leonardo da Vinci’s Mona Lisa is perhaps the best-known work of Renaissance art in the world. Millions of people, even those who aren’t art connoisseurs, would likely share the cultural opinion that the paintings in the chapel represent “great art” (as compared to a smaller subset of people who might feel the same way about Jackson Pollock’s drip paintings, or Marcel Duchamp’s upturned urinal).
Highlight similarities and obscure differences
Reynold’s analogy highlights, through implication, similarities between the heist and the chapel—both took meticulous planning and masterful execution. After all, stopping a train and stealing the equivalent of \$60m—and doing it without guns—does require a certain artistry. At the same time, the analogy obscures important differences. By invoking the image of a holy sanctuary, Reynolds triggers a host of associations in the audience’s mind—God, faith, morality, and forgiveness, among others—that camouflage the fact that he’s describing an action few would consider morally commendable, even if the artistry involved in robbing that train was admirable.
Identify useful abstractions
The analogy offers a subtle but useful abstraction: Genius is genius and art is art, no matter what the medium. The logic? If we believe that genius and artistry can transcend genre, we must concede that Reynolds, whose artful, ingenious theft netted millions, is an artist.
Tell a coherent story
The analogy offers a coherent narrative. Calling the Great Train Robbery his Sistine Chapel offers the audience a simple story that, at least on the surface makes sense: Just as Michelangelo was called by God, the pope, and history to create his greatest work, so too was Bruce Reynolds called by destiny to pull off the greatest robbery in history. And if the Sistine Chapel endures as an expression of genius, so too must the Great Train Robbery. Yes, robbing the train was wrong. But the public perceived it as largely a victimless crime, committed by renegades who were nothing if not audacious. And who but the most audacious in history ever create great art? Ergo, according to this narrative, Reynolds is an audacious genius, master of his chosen endeavor, and an artist to be admired in public.
There is an important point here. The narrative need not be accurate. It is the feelings and ideas the analogy evokes that make it powerful. Within the structure of the analogy, the argument rings true. The framing is enough to establish it succulently and subtly. That’s what makes it so powerful.
Resonate emotionally
The analogy resonates emotionally. To many people, mere mention of the Sistine Chapel brings an image to mind, perhaps the finger of Adam reaching out toward the finger of God, or perhaps just that of a lesser chapel with which they are personally familiar. Generally speaking, chapels are considered beautiful, and beauty is an idea that tends to evoke positive emotions. Such positive emotions, in turn, reinforce the argument that Reynolds is making—that there’s little difference between his work and that of a great artist.
## Jumping to Conclusions
Daniel Kahneman explains the two thinking structures that govern the way we think: System one and system two . In his book, Thinking Fast and Slow, he writes “Jumping to conclusions is efficient if the conclusions are likely to be correct and the costs of an occasional mistake are acceptable, and if the jump saves much time and effort.”
“A good analogy serves as an intellectual springboard that helps us jump to conclusions,” Pollack writes. He continues:
And once we’re in midair, flying through assumptions that reinforce our preconceptions and preferences, we’re well on our way to a phenomenon known as confirmation bias. When we encounter a statement and seek to understand it, we evaluate it by first assuming it is true and exploring the implications that result. We don’t even consider dismissing the statement as untrue unless enough of its implications don’t add up. And consider is the operative word. Studies suggest that most people seek out only information that confirms the beliefs they currently hold and often dismiss any contradictory evidence they encounter.
The ongoing battle between fact and fiction commonly takes place in our subconscious systems. In The Political Brain: The Role of Emotion in Deciding the Fate of the Nation, Drew Westen, an Emory University psychologist, writes: “Our brains have a remarkable capacity to find their way toward convenient truths—even if they are not all true.”
This also helps explain why getting promoted has almost nothing to do with your performance.
Remember Apollo Robbins? He’s a professional pickpocket. While he has unique skills, he succeeds largely through the choreography of people’s attention. “Attention,” he says “is like water. It flows. It’s liquid. You create channels to divert it, and you hope that it flows the right way.”
“Pickpocketing and analogies are in a sense the same,” Pollack concludes, “as the misleading analogy picks a listener’s mental pocket.”
And this is true whether someone else diverts our attention through a resonant but misleading analogy—“Judges are like umpires”—or we simply choose the wrong analogy all by ourselves.
## Reasoning by Analogy
We rarely stop to see how much of our reasoning is done by analogy. In a 2005 study published in the Harvard Business Review, Giovanni Gavettie and Jan Rivkin wrote: “Leaders tend to be so immersed in the specifics of strategy that they rarely stop to think how much of their reasoning is done by analogy.” As a result they miss things. They make connections that don’t exist. They don’t check assumptions. They miss useful insights. By contrast “Managers who pay attention to their own analogical thinking will make better strategic decisions and fewer mistakes.”
***
Shortcut goes on to explore when to use analogies and how to craft them to maximize persuasion.
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# Obtuse triangle
An obtuse triangle is a triangle that has one angle that has a measure greater than 90° but less than 180°. The other two angles in the triangle are acute angles (< 90°).
## What is an obtuse triangle
An obtuse triangle is a type of triangle characterized by having one interior angle that measures larger than 90°. The remaining two angles must be acute because a triangle's interior angles always sum to 180°. The other types of triangles are acute, right, equilateral, scalene, and isosceles triangles.
### Obtuse triangle definition
An obtuse triangle is a 3-sided polygon in which one of the interior angles measures greater than 90° but less than 180°. The other two angles in the triangle measure less than 90°, and the sum of the 3 angles must equal 180°. An example of an obtuse triangle in real life is part of the roof of a house, as shown in the figure below.
It is possible for an obtuse triangle to be scalene or isosceles, but it cannot be right, equilateral, or acute.
## Types of triangles
There are two different types of obtuse triangles: isosceles and scalene.
### Isosceles obtuse triangle
An isosceles obtuse triangle has two sides of equal length and one much longer side that is opposite the largest interior angle of the triangle. The figure below shows an isosceles obtuse triangle.
### Scalene obtuse triangle
A scalene triangle is a triangle in which none of the sides have equal length. A scalene obtuse triangle is a one in which none of the sides have equal length and one of the internal angles measures greater than 90°. The figure below shows a scalene obtuse triangle.
## Obtuse triangle properties
Below are some properties of obtuse triangles:
• One interior angle of an obtuse triangle measures between 90° and 180°. The other two interior angles measure less than 90° and the sum of the three interior angles is 180°.
• The orthocenter and circumcenter of an obtuse triangle lie outside of the triangle. Its incenter and centroid lie inside the triangle. These points are referred to as points of concurrency of a triangle.
• Given sides a, b, and c, where c is the longest side of an obtuse triangle: c2 > a2 + b2. In other words, the square of the longest side of an obtuse triangle is always greater than the sum of the squares of its shorter sides.
• The side opposite the largest angle measure is the longest side of the triangle.
• A triangle cannot be both obtuse and right (or acute). An obtuse triangle also cannot be equilateral.
### Properties of triangles
Below are properties of all triangles.
• A triangle is a polygon with 3 sides, 3 angles, 3 vertices.
• The lengths of the sides of a triangle correspond to the measures of their angles; the larger the angle, the larger the side; the smaller the angle, the smaller the side. The longest side of a triangle is opposite the angle with the largest measure and the shortest side of the triangle is opposite the smallest angle.
• Angle sum property - The sum of the interior angles of a triangle is always equal to 180°. The sum of the exterior angles of a triangle is always equal to 360°
• Triangle inequality - The sum of the lengths of two sides of a triangle is always larger than the length of the third side; the difference between the lengths of any two sides is less than the length of the third side.
• Exterior angle theorem - The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle.
• Congruent - Two triangles are congruent if all of their corresponding sides and angles are equal.
• Similar - Two triangles are similar if all their corresponding angles are equal and their corresponding sides have the same ratio. In other words, the shape of the triangle is identical, but the size of the triangles are different.
• The sum of consecutive interior and exterior angles of a triangle is supplementary (180°).
## Triangle formulas
Below are some formulas related to obtuse triangles.
### Perimeter of an obtuse triangle
The perimeter of an obtuse triangle is the sum of lengths of its sides. Given a triangle with sides a, b, and c, the perimeter is:
P = a + b + c
There are many different ways to find the lengths of the sides of a triangle given enough information. If the side lengths are known, the perimeter is straightforward to calculate.
### Area of an obtuse triangle
There are a number of different ways to find the area of an obtuse triangle. The typical triangle area formula is
where b is the base and h is the height of the triangle. Any side of the triangle can be chosen as the base; the height for an obtuse triangle can be found by extending the base and drawing the altitude from the vertex outside the triangle down to the extended base. Refer to the figure below.
### Heron's formula
Heron's formula is a formula for the area of a triangle given that all 3 sides of the triangle are known. Heron's formula for the area of a triangle is
where a, b, and c are the sides of the triangle and s is the semiperimeter of the triangle which is found as:
### Converse of the Pythagorean theorem
For a right triangle with hypotenuse of length c, and legs of lengths a and b, the Pythagorean Theorem states:
a2 + b2 = c2
For any triangle, if a2 + b2 < c2, where c is the longest side, the triangle is an obtuse triangle.
Example:
Is triangle △ABC an obtuse triangle?
Side AB above is the longest side. Plugging this into the Pythagorean Theorem where c = 9 and a and b = 6:
a2 + b2 = 62 + 62 = 72
c2 = 92 = 81
72 < 81
Since a2 + b2 < c2, △ABC is an obtuse triangle with obtuse angle C.
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CC-MAIN-2024-38
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https://digitalwhipped.com/list-minus-list-python/
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I don’t know what it is about python, but I love lists. And not just lists of lists of lists. I love the way they can be organized and the way they can be broken down into logical chunks. Python lists are a great way to make a lot of things on your list easier to find and read.
One of the few ways to make something easier to find and read is to put it into a list. Lists give you a lot of flexibility in how you organize the information and they are so easy to use.
The Python list is a great way to organize information. Just add a function to the end of the list to return the items in an efficient and consistent manner. If you ever find yourself reading a list of numbers and you know you can get this list in one line, you can make it easy by making it a list of lists.
There are lots of ways to organize information. Some of them are more efficient and easier to use, some are more efficient and more difficult, and some are just plain easier. For example, the list of numbers. If you can’t get your head around the fact that you have to take a list of numbers and make it a list of lists, you’re in for a big surprise.
If you try to make a list of lists, you will end up with a list of lists of lists. This is a bad idea. List of lists is not the way to go.
In this article, we will be taking a look at one of the more “obvious” methods for creating lists. This is list minus list python. In list minus list python you take a list of numbers and create a new list that contains all the numbers in the list minus the numbers in the previous list.
But what happens if you try to do this with just two sets of numbers? You get a list which is all the numbers in the first set minus the numbers in the second set. In this way, you create a new list with one set of numbers in the list and another set of numbers in the new list. This is why this method works. But it quickly becomes a bit of a headache.
The trick here is that you don’t need to use set(list1) before you use list1. This is because list1 is already a list and you can just take the value of each element from the first list and set the value of each element from the second list. But this is often a little confusing and can cause more headaches than it’s worth.
You’ll notice that the example is using python’s built-in list comprehension; the list is already a list, and the new list is already a list. This is because this is how you can use list comprehension to get the result you want.
The example below will create a new list using the first list and then assign each element from the first list to each element from the second list. This is called a list multiplication. The second list is called the list of lists and is passed as a list to the first method.
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http://botw.org/top/Science/Math/Calculus/
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# Calculus
#### Top/Science/Math/Calculus
A Physicist's View of Teaching Calculus
Information includes the disscusion of topics in exponentials and logarithms, expansions and approximations, and the multi variable calculus.
Automatic Calculus Solutions
Provides mathematical solutions, covering derivatives, integrals, graphs, matrices, determinants, and systems of linear equations.
Calculus Solutions
Includes links to typical calculus topics and sample problems.
Dan's Math: Calculus
Lessons covering limits, differential calculus, integral calculus and vector calculus.
Dr. Vogel's Gallery of Calculus Pathologies
Challenges and refines the intuition of better calculus students and students in advanced calculus. By Tom Vogel.
The Elementary Calculus Line
Information about specific math functions, domains and ranges, and integrals. With derivatives and graphing functions.
Fractional Calculus
Graphics for the Calculus Classroom
Contains excerpts from a collection of graphical demonstrations. With other discussions in math.
Harvey Mudd College Tutorials
Pre-calculus includes algebra review, binomial theorem, and complex numbers. Includes differential equation and single variable calculus.
Langara College: Calculus Resources
Topics in calculus include precalculus review, limits and continuity, derivatives, integration, and infinite series. With other internet resources for calculus and analysis.
MTH 251 Differential Calculus
From the Oregon State University. Includes links to course information and lesson plan.
S.O.S. Math - Calculus
Includes integration studies, clinical sample analysis, and general expression analysis. With contract manufacturing and testing services.
Technology Based Problems
Offers a complex, technology-based problems in calculus with applications in science and engineering. Includes information on how to search categories.
That's Calculus
Includes ordering details, curriculum projects, and contact information.
The University of Minnesota Calculus Initiative
Emphasizes geometric concepts of calculus while examining applications of mathematics to the physical and life sciences. With a contact information for comments.
World Web Math: Calculus Summary
Informs about the two main parts of this area of mathematics, differential and integral calculus. With derivatives and integrals information.
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https://jaxwebster.wordpress.com/2010/04/07/the-light-bulb-and-tower-problem/
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# The Light Bulb and Tower Problem
A puzzle interview question I was recently asked:
You have two identical light-bulbs. You need to find out how strong they are. There is a 100-floor building and you can drop the light-bulbs out of the window. You know that there is some floor such that it will break, and it will break for all above it, but you don’t know what floor that is. It will never break for anything less than that floor.
Find the method of doing this such that the worst-case is as small as possible.
Obviously it is possible to just start at floor 1 and go up until you find it. The worst case then is 100 drops, which is pretty poor. An improvement on that is to go up every other floor, using the second light-bulb to work out which floor it was when it breaks. That gives a worst case of 51 drops.
This can be generalised to stepping up N floors each time. Then the worst case would be:
This function can easily be minimised by taking its derivative and seeing where it is 0. This method gives N = 10, suggesting you should go up every 10th step. The worst case is floor 99, which takes 19 drops.
But if it took 19 drops worst case, why did we bother starting on floor 10? If we started on floor 19 we wouldn’t make things any worse, and we might make them better. The key is: If the worst case is k drops, the first level to try should be k.
What about after that? Well, we can go up k-1 more levels and still the worst case stays the same – you only need to check out the k-1 levels between the two points if it breaks, giving an overall of k drops.
So on the assumption that the worst case is k, the best method is to:
1. Try floor k. If it breaks, try the k-1 floors below it, thus there can be at most k drops. Else go to 2.
2. Try floor k+(k-1). If it breaks, try the k-2 floors below it. Combined with the two drops already done, there can be at most k drops. Else go to 3.
3. Try floor k+(k-1)+(k-2). If it breaks, try the k-3 floors below it. Combined with the three drops already done, there can be at most k drops. Else to go 4.
4. And so on
So what should this value of k be? Well it is the least k such that there is some r such that:
That r may as well be zero, so we just need to find the least k such that the sum of the first k integers is greater than 100. Using the formula, that is just:
As k and k+1 are next to each other, this means the answer has to be somewhere around the square root of 200, which is about 14. Indeed, 14 is the least value, as it is easily seen that 13 does not work.
Therefore the floors you should try are:
• 14, going back to floor 1 if it breaks
• 27, going back to floor 15 if it breaks
• 39, going back to floor 28 if it breaks
• 50, going back to floor 40 if it breaks
• 60, going back to floor 51 if it breaks
• 69, going back to floor 61 if it breaks
• 77, going back to floor 70 if it breaks
• 84, going back to floor 78 if it breaks
• 90, going back to floor 85 if it breaks
• 95, going back to floor 91 if it breaks
• 99, going back to floor 96 if it breaks
• 100, although by now you know it must be 100 without trying it.
I’ve written a little Java program (using Multimedia Fusion) to try out the two approaches. Check it out
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http://au.metamath.org/mpegif/divalglem10.html
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Metamath Proof Explorer < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > divalglem10 Unicode version
Theorem divalglem10 12849
Description: Lemma for divalg 12850. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
divalglem8.1
divalglem8.2
divalglem8.3
divalglem8.4
Assertion
Ref Expression
divalglem10
Distinct variable groups: ,, ,,
Allowed substitution hints: (,)
Proof of Theorem divalglem10
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 divalglem8.1 . . . 4
2 divalglem8.2 . . . 4
3 divalglem8.3 . . . 4
4 divalglem8.4 . . . 4
5 eqid 2387 . . . 4
61, 2, 3, 4, 5divalglem9 12848 . . 3
7 elnn0z 10226 . . . . . . . . . 10
87anbi2i 676 . . . . . . . . 9
9 an12 773 . . . . . . . . . 10
10 ancom 438 . . . . . . . . . . 11
1110anbi2i 676 . . . . . . . . . 10
129, 11bitri 241 . . . . . . . . 9
138, 12bitri 241 . . . . . . . 8
1413anbi1i 677 . . . . . . 7
15 anass 631 . . . . . . 7
1614, 15bitri 241 . . . . . 6
17 oveq2 6028 . . . . . . . . . . 11
1817eqeq2d 2398 . . . . . . . . . 10
1918rexbidv 2670 . . . . . . . . 9
201, 2, 3, 4divalglem4 12843 . . . . . . . . 9
2119, 20elrab2 3037 . . . . . . . 8
2221anbi2i 676 . . . . . . 7
23 ancom 438 . . . . . . 7
24 anass 631 . . . . . . 7
2522, 23, 243bitr4i 269 . . . . . 6
26 df-3an 938 . . . . . . . . 9
2726rexbii 2674 . . . . . . . 8
28 r19.42v 2805 . . . . . . . 8
2927, 28bitri 241 . . . . . . 7
3029anbi2i 676 . . . . . 6
3116, 25, 303bitr4i 269 . . . . 5
3231eubii 2247 . . . 4
33 df-reu 2656 . . . 4
34 df-reu 2656 . . . 4
3532, 33, 343bitr4i 269 . . 3
366, 35mpbi 200 . 2
37 breq2 4157 . . . . 5
38 breq1 4156 . . . . 5
39 oveq2 6028 . . . . . 6
4039eqeq2d 2398 . . . . 5
4137, 38, 403anbi123d 1254 . . . 4
4241rexbidv 2670 . . 3
4342cbvreuv 2877 . 2
4436, 43mpbi 200 1
Colors of variables: wff set class Syntax hints: wa 359 w3a 936 wceq 1649 wcel 1717 weu 2238 wne 2550 wrex 2650 wreu 2651 crab 2653 class class class wbr 4153 ccnv 4817 cfv 5394 (class class class)co 6020 csup 7380 cr 8922 cc0 8923 caddc 8926 cmul 8928 clt 9053 cle 9054 cmin 9223 cn0 10153 cz 10214 cabs 11966 cdivides 12779 This theorem is referenced by: divalg 12850 This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1661 ax-8 1682 ax-13 1719 ax-14 1721 ax-6 1736 ax-7 1741 ax-11 1753 ax-12 1939 ax-ext 2368 ax-sep 4271 ax-nul 4279 ax-pow 4318 ax-pr 4344 ax-un 4641 ax-cnex 8979 ax-resscn 8980 ax-1cn 8981 ax-icn 8982 ax-addcl 8983 ax-addrcl 8984 ax-mulcl 8985 ax-mulrcl 8986 ax-mulcom 8987 ax-addass 8988 ax-mulass 8989 ax-distr 8990 ax-i2m1 8991 ax-1ne0 8992 ax-1rid 8993 ax-rnegex 8994 ax-rrecex 8995 ax-cnre 8996 ax-pre-lttri 8997 ax-pre-lttrn 8998 ax-pre-ltadd 8999 ax-pre-mulgt0 9000 ax-pre-sup 9001 This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2242 df-mo 2243 df-clab 2374 df-cleq 2380 df-clel 2383 df-nfc 2512 df-ne 2552 df-nel 2553 df-ral 2654 df-rex 2655 df-reu 2656 df-rmo 2657 df-rab 2658 df-v 2901 df-sbc 3105 df-csb 3195 df-dif 3266 df-un 3268 df-in 3270 df-ss 3277 df-pss 3279 df-nul 3572 df-if 3683 df-pw 3744 df-sn 3763 df-pr 3764 df-tp 3765 df-op 3766 df-uni 3958 df-iun 4037 df-br 4154 df-opab 4208 df-mpt 4209 df-tr 4244 df-eprel 4435 df-id 4439 df-po 4444 df-so 4445 df-fr 4482 df-we 4484 df-ord 4525 df-on 4526 df-lim 4527 df-suc 4528 df-om 4786 df-xp 4824 df-rel 4825 df-cnv 4826 df-co 4827 df-dm 4828 df-rn 4829 df-res 4830 df-ima 4831 df-iota 5358 df-fun 5396 df-fn 5397 df-f 5398 df-f1 5399 df-fo 5400 df-f1o 5401 df-fv 5402 df-ov 6023 df-oprab 6024 df-mpt2 6025 df-1st 6288 df-2nd 6289 df-riota 6485 df-recs 6569 df-rdg 6604 df-er 6841 df-en 7046 df-dom 7047 df-sdom 7048 df-sup 7381 df-pnf 9055 df-mnf 9056 df-xr 9057 df-ltxr 9058 df-le 9059 df-sub 9225 df-neg 9226 df-div 9610 df-nn 9933 df-2 9990 df-3 9991 df-n0 10154 df-z 10215 df-uz 10421 df-rp 10545 df-fz 10976 df-seq 11251 df-exp 11310 df-cj 11831 df-re 11832 df-im 11833 df-sqr 11967 df-abs 11968 df-dvds 12780
Copyright terms: Public domain W3C validator
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https://convertoctopus.com/86-cubic-centimeters-to-fluid-ounces
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## Conversion formula
The conversion factor from cubic centimeters to fluid ounces is 0.033814022558919, which means that 1 cubic centimeter is equal to 0.033814022558919 fluid ounces:
1 cm3 = 0.033814022558919 fl oz
To convert 86 cubic centimeters into fluid ounces we have to multiply 86 by the conversion factor in order to get the volume amount from cubic centimeters to fluid ounces. We can also form a simple proportion to calculate the result:
1 cm3 → 0.033814022558919 fl oz
86 cm3 → V(fl oz)
Solve the above proportion to obtain the volume V in fluid ounces:
V(fl oz) = 86 cm3 × 0.033814022558919 fl oz
V(fl oz) = 2.9080059400671 fl oz
The final result is:
86 cm3 → 2.9080059400671 fl oz
We conclude that 86 cubic centimeters is equivalent to 2.9080059400671 fluid ounces:
86 cubic centimeters = 2.9080059400671 fluid ounces
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 fluid ounce is equal to 0.34387825218023 × 86 cubic centimeters.
Another way is saying that 86 cubic centimeters is equal to 1 ÷ 0.34387825218023 fluid ounces.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that eighty-six cubic centimeters is approximately two point nine zero eight fluid ounces:
86 cm3 ≅ 2.908 fl oz
An alternative is also that one fluid ounce is approximately zero point three four four times eighty-six cubic centimeters.
## Conversion table
### cubic centimeters to fluid ounces chart
For quick reference purposes, below is the conversion table you can use to convert from cubic centimeters to fluid ounces
cubic centimeters (cm3) fluid ounces (fl oz)
87 cubic centimeters 2.942 fluid ounces
88 cubic centimeters 2.976 fluid ounces
89 cubic centimeters 3.009 fluid ounces
90 cubic centimeters 3.043 fluid ounces
91 cubic centimeters 3.077 fluid ounces
92 cubic centimeters 3.111 fluid ounces
93 cubic centimeters 3.145 fluid ounces
94 cubic centimeters 3.179 fluid ounces
95 cubic centimeters 3.212 fluid ounces
96 cubic centimeters 3.246 fluid ounces
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en
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https://www.spdba.com.au/stround/
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crawl-data/CC-MAIN-2024-26/segments/1718198861606.63/warc/CC-MAIN-20240615190624-20240615220624-00365.warc.gz
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STRound
STRound — Function which rounds the ordinates of the supplied geometry.
Function Specification
```Function [dbo].[STRound] (
@p_geometry geometry,
@p_round_xy int = 3,
@p_round_zm int = 2
)
Returns geometry
```
Description
The result of many geoprocessing operations in any spatial type can be geometries whose ordinates (X, Y etc) have far more decimal digits of precision than the initial geometry.
Additionally, some input GIS formats, such as shapefiles (which has no associated precision model), when loaded, can show far more decimal digits of precision in the created ordinates misrepresenting the actual accuracy of the data.
STRound takes a geometry object and some specifications of the precision of any X, Y, Z or M ordinates, applies those specifications to the geometry and returns the corrected geometry.
The @p_round_xy/@p_round_zm values are decimal digits of precision, which are used in TSQL’s ROUND function to round each ordinate value.
Parameters
``` @p_geometry (geometry) - supplied geometry of any type.
@p_round_xy (int) - Decimal degrees of precision to which calculated ordinates are rounded.
@p_round_zm (int) - Decimal degrees of precision to which calculated ordinates are rounded.
```
Result
This function accesses all ordinates in the supplied geometry, rounding all ordinates using the supplied parameter values.
Examples
```-- Geometry
-- Point
SELECT [dbo].[STRound](geometry::STPointFromText('POINT(0.345 0.282)',0),1,1).STAsText() as RoundGeom
UNION ALL
-- MultiPoint
SELECT [dbo].[STRound](geometry::STGeomFromText('MULTIPOINT((100.12223 100.345456),(388.839 499.40400))',0),3,1).STAsText() as RoundGeom
UNION ALL
-- Linestring
SELECT [dbo].[STRound](geometry::STGeomFromText('LINESTRING(0.1 0.2,1.4 45.2)',0),2,1).STAsText() as RoundGeom
UNION ALL
-- LinestringZ
SELECT [dbo].[STRound](geometry::STGeomFromText('LINESTRING(0.1 0.2 0.312,1.4 45.2 1.5738)',0),2,1).AsTextZM() as RoundGeom
UNION ALL
-- Polygon
SELECT [dbo].[STRound](geometry::STGeomFromText('POLYGON((0 0,10 0,10 10,0 10,0 0))',0),2,1).STAsText() as RoundGeom
UNION ALL
-- MultiPolygon
SELECT [dbo].[STRound](
geometry::STGeomFromText('MULTIPOLYGON (((160 400, 200.00000000000088 400.00000000000045, 200.00000000000088 480.00000000000017, 160 480, 160 400)), ((100 200, 180.00000000000119 300.0000000000008, 100 300, 100 200)))',0),
2,1).STAsText() as RoundGeom
RoundGeom
POINT (0.3 0.3)
MULTIPOINT ((100.122 100.345), (388.839 499.404))
LINESTRING (0.1 0.2, 1.4 45.2)
LINESTRING (0.1 0.2 0.3, 1.4 45.2 1.6)
POLYGON ((0 0, 10 0, 10 10, 0 10, 0 0))
MULTIPOLYGON (((160 400, 200 400, 200 480, 160 480, 160 400)), ((100 200, 180 300, 100 300, 100 200)))
-- Geography
-- Can't overload existing STRound so have to use conversion functions.
SELECT [dbo].STToGeography(
[dbo].[STRound](
[dbo].STToGeometry(
geography::STGeomFromText('LINESTRING(141.29384764892390 -43.93834736282 234.82756,
141.93488793487934 -44.02323872332 235.26384)',
4326),
4326
),
7,
3
),
4326
).AsTextZM() as rGeom;
rGeom
LINESTRING (141.2938476 -43.9383474 234.828, 141.9348879 -44.0232387 235.264)
```
I hope this function is of use to someone.
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8
daniel-wu
340d
# Our job is safe, guys. :-)
Comments
• 1
I don't get it, GPT isn't wrong that 1 isn't standing..
• 4
@MammaNeedHummus pretty sure the others would fly off when they hear the shot
• 1
@MammaNeedHummus I think the remaining birds will fly away. There will be no birds left.
• 1
If your job requires to solve dumb riddles, then yes. But not for long.
• 1
@MammaNeedHummus Maybe it was a stupid question to begin with.
The individual already had the 'correct' answers in their head so no matter how ChatGTP answered, ChatGTP would be 'wrong'.
• 2
@PaperTrail if chatgpt had answered: "mathematically speaking there would be 2, but because birds perceive loud noise as potential danger they would fly away from the sound of the shot and there would be none." That would be good. And if you're a sufficiently skilled "proompt engineer" you could probably get there.
You need to provide enough context to steer chatGPT in the right direction.
• 2
In case some people didn't able to catch it: I did provide the exact direction for my question.
Yeah.. The original question of this 'dumb' riddle is: how many birds are left? This opens up many interpretation. Do you mean how many birds are left alive? How many birds are left standing? How many birds stays on the tree branch? It's always wrong no matter what you answer.
My riddle however, is not dumb, the question and the answer is obvious. How many birds are left standing on the tree branch? The answer is NONE. Null. Nada. There is no trick, mislead or confusion here. The only dumb one here is ChatGPT, for not being able to give the correct answer.
• 4
@sutekh42 I like to think the birds are so shocked they're stuck to the branch
• 1
@bigmonsterlover Dude.. you really go a long way to defend chatgpt huh. Would you listen to yourself? WTF do you mean by shoot down doesn't really mean shot down? I shot it, I didn't miss. Also there is a down word there. I downed it.
Also, since when dead birds can stand? Can you still stand up when you are dead??
• 0
@daniel-wu I think the point is rather that the statement is full of assumptions about the likelihood of the situation.
As humans having witnessed birds on branches a lot of times (which chatGPT has not) we know that the following scenario's are extremely unlikely:
- the birds could be aligned on the branch 1 behind the other, then the bullet would have a chance to pierce a second one's wing disabling its ability to fly away.
- the birds could be 2 stuffed animals glued to the branch and 1 real bird (that would make 3 dead and 2 on the branch)
Also how "many birds are left, alive, on the branch" doesn't specify when: at bullet contact, 1s later, 5s later, 10min later?
• 1
@webketje I don’t think that underspecification is the problem here.
It’s just that the whole scenario is purposely constructed so that the answer can be very different depending on if you just consider the technical situation or a real life situation with human experience.
• 3
really it depends what firmware version the birds are running
• 0
It’s like asking
"If you drink 1,000 liters of water each second, how many liters of water will you have in your stomach after 10 seconds?"
Technically, it would be 10,000 liters.
But you could also say that it’s impossible to drink that much water and start analyzing real world possibilities.
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# Regarding a consumption aggregator: How do I differentiate under the integral sign?
Let $\varepsilon>1$ and let $$C_t\equiv\left(\int_0^1C_t(i)^{(\varepsilon-1)/\varepsilon} \, di\right)^{\varepsilon/(\varepsilon-1)}$$ denote a consumption basket in time period $t$, where $C_t(i)$ is consumption of good $i\in [0,1]$. In e.g. new Keynesian models we want to differentiate $C_t$ with respect to $C_t(i)$ for some $i\in [0,1]$ so as to solve a utility optimization problem. In my lecture notes, and in many texts on this subject, it is said that $$\frac{\partial C_t}{\partial C_t(i)} = \frac{\varepsilon}{\varepsilon-1} C_t^{1/\varepsilon} \frac{\varepsilon-1}{\varepsilon} C_t(i)^{-1/\varepsilon}.$$ Does anyone know how this differentiation is accomplished? This is my question I want answered. Below I will outline how I have thought about this question.
I am prone to thinking that it is wrong. For using the chain rule I would say that the answer is the following. $$\frac{\partial C_t}{\partial C_t(i)} = \frac{\varepsilon}{\varepsilon-1} C_t^{1/\varepsilon} \left(\frac{\partial}{\partial C_t(i)} \int_0^1C_t(i)^{(\varepsilon-1)/\varepsilon} \, di\right),$$ which, when asssuming that the function is such that we may differentiate under the integral sign, I get $$\frac{\partial C_t}{\partial C_t(i)} = \frac\varepsilon {\varepsilon-1} C_t^{1/\varepsilon} \left(\int_0^1 \frac{\varepsilon-1} \varepsilon C_t(i)^{-1/\varepsilon} \, di\right).$$ Now, using the mean value theorem for integrals it would be possible to say that $$\int_0^1 C_t(i)^{-1/\varepsilon} \, di = C_t(j)^{-1/\varepsilon}(1-0)$$ for some $j\in (0,1)$, and insert this result above and then get a similar result to what was shown in my lecture notes. However, this would lead us to considering another good $j$ not necessarily equal to good $i$.
The reader may think that I am confusing the symbol '$i$' in the integral, for the same symbol used when differentiating with respect to $C_t(i)$, and that I should, when differentiating, consider a good $i_0$, and then perform the following differentiation: $$\frac{\partial C_t}{\partial C_t(i_0)} = \frac{\partial }{\partial C_t(i_0)} \left(\int_0^1 C_t(i)^{(\varepsilon-1)/\varepsilon} \, di\right)^{\varepsilon/(\varepsilon-1)}.$$ This may be so, but I do not know how to get the desired result from this, and if I take this approach, I would say that the derivative is equal to $0$ (!) as the integral is just a real constant if $t$ is fixed, which it is.
It is sometimes said that we may differentiate the integral just mentioned by looking at the integral as beeing a sum. What they mean by this, I do not know. Maybe they represent the integral as the limit of a Riemann sum, which it is, and write $$\frac{\partial }{\partial C_t(i_0)} \int_0^1 C_t(i)^{(\varepsilon-1/\varepsilon} \, di = \frac{\partial }{\partial C_t(i_0)} \lim_{n\to\infty} \sum_{k=1}^n C_t(\xi_k)^{(\varepsilon-1)/\varepsilon}(i_k-i_{k-1}),$$ with $i_0=0<i_1<\cdots < i_{n-1}<i_n=1$ and $i_{k-1}\leq\xi_k\leq i_k$ for each $k=1,2\ldots,n$. When the authors write that we should look at the integral as beeing a sum, this must be it. But differentiating this sum with respect to $C_t(i_0)$ would in the best cases (i.e., when we can do differentiation inside the limit) be equal to $\lim_{n\to\infty}\frac{\varepsilon-1} \varepsilon C_t(i_0)^{1/(\varepsilon-1)} \cdot (i_\alpha - i_{\alpha-1})$ for some $\alpha\in\{1,2,\ldots,n\}$ such that $i_{\alpha-1}\leq i_0\leq i_\alpha$; the problem now is that $\lim_{n\to\infty}\frac{\varepsilon-1} \varepsilon C_t(i_0)^{-1/\varepsilon}\cdot (i_\alpha - i_{\alpha-1})=0$, which is consistent with modern advanced real analysis (to my knowledge) in the sense that if we just increase or decrease the value of $C_t(i)$ at one $i=i_0$, then the value of the integral will not change, and hence the derivative should be $0$ (i.e., no change in the value of the integral for a change in $C_t(i_0)$).
Note: These problems occur when studying e.g. the so called "Dixit-Stiglitz aggregator".
• This answer has some related math literature, economics.stackexchange.com/a/231/61 Nov 5, 2016 at 19:14
• @AlecosPapadopoulos Thanks for the link! I've been thinking about that problem also. Nov 5, 2016 at 19:34
Using your formalism above, you can think (heuristically) of the integral as $$\sum_{i=1}^nC_t(i)^{(\epsilon-1)/\epsilon}$$
If we differentiate this with respect to $C_t(j)$, we get
$$\frac{\epsilon - 1}{\epsilon} C_t(j)^{-1/\epsilon}$$
Which is exactly what we needed. To do this rigorously, you need a notion of taking derivatives on function spaces. Look up the Gâteaux and Fréchet derivatives.
• Thanks for your answer! I am looking for a rigorous answer, how would you use the concept of a Gâteaux derivative to show the above result? As I said below, I think the right way to understand this is through an understaning of functional derivatives (e.g., Gâteaux derivatives), what do you think about my answer below? Is this how you think about it? Nov 5, 2016 at 18:02
• Yes, I am basically using the same notion of the functional derivative you give there. Here's something that I think should work; let me know if it doesn't. Set $\eta_t (i) = 1$ for $i = i_0$ and $0$ otherwise. The integral sign disappears, and we can differentiate $f(x)=x^{(\epsilon-1)/\epsilon}$ as we would normally. Nov 7, 2016 at 13:23
• Hmmm, I've tried to understand your answer, but I don't get it yet. From my perspective, your approach leads just to a vanishing integral in the sense that it equals 0. Can you explain your answer in more detail? Nov 8, 2016 at 20:50
• As I see it, if $F(C_t):=\int_0^1C_t(i)^{(\epsilon-1)/\epsilon}$, then $\lim_{\epsilon\to 0^+}\frac{F(C_t+\epsilon\eta_t)-F(C_t)}{\epsilon}=\lim_{\epsilon\to 0^+}\int_0^1 \frac{(C_t(i)+\epsilon\eta_t(i))^{(\epsilon-1)/\epsilon}-C_t(i)^{(\epsilon-1)/\epsilon}}{\epsilon}\, di$, then this becomes $\int_0^1\frac{\epsilon-1}{\epsilon}C_t(i)^{-1/\epsilon}\eta_t(i)\, di$. Hence, $\frac{\delta F}{\delta C_t(i)}=\frac{\epsilon-1}{\epsilon}C_t(i)^{-1/\epsilon}$. This argument is based on the strong assumption of taking the limit inside the integral, and my weak understanding of Gâteaux derivatives. Nov 8, 2016 at 20:58
• Ah, yes, you're right. My mistake. I suspect a slightly sneakier choice for $eta_t$ should work. Alternatively, assume that $C_t(i)$ is uniformly bounded (for all $i$ -- I suspect you need this in equilibrium anyway, if not for integrability). You can then appeal to the dominated convergence theorem. Nov 11, 2016 at 16:48
I think I can answer my own question, so I will answer it here so as to mark it as an answer.
I think the problem boils down to differentiating a functional $$F(C_t)=\int_0^1C_t(i) \, di$$ where I purposefully have ignored the exponent $\frac{\varepsilon-1}{\varepsilon}$ stated in my question. To give meaning to the notion of a partial derivate, see Functionals and the Functional Derivative. Basically, when studying the differential of a functional w.r.t to its argument, we study the directional derivative $$\lim_{\varepsilon\to 0⁺}\frac{F(C_t+\varepsilon\eta_t)-F(C_t)}{\varepsilon},$$ where $\eta$ is some continuous test function, and define the first partial derivative of the functional w.r.t. $C_t(i)$ for some given good $i\in [0,1]$, which I write as $\frac{\delta F(C_t)}{\delta C_t(i)}$, in such a way that $$\lim_{\varepsilon\to 0⁺}\frac{F(C_t+\varepsilon\eta_t)-F(C_t)}{\varepsilon}=:\int_0^1 \, di \frac{\delta F(C_t)}{\delta C_t(i)}\eta_t(i).$$
In our case, the directional derivative is $$\int_0^1 \eta_t(i) \, di$$ and so $\frac{\delta F(C_t)}{\delta C_t(i)}=1$. For the functional $G(C_t)=\int_0^1C_t(i)p_t(i) \, di$, where $p_t(i)$ is the price of good $i$ in time period $t$, we have the directional derivative $$\int_0^1\eta_t(i)p_t(i) \, di$$ and thus $\frac{\delta G(C_t)}{\delta C_t(i)}=p_t(i)$.
To generalize, I conjecture that, under some not so strong conditions, we have that for a functional $\int_0^1 H(C_t(i),i) \, di$, where $H$ is some continuously differentiable function, we get that $\frac{\partial H}{\partial C_t(i)}$ will be the partial derivative of the functional with respect to $C_t(i)$
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# SVGBasics
Turn SVG On
I find that the percentage method isolates thinking to the current gradient without concern about where it's being used. A radialGradient produces a ring (or rings) of colour instead of stripes. Instead of a line, it uses a circle whose center is (cx,cy) and radius is r. When using percentages, remember r=50% means the diameter of the circle will be 100% of the gradient.
<?xml version="1.0" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
<svg viewBox = "0 0 1100 400" version = "1.1">
<defs>
<rect id = "r1" width = "100" height = "100" stroke = "black" stroke-width = "1"/>
<circle id = "r2" cx = "100" cy = "100" r = "50" stroke = "black" stroke-width = "1"/>
<circle id = "r3" cx = "100" cy = "100" r = "100" stroke = "black" stroke-width = "1"/>
<radialGradient id = "g1" cx = "50%" cy = "50%" r = "50%">
<stop stop-color = "black" offset = "0%"/>
<stop stop-color = "teal" offset = "50%"/>
<stop stop-color = "white" offset = "100%"/>
</defs>
<use x = "350" y = "0" xlink:href = "#r1" fill = "url(#g1)"/>
<use x = "350" y = "150" xlink:href = "#r2" fill = "url(#g1)"/>
<use x = "580" y = "125" xlink:href = "#r3" fill = "url(#g1)"/>
</svg>
An interesting thing about the radial gradient is that the point from which it radiates can be moved from the center of the circle by changing the focal point (fx,fy). Here, fx=25% and fy=25%.
<?xml version="1.0" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
<svg viewBox = "0 0 1100 400" version = "1.1">
<defs>
<rect id = "r1" width = "100" height = "100" stroke = "black" stroke-width = "1"/>
<circle id = "r2" cx = "100" cy = "100" r = "50" stroke = "black" stroke-width = "1"/>
<circle id = "r3" cx = "100" cy = "100" r = "100" stroke = "black" stroke-width = "1"/>
<!-- note the 'fx' and 'fy' attributes -->
<radialGradient id = "g1" cx = "50%" cy = "50%" r = "50%" fx = "25%" fy = "25%">
<stop stop-color = "black" offset = "0%"/>
<stop stop-color = "teal" offset = "50%"/>
<stop stop-color = "white" offset = "100%"/>
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Alexei Averchenko
Reputation
3,255
Top tag
Next privilege 5,000 Rep.
Approve tag wiki edits
Jan21 awarded Popular Question Jan5 awarded Popular Question Nov21 awarded Yearling Nov13 awarded Necromancer Nov3 awarded Favorite Question Oct9 awarded Popular Question Sep30 awarded Explainer Sep24 awarded Autobiographer Aug14 revised How can I prove that $xy\leq x^2+y^2$? added 637 characters in body Aug14 comment How can I prove that $xy\leq x^2+y^2$? @ronno Not really, because $X := \mathbb{R}^2 \setminus \{x = -y\}$ is dense in $\mathbb{R}^2$, and $[0, +\infty)$ is closed in $\mathbb{R}$, therefore its pullback along $(x, y) \mapsto x^2 - xy + y^2$ must be $\operatorname{cl}X = \mathbb{R}^2$. Of course it is too technical for a precalculus-level question. Jul2 awarded Curious Jul2 awarded Inquisitive May10 awarded Good Question Mar30 revised How can I prove that $xy\leq x^2+y^2$? Removed dickishness Feb20 awarded Popular Question Nov21 awarded Yearling Nov16 awarded Notable Question Aug19 comment Rigour in mathematics @RobertIsrael Suppose there is a proposition, and you can prove that it is impossible to construct a counterexample to it. Is this proposition true? Aug17 awarded Nice Answer Aug11 answered Rigour in mathematics
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# Data structure recommendation?
Steven Clark steven.p.clark at gmail.com
Tue Apr 8 17:49:39 CEST 2008
```> bisect is definitely the way to go. You should take care with
> floating point precision, though. One way to do this is to choose a
> number of digits of precision that you want, and then internally to
> your class, multiply the keys by 10**precision and truncate, so that
> you are working with ints internal to the Foo class.
Thanks for the reply. Can you explain how I could be bitten by
floating point precision here?
I'm familiar with how&why 1.3*3 != 3.9, etc., but I'm not sure how it
applies here, or what you are gaining by converting to int.
What do you guys think of this approach which uses tuples:
from bisect import insort_right, bisect_right
class Heavy(object):
def __cmp__(self, other):
return 1
heavy = Heavy()
class Foo(object):
def __init__(self):
self.data = []
def __setitem__(self, k, v):
#if k's are the same, will be sorted by v's. may or may not be
desireable
insort_right(self.data, (k, v))
def __getitem__(self, k):
i = bisect_right(self.data, (k, heavy))
if i == 0:
return None
else:
return self.data[i-1][1]
def main():
foo = Foo()
assert(foo[1.5] == None)
foo[1.3] = 'a'
foo[2.6] = 'b'
assert(foo[1.2999] == None)
assert(foo[1.3] == 'a')
assert(foo[1.5] == 'a')
assert(foo[2.6] == 'b')
assert(foo[7.8] == 'b')
foo[5.0] = 'c'
assert(foo[7.8] == 'c')
print('Foo checks passed.')
if __name__ == '__main__':
main()
```
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A particular contour integral
Mathoverflow,
I'd like to carry out the following integral,
$$f(t) = \int_{- \infty}^{\infty}\frac{-i\Omega e^{i \Omega t}}{1-\sqrt{-i\Omega}\coth(\sqrt{-i\Omega})} d\Omega.$$
Here's what I've done so far. First, there is a removable singularity at the origin that is of no import. Second, there are infinitely many poles along the negative imaginary axis. These singularities are simple poles of order one and correspond to roots of the denominator of the integrand. We label these roots $\Omega_j=-ir_j^2$ with $r_j>0$. The locations of the simple poles are therefore obtained by solving for the $r_j$ such that $\tan(r_j) + r_j = 0$.
I take the contour $\gamma$ to be in the negative half plane, including the real axis and around all of these singularities.
We evaluate the residue of $f(\sqrt{\Omega})$ as $p/q^{\prime}$ where $p/q \equiv f$. This results in $$\text{Res}\left(f,\Omega_j\right) = \frac{-2 i (-i \Omega_j)^{3/2} e^{i\Omega_j t}}{\coth(\sqrt{-i \Omega_j})-\sqrt{-i \Omega_j}\text{csch}^2(\sqrt{-i \Omega_j})}$$ $$= \frac{-2 i (-i r_j)^{3} e^{r_j^2 t}}{i\cot r_j-ir_j\csc^2r_j}$$ $$= -2i\frac{ r_j^{3} e^{r_j^2 t}}{\cot r_j-r_j\csc^2r_j}$$ $$= 2i\frac{ r_j^{3} e^{r_j^2 t}}{1/r_j - r_j - 1/r_j}$$ $$= -2i r_j^2 e^{r_j^2 t}$$
This gives the integral as,
$$f(t) = 4\pi\sum_{j=0}^{\infty} r_j^2 e^{r_j^2 t}$$
My troubles are:
1. This series does not converge. I would like the exponential to have a negative power, but I need the integral to be in the negative half plane since this is where all of the singularities are. I'm not sure how to resolve this.
2. In the frequency domain, I take the limit as $\Omega\to0$ and find the integrand to be $-3$, suggesting the inverse to be $f(t) = -3\delta(t)$. I'm not able to recover this limit.
• you are closing the integral in the negative half plane, so you have assumed $t<0$ and then your sum over $j$ converges; if $t>0$, you will close the integral in the positive half plane, resulting in zero. – Carlo Beenakker Sep 27 '13 at 7:34
Carlo Beenakker's answer is right. When t>0, you cannot close the contour in the lower half-plane, because the exp in the numerator is large in the lower half-plane. You must close the contour in the upper half-plane. Then your integral is 0, for $t>0$. When $t<0$ your series computed with residues converges.
• I disagree (or I'm wrong and am confused): the origin is not a pole because $$\lim_{x\to0} \hat F(x) = \frac{-i x e^{i x t}}{1-\sqrt{-i x} \coth(\sqrt{-i x})} = 3$$ Furthermore, in the Taylor series expansion of $\hat F(x)$, the root in the denominator at the origin cancels. – DieLuftDerFreiheit Sep 27 '13 at 19:35
• I agree that the contour should be closed in the upper half plane. I now realize that there is a pole at infinity of order 1/2. How do I account for this pole at infinity when summing residues? – DieLuftDerFreiheit Sep 27 '13 at 19:36
• Sorry, 0 is indeed removable. Therefore, the integral is 0. – Alexandre Eremenko Sep 28 '13 at 2:10
• There is no "pole at infinity". Infinity is an essential singularity. But the function under the integral decreases sufficiently fast in the upper half-plane to close the contour. – Alexandre Eremenko Sep 28 '13 at 2:11
• I'm a bit confused: for $t>0$ the integrand is indeed zero, but why would it be zero for $t<0$? Then the contour has to be closed in the lower half-plane, resulting in this sum over residues $r_j$ (which converges for $t<0$). – Carlo Beenakker Sep 28 '13 at 13:48
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{{short description|Problem or enigma that tests the ingenuity of the solver}}{{Puzzles|Types}}{{other uses}}A puzzle is a game, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together in a logical way, in order to arrive at the correct or fun solution of the puzzle. There are different genres of puzzles, such as crossword puzzles, word-search puzzles, number puzzles, relational puzzles, or logic puzzles.Puzzles are often created to be a form of entertainment but they can also arise from serious mathematical or logistical problems. In such cases, their solution may be a significant contribution to mathematical research.Kendall G.; Parkes A.; and Spoerer K. (2008) A Survey of NP-Complete Puzzles, International Computer Games Association Journal, 31(1), pp 13–34.
## Etymology
The 1989 edition of the Oxford English Dictionary dates the word puzzle (as a verb) to the end of the 16th century. Its first documented use (to describe a new type of game) was in a book titled The Voyage of Robert Dudley...to the West Indies, 1594–95, narrated by Capt. Wyatt, by himself, and by Abram Kendall, master (published circa 1595). The word later came to be used as a noun.The word puzzle comes from pusle, meaning "bewilder, confound", which is a frequentive of the obsolete verb pose (from Medieval French aposer) in the sense of "perplex". The use of the word to mean "a toy contrived to test one's ingenuity" is relatively recent (within mid-19th century).
## Genres
(File:Set of various puzzles.jpg|thumb|Various puzzles)Puzzles can be divided into categories. For example, a maze is a type of tour puzzle. Some other categories are construction puzzles, stick puzzles, tiling puzzles, disentanglement puzzles, lock puzzles, folding puzzles, combination puzzles, and mechanical puzzles.
## Puzzle solving
{{Original research section|date=November 2018}}Solutions of puzzles often require the recognition of patterns and the adherence to a particular kind of ordering. People with a high level of inductive reasoning aptitude may be better at solving such puzzles than others. But puzzles based upon inquiry and discovery may be solved more easily by those with good deduction skills. Deductive reasoning improves with practice. Mathematical puzzles often involves BODMAS. BODMAS is an acronym and it stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. In certain regions, PEDMAS (Parentheses, Exponents, Division, Multiplication, Addition and Subtraction) is the synonym of BODMAS. It explains the order of operations to solve an expression. Some mathematical puzzle requires Top to Bottom convention to avoid the ambiguity in the order of operations. It is an elegantly simple idea that relies, as sudoku does, on the requirement that numbers appear only once starting from top to bottom as coming along.{{Citation needed|date=November 2018}}
## Puzzle makers
Puzzle makers are people who make puzzles. In general terms of occupation, a puzzler is someone who composes and/or solves puzzles.Some notable creators of puzzles are:
## History of jigsaw puzzles
Jigsaw puzzles are perhaps the most popular form of puzzle. Jigsaw puzzles were invented around 1760, when John Spilsbury, a British engraver and cartographer, mounted a map on a sheet of wood, which he then sawed around the outline of each individual country on the map. He then used the resulting pieces as an aid for the teaching of geography.After becoming popular among the public, this kind of teaching aid remained the primary use of jigsaw puzzles until about 1820.History of Jigsaw Puzzles The American Jigsaw Puzzle SocietyThe largest puzzle (40,320 pieces) is made by German game company Ravensburger.WEB,weblink The worlds biggest Puzzle {{!, Ravensburger|website=www.ravensburger.us|language=en|access-date=2018-06-23}} The smallest puzzle ever made was created at LaserZentrum Hannover. It is only five square millimetres, the size of a sand grain.
## History of other puzzles
By the early 20th century, magazines and newspapers had found that they could increase their readership by publishing puzzle contests, beginning with crosswords and in modern days sudoku.
## Organizations and events
There are organizations and events that cater to puzzle enthusiasts, such as:
• {{annotated link|List of impossible puzzles}}
• {{annotated link|List of Nikoli puzzle types}}
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# Cosmology Theory
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Clearly you missed my point.Lets try this - the volume of 10^-50 ccm is based on nothing physical. Your number of 10^50 JAH is based on nothing physical.You picked those 2 numbers because the cube root of the quotient equals the Plank Length. There must be some PHYSICAL REASON for picking those numbers or the rest of the calculations are just so much drivelAnd to top it all off the cube root of 10^-100 does not even equal the Planck Length.You should learn some real physics, it is much more interesting than this woo-woo stuff. <br />Posted by origin</DIV></p><p>origin:</p><p>take the cube root on your calculater and tell me your results.<br /></p>
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#### origin
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>origin:take the cube root on your calculater and tell me your results. <br />Posted by bechcube</DIV></p><p>I get a number. The number is 4.64 x 10^-34. This particular number is about 3 times smaller than the planck length.</p><p>What is the physical significance of this number?</p><p>What is the physical significance of 10^50.</p><p>What is the physical significance of 10^-50.</p> <div class="Discussion_UserSignature"> </div>
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#### origin
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Heheh! I laughed out loud when I saw you phrase it like that! I found myself wondering if it was more relevent if I am not a physicist, or if I am not sane... <br />Posted by SpeedFreek</DIV><br /><br />Glad you liked it, I was a bit uncomfortable putting words in your mouth so to speak... Like several here I thought your post was well written - regardless if a certain individual (who will remain unnamed) was not able to follow it.</p><p> </p> <div class="Discussion_UserSignature"> </div>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I get a number. The number is 4.64 x 10^-34. This particular number is about 3 times smaller than the planck length.What is the physical significance of this number?What is the physical significance of 10^50.What is the physical significance of 10^-50. <br />Posted by origin</DIV></p><p>origin:</p><p>Your analysis is true when accepting the coefficient 1.6.</p><p>However, my calculation is based on 1x10>-33 1/3 which my theory presents as the more correct value of the Plank length.</p><p>Further, this constant produces very close but more accurate values for the standard constants, that is, mass of the electron, proton and neutron. </p><p>Sure, they are slightly different than those calculated using the 1.6</p><p>coefficient but that is the entire purpose of the theory.</p><p>Why 10>50 and 10>-50? These numbers fit the perfect isosceles triangle having 10>50 at each junction and they fit the diameter of the circular strings or equator of the JAH, 1x10>-33cm. A circular string deletes the need for the fanciful 11 dimensions created by a linear string.</p><p> </p><p><br /><br /> </p>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Wrong. There is no edge to be close to. There is only an apparent 'edge' of the OBSERVABLE universe.That is only if your belief is correct about and edge moving out from a center, which all sane physicists and speedfreak do not agree withHe did not support it, you simply misunderstood - what a shock. Could you support the contention that galaxies are continuosly be formed throughout the universe?The ONLY galaxies which are NOT moving away from us are the ones that are very close to us and we are interacting with, which supports that theory that the universe (ie space) is expanding and there is no center (or edge).No, really wrong! <br />Posted by origin</DIV></p><p>Sorry! Who is writing and to whom?</p><p>Anyway, many galaxies are detected moving tangiential to the Milky Way motion.</p><p>Yet, they all, on average, move away from us and towards the observed edge of the universe.<br /></p>
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#### DrRocket
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>There is an edge to the observable universe, as light has only had 13.7 billion years in which to travel, and light travels at a finite speed. But we think the edge is relative to the observer. <br />Posted by SpeedFreek</DIV></p><p>Be careful with terminology here. That is as far as you can see because light can have traveled no further since time zero. But it is not more an edge (because that is as far as you can see) than the horizon is the edge of the Earth.</p><p>So far as anyone knows, and so far as the model used for a description of space-time the universe is a manifold without boundary -- no edge. <br /></p> <div class="Discussion_UserSignature"> </div>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Be careful with terminology here. That is as far as you can see because light can have traveled no further since time zero. But it is not more an edge (because that is as far as you can see) than the horizon is the edge of the Earth.So far as anyone knows, and so far as the model used for a description of space-time the universe is a manifold without boundary -- no edge. <br />Posted by DrRocket</DIV></p><p>Dr. Rocket:</p><p>What you say may be true for your accepted theory of the universe but common decency requires serious debaters to not demean or make fun of the opposition. Otherwise all advancements in science would have been delayed. . If you look back in history, there were scientist who made fun of, rather than presenting realistic opposition argument and they were sometimes proved wrong. So get serious! I am not hearing realistic opposition. Quote your beliefs not someone else. One example is the theory of Branes. It predicts that when two flat surface universes or Branes have one area protruding from one, it will collide with onother Brane and create a "Big Bang"</p><p>But what if there are more than one protrusion, will there be multiple "Bangs"?</p><p>If so where are they? I do not make fun of this theory but I do have this question to present. That is real scientific debate.<br /></p>
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#### SpeedFreek
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Be careful with terminology here. That is as far as you can see because light can have traveled no further since time zero. But it is not more an edge (because that is as far as you can see) than the horizon is the edge of the Earth.So far as anyone knows, and so far as the model used for a description of space-time the universe is a manifold without boundary -- no edge. <br /> Posted by DrRocket</DIV></p><p>Yes of course, I should have been clearer in defining what the <em>"observable"</em> universe actually means. The horizon of the Earth is a <em>great</em> analogy (which I intend to use in future - thank you)!</p> <div class="Discussion_UserSignature"> <p><font color="#ff0000">_______________________________________________<br /></font><font size="2"><em>SpeedFreek</em></font> </p> </div>
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#### MeteorWayne
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Dr. Rocket:What you say may be true for your accepted theory of the universe but common decency requires serious debaters to not demean or make fun of the opposition. Otherwise all advancements in science would have been delayed. . If you look back in history, there were scientist who made fun of, rather than presenting realistic opposition argument and they were sometimes proved wrong. So get serious! I am not hearing realistic opposition. Quote your beliefs not someone else. One example is the theory of Branes. It predicts that when two flat surface universes or Branes have one area protruding from one, it will collide with onother Brane and create a "Big Bang"But what if there are more than one protrusion, will there be multiple "Bangs"?If so where are they? I do not make fun of this theory but I do have this question to present. That is real scientific debate. <br />Posted by bechcube</DIV><br /><br />I saw nothing in what you quoted from Dr. Rocket that was demaning in any way. </p><p>To be honest, your ideas appear to have no foundation in physics, just number picked from the air. Bringing in isocoles triangles? Please...</p><p>It's hard to even make fun, or provide opposition to such a "theory"</p> <div class="Discussion_UserSignature"> <p><font color="#000080"><em><font color="#000000">But the Krell forgot one thing John. Monsters. Monsters from the Id.</font></em> </font></p><p><font color="#000080">I really, really, really, really miss the "first unread post" function</font><font color="#000080"> </font></p> </div>
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#### origin
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>origin:Your analysis is true when accepting the coefficient 1.6.However, my calculation is based on 1x10>-33 1/3 which my theory presents as the more correct value of the Plank length.Further, this constant produces very close but more accurate values for the standard constants, that is, mass of the electron, proton and neutron. Sure, they are slightly different than those calculated using the 1.6coefficient but that is the entire purpose of the theory.Why 10>50 and 10>-50? These numbers fit the perfect isosceles triangle having 10>50 at each junction and they fit the diameter of the circular strings or equator of the JAH, 1x10>-33cm. A circular string deletes the need for the fanciful 11 dimensions created by a linear string. <br />Posted by bechcube</DIV><br /><br />I was going to write some sort of involved response, but what is the point? You have no idea what you are talking about and don't even realize the implications of saying things like the mass of the electrons and protons currently used are wrong. You appear to have delusions of grandeur - I think you should just go ahead and enjoy your belief that you have the ANSWER. </p><p> </p> <div class="Discussion_UserSignature"> </div>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Clearly you missed my point.Lets try this - the volume of 10^-50 ccm is based on nothing physical. Your number of 10^50 JAH is based on nothing physical.You picked those 2 numbers because the cube root of the quotient equals the Plank Length. There must be some PHYSICAL REASON for picking those numbers or the rest of the calculations are just so much drivelAnd to top it all off the cube root of 10^-100 does not even equal the Planck Length.You should learn some real physics, it is much more interesting than this woo-woo stuff. <br />Posted by origin</DIV></p><p>origin:</p><p>Are you arguing that 1x10>{(-331/3)+(-331/3+)+(-331/2)} does not = 1x10>-100?<br /></p>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>SpeedFreek.Your argument is absolutely correct. It is logical, it is valid, and it is basically the argument that is often used by professional cosmologists.You have no chance whatever of convincing Bechcube. Logic and physics are not the point when one's mind is made up, one does not wish to be confused by facts, one's grasp of physics is minimal, and one is being driven by the delusion of having made a revolutionary discovery. Rational argument does not affect the irrational. <br />Posted by DrRocket</DIV></p><p>Dr. Rocket: I trust that you do have the education to earn the title you are using. If not, it would explain why you have difficulty grasping elemental ideas.</p><p>I suspect that you are a proponent of the theory of evolution. This theory purposes that all creation evolves into survival of the fittest. If you are, have you considered the question of why, if everything evolves, that at a distant of 13-14 bly's the creations are the same as in all other areas of the universe?</p><p>Further, it has been noted that my arguments are "Woo,Woo, etc..</p><p>However, my post is presented as a theory, not fact and therefore does not require proof by the initiator of the theory. That is left open to other educated interested scientist to delve into. Example: Dr. Einstein proposed that light curved when passing through an area of space-time which was allegedly curved by the presence of mass.</p><p>It was a theory, not a fact until a scientist measured the light during an eclipse.<br /></p>
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#### SpeedFreek
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I suspect that you are a proponent of the theory of evolution. This theory purposes that all creation evolves into survival of the fittest.</DIV></p><p>Well, I am. The theory of the evolution of species applies only to life and it is the survival of the <em>best fitted</em> to their environment. </p><p> </p><p>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>If you are, have you considered the question of why, if everything evolves, that at a distant of 13-14 bly's the creations are the same as in all other areas of the universe?</DIV></p><p>If you are now referring to how galaxies "evolve", which is a different kind use of the word evolution, then they evolve subject to the laws of physics, which are assumed to be the same throughout the universe. You might think of it like this - they "evolve" in the manner that best fits the laws of physics in their environment, the universe.<br /> </p><p>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Further, it has been noted that my arguments are "Woo,Woo, etc..However, my post is presented as a theory, not fact and therefore does not require proof by the initiator of the theory. That is left open to other educated interested scientist to delve into. Example: Dr. Einstein proposed that light curved when passing through an area of space-time which was allegedly curved by the presence of mass.It was a theory, not a fact until a scientist measured the light during an eclipse. <br /> Posted by bechcube</DIV></p><p>You started this when you challenged us to "disprove" your theory. It is not about proof but rather about the ability of a theory to make accurate predictions - a theory that can accurately describe different aspects of the universe around us. </p> <div class="Discussion_UserSignature"> <p><font color="#ff0000">_______________________________________________<br /></font><font size="2"><em>SpeedFreek</em></font> </p> </div>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I was going to write some sort of involved response, but what is the point? You have no idea what you are talking about and don't even realize the implications of saying things like the mass of the electrons and protons currently used are wrong. You appear to have delusions of grandeur - I think you should just go ahead and enjoy your belief that you have the ANSWER. <br />Posted by origin</DIV></p><p>orgin:If you read my theory, you would see that the calculations of the electron, proton and neutron masses are within three significant figures of the accepted values. Questioning 3rd, and above decimal places is hardly "delusions of grandeur" as they are spoken of as being questionabl in the standard model.<br /></p>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Well, I am. The theory of the evolution of species applies only to life and it is the survival of the best fitted to their environment. If you are now referring to how galaxies "evolve", which is a different kind use of the word evolution, then they evolve subject to the laws of physics, which are assumed to be the same throughout the universe. You might think of it like this - they "evolve" in the manner that best fits the laws of physics in their environment, the universe. You started this when you challenged us to "disprove" your theory. It is not about proof but rather about the ability of a theory to make accurate predictions - a theory that can accurately describe different aspects of the universe around us. <br />Posted by SpeedFreek</DIV></p><p>SpeedFreek:</p><p>Where, in the universe, do you see the creation applying only to one aspect of creation?</p><p>Further, evolutionist say the Rex dinosaur remained the same over millions of years. How did we go from amoebae-monkeys-humans in the same time period but Rex did not change?<br /></p>
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#### origin
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>origin:Are you arguing that 1x10>{(-331/3)+(-331/3+)+(-331/2)} does not = 1x10>-100? <br />Posted by bechcube</DIV></p><p>No, I am arguing that high school algebra combined with the complete lack of understanding basic physics = your theory.<br />But you believe so - enjoy and bask in your ignorance.</p><p> </p> <div class="Discussion_UserSignature"> </div>
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#### SpeedFreek
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Try multiple protrusions in our Brane colliding with another Brane. Would that resulting "''Bang" not be seen in our universe. Think about it. <br /> Posted by bechcube</DIV></p><p>No it would not be seen, as far as I can tell, from my scant knowledge of M-Theory. I'm not even sure if protrusions is the correct term, as I thought the universe was simply formed where 2 Branes intersect. If these Branes also intersect somewhere else on their "surface", I would assume that would form a separate universe to our own.</p><p> </p><p>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Where, in the universe, do you see the creation applying only to one aspect of creation? <br /> Posted by bechcube</DIV></p><p>You are talking about natural selection, if you refer to survival of the best fitted. You are talking about mutations that are beneficial to a form of life, that help it succeed in its current environment. The more successful a life-form, the more likely it will pass on the beneficial mutation. Thus, the best fitted life-form for its environment is the one that does the best. What has this to do with galaxies?</p><p> </p><p>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Further, evolutionist say the Rex dinosaur remained the same over millions of years. How did we go from amoebae-monkeys-humans in the same time period but Rex did not change? <br /> Posted by bechcube</DIV></p><p>Firstly we did not go from amoeba to human in that period. You really don't seem to understand evolution at all. We probably evolved from small mammals that were alive towards the end of the reign of the dinosaurs. The rex dinosaurs stayed the same for so long because they were the dominant species and were the best fitted for their environment during a period when the environment was pretty stable. When the environment changed radically, the dinosaurs all died out, as they were too specialised. We evolved from mammals that were left behind at that time, mammals that could now proliferate as the dinosaurs were finally out of the way. </p><p> </p> <div class="Discussion_UserSignature"> <p><font color="#ff0000">_______________________________________________<br /></font><font size="2"><em>SpeedFreek</em></font> </p> </div>
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#### bechcube
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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>No it would not be seen, as far as I can tell, from my scant knowledge of M-Theory. I'm not even sure if protrusions is the correct term, as I thought the universe was simply formed where 2 Branes intersect. If these Branes also intersect somewhere else on their "surface", I would assume that would form a separate universe to our own. You are talking about natural selection, if you refer to survival of the best fitted. You are talking about mutations that are beneficial to a form of life, that help it succeed in its current environment. The more successful a life-form, the more likely it will pass on the beneficial mutation. Thus, the best fitted life-form for its environment is the one that does the best. What has this to do with galaxies? Firstly we did not go from amoeba to human in that period. You really don't seem to understand evolution at all. We probably evolved from small mammals that were alive towards the end of the reign of the dinosaurs. The rex dinosaurs stayed the same for so long because they were the dominant species and were the best fitted for their environment during a period when the environment was pretty stable. When the environment changed radically, the dinosaurs all died out, as they were too specialised. We evolved from mammals that were left behind at that time, mammals that could now proliferate as the dinosaurs were finally out of the way. <br />Posted by SpeedFreek</DIV></p><p>SpeedFreek:</p><p>Good for you. At least you have a theory outside of the mainstream</p><p> I can understand how biology was led astray in believing that single cell entities changed into fish and then into amphibians.</p><p>They lost me when they taught that the fish left the oceans and embarked upon land.</p><p>I can understand fish for the seas, monkeys for the trees and man for the land. Beyond that, I loose reality.</p><p><br /><br /> </p>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>No, I am arguing that high school algebra combined with the complete lack of understanding basic physics = your theory.But you believe so - enjoy and bask in your ignorance. <br />Posted by origin</DIV></p><p>orgin: That was a game clean up. I expected better.<br /></p>
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#### SpeedFreek
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>SpeedFreek:Good for you. At least you have a theory outside of the mainstream I can understand how biology was led astray in believing that single cell entities changed into fish and then into amphibians.They lost me when they taught that the fish left the oceans and embarked upon land.I can understand fish for the seas, monkeys for the trees and man for the land. Beyond that, I loose reality. <br /> Posted by bechcube</DIV></p><p> <br /> <img src="http://sitelife.space.com/ver1.0/Content/images/store/6/13/666b0f52-dc76-4bbf-8dfe-48c5ec4801cb.Medium.jpg" alt="" /><br /> <img src="http://sitelife.space.com/ver1.0/Content/images/store/7/8/c7132689-1959-439f-8893-592dddc507f3.Medium.jpg" alt="" /></p><p>Nope, my view is the mainstream view as far as I know. You have shown here that you do not understand the theory of evolution, and yet you seem very sure it must be wrong, even though you do not actually understand it. Did you know that birds seem to have evolved from reptiles? Did you know that whales seem to have evolved from mammals that once walked the land?</p><p>By the way, this has nothing to do with galaxies disintegrating into GRBs at the edge of the universe. </p> <div class="Discussion_UserSignature"> <p><font color="#ff0000">_______________________________________________<br /></font><font size="2"><em>SpeedFreek</em></font> </p> </div>
M
#### MeteorWayne
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Dr. Rocket: I trust that you do have the education to earn the title you are using. If not, it would explain why you have difficulty grasping elemental ideas.I suspect that you are a proponent of the theory of evolution. This theory purposes that all creation evolves into survival of the fittest. If you are, have you considered the question of why, if everything evolves, that at a distant of 13-14 bly's the creations are the same as in all other areas of the universe?Further, it has been noted that my arguments are "Woo,Woo, etc..However, my post is presented as a theory, not fact and therefore does not require proof by the initiator of the theory. That is left open to other educated interested scientist to delve into. Example: Dr. Einstein proposed that light curved when passing through an area of space-time which was allegedly curved by the presence of mass.It was a theory, not a fact until a scientist measured the light during an eclipse. <br />Posted by bechcube</DIV><br /><br />And exactly what evidence do you have that the "creations" in the distant Universe are the same as here? I'll answer, None at all. </p><p>So your point (whatever it was) has no merit at all.</p> <div class="Discussion_UserSignature"> <p><font color="#000080"><em><font color="#000000">But the Krell forgot one thing John. Monsters. Monsters from the Id.</font></em> </font></p><p><font color="#000080">I really, really, really, really miss the "first unread post" function</font><font color="#000080"> </font></p> </div>
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#### bechcube
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>And exactly what evidence do you have that the "creations" in the distant Universe are the same as here? I'll answer, None at all. So your point (whatever it was) has no merit at all. <br />Posted by MeteorWayne</DIV>MeteorWayne:</p><p>The deep sky photos by the Hubble showing galaxies calculated to be 12billions ly's distant exist. Of course, you can say science is wrong, that's your privilege.<br /></p>
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#### SpeedFreek
##### Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>MeteorWayne:The deep sky photos by the Hubble showing galaxies calculated to be 12billions ly's distant exist. Of course, you can say science is wrong, that's your privilege. <br /> Posted by bechcube</DIV></p><p>MeteorWayne was, of course, correct.</p><p>From the digital universe wiki:</p><p>"Galaxies at high redshift often have peculiar shapes, in contrast to the well-developed spiral and elliptical galaxies seen in the local universe. At redshifts greater than 3 or 4, corresponding to when the universe was less than 2 billion years old, galaxies are primarily starburst systems."</p><p>We have seen galaxies with redshifts up to z=7, whose light has been travelling for 12.9 billion years, but they don't look anything like the well developed galaxies we see around us locally. The Hubble deep sky image shows galaxies <em><strong>up to</strong></em> 12 billion ly's away, but most of them are a lot closer than that.</p><p>You do not understand the science that you are trying to interpret in your own way. </p> <div class="Discussion_UserSignature"> <p><font color="#ff0000">_______________________________________________<br /></font><font size="2"><em>SpeedFreek</em></font> </p> </div>
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https://mathematica.stackexchange.com/questions/89183/factor-out-the-scalar-multiplier-for-the-dot-product-of-2x2-matrices?noredirect=1
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# Factor out the scalar multiplier for the dot product of 2x2 matrices
If yy and zz are 2x2 Hermitian matrices, is there a way that I can mark them (with a property?) as Hermitian so that Mathematica can assume that it can factor out and simplify scalar multipliers from a dot product expression? In this example, we have -1 * -1 as the multiplier:
ClearAll[a, yy, zz]
a = -(-yy.zz).zz
FullForm[a]
This gives:
-(-yy.zz).zz
Times[-1,Dot[Times[-1,Dot[yy,zz]],zz]]
Can it be made to simplify to just:
yy.zz.zz
• Dave, just a gentle reminder that, if one of the answers provided below solve your problem, you might want to accept it by clicking on the gray check mark next to it. Commented Jul 27, 2015 at 19:44
To factor out numeric factors in any argument of Dot:
(2 yy.(3 zz)).(4 zz) //. Dot[a___, d_?NumericQ b_, c___] :> d Dot[a, b, c]
24 yy.zz.zz
Edit: If you want this to happen automatically, you can add the rule as a new definition for Dot:
Unprotect[Dot];
Dot[a___, d_?NumericQ b_, c___] := d Dot[a, b, c]
Protect[Dot];
Now the factoring happens by itself:
(2 yy.(3 zz)).(4 zz)
24 yy.zz.zz
• Thanks; this is a nice and straightforward replacement that will do the job. I must say, though, that I was hoping that Mathematica could just "figure it out". Commented Jul 27, 2015 at 20:28
• @daveboden You can make it automatic, if you like -- see the edit. Commented Jul 27, 2015 at 21:21
• Please, try your code for this string J.Transpose[(-P)].x
– ayr
Commented Sep 9, 2022 at 4:21
• And for this: J.Transpose[1].x, J.Transpose[-1].x and J.Transpose[-P].(-1)
– ayr
Commented Sep 9, 2022 at 5:27
• @dtn, I see you have asked your question under multiple answers but I think it is unclear to people exactly what you mean. In your first problem, do you want Transpose[1] to return the identity, such that J.Transpose[1].x reduces to J.x? I think Mathematica can't do that because it doesn't know the dimensions of J and x so that it can't return identity matrix of proper dimensions. At least as far as I know Mathematica can't do that. Perhaps a Mathematica export can comment on this. Commented Sep 28, 2022 at 12:50
Is the following sufficiently general?
t[e_] := e /. Dot[Times[z1_ /;!ArrayQ[z1], Dot[z2__]], z3__] :> z1 Dot[z2, z3]
Simplify[a, TransformationFunctions -> {Automatic, t}]
(* yy.zz.zz *)
• Please, try your code for this string J.Transpose[(-P)].x
– ayr
Commented Sep 9, 2022 at 4:22
• @dtn, What are the three quantities in your expression? Commented Sep 9, 2022 at 4:56
• J,Transpose[-P] and x
– ayr
Commented Sep 9, 2022 at 5:00
Easier and more generally applicable is to use TensorExpand:
In[]: -(-yy.zz).zz//TensorExpand
Out[]: yy.zz.zz
• Your code work for this string J.Transpose[(-P)].x, but how to return the standard notation for transpose?
– ayr
Commented Sep 9, 2022 at 4:22
• Is your question that J.Transpose[(-P)].x//TensorExpand returns -J.Transpose[P, {2, 1}].x but you want J.Transpose[P].x? Besides what it would mean mathematically a way to get this output is to use a replacement rule in the following way (J.Transpose[(-P)].x//TensorExpand)//Transpose[a_, {2, 1}] :> Transpose[a]. For a better understanding of what the {2,1} means you can lookup the documentation of Transpose or have a look at this post. Commented Sep 28, 2022 at 12:34
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http://sqlmag.com/print/sql-server/trading-stocks
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Congratulations to Ahmed Mohamed, a DBA and developer for Ernst & Young in Cleveland, Ohio, and Alejandro Mesa, a software developer for ICCO in Coral Springs, Florida. Ahmed won first prize of \$100 for the best solution to the May Reader Challenge, "Trading Stocks." Alejandro won second prize of \$50. Here’s a recap of the problem and the solution to the May Reader Challenge.
### Problem:
Jane is a database programmer for a company that provides stock-trading services online. A SQL Server 2000 database stores the transactions users create. The database contains two tables, Stocks and TradeSummary, that contain information about traded stocks and the trade details. Here are the tables' schema with sample data:
```CREATE TABLE Stocks (
StockId int NOT NULL PRIMARY KEY CLUSTERED,
Market varchar(30) NOT NULL,
Ticker varchar(10) NOT NULL,
DisplayName varchar(30) NOT NULL,
UNIQUE(Market, Ticker)
)
INSERT INTO Stocks VALUES( 1, 'NASDAQ', 'MSFT', 'Microsoft' )
INSERT INTO Stocks VALUES( 2, 'NASDAQ', 'INTC', 'Intel' )
INSERT INTO Stocks VALUES( 3, 'Tokyo', '6758', 'Sony' )
INSERT INTO Stocks VALUES( 4, 'India', 'TTPW.BO', 'Tata' )
GO
CREATE TABLE TradeSummary (
StockId int NOT NULL REFERENCES Stocks( StockId ),
UserId int NOT NULL,
IsBuy char(1) NOT NULL CHECK( IsBuy in ('Y', 'N')),
TradeDt smalldatetime NOT NULL DEFAULT( current_timestamp ),
Price decimal(10, 2) NOT NULL,
Qty int NOT NULL
)
CREATE CLUSTERED INDEX TradeSummary_StockDt ON TradeSummary( StockId, TradeDt)
INSERT INTO TradeSummary VALUES( 1, 1, 'Y', '2004-1-1 10:00', 26.35, 100 )
INSERT INTO TradeSummary VALUES ( 1, 2, 'Y', '2004-1-1 11:00', 26.35, 50 )
INSERT INTO TradeSummary VALUES ( 1, 2, 'N', '2004-1-1 12:00', -26.35, 50 )
INSERT INTO TradeSummary VALUES ( 1, 3, 'Y', '2004-1-1 13:00', 26.35, 25 )
INSERT INTO TradeSummary VALUES ( 1, 4, 'Y', '2004-1-1 10:00', 26.35, 125 )
INSERT INTO TradeSummary VALUES ( 2, 1, 'N', '2004-1-1 09:00', -28.95, 100 )
INSERT INTO TradeSummary VALUES ( 2, 2, 'Y', '2004-1-1 10:00', 28.95, 50 )
INSERT INTO TradeSummary VALUES ( 2, 3, 'Y', '2004-1-1 11:00', 26.35, 25 )
INSERT INTO TradeSummary VALUES ( 1, 1, 'Y', '2004-2-1 23:00', 26.35, 200 )
INSERT INTO TradeSummary VALUES ( 1, 2, 'N', '2004-2-1 11:00', -26.35, 50 )
INSERT INTO TradeSummary VALUES ( 1, 3, 'Y', '2004-2-1 12:00', 26.35, 25 )
INSERT INTO TradeSummary VALUES ( 1, 4, 'Y', '2004-2-1 23:00', 26.35, 125 )
INSERT INTO TradeSummary VALUES ( 2, 2, 'Y', '2004-2-1 10:00', 28.95, 50 )
INSERT INTO TradeSummary VALUES ( 2, 3, 'Y', '2004-2-1 23:00', 26.35, 25 )
GO```
Jane needs to create a report that provides daily summary information about the various stock trades. For each stock, she wants daily stock details, the opening buy or sell price, the closing buy or sell price, the volume of stocks purchased, the highest buy or sell price, and the lowest buy or sell price. Help Jane write a query that provides these report details.
### Solution:
Here’s the SELECT statement for the query Jane uses get the results she wants:
```SELECT s.StockId, s.Market, s.Ticker, s.DisplayName,
ds.TradeDt, ds."Volume", ds."High", ds."Low", ds."Open", ds."Close"
FROM Stocks AS s
JOIN (
SELECT StockId,
CAST(CONVERT(varchar, TradeDt, 112) AS smalldatetime) AS TradeDt,
COALESCE(SUM(CASE IsBuy WHEN 'Y' THEN Qty END), 0) AS "Volume",
MIN(ABS(Price)) AS "Low",
MAX(ABS(Price)) AS "High",
CAST(SUBSTRING(MIN(CONVERT(char(8), TradeDt, 14) +
REPLACE(STR(ABS(Price), 10, 2),
SPACE(1), '0')), 9, 10) AS decimal(10,2)) AS "Open",
CAST(SUBSTRING(MAX(CONVERT(char(8), TradeDt, 14) +
REPLACE(STR(ABS(Price), 10, 2),
SPACE(1), '0')), 9, 10) AS decimal(10,2)) AS "Close"
FROM TradeSummary
GROUP BY StockId, CAST(CONVERT(varchar, TradeDt, 112) AS smalldatetime)
) AS ds
ON ds.StockId = s.StockId```
The Open column expression combines information from several columns into a string value, then aggregates the combined data. For example, this lets Jane get a particular stock’s lowest price for the day. TradeDt’s time portion is extracted in hh:mm:ss format (style 14) using the CONVERT() system function. The STR() system function converts the absolute numeric price value into a string. The REPLACE() function ensures the price’s string value contains zeroes instead of leading spaces. Then the MIN() aggregation function is applied to the combined data. Jane can extract the individual parts (time and price) using the SUBSTRING() system function. Depending on data type, Jane can also construct the aggregation’s expression as binary data (instead of string). The single-query approach performs best because it involves a single join between the Stocks table and the TradeSummary table and only one aggregation pass.
Jane can also create the view, DailyTradeSummary, which provides the basic calculations she needs for each stock. The view’s definition looks like the following:
```CREATE VIEW DailyTradeSummary
( StockId, TradeDt, "Volume", "Low", "High", MinTradeDt, MaxTradeDt )
AS
SELECT StockId,
CAST(CONVERT(varchar, TradeDt, 112) AS smalldatetime),
COALESCE(SUM(CASE IsBuy WHEN 'Y' THEN Qty END), 0),
CAST(MAX(ABS(Price*Qty)) AS decimal(15, 2)),
CAST(MIN(ABS(Price*Qty)) AS decimal(15, 2)),
MIN(TradeDt),
MAX(TradeDt)
FROM TradeSummary
GROUP BY StockId,
CAST(CONVERT(varchar, TradeDt, 112) AS smalldatetime)
GO```
For each stock, the DailyTradeSummary view provides the volume of stocks purchased, the lowest trade price, the highest trade price, and the timestamp for the day’s first and last transactions. The view aggregates the day’s measures for each stock. The TradeDt column contains date and time values, so for each day, every row will have a unique column value (like the different times in the sample data). For grouping purposes, Jane needs to get only the date value from the column, which is what she uses the CONVERT90 function for. The CONVERT() function converts the TradeDt column value to YYYYMMDD format (style 112) and the transaction’s time portion is ignored. Then, Jane uses the DailyTradeSummary view in the reporting query to obtain the stock details and the day’s opening and closing trade prices. Jane obtains the opening and closing trade prices by joining the DailyTradeSummary view with the TradeSummary table. She gets the rows corresponding to the day’s first and last transaction for each stock. The following query obtains the desired information:
```SELECT s.StockId, s.Market, s.Ticker, s.DisplayName,
ds.TradeDt, ds."Volume", ds."High", ds."Low", ds."Open", ds."Close"
FROM Stocks AS s
JOIN (
SELECT d.StockId, d.TradeDt, d."Volume", d."High", d."Low",
MIN(CASE t.TradeDt WHEN d.MinTradeDt THEN ABS(t.Price*t.Qty) END) AS "Open",
MIN(CASE t.TradeDt WHEN d.MaxTradeDt THEN ABS(t.Price*t.Qty) END) AS "Close"
FROM DailyTradeSummary AS d
JOIN TradeSummary AS t
ON t.StockId = d.StockId
AND t.TradeDt IN (d.MinTradeDt , d.MaxTradeDt)
GROUP BY d.StockId, d.TradeDt, d."Volume", d."High", d."Low"
) AS ds
ON ds.StockId = s.StockId```
Finally, to show the stock symbols without trading activity for the day, Jane can use a calendar table in the join. She gets the desired dates for each stock symbol, then performs an outer join against the trade summary data.
### JUNE READER CHALLENGE:
Now, test your SQL Server savvy in the June Reader Challenge, "Averaging Employee Salaries" (below). Submit your solution in an email message to challenge@sqlmag.com by May 20. Umachandar Jayachandran, a SQL Server Magazine technical editor, will evaluate the responses. We’ll announce the winner in an upcoming SQL Server Magazine UPDATE. The first-place winner will receive \$100, and the second-place winner will receive \$50.
### Problem:
Jon develops and maintains a SQL Server 2000 database for his company’s human resources department. The database contains a table that provides details about each employee in the company by department and branch. Jon needs to write a report that provides the average employee salary in each department and branch as well as the sum of the average employee salary for each branch. Resulting rows must be ordered by branch, by department, then by summary information. Help Jon write an efficient query. The following statements provide the table definition and a sample of the data:
```CREATE TABLE BranchDeptEmpSal (
BranchId char(3) NOT NULL,
DeptId int NOT NULL,
EmpId int NOT NULL,
Salary money NOT NULL
)
INSERT INTO BranchDeptEmpSal VALUES( '001', 1, 1, 1000 )
INSERT INTO BranchDeptEmpSal VALUES( '001', 1, 2, 500 )
INSERT INTO BranchDeptEmpSal VALUES( '001', 1, 3, 600 )
INSERT INTO BranchDeptEmpSal VALUES( '001', 2, 4, 2000 )
INSERT INTO BranchDeptEmpSal VALUES( '001', 2, 6, 1600 )
INSERT INTO BranchDeptEmpSal VALUES( '002', 1, 11, 400 )
INSERT INTO BranchDeptEmpSal VALUES( '002', 1, 12, 800 )
INSERT INTO BranchDeptEmpSal VALUES( '002', 2, 8, 400 )```
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http://www.enotes.com/homework-help/sketch-region-enclosed-by-given-curves-then-find-255554
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# Sketch the region enclosed by the given curves. Then find the are of the region: 1. y=sin x, y=x, x=PI/2, x=pi
sciencesolve | Teacher | (Level 3) Educator Emeritus
Posted on
You need to evaluate the functions `y = sin x` and `y = x` at `x = pi/2` and `x = pi` , such that:
`x = pi/2 => y = sin (pi/2) = 1`
`x= pi/2 => y = pi/2 ~~ 1.57`
`x = pi => y = sin pi = 0`
`x = pi => y = pi = 3.14`
Notice that the graph of the function `y = x` is located above the graph of the function `y = sin x` , over the interval `[pi/2, pi]` , hence, evaluating the graph of the region between `y = sin x, y = x` and `x = pi/2, x = pi` , yields:
`A = int_(pi/2)^pi (x - sin x) dx`
Using the property of linearity of integral yields:
`A = int_(pi/2)^pi x dx - int_(pi/2)^pi sin x dx`
`A = x^2/2|_(pi/2)^pi + cos x|_(pi/2)^pi`
Using the fundamental theorem of calculus yields:
`A = (pi^2/2 - pi^2/8) + (cos pi - cos (pi/2))`
Since `cos pi = -` 1 and `cos(pi/2) = 0` yields:
`A = (3pi^2)/8 - 1`
Hence, evaluating the area of the region enclosed by the given curves, using the process of integration, yields `A = (3pi^2)/8 - 1.`
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http://www.archive.org/stream/TheFlowOfGasesInFurnaces/TXT/00000387.txt
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# Full text of "The Flow Of Gases In Furnaces"
## See other formats
```366
APPENDIX X
TABLE 9
Heat capacity in calories of gases above 0° per gram molecule (or for
22.32 liters ==0.78822 cubic foot of gas).
Tempera-
N2 = 2S,
ture C.
*=r-273
O2 = 32, H2= 2,
H20 = 18
CO2-44
CH4 = 16
Molecular weights given
Degrees
CO = 28
in grams
0 200 400
0.00 1.39
2.82
0.00 1.73 3.69
0.00 1.85 3.99
0.00 2.19
4.85
B.t.u. per cubic foot = 5.03794X calories per gram molecule
600 800 1000
4.31
5.82 7.43
5.87 8.23 10.98
6.44 9.07 12.42
8.02 11.46 15.77
Ounce molecule = 22.32 cubic feet
1200 1400 1600 1800
9.05 10.73 12.46 14.21
13.87 17.00 20.35 23.86
15.55 19.18 23.10 27.21
20.37 22.44 30.99 35.86
B.t.u. per ounce molecule = 112.472077X calories per gram molecule
2000 2200 2400 2600
16.05 17.01 19.84 21.81
27.76 31.82 36.10 40.62
31.84 36.65 41.76 47.16
43.55 50.54 58.02 66.04
Calories per cubic meter =44.80287 X calorics per gram molecule
Heat capacity values are based on the formulas developed by Mallard
and Le Chatelier.
As hydrocarbon gases dissociate when exposed to high temperatures the
heat capacity values for CH4 have a theoretical value above 800°.
Simmersbach in Journal of Chemical Industry, Feb. 28, 1913, page 186,
and in Journal fur Gasbeleuchtung, Dec. 13, 1913, page 1242, gives data
covering his experiments in regard to the decrease in heating value of gases
containing hydrocarbons. ```
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http://burtleburtle.net/bob/knot/homfly.html
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## Computing the HOMFLY polynomial
My master's thesis was A Dynamic Approach to Calculating the HOMFLY Polynomial for Directed Knots and Links. The algorithm used is dynamic programming, running time is O((n!)(2n)(c3)) and the space used is O((n!)(c2)), where c is the number of crossings in the link and n is bounded above by sqrt(c)+1.
I have code for this:
The program is written to handle strings with cross sections of up to 24 strings (12 in, 12 out), but remember that it uses a factorial amount of space. The input file describing a knot needs to be created by hand, see knot.h for that format. Here are some sample input files.
1. boromean.txt, three strings, no two of which are knotted, but the three together are. This finishes instantaneously.
2. knotseven.txt, seven strings which are intertwined. This finished in 7 seconds with these results. (This is the smallest link with a cross section of 14 strings.)
3. knotnine.txt, nine strings which are intertwined. This ran for 8 hours before crashing my machine. I ran it on a 66mhz 486 with 16 meg ram and a 500 meg hard drive compiled optimized with a 32-bit gcc. (This is the smallest link with a cross section of 18 strings.)
KnotPlot is capable of generating this knot format when it computes the Homfly polynomial. On Windows, it places files in this format temporarily in the %TMP% directory. It pictorially displays the knots and can save them in this format; I don't know if it can read this format and display the knot a file represents.
thesis2.ps is a diagram showing how the algorithm works for one small knot.
thesis1.ps is a diagram showing how a simpler method (which I call the binary tree approach) works for that same small knot.
If you want just the Jones polynomial, and you want it for an alternating link, there is a very similar but faster algorithm (with no code available). The paper on it is "Computing the Tutte Polynomial of a Graph and Jones Polynomial of a Knot of Moderate Size" by K. Sekine, H. Imai, and S. Tani.
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# Math
posted by .
Contract the expressions.
In (3) - 2 In (4)+ In (8)=
In (3) - 2 In (4+8) =
In (3) - 2 (In (4) + In (8) ) =
• Math -
You will have to know the 3 main rules of logs
1. Log (AB) = logA + logB
2. log(A/B) = logA - logB
3. log (A^n) = nlogA
e.g, for
ln3 - 2ln(4+8)
= ln3 + 2ln12
= ln3 + ln(12^2)
= ln3 + ln144
= ln (3/144)
= ln (1/48)
= ln 1 - ln48
= 0 - ln48
= - ln48
do the others the same way
## Similar Questions
1. ### Business law questions? (multiple choice)?
Farmer J enters into a contract to sell 15,000 bushels of oats to ABC Mills. Flood destroys one-half of his crop. Farmer J a. Is excused from performance od contract because of impossibility b. Is excused from perf of contract because …
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Read the expressions in a loud voice. Read the expressions in a soft voice. Read the expressions loudly. Read the expressions out loud. Read the expressions aloud. Read the expressions loud. Read the expressions softly. Read the expressions …
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Contract the expressions. That is, use the properties of logarithms to write each expression as a single logarithm with a coefficient of 1. ln(3)-2ln(4+8)
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Contract the expressions. That is, use the properties of logarithms to write each expression as a single logarithm with a coefficient of 1. text ((a) ) ln\(3\)-2ln\(4\)+ln\(8\) ((b) ln\(3\)-2ln\(4+8\) (c) )ln\(3\)-2(ln\(4\)+ln\(8\))
5. ### Business Law
Regan marries at age 17. Then, she enters into a contract with Art to purchase an automobile for \$10,000. She later changes her mind and wants to void the contract. Which of the following is true?
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Regan marries at age 17. Then, she enters into a contract with Art to purchase an automobile for \$10,000. She later changes her mind and wants to void the contract. Which of the following is true?
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# BANK PO QUANT QUIZ 1
Attempt now to get your rank among 21 students!
## Question 1:
Direction: What should come in Place of Question Mark (?) in the following question?
## Question 2:
Direction: What should come in Place of Question Mark (?) in the following question?
$5760 \div 45 \times 15=X$
## Question 3:
Direction: What should come in Place of Question Mark (?) in the following question?
$6625-4242-1025=X$
## Question 4:
Direction: What should come in Place of Question Mark (?) in the following question?
$\sqrt{1369}+(6859)^{\frac{1}{3}}=(X)^{2}-\sqrt{625}$
## Question 5:
Direction: What should come in Place of Question Mark (?) in the following question?
## Question 6:
What should come in Place of Question Mark (?) in the following question?
## Question 7:
Direction: What should come in Place of Question Mark (?) in the following question?
## Question 8:
Direction: What should come in Place of Question Mark (?) in the following question?
$13 \times 25 \times 20+(X)^{2}=6,50,000$
## Question 9:
Direction: What should come in Place of Question Mark (?) in the following question?
$X-1.2 \times 8.5=3.5 \times 16.4$
## Question 10:
Direction: What should come in Place of Question Mark (?) in the following questions?
$5 \frac{2}{3}+4 \frac{5}{9}-4 \frac{1}{2}-3 \frac{1}{4}=X-16 \frac{2}{9}+12 \frac{1}{3}-5 \frac{1}{3}$
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## ››Convert milímetro to inch
milimetro inch
How many milimetro in 1 inch? The answer is 25.4.
We assume you are converting between milímetro and inch.
You can view more details on each measurement unit:
milimetro or inch
The SI base unit for length is the metre.
1 metre is equal to 1000 milimetro, or 39.370078740157 inch.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between milimetros and inches.
Type in your own numbers in the form to convert the units!
## ››Quick conversion chart of milimetro to inch
1 milimetro to inch = 0.03937 inch
10 milimetro to inch = 0.3937 inch
20 milimetro to inch = 0.7874 inch
30 milimetro to inch = 1.1811 inch
40 milimetro to inch = 1.5748 inch
50 milimetro to inch = 1.9685 inch
100 milimetro to inch = 3.93701 inch
200 milimetro to inch = 7.87402 inch
## ››Want other units?
You can do the reverse unit conversion from inch to milimetro, or enter any two units below:
## Enter two units to convert
From: To:
## ››Definition: Milimetro
El milímetro es una unidad de longitud. Es el tercer submúltiplo del metro y equivale a la milésima parte de él. Su abreviatura es mm.
## ››Definition: Inch
An inch is the name of a unit of length in a number of different systems, including Imperial units, and United States customary units. There are 36 inches in a yard and 12 inches in a foot. The inch is usually the universal unit of measurement in the United States, and is widely used in the United Kingdom, and Canada, despite the introduction of metric to the latter two in the 1960s and 1970s, respectively. The inch is still commonly used informally, although somewhat less, in other Commonwealth nations such as Australia; an example being the long standing tradition of measuring the height of newborn children in inches rather than centimetres. The international inch is defined to be equal to 25.4 millimeters.
## ››Metric conversions and more
ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!
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20Iterator
# Well use the value null to denote the situation where
• Notes
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We’ll use the value null to denote the situation where the iterator is positioned to the left of the list. The first time next() is called, we’ll move current to the first node and return its value. For each successive call, current is moved one position forward and the value found at that node is returned.
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head l B C A current Iterator i = l.iterator(); head l B C A current after i.next(); head l B C A current after i.next(); head l B C A current after i.next();
Implementing iterator(). public class LinkedList<E> implements List<E>, Iterator<E> { private static class Node<E> { ... } private Node<E> head; private Node<E> tail; private Node<E> current; // <--- public Iterator<E> iterator() { current = null; return this; } }
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But do we need iterator() ? Couldn’t we simply assign null in the constructor of the list? No, the method is necessary to traverse the list more than one time. int sum = 0; int n = 0; Iterator<Integer> i = l.iterator(); while ( i.hasNext() ) { Integer v = i.next(); sum += v.intValue(); n++; } i = l.iterator(); while ( i.hasNext() ) { Integer v = i.next(); if ( v.intValue() > ( sum / n ) ) { System.out.println( "is greater than average" ); } }
Implementing next . public class LinkedList<E> implements List<E>, Iterator<E> { private static class Node<E> { ... } private Node<E> head; private Node<E> current; public Iterator<E> iterator() { ... } public E next() { if ( current == null ) { current = head; } else { current = current.next; } if ( current == null ) { throw new NoSuchElementException(); } return current.value; } }
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head l B C A current
head l B C A current
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head l B C A current
head l B C A current
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Current limitation Our current implementation does not allow for multiple iterators! while ( i.hasNext() ) { oi = i.next(); while ( j.hasNext() ) { oj = i.next(); // process oi and oj } }
What’s needed then? We need as many references (variables) as iterators; An iterator has to have access to the implementation (Node) ; An iterator needs access to the instance variables of the class LinkedList .
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Allowing for multiple iterators Suggestions?
Implementation -1.5- Let’s create the following top-level class (in the same package as LinkedList ), this will require that i) Node is also a top-level class and ii) the variable head should be protected .
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class LinkedListIterator<E> implements Iterator<E> { private Node<E> current; private LinkedList<E> myList; LinkedListIterator( LinkedList<E> myList ) { this.myList = myList; current = null; } public boolean hasNext() { return ( ( current == null && myList.head != null ) || ( current != null && current.next != null) ); } public E next() { if ( current == null ) { current = myList.head; } else { current = current.next; } return current.value; } }
head l D B C A current i current j myList myList
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The method iterator() of the class LinkedList is defined as follows.
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How do you solve x^ { 2} - 6x = 7?
May 2, 2018
$x = 7 \mathmr{and} x = \left(- 1\right)$
Explanation:
Subtract 7 from both sides.
$\implies {x}^{2} - 6 x - 7 = 0$
We need to factorize this expression by splitting its middle term.
$\implies {x}^{2} - 7 x + x - 7 = 0$
$\implies x \left(x - 7\right) + 1 \left(x - 7\right) = 0$
$\implies \left(x - 7\right) \left(x + 1\right) = 0$
$\implies x - 7 = 0 \mathmr{and} x + 1 = 0$
$\implies x = 7 \mathmr{and} x = \left(- 1\right)$
May 2, 2018
$x = 7$
$x = - 1$
Explanation:
${x}^{2} - 6 x = 7$ Subtract $7$ from both sides:
${x}^{2} - 6 x - 7 = 0$
$- 7 , 1$ are factors of $- 7$ that add up to $6$, so they will be factors:
$\left(x - 7\right) \left(x + 1\right) = 0$ Using the $0$ product property:
$x - 7 = 0$
$x = 7$
and
$x + 1 = 0$
$x = - 1$
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# Calculate Critical Speed Ball Mill Philippines
• ### Ball Mill Specifications Ball Mill Speed Calculation
How To Calculate Critical Speed Of Ball Mill A ball mill critical speed actually ball rod or s is the speed at which the centrifugal forces equal gravitational forces at the mill shells inside surface and no balls will fall from its position onto the shellRolling mill speed calculation formula pdftorque speed calculation for ball mill designing calculation of critical speed of ball.
• ### Critical Speed Ball Mill Calculation
Commonly the Ball mill speed is calculated as 77 of Simulation studies on Energy Requirement Work Input and IJETAE grindability of ball mill have been studied by varying different operate at 65 to 75 of the critical speed and this.
• ### Critical Speed Ball Mill Calculation
Critical speed formula for ball millThe formula to calculate critical speed is given below n c sqtdd n c critical speed of the mill d mill diameter specified in meters d diameter of the ball in practice ball mills are driven at a speed of of the critical speed the factor.
• ### Calculate Critical Speed Of Ball Mill
Critical Speed Of Ball Mill Formula Derivation 201983 The formula to calculate critical speed is given below N c 42 305 sqtDd N c critical speed of the mill D mill diameter specified in meters d diameter of the ball In practice Ball Mills are driven at a speed of 5090 of the critical speed the factor being influenced by economic.
• ### Calculating Critical Speed In A Ball Mill
Ball mill design critical speed formula Critical speed of grinding mill The formula to calculate critical speed is given below n c = 42305 sqt (d d) n c = critical speed of the mill d = mill diameter specified in meters d = diameter of the ball in practice ball mills are driven at Online Chat.
• ### Critical Speed On A Ball Mill
Calculate Ball Mill Critical Speed Critical Speed Calculation For Ball Mill Procedure To Determine The Critical Speed Of Ball Mill Derivation to calculate critical speed of ball mill mar 17 2017 a ball mill critical speed actually ball rod ag or sag is the speed at which the centrifugal forces equal gravitational forces at the mill shells inside surface and no balls will fall from its.
• ### Calculation Of Ball Mill Critical Speed Ball Mill
Mill Critical Speed Calculation Effect of mill speed on the energy input in this experiment the overall motion of the assembly of 62 balls of two different sizes was studied the mill was rotated at 50 62 75 and 90 of the critical speed six lifter bars of rectangular crosssection were used at equal spacing the overall motion of the balls at the end of five revolutions is shown in figure 4 as.
• ### Calculate Critical Speed Of Ball Mill Nldcindia In
Calculate critical speed of ball mill Calculate critical speed of ball mill how to calculate critical speed of a ball mill ball mill critical speed mineral processing amp a ball mill critical speed actually ball rod ag or sag is the speed at which the centrifugal forces equal gravitational forces at the mill shells inside surface and no balls will fall from its position onto the shell.
• ### Ball Mill Critical Speed 911 Metallurgist
Jun 19 2015 Charge movement at Various Mill Critical Speed The Formula derivation ends up as follow Critical Speed is N c =76 6(D 0 5) where N c is the critical speed in revolutions per minute D is the mill effective inside diameter in feet Example a mill measuring 11’0” diameter inside of new shell liners operates at 17 3 rpm Critical speed is.
• ### Mill Critical Speed Calculation 911 Metallurgist
Oct 15 2015 The mill was rotated at 50 62 75 and 90 of the critical speed Six lifter bars of rectangular cross section were used at equal spacing The overall motion of the balls at the end of five revolutions is shown in Figure 4 As can be seen from the figure the overall motion of the balls changes with the mill speed inasmuch as the shoulder.
• ### Critical Speed Calculation Formula Of Ball Mill
Nov 18 2013 formula for calculating the critical speed of a ball mill ball mill speed newbie questions 2 most if the actual speed of a 6 ft diameter ball mill how to calculate critical speed of a ball mill YouTube.
• ### To Calculate Critical Speed Of Ball Mill Practical
Ball Nose Surface Finish Calculate Surface Finish when SAGMILLING COM Mill Critical Speed Determination Mill Critical Speed Determination This mill would need to spin at RPM to be at critical speed Result 2 This mill's measured RPM is of critical speed.
• ### Calculate Critical Speed Of Ball Mill Flyfilm Studio
How To Calculate Critical Speed Of A Ball Mill Ball mil design calculation yahoo answers nbsp there exists a speed of rotation the quot critical speed quot at which the contents of the mill would simply ride over the roof of the mill due to centrifugal action the critical speed rpm is given by nc d where d is the internal diameter in metres ball mills are normally operated.
• ### Calculate Critical Speed Of Ball Mill Flyfilm Studio
How to calculate critical speed of ball mill ball mill operating speed mechanical operations solved at what speed will the mill have to be run if the mm balls are replaced by mm balls all the other conditions remaining the same calculations the critical speed of ball mill is given by where r radius of ball mill r radius of ball for r mm and r mm n c rpm but the mill is.
• ### Critical Speed Of A Ball Mill
A ball mill grinds material by rotating a cylinder with steel or ceramic grinding balls causing the balls to fall back into the cylinder and onto the material to be groundll mills have been successfully run at speeds between 60 and 90 percent of critical speedwever most ball mills operate at speeds between 65 and 75 percent of critical speed.
• ### Critical Speed Of Ball Mill Formula Derivation
201983 The formula to calculate critical speed is given below N c 42 305 sqtDd N c critical speed of the mill D mill diameter specified in meters d diameter of the ball In practice Ball Mills are driven at a speed of 5090 of the critical speed the factor being influenced by economic consideration.
• ### Derivation For The Critical Speed Of Ball Mill Philippines
The critical speed of ball mill is given by where R = radius of ball mill r = radius of ball For R = 1000 mm and r = 50 mm n c = 307 rpm But the mill is operated at a speed of 15 rpm Therefore the mill is operated at 100 x 15 307 = 4886 of critical speed.
• ### Ball Mill Drum Speed Critical Solutions
150tph andesite crushing and reshaping production line Material andesite Output size 0 5 10 20 30mm Capacity 150tph Equipment ZSW1100X4200+PE750X1060+CSB160+VSI5X8522+3YZS2160.
• ### How To Calculate Mill Critcal Speed Ball Mill
The critical speed for a grinding mill is defined as the formula to calculate critical speed in ball mill and support online ball mills mine this formula calculates the critical speed of any ball mill most ball mills operate most efficiently between 65 and 75 of their critical speed and support online.
• ### Ball Mill Filling Degree Calculation
Ball Mills Ball Mills What Are These This formula calculates the critical speed of any ball mill Most ball mills operate most efficiently between 65 and 75 of their.
• ### Question On Critical Speed Of Ball Mill
2012 11 26The Critical Speed is used for the determination of ball mill ideal operating speed But for comparison rod mills would operate between 50 to 95 of the critical speed The faster the mill speed the greater the wear on the rods and liners Trade Assurance Pollo Mining Heavy Industries.
• ### Designing Calculation Of Critical Speed Of Mill
The formula to calculate critical speed is given below N c = 42 305 sqt (D d) N c = critical speed of the mill D = mill diameter specified in meters d = diameter of the ball In practice Ball Mills are driven at a speed of 50 90 of the critical speed the factor being influenced by economic consideration.
• ### Critical Speed Of The Ball Mill – Grinding Mill China
Critical speed of ball mill SBM supplies complete stone crushing screening plant grinding mill for quarrying mining industry in India China South Africa Germany Critical Speed Calculation Of Ball Mill – Raymond Grinding Mill Critical Speed Calculation Of Ball Mill BINQ provides both small ball mill and big ball mill each.
• ### How To Calculate Mill Critcal Speed
SAGMILLING COM Mill Critical Speed Determination The mill critical speed will be calculated based on the diameter (above) less twice this shell liner width Mill Actual RPM Enter the measured mill rotation in revolutions per minute Result 1 This mill would need to spin at RPM to be at 100 critical speed.
• ### How To Calculate Mill Critcal Speed
How To Calculate Critical Speed Of Ball Mill The critical speed of the mill c is defined as the speed at which a single ball in equation 814 d is the diameter inside the mill liners and le is the rod and ball mills in mular al and bhappu r b editors mineral processing plant design.
• ### Formula For Calculating The Critical Speed Of A Ball Mill
The formula to calculate critical speed is given below n c 42305 sqtdd n c critical speed of the mill d mill diameter specified in meters d diameter of the ball in practice ball mills are driven at a speed of 5090 of the critical speed the factor being influenced by economic tion for the.
• ### How To Calculate Mill Critcal Speed
Critical speed formula for ball mill the formula to calculate critical speed is given below n c 42305 sqtdd n c critical speed of the mill d mill diameter specified in meters d diameter of the ball in practice ball mills are driven at a speed of 5090 of the critical speed the factor being influenced by economic consideration details.
• ### Tubular Ball Mills Sciencedirect
Jan 01 2016 Substituting these values into Equation the critical speed will be given by (7 38) ν C = 42 3 D − d revs min for mill and ball diameters in metres In.
• ### Ball Mill Slideshare
Apr 24 2015 Critical speed R –radii of the mill r radii of the ball g –acceleration due to gravity 10 Critical speed of ball mill(in rps)= R= 45 2= 225m r=25 2=12 5m g=9 81m s2 Nc=1 08rps=64 8 rpm 11 r=60 2=30mm R=800 2=400mm Nc= 82 rpm Critical speed=1 4 operating speed Operating speed = 82 1 4= 586 rps=35 rpm.
• ### Critical Speed Formula For Ball Mill
The critical speed of ball mill is given by where R = radius of ball mill r = radius of ball For R = 1000 mm and r = 50 mm n c = 30 7 rpm But the mill is operated at a speed of 15 rpm Therefore the mill is operated at 100 x 15 30 7 = 48 86 of critical speed.
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# Convert Radians to Degrees and Vice-Versa in Python
Manav Narula Jul-18, 2021 Mar-27, 2021 Python Python Math
Degrees and radians are two of the highly used units to represent angles. The representation between these two units is shown below.
In this tutorial, we will discuss how to convert degree to radian and vice-versa.
## Use the `math` Module in Python to Convert Degree to Radian and Vice-Versa
It is straightforward to implement their relations manually in Python. We can use the `math` library for the required constants if we are not familiar with them. For example,
``````print((math.pi/2) * 180.0 / math.pi) #Rad to Deg
print(90 * math.pi / 180.0 ) #Deg to Rad
``````
Output:
``````90.0
1.5707963267948966
``````
Note that the `math.pi` returns the mathematical constant pi and can be replaced by its value 3.141592….
We can also use different functions from the `math` library to carry out these conversions.
The `math.degrees()` function will convert the radian value passed to it into degree. For example,
``````import math
print(math.degrees(math.pi/2))
``````
Output:
``````90.0
``````
The `math.radians()` function will do the opposite and convert the degree value to radians. For example,
``````import math
``````
Output:
``````1.5707963267948966
``````
## Use the `NumPy` Module to Convert Degree to Radian and Vice-Versa
The `NumPy` module is also equipped with different functions to convert between radian and degree values. The `numpy.degrees()` function converts radians to degree and can accept an array or list of values at once.
Similarly, the `numpy.radians()` function converts degree to radians and can also accept an array or list of values.
The following code shows an example of these two functions.
``````import numpy
lst1 = [math.pi/2, math.pi]
print(numpy.degrees(lst1)) #Rad to Deg
lst2 = [90,180]
``````
Output:
``````[ 90. 180.]
[1.57079633 3.14159265]
``````
This module also has `deg2rad()` and `rad2deg()` functions used to perform the same function but have a more descriptive name.
Author: Manav Narula
Manav is a IT Professional who has a lot of experience as a core developer in many live projects. He is an avid learner who enjoys learning new things and sharing his findings whenever possible.
## Related Article - Python Math
• Calculate Factorial in Python
• Calculate Inverse of Cosine in Python
• Calculate Modular Multiplicative Inverse in Python
• Fit Poisson Distribution to Different Datasets in Python
• Reduce Fractions in Python
• Define an Infinite Value in Python
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Problems & Puzzles: Puzzles
Puzzle 1078 Find the next prime of the form... Sebastián Martín Ruiz sent the following puzzle Let p3 or find a counterexample.
During the week 6-12 March, 2012, contributions came from Richard Chen, Alain Rochelli,
***
Richard Chen wrote:
p+(2^(p-2))(q-p) cannot be proven as composite for all primes p>3, thus it should eventually have a prime, but none is found for p<=prime(600), if a form can be proven as only contain composite numbers, then it either have covering congruence (e.g. 78557*2^n+1) or algebraic factorization (e.g. 4*9^n-1) or combine of them (e.g. 25*12^n-1), see section “proof” of this article, it has many examples and references.
If a form can be proven as only contain composite numbers by covering congruence, then every number of this form has small prime factors (usually < 10^4), and if a form can be proven as only contain composite numbers by algebraic factorization, then every number of this form has two factors with near size (for the case for difference-of-two-squares factorization) or a factor with near double the size of the other (for the case for difference-of-two-cubes factorization), if a for can be proven as only contain composite numbers by combine of them, then every number of this form meet at least one of these two conditions, but see the factorizations for n=15 and 32, they do not meet any of these two conditions, thus this form cannot be proven composite, other forms like this including 4*72^n-1, 2^n-n-2, n*13^n+1, (2^n-7)*2^n+1, 5*11^n+7*(11^n-1)/10, p*2^p+1 with prime p, (18^p-1)/17 with odd prime p, (32^p+5^p)/37 with prime p, all these forms do not have small primes, but cannot be proven as only contain composite numbers.
***
Alain wrote:
I am sending you a partial proof and a computed contribution for Puzzle 1078:
If q-p = 6*k+2 or 6*k+4 with k integer >= 0, based on congruence theory, we can prove that p+(q-p)*2^(p-2) is a multiple of 3.
In the other cases where q-p is a multiple of 6 (sequence OEIS A258578), using PARI application, I checked that there is no prime of the form p+(q-p)*2^(p-2) for p less than 75 000.
***
Records | Conjectures | Problems | Puzzles
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# Your Daily Equation #03: Lorentz Contraction
119,174 views | 00:25:16
202,992 views | 00:25:16
81,978 views | 00:02:19
#YourDailyEquation with Brian Greene offers brief and breezy discussions of the most pivotal equations of the ages. Even if your math is a bit rusty, these accessible and exciting stories of nature and numbers will allow you to see the universe in a new way.
The series includes live Q+As that explore many of the big questions that have occupied some of the greatest thinkers of our age and yielded some of the deepest insights into the nature of reality.
Episode 03: Last week, Brian spoke about time dilation and the impact of motion on the passage of time. Today, as the counterpart to time dilation, Brian will unpack length contraction or what is also known as the Lorentz contraction. If you want to hear more about the “weirdness of space” you’ll want to tune in to this episode of #YourDailyEquation.
Speaker 1:
Hey, everyone. Welcome to this next episode of your daily equation. In the last episode, we spoke about the impact of motion on the passage of time. And remember, it all came from the constant nature of the speed of light. If speed, according to Einstein, has strange properties at high speeds, namely near the speed of light, then since speed is nothing but space per time, then we learned that space and time have weird properties, and we worked out the weird properties of time in the last episode. Today as the counterpart to time dilation, what we did previously, we’re going to talk about the weirdness of space, which yields the equation as we will see, that is called length contraction, or Lorentz contraction. Lorentz after a famous physicist, who actually, strangely enough, even though we’re focusing on Einstein here, he actually came up with this equation first.
He didn’t completely interpret it correctly, and that’s really why these ideas are deeply associated with Einstein, but other people were thinking about these ideas as well. Okay, so let’s get into it, and I’m going to describe length contraction by using a concrete example first. But before I show you that little animation, let me just give you the basic idea, and then we’ll try to derive it first, intuitively, through animation, and then I’ll write down some equations that will capture this rigorously, mathematically. Okay. What’s the basic idea? The basic idea is, if I am watching an object race by me, and the canonical example that we will use is a train. If I watch a train race by me, and say, you are on that train, you will measure the length of the train, say, and get a particular value. If I then measure the length of the train that rushing by me, I will get a smaller value.
A shorter length only in the direction of motion. Lengths are contracted along the direction of motion according to an observer, in this case, me, watching that object in motion. That’s the basic idea. And how are we going to understand this? Where does it come from? Let’s get into a concrete example, and in fact, I’m going to use that example of the train. Let me bring up some animations, I think that will help make it clear. So, imagine that the train is rushing by me, but let’s focus upon you first. Imagine that you are on the train. That is you, generic you, right there. And how would you go about measuring the length of the train? Well, you pull out a tape measure, and you simply go from one end of the train all the way to the other end of the train, and you’d read off.
In this particular case, these numbers are completely made up. It’s 210 meters, according to your tape measure. How would I go about measuring the length of the train as it rushes by me? Well, I can’t really use a tape measure, at least not in any conventional way, because the train’s rushing by me. So, as I bring the tape measure up to the train, that’s going to rush away and I won’t be able to do the usual approach to measuring the length of an object with a ruler, with a measuring tape. Instead, there’s something clever that I can do, which is this. If I have a stopwatch, and if I know the speed, the velocity of the train along the track, here’s what I can do. As the train approaches me, right when the front of the train passes me, I turn on the stopwatch.
Okay? I let the walk go until the caboose, the very end of the train, goes by me, and then I click, I stop the watch. So, I get the elapsed time, from my perspective, that it took the train to rush by me, and then I simply use distance is velocity times time. I know the velocity of the train. I know the amount of time that elapsed between the front of the train passing me and the rear of the train passing me. I simply multiply those two together to get the length of the train that I would measure. Show that in a little visual here. So, there’s me, and there is where I’m going to stand. And when the front of the train passes me, I start the watch, I let it tick along. And then finally, when the back of the train passes, click. I stop the watch.
In this case, I got, say, 5.9 seconds. If the speed of the train was 30 meters per second, I would simply multiply those two numbers together. And the claim is that when I carry out that arithmetic, I will get a smaller number for the length of the train then you got using the tape measure approach. Again, these numbers, completely made up. This is not the amount of contraction at a slow speed of 30 meters per second, I should say. So, it’s really just illustrative of the qualitative effect that the length of an object in motion will be shrunken. Okay, so that’s the basic idea. Now, how do we argue for it?
And there are many ways that we can go about this, but the simplest is to make use of what we already derived, time dilation. And simply by using our earlier understanding of time dilation, we can get this result that I will measure a shorter length of the train. So, let’s do that. Again, I’ve got my handy iPad here to do that, and this should come up on your screen if the, yeah, the technology seems to be working. So, what did we learn about time dilation? Well, we learned that when someone is looking at a clock in motion from their perspective, they will say that that clock is ticking off time slowly compared to their clock. Now, I’m going to do something a little bit strange right now. I’m going to take your perspective on the train and consider Delta T according to you, versus the Delta T, the amount of time that you will claim elapses on my watch.
The reason why I’m doing this perspective, I’m looking at things from your perspective first, is a little bit subtle. Let’s do the calculation and then it’ll indicate why I had to do it this way for this particular derivation. But Delta T, right, the amount of time that will elapse on your watch compared to Delta T on my watch. We know the answer to that. You will say that more time elapses and you know the factor by which it is going to be greater. It’s one over the square root of one minus V squared over C squared from last time. In other words, the amount of time that elapses on my stop watch, compared to the amount of time that would elapse on your watch measuring the same events, would be given by squared of one minus V squared over C squared, times Delta T you.
So, less time on my clock compared to your clock. Why is that relevant? Well, if I consider the length of your train according to me, that’s my measurement of the length of your train. What am I doing? Well, as we described in that little animation, I’m taking the velocity of the train times the amount of time that goes by on my stopwatch. But now using the relationship between time according you, time according to me, I can write this as V times square root of one minus V squared over C squared times Delta T you. And then we know that if we write this essay, just move this guy over. One minus V squared over C squared, V Delta T you. This combination over here is just the length according to you, right? And therefore, length according to me is square root of one minus V squared over C squared times length according to you.
And so, there you have it, right? Because this factor over here, let me actually give it a little color to distinguish it. This guy over here is a number that will always be less than one, because it’s the reciprocal of gamma. In fact, I can write this if I would like as, equal to L U, divided by gamma. Gamma is always bigger than one, now that I’ve put it upside down there. And therefore, the length according to me will be less than the length according to you, who measures the length of the train while being on the train itself, being stationary with respect to the train. So, that’s the little derivation that the length of the train according to me will be less than the length of the train according to you.
Why do I have to play this funny game of going to your perspective, watching my clock, you might wonder. Well, couldn’t the person on the platform, namely me, say that the clock on the train is running slow and wouldn’t that give us the reverse result? If you think about it, if we tried to play this same game by using clocks on the train, as opposed to a clock on the platform, we’d have to make use of two such clocks. Because as your train is rushing by me, you could start your watch as you pass me, but you wouldn’t then pass me again to stop the watch.
Instead you’d need someone situated at the back of the train to click off when that person passes by me. There’s an asymmetry there, so you need to have two clocks in the train, and that yields a subtlety that we will come back to in one of the subsequent discussions. And that’s why I didn’t do it that way. So, this slightly circuitous approach, where I go from your view of my clock to my view of your length, is actually the shortest way to get to the result that we just derived. Now again, as with all things in special relativity, the effects are small in everyday life, because the factor of V over C is usually incredibly tiny, and therefore this gamma is often very, very close to one. It is very close to one at small speeds, but large speeds, it can make a really big difference.
So, let me just show you an example. Imagine that you have a taxi cab that is streaking down 5th Avenue in Manhattan at a speed very near the speed of light, and you’re watching this very fast moving taxi cab. What would that look like? Well, let me just show you a little animation of it. Of course, we’re imagining that the speed is close to the speed of light. That’s a little hard in everyday life, but we can do it in animation. And look at that taxi cab, isn’t that strange, right? The taxi cab is shrunken in the direction of motion only. The height of the taxi cab is unchanged. It’s that its length has been squeezed down by this factor of gamma. Now, you note something else. If you look at that picture a little bit more carefully, it’s not only that the taxi cab is squeezed along the direction of motion, it’s also twisted a bit, right?
We’re seeing the back bumper at a funny angle, relative to what you might expect. And the reason for that is that we are in a situation with relativity where there’s a difference between what’s actually happening out there in the world and what we perceive when we consider the rays of light bouncing off of an object. And if you consider the rays of light bouncing off of the taxi cab, you’re actually seeing the taxi cab at different moments in time, different points on it, because the light from different locations on the taxi cab have to travel different distances to your eyeball, and therefore you’re not seeing the taxi cab, the whole thing at one instant of time. You’re seeing different points on the taxi cab at different moments in time, dependent on how far away those points in the taxi cab are from your eyeball.
And when you take that complexity into account, you get that interesting twisting effect that you’re seeing in the animation. But the bottom line of what’s actually happening to the taxi cab from our perspective is what we derive mathematically, it’s length in the direction of motion is being shrunken by a factor of gamma. Now, imagine that you were inside of that taxi cab. How would things look from your perspective? Well, from your perspective, the taxi cab is not moving relative to you. In fact, as we’ve emphasized, if you’re moving at a fixed speed in a fixed direction, you can claim to be at rest, and it’s everything else that’s rushing by you in the opposite direction. So, from your perspective, it’s life as normal inside of the taxi cab. And if you look out the window, it’ll be the outside world that has all this weird stuff happening with lengths being contract, and again, based upon the light travel time, interesting twisting and curving from your perspective.
So, let me show you that alternative perspective. Here it is. So, there you are inside the taxi cab. Everything appears normal inside, but look at what things look like on the outside. Things are shrunken, they’re twisted, because of the weirdness of the rate at which different clocks are ticking and the different distances that the light has to travel, all folded into this length contraction in the direction of motion. So, that’s the bottom line of how motion affects space, shrunken in the direction of motion. The other perpendicular directions are not influenced at all. And as we’ve seen, we actually were able to derive it from our understanding of how clocks that are in relative motion will tick with respect to one another. Okay, so that’s today’s daily equation. Keep in mind that the length, me, being equal to length of you, divided by gamma, you have to interpret what these symbols mean.
It’s the length according to me of your length, as measured with respect to a stationary object. You are on the train itself, but if you keep the symbols in your mind straight, we now understand the relationship between time for you, time for me, length for you, length for me. I think next time we’re going to take up, I think I’m going to look at maybe relativistic mass, or the relativistic velocity combination formula. See, as I go forward, again, love to hear more of your suggestions, which I’m keeping a list of, and as we go forward, I’ll try to incorporate your suggestions into the equations that we discuss. Okay. But that’s it for today. That is your daily equation. Look forward to seeing you at the next episode. Take care.
# Your Daily Equation #03: Lorentz Contraction
#YourDailyEquation with Brian Greene offers brief and breezy discussions of the most pivotal equations of the ages. Even if your math is a bit rusty, these accessible and exciting stories of nature and numbers will allow you to see the universe in a new way.
The series includes live Q+As that explore many of the big questions that have occupied some of the greatest thinkers of our age and yielded some of the deepest insights into the nature of reality.
Episode 03: Last week, Brian spoke about time dilation and the impact of motion on the passage of time. Today, as the counterpart to time dilation, Brian will unpack length contraction or what is also known as the Lorentz contraction. If you want to hear more about the “weirdness of space” you’ll want to tune in to this episode of #YourDailyEquation.
### Transcription
Speaker 1:
Hey, everyone. Welcome to this next episode of your daily equation. In the last episode, we spoke about the impact of motion on the passage of time. And remember, it all came from the constant nature of the speed of light. If speed, according to Einstein, has strange properties at high speeds, namely near the speed of light, then since speed is nothing but space per time, then we learned that space and time have weird properties, and we worked out the weird properties of time in the last episode. Today as the counterpart to time dilation, what we did previously, we’re going to talk about the weirdness of space, which yields the equation as we will see, that is called length contraction, or Lorentz contraction. Lorentz after a famous physicist, who actually, strangely enough, even though we’re focusing on Einstein here, he actually came up with this equation first.
He didn’t completely interpret it correctly, and that’s really why these ideas are deeply associated with Einstein, but other people were thinking about these ideas as well. Okay, so let’s get into it, and I’m going to describe length contraction by using a concrete example first. But before I show you that little animation, let me just give you the basic idea, and then we’ll try to derive it first, intuitively, through animation, and then I’ll write down some equations that will capture this rigorously, mathematically. Okay. What’s the basic idea? The basic idea is, if I am watching an object race by me, and the canonical example that we will use is a train. If I watch a train race by me, and say, you are on that train, you will measure the length of the train, say, and get a particular value. If I then measure the length of the train that rushing by me, I will get a smaller value.
A shorter length only in the direction of motion. Lengths are contracted along the direction of motion according to an observer, in this case, me, watching that object in motion. That’s the basic idea. And how are we going to understand this? Where does it come from? Let’s get into a concrete example, and in fact, I’m going to use that example of the train. Let me bring up some animations, I think that will help make it clear. So, imagine that the train is rushing by me, but let’s focus upon you first. Imagine that you are on the train. That is you, generic you, right there. And how would you go about measuring the length of the train? Well, you pull out a tape measure, and you simply go from one end of the train all the way to the other end of the train, and you’d read off.
In this particular case, these numbers are completely made up. It’s 210 meters, according to your tape measure. How would I go about measuring the length of the train as it rushes by me? Well, I can’t really use a tape measure, at least not in any conventional way, because the train’s rushing by me. So, as I bring the tape measure up to the train, that’s going to rush away and I won’t be able to do the usual approach to measuring the length of an object with a ruler, with a measuring tape. Instead, there’s something clever that I can do, which is this. If I have a stopwatch, and if I know the speed, the velocity of the train along the track, here’s what I can do. As the train approaches me, right when the front of the train passes me, I turn on the stopwatch.
Okay? I let the walk go until the caboose, the very end of the train, goes by me, and then I click, I stop the watch. So, I get the elapsed time, from my perspective, that it took the train to rush by me, and then I simply use distance is velocity times time. I know the velocity of the train. I know the amount of time that elapsed between the front of the train passing me and the rear of the train passing me. I simply multiply those two together to get the length of the train that I would measure. Show that in a little visual here. So, there’s me, and there is where I’m going to stand. And when the front of the train passes me, I start the watch, I let it tick along. And then finally, when the back of the train passes, click. I stop the watch.
In this case, I got, say, 5.9 seconds. If the speed of the train was 30 meters per second, I would simply multiply those two numbers together. And the claim is that when I carry out that arithmetic, I will get a smaller number for the length of the train then you got using the tape measure approach. Again, these numbers, completely made up. This is not the amount of contraction at a slow speed of 30 meters per second, I should say. So, it’s really just illustrative of the qualitative effect that the length of an object in motion will be shrunken. Okay, so that’s the basic idea. Now, how do we argue for it?
And there are many ways that we can go about this, but the simplest is to make use of what we already derived, time dilation. And simply by using our earlier understanding of time dilation, we can get this result that I will measure a shorter length of the train. So, let’s do that. Again, I’ve got my handy iPad here to do that, and this should come up on your screen if the, yeah, the technology seems to be working. So, what did we learn about time dilation? Well, we learned that when someone is looking at a clock in motion from their perspective, they will say that that clock is ticking off time slowly compared to their clock. Now, I’m going to do something a little bit strange right now. I’m going to take your perspective on the train and consider Delta T according to you, versus the Delta T, the amount of time that you will claim elapses on my watch.
The reason why I’m doing this perspective, I’m looking at things from your perspective first, is a little bit subtle. Let’s do the calculation and then it’ll indicate why I had to do it this way for this particular derivation. But Delta T, right, the amount of time that will elapse on your watch compared to Delta T on my watch. We know the answer to that. You will say that more time elapses and you know the factor by which it is going to be greater. It’s one over the square root of one minus V squared over C squared from last time. In other words, the amount of time that elapses on my stop watch, compared to the amount of time that would elapse on your watch measuring the same events, would be given by squared of one minus V squared over C squared, times Delta T you.
So, less time on my clock compared to your clock. Why is that relevant? Well, if I consider the length of your train according to me, that’s my measurement of the length of your train. What am I doing? Well, as we described in that little animation, I’m taking the velocity of the train times the amount of time that goes by on my stopwatch. But now using the relationship between time according you, time according to me, I can write this as V times square root of one minus V squared over C squared times Delta T you. And then we know that if we write this essay, just move this guy over. One minus V squared over C squared, V Delta T you. This combination over here is just the length according to you, right? And therefore, length according to me is square root of one minus V squared over C squared times length according to you.
And so, there you have it, right? Because this factor over here, let me actually give it a little color to distinguish it. This guy over here is a number that will always be less than one, because it’s the reciprocal of gamma. In fact, I can write this if I would like as, equal to L U, divided by gamma. Gamma is always bigger than one, now that I’ve put it upside down there. And therefore, the length according to me will be less than the length according to you, who measures the length of the train while being on the train itself, being stationary with respect to the train. So, that’s the little derivation that the length of the train according to me will be less than the length of the train according to you.
Why do I have to play this funny game of going to your perspective, watching my clock, you might wonder. Well, couldn’t the person on the platform, namely me, say that the clock on the train is running slow and wouldn’t that give us the reverse result? If you think about it, if we tried to play this same game by using clocks on the train, as opposed to a clock on the platform, we’d have to make use of two such clocks. Because as your train is rushing by me, you could start your watch as you pass me, but you wouldn’t then pass me again to stop the watch.
Instead you’d need someone situated at the back of the train to click off when that person passes by me. There’s an asymmetry there, so you need to have two clocks in the train, and that yields a subtlety that we will come back to in one of the subsequent discussions. And that’s why I didn’t do it that way. So, this slightly circuitous approach, where I go from your view of my clock to my view of your length, is actually the shortest way to get to the result that we just derived. Now again, as with all things in special relativity, the effects are small in everyday life, because the factor of V over C is usually incredibly tiny, and therefore this gamma is often very, very close to one. It is very close to one at small speeds, but large speeds, it can make a really big difference.
So, let me just show you an example. Imagine that you have a taxi cab that is streaking down 5th Avenue in Manhattan at a speed very near the speed of light, and you’re watching this very fast moving taxi cab. What would that look like? Well, let me just show you a little animation of it. Of course, we’re imagining that the speed is close to the speed of light. That’s a little hard in everyday life, but we can do it in animation. And look at that taxi cab, isn’t that strange, right? The taxi cab is shrunken in the direction of motion only. The height of the taxi cab is unchanged. It’s that its length has been squeezed down by this factor of gamma. Now, you note something else. If you look at that picture a little bit more carefully, it’s not only that the taxi cab is squeezed along the direction of motion, it’s also twisted a bit, right?
We’re seeing the back bumper at a funny angle, relative to what you might expect. And the reason for that is that we are in a situation with relativity where there’s a difference between what’s actually happening out there in the world and what we perceive when we consider the rays of light bouncing off of an object. And if you consider the rays of light bouncing off of the taxi cab, you’re actually seeing the taxi cab at different moments in time, different points on it, because the light from different locations on the taxi cab have to travel different distances to your eyeball, and therefore you’re not seeing the taxi cab, the whole thing at one instant of time. You’re seeing different points on the taxi cab at different moments in time, dependent on how far away those points in the taxi cab are from your eyeball.
And when you take that complexity into account, you get that interesting twisting effect that you’re seeing in the animation. But the bottom line of what’s actually happening to the taxi cab from our perspective is what we derive mathematically, it’s length in the direction of motion is being shrunken by a factor of gamma. Now, imagine that you were inside of that taxi cab. How would things look from your perspective? Well, from your perspective, the taxi cab is not moving relative to you. In fact, as we’ve emphasized, if you’re moving at a fixed speed in a fixed direction, you can claim to be at rest, and it’s everything else that’s rushing by you in the opposite direction. So, from your perspective, it’s life as normal inside of the taxi cab. And if you look out the window, it’ll be the outside world that has all this weird stuff happening with lengths being contract, and again, based upon the light travel time, interesting twisting and curving from your perspective.
So, let me show you that alternative perspective. Here it is. So, there you are inside the taxi cab. Everything appears normal inside, but look at what things look like on the outside. Things are shrunken, they’re twisted, because of the weirdness of the rate at which different clocks are ticking and the different distances that the light has to travel, all folded into this length contraction in the direction of motion. So, that’s the bottom line of how motion affects space, shrunken in the direction of motion. The other perpendicular directions are not influenced at all. And as we’ve seen, we actually were able to derive it from our understanding of how clocks that are in relative motion will tick with respect to one another. Okay, so that’s today’s daily equation. Keep in mind that the length, me, being equal to length of you, divided by gamma, you have to interpret what these symbols mean.
It’s the length according to me of your length, as measured with respect to a stationary object. You are on the train itself, but if you keep the symbols in your mind straight, we now understand the relationship between time for you, time for me, length for you, length for me. I think next time we’re going to take up, I think I’m going to look at maybe relativistic mass, or the relativistic velocity combination formula. See, as I go forward, again, love to hear more of your suggestions, which I’m keeping a list of, and as we go forward, I’ll try to incorporate your suggestions into the equations that we discuss. Okay. But that’s it for today. That is your daily equation. Look forward to seeing you at the next episode. Take care.
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## SurvivalSignature.jl
Computation and numerical approximation of survival signatures.
Author FriesischScott
Popularity
5 Stars
Updated Last
1 Month Ago
Started In
January 2020
# SurvivalSignature.jl
Julia package for the computation of survival signatures as introduced by Coolen et al. (2013).
In addition to the regular analytical computations, this package contains a Monte Carlo simulation based algorithms to approximate the survival signature for systems where the computational demand for the standard approach is too high.
## Examples
### Exact computation
Consider a simple system of six components divided into two types.
Computing the survival signature for any system requires three definitions: the system, the component types, and a structure function. Start by defining the system as an adjacency matrix and the types as a dictionary.
```A = zeros(6, 6)
A[1, [2, 3]] .= 1.0
A[2, [4, 5]] .= 1.0
A[3, [4, 6]] .= 1.0
A[4, [5, 6]] .= 1.0
types = Dict(1 => [1, 2, 5], 2 => [3, 4, 6])```
`SurvivalSignature.jl` provides a simple structure function to check s-t-connectivity, suitable for reliability block diagrams. The function `s_t_connectivity(nodes, source, target)` returns new function which accepts a system and vector of functioning components as arguments.
```# returns a new function (system::Array{Float64,2}, x::Vector)
φ = s_t_connectivity([1:6;], [1], [5, 6])```
Next, the survival signature is calculated by running
`Φ = survivalsignature(A, types, φ)`
resulting in the following signature
```4×4 Matrix{Float64}:
0.0 0.0 0.0 0.0
0.0 0.0 0.111111 0.333333
0.0 0.0 0.444444 0.666667
1.0 1.0 1.0 1.0```
### Approximation
If exact computation of the survival signature is not possible for the desired system, it can be approximated using Monte Carlo simulation by providing a desired number of samples to use per entry of the survival signature and optional target coefficient of variation.
`Φ, cov = survivalsignature(A, types, φ, 10000, 0.001)`
In addition to the survival signature this will return the coefficients of variation for each entry.
### Preprocessing
Both the analytical solution and the Monte Carlo approximation accept an optional preprocessor to exclude entries of the survival signature based on some prior knowledge.
```Φ, cov = survivalsignature(A, types, φ, preprocessor)
Φ = survivalsignature(A, types, φ, 10000, 0.001, preprocessor)```
A valid preprocessor is a function which takes the survival signature and a system as arguments. At the time of preprocessing the signature passed to the function will be `Inf` for all entries. Entries are excluded from computation by setting them to zero. The preprocessor must modify the signature in place and return `nothing`.
```function preprocessor!(Φ, system)
# exclude entries by setting them to 0
return nothing
end```
A preprocessor using percolation is included as `percolation_preprocessor!(Φ, A)`.
### Reliability Analysis
If the cdfs of the failure time distributions for each component type are known, the reliability can be computed analytically using distributions from `Distributions.jl`.
```distributions = Dict(1 => Exponential(1), 2 => Weibull(2, 1))
time = [0:0.001:1;]
P = reliability(time, distributions, Φ)```
Alternatively, the reliability can be approximated by providing a `NxM Matrix{Float64}` of failure times where `N` is the number of samples and `M` the number of components.
`P = reliability(time, Φ, types, failures)`
### Interval Predictor Model
For complex examples with a lot of components and component types even the Monte Carlo approximation will have difficulties estimating the complete survival signature. For these cases we provide an alternative presented in Behrensdorf et al. (2024), where a surrogate model of the survival signature is used instead of the full approximation. This surrogate is based on radial basis function networks and interval predictor models (IPM). By strategically selecting a few values of the signature to approximate with the MC approximation it significantly reduces the numerical demand. The IPM captures the uncertainty of the approximation and as a result the surrogate returns imprecise bounds on the survival signature. The following code will compute the `IPMSurvivalSignature`.
```N = 1000
covtol = 1e-3
wtol = 1e-3
ci = [15, 15, 10]
ipmsignature = survivalsignature(A, types, φ, ci; samples=N, covtol=covtol, wtol=wtol)```
Here, `ci` refers to the number of center points used in each dimension of the underlying radial basis function network and `wtol` is the tolerance used to end the adaptive refinement of the surrogate. For more information please refer to the paper.
The reliability analysis returns the upper and lower bound of the reliability when used with the interval predictor model.
`P_l, P_u = reliability(time, distributions, ipmsignature)`
## References
Behrensdorf, J., Regenhardt, T.-E., Broggi, M., Beer, M. (2021) Numerically efficient computation of the survival signature for the reliability analysis of large networks, Reliability Engineering & System Safety, 107935, https://doi.org/10.1016/j.ress.2021.107935.
Behrensdorf, J., Broggi, M., & Beer, M. (2024). Interval Predictor Model for the Survival Signature Using Monotone Radial Basis Functions. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, 10(3), 04024034. https://doi.org/10.1061/AJRUA6.RUENG-1219
Coolen F.P.A., Coolen-Maturi T. (2013) Generalizing the Signature to Systems with Multiple Types of Components. In: Zamojski W., Mazurkiewicz J., Sugier J., Walkowiak T., Kacprzyk J. (eds) Complex Systems and Dependability. Advances in Intelligent and Soft Computing, 170. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30662-4_8
### Required Packages
View all packages
### Used By Packages
No packages found.
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# Is energy conserved?
1. Feb 16, 2015
My understanding at present is that if a system of interacting particles is analysed using classical physics or special relativity energy is conserved, but if that same system is analysed using general relativity energy is not conserved. So is it conserved or not?
Looking at it another way, energy is conserved if spacetime is static but not conserved if spacetime is evolving. It's apparently believed that spacetime is evolving so energy is not conserved. Or is it conserved? Help!
Is the conservation of energy principle an approximation only which works well in certain conditions such as smallish localised areas or is there some other resolution to this apparent paradox?
Thank you
2. Feb 16, 2015
### Orodruin
Staff Emeritus
Energy conservation is something local and in GR it is not necessarily true that it can be extended to a global concept. You might simply not be able to ask the question "what is the total energy of the Universe?" However, the conservation of energy is replaced by the divergence of the energy-momentum tensor being zero, so you cannot go all out crazy with energy non-conservation even in GR.
3. Feb 16, 2015
### HallsofIvy
Even in special relativity, it is "total mass-energy" that is conserved, not mass or energy separately.
4. Feb 16, 2015
### Orodruin
Staff Emeritus
I do not agree with this. Energy, including the rest energy due to mass, is conserved in special relativity and this has a well defined meaning. Simply take a surface of simultaneity in a given frame and integrate the time-time component of the energy-momentum tensor over it and it will be the same regardless of the time defining the surface. That you have to include the masses of your constituents in order to obtain this is a different issue altogether.
5. Feb 16, 2015
### DrGreg
6. Feb 16, 2015
### Staff: Mentor
This doesn't make sense. SR is just a special case of GR, so if SR applies to a given system, it must give the same answers as GR gives, since GR applied to that same system just is SR.
But this isn't the same system analyzed two different ways (with SR and GR). It's two different systems. A static spacetime is a different system from a non-static spacetime. So there's nothing mysterious about the fact that energy conservation works differently in the two systems; they're different systems.
7. Feb 17, 2015
### pervect
Staff Emeritus
What do you mean by energy? There are certain technical definitions of energy that ARE conserved in GR, but they have prerequisites (such as static spacetimes, or asymptotically flat spacetimes) before they are able to be calculated. There isn't a single universal definition of "energy" in GR that always gives a conserved quantity.
It would seem to me from the tone of your question that you're not familiar with the technicalities. I can't blame you for that, really, but I'm at a loss to answer a question about "energy being conserved" if we don't have a mutual understanding of what "energy" is.
Probably the most readable introduction is what Dr. Greg already quoted, the sci.physics.faq reference.
It might also be helpful to say things like "ADM energy and Bondi energy are defined and conserved in asymptotically flat space-times, while Komar energy is defined and conserved in static space-times.
I suppose it might be helpful to note that if you add up non-gravitational sources of energy, you won't get a conserved quantity unless you include something that's equivalent to the Newtonian idea of "gravitational potential energy". But GR doesn't have a single clear idea to replace this Newtonian idea, though it does have some ideas of how to define conserved energies in special circumstances.
8. Feb 17, 2015
Thank you very much Orodruin. That's largely clarified things but I still have some problems one of them best exemplified by the following question:
Is it true that there exists certain problems that can be solved but which require the application of the conservation of energy principle for their solution?
If it is true then I assume that GR can't be applied to the problem because it doesn't necessarily recognise energy conservation. Does this mean that such problems do not lie withinin the domain of applicability of GR or could there be some other reasons why GR does not work
9. Feb 17, 2015
Thank you Orodruin, Halls of Ivy and DrGreg. I need to do some more research on this and you have provided some pointers about where to look. The FAQ referred to by yourself DrGreg looks particularly promising.
10. Feb 17, 2015
Thanks for your reply PeterDonis but I am just learning this stuff and I find your replies to be a bit contradictory. I Might be misunderstanding the points you have made. You say that SR and GR "must give the same answers" but then say that "energy conservation works differently in the two systems". If it works differently does it still give the same answers?
The following is a quote from Sean Carrolls blog referring to GR: (google "energy is not conserved")
"When the space through which particles move is changing the total energy of those particles is not conserved"
That's the thing that confuses me because in SR energy is conserved evidenced, for example, by nuclear energy
(Sean Carrol is a theoretical cosmologist from Caltech who amongst other things specialises in GR)
11. Feb 17, 2015
Thank you pervect.
By energy I mean how it is defined in classical physics in terms of work done and how it relates to particle events and interactions. I'm interested in things such as potential/kinetic energy changes, particle collisions, nuclear reactions and so on. As an example consider a high energy electron electron collision. Can GR be used to analyse the event and would it give the same answers as SR?
12. Feb 17, 2015
### Staff: Mentor
Suppose you have three different situations, A, B, and C. And suppose that A and B are different scenarios without tidal gravity while C involves tidal gravity.
An SR energy analysis gets a certain result for A, and a SR energy analysis gets a certain result for B. The two results differ because A and B are different scenarios.
A GR energy analysis gets a certain result for A, and a GR energy analysis gets a certain result for B. Again, the two results differ as above. The GR result for A agrees with the SR result for A and the GR result for B agrees with the SR result for B.
Scenario C cannot be analyzed with SR at all, so GR is required. Depending on the details there may not be any globally conserved energy available for the analysis.
13. Feb 17, 2015
### Staff: Mentor
Perhaps this thread is the place to clarify a point that I always found cloudy. Conservation of energy in GR as discussed in the above referenced FAQ, and conservation of energy related to zero-point energy and increasing volume of space. Are those properly two separate subjects, or are they the same with the "cosmological constant" representing the average of what happens at the micro level?
14. Feb 17, 2015
### Staff: Mentor
Any time that SR can be used GR can also be used and will give the same answer.
Additionally, there are scenarios in which SR can't be used. In those GR can be used, but in some of them there is no globally conserved energy.
15. Feb 17, 2015
Thank you DaleSpam
I can see that the results differ for different scenarios but not that they agree for the same scenario. Here's another quote from the Sean Carroll blog:
"If that spacetime is standing completely still, the total energy is constant; if its evolving, the energy changes in a completely unambiguous way".
To me that suggests that the SR energy analysis and GR energy analysis give different results.
16. Feb 17, 2015
### Staff: Mentor
That is two different scenarios. A static spacetime and a non static spacetime. A vs B, not SR vs GR.
17. Feb 17, 2015
### Staff: Mentor
When they are used to analyze the same system, yes.
When the systems are different, yes.
That's because in any system that can be analyzed using SR, "the space through which particles move" is not changing. In such a system, as Carroll says, energy is conserved.
If "the space through which particles move" is changing (for example, in the universe as a whole, which is expanding), you can't use SR to analyze the system. You have to use GR, and you will find, as Carroll says, that energy is not conserved.
18. Feb 17, 2015
### DrStupid
What does that mean for mass and energy to be conserved separately or not?
19. Feb 17, 2015
Thank you both. The last comment above and Dalespams comment summarise the difficulty I'm having here. Basically I want to analyse a single scenario and not two scenarios. If I think about a moving proton I see that as a single scenario. I can imagine it moving through spacetime and not two different spacetimes.
Spacetime is whatever it is and the proton moves through it. How can spacetime be changing and not changing? How can energy be conserved and not conserved? (Reminds me of quantum superpositions.)
I'm still puzzling over this but it will start to make some sort of sense if, for example, SR and GR each had its own domain of applicabilty, perhaps to do with the scale of the event. If something like this is the case I still see difficulties.
20. Feb 17, 2015
### Orodruin
Staff Emeritus
This depends on what your actual spacetime is. If it is sufficiently close to a Minkowski spacetime, then SR will suffice and the GR approach will just mimic it. If it is not sufficiently close, then only GR will be applicable.
I want to make a comparison to parallel lines on a sphere. If you are studying a sufficiently small portion of the sphere, the deviations from Euclidean space will be small and parallel lines will not cross. Euclidean space would be fine for approximating this behaviour. But if you look at distances comparable to the curvature, then all straight lines will cross and Euclidean space is not sufficient to make the description.
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# nLab (infinity,1)-categorical hom-space
## Theorems
#### $\left(\infty ,1\right)$-Category theory
(∞,1)-category theory
model category
for ∞-groupoids
mapping space
# Contents
## Idea
Where an ordinary category has a hom-set, an (∞,1)-category has an ∞-groupoid of morphisms between any two objects, a hom-space.
There are several ways to present an (∞,1)-category $C$ by an ordinary category $C$ equipped with some extra structure: for instance $C$ may be a category with weak equivalences or a model category or even a simplicial model category. In all of these presentations, given two objects $X,Y\in C$, there is a way to construct a simplicial sets $ℝC\left(X,Y\right)$ that prsesents the hom-∞-groupoid $C\left(X,Y\right)$. This simplicial set – or rather its homotopy type – is called the derived hom space or homotopy function complex and denoted $R\mathrm{Hom}\left(X,Y\right)$ or similarly.
## Presentations
There are many ways to present an (∞,1)-category by category theoretic data, and for each of these there are corresponding tools for explicitly computing the derived hom spaces.
The most basic data is that of a category with weak equivalences. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called simplicial localization. This we discuss below in For a category with weak equivalences.
Particularly useful extra structure on a category with weak equivalences that helps with computing the derived hom spaces is the structure of a model category. Using this one can construc simplicial resolutions of objects – called framings – that generalize cylinder objects and path objects, and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in For a model category.
Still a bit more helpful structure on top of a bare model category is that of a simplicial model category. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the sSet-hom objects. This we discuss below in For a simplicial model category.
### For a category with weak equivalences
Let $\left(C,W\subset \mathrm{Mor}\left(C\right)\right)$ be a category with weak equivalences.
###### Definition
Fix $n\in ℕ$. For $X,Y\in \mathrm{Obj}\left(C\right)$, define a category ${\mathrm{wMor}}_{C}^{n}\left(X,Y\right)$
• whose objects are zig-zags of morphisms in $C$ of length $n$
$X={X}_{0}←{X}_{1}\to {X}_{2}←\cdots \to {X}_{n-1}←{X}_{n}=Y$X = X_0 \leftarrow X_1 \to X_2 \leftarrow \cdots \to X_{n-1} \leftarrow X_n = Y
such that each morphism going to the left, ${X}_{2k}←{X}_{2k+1}$, is a weak equivalence, an element in $W$;
• morphisms between such objects $\left(X,{X}_{i},Y\right)\to \left(X\prime ,X{\prime }_{i},Y\prime \right)$ are collections of weak equivalences $\left({X}_{i}\to X{\prime }_{i}\right)$ for all $0 such that all triangles and squares commute.
###### Definition
Write $N\left({\mathrm{wMor}}_{C}^{n}\left(X,Y\right)\right)$ for the nerve of this category, a simplicial set.
The hammock localization ${L}_{W}^{H}C$ of $C$ is the simplicially enriched category with objects those of $C$ and hom-objects given by the colimit over the length of these hammock hom-categories
${L}^{H}C\left(X,Y\right):=\underset{{\to }_{n}}{\mathrm{lim}}N\left({\mathrm{wMor}}_{C}^{n}\left(X,Y\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$L^H C(X,Y) := \lim_{\to_n} N(wMor_C^n(X,Y)) \,.
The Kan fibrant replacement of this simplicial set is the derived hom-space between $X$ and $Y$ of the $\left(\infty ,1\right)$-category modeled by $\left(C,W\right)$.
### For a model category
The derived hom spaces of a model category $C$ may always be computed in terms of simplicial resolutions with respect to the Reedy model structure $\left[{\Delta }^{\mathrm{op}},C{\right]}_{\mathrm{Reedy}}$. These resolutions are often called framings (Hovey). These constructions are originally due to (Dwyer-Hirschhorn-Kan).
Let $C$ be any model category.
###### Observation
$\left(\mathrm{const}⊣{\mathrm{ev}}_{0}⊣\left(-{\right)}^{{×}^{•}}\right):C\stackrel{\stackrel{\mathrm{const}}{\to }}{\stackrel{\stackrel{{\mathrm{ev}}_{0}}{←}}{\underset{\left(-{\right)}^{{×}^{•}}}{\to }}}\phantom{\rule{thinmathspace}{0ex}},\left[{\Delta }^{\mathrm{op}},C\right]\phantom{\rule{thinmathspace}{0ex}},$(const \dashv ev_0 \dashv (-)^{\times^\bullet}) : C \stackrel{\overset{const}{\to}}{\stackrel{\overset{ev_0}{\leftarrow}}{\underset{(-)^{\times^\bullet}}{\to}}} \,, [\Delta^{op}, C] \,,
where
1. $\mathrm{const}X:\left[n\right]↦X$;
2. ${\mathrm{ev}}_{0}{X}_{•}={X}_{0}$;
3. ${X}^{{×}^{•}}:\left[n\right]↦{X}^{{×}^{n}}$.
###### Remark
For $X\in C$ fibrant, ${X}^{{×}^{•}}$ is fibrant in the Reedy model structure $\left[{\Delta }^{\mathrm{op}},C{\right]}_{\mathrm{Reedy}}$.
###### Proof
The matching morphisms are in fact isomorphisms.
###### Definition
Let $C$ be a model category
1. For $X\in C$ any object, a simplicial frame on $X$ is a factorization of $\mathrm{const}X\to {X}^{{×}^{•}}$ into a weak equivalence followed by a fibration in the Reedy model structure $\left[{\Delta }^{\mathrm{op}},C{\right]}_{\mathrm{Reedy}}$.
2. A right framing in $C$ is a functor $\left(-{\right)}_{•}:C\to \left[{\Delta }^{\mathrm{op}},C\right]$ with a natural isomorphism $\left(X{\right)}_{0}\simeq X$ such that ${X}_{•}$ is a simplicial frame on $X$.
Dually for cosimplicial frames.
This appears as (Hovey, def. 5.2.7).
###### Remark
By remark 1 a simplicial frame ${X}_{•}$ in the above is in particular fibrant in $\left[{\Delta }^{\mathrm{op}},C{\right]}_{\mathrm{Reedy}}$.
###### Proposition
For $X\in C$ cofibrant and $A\in C$ fibrant, there are weak equivalences in ${\mathrm{sSet}}_{\mathrm{Quillen}}$
${\mathrm{Hom}}_{C}\left({X}^{•},A\right)\stackrel{\simeq }{\to }\mathrm{diag}{\mathrm{Hom}}_{C}\left({X}^{•},{A}_{•}\right)\stackrel{\simeq }{←}{\mathrm{Hom}}_{C}\left(X,{A}_{•}\right)\phantom{\rule{thinmathspace}{0ex}},$Hom_C(X^\bullet, A) \stackrel{\simeq}{\to} diag Hom_C(X^\bullet, A_\bullet) \stackrel{\simeq}{\leftarrow} Hom_C(X, A_\bullet) \,,
(where in the middle we have the diagonal of the bisimplicial set $\mathrm{Hom}\left({X}^{•},{A}_{•}\right)$).
This appears as (Hovey, prop. 5.4.7).
Either of these simplicial sets is a model for the derived hom-space $ℝ\mathrm{Hom}\left(X,A\right)$.
###### Remark
By developing these constructions further, one obtains a canonical simplicial model category-resolution of (left proper and combinatorial) model categories $C$, such that the simplicial resolutions given by framings are just the cofibrant$\to$fibrant $\mathrm{sSet}$-hom objects as discussed below.
This is discussed at Simplicial Quillen equivalent models.
### For a simplicial model category
We describe here in more detail properties of hom-objects in a simplicial model category for the case that the domain objects are cofibrant and the codomain objects are fibrant.
The crucial axiom used for this is the axiom of an enriched model category $C$ which says that
• $\cdot :C×\mathrm{SSet}\to C$\cdot : C \times SSet \to C
is a Quillen bifunctor;
• or equivalently that for $X\to Y$ a cofibration and $A\to B$ a fibration the induced morphism
$C\left(Y,A\right)\to C\left(X,A\right){×}_{C\left(X,B\right)}C\left(Y,B\right)$C(Y, A) \to C(X,A) \times_{C(X,B)} C(Y,B)
is a fibration, which is acyclic if either $X\to Y$ or $A\to B$ is.
First of all the first statement directly implies that for $\varnothing \in C$ the initial object and $A\in C$ any object, the simplicial set $C\left(\varnothing ,A\right)=*$ is the terminal simplicial set, because for any simplicial set $S$
$\begin{array}{rl}\mathrm{SSet}\left(S,C\left(\varnothing ,A\right)\right)& ={\mathrm{Hom}}_{C}\left(\varnothing \cdot S,A\right)\\ & ={\mathrm{Hom}}_{C}\left({\mathrm{colim}}_{\varnothing }\cdot S,A\right)\\ & ={\mathrm{Hom}}_{C}\left(\varnothing ,A\right)\\ & =*\end{array}\phantom{\rule{thinmathspace}{0ex}},$\begin{aligned} SSet(S,C(\emptyset, A)) & = Hom_C(\emptyset \cdot S, A) \\ & = Hom_C(colim_{\emptyset} \cdot S, A) \\ & = Hom_C(\emptyset, A) \\ &= {*} \end{aligned} \,,
where we use that the tensor Quillen bifunctor is required to respect colimits and that the empty colimit is the initial object. (All equality signs here denote isomorphisms, to distinguish them from weak equivalences.)
Similarly one has for all $X$ that $C\left(X,*\right)=*$.
Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.
###### Lemma
In a simplicial model category $C$, for $X\in C$ cofibrant and $A\in C$ fibrant, the simplicial set $C\left(X,A\right)$ is a Kan complex.
###### Proof
We apply the enriched model category axiom to the cofibration $\varnothing \to X$ and the fibration $A\to *$ to obtain a fibration
$C\left(X,A\right)\to C\left(\varnothing ,A\right){×}_{C\left(\varnothing ,*\right)}C\left(X,*\right)\phantom{\rule{thinmathspace}{0ex}}.$C(X,A) \to C(\emptyset, A) \times_{C(\emptyset,{*})} C(X,{*}) \,.
The right hand is the pullback of the terminal simplicial set $*={\Delta }^{0}$ to itself, hence is itself the point. So we have a fibration $C\left(X,A\right)\to *$ and $C\left(X,A\right)$ is a fibrant object in the standard model structure on simplicial sets, hence a Kan complex. .
###### Lemma
In a simplicial model category $C$, for $X\in C$ cofibrant and $f:A\to B$ a fibration, the morphism of simplicial sets $C\left(X,f\right):C\left(X,A\right)\to C\left(X,B\right)$ is a Kan fibration that is a weak homotopy equivalence if $f$ is acyclic.
Dually, for $i:X\to Y$ a cofibration and $A$ fibrant, the morphism $C\left(i,A\right):C\left(X,A\right)\to C\left(Y,A\right)$ is a cofibration of simplicial sets.
###### Proof
This is as before. Explicity, consider the first case, the second one is the formal dual of that:
We enter the enrichment axiom with the morphisms $\varnothing \to X$ and $A\to B$ and find for the required pullback that
$C\left(\varnothing ,A\right){×}_{C\left(\varnothing ,B\right)}C\left(X,B\right)=*{×}_{*}C\left(X,B\right)=C\left(X,B\right)$C(\emptyset,A) \times_{C(\emptyset, B)} C(X,B) = {*} \times_{*} C(X,B) = C(X,B)
and hence that $C\left(X,A\right)\to C\left(X,B\right)$ is, indeed, a fibration, which is acyclic if $A\to B$ is.
###### Proposition
Let $C$ be a simplicial model category.
Then for $X$ a cofibant object and
$f:A\stackrel{\simeq }{\to }B$f : A \stackrel{\simeq}{\to} B
a weak equivalence between fibrant objects, the enriched hom-functor
$C\left(X,f\right):C\left(X,A\right)\to C\left(X,B\right)$C(X,f) : C(X,A) \to C(X,B)
is a weak homotopy equivalence of Kan complexes.
Similarly, for $A$ a fibrant object and $j:X\stackrel{\simeq }{\to }Y$ a weak equivalence between cofibrant objects, the morphism
$C\left(j,A\right):C\left(X,A\right)\to C\left(Y,A\right)$C(j,A) : C(X,A) \to C(Y,A)
is a weak homotopy equivalence of Kan complexes.
###### Proof
The second case is formally dual to the first, so we restrict attention to the first one.
By the above, the axioms of an enriched model category ensure that the above statement is true when $f$ is in addition a fibration. So we reduce the situation to that case.
This is possible because both $A$ and $B$ are assumed to be fibrant. This allows to apply the factorization lemma that is described in great detail at category of fibrant objects. By this lemma, for every morphism $f:A\to B$ between fibrant objects there is a commutative diagram
$\begin{array}{ccc}& & {E}_{f}B\\ & {}^{\in \mathrm{fib}\cap W}↙& & {↘}^{\in \mathrm{fib}}\\ A& & \stackrel{\simeq }{\to }& & B\end{array}$\array{ && \mathbf{E}_f B \\ & {}^{\mathllap{\in fib \cap W}}\swarrow && \searrow^{\mathrlap{\in fib}} \\ A &&\stackrel{\simeq}{\to}&& B }
Since $f$ is assumed a weak equivalence it follows by 2-out-of-3 that ${E}_{f}B$ is also a weak equivalence.
Therefore by the above properties of simpliciall enriched categories we obtain a span of acyclic fibrations of Kan complexes
$C\left(X,A\right)\stackrel{\simeq }{←}C\left(X,{E}_{f}B\right)\stackrel{\simeq }{\to }C\left(X,B\right)\phantom{\rule{thinmathspace}{0ex}}.$C(X,A) \stackrel{\simeq}{\leftarrow} C(X, \mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B) \,.
By the Whitehead theorem every weak equivalence of Kan complexes is a homotopy equivalence, hence there is a weak equivalence
$C\left(X,A\right)\stackrel{\simeq }{\to }C\left(X,{E}_{f}B\right)\stackrel{\simeq }{\to }C\left(X,B\right)$C(X,A) \stackrel{\simeq}{\to} C(X,\mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B)
that is homotopic to our $C\left(X,f\right)$. Therefore this is also a weak equivalence.
### Comparison
Let $C$ be a model category. We discuss how its simplicial function complexes from prop. 2 are related to the simplicial localization from def. 1 and def. 2.
Suppose now that $Q:C\to C$ is a cofibrant replacement functor and $R:C\to C$ a fibrant replacement functor, ${\Gamma }^{•}:C\to \left(\mathrm{cC}{\right)}_{c}$ a cosimplicial resolution functor and ${\Lambda }_{•}:C\to \left(\mathrm{sC}{\right)}_{f}$ a simplicial resolution functor in the model category $C$.
###### Theorem
(Dwyer–Kan)
There are natural weak equivalences between the following equivalent realizations of this SSet hom-object:
$\begin{array}{ccccc}{\mathrm{Mor}}_{C}\left({\Gamma }^{•}X,RY\right)& \stackrel{\simeq }{\to }& \mathrm{diag}{\mathrm{Mor}}_{C}\left({\Gamma }^{•}X,{\Lambda }_{•}Y\right)& \stackrel{\simeq }{←}& {\mathrm{Mor}}_{C}\left(QX,{\Lambda }_{•}Y\right)\\ & & {↑}^{\simeq }\\ & & {\mathrm{hocolim}}_{p,q\in {\Delta }^{\mathrm{op}}×{\Delta }^{\mathrm{op}}}{\mathrm{Mor}}_{C}\left({\Gamma }^{p}X,{\Lambda }_{q}Y\right)\\ & & {↓}^{\simeq }\\ & & N{\mathrm{wMor}}_{C}^{3}\left(X,Y\right)\\ & & {↓}^{\simeq }\\ & & {\mathrm{Mor}}_{{L}^{H}C}\left(X,Y\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ Mor_C(\Gamma^\bullet X, R Y) &\stackrel{\simeq}{\to}& diag Mor_C(\Gamma^\bullet X, \Lambda_\bullet Y) &\stackrel{\simeq}{\leftarrow}& Mor_C(Q X, \Lambda_\bullet Y) \\ && \uparrow^\simeq \\ && hocolim_{p,q \in \Delta^{op} \times \Delta^{op}} Mor_C(\Gamma^p X, \Lambda_q Y) \\ &&\downarrow^\simeq \\ &&N wMor_C^3(X,Y) \\ &&\downarrow^\simeq \\ &&Mor_{L^H C}(X,Y) } \,.
The top row weak equivalences are those of prop. 2
### In a category of fibrant objects
There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a category of fibrant objects. This is described in (Cisinksi 10) and (Nikolaus-Schreiber-Stevenson 12, section 3.6.2).
## Properties
### Hom-spaces of equivalences
###### Theorem
For $C$ a simplicial model category and $X$ an object, the delooping of the automorphism ∞-group
${\mathrm{Aut}}_{W}\left(X\right)\subset ℝ\mathrm{Hom}\left(X,X\right)$Aut_W(X) \subset \mathbb{R}Hom(X,X)
has the homotopy type of the component on $X$ of the nerve $N\left({C}_{W}\right)$ of the subcategory of weak equivalences:
$B{\mathrm{Aut}}_{W}\left(X\right)\simeq N\left({C}_{W}{\right)}_{X}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{B} Aut_W(X) \simeq N(C_W)_X \,.
The equivalence is given by a finite sequence of zig-zags and is natural with respect to sSet-enriched functors of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for $X$ again to a fibrant cofibrant object.
This is Dwyer-Kan 84, 2.3, 2.4.
###### Corollary
For $C$ a model category, the simplicial set $N\left({C}_{W}\right)$ is a model for the core of the (∞,1)-category determined by $C$.
###### Proof
That core, like every ∞-groupoid is equivalent to the disjoint union over its connected components of the deloopings of the automorphism $\infty$-groups of any representatives in each connected component.
homotopycohomologyhomology
$\left[{S}^{n},-\right]$$\left[-,A\right]$$\left(-\right)\otimes A$
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space $ℝ\mathrm{Hom}\left({S}^{n},-\right)$cocycles $ℝ\mathrm{Hom}\left(-,A\right)$derived tensor product $\left(-\right){\otimes }^{𝕃}A$
## References
For some original references by William Dwyer and Dan Kan see simplicial localization. For instance
• William Dwyer, Dan Kan, A classication theorem for diagrams of simplicial sets, Topology 23 (1984), 139-155.
The theory of framings is due to
and in parallel section 5 of
and in sections 16, 17 of
A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of
Discussion of derived hom spaces for categories of fibrant objects is in
• Denis-Charles Cisinski, Invariance de la K-théorie par equivalences dérivées, J. K-theory, 6 (2010), 505–546.
and section 3.6.2 of
Revised on May 17, 2013 03:01:38 by Urs Schreiber (82.169.65.155)
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IIT-JEE
Apne doubts clear karein ab Whatsapp (8 400 400 400) par bhi. Try it now.
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Question From class 9 Chapter DEFAULT
# If and then
If and , then (a)35 (b) 58 (c) 127 (d) none of these
1:29
If and , find the value of
2:18
Prove that :
3:54
If then
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If then
2:00
Prove that: is divisible by and find the quotient.
5:44
4:22
(i) If . <br> (ii) If . <br> (iii) If a+b+c=11 and ab+bc+ca=25, then find the value of abc.
3:39
If and , then find .
1:08
if and then find
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Prove that :
4:38
सिद्ध कीजिए की
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### Problem Statement
```PROBLEM STATEMENT:
Class name: HoldCost
Method name: calcHoldCost
Parameters: int[], int, int, int
Returns: int
Material Requirements Planning is a process that plans out the production of a
product based on a sales forecast. An optimal production schedule helps lower
the costs of production. One key concept to this is holding costs. Holding
costs represent the cost of holding one unit for one period of time (such as
weekly, monthly or quarterly).
Implement a class HoldCost, which contains a method calcHoldCost. The method
returns the sum of monthly holding costs given a twelve month requirements
schedule (number of units required each month), an initial inventory level, the
number of units produced per month and the holding cost per unit per month.
The method signature is:
int calcHoldCost(int[] requirements, int initialInventory, int prodPerMonth,
int holdingCostPerUnit);
Be sure your method is public.
To calculate the holding cost for each month:
Inventory for the month = prior months inventory + production for the current
month - requirement for the current month
Holding Cost = (Inventory from "prior" month) * holdingCostPerUnit
TopCoder will ensure the validity of the inputs. Inputs are valid if all of the
following criteria are met:
*The requirements int[] will contain twelve months of information (twelve
entries). Month one will be index zero, month two will be index one, etc.
*Each requirement element will be an int between 0 and 1000 (inclusive)
*The initialInventory represents the inventory prior to month one
*The initialInventory will be an int between 0 and 1000 (inclusive)
*The prodPerMonth will be an int between 0 and 1000 (inclusive)
*The holdingCostPerUnit will be an int between 1 and 100 (inclusive)
Notes:
*Negative inventory levels should NOT be carried forward (see example below)
Examples:
--An input of ({110,70,40,20,10,100,15,40,120,70,10,80},70,50,15) would be
calculated as follows:
Month 1's holding cost would be 70 (initialInv) * 15 = 1050
Month 1's inventory level would be 70+50-110 = 10
Total holding cost thus far = 0 (since there is no total holding cost yet) +
1050 = 1050
Month 2's holding cost would be 10 (month 1 inventory) * 15 = 150
Month 2's inventory level would be 10+50-70 = -10
Total holding cost thus far = 1050 + 150 = 1200
Month 3's holding cost would be 0 (because there is a negative inventory
level)
Month 3's inventory level would be 0 (previous month had a negative
inventory) + 50 - 40 = 10
Total holding cost thus far = 1200 + 0 = 1200
... etc ...
Month 12's holding cost would be 40 (month 11's inventory) * 15 = 600
Month 12's inventory level would be 40 + 50 - 80 = 10
Total holding cost thus far = 5775 + 600 = 6375
The method then returns 6375
-- An input of ({10,10,10,10,10,10,10,10,10,10,10,10},0,0,15) would return 0
-- An input of ({10,10,10,10,10,10,10,10,10,10,10,10},10,10,15) would return
1800
-- An input of ({40,50,60,40,50,60,40,50,60,40,50,60},0,50,10) would return 800
```
### Definition
Class: HoldCost Method: calcHoldCost Parameters: int[], int, int, int Returns: int Method signature: int calcHoldCost(int[] param0, int param1, int param2, int param3) (be sure your method is public)
#### Problem url:
http://www.topcoder.com/stat?c=problem_statement&pm=183
#### Problem stats url:
http://www.topcoder.com/tc?module=ProblemDetail&rd=4005&pm=183
Pops
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# NAG FL Interfacef08quf (ztrsen)
## ▸▿ Contents
Settings help
FL Name Style:
FL Specification Language:
## 1Purpose
f08quf reorders the Schur factorization of a complex general matrix so that a selected cluster of eigenvalues appears in the leading elements on the diagonal of the Schur form. The routine also optionally computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.
## 2Specification
Fortran Interface
Subroutine f08quf ( job, n, t, ldt, q, ldq, w, m, s, sep, work, info)
Integer, Intent (In) :: n, ldt, ldq, lwork Integer, Intent (Out) :: m, info Real (Kind=nag_wp), Intent (Out) :: s, sep Complex (Kind=nag_wp), Intent (Inout) :: t(ldt,*), q(ldq,*), w(*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Logical, Intent (In) :: select(*) Character (1), Intent (In) :: job, compq
#include <nag.h>
void f08quf_ (const char *job, const char *compq, const logical sel[], const Integer *n, Complex t[], const Integer *ldt, Complex q[], const Integer *ldq, Complex w[], Integer *m, double *s, double *sep, Complex work[], const Integer *lwork, Integer *info, const Charlen length_job, const Charlen length_compq)
The routine may be called by the names f08quf, nagf_lapackeig_ztrsen or its LAPACK name ztrsen.
## 3Description
f08quf reorders the Schur factorization of a complex general matrix $A=QT{Q}^{\mathrm{H}}$, so that a selected cluster of eigenvalues appears in the leading diagonal elements of the Schur form.
The reordered Schur form $\stackrel{~}{T}$ is computed by a unitary similarity transformation: $\stackrel{~}{T}={Z}^{\mathrm{H}}TZ$. Optionally the updated matrix $\stackrel{~}{Q}$ of Schur vectors is computed as $\stackrel{~}{Q}=QZ$, giving $A=\stackrel{~}{Q}\stackrel{~}{T}{\stackrel{~}{Q}}^{\mathrm{H}}$.
Let $\stackrel{~}{T}=\left(\begin{array}{cc}{T}_{11}& {T}_{12}\\ 0& {T}_{22}\end{array}\right)$, where the selected eigenvalues are precisely the eigenvalues of the leading $m×m$ sub-matrix ${T}_{11}$. Let $\stackrel{~}{Q}$ be correspondingly partitioned as $\left(\begin{array}{cc}{Q}_{1}& {Q}_{2}\end{array}\right)$ where ${Q}_{1}$ consists of the first $m$ columns of $Q$. Then $A{Q}_{1}={Q}_{1}{T}_{11}$, and so the $m$ columns of ${Q}_{1}$ form an orthonormal basis for the invariant subspace corresponding to the selected cluster of eigenvalues.
Optionally the routine also computes estimates of the reciprocal condition numbers of the average of the cluster of eigenvalues and of the invariant subspace.
## 4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
## 5Arguments
1: $\mathbf{job}$Character(1) Input
On entry: indicates whether condition numbers are required for the cluster of eigenvalues and/or the invariant subspace.
${\mathbf{job}}=\text{'N'}$
No condition numbers are required.
${\mathbf{job}}=\text{'E'}$
Only the condition number for the cluster of eigenvalues is computed.
${\mathbf{job}}=\text{'V'}$
Only the condition number for the invariant subspace is computed.
${\mathbf{job}}=\text{'B'}$
Condition numbers for both the cluster of eigenvalues and the invariant subspace are computed.
Constraint: ${\mathbf{job}}=\text{'N'}$, $\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.
2: $\mathbf{compq}$Character(1) Input
On entry: indicates whether the matrix $Q$ of Schur vectors is to be updated.
${\mathbf{compq}}=\text{'V'}$
The matrix $Q$ of Schur vectors is updated.
${\mathbf{compq}}=\text{'N'}$
No Schur vectors are updated.
Constraint: ${\mathbf{compq}}=\text{'V'}$ or $\text{'N'}$.
3: $\mathbf{select}\left(*\right)$Logical array Input
Note: the dimension of the array select must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: specifies the eigenvalues in the selected cluster. To select a complex eigenvalue ${\lambda }_{j}$, ${\mathbf{select}}\left(j\right)$ must be set .TRUE..
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{t}\left({\mathbf{ldt}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array t must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ upper triangular matrix $T$, as returned by f08psf.
On exit: t is overwritten by the updated matrix $\stackrel{~}{T}$.
6: $\mathbf{ldt}$Integer Input
On entry: the first dimension of the array t as declared in the (sub)program from which f08quf is called.
Constraint: ${\mathbf{ldt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7: $\mathbf{q}\left({\mathbf{ldq}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{compq}}=\text{'V'}$ and at least $1$ if ${\mathbf{compq}}=\text{'N'}$.
On entry: if ${\mathbf{compq}}=\text{'V'}$, q must contain the $n×n$ unitary matrix $Q$ of Schur vectors, as returned by f08psf.
On exit: if ${\mathbf{compq}}=\text{'V'}$, q contains the updated matrix of Schur vectors; the first $m$ columns of $Q$ form an orthonormal basis for the specified invariant subspace.
If ${\mathbf{compq}}=\text{'N'}$, q is not referenced.
8: $\mathbf{ldq}$Integer Input
On entry: the first dimension of the array q as declared in the (sub)program from which f08quf is called.
Constraints:
• if ${\mathbf{compq}}=\text{'V'}$, ${\mathbf{ldq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compq}}=\text{'N'}$, ${\mathbf{ldq}}\ge 1$.
9: $\mathbf{w}\left(*\right)$Complex (Kind=nag_wp) array Output
Note: the dimension of the array w must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the reordered eigenvalues of $\stackrel{~}{T}$. The eigenvalues are stored in the same order as on the diagonal of $\stackrel{~}{T}$.
10: $\mathbf{m}$Integer Output
On exit: $m$, the dimension of the specified invariant subspace, which is the same as the number of selected eigenvalues (see select); $0\le m\le n$.
11: $\mathbf{s}$Real (Kind=nag_wp) Output
On exit: if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$, s is a lower bound on the reciprocal condition number of the average of the selected cluster of eigenvalues. If ${\mathbf{m}}=0$ or ${\mathbf{n}}$, ${\mathbf{s}}=1$.
If ${\mathbf{job}}=\text{'N'}$ or $\text{'V'}$, s is not referenced.
12: $\mathbf{sep}$Real (Kind=nag_wp) Output
On exit: if ${\mathbf{job}}=\text{'V'}$ or $\text{'B'}$, sep is the estimated reciprocal condition number of the specified invariant subspace. If ${\mathbf{m}}=0$ or ${\mathbf{n}}$, ${\mathbf{sep}}=‖T‖$.
If ${\mathbf{job}}=\text{'N'}$ or $\text{'E'}$, sep is not referenced.
13: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
14: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08quf is called, unless ${\mathbf{lwork}}=-1$, in which case a workspace query is assumed and the routine only calculates the minimum dimension of work.
Constraints:
• if ${\mathbf{job}}=\text{'N'}$, ${\mathbf{lwork}}\ge 1$ or ${\mathbf{lwork}}=-1$;
• if ${\mathbf{job}}=\text{'E'}$, ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{m}×\left({\mathbf{n}}-\mathit{m}\right)\right)$ or ${\mathbf{lwork}}=-1$;
• if ${\mathbf{job}}=\text{'V'}$ or $\text{'B'}$, ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\mathit{m}×\left({\mathbf{n}}-\mathit{m}\right)\right)$ or ${\mathbf{lwork}}=-1$.
The actual amount of workspace required cannot exceed ${{\mathbf{n}}}^{2}/4$ if ${\mathbf{job}}=\text{'E'}$ or ${{\mathbf{n}}}^{2}/2$ if ${\mathbf{job}}=\text{'V'}$ or $\text{'B'}$.
15: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).
## 6Error Indicators and Warnings
${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
## 7Accuracy
The computed matrix $\stackrel{~}{T}$ is similar to a matrix $\left(T+E\right)$, where
$‖E‖2 = O(ε) ‖T‖2 ,$
and $\epsilon$ is the machine precision.
s cannot underestimate the true reciprocal condition number by more than a factor of $\sqrt{\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n-m\right)}$. sep may differ from the true value by $\sqrt{m\left(n-m\right)}$. The angle between the computed invariant subspace and the true subspace is $\frac{\mathit{O}\left(\epsilon \right){‖A‖}_{2}}{\mathit{sep}}$.
The values of the eigenvalues are never changed by the reordering.
## 8Parallelism and Performance
f08quf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The real analogue of this routine is f08qgf.
## 10Example
This example reorders the Schur factorization of the matrix $A=QT{Q}^{\mathrm{H}}$ such that the eigenvalues stored in elements ${t}_{11}$ and ${t}_{44}$ appear as the leading elements on the diagonal of the reordered matrix $\stackrel{~}{T}$, where
$T = ( -6.0004-6.9999i 0.3637-0.3656i -0.1880+0.4787i 0.8785-0.2539i 0.0000+0.0000i -5.0000+2.0060i -0.0307-0.7217i -0.2290+0.1313i 0.0000+0.0000i 0.0000+0.0000i 7.9982-0.9964i 0.9357+0.5359i 0.0000+0.0000i 0.0000+0.0000i 0.0000+0.0000i 3.0023-3.9998i )$
and
$Q = ( -0.8347-0.1364i -0.0628+0.3806i 0.2765-0.0846i 0.0633-0.2199i 0.0664-0.2968i 0.2365+0.5240i -0.5877-0.4208i 0.0835+0.2183i -0.0362-0.3215i 0.3143-0.5473i 0.0576-0.5736i 0.0057-0.4058i 0.0086+0.2958i -0.3416-0.0757i -0.1900-0.1600i 0.8327-0.1868i ) .$
The original matrix $A$ is given in f08ntf.
### 10.1Program Text
Program Text (f08qufe.f90)
### 10.2Program Data
Program Data (f08qufe.d)
### 10.3Program Results
Program Results (f08qufe.r)
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# Homework 8 - CSE 3380 Homework#8 Assigned Due Wednesday...
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CSE 3380 – Homework #8 Assigned: Wednesday, March 23, 2016 Due: Monday, April 4, 2016 at 3:50pm (the end of class) Note the following about the homework: 1. You must show your work to receive credit. 2. If your submission has more than one page, staple the pages. Assignment: Process 1. Given vectors vector y = 1 2 3 4 and vectoru = 2 1 0 1 what is the orthogonal projection of vector y onto vectoru ? 2. Find the closest point to vector y in the subspace spanned by vectorv 1 and vectorv 2 . vector y = 2 1 3 , vectorv 1 = 1 0 1 , vectorv 2 = 1 1 - 1 This is asking for the orthogonal projection of vector y onto the subspace spanned by vectorv i . 3. Orthogonalize the following set of vectors using the Gram-Schmidt procedure. 4 - 10 - 4 - 8 , - 6 - 14 4 - 12 4. Orthogonalize the following set of vectors using the Gram-Schmidt procedure. 4 4 2 , - 4 2 4 , 36 0 0 5. Orthogonalize the following set of vectors using the Gram-Schmidt procedure. 3 3 3 3 , 3 0 0 3 , 0 6 3 - 3
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# Unicus Olympiads - UCTO PDF Sample Papers for Class 5
Class 5 sample paper & practice questions for Unicus Critical Thinking Olympiad (UCTO) PDF Sample Papers are given below. Syllabus for UCTO is also mentioned for these exams. You can refer these sample paper & quiz for preparing for the UCTO exam.
#### Resources:
##### Sample Questions from Olympiad Success:
Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9 Q.10
Q.1 If 1st January 2008 is Tuesday, then what day of the week is on 1st January 2009? a) Monday b) Wednesday c) Thursday d) Sunday
Q.2 Four boxes are labelled as 1, 2, 3, and 4. Each of them has a different weight. Box 2 is heavier than box 4. More than one boxes are heavier than box 1. Box 1 is not the lightest. Box 1 is heavier than box 3. Which of the following box is the lightest? a) 1 b) 2 c) 3 d) 4
Q.3 Find that letter which will end the first word and start the second word: B I R ? U C K a) T b) K c) L d) D
Q.4 How many such pairs of letters are there in the word "SIMPLE" which has as many letters between them as there are in the english alphabet series? a) 2 b) 3 c) 1 d) 0
Q.5 In the following question, find out the correct pair of words which will make an appropriate analogical relationship between the two words to the left of the sign of double colon (::) and the same relationship between the two words to the right of the sign of double colon (::): A : Ship :: Platform : B a) A. Caption, B. Coolie b) A. Port, B. Station c) A. Quay, B. Train d) A. Shore, B. Bench
Q.6 If 'VEHEMENT is written as 'VEHETNEM', then how 'MOURNFUL' be written in that code language? a) MOURLUFN b) MOUNULER c) OURMNFUL d) URNFULMO
Q.7 If 1st half of the English alphabet is written in reverse order, then find the 15th letter from right: a) A b) B c) C d) D
Q.8 Find the next three letters of the following series: A B C C D E F F G H I I J _ _ _? a) K K L b) K L M c) K L L d) L M M
Q.9 Which image most closely resembles the mirror image of the given word, if the mirror is placed vertically to the right? a) b) c) d)
Q.10 How many quadrilaterals are there in the given figure? a) 4 b) 2 c) 3 d) 5
Sample PDF of Unicus Olympiads - Unicus Critical Thinking Olympiad (UCTO) PDF Sample Papers for Class 5:
Q.1 )c Q.2 )c Q.3 )d Q.4 )a Q.5 )c Q.6 )a Q.7 )b Q.8 )c Q.9 )a Q.10 )d
Q.1 : c | Q.2 : c | Q.3 : d | Q.4 : a | Q.5 : c | Q.6 : a | Q.7 : b | Q.8 : c | Q.9 : a | Q.10 : d
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Switch to:
GuruFocus has detected 2 Warning Signs with Groupon Inc \$GRPN.
More than 500,000 people have already joined GuruFocus to track the stocks they follow and exchange investment ideas.
Groupon Inc (NAS:GRPN)
Net-Net Working Capital
\$-0.92 (As of Dec. 2016)
In calculating the Net-Net Working Capital (NNWC), Benjamin Graham assumed that a companys accounts receivable is only worth 75% its value, its inventory is only worth 50% of its value, but its liabilities have to be paid in full. In addition, Graham believed that preferred stock belongs on the liability side of the balance sheet, not as part of capital and surplus. This is a conservative way of estimating the companys value.
Groupon Inc's net-net working capital per share for the quarter that ended in Dec. 2016 was \$-0.92.
Definition
Groupon Inc's Net-Net Working Capital (NNWC) per share for the fiscal year that ended in Dec. 2016 is calculated as
Net-Net Working Capital Per Share (A: Dec. 2016 ) = (Cash And Cash Equivalents + 0.75 * Acct. Receivable + 0.5 * Inventory - Total Liabilities - Preferred Stock) / Shares Outstanding = (891.846 + 0.75 * 86.655 + 0.5 * 35.61 - 1496.957 - 0) / 564.84 = -0.92
Groupon Inc's Net-Net Working Capital (NNWC) per share for the quarter that ended in Dec. 2016 is calculated as
Net-Net Working Capital Per Share (Q: Dec. 2016 ) = (Cash And Cash Equivalents + 0.75 * Acct. Receivable + 0.5 * Inventory - Total Liabilities - Preferred Stock) / Shares Outstanding = (891.846 + 0.75 * 86.655 + 0.5 * 35.61 - 1496.957 - 0) / 564.84 = -0.92
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
In calculating the Net-Net Working Capital (NNWC), Benjamin Graham assumed that a companys accounts receivable is only worth 75% its value, its inventory is only worth 50% of its value, but its liabilities have to be paid in full.
In addition, Graham believed that preferred stock belongs on the liability side of the balance sheet, not as part of capital and surplus. In "Security Analysis", preferred stock is dubbed "an imperfect creditorship position" that is best placed on the balance sheet alongside funded debt.
This is a conservative way of estimating the companys value.
Explanation
One research study, covering the years 1970 through 1983 showed that portfolios picked at the beginning of each year, and held for one year, returned 29.4 percent, on average, over the 13-year period, compared to 11.5 percent for the S&P 500 Index. Other studies of Grahams strategy produced similar results.
Benjamin Graham looked for companies whose market values were less than two-thirds of their net-net value. They are collected under our Net-Net screener. GuruFocus also publishes a monthly Net-Net newsletter.
Related Terms
Historical Data
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Groupon Inc Annual Data
Dec09 Dec10 Dec11 Dec12 Dec13 Dec14 Dec15 Dec16 NNWC 0.00 0.00 0.00 -0.65 0.21 0.02 0.01 -0.53 -0.68 -0.92
Groupon Inc Quarterly Data
Sep14 Dec14 Mar15 Jun15 Sep15 Dec15 Mar16 Jun16 Sep16 Dec16 NNWC -0.55 -0.53 -0.51 -0.07 -0.61 -0.68 -0.82 -0.96 -0.90 -0.92
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# Simplify (2v^2-3v^2)-(-4v^2-9v+3)+(9v^2-8v+6)
(2v2-3v2)-(-4v2-9v+3)+(9v2-8v+6)
Remove parentheses.
(2v2-3v2)-(-4v2-9v+3)+9v2-8v+6
Simplify each term.
Apply the distributive property.
2v2-3v2-(-4v2)-(-9v)-1⋅3+9v2-8v+6
Simplify.
Multiply -4 by -1.
2v2-3v2+4v2-(-9v)-1⋅3+9v2-8v+6
Multiply -9 by -1.
2v2-3v2+4v2+9v-1⋅3+9v2-8v+6
Multiply -1 by 3.
2v2-3v2+4v2+9v-3+9v2-8v+6
2v2-3v2+4v2+9v-3+9v2-8v+6
2v2-3v2+4v2+9v-3+9v2-8v+6
Subtract 3v2 from 2v2.
-v2+4v2+9v-3+9v2-8v+6
3v2+9v-3+9v2-8v+6
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Definition of Problems. Meaning of Problems. Synonyms of Problems
# Definition of Problems. Meaning of Problems. Synonyms of Problems
Here you will find one or more explanations in English for the word Problems. Also in the bottom left of the page several parts of wikipedia pages related to the word Problems and, of course, Problems synonyms and on the right images related to the word Problems.
## Definition of Problems
Problem
Problem Prob"lem, n. [F. probl[`e]me, L. problema, fr. Gr. ? anything thrown forward, a question proposed for solution, fr. ? to throw or lay before; ? before, forward + ? to throw. Cf. Parable. ] 1. A question proposed for solution; a matter stated for examination or proof; hence, a matter difficult of solution or settlement; a doubtful case; a question involving doubt. --Bacon. 2. (Math.) Anything which is required to be done; as, in geometry, to bisect a line, to draw a perpendicular; or, in algebra, to find an unknown quantity. Note: Problem differs from theorem in this, that a problem is something to be done, as to bisect a triangle, to describe a circle, etc.; a theorem is something to be proved, as that all the angles of a triangle are equal to two right angles. Plane problem (Geom.), a problem that can be solved by the use of the rule and compass. Solid problem (Geom.), a problem requiring in its geometric solution the use of a conic section or higher curve.
## Meaning of Problems from wikipedia
- while ill-defined problems do not. Well-defined problems allow for more initial planning than ill-defined problems. Solving problems sometimes involves...
- Many mathematical problems have not been solved yet. These unsolved problems occur in multiple domains, including theoretical physics, computer science...
- Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute on May 24, 2000. The problems are the Birch...
- unsolved problems may refer to several notable conjectures or open problems in various academic fields: Unsolved problems in astronomy Unsolved problems in...
- Problems, Problems, Problems is a series of educational mathematics textbooks ranging from grade 7-12. The math questions are from previous math contests...
- Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time...
- refers to the commonplace problem of ****ng the most valuable or useful items without overloading the luggage. Knapsack problems appear in real-world decision-making...
- of social system must confront and solve the three fundamental economic problems: What kinds and quantities of goods shall be produced, "how much and which...
- wicked problems". Rittel and Melvin M. Webber formally described the concept of wicked problems in a 1973 treatise, contrasting "wicked" problems with relatively...
- region. Injuries can cause ankle problems, for example, tripping or going over on the ankle. Diagnosis of ankle problems, especially of osteoarthritis involves:...
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## How to vlookup values from right to left in Excel?
Vlookup is a useful function in Excel, we can use it to quickly return the corresponding data of the leftmost column in the table. However, if you want to look up a specific value in any other column and return the relative value to the left, the normal vlookup function will not work. Here, I can introduce you other formulas to solve this problem.
Vlookup values from right to left with VLOOKUP and IF function
Vlookup values from right to left with INDEX and MATCH function
Look for a value from left to right: With this formula of Kutools for Excel, you can quickly vlookup the exact value from a list without any formulas. Kutools for Excel: with more than 200 handy Excel add-ins, free to try with no limitation in 60 days.
#### Vlookup values from right to left with VLOOKUP and IF function
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To get the corresponding left value from the right specific data, the following vlookup function may help you.
Supposing you have a data range, now you know the age of the persons, and you want to get their relative name in the left Name column as following screenshot shown:
Please enter this formula into your needed cell: =VLOOKUP(F2,IF({1,0},\$D\$2:\$D\$10,\$B\$2:\$B\$10),2,0) and press Enter key, you will get the correct result you need, see screenshot:
And then drag the fill handle to the cells you want to apply this formula to get all the corresponding names of the specific age.
Notes:
1. In the above formula, F2 is the value which you want to return its relative information, D2:D10 is the column that you are looking for and B2:B10 is the list that contains the value you wish to return.
2. When you drag this formula down, the absolute references \$D\$2:\$D\$10 and \$B\$2:\$B\$10 stay the same, while the relative reference F2 changes to F3, F4, F5….
#### Vlookup values from right to left with INDEX and MATCH function
Except above formula, here is another formula mixed with INDEX and MATCH function also can do you a favor.
Type this formula: =INDEX(\$B\$2:\$B\$10,MATCH(F2,\$D\$2:\$D\$10,0)) and press Enter key to get the corresponding data you need, see screenshot:
And then drag the fill handle down to your cells that you want to contain this formula.
Note: In this formula, F2 is the value which you want to return its relative information, B2:B10 is the list that contains the value you want to return and D2:D10 is the column that you are looking for.
Related articles:
How to use vlookup exact and approximate match in Excel?
How to lookup value to match case sensitive in Excel?
How to vlookup to get the row number in Excel?
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· 3 months ago
Thanks.....its works
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· 4 months ago
Hi!
I'm trying to show a cell adjacent to a referenced cell when the referenced cell could be in one of two columns.
The referenced cell, M9, uses this function to find the upcoming date closest to today (i.e. which bill is due next):
=INDEX(\$K\$1:\$K\$160,MATCH(M9,\$L\$1:\$L\$160,0))
I want to cell M8 to show the AMOUNT due on that day, which is in the cell to the LEFT of the referenced cell in the list.
I figured out in O9 how to show it when M9 references a cell in a single column L:
=INDEX(\$K\$1:\$K\$160,MATCH(M9,\$L\$1:\$L\$160,0))
But I can't figure out how to have that apply when the referenced cell is in column N.
A few things I've tried in O10-O12 that didn't work:
=INDEX(\$K\$1:\$K\$160&\$M\$1:\$M\$160,MATCH(M9,\$L\$1:\$L\$160&\$N\$1:\$N\$160,0))
=INDEX(K1:K160,MATCH(M9,L1:L160,0))OR(M1:M160,MATCH(M9,N1:N160,0))
=INDEX(K1:M160,MATCH(M9,L1:N160,0))
Would love some help! Thanks!
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· 4 months ago
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· 4 months ago
Screenshot (not sure why it didn't attach above)
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· 11 months ago
• To post as a guest, your comment is unpublished.
· 1 years ago
Thank you very much...
• To post as a guest, your comment is unpublished.
· 1 years ago
Thank u thank u so much
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Sample_mid_1_271
# Sample_mid_1_271 - Sample Mid term 1(72-271-30 Ch 7 8 9 11...
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Sample Mid term 1 (72-271-30) Ch. 7, 8, 9, 11 Q. 1 Find the IRR for a project costing \$36,500 and returning \$5,000 annually for the first four years, followed by \$11,000 annually for three years. Hint: At a discount rate of 7 percent the project's NPV is \$2,458.91 and changes to -\$1,966.86 at a discount rate of 10 percent. Ans: \$36,500 = \$5,000 1 (1 \$11,000 1 (1 4 - + + - + - - i) i i) i 3 8.6022% = i Q. 2 A new machine will cost \$100,000 and generate after-tax cash inflows of \$35,000 for four years. Find the NPV if the firm uses a 12 percent opportunity cost of capital. What is the IRR? What is the payback period? Ans: NPV = \$35,000 1 .12 1 128(1.12) 4 - – \$100,000 = \$35,000 [8.33 – 5296] – 100,000 = \$35,000 [3.037] – \$100,000 = 106,307.23 – 100,000 = \$6,307.23 IRR = \$35,000 1 1 (1 + ) 4 i i i - – \$100,000 = 14.96% Payback Period = 100,000 35,000 = 2.857 years Q. 3 Calculate the NPV for the following capital budgeting proposal: \$100,000 initial cost, to be depreciated straight-line over five years to an expected salvage value of \$5,000, 35 percent tax rate, \$45,000 additional annual revenues, \$15,000 additional annual expense, \$8,000 additional investment in working capital, 11 percent cost of capital.
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# Midterm 1: Version C
1. Evaluate if and
2. Solve for in the equation
3. Isolate the variable in the equation
4. Solve for in the equation
5. Write the equation of the vertical line that passes through the point
6. Find the equation that has a slope of and passes through the point
7. Find the equation of the line passing through the points and
8. Graph the relation
For questions 9 to 11, find each solution set and graph it.
1. Graph the relation
2. Find two numbers such that 5 times the larger number plus 3 times the smaller is 49, and 4 times the larger minus twice the smaller is 26.
3. Karl is going to cut a 42 cm cable into 2 pieces. If the first piece is to be 5 times as long as the second piece, find the length of each piece.
4. varies jointly with and inversely with the square of If when and find the constant then use to find when and
Midterm 1: Version C Answer Key
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Modern physicians often employee laboratory tests, in : GMAT Critical Reasoning (CR)
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# Modern physicians often employee laboratory tests, in
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Modern physicians often employee laboratory tests, in addition to physical examinations, in order to diagnose diseases accurately. Insurance company regulations that deny coverage for certain laboratory tests therefore decrease the quality of medical care provided to patients.
Which one of the following is an assumption that would serve to justify the conclusion above?
(A) Physical examinations and the uncovered laboratory tests together provide a more accurate diagnosis of many diseases than do physical examinations alone.
(B) Many physicians generally oppose insurance company regulations that, in order to reduce costs, limit the use of laboratory tests.
(C) Many patients who might benefit from the uncovered laboratory tests do not have any form of health insurance.
(D) There are some illnesses that experienced physicians can diagnose accurately from physicians examination alone.
(E) Laboratory tests are more costly to perform than are physical examinations.
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Re: Aristotle LSAT Tough CR [#permalink]
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09 Feb 2011, 20:44
B ??
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Re: Aristotle LSAT Tough CR [#permalink]
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09 Feb 2011, 22:50
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IMO A.
the conclusion here is "Insurance company regulations that deny coverage for certain laboratory tests therefore decrease the quality of medical care provided to patients."
in order to justify this we must prove insurance regulations decrease medical care quality(quality would mean correct diagnosis, correct treatment etc...we do not know exactly what it is).
now
choice A says without lab tests there wont be an accurate diagnosis----good one, lets see others
choice B , so wat if physicians oppose?? doesn't mean a thing---- irrelevant
choice C, so they dont have insurance. ok, but if they DO have insurance wat wud happen? a lot of questions arise. will that insurance cover tests? is not having insurance affecting quality of healthcare? nah----inconclusive
choice D and E are obviously irrelavant
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Re: Aristotle LSAT Tough CR [#permalink]
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09 Feb 2011, 22:57
I will go with A.
B just indicates physician's opinion- which cannot be credited enough to be a justification.
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10 Feb 2011, 05:56
Looks like I got it wrong
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11 Feb 2011, 02:43
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Yes ans should be A!
use assumption negation
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Re: Aristotle LSAT Tough CR [#permalink]
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11 Feb 2011, 12:50
+1 A
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Re: Aristotle LSAT Tough CR [#permalink]
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12 Feb 2011, 19:46
A....if you negate this the conclusion falls apart.
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Re: Aristotle LSAT Tough CR [#permalink]
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13 Feb 2011, 23:01
The arg is dependent on the laborary test - its absence violates the conclusion. A more refined version of the assumption "uncovered laboratory tests are essential to diagnosis"
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Re: Aristotle LSAT Tough CR [#permalink]
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13 Feb 2011, 23:40
A correctly identifies the underlying assumption.
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Re: Aristotle LSAT Tough CR [#permalink]
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14 Feb 2011, 02:05
+1 for A
OA pls
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Re: Aristotle LSAT Tough CR [#permalink]
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01 Mar 2011, 06:02
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the conclusion is : Insurance company regulations that deny coverage for certain laboratory tests therefore decrease the quality of medical care provided to patients.
so the assumption obviously is that the laboratory tests which remain uncovered due to insurance company regulations, decrease the quality of medical care .
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Re: Aristotle LSAT Tough CR [#permalink]
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03 Mar 2011, 04:09
+ 1 A
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Re: Aristotle LSAT Tough CR [#permalink]
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06 Mar 2011, 10:23
IMO A.
A is the only option which is in line with the argument and correctly fits in the implied area.
(A) Physical examinations and the uncovered laboratory tests together provide a more accurate diagnosis of many diseases than do physical examinations alone.
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Re: Aristotle LSAT Tough CR [#permalink]
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06 Mar 2011, 23:11
shd be A.
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Re: Aristotle LSAT Tough CR [#permalink]
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08 Mar 2011, 03:32
Yup, "A" .
Re: Aristotle LSAT Tough CR [#permalink] 08 Mar 2011, 03:32
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# RDP 1999-11: A Structural Vector Autoregression Model of Monetary Policy in Australia 2. Monetary SVAR Models
## 2.1 SVAR Framework
The estimation structure is as follows.[5] Let xt be an n × 1 vector of variables and ut be an n× 1 vector of mean zero structural innovations. For simplicity of presentation, we ignore any constant terms in the model. The pth order structural VAR model is written as:
for t = −(p−1)…T. We condition on the first p observations, xp+1x0. B(L) is a pth order matrix polynomial in the lag operator L, B(L)= B0B1LB2L2−…−BpLp. B0 is a non-singular matrix and is normalised to have ones on the diagonal. This matrix summarises the contemporaneous relationships between the variables of the model and is most commonly where identification restrictions are imposed.
Associated with the structural model is the reduced form VAR representation:
where and .
To estimate the structural VAR model requires that the model be either exactly identified or over-identified. A necessary condition for the model to be exactly identified is that there must be the same number of parameters in B0 and D as there are in Σ, the covariance matrix from the reduced form. In other words, it must be possible to recover the structural parameters from the reduced form model. (This is the order condition. The model must also satisfy the rank condition that is more difficult to verify. We assume that this condition is met. For further discussion, see Hamilton (1994).)
From the above, the relationship between the reduced form and the structural model can be expressed as:
Exact identification requires that the parameters in B0 and D, of which there are 2n2n, be uniquely recoverable from the reduced form. Since Σ has n(n+1)/2 parameters, we require 2n2nn(n+1)/2 restrictions on B0 and D. It is standard in the SVAR literature to restrict D to be diagonal, imposing n(n−1) restrictions.[6] We require a further n(n−1)/2 restrictions on B0. For example, this can be accomplished by assuming that B0 is lower triangular; this is the standard recursive or Wold causal ordering often employed in SVAR studies.
For an exactly identified model with no restrictions on Aj, a simple two-step maximum likelihood estimation procedure can be employed, assuming the structural errors are jointly normal. This is the FIML estimator for the SVAR model.[7] First, Σ is estimated as , where are the OLS residuals from each equation of the reduced form model. Estimates of B0 and D are then obtained by maximising the log likelihood for the system conditional on . This amounts to finding the solution to the system of non-linear equations given in Equation (3).
When the model is over-identified, however, the two-step procedure is not the FIML estimator for the SVAR model. The estimates are consistent but not efficient (see the discussion in Judge et al (1985, p. 619)) since they do not take the over-identification restrictions into account when estimating the reduced form. Nevertheless, this is a common means to estimate these models, for example Sims (1986), Gordon and Leeper (1994) and Kim and Roubini (1999), and it is the method we use here.[8]
Ideally, the restrictions imposed to identify a SVAR model would result from a fully specified macroeconomic model. In practice, however, this is rarely done. (Gali (1992), Garratt et al (1998), Huh (1999) and, to a lesser extent, Sims and Zha (1998a) are studies that go some way toward this ideal.) Instead, the more common approach is to impose a set of identification restrictions that are broadly consistent with economic theory and provide sensible outcomes. Generally, the metric used is whether the behaviour of the dynamic responses of the model accords with economic theories.
Leeper, Sims and Zha (1996), LSZ, provide a defence of this approach, representing it as an informal means of applying more formal prior beliefs to the econometric modelling. They argue persuasively that such an approach is in principle no different from other specification methods used in modelling – as long as the modeller does not fail to disclose the methods used to select the model. In particular, they argue that specifications consistent with any reasonable economic theory should not be dismissed in favour of a specification that accords with the modeller's own prior beliefs. Nonetheless, there are still legitimate concerns about SVAR models and the identification restrictions that have been employed in the literature. One important issue is the robustness of the conclusions to alternative reasonable identification restrictions, see Faust (1998). Uhlig (1997) has raised a further issue concerning the approach advocated by LSZ. He argues that in most instances of VAR modelling, it is too difficult to document completely the model specification process undertaken making it unclear what aspects of the model arise from criteria imposed on the model and what arise from the data.[9]
These concerns are clearly important and provide a motivation for considering an existing structure that has proved successful, in some directions, for other countries. We can reasonably argue that the model has not been tailored in unspecified directions to provide desired responses for the Australian data. Of course, this does not provide any assurance that the identification structure is in some sense ‘correct’ or that our conclusions are not sensitive to the identification restrictions imposed. It merely makes our approach more transparent.
Given a set of variables of interest and criteria for model selection, identification restrictions can be imposed in a number of different ways. Most commonly, these involve restrictions on B0, or long-run restrictions imposed on B(1).[10] The KR model, consistent with much of the SVAR literature, imposes restrictions only on the B0 matrix, the contemporaneous relationship between the variables of the system. Generally, restrictions on B0 are motivated in one of the following ways.
First, with open economy models, it is common to identify an external sector that does not respond contemporaneously to movements in domestic variables so that the B0 matrix is block triangular. (This can be extended further by restricting the whole of B(L) to be block triangular as in Cushman and Zha (1997) and Dungey and Pagan (1998).) The second argument used to justify identification restrictions on the B0 matrix is the timing of information. If we think of an equation of the SVAR model as a behavioural equation, a policy response function for example, then we can impose zero restrictions based on the fact that certain variables are only available with a lag. For example, an output measure for time t is only available after one quarter, at time t + 1.[11]
A third type of argument is the imposition of behavioural assumptions. For example, KR imposes the restriction that domestic monetary authorities do not respond contemporaneously to movements in foreign interest rates. A further type of behavioural restriction often imposed is that certain variables respond slowly to movements in financial and policy variables. So, for example, output and prices do not respond contemporaneously to changes in domestic monetary policy variables.
## 2.2 The Kim and Roubini Model
The model specified by Kim and Roubini considers the following set of variables:
where all variables except interest rates are expressed in logarithms. is an oil price index in current US dollars, the Federal Funds rate, yt is domestic output, pt the domestic price level, mt a narrow monetary aggregate, it the domestic policy interest rate and et the exchange rate expressed in US dollars.
The oil price variable is included to capture anticipated inflation for the G6 countries. It is generally accepted that these models require such a variable in order to account for forward-looking monetary policy. Typically, we observe interest rates rising in advance of measured inflation. Without specifying the response of policy-makers to anticipated inflation, these models predict that an innovation to monetary policy leads to a rise in the price level (or inflation, depending upon the specification) – this is the price puzzle. KR includes the oil price index to resolve the price puzzle.
The Federal Funds rate is included to control for the response of domestic monetary policy to US financial variables. KR cites evidence in Grilli and Roubini (1995) that this is important for the G6 countries. For the sample we consider, Australia has had relatively open capital markets and it is also reasonable to assume that domestic interest rates are related to US interest rates.
The remaining variables are simply the standard variables of open economy monetary business cycle models: output, price, money, an interest rate, and the exchange rate. One point to note, however, is the focus on US interest rates and the USD exchange rate. This means that the US is serving as a proxy for the international economy. While this might not be ideal for all purposes, it has the advantage of being simple. Further, there is sufficient evidence to suggest that the US has an important influence on Australian financial variables and is likely to act as a reasonable proxy. (See de Roos and Russell (1996) and Dungey and Pagan (1998).)
The KR model restricts the elements of the B0 matrix as follows (with zeros above the diagonal suppressed):
Further, the structural variance covariance matrix D is assumed to be diagonal. The model is over-identified – there are five more restrictions than required to just identify the model. As well, written in this manner, it is easy to see that the model is largely recursive with the exception being the relationship between the domestic interest rate, the monetary aggregate and the nominal exchange rate.[12] To make the notation of Equation (5) more explicit, it is useful to consider an individual equation of the model in complete detail. For example, the domestic interest rate equation is:
The other equations of the model can be similarly expanded.
The explanation for the restrictions imposed is as follows. The first two variables are treated as external variables which are unaffected by contemporaneous movements in any domestic variable. The Federal Funds rate, denoted , depends contemporaneously on the oil price variable reflecting the role this variable plays as a proxy for measures of anticipated inflation. Domestic output is assumed to respond to changes in oil prices immediately, as are domestic prices. This is based on the assumption that oil prices are an important determinant of production and pricing decisions and firms respond quickly to any changes.[13] Otherwise, output is assumed to adjust slowly to the financial variables of the model. Similarly, the price level is assumed to adjust slowly to all variables except for movements in output (and oil prices as discussed).
The interest rate equation is interpreted as the policy reaction function of the central bank. The interest rate we use is the official cash rate, the interest rate in the overnight money market in Australia.[14] The policy reaction function of the central bank depends contemporaneously on three variables: the oil price variable, the domestic monetary aggregate and the nominal exchange rate. The oil price variable is included as a proxy for anticipated inflation. The justification for excluding output and the price level is based upon the timing of information; that is, measures of these variables are not available at the time policy is set. For the price level, this is reasonable since we have included a variable to act as a measure of anticipated inflation. For output, however, one should be aware we are restricting the monetary authorities from responding to any indicators of future output apart from those specified in the model.
Finally, an important omission from the policy reaction function is the US interest rate. Kim and Roubini justify this by arguing that for domestic monetary authorities the information in changes to the Federal Funds rate, expected or unexpected, is dominated by the information in the movements of the nominal exchange rate. This contrasts with Cushman and Zha's model for Canada, which includes the US Federal Funds rate in the domestic policy reaction function. Cushman and Zha (1997) (CZ hereafter) argue that inclusion of this variable is important in specifying their model. To foreshadow our results, we find it necessary to follow CZ and include the contemporaneous US Federal Funds rate in the domestic interest rate equation in order to obtain sensible dynamic responses. This is also a source of difference between our model and Dungey and Pagan as the latter also exclude the foreign interest rate (contemporaneously) from their domestic interest rate equation.
The monetary aggregate equation is specified as a standard money demand equation, dependent upon output, prices and interest rates. We use a measure of M1, consistent with KR. One aspect of this specification is that we do not restrict =1 so that we would be modelling a demand for real balances (as, for example, CZ do). Were we to impose this restriction we would also have to consider how we wish to model the dynamic adjustment of money demand – in terms of nominal or real balances. For simplicity, we follow KR and leave the contemporaneous and dynamic relationships unrestricted with a possible loss of efficiency.
Finally, the exchange rate is treated as dependent upon all innovations of the model. This reflects the fact that the exchange rate is a financial variable and reacts quickly to all information. A similar argument is employed in CZ.
It is instructive to highlight the similarities between the KR model and the CZ model. Although the KR model is much smaller in dimension, the two models share, to a considerable extent, the same structure for money demand and supply. For money demand, the specifications are identical in terms of exclusion restrictions. The only difference is that the coefficients on m and p are restricted to be equal and opposite signs in CZ. The money supply or interest rate equation is also nearly identical in structure, the only difference being that CZ include the Federal Funds rate, as already mentioned. So, to summarise, both KR and CZ impose effectively the same non-recursive structure for the monetary components of the model.
As both papers provide models with reasonable dynamic responses for monetary policy innovations, it suggests that this non-recursive monetary structure is the important element of the identification scheme for each model. Some sensitivity analysis presented below confirms that this is the case for Australia. It further suggests that the additional restrictions imposed on the dynamics of the model by CZ so that the B(L) matrix is block triangular, are not crucial for identifying a reasonable model for a small open economy and, at least in the first instance, may be set aside.
## Footnotes
A useful general reference for SVAR models is Hamilton (1994). [5]
Although it is standard to restrict D to be diagonal, it is not an innocuous restriction as it affects the interpretation of restrictions on B0. [6]
This follows the discussion in Hamilton (1994, pp. 331–332). Hausman, Newey and Taylor (1987) generalise the estimation of simultaneous equations models with covariance restrictions beyond the normality assumptions underlying MLE. [7]
Two alternatives are to estimate the model using instrumental variables, as discussed in Pagan and Robertson (1998) or to use the Bayesian estimation method suggested by Sims and Zha (1998b) and Zha (1999). [8]
This is based upon the discussion in Faust (1998). [9]
Gali (1992) is an example of a model that uses all of these types of restrictions. Pagan and Robertson (1998) provide a discussion of the different type of restrictions used in SVAR models and some estimation issues; similarly, Faust and Leeper (1997) consider the usefulness of long-run identification restrictions. Finally, it is also possible to impose identification restrictions on the cointegration matrix of a VECM; see for example Garratt et al (1998) and Fung and Kasumovich (1998). [10]
The model must have sufficient dimension to warrant restrictions based upon the timing of information and a diagonal structural covariance matrix. For example, while inflation measures may only be available with a lag, a leading indicator of inflation may inform monetary policy. In this case, the exclusion restrictions on B0 coupled with the diagonal covariance matrix are not valid restrictions. Fundamentally, the problem is one of an omitted variable leading to a mis-specified model. [11]
We order the model differently to Kim and Roubini. We do so to identify the features of the model which make it differ from the more standard recursive ordering. For example, this model is very similar to the recursive model in Eichenbaum and Evans' (1995) study for the US which documents evidence of the exchange rate and forward discount puzzle. [12]
Notice that the oil price variable is measured in USD. Domestic producers, however, are more likely to be concerned with the AUD value. Consequently, these identification restrictions are likely to be too simple, not accounting for the interaction between the USD oil price and the exchange rate. This is a limitation of the KR structure. (We are grateful to Adrian Pagan for bringing this to our attention.) As it turns out that the oil price variable plays very little role in the model when applied to Australian data, we do not amend this aspect of the KR model. [13]
For the whole of the sample period we consider, this interest rate has been the principal policy instrument of the Reserve Bank. While reserve ratios were also used to some extent in the early part of the sample, the cash rate was still an important policy instrument. For a more complete discussion of these issues, see Macfarlane (1984) and Rankin (1992). [14]
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https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition/chapter-7-trigonometric-identities-and-equations-7-4-double-angle-and-half-angle-identities-7-4-exercises-page-692/3
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## Precalculus (6th Edition)
$3\rightarrow B$
$2\sin \alpha \cos \alpha =\sin 2\alpha \Rightarrow 2\sin 22.5\cos 22.5=\sin \left( 2\times 22.5\right) =\sin 45=\dfrac {\sqrt {2}}{2}$ $3\rightarrow B$
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http://math.stackexchange.com/questions/609051/is-there-a-branch-of-mathematics-that-studies-the-factors-of-rational-numbers
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Is there a branch of mathematics that studies the factors of rational numbers?
Is there a branch of mathematics that studies the factors of rational numbers? I am imagining that defining this would work pretty much the same way as defining the factors x of an integer n:
$\{x \mid n\mod x\ = 0\}$
but maybe as something more like
$\{x^{-1} \mid n\mod x\ = 0\}$
giving the factors of $n^{-1}$.
-
The rational numbers form a field, so that every nonzero rational is invertible. This makes the notion of a "factor" less interesting. – Isaac Solomon Dec 16 '13 at 13:20
@IsaacSolomon, well you'd have to define it right. After all, every positive rational number can be expressed uniquely as the product of a $\mathbb{Z}$-valued multiset of prime numbers, where by prime number I just mean $2,3,5$ etc. – goblin Dec 16 '13 at 13:24
It seems you may be asking if we can extend the theory of divisibility from integers to rationals. One general way to do this is as follows. Consider the natural extension of the divisibility relation from integers to rationals: for rationals $\rm\:r,s,\:$ we define $\rm\:r\:$ divides $\rm\:s,\:$ if $\rm\ s/r\:$ is an integer, in symbols $\rm\:r\:|\:s\:$ $\!\iff\!$ $\rm\:s/r\in\mathbb Z.\:$ [divisibility relations induced by subrings are discussed further here]
Then we can obtain lcm and gcd of rationals by scaling the gcd arguments by a factor that yields a known gcd (of integers), then performing the inverse scaling back to rationals.
Even in more general number systems (integral domains), where gcds need not always exist, this scaling method still works to compute gcds from the value of a known scaled gcd, namely
$\rm{\bf Lemma}\ \ \ gcd(a,b)\ =\ gcd(ac,bc)/c\ \ \ if \ \ \ gcd(ac,bc)\$ exists $\rm\quad$
Therefore $\rm\ \ gcd(a,b)\, c = gcd(ac,bc) \ \ \ \ \ if\ \ \ \ gcd(ac,bc)\$ exists $\quad$ (GCD distributive law)
The reverse direction fails, i.e. $\rm\:gcd(a,b)\:$ exists does not generally imply that $\rm\:gcd(ac,bc)\:$ exists. $\$ For a counterexample see my post here, which includes further discussion and references.
Generally, as proved here, we have these dual formulas for $\rm\color{#c00}{reduced}$ fractions
$$\rm\ gcd\left(\frac{a}b,\frac{c}d\right) = \frac{gcd(a,c)}{lcm(b,d)}\ \ \ if\ \ \ \color{#c00}{\gcd(a,b) = 1 = \gcd(c,d)}$$
$$\rm\ lcm\left(\frac{a}b,\frac{c}d\right) = \frac{lcm(a,c)}{gcd(b,d)}\ \ \ if\ \ \ \color{#c00}{\gcd(a,b) = 1 = \gcd(c,d)}$$
Some of these ideas date to Euclid, who computed the greatest common measure of line segments, by anthyphairesis (continually subtract the smaller from the larger), i.e. the subtractive form of the Euclidean algorithm. The above methods work much more generally since they do not require the existence of a Euclidean (division) algorithm but, rather, only the existence of (certain) gcds.
gcds and lcms of rationals (fractions) are disscussed in a few prior posts here, e.g. see my post here, which gives a direct proof of the gcd * lcm formula using universal gcd laws.
-
Just like every integer $n\in \mathbb{Z}$ has a unique representation as a product of primes (up to multiplication by $-1$), you could say the same for every rational number, accept now we allow negative powers in the "product of primes" representation.
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If $a = \prod_{p \text{ prime}} p^{n_p}$ and $b = \prod_{p \text{ prime}} p^{m_p}$ then $a/b =\prod_{p \text{ prime}} p^{n_p -m_p}$.
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# basic linear algebra questions
• Apr 17th 2009, 07:32 PM
Hikari Clover
basic linear algebra questions
let ABC be a triangle and let p be the midpoint of side AB
let http://i52.photobucket.com/albums/g37/mmmmms/1-7.jpg and http://i52.photobucket.com/albums/g37/mmmmms/2-6.jpg
express http://i52.photobucket.com/albums/g37/mmmmms/5-3.jpg and http://i52.photobucket.com/albums/g37/mmmmms/6-2.jpg in terms of http://i52.photobucket.com/albums/g37/mmmmms/7-1.jpg and http://i52.photobucket.com/albums/g37/mmmmms/8-1.jpg and http://i52.photobucket.com/albums/g37/mmmmms/9-1.jpg
hence prove that the midpoint of the hypotenuse of a right-angle triangle is equidistant from the three vertices
==================================================
let p be the plane with cartesian equation 2x+y-3z=9 and let A,B be points with coordinates (1,-2,3) and (-2,1,-1)
l is the line passing through the point A and B
find the coordinates of the point C where the line l intersects the plane p
i have no ideas how to find the intersection between a line and a plane
thx
• Apr 17th 2009, 11:43 PM
running-gag
Quote:
Originally Posted by Hikari Clover
Hi
$\displaystyle \overrightarrow{AP}^2 = \frac12\:\overrightarrow{AB}\cdot\frac12\:\overrig htarrow{AB}\cdot = \frac14\:\overrightarrow{AB}\cdot\overrightarrow{A B}$
$\displaystyle \overrightarrow{AB} = \overrightarrow{AC} + \overrightarrow{CB} = -\vec{a} + \vec{b}$
$\displaystyle \overrightarrow{AP}^2 = \frac14\:\left(-\vec{a} + \vec{b})^2\right) = \frac14\:\left(\vec{a}^2 + \vec{b}^2 -2 \vec{a} \cdot \vec{b}\right)$
$\displaystyle \overrightarrow{CP} = \frac12\:\left(\overrightarrow{CA} + \overrightarrow{CB}\right) = \frac12\:\left(\vec{a} + \vec{b}\right)$
==================================================
Quote:
Originally Posted by Hikari Clover
let p be the plane with cartesian equation 2x+y-3z=9 and let A,B be points with coordinates (1,-2,3) and (-2,1,-1)
l is the line passing through the point A and B
find the coordinates of the point C where the line l intersects the plane p
i have no ideas how to find the intersection between a line and a plane
thx
AB coordinates are (-3,3,-4)
Therefore one parametric equation of l is :
x = -3t + 1
y = 3t -2
z = -4t +3
C(x,y,z) is on the plane iff 2x+y-3z=9
C is on l iff there exists t such that
x = -3t + 1
y = 3t -2
z = -4t +3
Substitute x,y,z in the Cartesian equation of the plane to get one linear equation. Solve for t and substitute in x,y,z expressions.
Spoiler:
2(-3t + 1)+(3t -2)-3(-4t +3)=9 gives t=2
x = -5
y = 4
z = -5
C(-5,4,-5)
• Apr 18th 2009, 04:20 AM
Hikari Clover
Quote:
Originally Posted by running-gag
Hi
$\displaystyle \overrightarrow{AP}^2 = \frac12\:\overrightarrow{AB}\cdot\frac12\:\overrig htarrow{AB}\cdot = \frac14\:\overrightarrow{AB}\cdot\overrightarrow{A B}$
$\displaystyle \overrightarrow{AB} = \overrightarrow{AC} + \overrightarrow{CB} = -\vec{a} + \vec{b}$
$\displaystyle \overrightarrow{AP}^2 = \frac14\:\left(-\vec{a} + \vec{b})^2\right) = \frac14\:\left(\vec{a}^2 + \vec{b}^2 -2 \vec{a} \cdot \vec{b}\right)$
$\displaystyle \overrightarrow{CP} = \frac12\:\left(\overrightarrow{CA} + \overrightarrow{CB}\right) = \frac12\:\left(\vec{a} + \vec{b}\right)$
==================================================
AB coordinates are (-3,3,-4)
Therefore one parametric equation of l is :
x = -3t + 1
y = 3t -2
z = -4t +3
C(x,y,z) is on the plane iff 2x+y-3z=9
C is on l iff there exists t such that
x = -3t + 1
y = 3t -2
z = -4t +3
Substitute x,y,z in the Cartesian equation of the plane to get one linear equation. Solve for t and substitute in x,y,z expressions.
Spoiler:
2(-3t + 1)+(3t -2)-3(-4t +3)=9 gives t=2
x = -5
y = 4
z = -5
C(-5,4,-5)
hey , thx for ur replying(Clapping)
but for the first question,did u forget to put that absolute value sign? or it doesnot matter?
• Apr 18th 2009, 05:58 AM
running-gag
Quote:
Originally Posted by Hikari Clover
hey , thx for ur replying(Clapping)
but for the first question,did u forget to put that absolute value sign? or it doesnot matter?
Are you talking about the modulus ?
$\displaystyle ||\overrightarrow{AB}||^2 = \overrightarrow{AB}\cdot\overrightarrow{AB} = \overrightarrow{AB}^2 = AB^2$
• Apr 18th 2009, 06:17 AM
Hikari Clover
oh i got it
thanks so much ^_^
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# Lab-0012
## River Discharge Data Analysis
Model
n/a
Duration
3.0 hrs
Updated
2022-04-14
Download
download
Run online using:
Contributor(s)
Irina Overeem at INSTAAR - University of Colorado Boulder.
Introduction
In this lab, we learn about river stage and discharge, using gage height data downloaded from the USGS for the upper Colorado River. We use standard Python libraries to read, analyze, and visualize the data.
Classroom organization
This lab is appropriate for advanced undergraduates and graduate students majoring in earth science/engineering.
We will be looking at data on river discharge--the volume of water transported through a given cross section per time--in the Colorado River. This Jupyter Notebook lends itself to a short introduction on the concept of river discharge and how it is measured, as well as an introduction to the gauging stations of the USGS. The data analysis requires basic Python data handling skills, but the coding is introductory level.
Students can run the Notebook, and they're encouraged to do assignments on their own or as homework. A review and discussion of solutions by the instructor after completion by the participants is recommended.
Download associated file: RiverStageDischargeIntroduction.pdf
Concept Diagrams of Stage and Discharge Measurements
Learning objectives
Skills
• Load data from a CSV file using the pandas library
• Access data in a DataFrame
• Create plots of data from a DataFrame
• Save plots to a file
Key concepts
• River discharge and stage
• Stage-discharge relationship
• Difficulties in relating stage to discharge
Lab notes
River discharge data for many US rivers is available from the USGS WaterWatch website:
River stage data is typically measured by keeping track of the water surface height over time, i.e. stage, and this needs to be converted to discharge through a stage-discharge relationship.
Tabular data like these, with a combination of dates, name and data quality strings, and numbers are best handled by spreadsheets where entries such as dates and times are in some useful format. In Python the Python Data Analysis Library (a.k.a. Pandas) is really useful for this purpose.
We use one discharge data file downloaded for the USGS station at Kremmling, CO, for the Upper Colorado.
This lab can be run on either the lab (for educators) or jupyter (for general use) instance of the OpenEarthscape JupyterHub: just click one of the links under the Run online using heading at the top of this page, then run the notebook in the "CSDMS" kernel.
If you don't already have a JupyterHub account, follow the instructions to sign up at https://csdms.colorado.edu/wiki/JupyterHub. If you're an educator, you can get JupyterHub accounts for students--please contact us through the CSDMS Help Desk: https://csdms.github.io/help-desk.
Requirements
If run locally, this lab requires the installtion of the Python packages matplotlib and pandas.
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant No. 1831623, Community Facility Support: The Community Surface Dynamics Modeling System (CSDMS).
References
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Metamath Proof Explorer < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > nbuhgr Structured version Visualization version GIF version
Theorem nbuhgr 27124
Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 27117 (with 𝑁 being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
Hypotheses
Ref Expression
nbuhgr.v 𝑉 = (Vtx‘𝐺)
nbuhgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbuhgr ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Distinct variable groups: 𝑒,𝐸 𝑒,𝐺,𝑛 𝑒,𝑁,𝑛 𝑒,𝑉,𝑛 𝑒,𝑋,𝑛
Allowed substitution hint: 𝐸(𝑛)
Proof of Theorem nbuhgr
StepHypRef Expression
1 nbuhgr.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbuhgr.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbgrval 27117 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
43a1d 25 . 2 (𝑁𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
5 df-nel 3118 . . . . . 6 (𝑁𝑉 ↔ ¬ 𝑁𝑉)
61nbgrnvtx0 27120 . . . . . 6 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
75, 6sylbir 238 . . . . 5 𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
87adantr 484 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9 simpl 486 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → 𝐺 ∈ UHGraph)
109adantr 484 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph)
112eleq2i 2907 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
1211biimpi 219 . . . . . . . . . . 11 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
13 edguhgr 26913 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
1410, 12, 13syl2an 598 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15 velpw 4525 . . . . . . . . . . . 12 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
161eqcomi 2833 . . . . . . . . . . . . 13 (Vtx‘𝐺) = 𝑉
1716sseq2i 3980 . . . . . . . . . . . 12 (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒𝑉)
1815, 17bitri 278 . . . . . . . . . . 11 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒𝑉)
19 sstr 3959 . . . . . . . . . . . . . . 15 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → {𝑁, 𝑛} ⊆ 𝑉)
20 prssg 4735 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑋𝑛 ∈ V) → ((𝑁𝑉𝑛𝑉) ↔ {𝑁, 𝑛} ⊆ 𝑉))
2120bicomd 226 . . . . . . . . . . . . . . . . 17 ((𝑁𝑋𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
2221elvd 3485 . . . . . . . . . . . . . . . 16 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
23 simpl 486 . . . . . . . . . . . . . . . 16 ((𝑁𝑉𝑛𝑉) → 𝑁𝑉)
2422, 23syl6bi 256 . . . . . . . . . . . . . . 15 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉𝑁𝑉))
2519, 24syl5com 31 . . . . . . . . . . . . . 14 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → (𝑁𝑋𝑁𝑉))
2625ex 416 . . . . . . . . . . . . 13 ({𝑁, 𝑛} ⊆ 𝑒 → (𝑒𝑉 → (𝑁𝑋𝑁𝑉)))
2726com13 88 . . . . . . . . . . . 12 (𝑁𝑋 → (𝑒𝑉 → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
2827ad3antlr 730 . . . . . . . . . . 11 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑒𝑉 → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
2918, 28syl5bi 245 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
3014, 29mpd 15 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3130rexlimdva 3276 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3231con3rr3 158 . . . . . . 7 𝑁𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3332expdimp 456 . . . . . 6 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3433ralrimiv 3175 . . . . 5 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
35 rabeq0 4319 . . . . 5 ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
3634, 35sylibr 237 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
378, 36eqtr4d 2862 . . 3 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
3837ex 416 . 2 𝑁𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
394, 38pm2.61i 185 1 ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Colors of variables: wff setvar class Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2115 ∉ wnel 3117 ∀wral 3132 ∃wrex 3133 {crab 3136 Vcvv 3479 ∖ cdif 3915 ⊆ wss 3918 ∅c0 4274 𝒫 cpw 4520 {csn 4548 {cpr 4550 ‘cfv 6338 (class class class)co 7140 Vtxcvtx 26780 Edgcedg 26831 UHGraphcuhgr 26840 NeighbVtx cnbgr 27113 This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2117 ax-9 2125 ax-10 2146 ax-11 2162 ax-12 2179 ax-ext 2796 ax-sep 5186 ax-nul 5193 ax-pow 5249 ax-pr 5313 ax-un 7446 This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2071 df-mo 2624 df-eu 2655 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2964 df-nel 3118 df-ral 3137 df-rex 3138 df-rab 3141 df-v 3481 df-sbc 3758 df-csb 3866 df-dif 3921 df-un 3923 df-in 3925 df-ss 3935 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4822 df-iun 4904 df-br 5050 df-opab 5112 df-mpt 5130 df-id 5443 df-xp 5544 df-rel 5545 df-cnv 5546 df-co 5547 df-dm 5548 df-rn 5549 df-res 5550 df-ima 5551 df-iota 6297 df-fun 6340 df-fn 6341 df-f 6342 df-fv 6346 df-ov 7143 df-oprab 7144 df-mpo 7145 df-1st 7674 df-2nd 7675 df-edg 26832 df-uhgr 26842 df-nbgr 27114 This theorem is referenced by: uhgrnbgr0nb 27135
Copyright terms: Public domain W3C validator
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# 2483: Linked List Interview Problem
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Linked List Interview Problem Title text: I'd traverse it myself, but it's singly linked, so I'm worried that I won't be able to find my way back to 2021.
## Explanation
This is another one of Randall's Tips, this time a coding interview tip.
In computer programming, a linked list is a type of data structure that stores data throughout memory accompanied with memory addresses of the next, and potentially previous data point, establishing a relative ordering for a collection of data. Several common software engineering interview questions involve manipulating or otherwise interacting with linked lists. Possibly because programmers in the current day rarely work with linked lists directly, Randall suggests that such structures belong in a "technology museum," and thinks it would be more beneficial to mankind to email the list to such a museum rather than perform any useful work with it.
A linked list is a way to store sequential data in computer memory. Each piece of data is stored with a pointer to the next piece. This makes it very easy to add new data in the middle, since only one existing pointer must change to point to the new data. The drawback of a naive implementation can be that finding data may require following the entire chain. Technical programming interviewers like to see if applicants are familiar with the structure and the computational complexity concept itself.
Linked lists are, historically, one of the two main data structures that represent sequential data, along with arrays. Unlike arrays, they have the theoretical advantage of O(1) insertions and deletions thanks to not needing to reallocate the entire structure, but have O(n) random access (see comparisons). However, modern processors' cache structure favors data that are located next to each other, pre-fetching the adjacent items, and modern processors can perform bulk memory moves, making resize operations faster. Finally, using linked lists usually implies dynamic allocation of each list member as opposed to reserving memory for a bunch of items in a bulk and then using that memory once an item has to be added. Memory allocation tends to be slow on modern systems and adds overhead for managing the information, which byte is allocated for what item, which can be significant, particularly for smaller data items; many small allocations also tend to fragment memory, which can lead to it being wasted and unavailable to the app later, particularly in long-running processes such as web servers. These properties tend to make linked lists poorly suited for most system programming applications in which a programmer might write algorithms to manipulate data structures, instead of using existing libraries.
Modern programming languages usually provide abstractions (often named "array," "vector" or "list") which interact with the sequential data at the memory level, providing access to this data while using arrays, linked lists, hybrids of the aforementioned technologies, or other approaches, and the programmer doesn't necessarily need to care one way or another. Knowing the underlying concepts is still useful, however, when creating fast running code which scales well to large data, avoiding (e.g.) traversing the list over and over again, or performing particularly inefficient operations.
Cueball's code implements a routine whose name implies that it does a mundane task, specifically traversing a linked list, but in fact emails the contents of the list to a technology museum. This could reveal private data that might be stored in a linked list, such as bank account numbers, medical information, passwords, etc., and would thus be a terrible idea. This is why interviewers - presumably job interviewers - would "get really mad".
In the title text, a singly linked list contains pointers to traverse the list in only one direction; namely, from the head to the end. By contrast, each element in a doubly linked list contains pointers to both the "next" and "previous" elements, enabling traversal in either direction. Randall continues the implication that such lists are obsolete by implying that traversing such a list would be akin to time travel. Without the "previous element" pointers, Randall is concerned he would not be able to reverse the time travel, as he could not traverse the list in the reverse direction.
## Transcript
[Cueball is writing on a whiteboard with a blue pen with Ponytail looking over his shoulder. The text on the board is unreadable, but it is is written in blue above them. It is a piece of code and it reads:]
``` define traverseLinkedList(headPointer):
myId="<illegible scribbling>"
authToken="<illegible>"
client=mailRestClient(myID, authToken)
subj="Item donation?", body="Thought you
return
```
Ponytail: Hey.
[Caption beneath the panel:]
Coding interview tip: Interviewers get really mad when you try to donate their linked lists to a technology museum.
# Discussion
Assuming not everyone understands O notation: O(1) means that it always takes the same time, no matter how much data is stored. O(n) means the time is proportional to the amount of data stored - if you have 10 times the data, it takes 10 times as long to find the one you want. 108.162.221.84 (talk) (please sign your comments with ~~~~)
This code won't mail the linked list to a museum - it will mail the memory location of the head of the list to a museum. 172.70.130.192 (talk) (please sign your comments with ~~~~)
I think part of the joke might be that the high-level language being used will actually spit out a representation of the entire list when using the str function. So it actually does all the traversing and abstracts it away, again making the interview question seem redundant! 162.158.159.48 10:40, 1 July 2021 (UTC)
The language looks almost like Python -- the only difference being the keyword define instead of def. Lisp is the only family of languages I can think of that automatically converts linked lists to a representation of all the elements, since the linked list is its fundamental data structure. Barmar (talk) 14:06, 1 July 2021 (UTC)
At the lowest level of access, such an array would be like the sequence "1234" (analogising to a simple string/char-array), asking for the nth-element quickly gets the nth-character by offset plus suitably multiplied memory reference). Inserting ("12a34") or deleting ("124") needs at least partial shuffling and resizing, while switching ("1324") or other internal re-ordering has widely variable overheads.
A linked-list could be thought of as defining as "¹" with ¹="1²", ²="2³", ³="3⁴" and ⁴="4∅", taking up more initial memory, and effort to discover the nth item. But, done right and for the right reasons, additions (²="2⁵", ⁵="a³"), removals (²="2⁴", dump/reuse ³) and switches (either ²="3³", ³="2⁴" or ¹="1³", ³="3²", ²="2⁴") can be as efficient as possible once the splice-and-switch process knows which points to work with.
(A linked-list sorter/editor will probably traverse the list, not worrying what 'offset' it is at, but holding an ⁿ pointer address for at least two adjacent items, ready to alter their ⁿs-as-reference to fulfil the change required, without worrying which ⁿs they were, and when created in whatever the next memory slot is.)
Doubly-linked might be list header "¹" where ¹="∅1²", ²="¹2³", ³="²3⁴" and ⁴="³4∅" and is heavier in storage (though often balanced by the "1234" being much more complex as actual data (e.g. multi-word, possibly variable-length records) than the simple ⁿs, that in an array-accessed form would include far too much padding and wasting storage (or too little, requiring optionally-defined ⁿs at the end of each fixed-length record to direct to an 'overflow' memory location, effectively LLing) thus justifying the potential LL packing overheads.
For further hybrid fun, nothing stops you having a fixed array "¹²³⁴∅∅∅" and define ¹="1", etc, then change the array-of-references accordingly ("¹²⁵³⁴∅∅", "¹²⁴∅∅∅∅", "¹³²⁴∅∅∅" or - if it's sensible - "¹²³⁴³²¹" which actually does something the LL would be hard-pressed to achieve for you without further structural overheads specifically designed for beyond-linear traversal).
That it potentially becomes spaghetti-data should not concern you so long as you don't have spaghetti-code as well which causes some oversight of data-mangling to mess things up. And you'll probably want to maintain a custom data-dumper/collator/formatter capability to keep an eye on things as you're debugging the inevitably miswritten shuffle-function, and/or do battle with the compiler's garbage-handling insertions when you confuse it beyond reasonable limits. (No, wait, did you do full low-level garbage-handling yourself? Did you do it properly? ;) )
...but I must say I'm not overly keen to abandon modern inbuilt splice-functions (for arrays/otherwise) doing all this hard work for me. Only if I'm looking at something of more of a net-/tree-like relationship (esp. non-Euclidean), or something with complicated multi-layered disparity of pointed-at data might I design up from such basic foundations. But I can also be nostalgic about when it was far more necessary! 162.158.159.48 10:18, 1 July 2021 (UTC)
I'm sorry, but I found this *really* hard to understand, despite already knowing what linked lists are and how they work. Beanie talk 13:20, 3 April 2023 (UTC)
Being the one who wrote that, I can see what I was explaining but I'm not right now sure why I did... ;) So, for the latter, I do apologise. As for it being complicated, well... Linked Lists/etc are often somewhat complicated to implement/document, so I can't take any blame for that particular aspect of the universe. :P 14:01, 3 April 2023 (UTC)
Does anyone know when the last comic was that used colors? Is this something worth mentioning? --162.158.88.42 06:11, 1 July 2021 (UTC)
I found the category: Category:Comics with color. --162.158.93.153 06:17, 1 July 2021 (UTC)
I added some words regarding the title text. Feel free to expand/clarify/correct as necessary. 172.69.35.209 06:57, 1 July 2021 (UTC)
The comic could also be a reference to the British Museum Algorithm. --162.158.88.110 09:09, 1 July 2021 (UTC)
I second a previous comment, the code *does not* send the list to the museum, only the string representation of the head pointer. So the examiner may be rightully pissed off because both can be true: the candidate is trying to make fun of list algorithms and he doesn't know how to deal with a list. (Unsure of what follows: given that the code looks like python, this may also be sarcasm about the style of (not only) python programming that always resorts to some external code module instead of defining new data structures and coding related methods. In this case, the external module is a museum :-) ). Xkcdmax (talk)
Those wondering why linked lists are considered obsolete: insertion and deletion performance is rarely the issue these days. It's the cost of enumerating over all elements in the list. Both arrays and linked lists have O(n) complexity there, but arrays have the lower cost. And that's before we get into stuff like caches liking predictable access patterns (pointer chasing is not predictable) and all those pointers costing precious cache memory space.--Henke37 (talk) 09:45, 1 July 2021 (UTC)
If the elements are simpler and relatively constant in individual storage demands (regardless of total numbers to store), arrays and bulk-caching work well. If they're more convoluted records (e.g. up to 64 characters as element name, 256 characters for a description, version 'number' that's another string, a notes field that is a pointer to an arbitrary chain of formatted/markupped punctuated character-storing freetext variable slots, any number of other object properties you find useful) then most of the advantages of indexable layout for lookahead loading are lost. If you're writing at significantly low-level of code, already, then you could still possibly see an advantage to implementing linked-list structures and not lose out enough to the advantages you'd get for an array implementation.
Though these days you're not encouraged to tunnel past the abstractions the higher-level compiler/interpreter will present to you. You could be hard pressed to do anything efficient yourself (like an array-of-pointers approach, or using XOR packing to cut down on memory requirements in a doubly-linked list) and must blindly trust that the original authors of the intermediate builder gave it the wisdom to not be too bad trying to match what you input to a suitably workable pre-anticipated family of data-series methodologies by the time it gets to runtime.
And there's so much power in a modern computer core that, even with a resource-hogging OS, you're probably not going to break it by manually forcing the worst option, unless you're already in danger of stressing the system even with the truly best one. 141.101.99.93 23:44, 1 July 2021 (UTC)
Anyone else think the chosen color might be relevant? We're talking about **link**ed lists and the text is written in blue, the traditional color of hyper**link**s. In any other comic, I might think it a coincidence, but this is a comic that rarely uses color, and never without a purpose. Trlkly (talk) 07:15, 3 July 2021 (UTC)
Blue whiteboard pens are probably the more used 'not black' (because easier on the eye?) but not specifically hued (red for important/'do not do' information, green for softer suggestions or else with comparative 'do do' positive stuff). From personal experience. Not sure if this is relevent, maybe it's just that blue-on-white is what Randall overwhelmingly experiences when he casually wanders in to NASA, JPL, Cern, NIF, Alphabet Inc, Apple Park, Redmond Campus, etc, and looks for casual inspiration on their various walls. 141.101.98.206 18:33, 3 July 2021 (UTC)
I think you're all missing the point of the joke: it's not the linked list itself but the interview question about linked lists that should be donated to the museum. A typical interview question is "how do you reverse a linked list?", with the interviewer expecting you to write down the algorithm where you walk down the list while creating a new linked list in the process, wiring up its "next" pointer to the previously visited element. For the first element you traverse, you set the "next" pointer of that element in the reversed list to nil, because it will be the last element in the reversed list. The final result is a pointer to the last visited element, which becomes the head of the reversed list. These kind of questions are stereotypical for programmer interviews (just like "how do you swap to numbers without using a temporary variable?") and therefore Cueball makes a snarky remark that this question is now so archaic that it should be in a historical museum of sorts.162.158.88.88 14:22, 5 July 2021 (UTC)
The text below the comment ("... donate their linked list ...") suggests the reading others have taken...
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## Philosophy question assignment help
### philosophy
##### Description
The teacher shows students the inside of the box. It contains three blue cups, two red
cups and nothing else. (It contains three Cubs caps, two Cardinals caps, and nothing
else: the Cubs caps are true blue (for goodness), the Cardinals ones fire red (for evil)).
There are only three students in classroom and teacher is going to blindfold each of
them and
place one cup of five on top of the head. The remaining two cups are left in the box
and no one knows the color of the cup after the blindfold is removed. The teacher said
student cannot guess color of cup and they have to prove they have that cap on.
The teacher removes the first student’s blindfold and student now can see the color of
cups on other two student’s heads but he cannot see his cup’s color. First student look
other two student’s cups and thinks for seconds and he said he does not know his
color of hat. (The first student does not say loud about other student’s color of hats)
The teacher removes the second student’s blindfold and student now can see the color
of cups on other two student’s heads. He said he does not know his color of cap either
(the second student does not say loud about other student’s color of hats like first
student)
Just as teacher remove third student’s blindfold, the third student says that she knows
exactly color of the cap on her head and she does not need to see other student’s color
of caps.
Question – three possibilities which one is correct?
1.She cannot know what color of cap on her head
2.she has blues cap and also prove it
3.she has red cap and also prove it
## Related Questions in philosophy category
##### Disclaimer
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# How many pints in a half quart?
He wont show up how many pints in one quart here, never; an he aint likely to show up anywheres else that hes known. It was close by, ... - Read more
... I can never seem to remember how many ounces are in a cup, cups are in a ... two and a half cups. From there, the quart is 2 pints = 5 cups = 40 oz, ... - Read more
## How many pints in a half quart? resources
### How many pints in a quart? - Evi - Evi | Ask me anything
... values varying from state to state from less than half a litre to over one ... how many pints in a quart ... how many imperial pints in 1 us quart ...
### How many cups in a pint? How many pints in a quart? How ...
Now determine how many half pints you want to make. That's by how much you will have to reduce the ingredients. Say the recipe says it will make 3 quarts.
### Pint - Wikipedia, the free encyclopedia
... but will have "568 ml 1 pint", or just "568 ml" on the label. Many recipes published ... harking back to the days when liquor came in US pints, quarts, and half ...
### Quarts to Pints - How many pints in a quart?
... is 2. To find out how many pints in quarts, multiply by the factor or simply use the converter below. 1 Quart = 2 Pints [US, UK, Fluid or Dry]
### Pints to Quarts (pt to qt) conversion - Ask Numbers
Pint is half size of a quart, so the conversion factor from pints to quarts is 0.5 (It's the same for fluid and dry measurement). To find out how many quarts in pints ...
### How many quarts are in a half pint? | Answerbag
How many quarts are in a half pint? One quart is equivalent to two pints, so half a quart is one pint. A half of a pint would therefore be equal to one ...
### gallon, quart und pint - Englische Volumeneinheiten
... , genannt quart, und eine achtel Gallone, das pint ... Apotheken und Parfümerien interessieren sich nicht so sehr für pints und gallons. Hier braucht man ein ...
### 1 Quart How Many Pints | Keywordslanding.com
How many half pints in a quart? 2 pints = 1 quart so 4 half pints = 1 quart. How many pints are in 24 quarts? The correct answer is 48. How many quarts are there ...
### How Many Pints Are In One Quart The Qa Wiki | Autos Weblog
How many half pints in a quart? 2 pints = 1 quart so 4 half pints = 1 quart. how many pints are in 24 quarts? the correct answer is 48. how many quarts are there in ...
### Number of Pints in a Quart | Number Of | How Many
How many pints are there in 1 quart? 2. 1 quart = 2 pints. ... • Blog > Measurement > Number of Pints in a Quart | How many pints are there in 1 quart? 2.
### How many Quarts are there in a Gallon? How many Pints are ...
How many Quarts are there in a Gallon? How many Pints are there in a Quart? How many Cups in a Pint? If you don't know the answer, this little story ...
### How many pints in a quart? 2 4 8 16
User: How many pints in a quart? 2 4 8 16 Weegy: 3:45 justpretty|Points 1771| ... User: How many feet in a rod? 2.5 4.5 8.5 16.5 Weegy: 16.5.
### Cups, Pints, Quarts, Half Gallons and Gallons: Song For ...
... Pints, Quarts, Half Gallons and Gallons ... Two quarts is a half-gallon, two half-gallons make ... Many thanks to Kathleen Wiley for permission to publish these ...
### BrainPOP Jr. | Math | Learn about Cups, Pints, Quarts, Gallons
... including cups, pints, quarts, and gallons. ... and four quarts in a gallon. But, how many cups are in a gallon? Watch the movie to find out!
### Liter to Gallon / Quart / Pint / Cup / Ounce
Liter to Gallon / Quart / Pint / Cup / Ounce ... LITER: GALLON: QUART: PINT: CUP: OUNCE: NOTE: Entering large positive or negative values ...
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# Help defining Functor type
I am almost there, but this is my first time using `StaticArrays.jl` and I am not sure I got the concepts right:
``````using StaticArrays: SMatrix
"""
EllipsoidDistance([a₁, a₂, ...], [θ₁, θ₂, ...])
A distance defined by an ellipsoid with given semiaxes `a₁, a₂, ...`
and rotation angles `θ₁, θ₂, ...`.
The positive definite matrix A = PΛP' representing the ellipsoid is
assembled once and cached in the object as an static array. Calls to
(x-y)'*A*(x-y) are therefore very efficient.
"""
immutable EllipsoidDistance{N,T<:Real} <: AbstractDistance
# state fields
A::SMatrix{N}
function EllipsoidDistance{N,T}(as, θs) where {N,T<:Real}
@assert length(as) == length(θs) == N "number of semiaxes and rotations must match spatial dimension"
Λ = spdiagm(one(T)./as.^2)
P = eye(T, N) # TODO: define rotation matrix, probably using Rotations.jl
A = P*Λ*P'
new(A) # this is causing the error as expected since the local variable A is not static
end
end
EllipsoidDistance(as::Vector{T}, θs::Vector{T}) where {T<:Real} = EllipsoidDistance{length(as),T}(as, θs)
(d::EllipsoidDistance)(x, y) = begin
z = x - y
z'*A*z
end
``````
The error is caused by the `new` operator because I am passing a non-static array to it. What would be the appropriate way of caching `A`?
Just convert `A` to a `StaticArray`? Also, you probably want to have `A` concretely typed, aka `SArray{Tuple{M, N}, T, 2, M*N}` where `M` is number of rows and `N` number of columns.
1 Like
Thank you @kristoffer.carlsson, I replaced `SMatrix{N}` by the concrete type `SMatrix{N,N,T}`, but how to convert `A` to this type? Also, is this a good thing to do in general? Will the functor be much faster for 2D and 3D vectors for example?
Got it, just calling `SMatrix{N,N,T}(A)` will do it. I had issues with the inferred type `T`, that is why it was not working.
I am still wondering if the performance gain will be visible, starting some benchmarks.
What size is `N`? It should be significantly faster if `N` is smallish (assuming `x` and `y` are `StaticVector`s).
The dimension will be usually 2D or 3D (at most 4D), but I have no guarantee on the vectors `x` and `y` multiplying the matrix, they are general heap allocated arrays. Should I forget about making `A` static in this case?
Variables ending with `s` are Static (dim = 4):
``````julia> f(x, A, y) = (z = x-y; z' * A * z)
f (generic function with 1 method)
julia> @btime f(\$x, \$A, \$y);
156.315 ns (2 allocations: 224 bytes)
julia> @btime f(\$x, \$As, \$y);
93.785 ns (2 allocations: 224 bytes)
julia> @btime f(\$xs, \$As, \$ys);
6.540 ns (0 allocations: 0 bytes)
``````
3 Likes
I think I will stick with heap-allocated arrays for now. If this becomes a bottleneck at some point, I will hack my way with StaticArrays.
As a reference, for me, swapping to static arrays has lead to cleaner code with less bugs (because of array sizes in the type and immutability) as well as tremendous performance improvements. Depending on the performance characteristics of your code, the performance improvements might of course not carry over for you.
1 Like
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# SAVY 2019: Session 5, Day 2 – Math and Music (Rising 5th/6th)
Posted by on Tuesday, July 16, 2019 in Grade 5, Grade 6, SAVY.
We began Math and Music today by splitting up into two teams and laying the groundwork for our John Cage trial later in the week. I was impressed with how well the students worked to come up with their respective definitions of music! Of course the two sides will be presenting different definitions as they try to argue either in favor of or against “4’33”” being music, and I look forward to seeing how it plays out on Friday! We will begin each day with a period of trial research and preparations.
The rest of the morning was spent learning about musical notation and intervals. In addition to the treble and bass clefs, students learned that there are several other clefs that exist but are now rarely used. Unfortunately, we did not have any violists in the room to argue for the necessity of alto clef!
In the afternoon, we read our first excerpt from Godel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. We will be looking at three of the dialogues from this book as jumping off places for musical and mathematical discussions. I was quite amazed by student enthusiasm for the Desmos graphing calculator (https://www.desmos.com/calculator). Somehow we ended up making equations that spelled YMCA, all of this out of a desire to understand how to move things left and right!
The end of this particular reading gives an assertion of how Johann Sebastian Bach might have felt about the music of John Cage, specifically “4’33.”” Students wrote a brief response at the end of the day that asked them to address what they thought Bach would think of Cage. I am interested to read their thoughts tonight!
We wrapped up the day by beginning to talk about musical set theory. We will talk about mathematical set theory later this week. Today we worked on classifying sets of pitches into their “normal order,” and we will continue on to learn about a set’s “prime form,” which will also give us a chance to talk about prime numbers mathematically.
All in all, I was quite pleased with the day’s work! It was, in many ways, a more challenging day than day one, but it was so interesting to see the things that they were interested in and wanted to explore. I’m looking forward to exploring more of these things with them throughout the week!
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# k-e model
Register Blogs Members List Search Today's Posts Mark Forums Read
August 17, 1998, 03:29 k-e model #1 Anil Marathe Guest Posts: n/a I am working on N-S codes using Baldwin-Lomax turbulence model for last five years. I would like to try my codes with k-epsilon models. Can any CFD member help me by giving more details of k-e model, preferably FORTRAN source code. Otherwise, I have to write the code, debug and validate, which take time, which I feel unnecessary. Regards A.G.Marathe IIT Mumbai India
August 20, 1998, 12:21 Re: k-e model #2 John C. Chien Guest Posts: n/a First of all, you can find these two equations in any CFD related journal papers. Whether it is 2-D form, 3-D form, Cartesian coordinates, cylindrical coordinates, incompressible, compressible form. That is the starting point. Then you need to know what Navier-Stokes equations you are trying to solve. Whether it is steady state, or un-steady state, compressible of incompressible. Then you need to decide what numerical method you are going to use. This will determine how the two-equation model will be solved. As you see, the method of solving the two-equation model depends on the problem formulation. The programming is straightforward, but the source terms in the two-equation model can create some convergence problem for you ,if you are not carful.
August 21, 1998, 05:11 Re: k-e model #3 Vijay Thaker Guest Posts: n/a Why don't u look at this software called PHOENICS by CHAM Pte. Lmtd. They use a k-e model for their soultion. I'm currently using their PHOENICS1.5EO in my final year project. I'm currently pursuing an aeronautical degree from University Technology of Malaysia
August 21, 1998, 09:50 Re: k-e model #4 John C. Chien Guest Posts: n/a There is a TEACH program,with source code available,using staggered grid approach. This maye a good approach to learn what other people are doing. In general, this is a high Reynolds number model with special wall function treatment. The algebraic model normally solve the flow field all the way to the wall. You also need to worry about the single grid vs staggered grid issue,
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# Free help with homework
## Why join Brainly?
• find similar questions
# Which expressions could complete this equation so that it has one solution?
4(4x-3)-6x
Choose exactly two answers that are correct.
A. 12x-12-2x
B. 9x-12
C. 4x+2(3x-6)
D. 4(2x+3)-24
2
by alexandriasmith
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### Module Details
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title Singularity Theory of Differentiable Mappings Code MATH455 Coordinator Professor VV Goryunov Mathematical Sciences Victor.Goryunov@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 7 FHEQ First Semester 15
### Aims
To give an introduction to the study of local singularities of differentiable functions and mappings.
### Learning Outcomes
(LO1) To know and be able to apply the technique of reducing functions to local normal forms.
(LO2) To understand the concept of stability of mappings and its applications.
(LO3) To be able to construct versal deformations of isolated function singularities.
(S1) Problem solving skills
(S2) Numeracy
### Syllabus
Inverse and implicit function theorems; Morse Lemma;
Manifolds; tangent bundles; vector fields;
Germs of functions and mappings;
Derivative of a mapping between manifolds;
Critical points and critical values of mappings; Sard's lemma.
Equivalence of map-germs; stable map-germs of a plane into a plane; transversality; jet spaces; Thom's transversality theorem.
Local algebra of a singularity; local multiplicity of a mapping; Preparation theorem.
Stability and infinitesimal stability; finite determinacy; versal deformations of functions.
Beginning of the classification of function singularities; Newton diagram; ruler rotation method; simple functions; boundary function singularities.
### Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
MATH101 Calculus I 2020-21; MATH102 CALCULUS II 2020-21; MATH103 Introduction to Linear Algebra 2020-21
### Assessment
EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
final assessment 120 70
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Homework 1 Standard UoL penalty applies for late submissions 10
Homework 2 Standard UoL penalty applies for late submissions 10
Homework 3 Standard UoL penalty applies for late submissions 10
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With three simple steps you can calculate all the details related to lease price, sqft on rental and sqft on purchase.
## Rental Rate Calculations - Thornton Oliver Keller
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Calculate monthly rent for commercial space. ... Below is a simplified summary of rental rates and calculating the monthly cost of a space. Other factors such as tenant ... Industrial space is quoted on a per square foot, per month basis. Also ...
## Cost per square foot - Centrale des maths - University of Regina
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search. Subject: square footages. Name: llinden. Who are you: Other. how do you figure out the cost of square footage? the amount per sq ft is \$28. Thank you ...
## Square Footage Calculator - Calculator Soup
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How to calculate square footage for rectangular, round and bordered areas. Calculate project cost based on price per square foot, square yard or square meter.
## Home Buying: How do I determine the price per square foot of my ...
May 23, 2009 ... Take 300,000 and divide by 3000 = a price per square foot of \$100. ... the suare footage of the house and that figure is the price per sqare foot.
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## Is Price Per Square Foot the Best Way to Value a Home? | Joe ...
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Nov 18, 2012 ... First, let's define the method for calculating price per square foot to begin our discussion. Price per square foot is calculated by dividing the ...
## How do you determine price per square foot? | Zillow
Jul 16, 2012 ... Investing in Bonney Lake, WA - Hello, Just curious on how you calculate the value of a home? I noticed the square foot value ...
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Calculate building costs: Figure out the basic structure of the house ... more complex the shape, the more expensive the structure per square foot of floor area .
## How Do I Calculate Price Per Square Foot for House Painting ...
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Paint contractors usually price projects on a per square foot basis, although they may quote you a flat rate. Square foot pricing allocates a portion of labor and ...
How to Calculate Price Per Square Foot
Comparing the price per square foot of various homes lets you quickly and efficiently determine whether the home you're buying, selling or building is a good value. When the location and other amenities of several homes are similar, the property that has... More »
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To calculate cost per square foot, divide the purchase price of a house by its total square footage. For instance, if a house with 2000 square feet sells for ...
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Price. Per square foot (Cost / ft. calculated from Cost / Yard is rounded up to the nearest ... and automatically view the conversion (feet); To calculate the total cost
### How to Calculate Price Per Square Foot | eHow
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How to Calculate Price Per Square Foot. Comparing the price per square foot of various homes lets you quickly and efficiently determine whether the home ...
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# Constructing a python set from a numpy matrix
I'm trying to execute the following
``````>> from numpy import *
>> x = array([[3,2,3],[4,4,4]])
>> y = set(x)
TypeError: unhashable type: 'numpy.ndarray'
``````
How can I easily and efficiently create a set from a numpy array?
-
If you want a set of the elements, here is another, probably faster way:
``````y = set(x.flatten())
``````
PS: after performing comparisons between `x.flat`, `x.flatten()`, and `x.ravel()` on a 10x100 array, I found out that they all perform at about the same speed. For a 3x3 array, the fastest version is the iterator version:
``````y = set(x.flat)
``````
which I would recommend because it is the less memory expensive version (it scales up well with the size of the array).
PS: There is also a NumPy function that does something similar:
``````y = numpy.unique(x)
``````
This does produce a NumPy array with the same element as `set(x.flat)`, but as a NumPy array. This is very fast (almost 10 times faster), but if you need a `set`, then doing `set(numpy.unique(x))` is a bit slower than the other procedures (building a set comes with a large overhead).
-
Good suggestion! You could also use set(x.ravel()), which does the same thing but creates a copy only if needed. Or, better, use set(x.flat). x.flat is an iterator over the elements of the flattened array, but does not waste time actually flattening the array – musicinmybrain Dec 21 '09 at 12:11
@musicinmybrain: very good points! Thank you! – EOL Dec 21 '09 at 14:23
WARNING: this answer will not give you a set of vectors, but rather a set of numbers. If you want a set of vectors then see miku's answer below which converts the vectors to tuples – conradlee Aug 2 '11 at 11:30
@conradlee: This solution is indeed designed to give the set of all the numbers found in the array. – EOL Aug 3 '11 at 2:01
The immutable counterpart to an array is the tuple, hence, try convert the array of arrays into an array of tuples:
``````>> from numpy import *
>> x = array([[3,2,3],[4,4,4]])
>> x_hashable = map(tuple, x)
>> y = set(x_hashable)
set([(3, 2, 3), (4, 4, 4)])
``````
-
and how to I easily/efficiently transform back to a list? – user989762 Feb 18 '14 at 5:57
`map(array, y)` – Manuel Mar 26 '14 at 12:07
The above answers work if you want to create a set out of the elements contained in an `ndarray`, but if you want to create a set of `ndarray` objects – or use `ndarray` objects as keys in a dictionary – then you'll have to provide a hashable wrapper for them. See the code below for a simple example:
``````from hashlib import sha1
from numpy import all, array, uint8
class hashable(object):
r'''Hashable wrapper for ndarray objects.
Instances of ndarray are not hashable, meaning they cannot be added to
sets, nor used as keys in dictionaries. This is by design - ndarray
objects are mutable, and therefore cannot reliably implement the
__hash__() method.
The hashable class allows a way around this limitation. It implements
the required methods for hashable objects in terms of an encapsulated
ndarray object. This can be either a copied instance (which is safer)
or the original object (which requires the user to be careful enough
not to modify it).
'''
def __init__(self, wrapped, tight=False):
r'''Creates a new hashable object encapsulating an ndarray.
wrapped
The wrapped ndarray.
tight
Optional. If True, a copy of the input ndaray is created.
Defaults to False.
'''
self.__tight = tight
self.__wrapped = array(wrapped) if tight else wrapped
self.__hash = int(sha1(wrapped.view(uint8)).hexdigest(), 16)
def __eq__(self, other):
return all(self.__wrapped == other.__wrapped)
def __hash__(self):
return self.__hash
def unwrap(self):
r'''Returns the encapsulated ndarray.
If the wrapper is "tight", a copy of the encapsulated ndarray is
returned. Otherwise, the encapsulated ndarray itself is returned.
'''
if self.__tight:
return array(self.__wrapped)
return self.__wrapped
``````
Using the wrapper class is simple enough:
``````>>> from numpy import arange
>>> a = arange(0, 1024)
>>> d = {}
>>> d[a] = 'foo'
Traceback (most recent call last):
File "<input>", line 1, in <module>
TypeError: unhashable type: 'numpy.ndarray'
>>> b = hashable(a)
>>> d[b] = 'bar'
>>> d[b]
'bar'
``````
-
If you want a set of the elements:
``````>> y = set(e for r in x
for e in r)
set([2, 3, 4])
``````
For a set of the rows:
``````>> y = set(tuple(r) for r in x)
set([(3, 2, 3), (4, 4, 4)])
``````
-
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1. Asides from Maxwell's equations, which are too complicated for me, I have found these two. If anyone is familiar with this, I would greatly appreciate your help.
Now, an equation to find the force of a magnetic field and electric field are as follows, repectively:
F = qv x B
where F is the force;
q is the value of the test charge;
v is the velocity of the charge;
B is the vector magnetic field
And x is as follows: "The "x" indicates a vector cross-product, which in this case means a magnetic force will only arise if the velocity of the charge is perpendicular to the magnetic field. The mathematical cross product determines the direction of the magnetic force."
and
F = qE
where F is the force;
q is the value of the test charge;
E is the vector electric field.
Anyone know? In fact, is it even possible to solve these without having proper equipment? Also, is anyone familiar with tesla equations? As in finding the tesla value of an electromagnetic field?
I suppose anything else anyone may now has the potential to be helpful; follow your whims, as long as those whims pertain to electromagnetic fields.
2.
3. What exactly are you planning to do?
4. I am testing the effects of electromagnetic field on silk worm and spiders in connection to the uncertainty of their effects on health. I have done some reading, and I am unclear of the differences between thermal electromagnetic radiation and non-thermal electromagnetic radiation. Is thermal radiation that which is characterized by heat which is able to break down chemical bonds? And what of non-thermal radiation? Also, an equation to find the tesla amount would be most beneficiary so I would be able to compare the electromagnetic field I created to those that are created by power lines, cell phones, electric wiring, et cetera.
5. Well, I'm not sure about the difference between thermal and non-thermal radiation. I'd guess that it has something to do with the components of the radiation, like matter particles and energy particles but I don't really know. Faraday's Law of induction is just:
EMF = -d(magnetic flux)/dt
so:
magnetic flux = -int(EMF dt)
so basically, the magnetic field is greater, the faster the electric field changes.
6. Originally Posted by IllusionConcerto
I am testing the effects of electromagnetic field on silk worm and spiders in connection to the uncertainty of their effects on health. I have done some reading, and I am unclear of the differences between thermal electromagnetic radiation and non-thermal electromagnetic radiation. Is thermal radiation that which is characterized by heat which is able to break down chemical bonds? And what of non-thermal radiation? Also, an equation to find the tesla amount would be most beneficiary so I would be able to compare the electromagnetic field I created to those that are created by power lines, cell phones, electric wiring, et cetera.
biological biengs are uneffected by magnetic and electrical fields. They do not take anykind of damage in such. But electromagnetic radiation can cause destruction on their genetic code
7. Originally Posted by Zelos
Originally Posted by IllusionConcerto
I am testing the effects of electromagnetic field on silk worm and spiders in connection to the uncertainty of their effects on health. I have done some reading, and I am unclear of the differences between thermal electromagnetic radiation and non-thermal electromagnetic radiation. Is thermal radiation that which is characterized by heat which is able to break down chemical bonds? And what of non-thermal radiation? Also, an equation to find the tesla amount would be most beneficiary so I would be able to compare the electromagnetic field I created to those that are created by power lines, cell phones, electric wiring, et cetera.
biological biengs are uneffected by magnetic and electrical fields. They do not take anykind of damage in such. But electromagnetic radiation can cause destruction on their genetic code
1) Many specied of birds and bats navigate using the Earth's magnetic field.
2) Where do you think electromagnetic radiation comes from ?
8. 1: yes but they dont take harm they just get confused, maybe they hit a wall but thats just cause they are stupid
2: shifting electrical or magnetical fields
and other soruces aswell
9. Ok, well, I found this equation:
B=F/IL
which is what I was looking for. However, F is the force experienced by a wire, measured in Newtons. How would I find this?
Also, how woulud I find the hertz value of an electromagnetic field? And, the velocity of an electron traveling through a copper wire? Which is incorporated in the second part of that equation F=BQv
10. you find it by measuring it in different ways, netwonmeter using gravity etc
11. This is a tough subject, I've done a little research for a project in the past, but as I understand it:
Static electric fields are distribution of electrically charged particles in space. The movement of these particles (electrons) create a magnetic field. Thus a purely static field has an electric field component but no magnetic field component. The magnetic field component is perpendicular to the electric field and increases with frequency (rapidity of change in electric field). This is an electromagnetic field.
At higher frequencies (20-30kHz) the electric and magnetic fields are tightly coupled and form an electromagnetic wave the propogates in a direction orthogonal to the electric and magnetic fields.
High frequency magnetic waves are called electromagnetic radiation and are composed of discrete packets of energy called photons. The energy of a photon is proportional to and increases with the wave frequency.
At some point (i don't know where off hand) the frequency and hence energy is great enough to break molecular bonds and is said to be ionizing. Only very high frequency radiation such as gamma rays are ionizing, radio waves used in communications and 50Hz power lines are non-ionizing.
Radio frequency waves can cause a heating effect (perhaps what you mean by thermal radiation?) though and this is usually what health concerns arise from. RF cannot directly damage DNA though some people think it may disrupt some processes such as cell division. It has also been suggested that a resonance between some electrochemical processes in the brain and RF can cause adverse health effects though this is not generally accepted.
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# Why the canonical bundle of a complex manifold is a line bundle?
I think that I do not understand something in the definition of the line bundles. Line bundles have fibers of rank 1, that is isomorphic to $\mathbb{C}$. But I do not know how to connect this vector space with elements like $dz_i$, $\,dz_{i_1}\wedge dz_{i_{2}},$ etc. Here, $dz_i$ are locally defined one-forms for the coordinates $\{z_i\}$ of some patch of the manifold of complex dimension $m$.
So, why is the $m$-th product of forms a line bundle? What is the correspondence with $\mathbb{C}$? Can you give an example?
Also, what kind of bundle is then the collection of elements $\{dz_i\}$, $\{dz_{i_1}\wedge dz_{i_2} \}$ and so on?
• Presumably you mean the $m^{\text{th}}$ exterior product of the tangent bundle of an $m$-dimensional complex manifold forms a line bundle. This is really a linear algebra problem on the level of tangent spaces: for a vector space $V$, you define the exterior powers $\Lambda^k V$. If $V$ has dimension $n$, then $\Lambda^k V$ can readily be seen to have dimension $\binom{n}{k}$. – Dustan Levenstein Feb 4 '16 at 15:21
• Yes this is what I mean. But I do not see why the space $\Lambda^k V$ has the dimension $\frac{n!}{k!(n-k)!}$. Would you mind to expand slightly or to direct me to some specific reference? – Marion Feb 4 '16 at 15:31
• Well, the Wikipedia article on this topic does give the definitions and basis. Specifically, if $\{e_1, \ldots, e_n\}$ is a basis for $V$, then $\{e_{i_1} \land \cdots \land e_{i_k} \mid 1 \le i_1 < \cdots < i_k \le n\}$ forms a basis for $\Lambda^k V$. Since the basis is enumerated by ordered sets of $k$ distinct elements from the basis of $V$, the dimension formula follows. – Dustan Levenstein Feb 4 '16 at 15:41
• You should really read in detail some treatment of multilinear algebra and exterior powers in particular before moving any further with complex manifolds, vector bundles and what not! – Mariano Suárez-Álvarez Feb 27 '16 at 22:01
• Thanks that helps a lot LOL – Marion Feb 28 '16 at 3:17
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# Pilbeam CHAPTER 6
### Student Study Note Cards
QuestionAnswer
Settings for Vt and f should reflect a Ve that is derived from the initial calculation based on _______ ,________ and _______.(ApQ)(TM) Gender, BSA, pts pathology(TM)(ApQ)
The advantage of VS is?(RQ)(TM) Spontaneously breathing pts can establish their own RR and Vt.(TM)(RQ)
A pt with ARDS who is 5'4" and weighs 195 lbs needs to be changed from a CPAP of 10 cmH2O to VC-CMV. What tidal volume and rate would you set and why?(AzQ)(TM) 4-8ml/kg and RR 15-30bpm because these are appropriate for ARDS pts(AzQ)(TM)
Define Mechanical Tidal Volume. (RQ) Set volume delivered to the patient; amount breathed in during 1 breath. (CG)
5'2" pt. What is her VT range? (ApQ) 418.2mL-627.3mL (CG)
5'7" pt on pressure ventilation with PIP of 15 cmH2O, PEEP 5cmH2O, RR 14bpm. Actual Vt is 350ml. What setting should you modify to achieve appropriate Vt? (AzQ) Increase PIP to give bigger breaths (CG)
What is the IBW for a 6'1" male in kg? And what would you expect his VT, RR, and VE(min) to be? (BSA = 2.08) (Az Q) 83.6 kg, 585 ml - 836 ml, 12-18 bpm, 8.3 Lpm (ACE)
What is mechanical dead space? (RQ) The volume of gas that is rebreathed during ventilation (ACE)
What's the equation for respiratory frequency? (AP Q) f= VE(min)/VT (ACE)
What basic settings help improve ventilation? (RQ) RR, Vt or PC level, PS (KMH)
5'6"female patient--what is her Vt range? (ApQ) 488-732ml (KMH)
What is a normal flow range? Do COPDers like a high flow or low flow and why? (AzQ) 40-60lpm--high flow--they like their flows given to them fast with a short I time to increase E time (KMH)
What will improve a pts VE leading to a decrease in CO2 and improves ventilation? (RQ) RR (MK)
A 5'8' pt has a RR of 15bpm. What is her TCT? (ApQ) 60/15=4 seconds (MK)
A pt has a TCT of 7 seconds. They have an I Time % of 30%. What is their I Time, E time and I:E ratio? Is their I:E ratio normal? (AzQ) I time=2.1 sec E time=4.9 sec I:E=1:2.3--yes (MK)
We will use Itime to manipulate what other value? KRM Flow
Pt has the following: RR 15, Itime % of 35% and Flow 45 Lpm. Find the TCT, Itime, and Etime. KRM TCT=4 seconds, Itime=1.4, Etime=2.6 second
What are 3 factors we look for to check the body's ability ot oxygenate the tissues? KRM Hemoglobin, Circulation and adequate ventilation.
What would be the first setting you would change on a ventilator to improve ventilation? RQ RR(JB)
Increasing PEEP has what kind of side effects? Apq increase in intrathoracic pressure, decrease venous return, decreased bp, increase in ICP, (JB)
If you have a TCT of 7 and an itime of 20%, what would be your Itime set in seconds? AzQ intime=TCTXItime%. 7X.20= 1.4s (JB)
A female 5'4" patient is on a ventilator. You check her values and her Vt is is 354 ml. What is her tidal volume range and is hers adequate right now?(AzQ) 454 to 682 mL. NO. (AT)
What is the normal PS range? (RQ) 5 to 15 cmH2O (AT)
Find the tidal volume if you have a flow of 30 lpm and an I-time of 0.8 (ApQ) 400 mL (AT)
Pressure support only effects what type of Pts?RQ Spontaneously breathing (MC)
What is the goal Vt for a Pt that is 6 foot 2inches? ApQ 687-1032 mL (MC)
If you have a Pt who has CHF and you notice that their PEEP is at 10 cmH2O and their heart rate begins to rise and their BP begins to drop what should you do and why is this occurring?AZQ The pt is having to much intrathorasic pressure causing restriction on the heart. Decrease PEEP and notify the MD. (MC)
Basic settings for 100: Respiratory rate falls under this categorize (RK) What is Ventilation (RQ)
Basic settings for 250: This oxygenation setting holds the alveoli open. there normal values are..(RK) What is PEEP, normal 5-15cmH2O good starting point around 10cmH2O (AqQ)
Basic settings for 500: These are all normal ranges for all basic settings to start mechanical ventilation. (RK) What is RR:12-20bpm Vt:8-12ml/kg PC:15-20cmH2O FiO2:SpO2 >90% PEEP:5-15cmH2O Itime:.8-1.2sec (Azq)
What would be some settings to change on a vent to help improve oxygenation? (RQ) FiO2, PEEP, Itime (KAH)
If your patient in ICU on a vent had a change in FiO2 of 15% over your 8 hour shift what would you do? (AzQ) The physician should be contacted, this pt could be having a worsening disease process or a diffusion problem (KAH)
What is the "PEEP rule?" (RQ) If FiO2 is at 60% or higher and there is little to no change in PaO2 then PEEP maybe be needed to fight refractory hypoxemia.
List the settings that affect ventilation, and then those that affect oxygenation. (ApQ) -Ventilation: RR, Vt or PC, and PS -Oxygenation: FiO2, PEEP, Itime, Hgb, circulation, adequate ventilation (BH)
You are giving an aerosol to a ventilated patient and they look like they are having a hard time taking a breath when they want to. What is the problem and how would you fix it? (AzQ) The patient's vent is on a flow trigger setting and the extra flow being added to the circuit by the aerosol causes it to be too much for the patient to trigger. While delivering the aerosol the vent should be changed to pressure triggering. (BH)
How does PEEP affect the lungs? (ApQ) It applies pressure into the lungs, and holds the alveoli open to improve oxygenation. (KJ)
When you are in _____ ______, you can't have a ____ and a _____ set at the same time. This setting will do what for your patient, and will over come _____. (AZQ) Pressure Support. Pressure Control, Tidal Volume. This setting will provide a boost to overcome AW resistance and compliance of the ET Tube. (KJ)
On Vt, the faster the Itime, the bigger the ____. When increasing the Vt, we want to be careful of what? Surface Area, slower flow. Barotrauma, which can lead to damage of the lungs. It stretches the alveoli. (KJ)
Pressure control level is only set in which type of ventilation? (RQ) BL Pressure ventilation
When setting the Itime for a COPD or asthma patients, where would you set it at? What does this do for them? (AzQ) BL 0.8 cmH2O or the lower side because it gives them higher flows and gives them a longer Etime for their airtrapping.
what would the tidal volume range for a 5'9 male? What two ways can you calculate this? (ApQ) BL 582-873mL or 728mL Find by IBW in kg multiplied by 8 and 12ml OR find IBW in kg and multiply by ten for quick reference.
What are three setting that would effect Oxygenation on a ventilated patient? (RQ) MB FiO2, PEEP, Itime. MB
You have a 5'2" male what would be his Vt range? (RpQ) MB 430-644 mL MB
Where would you most likely set the Itime for a COPD patient? (RpQ) MB Most likely 0.8-1 sec MB
Which setting would you change first to adjust ventilation? (AH) RR
When would you not want to set a Pressure Support? (AH) When pt is not spontaneously breathing.
A patient'a mechanical ventilator settings are: RR 16 Vt 650ml, PEEP 15cmH20, Itime 1sec, FiO2 50%, PS 0. Pt's blood pressure is 75/43. What can be causing the low blood pressure? (AH) PEEP of 15cmH2O. Peep causes increased ICP and decreased BP.
What is mechanical deadspace? (Marianne B.) It is the volume of gas that is rebreathed during ventilation. (RQ)
Calculate CT with a volume change of 50 ml and a pressure change of 25 cm H2O. (Marianne B.) It is 2 ml/cm H2O. (ApQ)
If a physician orders a Vt of 600 ml and 14 bpm for a 25 year old female with a BSA of 3.0. Is the ordered VE adequate? (Marianne B.) No it is not. The ordered VE is 8.4 L and the estimated VE is 10.5 L. So the patient would have to take additional breaths. (AzQ)
What is your first choice in settings to change when you want to change ventilation? (ApQ)(CZ) Respiratory rate
What is one setting we can change without a physician's order? (RQ)(CZ) FiO2It is ordered most often in order to keep the patient's SpO2 over 90% Itime, Flow, and Trigger could also be changed without a MD order.
Your patient is set at 12 RR. Her I time is 1 sec and tidal volume is 450 ml. What is her TCT? Etime? flow? (AzQ) (CZ) TCT is 5, Etime is 4 sec, flow is 27 lpm
6'0" male, what is his Vt and IBW? ApQ (JM) IBW is 178 lbs. 80.9 kg Vt on vent is 647 to 970 ml
Define respiratory distress.RQ (JM) This has a sudden onset, pt appears alarmed, sweating, flushed, anxious, or panicked. Pt can talk in short broken sentences. HR will be increased and irregular.
What is this pts I:E ratio and why is that important to know? Itime=1.8 sec, TCT is 3.4 AzQ (JM) This is an inverse I:E ratio and indicated a severe obstruction. Itime should be less than Etime.
Your pt is showing increased use of the sternocleidomastoid muscle during inspiration on a PSV mode. What does this indicate & how might you adjust your vent? (NMB) May indicate that the level of pressure support is not high enough for the pt. Adjust by increasing your patients pressure support value. (ApQ)
What mode of ventilation provides closed-loop pressure breaths & targets the pressure to achieve the set volume? This mode is also a pressure-limited & time-cycled mode. (NMB) PRVC (Pressure Regulated Volume Control) (RQ)
Your pt has set PS of 10cmH2O and a PEEP of 5cmH20. How much total pressure are you delivering to your pt? (NMB) 10cmH2O + 5cmH20 = 15cmH20 (AzQ)
A ventilated patient has been transferred from an acute care facility to a long-term care patient. What pressures would you use? (RQ) rt 1/3 of previous PIP or 15-20 cwp. rt
What two functions does setting the RR do? ApQ rt 1) provide a trigger 2) provide cycling (rt)
A patient with pneumonia on volume control has a PIP of 32; ABGs show that the patient is growing hypercapnic. What can you do to improve ventilation? AzQ-rt Decrease Vt and increase RR. (rt)
An unresponsive person was brought in by a family member. They are breathing shallow with periods of apnea. What Fio2 would you put them on? (Jenn B) If a previous FiO2 is unknown then start the patient at 100% and wean them down.
What is the volume of gas that is rebreathed during mechanical ventilation? (Jenn B) Mechanical dead space (Vdmech).
A patient is breathing 15 bpm with Vt 500ml and Flow of 35 lpm. What is their Itime? What is thier I:E? (Jenn B) Itime=.86, I:E= 1:3.65
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# 1 PHYSICS FOR ENGINEERS. B-EXAMS 2006-2007. B1B2B3B4.
## Presentation on theme: "1 PHYSICS FOR ENGINEERS. B-EXAMS 2006-2007. B1B2B3B4."— Presentation transcript:
1 PHYSICS FOR ENGINEERS. B-EXAMS 2006-2007. B1B2B3B4
2 B1 (a)(b)(c) You have a spring whose natural length is L 0 = 10 cm -figure (a)-. When a mass M = 250 g is hung on the spring, its length increases by L = 40 cm –figure (b)-. Finally, the hanging mass oscillates after the spring is stretched A = 10 cm and then released –figure (c)-. Answer the following questions: What is the constant of the spring? Find the period of the oscillation Find the position of the mass 6.98 s after the oscillations start. Find the period of the oscillation if you had hung the same mass M from two identical springs like this one disposed in a paralell way. a) b) c) d) 0.5 p 1.0 p 2.0 p PROBLEM 1 07B1EngHOME EXAM B1 The mechanism depicted in the figure is composed by two rigid rods, each of lenght L. Rod AB girates around toggle A, whereas rod BC has a joint in point B and its end C slides on the floor. You know the lenght L, the angular velocity of rod AB, AB, and the angles 1 and 2. Find: b) The angular velocity of rod BC, BC, and the velocity of point C, v C. 4.0 p a) The velocity of point B, v B, and the angle formed by v B with the horizontal. 2.0 p PROBLEM 2 A C 11 22 B
3 B1 (a)(b)(c) You have a spring whose natural length is L 0 = 10 cm -figure (a)-. When a mass M = 250 g is hung on the spring, its length increases by L = 40 cm –figure (b)-. Finally, the hanging mass oscillates after the spring is stretched A = 10 cm and then released –figure (c)-. Answer the following questions: What is the constant of the spring? Find the period of the oscillation Find the position of the mass 6.98 s after the oscillations start. Find the period of the oscillation if you had hung the same mass M from two identical springs like this one disposed in a paralell way. a) b) c) d) From Hooke’s law: From Newton’s 2 nd law: a) b) c) 0.5 p 1.0 p 2.0 p We choose t = 0 when y = A That implies = 0 PROBLEM 1 07B1_EngHOME EXAM B1
4 B1 From Newton’s 2 nd law: d) From Hooke’s law: Equation of the oscillation driven by two identical springs: The solution of this equation is Let us write the equation as where The period is The set of two identical springs disposed in a parallel way (each constant = k) behaves as a single spring of constant 2 k. (The springs are identical) PROBLEM 1 (SOLUTION CONTINUED)
5 B1 A B C 11 22 B The mechanism depicted in the figure is composed by two rigid rods, each of lenght L. Rod AB girates around toggle A, whereas rod BC has a joint in point B and its end C slides on the floor. You know the lenght L, the angular velocity of rod AB, AB, and the angles 1 and 2. Find: b) The angular velocity of rod BC, BC, and the velocity of point C, v C. 4.0 p a) The velocity of point B, v B, and the angle formed by v B with the horizontal. 2.0 p PROBLEM 2
6 B1 A 11 B Rod AB See that From dot product definition: a) Find: The velocity of point B, v B, and the angle formed by v B with the horizontal. PROBLEM 2 (solution continued)
7 B1 C 22 b) Find: The angular velocity of rod BC, BC, and the velocity of point C, v C. We know: A 11 B Vector Although we don’t know their values, we can write for angular velocity and velocity of point C: C end slides on the floor BC and v C are the quantities to find. Let’s call This vectorial equation can be splitted into two equations, one for each component:
8 B2 A rigid thin rod of lenght L = 1.80 m, mass M = 6 kg is articulated on a toggle (point O in the figure). The rod is kept tilted by an steel towline attached to the wall. The angles between the towline, the rod and the wall are 1 = 60º and 2 = 50º respectively. A counterweight m = 4 kg hangs from the other end of the rod. a) Draw the free body diagram for the rod (2 p). b) Find the tension on the towline and the rectangular components of the reaction in the point O (2 p). PROBLEM 1 HOME EXAM B2 (2007) MODEL A The picture shows a disc of radious 3R, with four circular holes, each of radious R, lying on the indicated positions. The surface density of the disc is (kg·m -2 ). The disc moves on the floor. Answer the following questions for the numerical values given below: a) Calculate the I zz moment of inertia of the disc, where Z is the perpendicular axis which passes through its simmetry center (not shown in the figure). (2 p) b) The initial angular velocity of the disc when it takes contact with the floor is 0 (clockwise direction), meanwhile the initial linear velocity of its center of mass is zero. The dynamics friction coefficient is . Make a FBD taking into account the external forces acting on the disc, and find the time it takes the disc to roll without slipping, the linear velocity of its center of mass and the angular velocity of the solid at the moment in which rolling without slipping begins. (4 p) Find the numerical results for the above questions, being PROBLEM 2 R = 14,7 cm, = 50 kg·m -2, 0 = 0,60 rad/s, = 0.15
9 B2 A rigid thin rod of lenght L = 1.80 m, mass M = 6 kg is articulated on a toggle (point O in the figure). The rod is kept tilted by an steel towline attached to the wall. The angles between the towline, the rod and the wall are 1 = 60º and 2 = 50º respectively. A counterweight m = 4 kg hangs from the other end of the rod. a) Draw the free body diagram for the rod (2 p). b) Find the tension on the towline and the rectangular components of the reaction in the point O (2 p). FBD PROBLEM 1 Solution HOME EXAM B2 (2007) MODEL A
10 B2 Z axis normal to the plane, not shown Moment of inertia of a disc ( surface density, a radious) About to a normal axis passing through its simmetry center (I zz ) In our problem, we have two different types of disc: 1. A solid disc of radious a = 3R and surface density = . All of them are simmetrically disposed around, being 2R the distance between each center and the center of the body. Momentum of inertia of every hole about to the Z axis passing through its simmetry center We have now to calculate the momentum of inertia of every hole about to the Z axis passing through the center of the body. We apply Steiner theorem: Z axis 3R3R 2R2R R 2R2R 2R2R PROBLEM 2 Solution (continued)
11 B2 t > 0, but before rolling without slipping Initial situation C X Y Z This means that initially point C moves towards the left, therefore the friction force goes to the right. CM Initial picture of the problem 2 nd Newton’s law: Rotation of the disk: The moment of inertia I CM is the same I ZZ previously found, because our disc is a flat figure, so the angular acceleration can be written as: Where the mass is given by and the moment of inertia is Anticlockwise Angular velocity decreases at the same time that velocity of the center of mass increases PROBLEM 2 Solution (continued)
12 B2 Translation & Rotation equations t > 0, but before rolling without slipping Rolling condition When rolling without slipping begins, we have PROBLEM 2 Resolution R = 14,7 cm, = 50 kg·m -2, 0 = 0,60 rad/s, = 0.15 I ZZ = 1.65 kg·m 2 = 6.67 rad·s -2 t f = 0.060 s v CM (t f ) = 8.82·10 -2 m/s (t f ) = 0.20 rad/s
13 B3 07B3_EngHOME EXAM B3 PROBLEM 1 (4 p) A hollow sphere (inner radius R 1 = 10 cm, outer radius R 2 = 20 cm) is made of material of density 0 = 0.80 g·cm -3 and is floating in a liquid of density L = 1.60 g·cm -3. The interior is now filled with material of density m so that the sphere just floats completely submerged. (a) Find the volume fraction of the floating hollow sphere under the liquid surface level before filling its interior. (b) Find density m. PROBLEM 2 (6 p) A cylindrical-shaped tank is used in a farm as water reservoir. The area of the base of the tank is 10 m 2. A faucet on the top lets go in 0.60 kg of water per second, meanwhile the outcoming flux across the sink lying on the bottom pours 0,50y kg/s outside, where y means the height of the liquid above the flat bottom of the tank. The tank has also a spillway 1 m above the bottom. Assuming that at the beginning of the day the tank contains 100 liters, then we open at the same time the incoming faucet and the sink, find: b) In case it would not have any spillway, find the maximum height the surface of the water could reach. c) Plot the height of the surface of the water versus time, and mark in your plot the values you obtained as results for the paragraphs a) and b). a) How long does the surface of the water take to reach the spillway (in case it reaches).
14 B3 A hollow sphere (inner radius R 1, outer radius R 2 ) is made of material of density 0 and is floating in a liquid of density L. The interior is now filled with material of density m so that the sphere just floats completely submerged. (a) Find the volume fraction of the floating hollow sphere under the liquid surface level before filling its interior. (b) Find density m. From Archimedes’ Principle, the sphere is buoyed up by a force E equal to the weight of the displaced fluid. As the sphere floats, E should be equal to its weight M 0 g. V L is the volume of the sphere’s submerged portion Setting E equal to the weight, we find V L (a) Volume and mass of the hollow sphere: Submerged fraction PROBLEM 1 SOLUTION
15 B3 (b) When the interior of the sphere is filled with material of density m, the sphere just floats completely submerged. Now the sphere is buoyed up by a force E’ equal to the weight of the displaced fluid, which coincides with the weight of an amount of fluid whose volume is the same that the sphere’s volume. The mass of the material filling the hollow is Setting equal both expressions for M’ we have PROBLEM 1 SOLUTION Numerical solutions
16 B3 A cylindrical-shaped tank is used in a farm as water reservoir. The area of the base of the tank is 10 m 2. A faucet on the top lets go in 0.60 kg of water per second, meanwhile the outcoming flux across the sink lying on the bottom pours 0,50y kg/s outside, where y means the height of the liquid above the flat bottom of the tank. The tank has also a spillway 1 m above the bottom. Assuming that at the beginning of the day the tank contains 100 liters, then we open at the same time the incoming faucet and the sink, find: a) How long it takes the surface of the water to reach the spillway (in case it reaches). b) In case it would not have any spillway, find the maximum height the surface of the water could reach. c) Plot the height of the surface of the water versus time, and mark in your plot the values you obtained as results for the paragraphs a) and b). Spillway Continuity equation: 0.500.60 PROBLEM 2 SOLUTION
17 B3 Spillway a) Find how long takes the surface of the water to reach the spillway (in case it reaches). b) In case it would not have any spillway, find the maximum height the surface of the watercould reach. Numerical values h (m) =1 PROBLEM 2 SOLUTION
18 B3 c) Plotting: level of water versus time t (s) y (m) The level of the water reaches the spillway y = h = 1 m t = 35668 s Maximum height in case there weren’t any spillway 1.20 m PROBLEM 2 SOLUTION
19 B4 07B4HOME EXAM B3Only spanish version available Un gas ideal de coeficiente adiabático = 1.4 ejecuta un ciclo de potencia formado por las siguientes etapas: 1→2. El gas se expande politrópicamente (índice de politropía k 1 = 1.35) desde las condiciones V 1 = 1 litro, P 1 = 7.87 bar, hasta que su volumen se duplica. 2→3. El gas se enfría a volumen constante, hasta que su temperatura es T 3 = 280 K. 3→1. El gas se comprime politrópicamente hasta restituir las condiciones iniciales (sea k 2 el índice de politropía de este proceso, que deberá determinarse). Se supone que todas las etapas son reversibles. La masa de gas es n = 0.20 moles, y la constante universal de los gases es R = 8,314 J/(K·mol). Se pide: A) Calcular las coordenadas de presión y temperatura en todos los puntos notables del ciclo (2 p). C) Calcular el trabajo asociado con cada una de las etapas del ciclo, discutiendo su signo (2 p). D) Calcular el calor asociado con cada una de las etapas del ciclo, discutiendo su signo (2 p). E) Determinar el rendimiento del ciclo (1 p). F) Calcular la variación de entropía de cada una de las etapas del ciclo (2 p). B) Determinar el índice k 2 y representar gráficamente el ciclo en un diagrama de Clapeyron (P-V) (2 p).
20 B4 1 2 3 1. + - 2. + - + - Rendimiento donde Incremento de entropía (politrópicas) Incremento de entropía (isocórica) EXAMEN DE CASA B4 (2007)
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# Inequalities HELP!!
Show 40 post(s) from this thread on one page
Page 2 of 2 First 12
• Aug 26th 2012, 07:17 AM
Prove It
Re: Inequalities HELP!!
Quote:
Originally Posted by zales127
Attachment 24603
The reason I asked the above question is because it states "Which of the following has the solution shown on the number line?
The answer that I came up with was |x + 1| ≥ 7 (Sorry the picture in my above post was really tiny..you have to click on the image)
That is correct, \displaystyle \begin{align*} |x + 1| \geq 7 \implies |x - (-1)| \geq 7 \end{align*} means all the points that are 7 or more units away from -1.
• Aug 26th 2012, 07:33 AM
zales127
Re: Inequalities HELP!!
Thank you. I think i got it now! Now I move on to the "Solving Sytems"..LORD help Me!! lol
Show 40 post(s) from this thread on one page
Page 2 of 2 First 12
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# [curves] PAKE questions
Brian Warner warner at lothar.com
Fri Feb 6 19:27:01 PST 2015
```I've been working on a (python) PAKE library, and as usually happens
with these things, there are a number of questions that don't really
surface until the bits meet the road. I talked to Trevor about it last
week, and we came up with a short list of things that we'd really like
to figure out or resolve.
I'm implementing SPAKE2, first with an integer group (prime-order
subgroup of a larger Zp* group), then I want to move to one of the 25519
groups. Portability matters more to me than speed, hence the
pure-python, but it'd be nice to not be painfully slow. I've got a
2048-bit integer group running in 16ms on a fast computer, 110ms on a
slower one, and it'd be nice to bring that down a little bit.
For reference, SPAKE2 (in additive notation, with scalars in lowercase
and group elements in caps) is:
public elements: M and N (one per role), generator G
private scalar: pw
Alice:
x = random_scalar()
X = G*x
MSG_X = X + M*pw , send MSG_X
Bob:
y = random_scalar()
Y = G*y
MSG_Y = Y + N*pw, send MSG_Y
Z = (MSG_X - M*pw) * y
key = H("Alice", "Bob", MSG_X, MSG_Y, Z)
Alice:
Z = (MSG_Y - N*pw) * x
key = H("Alice", "Bob", MSG_X, MSG_Y, Z)
1: Dealing with the cofactor
I'm planning to grab the group-operation algorithms from the Ed25519
reference implementation, because SPAKE2 needs both point-addition and
scalar-multiplication (so Curve25519's DH-specific X-only code won't
help me). I think I need both X and Y coordinates, but I haven't studied
the algorithm enough to know if maybe I can get away with just the X
coord (maybe trying both sign bits and accepting one of two different
Both Curve25519 and Ed25519 reference implementations clamp the secret
scalars (clearing the low three bits, setting the high bit). My partial
understanding is that this clears the cofactor and makes it safe to not
do as much checking on the received public point (the DH public
parameter for Curve25519/DH, or the public verifying key for
Ed25519/Schnorr). If you don't check that the received point is part of
the right group (and that it has the correct large order), then you
might end up revealing the low-order bits of the private key, and so
fixing them to zero means you're not revealing anything.
So I think I either must spend the time to verify the incoming point for
subgroup membership, or find a way to clear the cofactor in SPAKE2. Any
idea how to do this correctly? We figured that, from Bob's point of
view, the concern is that MSG_X is not really a point on the right
subgroup, but MSG_X*cofactor would be. So maybe do something like:
Z = (MSG_X*cofactor - N*pw*cofactor) * y
Are we on the right track?
2: Augmentation
Boneh/Shoup's "cryptobook" defines an additional algorithm, "PAKE2+",
which includes an augmentation step. I think this is beyond what Abdalla
and Pointcheval described in their 2005 RSA paper. Do people generally
agree that PAKE2+ is a good approach? I don't know how much review it's
(also, are "Balanced" and "Augmented" the modern terms for these two
modes? or has this changed?)
3: Simultaneous Init (M==N)
I don't know what the best term is for this, but most of the PAKE
algorithms I've seen assign roles to the two parties, and bake those
into the messages. I'm sure this makes the proofs easier to build, and
might block some sort of reflection attack (what happens if the attacker
echoes Alice's message back to her.. could that help them learn the key
But it's a nuisance for certain peer-to-peer use cases, like the
Petmail/Panda introduction protocol, where there's no easy way to give a
consistent name to each side. In the scheme I'm building for Petmail,
the two humans can pick a string and type it into their agents at
roughly the same time; I then want PAKE to amplify that into strong key
exchange. There's no client-vs-server, or first-vs-second distinction.
I'd have to explain to users that, in addition to remembering a random
string, they also have to pick sides, and that'd be a drag.
So what happens if you take SPAKE2 and set M==N? Does it break horribly?
I had another idea, which I *think* is reasonable:
Alice runs two instances in parallel, with the same password, one as A
and one as B. She glues the A and B messages together and sends both to
Bob.
Bob does the same, but swaps Alice's A/B messages before feeding them
into his A/B instances. Alice swaps Bob's A/B messages upon receipt.
Alice then XORs the keys that come out of her two instances. Bob does
the same.
This takes twice as much CPU and bandwidth, but:
* combining the two keys before revealing any derivative means an
active attacker still only gets one guess, not two
* using XOR doesn't require deciding upon a canonical order for
anything (like sorting them lexicographically, concatenating, then
feeding into HKDF)
* I think it preserves the validity of SPAKE2's security proof
* it's pretty easy to implement
4: Choosing M and N (aka U/V in PAKE2)
SPAKE2 requires two "arbitrary public elements" to blind the passwords,
but what wasn't obvious to me when I first started was that it's
important that nobody knows the discrete log of either one (otherwise an
active attacker gets a dictionary attack, I think). Since the easiest
way to pick a random element is to start with a random scalar, it's a
subtle failure mode for implementors.
Picking an arbitrary member of a an elliptic curve subgroup usually
means picking a random X coordinate, finding the 0 or 2 points that
match (50/50 chance), choose one, then see if it's in the subgroup.
Small cofactors make this probable(?) enough that random hunt-and-peck
will yield a result quickly, and you can make it more rigid/"sleeveless"
by starting your random search at SHA256("") or something.
Doing this for a q=~160-bit prime-order subgroup of a p=~4kbit Zp* group
doesn't work, because the cofactor (r=(p-1)/q) is gigantic: you'll never
randomly discover a subgroup member this way. I learned/believe that the
right approach is to pick a uniform integer h in Zp, then compute h^r,
which will be a uniform member of the subgroup. My more-rigid algorithm
is to hash a seed into enough bits to Zp, turn it into an integer, then
take it modulo p (introducing some bias), before raising to the cofactor
r.
Is this right?
This is one of the things we'd need to nail down so implementors can
The use cases that want Augmented PAKE really want something stronger
than a single scalar multiplication. PAKE appears to be at odds with
protecting server-side data against dictionary attacks, which is a pity
because it does a great job of protecting against network-side attacks.
Can we get both?
Trevor floated a crazy idea that involved only masking one side of the
exchange, and then being very careful about who reveals a derived key
first. We thought this might be useful for something, but tough to keep
it safe.
thoughts?
-Brian
```
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Lesson Plan: Dividing Two-Digit Numbers by One-Digit Numbers: Long Division with Remainders | Nagwa Lesson Plan: Dividing Two-Digit Numbers by One-Digit Numbers: Long Division with Remainders | Nagwa
# Lesson Plan: Dividing Two-Digit Numbers by One-Digit Numbers: Long Division with Remainders Mathematics • 4th Grade
This lesson plan includes the objectives, prerequisites, and exclusions of the lesson teaching students how to use the standard long division algorithm to divide a two-digit number by a one-digit number to give answers with remainders.
#### Objectives
Students will be able to
• use the standard long division algorithm to divide a two-digit number by a one-digit number to give answers with a remainder,
• find missing numbers in partially complete solutions,
• identify errors in solutions.
#### Prerequisites
Students should already be familiar with
• multiplication and division facts within multiplication tables up to .
#### Exclusions
Students will not cover
• division problems where the divisor is greater than 9.
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Percentages, Fractions, and Decimal Values
What is a number that can be put in fraction form?
Wiki User
Any number can be put in fraction form
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Related Questions
A rational number is one that can be put into a fraction form.
You cannot put a whole number into simplest form. A number has to a fraction to be simplified.
You put a fraction in simplest form by dividing the numerator and denominator by their greatest common factor. If the either the numerator or denominator is a prime number, the fraction cannot be simplified.
To convert an improper fraction to a mixed number, divide the denominator into the numerator. The answer is the whole number. Put any remainder over the original denominator to create the fraction part. Any fraction can be put into simplest form by finding the GCF of the numerator and denominator and dividing them both by it. If the GCF is 1, the fraction is in its simplest form.
To convert a mixed number to an improper fraction, multiply the denominator by the whole number, add that total to the numerator and put the whole thing over the original denominator. To put a fraction into its simplest form, find the GCF of the numerator and the denominator and divide them both by it. If the GCF is 1, the fraction is in its simplest form.
it is a rational number because it can be put into fraction form
Yes. Any number that can be put into fraction form with a/b, implying a and b are integers is a rational number
21/1 It is a whole number so it doesn't need to be put in simplest form.
12 is an integer, not a fraction.
To create a fraction you need a numerator and a denominator - the numerator is the number above the line, and the denominator is the number below the line. 70000 is an integer - to convert it to a fraction in the simplest form simply put it over 1 = 70000/1
you would see where the number is what place value then you put that number over the place value so 0.004 would be 4/1000 in fraction form.
In fraction form, that number would be 137662.5/100
To put a fraction in simplest form , simply divide the numerator and denominator by their greatest common factor(GCF) and the answer is its simplest form.
You change it into an improper fraction, then simplify it, then if you want to, change it back to a mixed number.
To convert an improper fraction to a mixed number, divide the denominator into the numerator. The answer is the whole number. Put any remainder over the original denominator to create the fraction part.
41.66 (repeating) as a fraction is 125/3 in improper fraction form. In mixed number form, the number is 41 and 2/3.
The fraction 44 over 99 put in its simplest form is 4/9
A negative exponent is put into fraction form because it is a way of writing powers of fractions or decimals.
Put the number being divided on top of the fraction (the numerator) and put the number you are dividing that number by at the bottom (the denominator).
358 is in its simplest form, but it is a whole number and not a fraction.
mm is not a number and so, in itself, it cannot have a fraction form.
4.75 as a mixed number with a fraction in simplest form = 43/4
It is already simplified, since thirty seven is a prime number.
any percent as a fraction is put over 100 so: 45 __ 100 now find the fraction in simplest form so take out 5 from each number 9_ 20
Percentages, Fractions, and Decimal ValuesMath and ArithmeticNumbers Algebra
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2004-05-10
Categories: Blog post Tags: Information theory
The formula for uncertainty (sometimes referred to as entropy) is
By convention, if any probability is zero, we set the product term to zero as well. Those of you who are familiar with calculus could use limits to establish that zero is a reasonable value for this product.
Data compression
Consider this distribution of ratings:
The uncertainty calculation would be
Suppose we wanted to send a coded binary message that would indicate what category the rater chose. A simple approach would use three binary digits
• A=000,
• B=001,
• C=010,
• D=011,
• E=100.
to send the message. But you could use the probabilities involved to save a bit of time. Set the most common category to a shorter binary string, and longer strings to the less common categories
• A=1,
• B=000,
• C=001,
• D=010,
• E=011.
Then half of the time we would only use one bit to send the message, the other half of the time, we’d need three bits. On average, you would use two bits, which is exactly the number produced by our uncertainty calculation.
This example is a bit tricky because you might ask, why couldn’t I use two bits for B, C, D, and E. This would lead to a coding of
• A=1,
• B=00,
• C=01,
• D=10,
• E=11.
The problem with this is that unless you could separate the bits, then you couldn’t distinguish between AA (11) and E (11). You’d need a stop bit or some other type of separator, which would add to the average length of a message.
Morse code also takes advantage of probabilities, with common letters being shorter combinations of dots and dashes. The single dot and single dash are E and T, while the dot-dot, dot-dash, dash-dot, and dash-dash are I, A, N, and M, respectively. The letters that require four dots/dashes are B, C, F, H, J, L, P, Q, V, X, Y, and Z. Morse code, of course, was developed long before we understood concepts like information theory, so it is not an optimal coding. Still it represents an interesting example.
A B C D E F G H I J K L M .- -… -.-. -.. . ..-. –. …. .. .— -.- .-.. N O P Q R S T U V W X Y Z -. — .–. –.- .-. … - ..- …- .– -..- -.– –..
In general, there is not always a coding that gives the optimal number of bits on average. The uncertainty represents the lower bound on average compression rate.
Huffman encoding
One of the compression methods used by various Zip programs, Huffman encoding, takes advantage of this feature. Instead of allocating 7/8 bits to each character, it will remap the characters with the most frequently occurring characters having short bit lengths and the least frequently occurring characters having long bit lengths.
The following example appears on a rather amazing page that animates the Huffman encoding algorithm. If you were to use a simple binary coding, it would require 5 bits for each character of the alphabet A-Z. But Huffman Encoding uses a variable number of bits to represent each letter. Notice that
• E and T use 3 bits,
• A, I, L, N, O, R, and S use 4 bits,
• B, C, D, F, G, H, M, P, U, W, and Y use 5-6 bits, and
• J, K, Q, V, X and Z use 7-12 bits.
The table shows that the surprisal values are close to the number of digits used in Huffman encoding. The weighted average surprisal is 4.167, which tells us that it takes slightly more than 4 bits of information to code this data. This is confirmed by the weighted average number of bits, 4.196, which is just a tiny bit higher than the uncertainty.
Application to genetics
This is a group of DNA segments representing Lambda cI and cro binding sites.
Notice that there are some common patterns to these DNA segments but also some uncertainty, especially at location zero. Information theory helps define the degree to which these DNA segments have common features.
First we need to compute the surprisal values.
If a letter has probability zero, that equals an infinite surprisal value and is dentoed by Inf in the table. Notice that letters A and G have high surprisal values in the -9 position because they only appear rarely.
The weighted surprisals appear below.
The uncertainty for this group of DNA segments is 17.5. The maximum possible uncertainty is 38 (2 bits times 19 base pairs). The information is 10.5 which represents the maximum possible uncertainty minus the uncertainty observed in this group.
One way of thinking of this is that if you knew nothing about the 19 base pair DNA segment, you would have a large amount of uncertainty (38 bits). If instead you restricted the DNA segments to the 12 listed above, you would have a lot less uncertainty. The information contained in these DNA segments represents the decrease in uncertainty.
You can also compute the information at any given base pair. The information is highest at -7, -5, and +5, because these are the base pairs where there is no variation. Information is also high at -3, +3, because the probabilities of C and G respectively are almost 100%.
The following graph illustrates the information at each base pair with the heights of each letter indicating how much that particular letter contributes to the information.
Evaluating new DNA segments
There’s another way to calculate uncertainty for these DNA segments, and although it is a bit more work, it provides a clue for how to evaluate new DNA segments. Notice that for each base pair location, we have surprisal values for A, C, G, and T. We can evaluate those surprisal values for each DNA segment.
Now we can average these values across all 12 DNA segments. This gives us the same weighted surprisals and the same uncertainty as before.
This could help us understand whether one or more of the DNA segments does not fit in with the others, but more importantly, it can help us identify new DNA segments that might fit into the same class of binding sites. If a DNA segment has a reasonably low total surprisal, then that indicates that this DNA segment might fit in, but if the surprisal is too high, then this DNA segment is too different to be in this class of binding sites.
Information theory applied to sperm morphology classifications
One of the more fascinating applications of information theory is in the study of sperm morphology, the classification of sperm cells into various size and shape categories.
Human sperm exist in many different shapes and sizes. There are many irregular and abnormal forms. Forms display continuous variation and there are lots of gradual transitions from one form to another. As a result, sperm morphology has greater inter- and intra-laboratory variation than many other tests.
Although sperm morphology is a tool used in fertility assessment, fertility is a complex process that involves multiple organs of both the male and the female. Given this complexity, it would be a mistake to think that a sperm analysis is going to be a valid maker of fertility. Instead, semen analysis is a “gateway” test for fertility. It determines the path of further testing and which partner should receive greater scrutiny.
There are many sperm morphology classification schemes, and much controversy over what to use.
A good friend of mine, Susan Rothmann has an NIH grant to review morphology classification as it is currently practiced. The long term goal of the grant is to get greater consistency within and between laboratories. The preliminary data from this grant involved the classification of 160 sperm images by 99 trained raters.
• the sperm head,
• the acrosome region of the head,
• the postacrosomal regions of the head,
• the sperm midpiece, and
• the sperm tail.
The following picture describes the various parts of the sperm cell:
In addition, the raters were asked to classify non-sperm cells such as monocytes and macrophages that appeared in the sample. Here is a list of the classifications available in the Rothmann study:
• Acrosome large (AcLrg)
• Acrosome small (AcSml)
• Acrosome deformed (AcDef)
• Acrosome missing (AcMis)
• Postacrosomal abnormal size (PoSiz)
• Postacrosomal abnormal shape (PoShp)
• Vacuoles (Vacuo)
• Midpiece thick (MdThk)
• Midpiece very thick (MdVTh)
• Midpiece irregular (MdIrr)
• Midpiece length (MdLen)
• Midpiece bent (MdBnt)
• Midpiece missing (MdMis)
• Midpiece droplet (MdCyt)
• Midpiece incorrect insertion(MdIns)
• Tail coiled (TCoil)
• Tail short (TShrt)
• Tail broken (TBrok)
• Multiple tails (TMult)
• Hairpin tail (THpin)
• Tail width (TWdth)
• Tail droplet (TDrop)
• Tail bent (TBent)
• Leukocyte (Leuko)
• Polymononuclear leukocyte (P.M.N)
• Monocyte (Moncy)
• Macrophage (Mcrph)
• Immature sperm cell (Immat)
• Spermatid (Sptid)
• Spermatocyte (Spcyt)
Not every person will use every one of these abnormal categories. The list was intended to be the collection of classifications across all systems. In addition to the specific rating categories shown above, each rater was asked to provide an overall classification:
• Definitely normal,
• Possibly abnormal,
• Definitely abnormal, and
• Non-sperm cell.
Here are two examples of the micrographs of sperm cells that the raters were asked to evaluate.
This picture and the one below it are not as good quality as the ones that the raters used, but I just wanted you to see the general type of images they were asked to rate.
What information theory can tell us about sperm classification
Certain sperm images are easier to classify and others are more difficult to classify. The former represent exemplars that can be displayed as model images of a particular classification. The latter represent areas of controversy and disagreement that need to be examined, perhaps by a consensus group, to resolve the ambiguity. Identifying the easy and difficult images is difficult by visual inspection, because of the large number of images and the variety of classifications available. We plan to use uncertainty to sift through those images and rapidly identify images both easy and difficult for further study.
Uncertainty for overall classification
The table below shows the overall classification and the uncertainty calculated on that classification.
Cell Norm Bord Abno NonS NotC Unce 1 24 34 41 0 0 1.6 7 62 18 19 0 0 1.3 2 79 12 7 1 0 1.0 5 13 8 78 0 0 0.9 3 4 10 85 0 0 0.7 4 0 5 94 0 0 0.3 6 1 2 96 0 0 0.2
The first column identifies the particular cell, and the possible classifications are abbreviated as
• Norm (Definitely normal)
• Bord (Borderline abnormal)
• Abno (Definitely abnormal)
• NonS (Non-Sperm Cell)
• NotC (Not Classified)
and “Entr” represents the calculated uncertainty. The cell with the greatest uncertainty (among the first seven cells) is cell #1. You can see by the rater counts that there was very little agreement about the overall classification of this cell.
The two cells with the least uncertainty are cells 4 and 6, and these are cells that pretty much everyone agrees are abnormal.
Information theory for individual classification
It is also possible to calculate uncertainty for each individual classification and then add these uncertainty values together. Cells with high total uncertainty represent cells with uncertainty across several classification categories. Here are the results for the first seven sperm cells with the percentages shown for any category selected by 5% or more of the raters:
Sperm #6. Uncertainty = 8.1
• TCoil: 53%
• TShrt: 44%
• MdThk: 31%
• MdVTh: 23%
• THpin: 17%
• MdLen: 13%
• MdIrr: 11%
• TWdth: 10%
• HdNOv: 8%
• HdPyr: 6%
Sperm #5. Uncertainty = 7.6
• Vacuo: 57%
• HdNOv: 32%
• HdRnd: 25%
• THpin: 20%
• AcSml: 12%
• HdSml: 11%
• HdRat: 10%
• TCoil: 9%
Sperm #3. Uncertainty = 6.4
• MdCyt: 64%
• HdNOv: 17%
• MdIrr: 16%
• HdTap: 14%
• MdVTh: 10%
• AcSml: 10%
• HdRat: 9%
• HdSml: 9%
• MdThk: 7%
• HdPyr: 5%
Sperm #1. Uncertainty = 5.8
• HdNOv: 33%
• HdPyr: 29%
• PoShp: 14%
• MdIns: 9%
• MdThk: 9%
• HdLrg: 9%
• HdRat: 8%
• HdRnd: 7%
• HdTap: 6%
Sperm #4. Uncertainty = 5.2
• MdVTh: 58%
• MdCyt: 52%
• HdNOv: 12%
• MdIrr: 9%
• HdLrg: 7%
• HdRat: 6%
Sperm #7. Uncertainty = 2.4
• TBrok: 10%
• TShrt: 9%
• HdNOv: 7%
Sperm #2. Uncertainty = 1.7
• HdNOv: 8%
Notice that cell #6, the cell with the lowest uncertainty on the overall classification, has the highest uncertainty among the seven cells when looking at the individual classification levels. This reflects the fact that raters agree that the cell is abnormal, but they can’t agree on the type of abnormality.
Future work–joint and conditional uncertainty
To calculate the total uncertainty in the sperm classification system above, I oversimplified a bit by adding the individual entropies for each individual classification system. This ignores some possibly important information. Let’s suppose that we have a two classifications: Small head (HdSml) and Missing Acrosome (AcMis). Let’s suppose there is a high amount of uncertainty in both categories:
Yes No
HdSml 50% 50%
and
Yes No
AcMis 50% 50%
There would be one full bit of uncertainty with each classification. But how do these classifications behave jointly? Here are two possibilities:
Yes No
Yes 0% 50%
No 50% 0%
and
Yes No
Yes 25% 25%
No 25% 25%
The first situation reflects a case where everyone agrees the cell is abnormal but half of the raters call it a small head and half of the raters call it a missing acrosome. The only uncertainty is in the use of labels.
The second situation is much more uncertain. A quarter of the raters say the cell has no defects, a quarter say is has two defects, and the remaining raters are split between calling the cell a small head only or a missing acrosome only.
You could calculate joint uncertainty by just treating the 2 by 2 table as a 4 by1 set of probabilities:
Yes/Yes Yes/No No/Yes No/No 0% 50% 50% 0%
and
Yes/Yes Yes/No No/Yes No/No 25% 25% 25% 25%
This would then allow you to calculate the joint uncertainty, and it would be 1 bit and 2 bits, respectively. The second situation, has much greater uncertainty, even though the marginal probabilities are identical to the first situation.
You can also define uncertainty using conditional probabilities, and the quantity produced, conditional uncertainty, represents how much the uncertainty in one variable is reduced when another variable is known.
There are several sperm morphology classification systems in use. Two commonly used systems are WHO-3 classification and strict classification. Strict classification works like it sounds, with borderline defects being classified as abnormal. In contrast, WHO-3 tends to classify borderline defects as normal.
Each of the 160 raters identified the system that they use. One area I want to look at is how the uncertainty decreases when you condition on rating system.
[Coming soon!]
Acknowledgements
The Huffman coding example comes from John Morris’s web pages, in particular, http://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/huffman.html
The genetics example comes from Tom Schneider’s web site and the sequence logo graph appears on the page http://www.lecb.ncifcrf.gov/~toms/paper/trieste1996/latex/node1.html
The diagram showing the regions of the sperm cell is reproduced with permission from The Andrology Trainer, Second Edition, Donna Kinzer and Susan Rothmann. 1998. Fertility Solutions, Inc. Cleveland, OH. Page 1-8. The two micrographs of sperm cells are also reproduced with permission from Fertility Solutions, Inc.
Peter Rogan is doing some really interesting work on information theory and has inspired me to look into this more deeply. Some of his work is reflected in the software available through https://splice.cmh.edu.
You can find an earlier version of this page on my original website.
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# Saving up for an expensive car
Let's say I have a salary of \$80,000 after tax.
My goal would be to buy a \$100,000 car.
I've read online that 20% is a reasonable amount to pay for a car each month.
20% x \$6,666 monthly pay = \$1,333 towards the car each month.
This means I can save 16,000\$ towards the car each year.
If I wanted my \$100,000 car I would have to save for about 6.5 years.
Is my logic reasonable? Is there anything I'm missing?
This 20% is purely for saving for the car. 30% goes towards everyday expenses like gas and food, 20% goes towards morgage and the other 30% goes towards emergency fund and retirement
• Note that this much money would buy a house in many areas. Also check what it's going to cost you to insure that monster. Dec 20, 2015 at 18:24
• I didn't work through the math, but I can tell you right away: If you make \$80k/yr in salary, then you should not be buying a car that costs \$100k. Economically that cannot make sense. If you want to do it anyway, that's up to you but it's not really on topic here anymore because it's in the category of frivolous spending.
– user32479
Dec 21, 2015 at 0:00
• Are these amounts in U.S. dollars? Dec 21, 2015 at 6:05
• @Ben Miller Yes Dec 21, 2015 at 20:59
• 20% of salary for car? That seems... high.
– Joe
Dec 21, 2015 at 22:21
The question is how does \$16,000/year for 6.5 years fit into your budget. Or to put it another way, what won't you be spending that money on? Housing, food, vacations, retirement fund, investments (though you can invest your car fund in the meantime), building a hefty emergency fund, kids college funds, saving for a down payment on a home, charity, etc... are all other places that money could go. I don't know what your needs are today let alone 6.5 years into the future, but I'd encourage you to consider all your financial goals and evaluate where this expense would fit.
It seems your plan is to save up to the total cost of the car and then buy it in cash. That's a valid strategy, but it means you'll have no car (unless you already own one) for 6.5 years.
Do you need a car? If so, what will you drive in the meantime (and even if you already own another car outright, you'll have gas and maintenance expenses)? If you don't need a car, then \$100,000 is a rather extravagant purchase for something we just established you don't need. Would you be happier having this expensive car in 6.5 years, or having a series of less expensive cars starting now? Or buying a used model of the expensive car sooner? Or having no car at all?
Also, a lot can change in 6.5 years. Cars will evolve and there'll be different models and options available. Maybe your salary will have doubled, or maybe you'll be unemployed. You could be living in a different city, have a different commute, and maybe you'll need a minivan to haul kids around or live in a place with bad winters and want a 4-wheel-drive. You'll also need to be prepared for the additional expenses that generally come with expensive cars, such as higher insurance and maintenance rates, and parking could be costly if you live in an expensive city.
The other option, of course, if the car is truly something you need, want, and can afford, would be to save up a sizable down payment and finance the rest so you can get the car sooner.
Finally, there's nothing wrong with saving your money for 6.5 years, building up that fund, and then reevaluating what makes the most sense for you at that time. Maybe it will the car, maybe something else, but the nice thing about having savings is that it gives you more options.
"I've read online that 20% is a reasonable amount to pay for a car each month" - Don't believe everything you read on the internet.
But, let me ask, does your current car have zero expense? No fuel, no oil change, no repairs, no insurance? If the 20% is true, you are already spending a good chunk of it each month. My car just celebrated her 8th birthday. And at 125,000 miles, needed \$3000 worth of maintenance repairs.
The issue isn't with buying the expensive car, you can buy whatever you can afford, that's a personal preference. It's how you propose to budget for it that seems to be bad math. Other members here have already pointed out that this financial decision might not be so wise.
• Happy Birthday, @JoeTaxpayer's Car... Mine just hit her 6th birthday as well :)
– Joe
Dec 21, 2015 at 22:26
If you can afford to put \$1,333 towards saving for a new car each month, then there is nothing wrong with your logic You should be aware that your car will probably cost around \$110,000 in 6.5 years, but other than that the logic is fine. However...
1. Check that you really can afford to save this amount of money. Just because some website says it's a reasonable amount in general doesn't mean its a reasonable amount for you.
2. Don't forget that expensive cars usually have higher running costs, like maintenance and insurance. Can you afford those?
3. What are you going to be driving for the next 6.5 years? Can you afford to buy, maintain, insure etc. your current car as well as putting \$1,333 away in savings?
Any way you look at it, this is a terrible idea.
Cars lose value. They are a disposable item that gets used up. The more expensive the car, the more value they lose.
If you spend \$100,000 on a new car, in four years it will be worth less than \$50,000.* That is a lot of money to lose in four years.
In addition to the loss of value, you will need to buy insurance, which, for a \$100,000 car, is incredible.
If your heart is set on this kind of car, you should definitely save up the cash and wait to buy the car. Do not get a loan. Here is why: Your plan has you saving \$1,300 a month (\$16,000 a year) for 6.5 years before you will be able to buy this car. That is a lot of money for a long range goal. If you faithfully save this money that long, and at the end of the 6.5 years you still want this car, it is your money to spend as you want. You will have had a long time to reconsider your course of action, but you will have sacrificed for a long time, and you will have the money to lose. However, you may find out a year into this process that you are spending too much money saving for this car, and reconsider.
If, instead, you take out a loan for this car, then by the time you decide the car was too much of a stretch financially, it will be too late. You will be upside down on the loan, and it will cost you thousands to sell the car.
So go ahead and start saving. If you haven't given up before you reach your goal, you may find that in 6.5 years when it is time to write that check, you will look back at the sacrifices you have made and decide that you don't want to simply blow that money on a car.
Consider a different goal. If you invest this \$1300 a month and achieve 8% growth, you will be a millionaire in 23 years.
* You don't need to take my word for it. Look at the car you are interested in, go to kbb.com, select the 2012 version of the car, and look up the private sale value. You'll most likely see a price that is about half of what a new one costs.
• The problem here is we don't have any other details from OP. If he rents with a roommate, and saves 20% for retirement but still has this money to burn, who are we to judge? People have a pool of discretionary income, for some, it's a \$10K vacation every year. Others, it's a \$500 dinner and night out every Saturday. This guy wants a car I wouldn't care to own. Dec 21, 2015 at 17:12
• "If you spend \$100,000 on a new car, in four years it will be worth less than \$50,000." Heck - the average car loses a significant chunk of its value literally the second you purchase it. Dec 21, 2015 at 19:15
• The car would be a 2006 Lamborghini Gallardo so it would have already done the bulk of its depreciation Dec 21, 2015 at 21:00
This seems really simple to me.
1. Don't sell current car
2. Save the amount you have planned to save to reach \$100,000 in 6.5 years
3. Re-evaluate in 6.5 years
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Search
# Mastering Regression Analysis for Class 12 Math
Updated: Oct 15
## How to learn Regression Analysis for Class 12 Math?
Welcome to Regression Analysis. This topic is used extensively in ISC Maths, Statistics and in Engineering Mathematics. So here's giving you an idea of how to go about studying this chapter.
There are a few free resources such as youtube videos here. Feel free to use them. In case you still need that extra help, you can contact me for online tutoring.
The statistical method which helps us in estimating the value of one variable, given the other variable is called regression. To learn regression, you'll need to have an idea of Correlation , Covariance and standard deviation. We can plot a scatter diagram for the data and then plot a smooth curve representing the data. This method is called curve fitting.
### What are dependent and independent variables?
There are two regression lines for two variables x and y. Firstly, we have the line of regression of y on x where y is the dependent variable and x is the independent variable. Next, we have the line of regression of x on y where x is the dependent variable and y is the independent variable.
Basics of Regression analysis-Lesson 1
#### How do you calculate the Regression coefficients?
We also have two regression coefficients, the regression coefficient of x on y which is also the slope of the regression line of x on y. Next, we have the regression coefficient of y on x which is the slope of the regression equation of y on x. There are multiple methods of calculating the regression coefficients.
You can calculate the regression coefficients from the two regression equations or there are separate formulas to calculate them. There are also formulas which show the relation between the regression coefficients, the correlation coefficient and the standard deviations of the variables. Note the the correlation coefficient r takes the same sign as the regression coefficients.
##### What is the relation between the correlation coefficients and the regression coefficients?
The correlation coefficient r is the geometric mean of the two regression coefficients. The point of intersection of the two regression lines is the arithmetic mean of the variables. The two regression lines coincide if and only if the correlation coefficient r is plus or minus one.
If the correlation coefficient is equal to zero, then the lines of regression are parallel to the coordinate axes. You can also calculate the acute angle between the two lines of regression. This is Regression Analysis Class 12 Math.
###### Do you still need extra help?
All these topics and more will be taught in my online classes. Learn formulas, concept based questions, case study based questions and basic problem solving skills. Share this with high school mathematics students.
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# Thread: Any chance I could get a little help with this puzzle?
1. Needs to say Hello!
Join Date
Apr 2010
Posts
9
## Any chance I could get a little help with this puzzle?
Hello all Rather new to this site, but not to puzzles :P
Recently, I have had a puzzle that stumped the ever-loving heck out of me, and I was wondering if I could pick your brains on it? Lol...
The puzzle is as such:
8 Words
A: LNR
B: AR
C: H
D: L
E: BNRSV
G: AL
I: AMS
L: EIR
M: P
N: DG
O: ENORT
P: E
R: IOSU
S: EH
T: HO
U: NS
V: R
W: A
Now, I have had it explained to me, thus:
The short form of the instructions is that there are eight words (as shown by the list header). These words share a common category; that category is the answer to the puzzle. The words are composed from the letters down the left-hand column, and are followed by the letters to their right. Letters on the rightmay be reused. For example, if the words were DOLLAR and DRAMA, the list would look like:
A - MR
D - OR
L - AL
M - A
R - A
Well, I have never some across one like this before, and while I can definitely see that certain pairs must go together (there being only 1 option), overall, I cannot figure out a word list. Apparently, there are 8 words total, but I dunno length, nor content.
The list of 2-letter pairs is:
al an ar
ba br
ch
dl
eb en er es ev
ga gl
ha hd he ho hu
ia im is
le li lr
mp
nd ng
oe on oo or ot
pe
ri ro rs ru
se sh
th to
un us
vr
wa
Could anyone PLEASE help me find a list of 8 words that use this whole list? I dunno what this kind of puzzle is even called :P
2. There are a few things that are unclear to me from your explanation.
First, must a letter from the right follow a letter from the left? If so you would never get a word like drama. All words would have an even number of letters.
Second, are letters from the left used only once? And are all letters supposed to be used? I don't see how letters from the right can be reused if this is the case. If letters can be reused somehow, then this is an ill-constrained problem.
ETA: What I'm asking is if each digraph is used only once or can they be re-used?
Third, is there a specific language that the words are from? If English, is it British English or American English? The letter combination 'vr' is not very common in American English, but moreso in British English. Characteristics like this can be useful in narrowing your word pool.
3. Needs to say Hello!
Join Date
Apr 2010
Posts
9
I am 99.9% sure it is english... and probably american english.
From how I understood it, every letter on the left is followed, in at least one of the words, by each letter on the right. So, for example, "W" can only be the START of a word, since it isn't listed on the right anywhere.
They CAN be odd-numbered... it's like this (I believe):
Take the word "American" for example. Your letter pairs would be "am, me, er, ri, ic, ca, an", and the puzzle would be written:
a: m, n
m: e
e: r
r: i
i: c
c: a
which would be put alphabetically, as:
a: m, n
c: a
e: r
i: c
m: e
r: i
This puzzle, though, is a total of 8 words. I do NOT think they are a phrase or sentence... just 8 words.
What I have figured out are probable word-parts, based on only appearing once (or so) in the puzzle, are these:
ch-
wa-
-evr-
-mpe-
-imp-
-hdl-
-nd OR -ndl-
etc.
Letters from the left are used as many times as there are letters on the right on each row. Letters on the right are all used. I believe that, if you have 2 words with the same letter pair in it somewhere, like "will" and "wind", a particular pair can be used more than once; such as:
i: n,l
l: l
n: d
w: i
This is how *I* understand the puzzle. But I am the one trying to solve it, not make it, so I am at a loss, myself :P
4. Looks a lot like this one from pianoman:
Amusement Park
I'll look at it when I get my computer turn in a bit.
5. I've solved it, would you like me to post it? Or give you a hint?
6. Hint if you'd like:
Solution if you'd like:
7. Needs to say Hello!
Join Date
Apr 2010
Posts
9
Oh, by all means, *I* would love the answer I gave up quite a while ago, myself. You can just PM me, if you think others would like to figure it out themselves; but at this point, for me, I just want the answer :P Lol!
And thank you, by the way!
8. You're welcome! Thanks for the diversion! It was a nice challenge.
#### Posting Permissions
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# Geometry
posted by on .
J is the midpoint of Gk, H is the midpoint of GJ and I is the midpoint of Hj. Suppose GK=64. Find the measure of HJ
• Geometry - ,
AB = 5x+3 and aback =3x + 19
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Wiki
# Best 14 What Time Is 19 38
Below is the best information and knowledge about what time is 19 38 compiled and compiled by the bmr.edu.vn team, along with other related topics such as: 19:38 meaning, 22/38, 19:40 12 hour time, 38/19, 19:38 utc to est, 19:39, 16:38, 21/38
Image for keyword: what time is 19 38
The most popular articles about what time is 19 38
## 1. What time is 19:38? – MathLearnIt.com
• Author: www.mathlearnit.com
• Evaluate 4 ⭐ (30862 Ratings)
• Top rated: 4 ⭐
• Lowest rating: 2 ⭐
• Summary: Articles about What time is 19:38? – MathLearnIt.com Convert 19:38 to AM/PM format ; 24 hour clock, 19:38 ; 12 hour clock (AM/PM), 7:38 PM …
• Match the search results: The conversion from 24-hour (or military) time to 12-hour (AM/PM) time is quite simple. In 12-hour time, the day is split into two distinct parts: before noon and after noon. When time is before noon (12:00 in 24-hour time), we append “AM” to the time. When time is after noon (after midday), we appe…
• Quote from the source:
• [browser-shot url=”https://www.mathlearnit.com/what-time-is-19-38″ width=”600″]
## 2. 1938 Military Time – What Time is 19:38? – Time Calculator
• Author: time-calculator.org
• Evaluate 3 ⭐ (11046 Ratings)
• Top rated: 3 ⭐
• Lowest rating: 1 ⭐
• Summary: Articles about 1938 Military Time – What Time is 19:38? – Time Calculator What time is 19:38? – 19:38 military time is equivalent to 7:38 PM in standard time or the 12 hour clock. Use our military time converter to convert any …
• Match the search results: What time is 19:38? – 19:38 military time is equivalent to 7:38 PM in standard time or the 12 hour clock. Use our military time converter to convert any military time to regular time and vice versa.
• Quote from the source:
• [browser-shot url=”https://time-calculator.org/1938-military-time” width=”600″]
## 3. What time is 19:38 ? is 07:38 PM
• Author: time.electronicbub.com
• Evaluate 4 ⭐ (24782 Ratings)
• Top rated: 4 ⭐
• Lowest rating: 2 ⭐
• Summary: Articles about What time is 19:38 ? is 07:38 PM 19:38 nineteen thirty-eight, What time is 19:38 military time?, or what is 19 hr 38 min in a 12-hour system.
• Match the search results: 1938 in military time is 07:38 PM in regular time
• Quote from the source:
## 4. Military Time 1938 is: 7:38 PM using 12-hour clock notation …
• Author: onlinemilitaryeducation.org
• Evaluate 3 ⭐ (2922 Ratings)
• Top rated: 3 ⭐
• Lowest rating: 1 ⭐
• Summary: Articles about Military Time 1938 is: 7:38 PM using 12-hour clock notation … Xem thêm 22 hàng
• Match the search results: See, what time is in the other military time zones at XXXXZ (Zulu Time Zone). Click on the table row to change the time zone.
• Quote from the source:
• [browser-shot url=”https://onlinemilitaryeducation.org/1938-military-time.html” width=”600″]
## 5. 19:38 Time | Conversion and Information | Telegram Guides
• Author: telgram.cn
• Evaluate 4 ⭐ (20743 Ratings)
• Top rated: 4 ⭐
• Lowest rating: 2 ⭐
• Summary: Articles about 19:38 Time | Conversion and Information | Telegram Guides 19:38 time in the 12-hour clock time convention is 07:38 pm. Regular time, normal time and standard time are synonym with the twelve hour …
• Match the search results: Telegram中文 是一款免費、安全、快速的即時通訊應用程序。
• Quote from the source:
• [browser-shot url=”https://telgram.cn/1938-time-conversion-and-information-telegram-guides.html” width=”600″]
## 6. Military Time Converter. Charts & How to Read
• Author: www.omnicalculator.com
• Evaluate 4 ⭐ (29209 Ratings)
• Top rated: 4 ⭐
• Lowest rating: 2 ⭐
• Summary: Articles about Military Time Converter. Charts & How to Read This military time converter is a must for all those searching for quick and easy … 7:00 p.m., 19:00, 1900, Nineteen hundred (hours).
• Match the search results: With our military time converter (also known as an army time converter), you’ll quickly find out what time it is in military time. Not only can you convert military time to standard time and vice versa, but you can also check what is the military time right now. For you visual learners out there, we…
• Quote from the source:
• [browser-shot url=”https://www.omnicalculator.com/conversion/military-time” width=”600″]
## 7. Military Time Chart – The 24 Hour Clock – Converter Tool
• Author: www.thetimenow.com
• Evaluate 3 ⭐ (10200 Ratings)
• Top rated: 3 ⭐
• Lowest rating: 1 ⭐
• Summary: Articles about Military Time Chart – The 24 Hour Clock – Converter Tool Quickly and easily read or convert military time with our Standard Time to Military Time … 19:00. “Nineteen hundred hours”. 8:00 p.m.. 2000 or 2000 hours.
• Match the search results: Welcome to TheTimeNow Military Time Converter and Military Time Chart! Below you will see your current time and military time compared. Also we have provided you with an easy to use military time converter tool so you can easily see the conversions from standard time to military time. Below this, yo…
• Quote from the source:
• [browser-shot url=”https://www.thetimenow.com/military-time-converter.php” width=”600″]
## 8. 1900 Military Time – 1900 24 Hour Time – Online Clock
• Author: online-clock.org
• Evaluate 3 ⭐ (10075 Ratings)
• Top rated: 3 ⭐
• Lowest rating: 1 ⭐
• Summary: Articles about 1900 Military Time – 1900 24 Hour Time – Online Clock 1900 Military Time to Standard Time ; 19:35, 7:35 PM ; 19:36, 7:36 PM ; 19:37, 7:37 PM ; 19:38, 7:38 PM.
• Match the search results: What time is 19:00 in 12 hour time? – 19:00 military time equals 7:00 PM in standard time or the 12 hour clock.
1900 24 hour time is used to convert military time to regular time.
• Quote from the source:
• [browser-shot url=”https://online-clock.org/1900-military-time” width=”600″]
## 9. Military Time | Conversion & How To Read – Tutors.com
• Author: tutors.com
• Evaluate 4 ⭐ (30192 Ratings)
• Top rated: 4 ⭐
• Lowest rating: 2 ⭐
• Summary: Articles about Military Time | Conversion & How To Read – Tutors.com What is military time? Learn how military time works, how to read, and write it. See a military time conversion chart and learn about midnight in military …
• Match the search results: Military time is a day clock that measures hours to 24 instead of 12 hours like civilian time. Military time is used to avoid confusion between a.m. and p.m. hours. The 24-clock is used by militaries, most European countries, and businesses engaged in 24-hour operations, like airlines and railroads.
• Quote from the source:
• [browser-shot url=”https://tutors.com/lesson/military-time” width=”600″]
## 10. Current Local Time in Hanoi, Vietnam
• Author: www.timeanddate.com
• Evaluate 3 ⭐ (15391 Ratings)
• Top rated: 3 ⭐
• Lowest rating: 1 ⭐
• Summary: Articles about Current Local Time in Hanoi, Vietnam Current local time in Vietnam – Hanoi. Get Hanoi’s weather and area codes, time zone and DST. … 12 hours, 38 minutes -1m 7s shorter …
• Match the search results:
© Time and Date AS 1995–2022.
Privacy & Terms
• Quote from the source:
• [browser-shot url=”https://www.timeanddate.com/worldclock/vietnam/hanoi” width=”600″]
## 11. Military Time Conversion – Alltrack USA
• Author: www.alltrackusa.com
• Evaluate 3 ⭐ (12414 Ratings)
• Top rated: 3 ⭐
• Lowest rating: 1 ⭐
• Summary: Articles about Military Time Conversion – Alltrack USA Convert Military Time to Civilian Time. And vice versa. Free utility to use. … Examples: 11:30, 5:45, 1:38, 12:15 … 7:00 a.m., 07:00, 7:00 p.m., 19:00.
• Match the search results: Type Military time and then click “Convert”
Examples: 1430, 545, 1938, 815
• Quote from the source:
• [browser-shot url=”http://www.alltrackusa.com/Military_Time_Conversion.htm” width=”600″]
## 12. What is 19.38 hours in hours, minutes, seconds?
• Author: beepmyclock.com
• Evaluate 3 ⭐ (15607 Ratings)
• Top rated: 3 ⭐
• Lowest rating: 1 ⭐
• Summary: Articles about What is 19.38 hours in hours, minutes, seconds? 19.38 hours is 19 hours, 22 minutes and 48 seconds. … Decimal to Time conversion. To convert to seconds, simply multiply the decimal hours by 60×60.
• Match the search results: To convert to seconds, simply multiply the decimal hours by 60×60. So, 19.38×60×60 = 69768 seconds.
• Quote from the source:
• [browser-shot url=”https://beepmyclock.com/decimal-to-hms/19.38″ width=”600″]
## 13. How to Tell Military Time: 7 Steps (with Pictures) – wikiHow
• Author: www.wikihow.com
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• Summary: Articles about How to Tell Military Time: 7 Steps (with Pictures) – wikiHow The military clock starts at midnight, known as 0000 hours. This is called “Zero Hundred Hours.” Instead of having a twelve-hour clock that resets twice, in …
• Match the search results: To tell military time, keep in mind that the military clock starts at 0, or midnight, and counts to 2359, or 11:59 pm. At midnight, the clock resets back to 0. For times between midnight and noon, simply remove the colon and add a 0 before the time. For example, 1:15 am is 0115 hours, pronounced “Ze…
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## 14. 3.4 Reading and calculating time | Conversions and time
• Author: www.siyavula.com
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• Summary: Articles about 3.4 Reading and calculating time | Conversions and time Being able to convert between different time formats and units and being able to … Mrs Gwayi has morning tea at 10:25; The Dube family eat dinner at 19:35.
• Match the search results: Being able to do calculations with time is a very useful skill to have. It is important to know how to plan and organise your time on a daily basis. For example, if it takes you a certain amount of time to walk to school, what time must you leave home in the morning to arrive in time for class? Or, …
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